{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,120.54,135.63,17.22],\"bbox_pdf\":[110.85,654.25,246.49,671.46],\"fragments\":[{\"bbox_pdf\":[110.85,654.25,246.49,671.46],\"words\":[127,128]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"heading\",\"source_ref\":\"title11\",\"source_text\":\"1 Introduction\",\"tag\":null,\"text\":\"1 Introduction\"},{\"bbox\":[110.85,152.08,388.55,170.86],\"bbox_pdf\":[110.85,469.06,499.4,639.92],\"fragments\":[{\"bbox_pdf\":[110.85,469.06,499.4,639.92],\"words\":[129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph12\",\"source_text\":\"Immediately after Einstein presented the field equations of General Relativity (GR) in 1915, Schwarzschild obtained the first exact solution to them: the spacetime outside a spherical and non-rotating object. It allowed for the description, to a good approximation, of the geometry outside astrophysical objects like planets and stars. It was also the prelude for describing the simplest imaginable black hole [1]. In the following decades, alternative descriptions of it were discovered and analyzed, while parallel developments in the realm of cosmology were made [2][3]. In black hole theory and cosmology, it became mandatory to explore the subtleties related to the covariance of GR, the equivalence principle, and the meaning of specific coordinate descriptions in analytic solutions [4]-[10]. The issue of coordinates and representations continues to be a relevant topic on both fronts [11]-[14].\",\"tag\":null,\"text\":\"Immediately after Einstein presented the field equations of General Relativity (GR) in 1915, Schwarzschild obtained the first exact solution to them: the spacetime outside a spherical and non-rotating object. It allowed for the description, to a good approximation, of the geometry outside astrophysical objects like planets and stars. It was also the prelude for describing the simplest imaginable black hole [1]. In the following decades, alternative descriptions of it were discovered and analyzed, while parallel developments in the realm of cosmology were made [2][3]. In black hole theory and cosmology, it became mandatory to explore the subtleties related to the covariance of GR, the equivalence principle, and the meaning of specific coordinate descriptions in analytic solutions [4]-[10]. The issue of coordinates and representations continues to be a relevant topic on both fronts [11]-[14].\"},{\"bbox\":[110.85,325.43,388.55,170.86],\"bbox_pdf\":[110.85,295.71,499.4,466.57],\"fragments\":[{\"bbox_pdf\":[110.85,295.71,499.4,466.57],\"words\":[266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph13\",\"source_text\":\"In this article we analyze different representations of the Schwarzschild spacetime. We start by recalling simple properties that can be extracted from Schwarzschild coordinates. Then we present some classical representations, such as the near-horizon approximation, the embedding of spatial sections, and the Tortoise and Kruskal coordinates. We also present two novel representations. The Diamond representation, which was recently introduced as an extension of a unified scheme for obtaining the Tortoise and Kruskal coordinates [15]. And the Golden representation is introduced here, inspired by previous work highlighting the appearance of the golden ratio in black hole solutions [16]-[18]. The transformations involved beautifully map the Schwarzschild spacetime to a fictional spacetime analogous to a relativistic point mass, where the golden ratio appears ubiquitous.\",\"tag\":null,\"text\":\"In this article we analyze different representations of the Schwarzschild spacetime. We start by recalling simple properties that can be extracted from Schwarzschild coordinates. Then we present some classical representations, such as the near-horizon approximation, the embedding of spatial sections, and the Tortoise and Kruskal coordinates. We also present two novel representations. The Diamond representation, which was recently introduced as an extension of a unified scheme for obtaining the Tortoise and Kruskal coordinates [15]. And the Golden representation is introduced here, inspired by previous work highlighting the appearance of the golden ratio in black hole solutions [16]-[18]. The transformations involved beautifully map the Schwarzschild spacetime to a fictional spacetime analogous to a relativistic point mass, where the golden ratio appears ubiquitous.\"},{\"bbox\":[110.85,498.78,388.55,170.86],\"bbox_pdf\":[110.85,122.36,499.4,293.22],\"fragments\":[{\"bbox_pdf\":[110.85,122.36,499.4,293.22],\"words\":[388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph14\",\"source_text\":\"The rest of the article is structured as follows: In Section 2 we present the Schwarzschild solution and briefly review some direct features that can be extracted from it. Then we obtain the near-horizon approximation alongside its connection with accelerated frames and the Flamm representation. Section 3 presents a derivation of the Tortoise and the Kruskal representations using the same procedure. Then we show the Diamond representation, which shares a common feature with the former two: they are all (1+1)-conformally flat to hypersurfaces of the Schwarzschild spacetime at constant angles. In Section 4 we introduce the Golden representation for the Schwarzschild spacetime, by slightly deforming the customary Schwarzschild ansatz and dimensionally reducing the Einstein-Hilbert action, until the Golden coordinates are defined. In Section 5 we show how the kinematics of the (1+1)-dimensional Golden representation yield the same dynamics as the Schwarzschild solution. The arguments of the previous sections allow us to present different visual Golden representations of the Schwarzschild geometry in Section 6. We end with a summary of the work and some final comments in Section 7. In the Appendix we present some properties of the Golden ratio and of what we call the Golden radius, relating it with the different representations.\",\"tag\":null,\"text\":\"The rest of the article is structured as follows: In Section 2 we present the Schwarzschild solution and briefly review some direct features that can be extracted from it. Then we obtain the near-horizon approximation alongside its connection with accelerated frames and the Flamm representation. Section 3 presents a derivation of the Tortoise and the Kruskal representations using the same procedure. Then we show the Diamond representation, which shares a common feature with the former two: they are all (1+1)-conformally flat to hypersurfaces of the Schwarzschild spacetime at constant angles. In Section 4 we introduce the Golden representation for the Schwarzschild spacetime, by slightly deforming the customary Schwarzschild ansatz and dimensionally reducing the Einstein-Hilbert action, until the Golden coordinates are defined. In Section 5 we show how the kinemat-\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0001.pdf\",\"page_no\":2,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"c074303ed71d6de5cb4fb16042183e63d6f61b81636482dff95e955e69ceab8c\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/c074303ed71d6de5cb4fb16042183e63d6f61b81636482dff95e955e69ceab8c.pdf","asset_sha256":"c074303ed71d6de5cb4fb16042183e63d6f61b81636482dff95e955e69ceab8c","document_id":"arxiv:2608.07909","kind":"page","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,125.8,388.55,98.63],\"bbox_pdf\":[110.85,567.57,499.4,666.2],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[110.85,567.57,499.4,666.2],\"words\":[520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph14\",\"source_text\":\"The rest of the article is structured as follows: In Section 2 we present the Schwarzschild solution and briefly review some direct features that can be extracted from it. Then we obtain the near-horizon approximation alongside its connection with accelerated frames and the Flamm representation. Section 3 presents a derivation of the Tortoise and the Kruskal representations using the same procedure. Then we show the Diamond representation, which shares a common feature with the former two: they are all (1+1)-conformally flat to hypersurfaces of the Schwarzschild spacetime at constant angles. In Section 4 we introduce the Golden representation for the Schwarzschild spacetime, by slightly deforming the customary Schwarzschild ansatz and dimensionally reducing the Einstein-Hilbert action, until the Golden coordinates are defined. In Section 5 we show how the kinematics of the (1+1)-dimensional Golden representation yield the same dynamics as the Schwarzschild solution. The arguments of the previous sections allow us to present different visual Golden representations of the Schwarzschild geometry in Section 6. We end with a summary of the work and some final comments in Section 7. In the Appendix we present some properties of the Golden ratio and of what we call the Golden radius, relating it with the different representations.\",\"tag\":null,\"text\":\"ics of the (1+1)-dimensional Golden representation yield the same dynamics as the Schwarzschild solution. The arguments of the previous sections allow us to present different visual Golden representations of the Schwarzschild geometry in Section 6. We end with a summary of the work and some final comments in Section 7. In the Appendix we present some properties of the Golden ratio and of what we call the Golden radius, relating it with the different representations.\"},{\"bbox\":[110.85,247.15,362.32,17.22],\"bbox_pdf\":[110.85,527.64,473.18,544.85],\"fragments\":[{\"bbox_pdf\":[110.85,527.64,473.18,544.85],\"words\":[596,597,598,599,600,601,602]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"heading\",\"source_ref\":\"title15\",\"source_text\":\"2 Near Horizon and the Flamm Parabola\",\"tag\":null,\"text\":\"2 Near Horizon and the Flamm Parabola\"},{\"bbox\":[110.85,278.69,332.15,11.96],\"bbox_pdf\":[110.85,501.35,443.01,513.31],\"fragments\":[{\"bbox_pdf\":[110.85,501.35,443.01,513.31],\"words\":[603,604,605,606,607,608,609,610,611]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph16\",\"source_text\":\"Many analytic solutions to the Einstein’s field equations, namely\",\"tag\":null,\"text\":\"Many analytic solutions to the Einstein’s field equations, namely\"},{\"bbox\":[247.13,309.76,252.26,13.75],\"bbox_pdf\":[247.13,468.49,499.4,482.24],\"fragments\":[{\"bbox_pdf\":[247.13,468.49,499.4,482.24],\"words\":[612,613,614,615,616,617,618,619]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"R_{\\\\mu\\\\nu}-\\\\frac{1}{2}g_{\\\\mu\\\\nu}R=8\\\\pi T_{\\\\mu\\\\nu},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S2.Ex1\",\"source_text\":\"$R_{\\\\mu\\\\nu}-\\\\frac{1}{2}g_{\\\\mu\\\\nu}R=8\\\\pi T_{\\\\mu\\\\nu},$ (2.1)\",\"tag\":\"(2.1)\",\"text\":\"1 Rµν − gµνR = 8πTµν, (2.1) 2\"},{\"bbox\":[110.85,339.88,222.22,11.96],\"bbox_pdf\":[110.85,440.17,333.07,452.12],\"fragments\":[{\"bbox_pdf\":[110.85,440.17,333.07,452.12],\"words\":[620,621,622,623,624,625]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph18\",\"source_text\":\"take the spherically symmetric form [1][19]:\",\"tag\":null,\"text\":\"take the spherically symmetric form [1][19]:\"},{\"bbox\":[211.76,367.82,287.64,12.91],\"bbox_pdf\":[211.76,411.28,499.4,424.18],\"fragments\":[{\"bbox_pdf\":[211.76,411.28,499.4,424.18],\"words\":[626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"ds^{2}=-e^{2\\\\alpha(r)}dt^{2}+e^{2\\\\beta(r)}dr^{2}+r^{2}d\\\\Omega^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S2.Ex2\",\"source_text\":\"$ds^{2}=-e^{2\\\\alpha(r)}dt^{2}+e^{2\\\\beta(r)}dr^{2}+r^{2}d\\\\Omega^{2},$ (2.2)\",\"tag\":\"(2.2)\",\"text\":\"ds 2 = −e 2α(r) dt 2 + e 2β(r) dr 2 + r 2 dΩ 2 , (2.2)\"},{\"bbox\":[110.85,388.9,388.55,173.45],\"bbox_pdf\":[110.85,229.66,499.4,403.1],\"fragments\":[{\"bbox_pdf\":[110.85,229.66,499.4,403.1],\"words\":[645,646,647,648,649,650,651,652,653,654,655,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.986,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph20\",\"source_text\":\"where $d\\\\Omega^{2}=d\\\\theta^{2}+\\\\sin^{2}\\\\theta d\\\\phi^{2}$ . We use units where $c=G=1$ and Greek indices like $\\\\mu,\\\\nu$ run from $0$ to $3$ , while Latin indices like $a,b$ run from $1$ to $2$ . Many of the solutions share the property $\\\\beta=-\\\\alpha$ [20]-[23]. The interpretation of a cosmological or black hole origin of the solutions may be subtle. For instance, if the energy-momentum tensor comes from a constant energy-density filling an isotropic and homogeneous space, the de Sitter or two Lanczos universes can take the same static form. But having distinct spatial curvature, they take a specific FLRW form in the comoving frame [23]. In a similar vein, assuming a central spherical source can lead to either the Schwarzschild-de-Sitter or the Schwarzschild solution [3]. The latter is the simplest of all, where $T_{\\\\mu\\\\nu}=0$ in (2.1) outside a certain radius, implying $R=0$ . Einstein’s equations reduce to $R_{\\\\mu\\\\nu}=0$ , and the metric solution is:\",\"tag\":null,\"text\":\"where dΩ 2 = dθ 2 + sin 2 θdϕ 2 We use units where c = G =1and Greek indices like µ, ν run from 0 to 3, while Latin indices like a, b run from 1 to 2. Many of the solutions share the property β = −α [20]-[23]. The interpretation of a cosmological or black hole origin of the solutions may be subtle. For instance, if the energy-momentum tensor comes from a constant energy-density filling an isotropic and homogeneous space, the de Sitter or two Lanczos universes can take the same static form. But having distinct spatial curvature, they take a specific FLRW form in the comoving frame [23]. In a similar vein, assuming a central spherical source can lead to either the Schwarzschild-de-Sitter or the Schwarzschild solution [3]. The latter is the simplest of all, where Tµν = 0 in (2.1) outside a certain radius, implying R = 0. Einstein’s equations reduce to Rµν = 0, and the metric solution is:\"},{\"bbox\":[204.14,569.18,295.26,37.55],\"bbox_pdf\":[204.14,185.27,499.4,222.82],\"fragments\":[{\"bbox_pdf\":[204.14,185.27,499.4,222.82],\"words\":[811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"ds^{2}=-\\\\left(1-\\\\frac{r_{s}}{r}\\\\right)dt^{2}+\\\\frac{dr^{2}}{1-\\\\frac{r_{s}}{r}}+%\\nr^{2}d\\\\Omega^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"S2.Ex3\",\"source_text\":\"$ds^{2}=-\\\\left(1-\\\\frac{r_{s}}{r}\\\\right)dt^{2}+\\\\frac{dr^{2}}{1-\\\\frac{r_{s}}{r}}+% r^{2}d\\\\Omega^{2},$ (2.3)\",\"tag\":\"(2.3)\",\"text\":\"(︂ rs )︂ dr 2 ds 2 = − 1 − dt 2 + + r 2 dΩ 2 , (2.3) r 1 − rs r\"},{\"bbox\":[110.85,616.03,388.55,26.4],\"bbox_pdf\":[110.85,149.57,499.4,175.97],\"fragments\":[{\"bbox_pdf\":[110.85,149.57,499.4,175.97],\"words\":[837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph22\",\"source_text\":\"where $r_{s}$ is the Schwarzschild radius. This is an appropriate solution outside massive spherical objects, when their angular momentum can be neglected.\",\"tag\":null,\"text\":\"where rs is the Schwarzschild radius. This is an appropriate solution outside massive spherical objects, when their angular momentum can be neglected.\"},{\"bbox\":[110.85,644.92,388.55,28.19],\"bbox_pdf\":[110.85,118.89,499.4,147.08],\"fragments\":[{\"bbox_pdf\":[110.85,118.89,499.4,147.08],\"words\":[859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,880,881,882,883,884,885,886,887,888,889]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph23\",\"source_text\":\"A distant observer will observe a time dilation for an event at $r$ given by $dt=\\\\left(-g_{00}\\\\right)^{-1/2}d\\\\tau$ . Then, when a photon travels along a radial geodesic from a given $r$ , a faraway observer measures that it is gravitationally redshifted by $\\\\omega^{\\\\prime}=\\\\omega_{0}\\\\sqrt{-g_{00}}=\\\\omega_{0}\\\\sqrt{1-r_{s}/r}$ . We shall refer to $\\\\sqrt{1-r_{s}/r}$ as the redshift factor [19]. Also, there is a complementary length contraction for the proper radial distance $d\\\\varrho$ :\",\"tag\":null,\"text\":\"A distant observer will observe a time dilation for an event at r given by −1/2 dt = (−g00) dτ Then, when a photon travels along a radial geodesic from\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0002.pdf\",\"page_no\":3,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph20\",\"c_block\":0.985,\"c_lines\":1.0,\"c_pdf\":0.999,\"c_region\":1.0,\"c_src\":0.993,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.985,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"7b1b3d42c9157de62cfdf519d9c106ff006ee87f6303ebf3397b0dec5cb118ec\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/7b1b3d42c9157de62cfdf519d9c106ff006ee87f6303ebf3397b0dec5cb118ec.pdf","asset_sha256":"7b1b3d42c9157de62cfdf519d9c106ff006ee87f6303ebf3397b0dec5cb118ec","document_id":"arxiv:2608.07909","kind":"page","page_number":3,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[247.13,309.76,252.26,13.75],\"bbox_pdf\":[247.13,468.49,499.4,482.24],\"fragments\":[{\"bbox_pdf\":[247.13,468.49,499.4,482.24],\"words\":[612,613,614,615,616,617,618,619]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"R_{\\\\mu\\\\nu}-\\\\frac{1}{2}g_{\\\\mu\\\\nu}R=8\\\\pi T_{\\\\mu\\\\nu},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S2.Ex1\",\"source_text\":\"$R_{\\\\mu\\\\nu}-\\\\frac{1}{2}g_{\\\\mu\\\\nu}R=8\\\\pi T_{\\\\mu\\\\nu},$ (2.1)\",\"tag\":\"(2.1)\",\"text\":\"1 Rµν − gµνR = 8πTµν, (2.1) 2\"},\"formula_index\":3,\"parent_page_sha256\":\"7b1b3d42c9157de62cfdf519d9c106ff006ee87f6303ebf3397b0dec5cb118ec\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/7b1b3d42c9157de62cfdf519d9c106ff006ee87f6303ebf3397b0dec5cb118ec.pdf","asset_sha256":"7b1b3d42c9157de62cfdf519d9c106ff006ee87f6303ebf3397b0dec5cb118ec","document_id":"arxiv:2608.07909","kind":"formula","page_number":3,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[211.76,367.82,287.64,12.91],\"bbox_pdf\":[211.76,411.28,499.4,424.18],\"fragments\":[{\"bbox_pdf\":[211.76,411.28,499.4,424.18],\"words\":[626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"ds^{2}=-e^{2\\\\alpha(r)}dt^{2}+e^{2\\\\beta(r)}dr^{2}+r^{2}d\\\\Omega^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S2.Ex2\",\"source_text\":\"$ds^{2}=-e^{2\\\\alpha(r)}dt^{2}+e^{2\\\\beta(r)}dr^{2}+r^{2}d\\\\Omega^{2},$ (2.2)\",\"tag\":\"(2.2)\",\"text\":\"ds 2 = −e 2α(r) dt 2 + e 2β(r) dr 2 + r 2 dΩ 2 , (2.2)\"},\"formula_index\":5,\"parent_page_sha256\":\"7b1b3d42c9157de62cfdf519d9c106ff006ee87f6303ebf3397b0dec5cb118ec\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/7b1b3d42c9157de62cfdf519d9c106ff006ee87f6303ebf3397b0dec5cb118ec.pdf","asset_sha256":"7b1b3d42c9157de62cfdf519d9c106ff006ee87f6303ebf3397b0dec5cb118ec","document_id":"arxiv:2608.07909","kind":"formula","page_number":3,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[204.14,569.18,295.26,37.55],\"bbox_pdf\":[204.14,185.27,499.4,222.82],\"fragments\":[{\"bbox_pdf\":[204.14,185.27,499.4,222.82],\"words\":[811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"ds^{2}=-\\\\left(1-\\\\frac{r_{s}}{r}\\\\right)dt^{2}+\\\\frac{dr^{2}}{1-\\\\frac{r_{s}}{r}}+%\\nr^{2}d\\\\Omega^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"S2.Ex3\",\"source_text\":\"$ds^{2}=-\\\\left(1-\\\\frac{r_{s}}{r}\\\\right)dt^{2}+\\\\frac{dr^{2}}{1-\\\\frac{r_{s}}{r}}+% r^{2}d\\\\Omega^{2},$ (2.3)\",\"tag\":\"(2.3)\",\"text\":\"(︂ rs )︂ dr 2 ds 2 = − 1 − dt 2 + + r 2 dΩ 2 , (2.3) r 1 − rs r\"},\"formula_index\":7,\"parent_page_sha256\":\"7b1b3d42c9157de62cfdf519d9c106ff006ee87f6303ebf3397b0dec5cb118ec\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/7b1b3d42c9157de62cfdf519d9c106ff006ee87f6303ebf3397b0dec5cb118ec.pdf","asset_sha256":"7b1b3d42c9157de62cfdf519d9c106ff006ee87f6303ebf3397b0dec5cb118ec","document_id":"arxiv:2608.07909","kind":"formula","page_number":3,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,125.8,388.55,55.29],\"bbox_pdf\":[110.85,610.91,499.4,666.2],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[110.85,610.91,499.4,666.2],\"words\":[890,891,892,893,894,903,895,896,897,898,899,900,904,901,902,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph23\",\"source_text\":\"A distant observer will observe a time dilation for an event at $r$ given by $dt=\\\\left(-g_{00}\\\\right)^{-1/2}d\\\\tau$ . Then, when a photon travels along a radial geodesic from a given $r$ , a faraway observer measures that it is gravitationally redshifted by $\\\\omega^{\\\\prime}=\\\\omega_{0}\\\\sqrt{-g_{00}}=\\\\omega_{0}\\\\sqrt{1-r_{s}/r}$ . We shall refer to $\\\\sqrt{1-r_{s}/r}$ as the redshift factor [19]. Also, there is a complementary length contraction for the proper radial distance $d\\\\varrho$ :\",\"tag\":null,\"text\":\"a given r, a faraway √︁ observer measures that it is gravitationally √︁ redshifted by √ ω ′ = ω0 −g00 = ω0 1 − rs/r. We shall refer to 1 − rs/r as the redshift factor [19]. Also, there is a complementary length contraction for the proper radial distance dϱ:\"},{\"bbox\":[261.48,185.53,237.92,37.66],\"bbox_pdf\":[261.48,568.8,499.4,606.47],\"fragments\":[{\"bbox_pdf\":[261.48,568.8,499.4,606.47],\"words\":[941,942,943,944,945,946,947,948,949,950]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"dr=d\\\\varrho\\\\sqrt{1-\\\\frac{r_{s}}{r}}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S2.Ex4\",\"source_text\":\"$dr=d\\\\varrho\\\\sqrt{1-\\\\frac{r_{s}}{r}}.$ (2.4)\",\"tag\":\"(2.4)\",\"text\":\"√︃ rs dr = dϱ 1 − . (2.4) r\"},{\"bbox\":[110.85,229.95,388.55,87.12],\"bbox_pdf\":[110.85,474.94,499.4,562.05],\"fragments\":[{\"bbox_pdf\":[110.85,474.94,499.4,562.05],\"words\":[951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1030,1032,1033,1034,1035,1036,1037,1038,1040]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.968,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph25\",\"source_text\":\"Thus, the $(t,r)$ Schwarzschild coordinates are the time and radial coordinates measured by an observer at infinity. The isometries in Eq. (2.3) imply the two Killing vectors $T=\\\\partial_{t}$ and $\\\\Phi=\\\\partial_{\\\\phi}$ . Spherical symmetry ensures two more independent rotational isometries, completing a total of four Killing vectors for the Schwarzschild spacetime. Given $T^{\\\\mu}=\\\\delta_{0}^{\\\\mu}$ , a static observer has four-velocity $u^{\\\\mu}=(-g_{a\\\\beta}T^{\\\\alpha}T^{\\\\beta})^{-1/2}T^{\\\\mu}$ . That is, we have $u^{\\\\mu}=\\\\delta_{0}^{\\\\mu}/\\\\sqrt{-g_{00}}$ .\",\"tag\":null,\"text\":\"Thus, the (t, r) Schwarzschild coordinates are the time and radial coordinates measured by an observer at infinity. The isometries in Eq. (2.3) imply the two Killing vectors T = ∂t and Φ = ∂ϕ. Spherical symmetry ensures two more independent rotational isometries, completing a total of four Killing µ µ vectors for the Schwarzschild spacetime. Given T = δ0 a static observer µ √ µ α β −1/2 µ µ has four-velocity u = (−gaβT T T That is, we have u = δ0 −g00.\"},{\"bbox\":[110.85,316.62,388.54,26.4],\"bbox_pdf\":[110.85,448.98,499.4,475.38],\"fragments\":[{\"bbox_pdf\":[110.85,448.98,499.4,475.38],\"words\":[1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph26\",\"source_text\":\"Let us approximate the geometry just above $r_{s}$ . Since $d\\\\varrho$ in (2.4) represents the proper radial infinitesimal distance at a fixed $r$ , we integrate it:\",\"tag\":null,\"text\":\"Let us approximate the geometry just above rs. Since dϱ in (2.4) represents the proper radial infinitesimal distance at a fixed r, we integrate it:\"},{\"bbox\":[202.35,346.26,297.05,38.81],\"bbox_pdf\":[202.35,406.93,499.4,445.74],\"fragments\":[{\"bbox_pdf\":[202.35,406.93,499.4,445.74],\"words\":[1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\varrho=\\\\sqrt{r\\\\left(r-r_{s}\\\\right)}+\\\\frac{r_{s}}{2}\\\\cosh^{-1}\\\\left(2\\\\frac{r}{%\\nr_{s}}-1\\\\right).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S2.Ex5\",\"source_text\":\"$\\\\varrho=\\\\sqrt{r\\\\left(r-r_{s}\\\\right)}+\\\\frac{r_{s}}{2}\\\\cosh^{-1}\\\\left(2\\\\frac{r}{% r_{s}}-1\\\\right).$ (2.5)\",\"tag\":\"(2.5)\",\"text\":\"(︃ )︃ √︁ rs r −1 ϱ = r (r − rs) + cosh 2 − 1 . (2.5) 2 rs\"},{\"bbox\":[110.85,398.05,388.54,45.39],\"bbox_pdf\":[110.85,348.56,499.4,393.95],\"fragments\":[{\"bbox_pdf\":[110.85,348.56,499.4,393.95],\"words\":[1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1104,1101,1102,1103,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.818,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph28\",\"source_text\":\"For $r>r_{s}$ , the identity $\\\\cosh^{-1}(2r/r_{s}-1)=2\\\\sinh^{-1}\\\\sqrt{r/r_{s}-1}$ is valid. Then, for $r\\\\cong r_{s}$ one can approximate $\\\\sqrt{r\\\\left(r-r_{s}\\\\right)}\\\\approx\\\\sqrt{r_{s}\\\\left(r-r_{s}\\\\right)}$ . Since $\\\\sinh^{-1}z\\\\approx z$ for $z\\\\ll 1$ , this yields $\\\\varrho\\\\approx 2r_{s}\\\\sqrt{r/r_{s}-1}$ , or\",\"tag\":null,\"text\":\"√︁ −1 −1 For r > rs, the identity cosh (2r/rs − 1) √︁ = 2 sinh √︁ r/rs − 1 is valid. Then, for r ∼= rs one can approximate √︁ r (r − rs) ≈ rs (r − rs). Since −1 sinh z ≈ z for z ≪ 1, this yields ϱ ≈ 2rs r/rs − 1, or\"},{\"bbox\":[271.81,463.29,227.59,21.95],\"bbox_pdf\":[271.81,306.76,499.4,328.71],\"fragments\":[{\"bbox_pdf\":[271.81,306.76,499.4,328.71],\"words\":[1148,1149,1150,1151,1152,1153,1154,1155,1156]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"r=r_{s}+\\\\frac{\\\\varrho^{2}}{4r_{s}}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S2.Ex6\",\"source_text\":\"$r=r_{s}+\\\\frac{\\\\varrho^{2}}{4r_{s}}.$ (2.6)\",\"tag\":\"(2.6)\",\"text\":\"ϱ 2 r = rs + . (2.6) 4rs\"},{\"bbox\":[110.85,495.15,388.55,28.19],\"bbox_pdf\":[110.85,268.65,499.4,296.85],\"fragments\":[{\"bbox_pdf\":[110.85,268.65,499.4,296.85],\"words\":[1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.8,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph30\",\"source_text\":\"Also, since $\\\\varrho\\\\ll 2r_{s}$ , then $1-r_{s}/r=\\\\varrho^{2}/\\\\left(\\\\varrho^{2}+4r_{s}^{2}\\\\right)\\\\approx\\\\varrho^{2}/(4% r_{s}^{2})$ . By rescaling the time as $t=2r_{s}T$ , the first two terms in Eq. (2.3) become $-\\\\varrho^{2}dT^{2}+d\\\\varrho^{2}$ .\",\"tag\":null,\"text\":\"Also, since ϱ ≪ 2rs, then 1 − rs/r = ϱ 2 / (ϱ 2 + 4r 2 2 /(4r 2 s) ≈ ϱ s). By rescaling the time as t = 2rsT the first two terms in Eq. (2.3) become −ϱ 2 dT 2 + dϱ 2\"},{\"bbox\":[110.85,524.04,388.54,40.85],\"bbox_pdf\":[110.85,227.11,499.4,267.96],\"fragments\":[{\"bbox_pdf\":[110.85,227.11,499.4,267.96],\"words\":[1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.969,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph31\",\"source_text\":\"Consider very small variations in the angle $\\\\theta$ respect to the (arbitrary) $z$ -direction. Then $x\\\\cong r_{s}\\\\theta\\\\cos\\\\phi$ and $y\\\\cong r_{s}\\\\theta\\\\sin\\\\phi$ . Thus, $r_{s}^{2}(d\\\\theta^{2}+\\\\theta^{2}d\\\\phi^{2})=dx^{2}+dy^{2}$ can be substituted into the last terms of Eq. (2.3), resulting [24]:\",\"tag\":null,\"text\":\"Consider very small variations in the angle θ respect to the (arbitrary) z-direction. Then x ∼= rsθ cos ϕ and y ∼= rsθ sin ϕ. Thus, r 2 2 + θ 2 dϕ 2 = s(dθ dx 2 + dy 2 can be substituted into the last terms of Eq. (2.3), resulting [24]:\"},{\"bbox\":[220.51,580.88,278.89,12.91],\"bbox_pdf\":[220.51,198.22,499.4,211.12],\"fragments\":[{\"bbox_pdf\":[220.51,198.22,499.4,211.12],\"words\":[1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"ds^{2}=-\\\\varrho^{2}dT^{2}+d\\\\varrho^{2}+dx^{2}+dy^{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S2.Ex7\",\"source_text\":\"$ds^{2}=-\\\\varrho^{2}dT^{2}+d\\\\varrho^{2}+dx^{2}+dy^{2}.$ (2.7)\",\"tag\":\"(2.7)\",\"text\":\"ds 2 = −ϱ 2 dT 2 + dϱ 2 + dx 2 + dy 2 . (2.7)\"},{\"bbox\":[110.85,602.51,388.55,69.74],\"bbox_pdf\":[110.85,119.75,499.4,189.49],\"fragments\":[{\"bbox_pdf\":[110.85,119.75,499.4,189.49],\"words\":[1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"paragraph\",\"source_ref\":\"paragraph33\",\"source_text\":\"This approximation is Minkowski spacetime, given the Rindler transformations $t=\\\\varrho\\\\sinh T$ and $z=\\\\varrho\\\\cosh T$ . The sections of hyperbolas induced by $z^{2}-t^{2}=\\\\varrho^{2}$ when $\\\\varrho=const.$ have slope less than $1$ in a $t-z$ diagram. They are timelike trajectories with constant acceleration $a=\\\\varrho^{-1}$ . Because Eq. (2.6) implies that $\\\\varrho\\\\rightarrow 0$ as $r\\\\rightarrow r_{s}$ , there is a direct association between the Rindler horizon for a highly accelerated (flat) frame in a radial direction and the Schwarzschild (curved) event horizon [25][26].\",\"tag\":null,\"text\":\"This approximation is Minkowski spacetime, given the Rindler transformations t = ϱ sinh T and z = ϱ cosh T The sections of hyperbolas induced by z 2 − t 2 = ϱ 2 when ϱ = const. have slope less than 1 in a t − z diagram. They are timelike trajectories with constant acceleration a = ϱ −1 Because Eq. (2.6) implies that ϱ → 0 as r → rs, there is a direct association between the\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0003.pdf\",\"page_no\":4,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph28\",\"c_block\":0.8,\"c_lines\":1.0,\"c_pdf\":0.999,\"c_region\":1.0,\"c_src\":0.953,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.8,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9.pdf","asset_sha256":"5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9","document_id":"arxiv:2608.07909","kind":"page","page_number":4,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[261.48,185.53,237.92,37.66],\"bbox_pdf\":[261.48,568.8,499.4,606.47],\"fragments\":[{\"bbox_pdf\":[261.48,568.8,499.4,606.47],\"words\":[941,942,943,944,945,946,947,948,949,950]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"dr=d\\\\varrho\\\\sqrt{1-\\\\frac{r_{s}}{r}}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S2.Ex4\",\"source_text\":\"$dr=d\\\\varrho\\\\sqrt{1-\\\\frac{r_{s}}{r}}.$ (2.4)\",\"tag\":\"(2.4)\",\"text\":\"√︃ rs dr = dϱ 1 − . (2.4) r\"},\"formula_index\":1,\"parent_page_sha256\":\"5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9.pdf","asset_sha256":"5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9","document_id":"arxiv:2608.07909","kind":"formula","page_number":4,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[202.35,346.26,297.05,38.81],\"bbox_pdf\":[202.35,406.93,499.4,445.74],\"fragments\":[{\"bbox_pdf\":[202.35,406.93,499.4,445.74],\"words\":[1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\varrho=\\\\sqrt{r\\\\left(r-r_{s}\\\\right)}+\\\\frac{r_{s}}{2}\\\\cosh^{-1}\\\\left(2\\\\frac{r}{%\\nr_{s}}-1\\\\right).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S2.Ex5\",\"source_text\":\"$\\\\varrho=\\\\sqrt{r\\\\left(r-r_{s}\\\\right)}+\\\\frac{r_{s}}{2}\\\\cosh^{-1}\\\\left(2\\\\frac{r}{% r_{s}}-1\\\\right).$ (2.5)\",\"tag\":\"(2.5)\",\"text\":\"(︃ )︃ √︁ rs r −1 ϱ = r (r − rs) + cosh 2 − 1 . (2.5) 2 rs\"},\"formula_index\":4,\"parent_page_sha256\":\"5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9.pdf","asset_sha256":"5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9","document_id":"arxiv:2608.07909","kind":"formula","page_number":4,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[271.81,463.29,227.59,21.95],\"bbox_pdf\":[271.81,306.76,499.4,328.71],\"fragments\":[{\"bbox_pdf\":[271.81,306.76,499.4,328.71],\"words\":[1148,1149,1150,1151,1152,1153,1154,1155,1156]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"r=r_{s}+\\\\frac{\\\\varrho^{2}}{4r_{s}}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S2.Ex6\",\"source_text\":\"$r=r_{s}+\\\\frac{\\\\varrho^{2}}{4r_{s}}.$ (2.6)\",\"tag\":\"(2.6)\",\"text\":\"ϱ 2 r = rs + . (2.6) 4rs\"},\"formula_index\":6,\"parent_page_sha256\":\"5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9.pdf","asset_sha256":"5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9","document_id":"arxiv:2608.07909","kind":"formula","page_number":4,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[220.51,580.88,278.89,12.91],\"bbox_pdf\":[220.51,198.22,499.4,211.12],\"fragments\":[{\"bbox_pdf\":[220.51,198.22,499.4,211.12],\"words\":[1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"ds^{2}=-\\\\varrho^{2}dT^{2}+d\\\\varrho^{2}+dx^{2}+dy^{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S2.Ex7\",\"source_text\":\"$ds^{2}=-\\\\varrho^{2}dT^{2}+d\\\\varrho^{2}+dx^{2}+dy^{2}.$ (2.7)\",\"tag\":\"(2.7)\",\"text\":\"ds 2 = −ϱ 2 dT 2 + dϱ 2 + dx 2 + dy 2 . (2.7)\"},\"formula_index\":9,\"parent_page_sha256\":\"5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9.pdf","asset_sha256":"5fb2763fa9bbe16b1811671f331bccf4dc66937954e367297b3d24984461cef9","document_id":"arxiv:2608.07909","kind":"formula","page_number":4,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,125.8,388.54,26.4],\"bbox_pdf\":[110.85,639.8,499.4,666.2],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[110.85,639.8,499.4,666.2],\"words\":[1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph33\",\"source_text\":\"This approximation is Minkowski spacetime, given the Rindler transformations $t=\\\\varrho\\\\sinh T$ and $z=\\\\varrho\\\\cosh T$ . The sections of hyperbolas induced by $z^{2}-t^{2}=\\\\varrho^{2}$ when $\\\\varrho=const.$ have slope less than $1$ in a $t-z$ diagram. They are timelike trajectories with constant acceleration $a=\\\\varrho^{-1}$ . Because Eq. (2.6) implies that $\\\\varrho\\\\rightarrow 0$ as $r\\\\rightarrow r_{s}$ , there is a direct association between the Rindler horizon for a highly accelerated (flat) frame in a radial direction and the Schwarzschild (curved) event horizon [25][26].\",\"tag\":null,\"text\":\"Rindler horizon for a highly accelerated (flat) frame in a radial direction and the Schwarzschild (curved) event horizon [25][26].\"},{\"bbox\":[110.85,154.69,388.55,56.31],\"bbox_pdf\":[110.85,581.0,499.4,637.31],\"fragments\":[{\"bbox_pdf\":[110.85,581.0,499.4,637.31],\"words\":[1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426,1427,1428,1429,1430,1431,1432,1433,1434,1435]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.95,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph34\",\"source_text\":\"Embedding diagram. By setting $dt=d\\\\theta=0$ in Eq. (2.3), we obtain a dimensionally reduced version that can be embedded into a 3D Euclidean space, for which cylindrical coordinates are appropriate. That is, one embeds the metric $\\\\left(1-\\\\frac{r_{s}}{r}\\\\right)^{-1}dr^{2}+r^{2}d\\\\phi^{2}$ into the target space:\",\"tag\":null,\"text\":\"Embedding diagram. By setting dt = dθ = 0 in Eq. (2.3), we obtain a dimensionally reduced version that can be embedded into a 3D Euclidean space, for which cylindrical coordinates are appropriate. That is, one embeds (︁ )︁−1 rs 2 2 2 the metric 1 − dr + r dϕ into the target space: r\"},{\"bbox\":[240.5,222.3,258.89,12.91],\"bbox_pdf\":[240.5,556.8,499.39,569.7],\"fragments\":[{\"bbox_pdf\":[240.5,556.8,499.39,569.7],\"words\":[1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"d\\\\Sigma^{2}=d\\\\rho^{2}+\\\\rho^{2}d\\\\phi^{2}+dz^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S2.Ex8\",\"source_text\":\"$d\\\\Sigma^{2}=d\\\\rho^{2}+\\\\rho^{2}d\\\\phi^{2}+dz^{2},$ (2.8)\",\"tag\":\"(2.8)\",\"text\":\"dΣ 2 = dρ 2 + ρ 2 dϕ 2 + dz 2 , (2.8)\"},{\"bbox\":[110.85,247.45,388.54,26.4],\"bbox_pdf\":[110.85,518.15,499.39,544.55],\"fragments\":[{\"bbox_pdf\":[110.85,518.15,499.39,544.55],\"words\":[1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph36\",\"source_text\":\"where $(\\\\rho,\\\\phi,z)$ are cylindrical coordinates. By identifying the azimuthal angles, one has $\\\\rho=r$ . The two-dimensional surface induces the dependence:\",\"tag\":null,\"text\":\"where (ρ, ϕ, z) are cylindrical coordinates. By identifying the azimuthal angles, one has ρ = r. The two-dimensional surface induces the dependence:\"},{\"bbox\":[261.96,275.93,237.43,38.56],\"bbox_pdf\":[261.96,477.51,499.39,516.07],\"fragments\":[{\"bbox_pdf\":[261.96,477.51,499.39,516.07],\"words\":[1475,1476,1477,1478,1479,1480,1481,1482,1483]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"dz=\\\\sqrt{\\\\frac{r_{s}}{r-r_{s}}}dr.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S2.Ex9\",\"source_text\":\"$dz=\\\\sqrt{\\\\frac{r_{s}}{r-r_{s}}}dr.$ (2.9)\",\"tag\":\"(2.9)\",\"text\":\"√︃ rs dz = dr. (2.9) r − rs\"},{\"bbox\":[110.85,323.2,388.54,28.19],\"bbox_pdf\":[110.85,440.61,499.4,468.8],\"fragments\":[{\"bbox_pdf\":[110.85,440.61,499.4,468.8],\"words\":[1484,1485,1486,1487,1488,1489,1490,1491,1498,1492,1493,1494,1495,1496,1497,1499,1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph38\",\"source_text\":\"We chose the positive solution, implying a positive slope for the graph $z=z(r)$ . This can be integrated to yield $z=\\\\sqrt{4r_{s}\\\\left(r-r_{s}\\\\right)}$ , or\",\"tag\":null,\"text\":\"We chose the positive solution, implying a positive √︁ slope for the graph z = z(r). This can be integrated to yield z = 4rs (r − rs), or\"},{\"bbox\":[271.81,369.61,227.6,21.95],\"bbox_pdf\":[271.81,400.44,499.4,422.39],\"fragments\":[{\"bbox_pdf\":[271.81,400.44,499.4,422.39],\"words\":[1513,1514,1515,1516,1517,1518,1519,1520]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"r=\\\\frac{z^{2}}{4r_{s}}+r_{s}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S2.Ex10\",\"source_text\":\"$r=\\\\frac{z^{2}}{4r_{s}}+r_{s}.$ (2.10)\",\"tag\":\"(2.10)\",\"text\":\"z 2 r = + rs. (2.10) 4rs\"},{\"bbox\":[110.85,398.66,388.54,26.4],\"bbox_pdf\":[110.85,366.94,499.39,393.34],\"fragments\":[{\"bbox_pdf\":[110.85,366.94,499.39,393.34],\"words\":[1521,1522,1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534,1535,1536,1537,1538,1539,1540,1541,1542,1543]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph40\",\"source_text\":\"By allowing negative values of $z$ and recalling the azimuthal symmetry in Eq. (2.8), the familiar Flamm paraboloid is obtained (Fig. 1) [27].\",\"tag\":null,\"text\":\"By allowing negative values of z and recalling the azimuthal symmetry in Eq. (2.8), the familiar Flamm paraboloid is obtained (Fig. 1) [27].\"},{\"bbox\":[169.14,441.73,271.99,203.78],\"bbox_pdf\":[169.14,146.48,441.13,350.27],\"float_ref\":\"S2.F1\",\"fragments\":[{\"bbox_pdf\":[169.14,146.48,441.13,350.27],\"words\":[]}],\"html\":null,\"kind\":\"figure\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"figure\",\"source_ref\":\"S2.F1\",\"source_text\":\"\",\"tag\":null,\"text\":\"\"},{\"bbox\":[123.68,657.97,362.89,11.96],\"bbox_pdf\":[123.68,122.08,486.57,134.03],\"float_ref\":\"S2.F1\",\"fragments\":[{\"bbox_pdf\":[123.68,122.08,486.57,134.03],\"words\":[1544,1545,1546,1547,1548,1549,1550,1551,1552,1553]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"caption\",\"source_ref\":\"caption42\",\"source_text\":\"Figure 1: Flamm Paraboloid. Spatial Embedding of the Schwarzschild Geometry.\",\"tag\":null,\"text\":\"Figure 1: Flamm Paraboloid. Spatial Embedding of the Schwarzschild Geometry.\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0004.pdf\",\"page_no\":5,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph34\",\"c_block\":0.947,\"c_lines\":1.0,\"c_pdf\":0.999,\"c_region\":1.0,\"c_src\":0.986,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.947,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"e50573b509be83fd0f4fff98fee9e8e630ccd7904c506b4ee9c841291f85381f\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/e50573b509be83fd0f4fff98fee9e8e630ccd7904c506b4ee9c841291f85381f.pdf","asset_sha256":"e50573b509be83fd0f4fff98fee9e8e630ccd7904c506b4ee9c841291f85381f","document_id":"arxiv:2608.07909","kind":"page","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[240.5,222.3,258.89,12.91],\"bbox_pdf\":[240.5,556.8,499.39,569.7],\"fragments\":[{\"bbox_pdf\":[240.5,556.8,499.39,569.7],\"words\":[1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"d\\\\Sigma^{2}=d\\\\rho^{2}+\\\\rho^{2}d\\\\phi^{2}+dz^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S2.Ex8\",\"source_text\":\"$d\\\\Sigma^{2}=d\\\\rho^{2}+\\\\rho^{2}d\\\\phi^{2}+dz^{2},$ (2.8)\",\"tag\":\"(2.8)\",\"text\":\"dΣ 2 = dρ 2 + ρ 2 dϕ 2 + dz 2 , (2.8)\"},\"formula_index\":2,\"parent_page_sha256\":\"e50573b509be83fd0f4fff98fee9e8e630ccd7904c506b4ee9c841291f85381f\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/e50573b509be83fd0f4fff98fee9e8e630ccd7904c506b4ee9c841291f85381f.pdf","asset_sha256":"e50573b509be83fd0f4fff98fee9e8e630ccd7904c506b4ee9c841291f85381f","document_id":"arxiv:2608.07909","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[261.96,275.93,237.43,38.56],\"bbox_pdf\":[261.96,477.51,499.39,516.07],\"fragments\":[{\"bbox_pdf\":[261.96,477.51,499.39,516.07],\"words\":[1475,1476,1477,1478,1479,1480,1481,1482,1483]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"dz=\\\\sqrt{\\\\frac{r_{s}}{r-r_{s}}}dr.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S2.Ex9\",\"source_text\":\"$dz=\\\\sqrt{\\\\frac{r_{s}}{r-r_{s}}}dr.$ (2.9)\",\"tag\":\"(2.9)\",\"text\":\"√︃ rs dz = dr. (2.9) r − rs\"},\"formula_index\":4,\"parent_page_sha256\":\"e50573b509be83fd0f4fff98fee9e8e630ccd7904c506b4ee9c841291f85381f\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/e50573b509be83fd0f4fff98fee9e8e630ccd7904c506b4ee9c841291f85381f.pdf","asset_sha256":"e50573b509be83fd0f4fff98fee9e8e630ccd7904c506b4ee9c841291f85381f","document_id":"arxiv:2608.07909","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[271.81,369.61,227.6,21.95],\"bbox_pdf\":[271.81,400.44,499.4,422.39],\"fragments\":[{\"bbox_pdf\":[271.81,400.44,499.4,422.39],\"words\":[1513,1514,1515,1516,1517,1518,1519,1520]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"r=\\\\frac{z^{2}}{4r_{s}}+r_{s}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S2.Ex10\",\"source_text\":\"$r=\\\\frac{z^{2}}{4r_{s}}+r_{s}.$ (2.10)\",\"tag\":\"(2.10)\",\"text\":\"z 2 r = + rs. (2.10) 4rs\"},\"formula_index\":6,\"parent_page_sha256\":\"e50573b509be83fd0f4fff98fee9e8e630ccd7904c506b4ee9c841291f85381f\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/e50573b509be83fd0f4fff98fee9e8e630ccd7904c506b4ee9c841291f85381f.pdf","asset_sha256":"e50573b509be83fd0f4fff98fee9e8e630ccd7904c506b4ee9c841291f85381f","document_id":"arxiv:2608.07909","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,125.44,388.54,42.99],\"bbox_pdf\":[110.85,623.56,499.4,666.56],\"fragments\":[{\"bbox_pdf\":[110.85,623.56,499.4,666.56],\"words\":[2362,2363,2364,2365,2366,2367,2368,2369,2370,2371,2373,2374,2375,2376,2377,2378,2379,2380,2381,2382,2383,2384,2385,2386,2387,2388,2389,2390,2391,2392,2393,2394,2395,2396,2397,2398,2399,2400,2401,2403,2404,2405,2406,2407,2408,2409,2410]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph65\",\"source_text\":\"The conformal factor now is $\\\\omega^{-2}=\\\\gamma(d\\\\rho/dr)^{2}$ . Performing the derivative yields $\\\\omega^{2}=4r_{s}^{3}r^{-1}e^{-r/r_{s}}$ . By using the Rindler transformations $\\\\zeta=\\\\rho\\\\sinh\\\\tau$ and $X=\\\\rho\\\\cosh\\\\tau$ . Given (3.9) and $\\\\tau=t/2r_{s}$ , one obtains:\",\"tag\":null,\"text\":\"The conformal factor now is ω −2 = γ(dρ/dr) 2 Performing the derivative yields ω 2 = 4r 3 −1 e −r/rs. By using the Rindler transformations ζ = ρ sinh τ sr and X = ρ cosh τ Given (3.9) and τ = t/2rs, one obtains:\"},{\"bbox\":[238.97,167.29,260.42,56.63],\"bbox_pdf\":[238.97,568.08,499.4,624.71],\"fragments\":[{\"bbox_pdf\":[238.97,568.08,499.4,624.71],\"words\":[2411,2412,2413,2414,2415,2416,2417,2418,2419,2420,2421,2422,2423,2424,2425,2426,2427,2428,2429,2430,2431,2432,2433,2434,2435,2436,2437,2438,2439]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{array}[]{c}\\\\zeta=e^{\\\\frac{r}{2r_{s}}}\\\\sqrt{\\\\frac{r}{r_{s}}-1}\\\\sinh\\\\frac%\\n{t}{2r_{s}},\\\\\\\\\\n\\\\\\\\\\nX=e^{\\\\frac{r}{2r_{s}}}\\\\sqrt{\\\\frac{r}{r_{s}}-1}\\\\cosh\\\\frac{t}{2r_{s}}.\\\\end{array}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S3.Ex10\",\"source_text\":\"$\\\\begin{array}[]{c}\\\\zeta=e^{\\\\frac{r}{2r_{s}}}\\\\sqrt{\\\\frac{r}{r_{s}}-1}\\\\sinh\\\\frac% {t}{2r_{s}},\\\\\\\\ \\\\\\\\ X=e^{\\\\frac{r}{2r_{s}}}\\\\sqrt{\\\\frac{r}{r_{s}}-1}\\\\cosh\\\\frac{t}{2r_{s}}.\\\\end{array}$ (3.10)\",\"tag\":\"(3.10)\",\"text\":\"r √︁ 2rs r t ζ = e − 1 sinh , rs 2rs (3.10) r √︁ 2rs r t X = e − 1 cosh . rs 2rs\"},{\"bbox\":[110.85,228.04,301.73,11.96],\"bbox_pdf\":[110.85,552.0,412.58,563.96],\"fragments\":[{\"bbox_pdf\":[110.85,552.0,412.58,563.96],\"words\":[2440,2441,2442,2443,2444,2445,2446,2447,2448,2449]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph67\",\"source_text\":\"The complete metric in Eq. (2.3) has been transformed to:\",\"tag\":null,\"text\":\"The complete metric in Eq. (2.3) has been transformed to:\"},{\"bbox\":[204.37,246.86,295.03,29.84],\"bbox_pdf\":[204.37,515.3,499.4,545.14],\"fragments\":[{\"bbox_pdf\":[204.37,515.3,499.4,545.14],\"words\":[2450,2451,2452,2453,2454,2455,2456,2457,2458,2459,2460,2461,2462,2463,2464,2465,2466,2467,2468,2469,2470,2471]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"ds^{2}=\\\\frac{4r_{s}^{3}e^{-r/r_{s}}}{r}\\\\left(-d\\\\zeta^{2}+dX^{2}\\\\right)+r^{2}d%\\n\\\\Omega^{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S3.Ex11\",\"source_text\":\"$ds^{2}=\\\\frac{4r_{s}^{3}e^{-r/r_{s}}}{r}\\\\left(-d\\\\zeta^{2}+dX^{2}\\\\right)+r^{2}d% \\\\Omega^{2}.$ (3.11)\",\"tag\":\"(3.11)\",\"text\":\"4r 3 −r/rs (︁ )︁ ds 2 = se −dζ 2 + dX 2 + r 2 dΩ 2 . (3.11) r\"},{\"bbox\":[128.41,280.82,185.3,11.96],\"bbox_pdf\":[128.41,499.22,313.71,511.18],\"fragments\":[{\"bbox_pdf\":[128.41,499.22,313.71,511.18],\"words\":[2472,2473,2474,2475,2476,2477,2478]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph69\",\"source_text\":\"Also, from Eq. (3.10) it follows that\",\"tag\":null,\"text\":\"Also, from Eq. (3.10) it follows that\"},{\"bbox\":[240.3,300.17,259.09,30.04],\"bbox_pdf\":[240.3,461.79,499.39,491.83],\"fragments\":[{\"bbox_pdf\":[240.3,461.79,499.39,491.83],\"words\":[2479,2480,2481,2482,2483,2484,2485,2486,2487,2488,2489,2490,2491,2492,2493,2494,2495]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"X^{2}-\\\\zeta^{2}=e^{\\\\frac{r}{r_{s}}}\\\\left(\\\\frac{r}{r_{s}}-1\\\\right),\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S3.Ex12\",\"source_text\":\"$X^{2}-\\\\zeta^{2}=e^{\\\\frac{r}{r_{s}}}\\\\left(\\\\frac{r}{r_{s}}-1\\\\right),$ (3.12)\",\"tag\":\"(3.12)\",\"text\":\"(︃ )︃ 2 2 r r rs X − ζ = e − 1 , (3.12) rs\"},{\"bbox\":[110.85,335.03,388.54,26.4],\"bbox_pdf\":[110.85,430.57,499.4,456.97],\"fragments\":[{\"bbox_pdf\":[110.85,430.57,499.4,456.97],\"words\":[2496,2497,2498,2499,2500,2501,2502,2503,2504,2505,2506,2507,2508,2509,2510,2511,2512,2513,2514,2515,2516,2517,2518,2519]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph71\",\"source_text\":\"which relates to the concept of maximal extension. The transformations (3.10) are associated to the exterior region of the black hole (see Fig. 2).\",\"tag\":null,\"text\":\"which relates to the concept of maximal extension. The transformations (3.10) are associated to the exterior region of the black hole (see Fig. 2).\"},{\"bbox\":[169.14,376.02,271.98,271.98],\"bbox_pdf\":[169.14,143.99,441.12,415.98],\"float_ref\":\"S3.F2\",\"fragments\":[{\"bbox_pdf\":[169.14,143.99,441.12,415.98],\"words\":[]}],\"html\":null,\"kind\":\"figure\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"figure\",\"source_ref\":\"S3.F2\",\"source_text\":\"\",\"tag\":null,\"text\":\"\"},{\"bbox\":[110.85,657.97,388.54,11.96],\"bbox_pdf\":[110.85,122.08,499.4,134.03],\"float_ref\":\"S3.F2\",\"fragments\":[{\"bbox_pdf\":[110.85,122.08,499.4,134.03],\"words\":[2520,2521,2522,2523,2524,2525,2526,2527,2528,2529]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"caption\",\"source_ref\":\"caption73\",\"source_text\":\"Figure 2: Maximally extended Schwarzschild spacetime, from the Kruskal representation.\",\"tag\":null,\"text\":\"Figure 2: Maximally extended Schwarzschild spacetime, from the Kruskal representation.\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0007.pdf\",\"page_no\":8,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":0.998,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.998,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"12ef9412ab4ca9707f6e49bee78e3f62e59420ab92b0bd9bc236578b52709104\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/12ef9412ab4ca9707f6e49bee78e3f62e59420ab92b0bd9bc236578b52709104.pdf","asset_sha256":"12ef9412ab4ca9707f6e49bee78e3f62e59420ab92b0bd9bc236578b52709104","document_id":"arxiv:2608.07909","kind":"page","page_number":8,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[238.97,167.29,260.42,56.63],\"bbox_pdf\":[238.97,568.08,499.4,624.71],\"fragments\":[{\"bbox_pdf\":[238.97,568.08,499.4,624.71],\"words\":[2411,2412,2413,2414,2415,2416,2417,2418,2419,2420,2421,2422,2423,2424,2425,2426,2427,2428,2429,2430,2431,2432,2433,2434,2435,2436,2437,2438,2439]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{array}[]{c}\\\\zeta=e^{\\\\frac{r}{2r_{s}}}\\\\sqrt{\\\\frac{r}{r_{s}}-1}\\\\sinh\\\\frac%\\n{t}{2r_{s}},\\\\\\\\\\n\\\\\\\\\\nX=e^{\\\\frac{r}{2r_{s}}}\\\\sqrt{\\\\frac{r}{r_{s}}-1}\\\\cosh\\\\frac{t}{2r_{s}}.\\\\end{array}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S3.Ex10\",\"source_text\":\"$\\\\begin{array}[]{c}\\\\zeta=e^{\\\\frac{r}{2r_{s}}}\\\\sqrt{\\\\frac{r}{r_{s}}-1}\\\\sinh\\\\frac% {t}{2r_{s}},\\\\\\\\ \\\\\\\\ X=e^{\\\\frac{r}{2r_{s}}}\\\\sqrt{\\\\frac{r}{r_{s}}-1}\\\\cosh\\\\frac{t}{2r_{s}}.\\\\end{array}$ (3.10)\",\"tag\":\"(3.10)\",\"text\":\"r √︁ 2rs r t ζ = e − 1 sinh , rs 2rs (3.10) r √︁ 2rs r t X = e − 1 cosh . rs 2rs\"},\"formula_index\":1,\"parent_page_sha256\":\"12ef9412ab4ca9707f6e49bee78e3f62e59420ab92b0bd9bc236578b52709104\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/12ef9412ab4ca9707f6e49bee78e3f62e59420ab92b0bd9bc236578b52709104.pdf","asset_sha256":"12ef9412ab4ca9707f6e49bee78e3f62e59420ab92b0bd9bc236578b52709104","document_id":"arxiv:2608.07909","kind":"formula","page_number":8,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[204.37,246.86,295.03,29.84],\"bbox_pdf\":[204.37,515.3,499.4,545.14],\"fragments\":[{\"bbox_pdf\":[204.37,515.3,499.4,545.14],\"words\":[2450,2451,2452,2453,2454,2455,2456,2457,2458,2459,2460,2461,2462,2463,2464,2465,2466,2467,2468,2469,2470,2471]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"ds^{2}=\\\\frac{4r_{s}^{3}e^{-r/r_{s}}}{r}\\\\left(-d\\\\zeta^{2}+dX^{2}\\\\right)+r^{2}d%\\n\\\\Omega^{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S3.Ex11\",\"source_text\":\"$ds^{2}=\\\\frac{4r_{s}^{3}e^{-r/r_{s}}}{r}\\\\left(-d\\\\zeta^{2}+dX^{2}\\\\right)+r^{2}d% \\\\Omega^{2}.$ (3.11)\",\"tag\":\"(3.11)\",\"text\":\"4r 3 −r/rs (︁ )︁ ds 2 = se −dζ 2 + dX 2 + r 2 dΩ 2 . (3.11) r\"},\"formula_index\":3,\"parent_page_sha256\":\"12ef9412ab4ca9707f6e49bee78e3f62e59420ab92b0bd9bc236578b52709104\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/12ef9412ab4ca9707f6e49bee78e3f62e59420ab92b0bd9bc236578b52709104.pdf","asset_sha256":"12ef9412ab4ca9707f6e49bee78e3f62e59420ab92b0bd9bc236578b52709104","document_id":"arxiv:2608.07909","kind":"formula","page_number":8,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[240.3,300.17,259.09,30.04],\"bbox_pdf\":[240.3,461.79,499.39,491.83],\"fragments\":[{\"bbox_pdf\":[240.3,461.79,499.39,491.83],\"words\":[2479,2480,2481,2482,2483,2484,2485,2486,2487,2488,2489,2490,2491,2492,2493,2494,2495]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"X^{2}-\\\\zeta^{2}=e^{\\\\frac{r}{r_{s}}}\\\\left(\\\\frac{r}{r_{s}}-1\\\\right),\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S3.Ex12\",\"source_text\":\"$X^{2}-\\\\zeta^{2}=e^{\\\\frac{r}{r_{s}}}\\\\left(\\\\frac{r}{r_{s}}-1\\\\right),$ (3.12)\",\"tag\":\"(3.12)\",\"text\":\"(︃ )︃ 2 2 r r rs X − ζ = e − 1 , (3.12) rs\"},\"formula_index\":5,\"parent_page_sha256\":\"12ef9412ab4ca9707f6e49bee78e3f62e59420ab92b0bd9bc236578b52709104\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/12ef9412ab4ca9707f6e49bee78e3f62e59420ab92b0bd9bc236578b52709104.pdf","asset_sha256":"12ef9412ab4ca9707f6e49bee78e3f62e59420ab92b0bd9bc236578b52709104","document_id":"arxiv:2608.07909","kind":"formula","page_number":8,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,125.8,388.55,98.63],\"bbox_pdf\":[110.85,567.57,499.4,666.2],\"fragments\":[{\"bbox_pdf\":[110.85,567.57,499.4,666.2],\"words\":[2530,2531,2532,2533,2534,2535,2536,2537,2538,2539,2540,2541,2542,2543,2544,2545,2546,2547,2548,2549,2550,2551,2552,2553,2554,2555,2556,2557,2558,2559,2560,2561,2562,2563,2564,2565,2566,2567,2568,2569,2570,2571,2572,2573,2574,2575,2576,2577,2578,2579,2580,2581,2582,2583,2584,2585,2586,2587,2588,2589,2590,2591,2592,2593,2594,2595,2596,2597,2598,2599]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph74\",\"source_text\":\"Note the striking similarities when deriving the Tortoise and Kruskal coordinates. Both describe conformally flat transformations for the first two terms of Eq. (2.3). There are other possible representations with this feature [15]. In particular, it is possible to obtain a compact representation of the Schwarzschild spacetime, by means of the pulsating coordinates [15]. Instead of hyperbolic trigonometric functions, they are defined completely in terms of trigonometric functions, as:\",\"tag\":null,\"text\":\"Note the striking similarities when deriving the Tortoise and Kruskal coordinates. Both describe conformally flat transformations for the first two terms of Eq. (2.3). There are other possible representations with this feature [15]. In particular, it is possible to obtain a compact representation of the Schwarzschild spacetime, by means of the pulsating coordinates [15]. Instead of hyperbolic trigonometric functions, they are defined completely in terms of trigonometric functions, as:\"},{\"bbox\":[246.68,228.53,252.72,54.84],\"bbox_pdf\":[246.68,508.62,499.4,563.47],\"fragments\":[{\"bbox_pdf\":[246.68,508.62,499.4,563.47],\"words\":[2600,2601,2602,2603,2604,2605,2606,2607,2608,2609,2610,2611,2612,2613,2614,2615,2616,2617,2618,2619,2620,2621,2622,2623,2624,2625,2626,2627,2628,2629]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{array}[]{c}\\\\xi=D\\\\sin\\\\left(\\\\frac{t}{\\\\alpha}\\\\right)\\\\cos\\\\left(\\\\frac{r^{%\\n\\\\ast}}{\\\\alpha}\\\\right),\\\\\\\\\\n\\\\\\\\\\n\\\\chi=D\\\\cos\\\\left(\\\\frac{t}{\\\\alpha}\\\\right)\\\\sin\\\\left(\\\\frac{r^{\\\\ast}}{\\\\alpha}\\\\right%\\n),\\\\end{array}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S3.Ex13\",\"source_text\":\"$\\\\begin{array}[]{c}\\\\xi=D\\\\sin\\\\left(\\\\frac{t}{\\\\alpha}\\\\right)\\\\cos\\\\left(\\\\frac{r^{% \\\\ast}}{\\\\alpha}\\\\right),\\\\\\\\ \\\\\\\\ \\\\chi=D\\\\cos\\\\left(\\\\frac{t}{\\\\alpha}\\\\right)\\\\sin\\\\left(\\\\frac{r^{\\\\ast}}{\\\\alpha}\\\\right% ),\\\\end{array}$ (3.13)\",\"tag\":\"(3.13)\",\"text\":\"(︁ )︁ (︁ ∗)︁ t r ξ = D sin cos , α α (︁ )︁ (︁ )︁ (3.13) ∗ χ = D cos t sin r , α α\"},{\"bbox\":[110.85,287.14,388.55,26.4],\"bbox_pdf\":[110.85,478.46,499.4,504.86],\"fragments\":[{\"bbox_pdf\":[110.85,478.46,499.4,504.86],\"words\":[2630,2631,2632,2633,2634,2635,2636,2637,2638,2639,2640,2641,2642,2643,2644,2645,2646,2647,2648,2649,2650,2651]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph76\",\"source_text\":\"where $D$ and $\\\\alpha$ are constants with units of distance. Under these transformations, the metric in Eq. (2.3) becomes equivalent to\",\"tag\":null,\"text\":\"where D and α are constants with units of distance. Under these transformations, the metric in Eq. (2.3) becomes equivalent to\"},{\"bbox\":[226.42,317.67,272.98,21.64],\"bbox_pdf\":[226.42,452.69,499.4,474.33],\"fragments\":[{\"bbox_pdf\":[226.42,452.69,499.4,474.33],\"words\":[2652,2653,2654,2655,2656,2657,2658,2659,2660,2661,2662,2663,2664,2665,2666,2667,2668]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"ds^{2}=\\\\varpi\\\\left(-d\\\\xi^{2}+d\\\\chi^{2}\\\\right)+r^{2}d\\\\Omega^{2}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S3.Ex14\",\"source_text\":\"$ds^{2}=\\\\varpi\\\\left(-d\\\\xi^{2}+d\\\\chi^{2}\\\\right)+r^{2}d\\\\Omega^{2}$ (3.14)\",\"tag\":\"(3.14)\",\"text\":\"(︁ )︁ 2 2 2 2 2 ds = ϖ −dξ + dχ + r dΩ (3.14)\"},{\"bbox\":[110.85,353.13,207.84,11.96],\"bbox_pdf\":[110.85,426.91,318.7,438.87],\"fragments\":[{\"bbox_pdf\":[110.85,426.91,318.7,438.87],\"words\":[2669,2670,2671,2672,2673,2674,2675]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.857,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph78\",\"source_text\":\"where the (1+1) conformal factor is now\",\"tag\":null,\"text\":\"where the (1+1) conformal factor is now\"},{\"bbox\":[205.06,379.32,294.34,28.6],\"bbox_pdf\":[205.06,384.08,499.39,412.68],\"fragments\":[{\"bbox_pdf\":[205.06,384.08,499.39,412.68],\"words\":[2676,2677,2678,2679,2680,2681,2682,2683,2684,2685,2686,2687,2688,2689,2690,2691,2693,2694,2695,2696,2697]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\varpi=\\\\frac{2\\\\alpha^{2}\\\\gamma}{D^{2}}\\\\left[\\\\cos\\\\left(\\\\frac{2t}{\\\\alpha}\\\\right)%\\n+\\\\cos\\\\left(\\\\frac{2r^{\\\\ast}}{\\\\alpha}\\\\right)\\\\right]^{-1}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S3.Ex15\",\"source_text\":\"$\\\\varpi=\\\\frac{2\\\\alpha^{2}\\\\gamma}{D^{2}}\\\\left[\\\\cos\\\\left(\\\\frac{2t}{\\\\alpha}\\\\right)% +\\\\cos\\\\left(\\\\frac{2r^{\\\\ast}}{\\\\alpha}\\\\right)\\\\right]^{-1}.$ (3.15)\",\"tag\":\"(3.15)\",\"text\":\"[︃ (︃ )︃ (︃ )︃]︃−1 2 ∗ 2α γ 2t 2r ϖ = cos + cos (3.15) 2 D α α\"},{\"bbox\":[110.85,420.41,221.15,11.96],\"bbox_pdf\":[110.85,359.63,332.0,371.59],\"fragments\":[{\"bbox_pdf\":[110.85,359.63,332.0,371.59],\"words\":[2698,2699,2700,2701,2702,2703]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph80\",\"source_text\":\"From the transformations (3.13), one finds:\",\"tag\":null,\"text\":\"From the transformations (3.13), one finds:\"},{\"bbox\":[236.36,444.84,263.04,33.37],\"bbox_pdf\":[236.36,313.79,499.4,347.16],\"fragments\":[{\"bbox_pdf\":[236.36,313.79,499.4,347.16],\"words\":[2704,2705,2706,2707,2708,2709,2710,2711,2712,2713,2714,2715,2717,2718,2719,2720,2721,2722,2723,2724,2725]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{\\\\xi^{2}}{\\\\cos^{2}\\\\left(\\\\frac{r^{\\\\ast}}{\\\\alpha}\\\\right)}+\\\\frac{\\\\chi^{2}}{%\\n\\\\sin^{2}\\\\left(\\\\frac{r^{\\\\ast}}{\\\\alpha}\\\\right)}=D^{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"S3.Ex16\",\"source_text\":\"$\\\\frac{\\\\xi^{2}}{\\\\cos^{2}\\\\left(\\\\frac{r^{\\\\ast}}{\\\\alpha}\\\\right)}+\\\\frac{\\\\chi^{2}}{% \\\\sin^{2}\\\\left(\\\\frac{r^{\\\\ast}}{\\\\alpha}\\\\right)}=D^{2}.$ (3.16)\",\"tag\":\"(3.16)\",\"text\":\"ξ 2 χ 2 (︁ )︁ + (︁ )︁ = D 2 (3.16) 2 r∗ 2 r∗ cos sin α α\"},{\"bbox\":[110.85,486.94,388.55,127.52],\"bbox_pdf\":[110.85,177.53,499.4,305.06],\"fragments\":[{\"bbox_pdf\":[110.85,177.53,499.4,305.06],\"words\":[2726,2727,2728,2729,2730,2731,2732,2733,2734,2735,2736,2737,2738,2739,2740,2741,2742,2743,2744,2745,2746,2747,2748,2749,2750,2751,2752,2753,2754,2755,2756,2757,2758,2759,2760,2761,2762,2763,2764,2765,2766,2767,2768,2769,2770,2771,2772,2773,2774,2775,2776,2777,2778,2779,2780,2781,2782,2783,2784,2785,2786,2787,2788,2789,2790,2791,2792,2793,2794,2795,2796,2797,2798,2799,2800,2801,2802,2803,2804,2805,2806,2807,2808,2809,2810,2811,2812,2813,2814,2815,2816,2817,2818,2819,2820,2821,2822,2823,2824,2825,2826,2827,2828,2829,2830,2831,2832,2833,2834,2835,2836]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph82\",\"source_text\":\"Here, we generally have ellipses instead of hyperbolas. The constants $D$ and $\\\\alpha$ govern the amplitude of the representation in a $(\\\\xi,\\\\chi)$ diagram and the modulation of how rapidly the ellipses change as $r$ varies. In the Kruskal representation there is a similar interpretation: the amplitude of the representing hyperbolas determines the placement of the upper hyperbola $\\\\zeta^{2}-X^{2}=1$ in Fig. 2, while the constant $2r_{s}$ inside the hyperbolic trigonometric functions of Eq. (3.10) determines the pace at which the hyperbolas evolve as they approach the event horizon. In this sense, we set $D=1$ to get the representation in Fig. 3.\",\"tag\":null,\"text\":\"Here, we generally have ellipses instead of hyperbolas. The constants D and α govern the amplitude of the representation in a (ξ, χ) diagram and the modulation of how rapidly the ellipses change as r varies. In the Kruskal representation there is a similar interpretation: the amplitude of the representing hyperbolas determines the placement of the upper hyperbola ζ 2 − X 2 = 1 in Fig. 2, while the constant 2rs inside the hyperbolic trigonometric functions of Eq. (3.10) determines the pace at which the hyperbolas evolve as they approach the event horizon. In this sense, we set D = 1 to get the representation in Fig. 3.\"},{\"bbox\":[110.85,616.96,388.55,57.09],\"bbox_pdf\":[110.85,117.96,499.4,175.04],\"fragments\":[{\"bbox_pdf\":[110.85,117.96,499.4,175.04],\"words\":[2837,2838,2839,2840,2841,2842,2843,2844,2845,2846,2847,2848,2849,2850,2851,2852,2853,2854,2855,2856,2857,2858,2859,2860,2861,2862,2863,2864,2865,2866,2867,2868,2869,2870,2871,2872,2873,2874,2875,2876,2877,2878,2879,2880,2882,2883,2884,2885,2886,2887,2888,2889,2890,2891,2892,2893,2894,2895,2896,2897,2898,2899,2900,2901,2902,2903,2904,2905,2906,2907]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph83\",\"source_text\":\"The four corners of the diamond region are located at $(\\\\xi,\\\\chi)=(\\\\pm 1,\\\\pm 1)$ . Equation (3.16) allows us to trace changes in the geometry, starting from a positive value of $r^{\\\\ast}/\\\\alpha$ . We choose $\\\\alpha=(2/\\\\pi)(\\\\varphi^{+}-\\\\ln\\\\varphi^{+})~{}r_{s}$ , where $\\\\varphi^{+}=(1+\\\\sqrt{5})/2$ . This constant is chosen so that when $r^{\\\\ast}/r_{s}=\\\\varphi^{+}-\\\\ln\\\\varphi^{+}\\\\approx 1.1368$ , we have $r^{\\\\ast}/\\\\alpha=\\\\pi/2$ , which yields the horizontal degenerate ellipse (a line segment) at $\\\\xi=0$ , with $-1\\\\leq\\\\chi\\\\leq 1$ . This somewhat arbitrary initial selection corresponds to the location $r=\\\\varphi^{+}r_{s}\\\\approx 1.618$ , appearing in the Golden representation discussed below in the following Sections.\",\"tag\":null,\"text\":\"The four corners of the diamond region are located at (ξ, χ) = (±1, ±1). Equation (3.16) allows us to trace changes in the geometry, starting from a positive value of r ∗ /α. We choose α = (2/π)(φ + − ln φ + rs, where φ + = √ (1+ 5)/2. This constant is chosen so that when r ∗ /rs = φ + −ln φ + ≈ 1.1368,\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0008.pdf\",\"page_no\":9,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph78\",\"c_block\":0.857,\"c_lines\":1.0,\"c_pdf\":0.999,\"c_region\":1.0,\"c_src\":0.998,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.857,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0.pdf","asset_sha256":"d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0","document_id":"arxiv:2608.07909","kind":"page","page_number":9,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[246.68,228.53,252.72,54.84],\"bbox_pdf\":[246.68,508.62,499.4,563.47],\"fragments\":[{\"bbox_pdf\":[246.68,508.62,499.4,563.47],\"words\":[2600,2601,2602,2603,2604,2605,2606,2607,2608,2609,2610,2611,2612,2613,2614,2615,2616,2617,2618,2619,2620,2621,2622,2623,2624,2625,2626,2627,2628,2629]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{array}[]{c}\\\\xi=D\\\\sin\\\\left(\\\\frac{t}{\\\\alpha}\\\\right)\\\\cos\\\\left(\\\\frac{r^{%\\n\\\\ast}}{\\\\alpha}\\\\right),\\\\\\\\\\n\\\\\\\\\\n\\\\chi=D\\\\cos\\\\left(\\\\frac{t}{\\\\alpha}\\\\right)\\\\sin\\\\left(\\\\frac{r^{\\\\ast}}{\\\\alpha}\\\\right%\\n),\\\\end{array}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S3.Ex13\",\"source_text\":\"$\\\\begin{array}[]{c}\\\\xi=D\\\\sin\\\\left(\\\\frac{t}{\\\\alpha}\\\\right)\\\\cos\\\\left(\\\\frac{r^{% \\\\ast}}{\\\\alpha}\\\\right),\\\\\\\\ \\\\\\\\ \\\\chi=D\\\\cos\\\\left(\\\\frac{t}{\\\\alpha}\\\\right)\\\\sin\\\\left(\\\\frac{r^{\\\\ast}}{\\\\alpha}\\\\right% ),\\\\end{array}$ (3.13)\",\"tag\":\"(3.13)\",\"text\":\"(︁ )︁ (︁ ∗)︁ t r ξ = D sin cos , α α (︁ )︁ (︁ )︁ (3.13) ∗ χ = D cos t sin r , α α\"},\"formula_index\":1,\"parent_page_sha256\":\"d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0.pdf","asset_sha256":"d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0","document_id":"arxiv:2608.07909","kind":"formula","page_number":9,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[226.42,317.67,272.98,21.64],\"bbox_pdf\":[226.42,452.69,499.4,474.33],\"fragments\":[{\"bbox_pdf\":[226.42,452.69,499.4,474.33],\"words\":[2652,2653,2654,2655,2656,2657,2658,2659,2660,2661,2662,2663,2664,2665,2666,2667,2668]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"ds^{2}=\\\\varpi\\\\left(-d\\\\xi^{2}+d\\\\chi^{2}\\\\right)+r^{2}d\\\\Omega^{2}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S3.Ex14\",\"source_text\":\"$ds^{2}=\\\\varpi\\\\left(-d\\\\xi^{2}+d\\\\chi^{2}\\\\right)+r^{2}d\\\\Omega^{2}$ (3.14)\",\"tag\":\"(3.14)\",\"text\":\"(︁ )︁ 2 2 2 2 2 ds = ϖ −dξ + dχ + r dΩ (3.14)\"},\"formula_index\":3,\"parent_page_sha256\":\"d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0.pdf","asset_sha256":"d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0","document_id":"arxiv:2608.07909","kind":"formula","page_number":9,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[205.06,379.32,294.34,28.6],\"bbox_pdf\":[205.06,384.08,499.39,412.68],\"fragments\":[{\"bbox_pdf\":[205.06,384.08,499.39,412.68],\"words\":[2676,2677,2678,2679,2680,2681,2682,2683,2684,2685,2686,2687,2688,2689,2690,2691,2693,2694,2695,2696,2697]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\varpi=\\\\frac{2\\\\alpha^{2}\\\\gamma}{D^{2}}\\\\left[\\\\cos\\\\left(\\\\frac{2t}{\\\\alpha}\\\\right)%\\n+\\\\cos\\\\left(\\\\frac{2r^{\\\\ast}}{\\\\alpha}\\\\right)\\\\right]^{-1}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S3.Ex15\",\"source_text\":\"$\\\\varpi=\\\\frac{2\\\\alpha^{2}\\\\gamma}{D^{2}}\\\\left[\\\\cos\\\\left(\\\\frac{2t}{\\\\alpha}\\\\right)% +\\\\cos\\\\left(\\\\frac{2r^{\\\\ast}}{\\\\alpha}\\\\right)\\\\right]^{-1}.$ (3.15)\",\"tag\":\"(3.15)\",\"text\":\"[︃ (︃ )︃ (︃ )︃]︃−1 2 ∗ 2α γ 2t 2r ϖ = cos + cos (3.15) 2 D α α\"},\"formula_index\":5,\"parent_page_sha256\":\"d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0.pdf","asset_sha256":"d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0","document_id":"arxiv:2608.07909","kind":"formula","page_number":9,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[236.36,444.84,263.04,33.37],\"bbox_pdf\":[236.36,313.79,499.4,347.16],\"fragments\":[{\"bbox_pdf\":[236.36,313.79,499.4,347.16],\"words\":[2704,2705,2706,2707,2708,2709,2710,2711,2712,2713,2714,2715,2717,2718,2719,2720,2721,2722,2723,2724,2725]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{\\\\xi^{2}}{\\\\cos^{2}\\\\left(\\\\frac{r^{\\\\ast}}{\\\\alpha}\\\\right)}+\\\\frac{\\\\chi^{2}}{%\\n\\\\sin^{2}\\\\left(\\\\frac{r^{\\\\ast}}{\\\\alpha}\\\\right)}=D^{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"S3.Ex16\",\"source_text\":\"$\\\\frac{\\\\xi^{2}}{\\\\cos^{2}\\\\left(\\\\frac{r^{\\\\ast}}{\\\\alpha}\\\\right)}+\\\\frac{\\\\chi^{2}}{% \\\\sin^{2}\\\\left(\\\\frac{r^{\\\\ast}}{\\\\alpha}\\\\right)}=D^{2}.$ (3.16)\",\"tag\":\"(3.16)\",\"text\":\"ξ 2 χ 2 (︁ )︁ + (︁ )︁ = D 2 (3.16) 2 r∗ 2 r∗ cos sin α α\"},\"formula_index\":7,\"parent_page_sha256\":\"d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0.pdf","asset_sha256":"d021a1dede0ac2e0a6ffeb387a050bec9276ae15c130d3ebf3b0f36ceacbb7a0","document_id":"arxiv:2608.07909","kind":"formula","page_number":9,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,125.44,388.55,55.65],\"bbox_pdf\":[110.85,610.91,499.4,666.56],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[110.85,610.91,499.4,666.56],\"words\":[2908,2909,2910,2911,2912,2913,2914,2915,2916,2917,2918,2919,2920,2921,2922,2923,2924,2925,2926,2927,2928,2929,2930,2931,2932,2933,2934,2935,2936,2937,2938,2939,2940,2941,2942,2943,2944,2945,2946,2947,2948,2949,2950,2951,2952,2953,2954,2955,2956,2957,2958,2959,2960]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph83\",\"source_text\":\"The four corners of the diamond region are located at $(\\\\xi,\\\\chi)=(\\\\pm 1,\\\\pm 1)$ . Equation (3.16) allows us to trace changes in the geometry, starting from a positive value of $r^{\\\\ast}/\\\\alpha$ . We choose $\\\\alpha=(2/\\\\pi)(\\\\varphi^{+}-\\\\ln\\\\varphi^{+})~{}r_{s}$ , where $\\\\varphi^{+}=(1+\\\\sqrt{5})/2$ . This constant is chosen so that when $r^{\\\\ast}/r_{s}=\\\\varphi^{+}-\\\\ln\\\\varphi^{+}\\\\approx 1.1368$ , we have $r^{\\\\ast}/\\\\alpha=\\\\pi/2$ , which yields the horizontal degenerate ellipse (a line segment) at $\\\\xi=0$ , with $-1\\\\leq\\\\chi\\\\leq 1$ . This somewhat arbitrary initial selection corresponds to the location $r=\\\\varphi^{+}r_{s}\\\\approx 1.618$ , appearing in the Golden representation discussed below in the following Sections.\",\"tag\":null,\"text\":\"we have r ∗ /α = π/2, which yields the horizontal degenerate ellipse (a line segment) at ξ = 0, with −1 ≤ χ ≤ 1. This somewhat arbitrary initial selection corresponds to the location r = φ + rs ≈ 1.618, appearing in the Golden representation discussed below in the following Sections.\"},{\"bbox\":[169.14,198.36,271.98,271.07],\"bbox_pdf\":[169.14,322.57,441.12,593.64],\"float_ref\":\"S3.F3\",\"fragments\":[{\"bbox_pdf\":[169.14,322.57,441.12,593.64],\"words\":[]}],\"html\":null,\"kind\":\"figure\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"figure\",\"source_ref\":\"S3.F3\",\"source_text\":\"\",\"tag\":null,\"text\":\"\"},{\"bbox\":[126.43,481.88,357.4,11.96],\"bbox_pdf\":[126.43,298.16,483.83,310.12],\"float_ref\":\"S3.F3\",\"fragments\":[{\"bbox_pdf\":[126.43,298.16,483.83,310.12],\"words\":[2961,2962,2963,2964,2965,2966,2967,2968,2969]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"caption\",\"source_ref\":\"caption85\",\"source_text\":\"Figure 3: Compactified Diamond representation of the Schwarzschild spacetime.\",\"tag\":null,\"text\":\"Figure 3: Compactified Diamond representation of the Schwarzschild spacetime.\"},{\"bbox\":[110.85,512.6,388.55,100.42],\"bbox_pdf\":[110.85,178.98,499.4,279.4],\"fragments\":[{\"bbox_pdf\":[110.85,178.98,499.4,279.4],\"words\":[2970,2971,2972,2973,2974,2975,2976,2977,2978,2979,2980,2981,2982,2983,2984,2985,2986,2987,2988,2989,2990,2991,2992,2993,2994,2995,2996,2997,2998,2999,3000,3001,3002,3003,3004,3005,3006,3007,3008,3009,3010,3011,3012,3013,3014,3015,3016,3017,3018,3019,3020,3021,3022,3023,3024,3025,3026,3027,3028,3029,3030,3031,3032,3033,3034,3035,3036,3037,3038,3039,3040,3041,3042,3043,3044,3045,3046,3047,3048,3049,3050,3051,3052,3053,3054,3055,3056,3057,3058,3059,3060,3061,3062]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph86\",\"source_text\":\"Decreasing $r$ yields horizontal ellipses that become progressively less elongated. When $r^{\\\\ast}/\\\\alpha=\\\\pi/4$ —which in our selection corresponds to the sphere at $r\\\\approx 1.5922~{}r_{s}$ — the geometry is represented in the diagram by a circle. Continuing to lower values of $r$ produces vertical ellipses that become increasingly elongated until reaching the vertical degenerate ellipse bounded by $\\\\xi=\\\\pm 1$ at $r^{\\\\ast}=0$ . Continuing to decrease $r$ repeats the inverse process, and this cycle continues infinitely many times as $r\\\\rightarrow r_{s}$ , since $r^{\\\\ast}\\\\rightarrow-\\\\infty$ .\",\"tag\":null,\"text\":\"Decreasing r yields horizontal ellipses that become progressively less elongated. When r ∗ /α = π/4—which in our selection corresponds to the sphere at r ≈ 1.5922 rs— the geometry is represented in the diagram by a circle. Continuing to lower values of r produces vertical ellipses that become increasingly elongated until reaching the vertical degenerate ellipse bounded by ξ = ±1 at r ∗ = 0. Continuing to decrease r repeats the inverse process, and this cycle continues infinitely many times as r → rs, since r ∗ → −∞.\"},{\"bbox\":[110.85,613.72,388.55,55.29],\"bbox_pdf\":[110.85,122.99,499.4,178.28],\"fragments\":[{\"bbox_pdf\":[110.85,122.99,499.4,178.28],\"words\":[3063,3064,3065,3066,3067,3068,3069,3070,3071,3072,3073,3074,3075,3076,3077,3078,3079,3080,3081,3082,3083,3084,3085,3086,3087,3088,3089,3090,3091,3092,3093,3094,3095,3096,3097,3098,3099,3100,3101,3102,3103,3104,3105,3106]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph87\",\"source_text\":\"We have reviewed some classical representations of the Schwarzschild spacetime, such as the Flamm paraboloid and the Tortoise and Kruskal transformations. We also presented the pulsating coordinates, recently proposed as a natural extension of these classical mappings [15]. In turn, the pulsating coordinates have motivated the Causal Diamond coordinates as an alternative to the customary Rindler representation of Minkowski spacetime [35]. Let us now explore the Golden representation.\",\"tag\":null,\"text\":\"We have reviewed some classical representations of the Schwarzschild spacetime, such as the Flamm paraboloid and the Tortoise and Kruskal transformations. We also presented the pulsating coordinates, recently proposed as a natural extension of these classical mappings [15]. In turn, the pulsating\"},{\"bbox\":[299.27,690.18,11.71,11.96],\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"fragments\":[{\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"words\":[10385]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":5,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"10\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0009.pdf\",\"page_no\":10,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"b0a618c1830e038664076b1e97fc7c40d306582224e17bd405661dced1ec71d0\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/b0a618c1830e038664076b1e97fc7c40d306582224e17bd405661dced1ec71d0.pdf","asset_sha256":"b0a618c1830e038664076b1e97fc7c40d306582224e17bd405661dced1ec71d0","document_id":"arxiv:2608.07909","kind":"page","page_number":10,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,125.8,388.54,40.85],\"bbox_pdf\":[110.85,625.35,499.4,666.2],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[110.85,625.35,499.4,666.2],\"words\":[3107,3108,3109,3110,3111,3112,3113,3114,3115,3116,3117,3118,3119,3120,3121,3122,3123,3124,3125,3126,3127,3128,3129,3130,3131,3132,3133]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph87\",\"source_text\":\"We have reviewed some classical representations of the Schwarzschild spacetime, such as the Flamm paraboloid and the Tortoise and Kruskal transformations. We also presented the pulsating coordinates, recently proposed as a natural extension of these classical mappings [15]. In turn, the pulsating coordinates have motivated the Causal Diamond coordinates as an alternative to the customary Rindler representation of Minkowski spacetime [35]. Let us now explore the Golden representation.\",\"tag\":null,\"text\":\"coordinates have motivated the Causal Diamond coordinates as an alternative to the customary Rindler representation of Minkowski spacetime [35]. Let us now explore the Golden representation.\"},{\"bbox\":[110.85,189.35,322.53,17.22],\"bbox_pdf\":[110.85,585.44,433.38,602.65],\"fragments\":[{\"bbox_pdf\":[110.85,585.44,433.38,602.65],\"words\":[3134,3135,3136,3137,3138]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"heading\",\"source_ref\":\"title88\",\"source_text\":\"4 Obtaining the Golden Coordinates\",\"tag\":null,\"text\":\"4 Obtaining the Golden Coordinates\"},{\"bbox\":[110.85,220.89,330.24,11.96],\"bbox_pdf\":[110.85,559.15,441.09,571.11],\"fragments\":[{\"bbox_pdf\":[110.85,559.15,441.09,571.11],\"words\":[3139,3140,3141,3142,3143,3144,3145,3146,3147,3148]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph89\",\"source_text\":\"Start with a generic spherically symmetric static form, given by:\",\"tag\":null,\"text\":\"Start with a generic spherically symmetric static form, given by:\"},{\"bbox\":[204.11,248.83,295.28,12.91],\"bbox_pdf\":[204.11,530.26,499.4,543.17],\"fragments\":[{\"bbox_pdf\":[204.11,530.26,499.4,543.17],\"words\":[3149,3150,3151,3152,3153,3154,3155,3156,3157,3158,3159,3160,3161,3162,3163,3164,3165,3166,3167]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"ds^{2}=-e^{2\\\\alpha(r)}dt^{2}+e^{2\\\\beta(r)}dr^{2}+e^{2\\\\gamma(r)}d\\\\Omega^{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S4.Ex1\",\"source_text\":\"$ds^{2}=-e^{2\\\\alpha(r)}dt^{2}+e^{2\\\\beta(r)}dr^{2}+e^{2\\\\gamma(r)}d\\\\Omega^{2}.$ (4.1)\",\"tag\":\"(4.1)\",\"text\":\"ds 2 = −e 2α(r) dt 2 + e 2β(r) dr 2 + e 2γ(r) dΩ 2 . (4.1)\"},{\"bbox\":[110.85,270.65,388.54,28.2],\"bbox_pdf\":[110.85,493.15,499.4,521.35],\"fragments\":[{\"bbox_pdf\":[110.85,493.15,499.4,521.35],\"words\":[3168,3169,3170,3171,3172,3173,3174,3175,3176,3177,3178,3179,3180,3181,3182,3183,3184,3185,3186,3187,3188,3189,3190,3191]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph91\",\"source_text\":\"In this section we are considering dimensionless units, scaling the coordinates with respect to $r_{s}=1$ , for reasons that will be clarified below.\",\"tag\":null,\"text\":\"In this section we are considering dimensionless units, scaling the coordinates with respect to rs = 1, for reasons that will be clarified below.\"},{\"bbox\":[128.41,299.54,224.23,11.96],\"bbox_pdf\":[128.41,480.5,352.64,492.46],\"fragments\":[{\"bbox_pdf\":[128.41,480.5,352.64,492.46],\"words\":[3192,3193,3194,3195,3196,3197]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph92\",\"source_text\":\"The Christoffel symbols are calculated from\",\"tag\":null,\"text\":\"The Christoffel symbols are calculated from\"},{\"bbox\":[213.94,322.69,285.46,23.0],\"bbox_pdf\":[213.94,446.31,499.4,469.31],\"fragments\":[{\"bbox_pdf\":[213.94,446.31,499.4,469.31],\"words\":[3198,3199,3200,3201,3202,3203,3204,3205,3206,3207,3208,3209,3210]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\Gamma_{\\\\mu\\\\nu}^{\\\\alpha}=\\\\frac{1}{2}g^{\\\\alpha\\\\lambda}(\\\\partial_{\\\\mu}g_{\\\\lambda%\\n\\\\nu}+\\\\partial_{\\\\nu}g_{\\\\mu\\\\lambda}-\\\\partial_{\\\\lambda}g_{\\\\mu\\\\nu}).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S4.Ex2\",\"source_text\":\"$\\\\Gamma_{\\\\mu\\\\nu}^{\\\\alpha}=\\\\frac{1}{2}g^{\\\\alpha\\\\lambda}(\\\\partial_{\\\\mu}g_{\\\\lambda% \\\\nu}+\\\\partial_{\\\\nu}g_{\\\\mu\\\\lambda}-\\\\partial_{\\\\lambda}g_{\\\\mu\\\\nu}).$ (4.2)\",\"tag\":\"(4.2)\",\"text\":\"α 1 αλ Γµν = g (∂µgλν + ∂νgµλ − ∂λgµν). (4.2) 2\"},{\"bbox\":[110.85,354.7,126.56,11.96],\"bbox_pdf\":[110.85,425.34,237.41,437.3],\"fragments\":[{\"bbox_pdf\":[110.85,425.34,237.41,437.3],\"words\":[3211,3212,3213,3214]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph94\",\"source_text\":\"The non-trivial ones are:\",\"tag\":null,\"text\":\"The non-trivial ones are:\"},{\"bbox\":[145.56,391.5,353.84,47.6],\"bbox_pdf\":[145.56,352.9,499.4,400.5],\"fragments\":[{\"bbox_pdf\":[145.56,352.9,499.4,400.5],\"words\":[3215,3216,3217,3218,3219,3220,3221,3222,3223,3224,3225,3226,3227,3229,3230,3231,3232,3234,3235,3236,3238,3239,3240,3241,3242,3243,3244,3245,3246,3247,3248,3249,3250,3251,3252,3253,3254,3255,3256,3257,3258,3259,3260,3262,3263,3264,3266,3267,3268,3269,3270,3271,3272,3273,3274,3275,3276,3277,3278,3279,3280,3281,3282,3284,3285,3286,3287]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\Gamma_{01}^{0}=\\\\frac{d\\\\alpha}{dr}, \\\\Gamma_{00}^{1}=e^{2(\\\\alpha-\\\\beta)}\\\\frac{d\\\\alpha}{dr}, \\\\Gamma_{11}^{1}=\\\\frac{d\\\\beta}{dr}, \\\\Gamma_{22}^{1}=-e^{2(\\\\gamma-\\\\beta)}\\\\frac{d\\\\gamma}{dr} \\\\Gamma_{33}^{1}=-\\\\sin^{2}\\\\theta e^{2(\\\\gamma-\\\\beta)}\\\\frac{d\\\\gamma}{dr}, \\\\Gamma_{12}^{2}=\\\\frac{d\\\\gamma}{dr}, \\\\Gamma_{33}^{2}=-\\\\sin\\\\theta\\\\cos\\\\theta, \\\\Gamma_{13}^{3}=\\\\frac{d\\\\gamma}{dr}, \\\\Gamma_{23}^{3}=\\\\cot\\\\theta,\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S4.Ex3\",\"source_text\":\"$\\\\Gamma_{01}^{0}=\\\\frac{d\\\\alpha}{dr},$ $\\\\Gamma_{00}^{1}=e^{2(\\\\alpha-\\\\beta)}\\\\frac{d\\\\alpha}{dr},$ $\\\\Gamma_{11}^{1}=\\\\frac{d\\\\beta}{dr},$ $\\\\Gamma_{22}^{1}=-e^{2(\\\\gamma-\\\\beta)}\\\\frac{d\\\\gamma}{dr}$ $\\\\Gamma_{33}^{1}=-\\\\sin^{2}\\\\theta e^{2(\\\\gamma-\\\\beta)}\\\\frac{d\\\\gamma}{dr},$ $\\\\Gamma_{12}^{2}=\\\\frac{d\\\\gamma}{dr},$ $\\\\Gamma_{33}^{2}=-\\\\sin\\\\theta\\\\cos\\\\theta,$ $\\\\Gamma_{13}^{3}=\\\\frac{d\\\\gamma}{dr},$ $\\\\Gamma_{23}^{3}=\\\\cot\\\\theta,$ (4.3)\",\"tag\":\"(4.3)\",\"text\":\"Γ 0 dα Γ 1 2(α−β) dα Γ 1 dβ 01 = dr 00 = e dr 11 = dr Γ 1 2(γ−β) dγ Γ 1 2 θe 2(γ−β) dγ Γ 2 dγ (4.3) 22 = −e dr 33 = − sin dr 12 = dr Γ 2 Γ 3 dγ Γ 3 33 = − sin θ cos θ, 13 = dr 23 = cot θ,\"},{\"bbox\":[110.85,449.29,347.82,15.29],\"bbox_pdf\":[110.85,327.43,458.68,342.71],\"fragments\":[{\"bbox_pdf\":[110.85,327.43,458.68,342.71],\"words\":[3288,3289,3290,3291,3292,3293,3294,3295,3296,3297,3298,3299,3300,3301,3302,3303]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph96\",\"source_text\":\"as well as those related by $\\\\Gamma_{\\\\alpha\\\\beta}^{\\\\mu}=\\\\Gamma_{\\\\beta\\\\alpha}^{\\\\mu}$ . Now we substitute them into\",\"tag\":null,\"text\":\"µ µ as well as those related by Γαβ = Γβα. Now we substitute them into\"},{\"bbox\":[198.72,476.52,300.67,15.86],\"bbox_pdf\":[198.72,299.62,499.4,315.48],\"fragments\":[{\"bbox_pdf\":[198.72,299.62,499.4,315.48],\"words\":[3304,3305,3306,3307,3308,3309,3310,3311,3312,3313,3314,3315,3316,3317,3318,3319,3320]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"R_{\\\\mu\\\\nu}=\\\\partial_{\\\\alpha}\\\\Gamma_{\\\\mu\\\\nu}^{\\\\alpha}-\\\\partial_{\\\\nu}\\\\Gamma_{\\\\mu%\\n\\\\alpha}^{\\\\alpha}+\\\\Gamma_{\\\\alpha\\\\lambda}^{\\\\alpha}\\\\Gamma_{\\\\mu\\\\nu}^{\\\\lambda}-%\\n\\\\Gamma_{\\\\nu\\\\lambda}^{\\\\alpha}\\\\Gamma_{\\\\mu\\\\alpha}^{\\\\lambda}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"S4.Ex4\",\"source_text\":\"$R_{\\\\mu\\\\nu}=\\\\partial_{\\\\alpha}\\\\Gamma_{\\\\mu\\\\nu}^{\\\\alpha}-\\\\partial_{\\\\nu}\\\\Gamma_{\\\\mu% \\\\alpha}^{\\\\alpha}+\\\\Gamma_{\\\\alpha\\\\lambda}^{\\\\alpha}\\\\Gamma_{\\\\mu\\\\nu}^{\\\\lambda}-% \\\\Gamma_{\\\\nu\\\\lambda}^{\\\\alpha}\\\\Gamma_{\\\\mu\\\\alpha}^{\\\\lambda}.$ (4.4)\",\"tag\":\"(4.4)\",\"text\":\"Rµν = ∂αΓ α α α λ α λ (4.4) µν − ∂νΓµα + ΓαλΓµν − ΓνλΓµα.\"},{\"bbox\":[110.85,503.75,252.95,11.96],\"bbox_pdf\":[110.85,276.29,363.8,288.25],\"fragments\":[{\"bbox_pdf\":[110.85,276.29,363.8,288.25],\"words\":[3321,3322,3323,3324,3325,3326,3327]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"paragraph\",\"source_ref\":\"paragraph98\",\"source_text\":\"The non-trivial Ricci tensor components are then\",\"tag\":null,\"text\":\"The non-trivial Ricci tensor components are then\"},{\"bbox\":[207.86,519.39,291.54,83.12],\"bbox_pdf\":[207.86,189.49,499.4,272.61],\"fragments\":[{\"bbox_pdf\":[207.86,189.49,499.4,272.61],\"words\":[3328,3329,3330,3331,3332,3333,3334,3335,3336,3337,3338,3339,3340,3341,3342,3343,3344,3345,3346,3347,3348,3349,3350,3351,3352,3353,3354,3355,3356,3357,3358,3359,3360,3361,3362,3363,3364,3365,3366,3367,3368,3369,3370,3371,3372,3373,3374,3375,3376,3377,3378,3379,3380,3381,3382,3383,3384,3385,3386,3387,3388]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{array}[]{c}R_{00}=e^{2(\\\\alpha-\\\\beta)}\\\\left[\\\\frac{d^{2}\\\\alpha}{dr^{2}}+%\\n\\\\frac{d\\\\alpha}{dr}\\\\frac{d(\\\\alpha-\\\\beta+2\\\\gamma)}{dr}\\\\right],\\\\\\\\\\nR_{11}=\\\\frac{d\\\\alpha}{dr}\\\\frac{d(\\\\beta-\\\\alpha)}{dr}+2\\\\frac{d\\\\gamma}{dr}\\\\frac{d%\\n(\\\\beta-\\\\gamma)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}.\\\\\\\\\\nR_{22}=1+e^{2(\\\\gamma-\\\\beta)}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha-2\\\\gamma)}{dr}\\\\frac{d%\\n\\\\gamma}{dr}-\\\\frac{d^{2}\\\\gamma}{dr^{2}}\\\\right],\\\\\\\\\\nR_{33}=\\\\sin^{2}\\\\theta R_{22}.\\\\end{array}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"S4.Ex5\",\"source_text\":\"$\\\\begin{array}[]{c}R_{00}=e^{2(\\\\alpha-\\\\beta)}\\\\left[\\\\frac{d^{2}\\\\alpha}{dr^{2}}+% \\\\frac{d\\\\alpha}{dr}\\\\frac{d(\\\\alpha-\\\\beta+2\\\\gamma)}{dr}\\\\right],\\\\\\\\ R_{11}=\\\\frac{d\\\\alpha}{dr}\\\\frac{d(\\\\beta-\\\\alpha)}{dr}+2\\\\frac{d\\\\gamma}{dr}\\\\frac{d% (\\\\beta-\\\\gamma)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}.\\\\\\\\ R_{22}=1+e^{2(\\\\gamma-\\\\beta)}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha-2\\\\gamma)}{dr}\\\\frac{d% \\\\gamma}{dr}-\\\\frac{d^{2}\\\\gamma}{dr^{2}}\\\\right],\\\\\\\\ R_{33}=\\\\sin^{2}\\\\theta R_{22}.\\\\end{array}$ (4.5)\",\"tag\":\"(4.5)\",\"text\":\"[︂ ]︂ 2(α−β) d 2 α dα d(α−β+2γ) R00 = e 2 + , dr dr dr dα d(β−α) dγ d(β−γ) d 2 (α+2γ) R11 = + 2 − 2 dr dr [︂ dr dr dr ]︂ . (4.5) 2(γ−β) d(β−α−2γ) dγ d 2 γ R22 = 1 + e 2 , dr dr − dr R33 = sin 2 θR22.\"},{\"bbox\":[110.85,613.88,258.96,14.1],\"bbox_pdf\":[110.85,164.02,369.81,178.12],\"fragments\":[{\"bbox_pdf\":[110.85,164.02,369.81,178.12],\"words\":[3389,3390,3391,3392,3393,3394,3395,3396,3397,3398,3399]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.857,\"verified\":true},\"order\":13,\"role\":\"paragraph\",\"source_ref\":\"paragraph100\",\"source_text\":\"Finally, the scalar curvature $R=g^{\\\\mu\\\\nu}R_{\\\\mu\\\\nu}$ results in\",\"tag\":null,\"text\":\"Finally, the scalar curvature R = g µν Rµν results in\"},{\"bbox\":[119.08,630.88,380.33,40.6],\"bbox_pdf\":[119.08,120.52,499.4,161.12],\"fragments\":[{\"bbox_pdf\":[119.08,120.52,499.4,161.12],\"words\":[3400,3401,3402,3403,3404,3405,3406,3407,3408,3409,3410,3411,3412,3413,3414,3415,3416,3417,3418,3419,3420,3421,3422,3423,3424,3425,3426,3427,3428,3429,3430,3432]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"R=2e^{-2\\\\beta}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha)}{dr}\\\\frac{d\\\\left(\\\\alpha+2\\\\gamma%\\n\\\\right)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}-3\\\\left(\\\\frac{d%\\n\\\\gamma}{dr}\\\\right)^{2}\\\\right]+2e^{-2\\\\gamma}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"S4.Ex6\",\"source_text\":\"$R=2e^{-2\\\\beta}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha)}{dr}\\\\frac{d\\\\left(\\\\alpha+2\\\\gamma% \\\\right)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}-3\\\\left(\\\\frac{d% \\\\gamma}{dr}\\\\right)^{2}\\\\right]+2e^{-2\\\\gamma}.$ (4.6)\",\"tag\":\"(4.6)\",\"text\":\"[︄ (︃ )︃2 ]︄ d(β − α) d (α + 2γ) d 2 (α + 2γ) dγ −2β −2γ R = 2e dr dr − dr2 − 3 dr + 2e (4.6)\"},{\"bbox\":[299.27,690.18,11.71,11.96],\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"fragments\":[{\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"words\":[10386]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":15,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"11\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0010.pdf\",\"page_no\":11,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph100\",\"c_block\":0.833,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.987,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.833,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7.pdf","asset_sha256":"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7","document_id":"arxiv:2608.07909","kind":"page","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[204.11,248.83,295.28,12.91],\"bbox_pdf\":[204.11,530.26,499.4,543.17],\"fragments\":[{\"bbox_pdf\":[204.11,530.26,499.4,543.17],\"words\":[3149,3150,3151,3152,3153,3154,3155,3156,3157,3158,3159,3160,3161,3162,3163,3164,3165,3166,3167]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"ds^{2}=-e^{2\\\\alpha(r)}dt^{2}+e^{2\\\\beta(r)}dr^{2}+e^{2\\\\gamma(r)}d\\\\Omega^{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S4.Ex1\",\"source_text\":\"$ds^{2}=-e^{2\\\\alpha(r)}dt^{2}+e^{2\\\\beta(r)}dr^{2}+e^{2\\\\gamma(r)}d\\\\Omega^{2}.$ (4.1)\",\"tag\":\"(4.1)\",\"text\":\"ds 2 = −e 2α(r) dt 2 + e 2β(r) dr 2 + e 2γ(r) dΩ 2 . (4.1)\"},\"formula_index\":3,\"parent_page_sha256\":\"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7.pdf","asset_sha256":"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7","document_id":"arxiv:2608.07909","kind":"formula","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[213.94,322.69,285.46,23.0],\"bbox_pdf\":[213.94,446.31,499.4,469.31],\"fragments\":[{\"bbox_pdf\":[213.94,446.31,499.4,469.31],\"words\":[3198,3199,3200,3201,3202,3203,3204,3205,3206,3207,3208,3209,3210]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\Gamma_{\\\\mu\\\\nu}^{\\\\alpha}=\\\\frac{1}{2}g^{\\\\alpha\\\\lambda}(\\\\partial_{\\\\mu}g_{\\\\lambda%\\n\\\\nu}+\\\\partial_{\\\\nu}g_{\\\\mu\\\\lambda}-\\\\partial_{\\\\lambda}g_{\\\\mu\\\\nu}).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S4.Ex2\",\"source_text\":\"$\\\\Gamma_{\\\\mu\\\\nu}^{\\\\alpha}=\\\\frac{1}{2}g^{\\\\alpha\\\\lambda}(\\\\partial_{\\\\mu}g_{\\\\lambda% \\\\nu}+\\\\partial_{\\\\nu}g_{\\\\mu\\\\lambda}-\\\\partial_{\\\\lambda}g_{\\\\mu\\\\nu}).$ (4.2)\",\"tag\":\"(4.2)\",\"text\":\"α 1 αλ Γµν = g (∂µgλν + ∂νgµλ − ∂λgµν). (4.2) 2\"},\"formula_index\":6,\"parent_page_sha256\":\"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7.pdf","asset_sha256":"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7","document_id":"arxiv:2608.07909","kind":"formula","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[145.56,391.5,353.84,47.6],\"bbox_pdf\":[145.56,352.9,499.4,400.5],\"fragments\":[{\"bbox_pdf\":[145.56,352.9,499.4,400.5],\"words\":[3215,3216,3217,3218,3219,3220,3221,3222,3223,3224,3225,3226,3227,3229,3230,3231,3232,3234,3235,3236,3238,3239,3240,3241,3242,3243,3244,3245,3246,3247,3248,3249,3250,3251,3252,3253,3254,3255,3256,3257,3258,3259,3260,3262,3263,3264,3266,3267,3268,3269,3270,3271,3272,3273,3274,3275,3276,3277,3278,3279,3280,3281,3282,3284,3285,3286,3287]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\Gamma_{01}^{0}=\\\\frac{d\\\\alpha}{dr}, \\\\Gamma_{00}^{1}=e^{2(\\\\alpha-\\\\beta)}\\\\frac{d\\\\alpha}{dr}, \\\\Gamma_{11}^{1}=\\\\frac{d\\\\beta}{dr}, \\\\Gamma_{22}^{1}=-e^{2(\\\\gamma-\\\\beta)}\\\\frac{d\\\\gamma}{dr} \\\\Gamma_{33}^{1}=-\\\\sin^{2}\\\\theta e^{2(\\\\gamma-\\\\beta)}\\\\frac{d\\\\gamma}{dr}, \\\\Gamma_{12}^{2}=\\\\frac{d\\\\gamma}{dr}, \\\\Gamma_{33}^{2}=-\\\\sin\\\\theta\\\\cos\\\\theta, \\\\Gamma_{13}^{3}=\\\\frac{d\\\\gamma}{dr}, \\\\Gamma_{23}^{3}=\\\\cot\\\\theta,\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S4.Ex3\",\"source_text\":\"$\\\\Gamma_{01}^{0}=\\\\frac{d\\\\alpha}{dr},$ $\\\\Gamma_{00}^{1}=e^{2(\\\\alpha-\\\\beta)}\\\\frac{d\\\\alpha}{dr},$ $\\\\Gamma_{11}^{1}=\\\\frac{d\\\\beta}{dr},$ $\\\\Gamma_{22}^{1}=-e^{2(\\\\gamma-\\\\beta)}\\\\frac{d\\\\gamma}{dr}$ $\\\\Gamma_{33}^{1}=-\\\\sin^{2}\\\\theta e^{2(\\\\gamma-\\\\beta)}\\\\frac{d\\\\gamma}{dr},$ $\\\\Gamma_{12}^{2}=\\\\frac{d\\\\gamma}{dr},$ $\\\\Gamma_{33}^{2}=-\\\\sin\\\\theta\\\\cos\\\\theta,$ $\\\\Gamma_{13}^{3}=\\\\frac{d\\\\gamma}{dr},$ $\\\\Gamma_{23}^{3}=\\\\cot\\\\theta,$ (4.3)\",\"tag\":\"(4.3)\",\"text\":\"Γ 0 dα Γ 1 2(α−β) dα Γ 1 dβ 01 = dr 00 = e dr 11 = dr Γ 1 2(γ−β) dγ Γ 1 2 θe 2(γ−β) dγ Γ 2 dγ (4.3) 22 = −e dr 33 = − sin dr 12 = dr Γ 2 Γ 3 dγ Γ 3 33 = − sin θ cos θ, 13 = dr 23 = cot θ,\"},\"formula_index\":8,\"parent_page_sha256\":\"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7.pdf","asset_sha256":"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7","document_id":"arxiv:2608.07909","kind":"formula","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[198.72,476.52,300.67,15.86],\"bbox_pdf\":[198.72,299.62,499.4,315.48],\"fragments\":[{\"bbox_pdf\":[198.72,299.62,499.4,315.48],\"words\":[3304,3305,3306,3307,3308,3309,3310,3311,3312,3313,3314,3315,3316,3317,3318,3319,3320]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"R_{\\\\mu\\\\nu}=\\\\partial_{\\\\alpha}\\\\Gamma_{\\\\mu\\\\nu}^{\\\\alpha}-\\\\partial_{\\\\nu}\\\\Gamma_{\\\\mu%\\n\\\\alpha}^{\\\\alpha}+\\\\Gamma_{\\\\alpha\\\\lambda}^{\\\\alpha}\\\\Gamma_{\\\\mu\\\\nu}^{\\\\lambda}-%\\n\\\\Gamma_{\\\\nu\\\\lambda}^{\\\\alpha}\\\\Gamma_{\\\\mu\\\\alpha}^{\\\\lambda}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"S4.Ex4\",\"source_text\":\"$R_{\\\\mu\\\\nu}=\\\\partial_{\\\\alpha}\\\\Gamma_{\\\\mu\\\\nu}^{\\\\alpha}-\\\\partial_{\\\\nu}\\\\Gamma_{\\\\mu% \\\\alpha}^{\\\\alpha}+\\\\Gamma_{\\\\alpha\\\\lambda}^{\\\\alpha}\\\\Gamma_{\\\\mu\\\\nu}^{\\\\lambda}-% \\\\Gamma_{\\\\nu\\\\lambda}^{\\\\alpha}\\\\Gamma_{\\\\mu\\\\alpha}^{\\\\lambda}.$ (4.4)\",\"tag\":\"(4.4)\",\"text\":\"Rµν = ∂αΓ α α α λ α λ (4.4) µν − ∂νΓµα + ΓαλΓµν − ΓνλΓµα.\"},\"formula_index\":10,\"parent_page_sha256\":\"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7.pdf","asset_sha256":"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7","document_id":"arxiv:2608.07909","kind":"formula","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[207.86,519.39,291.54,83.12],\"bbox_pdf\":[207.86,189.49,499.4,272.61],\"fragments\":[{\"bbox_pdf\":[207.86,189.49,499.4,272.61],\"words\":[3328,3329,3330,3331,3332,3333,3334,3335,3336,3337,3338,3339,3340,3341,3342,3343,3344,3345,3346,3347,3348,3349,3350,3351,3352,3353,3354,3355,3356,3357,3358,3359,3360,3361,3362,3363,3364,3365,3366,3367,3368,3369,3370,3371,3372,3373,3374,3375,3376,3377,3378,3379,3380,3381,3382,3383,3384,3385,3386,3387,3388]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{array}[]{c}R_{00}=e^{2(\\\\alpha-\\\\beta)}\\\\left[\\\\frac{d^{2}\\\\alpha}{dr^{2}}+%\\n\\\\frac{d\\\\alpha}{dr}\\\\frac{d(\\\\alpha-\\\\beta+2\\\\gamma)}{dr}\\\\right],\\\\\\\\\\nR_{11}=\\\\frac{d\\\\alpha}{dr}\\\\frac{d(\\\\beta-\\\\alpha)}{dr}+2\\\\frac{d\\\\gamma}{dr}\\\\frac{d%\\n(\\\\beta-\\\\gamma)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}.\\\\\\\\\\nR_{22}=1+e^{2(\\\\gamma-\\\\beta)}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha-2\\\\gamma)}{dr}\\\\frac{d%\\n\\\\gamma}{dr}-\\\\frac{d^{2}\\\\gamma}{dr^{2}}\\\\right],\\\\\\\\\\nR_{33}=\\\\sin^{2}\\\\theta R_{22}.\\\\end{array}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"S4.Ex5\",\"source_text\":\"$\\\\begin{array}[]{c}R_{00}=e^{2(\\\\alpha-\\\\beta)}\\\\left[\\\\frac{d^{2}\\\\alpha}{dr^{2}}+% \\\\frac{d\\\\alpha}{dr}\\\\frac{d(\\\\alpha-\\\\beta+2\\\\gamma)}{dr}\\\\right],\\\\\\\\ R_{11}=\\\\frac{d\\\\alpha}{dr}\\\\frac{d(\\\\beta-\\\\alpha)}{dr}+2\\\\frac{d\\\\gamma}{dr}\\\\frac{d% (\\\\beta-\\\\gamma)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}.\\\\\\\\ R_{22}=1+e^{2(\\\\gamma-\\\\beta)}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha-2\\\\gamma)}{dr}\\\\frac{d% \\\\gamma}{dr}-\\\\frac{d^{2}\\\\gamma}{dr^{2}}\\\\right],\\\\\\\\ R_{33}=\\\\sin^{2}\\\\theta R_{22}.\\\\end{array}$ (4.5)\",\"tag\":\"(4.5)\",\"text\":\"[︂ ]︂ 2(α−β) d 2 α dα d(α−β+2γ) R00 = e 2 + , dr dr dr dα d(β−α) dγ d(β−γ) d 2 (α+2γ) R11 = + 2 − 2 dr dr [︂ dr dr dr ]︂ . (4.5) 2(γ−β) d(β−α−2γ) dγ d 2 γ R22 = 1 + e 2 , dr dr − dr R33 = sin 2 θR22.\"},\"formula_index\":12,\"parent_page_sha256\":\"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7.pdf","asset_sha256":"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7","document_id":"arxiv:2608.07909","kind":"formula","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[119.08,630.88,380.33,40.6],\"bbox_pdf\":[119.08,120.52,499.4,161.12],\"fragments\":[{\"bbox_pdf\":[119.08,120.52,499.4,161.12],\"words\":[3400,3401,3402,3403,3404,3405,3406,3407,3408,3409,3410,3411,3412,3413,3414,3415,3416,3417,3418,3419,3420,3421,3422,3423,3424,3425,3426,3427,3428,3429,3430,3432]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"R=2e^{-2\\\\beta}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha)}{dr}\\\\frac{d\\\\left(\\\\alpha+2\\\\gamma%\\n\\\\right)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}-3\\\\left(\\\\frac{d%\\n\\\\gamma}{dr}\\\\right)^{2}\\\\right]+2e^{-2\\\\gamma}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"S4.Ex6\",\"source_text\":\"$R=2e^{-2\\\\beta}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha)}{dr}\\\\frac{d\\\\left(\\\\alpha+2\\\\gamma% \\\\right)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}-3\\\\left(\\\\frac{d% \\\\gamma}{dr}\\\\right)^{2}\\\\right]+2e^{-2\\\\gamma}.$ (4.6)\",\"tag\":\"(4.6)\",\"text\":\"[︄ (︃ )︃2 ]︄ d(β − α) d (α + 2γ) d 2 (α + 2γ) dγ −2β −2γ R = 2e dr dr − dr2 − 3 dr + 2e (4.6)\"},\"formula_index\":14,\"parent_page_sha256\":\"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7.pdf","asset_sha256":"82e89366213801452169f9b24d08ce0d29426b99f65a55f2f76ff6339c071fa7","document_id":"arxiv:2608.07909","kind":"formula","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,125.8,388.55,40.85],\"bbox_pdf\":[110.85,625.35,499.4,666.2],\"fragments\":[{\"bbox_pdf\":[110.85,625.35,499.4,666.2],\"words\":[3433,3434,3435,3436,3437,3438,3439,3440,3441,3442,3443,3444,3445,3446,3447,3448,3449,3450,3451,3452,3453,3454,3455,3456,3457,3458,3459,3460,3461,3462,3463,3464,3465,3466]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph102\",\"source_text\":\"The Einstein-Hilbert action is $16\\\\pi GS=\\\\int\\\\sqrt{-g}Rd^{4}x$ . Since $\\\\sqrt{-g}=e^{\\\\alpha+\\\\beta+2\\\\gamma}\\\\sin\\\\theta$ , omitting a constant factor, we are interested in the integral with respect to $r$ of\",\"tag\":null,\"text\":\"∫︁ √ √ 4 The Einstein-Hilbert action is 16πGS = −gRd x. Since −g = e α+β+2γ sin θ, omitting a constant factor, we are interested in the integral with respect to r of\"},{\"bbox\":[126.91,181.75,328.2,40.6],\"bbox_pdf\":[126.91,569.65,455.12,610.25],\"fragments\":[{\"bbox_pdf\":[126.91,569.65,455.12,610.25],\"words\":[3467,3468,3469,3470,3471,3472,3473,3474,3475,3476,3477,3478,3479,3480,3481,3482,3483,3484,3485,3486,3487,3488,3489,3490,3491,3492,3493,3494,3495,3496,3498]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{e^{\\\\alpha-\\\\beta+2\\\\gamma}}{2}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha)}{dr}\\\\frac{d\\\\left%\\n(\\\\alpha+2\\\\gamma\\\\right)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}-3%\\n\\\\left(\\\\frac{d\\\\gamma}{dr}\\\\right)^{2}\\\\right]+\\\\frac{e^{\\\\alpha+\\\\beta}}{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S4.Ex7\",\"source_text\":\"$\\\\frac{e^{\\\\alpha-\\\\beta+2\\\\gamma}}{2}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha)}{dr}\\\\frac{d\\\\left% (\\\\alpha+2\\\\gamma\\\\right)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}-3% \\\\left(\\\\frac{d\\\\gamma}{dr}\\\\right)^{2}\\\\right]+\\\\frac{e^{\\\\alpha+\\\\beta}}{2}.$ (4.7)\",\"tag\":\"(4.7)\",\"text\":\"[︄ (︃ )︃2 ]︄ e α−β+2γ d(β − α) d (α + 2γ) d 2 (α + 2γ) dγ e α+β 2 dr dr − dr2 − 3 dr + 2\"},{\"bbox\":[475.33,202.2,24.06,11.96],\"bbox_pdf\":[475.33,577.85,499.39,589.8],\"fragments\":[{\"bbox_pdf\":[475.33,577.85,499.39,589.8],\"words\":[3499]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{e^{\\\\alpha-\\\\beta+2\\\\gamma}}{2}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha)}{dr}\\\\frac{d\\\\left%\\n(\\\\alpha+2\\\\gamma\\\\right)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}-3%\\n\\\\left(\\\\frac{d\\\\gamma}{dr}\\\\right)^{2}\\\\right]+\\\\frac{e^{\\\\alpha+\\\\beta}}{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S4.Ex7\",\"source_text\":\"$\\\\frac{e^{\\\\alpha-\\\\beta+2\\\\gamma}}{2}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha)}{dr}\\\\frac{d\\\\left% (\\\\alpha+2\\\\gamma\\\\right)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}-3% \\\\left(\\\\frac{d\\\\gamma}{dr}\\\\right)^{2}\\\\right]+\\\\frac{e^{\\\\alpha+\\\\beta}}{2}.$ (4.7)\",\"tag\":\"(4.7)\",\"text\":\"(4.7)\"},{\"bbox\":[110.85,237.58,388.55,26.4],\"bbox_pdf\":[110.85,528.02,499.4,554.42],\"fragments\":[{\"bbox_pdf\":[110.85,528.02,499.4,554.42],\"words\":[3500,3501,3502,3503,3504,3505,3506,3507,3508,3509,3510,3511,3512,3513,3514,3515,3516,3517,3518,3519,3520]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph104\",\"source_text\":\"This functional allows us to use the Euler-Lagrange equations. Up to a total derivative, it is equivalent to the (radial) Lagrangian:\",\"tag\":null,\"text\":\"This functional allows us to use the Euler-Lagrange equations. Up to a total derivative, it is equivalent to the (radial) Lagrangian:\"},{\"bbox\":[196.37,267.95,303.03,40.6],\"bbox_pdf\":[196.37,483.45,499.4,524.05],\"fragments\":[{\"bbox_pdf\":[196.37,483.45,499.4,524.05],\"words\":[3521,3522,3523,3524,3525,3526,3527,3528,3529,3530,3531,3532,3533,3534,3535,3536,3538,3539,3540,3541,3542,3543]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"L_{r}=\\\\frac{e^{\\\\alpha-\\\\beta+2\\\\gamma}}{2}\\\\left[2\\\\frac{d\\\\alpha}{dr}\\\\frac{d\\\\gamma%\\n}{dr}+\\\\left(\\\\frac{d\\\\gamma}{dr}\\\\right)^{2}\\\\right]+\\\\frac{e^{\\\\alpha+\\\\beta}}{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S4.Ex8\",\"source_text\":\"$L_{r}=\\\\frac{e^{\\\\alpha-\\\\beta+2\\\\gamma}}{2}\\\\left[2\\\\frac{d\\\\alpha}{dr}\\\\frac{d\\\\gamma% }{dr}+\\\\left(\\\\frac{d\\\\gamma}{dr}\\\\right)^{2}\\\\right]+\\\\frac{e^{\\\\alpha+\\\\beta}}{2}.$ (4.8)\",\"tag\":\"(4.8)\",\"text\":\"[︄ (︃ )︃2 ]︄ e α−β+2γ dα dγ dγ e α+β Lr = 2 + + (4.8) 2 dr dr dr 2\"},{\"bbox\":[128.41,319.58,363.93,14.1],\"bbox_pdf\":[128.41,458.32,492.34,472.42],\"fragments\":[{\"bbox_pdf\":[128.41,458.32,492.34,472.42],\"words\":[3544,3545,3546,3547,3548,3549,3550,3551,3552,3553,3554,3555,3556,3557,3558,3559,3560,3561]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph106\",\"source_text\":\"We define the “coordinates” $q^{1}=\\\\gamma$ and $q^{2}=\\\\alpha$ . Then we rewrite $L_{r}$ as\",\"tag\":null,\"text\":\"We define the “coordinates” q 1 = γ and q 2 = α. Then we rewrite Lr as\"},{\"bbox\":[220.75,338.69,278.65,33.43],\"bbox_pdf\":[220.75,419.88,499.4,453.31],\"fragments\":[{\"bbox_pdf\":[220.75,419.88,499.4,453.31],\"words\":[3562,3563,3564,3565,3566,3567,3568,3569,3570,3571,3572,3573,3574,3575,3576,3577,3578,3579,3580,3581,3582,3583,3584]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"L=\\\\frac{\\\\lambda^{-1}}{2}\\\\left[\\\\left(\\\\dot{q}^{1}\\\\right)^{2}+2\\\\dot{q}^{1}\\\\dot{q}%\\n^{2}\\\\right]+\\\\frac{\\\\lambda}{2}M^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S4.Ex9\",\"source_text\":\"$L=\\\\frac{\\\\lambda^{-1}}{2}\\\\left[\\\\left(\\\\dot{q}^{1}\\\\right)^{2}+2\\\\dot{q}^{1}\\\\dot{q}% ^{2}\\\\right]+\\\\frac{\\\\lambda}{2}M^{2},$ (4.9)\",\"tag\":\"(4.9)\",\"text\":\"λ −1 [︂(︁ )︁2 ]︂ λ L = q˙ 1 + 2 ˙q 1 q˙ 2 + M 2 , (4.9) 2 2\"},{\"bbox\":[110.85,381.82,388.55,84.54],\"bbox_pdf\":[110.85,325.64,499.4,410.18],\"fragments\":[{\"bbox_pdf\":[110.85,325.64,499.4,410.18],\"words\":[3585,3586,3587,3588,3589,3590,3591,3592,3593,3594,3595,3596,3597,3598,3599,3600,3601,3602,3603,3604,3605,3607,3608,3609,3610,3611,3612,3613,3614,3615,3616,3617,3618,3619,3620,3621,3622,3623,3624,3625,3626,3627,3628,3629,3630,3631,3632,3633,3634,3635,3636,3637,3638,3648,3639,3640,3641,3649,3642,3643,3644,3645,3646,3647,3650,3651,3652,3653,3654,3655,3656,3657,3658,3659,3660,3661,3662,3663,3664,3665,3666,3667,3668,3669,3670,3671,3672,3673,3674,3675,3676,3677,3678]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph108\",\"source_text\":\"where we have further defined $\\\\lambda^{-1}=e^{\\\\alpha-\\\\beta+2\\\\gamma}$ and $M^{2}=\\\\lambda^{-1}e^{\\\\alpha+\\\\beta}=e^{2\\\\alpha+2\\\\gamma}$ . Additionally, for simplicity, we use dot notation for total derivatives with respect to $r$ , such as in $\\\\dot{q}^{1}=dq^{1}/dr$ . This suggestive form allows us to treat this Lagrangian as describing a relativistic point particle in two dimensions in the form $\\\\left(\\\\lambda^{-1}Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}+\\\\lambda M^{2}\\\\right)/2$ , where $a,b$ run over $\\\\{1,2\\\\}$ . We call $Q_{ab}$ the Fibonacci Q-metric, with components\",\"tag\":null,\"text\":\"where we have further defined λ −1 = e α−β+2γ and M 2 = λ −1 e α+β = e 2α+2γ Additionally, for simplicity, we use dot notation for total derivatives with respect to r, such as in q˙ 1 = dq 1 /dr. This suggestive form allows us to treat this Lagrangian (︁ as describing a )︁ relativistic point particle in two dimensions in the form λ −1 Qab ˙q a q˙ b + λM 2 /2, where a, b run over {1, 2}. We call Qab the Fibonacci Q-metric, with components\"},{\"bbox\":[266.05,469.44,233.35,30.61],\"bbox_pdf\":[266.05,291.95,499.4,322.56],\"fragments\":[{\"bbox_pdf\":[266.05,291.95,499.4,322.56],\"words\":[3679,3680,3681,3682,3683,3684,3685,3686,3687,3688]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"Q_{ab}=\\\\begin{pmatrix}1&1\\\\\\\\\\n1&0\\\\end{pmatrix}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S4.Ex10\",\"source_text\":\"$Q_{ab}=\\\\begin{pmatrix}1&1\\\\\\\\ 1&0\\\\end{pmatrix}.$ (4.10)\",\"tag\":\"(4.10)\",\"text\":\"(︃ )︃ 1 1 Qab = . (4.10) 1 0\"},{\"bbox\":[110.85,519.02,388.54,26.75],\"bbox_pdf\":[110.85,246.23,499.4,272.98],\"fragments\":[{\"bbox_pdf\":[110.85,246.23,499.4,272.98],\"words\":[3689,3690,3691,3692,3693,3695,3696,3697,3698,3699,3700,3701,3702,3703,3704,3705,3706,3707,3708,3709,3710,3711,3713,3714,3715,3716,3717,3718,3719,3720,3722,3723]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph110\",\"source_text\":\"We will use $q^{1}$ , $q^{2}$ and $\\\\lambda$ as independent variables. The variations with respect to $q^{1}$ and $q^{2}$ , with $\\\\partial M^{2}/\\\\partial q^{a}=2M^{2}$ , lead to\",\"tag\":null,\"text\":\"We will use q 1 q 2 and λ as independent variables. The variations with respect to q 1 and q 2 with ∂M 2 /∂q a = 2M 2 lead to\"},{\"bbox\":[238.63,564.7,260.77,21.11],\"bbox_pdf\":[238.63,206.19,499.4,227.3],\"fragments\":[{\"bbox_pdf\":[238.63,206.19,499.4,227.3],\"words\":[3724,3725,3726,3727,3728,3729,3730,3731,3732,3733,3734,3735,3736,3737,3738,3739,3740]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{d}{dr}\\\\left[\\\\lambda^{-1}\\\\left(\\\\dot{q}^{1}+\\\\dot{q}^{2}\\\\right)\\\\right]=%\\n\\\\lambda M^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"S4.Ex11\",\"source_text\":\"$\\\\frac{d}{dr}\\\\left[\\\\lambda^{-1}\\\\left(\\\\dot{q}^{1}+\\\\dot{q}^{2}\\\\right)\\\\right]=% \\\\lambda M^{2},$ (4.11)\",\"tag\":\"(4.11)\",\"text\":\"d [︁ (︁ )︁]︁ −1 1 2 2 λ q˙ + ˙q = λM , (4.11) dr\"},{\"bbox\":[110.85,589.28,18.86,11.96],\"bbox_pdf\":[110.85,190.77,129.71,202.72],\"fragments\":[{\"bbox_pdf\":[110.85,190.77,129.71,202.72],\"words\":[3741]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"paragraph\",\"source_ref\":\"paragraph112\",\"source_text\":\"and\",\"tag\":null,\"text\":\"and\"},{\"bbox\":[256.99,605.71,242.41,21.11],\"bbox_pdf\":[256.99,165.18,499.4,186.29],\"fragments\":[{\"bbox_pdf\":[256.99,165.18,499.4,186.29],\"words\":[3742,3743,3744,3745,3746,3747,3748,3749,3750,3751,3752,3753,3754]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{d}{dr}\\\\left(\\\\lambda^{-1}\\\\dot{q}^{1}\\\\right)=\\\\lambda M^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"S4.Ex12\",\"source_text\":\"$\\\\frac{d}{dr}\\\\left(\\\\lambda^{-1}\\\\dot{q}^{1}\\\\right)=\\\\lambda M^{2},$ (4.12)\",\"tag\":\"(4.12)\",\"text\":\"d (︁ )︁ −1 1 2 λ q˙ = λM , (4.12) dr\"},{\"bbox\":[110.85,630.29,261.3,11.96],\"bbox_pdf\":[110.85,149.76,372.16,161.71],\"fragments\":[{\"bbox_pdf\":[110.85,149.76,372.16,161.71],\"words\":[3755,3756,3757,3758,3759,3760,3761,3762]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":13,\"role\":\"paragraph\",\"source_ref\":\"paragraph114\",\"source_text\":\"respectively. The variation with respect to $\\\\lambda$ yields\",\"tag\":null,\"text\":\"respectively. The variation with respect to λ yields\"},{\"bbox\":[237.75,645.23,261.65,25.23],\"bbox_pdf\":[237.75,121.54,499.4,146.77],\"fragments\":[{\"bbox_pdf\":[237.75,121.54,499.4,146.77],\"words\":[3763,3764,3765,3766,3767,3768,3769,3770,3771,3772,3773,3774,3775,3776,3777,3778,3779,3780]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\lambda^{-2}\\\\left[\\\\left(\\\\dot{q}^{1}\\\\right)^{2}+2\\\\dot{q}^{1}\\\\dot{q}^{2}\\\\right]=%\\nM^{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"S4.Ex13\",\"source_text\":\"$\\\\lambda^{-2}\\\\left[\\\\left(\\\\dot{q}^{1}\\\\right)^{2}+2\\\\dot{q}^{1}\\\\dot{q}^{2}\\\\right]=% M^{2}.$ (4.13)\",\"tag\":\"(4.13)\",\"text\":\"[︂(︁ )︁2 ]︂ λ −2 q˙ 1 + 2 ˙q 1 q˙ 2 = M 2 . (4.13)\"},{\"bbox\":[299.27,690.18,11.71,11.96],\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"fragments\":[{\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"words\":[3781]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":15,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"12\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0011.pdf\",\"page_no\":12,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1.pdf","asset_sha256":"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1","document_id":"arxiv:2608.07909","kind":"page","page_number":12,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[126.91,181.75,328.2,40.6],\"bbox_pdf\":[126.91,569.65,455.12,610.25],\"fragments\":[{\"bbox_pdf\":[126.91,569.65,455.12,610.25],\"words\":[3467,3468,3469,3470,3471,3472,3473,3474,3475,3476,3477,3478,3479,3480,3481,3482,3483,3484,3485,3486,3487,3488,3489,3490,3491,3492,3493,3494,3495,3496,3498]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{e^{\\\\alpha-\\\\beta+2\\\\gamma}}{2}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha)}{dr}\\\\frac{d\\\\left%\\n(\\\\alpha+2\\\\gamma\\\\right)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}-3%\\n\\\\left(\\\\frac{d\\\\gamma}{dr}\\\\right)^{2}\\\\right]+\\\\frac{e^{\\\\alpha+\\\\beta}}{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S4.Ex7\",\"source_text\":\"$\\\\frac{e^{\\\\alpha-\\\\beta+2\\\\gamma}}{2}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha)}{dr}\\\\frac{d\\\\left% (\\\\alpha+2\\\\gamma\\\\right)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}-3% \\\\left(\\\\frac{d\\\\gamma}{dr}\\\\right)^{2}\\\\right]+\\\\frac{e^{\\\\alpha+\\\\beta}}{2}.$ (4.7)\",\"tag\":\"(4.7)\",\"text\":\"[︄ (︃ )︃2 ]︄ e α−β+2γ d(β − α) d (α + 2γ) d 2 (α + 2γ) dγ e α+β 2 dr dr − dr2 − 3 dr + 2\"},\"formula_index\":1,\"parent_page_sha256\":\"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1.pdf","asset_sha256":"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1","document_id":"arxiv:2608.07909","kind":"formula","page_number":12,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[475.33,202.2,24.06,11.96],\"bbox_pdf\":[475.33,577.85,499.39,589.8],\"fragments\":[{\"bbox_pdf\":[475.33,577.85,499.39,589.8],\"words\":[3499]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{e^{\\\\alpha-\\\\beta+2\\\\gamma}}{2}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha)}{dr}\\\\frac{d\\\\left%\\n(\\\\alpha+2\\\\gamma\\\\right)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}-3%\\n\\\\left(\\\\frac{d\\\\gamma}{dr}\\\\right)^{2}\\\\right]+\\\\frac{e^{\\\\alpha+\\\\beta}}{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S4.Ex7\",\"source_text\":\"$\\\\frac{e^{\\\\alpha-\\\\beta+2\\\\gamma}}{2}\\\\left[\\\\frac{d(\\\\beta-\\\\alpha)}{dr}\\\\frac{d\\\\left% (\\\\alpha+2\\\\gamma\\\\right)}{dr}-\\\\frac{d^{2}\\\\left(\\\\alpha+2\\\\gamma\\\\right)}{dr^{2}}-3% \\\\left(\\\\frac{d\\\\gamma}{dr}\\\\right)^{2}\\\\right]+\\\\frac{e^{\\\\alpha+\\\\beta}}{2}.$ (4.7)\",\"tag\":\"(4.7)\",\"text\":\"(4.7)\"},\"formula_index\":2,\"parent_page_sha256\":\"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1.pdf","asset_sha256":"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1","document_id":"arxiv:2608.07909","kind":"formula","page_number":12,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[196.37,267.95,303.03,40.6],\"bbox_pdf\":[196.37,483.45,499.4,524.05],\"fragments\":[{\"bbox_pdf\":[196.37,483.45,499.4,524.05],\"words\":[3521,3522,3523,3524,3525,3526,3527,3528,3529,3530,3531,3532,3533,3534,3535,3536,3538,3539,3540,3541,3542,3543]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"L_{r}=\\\\frac{e^{\\\\alpha-\\\\beta+2\\\\gamma}}{2}\\\\left[2\\\\frac{d\\\\alpha}{dr}\\\\frac{d\\\\gamma%\\n}{dr}+\\\\left(\\\\frac{d\\\\gamma}{dr}\\\\right)^{2}\\\\right]+\\\\frac{e^{\\\\alpha+\\\\beta}}{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S4.Ex8\",\"source_text\":\"$L_{r}=\\\\frac{e^{\\\\alpha-\\\\beta+2\\\\gamma}}{2}\\\\left[2\\\\frac{d\\\\alpha}{dr}\\\\frac{d\\\\gamma% }{dr}+\\\\left(\\\\frac{d\\\\gamma}{dr}\\\\right)^{2}\\\\right]+\\\\frac{e^{\\\\alpha+\\\\beta}}{2}.$ (4.8)\",\"tag\":\"(4.8)\",\"text\":\"[︄ (︃ )︃2 ]︄ e α−β+2γ dα dγ dγ e α+β Lr = 2 + + (4.8) 2 dr dr dr 2\"},\"formula_index\":4,\"parent_page_sha256\":\"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1.pdf","asset_sha256":"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1","document_id":"arxiv:2608.07909","kind":"formula","page_number":12,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[220.75,338.69,278.65,33.43],\"bbox_pdf\":[220.75,419.88,499.4,453.31],\"fragments\":[{\"bbox_pdf\":[220.75,419.88,499.4,453.31],\"words\":[3562,3563,3564,3565,3566,3567,3568,3569,3570,3571,3572,3573,3574,3575,3576,3577,3578,3579,3580,3581,3582,3583,3584]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"L=\\\\frac{\\\\lambda^{-1}}{2}\\\\left[\\\\left(\\\\dot{q}^{1}\\\\right)^{2}+2\\\\dot{q}^{1}\\\\dot{q}%\\n^{2}\\\\right]+\\\\frac{\\\\lambda}{2}M^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S4.Ex9\",\"source_text\":\"$L=\\\\frac{\\\\lambda^{-1}}{2}\\\\left[\\\\left(\\\\dot{q}^{1}\\\\right)^{2}+2\\\\dot{q}^{1}\\\\dot{q}% ^{2}\\\\right]+\\\\frac{\\\\lambda}{2}M^{2},$ (4.9)\",\"tag\":\"(4.9)\",\"text\":\"λ −1 [︂(︁ )︁2 ]︂ λ L = q˙ 1 + 2 ˙q 1 q˙ 2 + M 2 , (4.9) 2 2\"},\"formula_index\":6,\"parent_page_sha256\":\"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1.pdf","asset_sha256":"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1","document_id":"arxiv:2608.07909","kind":"formula","page_number":12,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[266.05,469.44,233.35,30.61],\"bbox_pdf\":[266.05,291.95,499.4,322.56],\"fragments\":[{\"bbox_pdf\":[266.05,291.95,499.4,322.56],\"words\":[3679,3680,3681,3682,3683,3684,3685,3686,3687,3688]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"Q_{ab}=\\\\begin{pmatrix}1&1\\\\\\\\\\n1&0\\\\end{pmatrix}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S4.Ex10\",\"source_text\":\"$Q_{ab}=\\\\begin{pmatrix}1&1\\\\\\\\ 1&0\\\\end{pmatrix}.$ (4.10)\",\"tag\":\"(4.10)\",\"text\":\"(︃ )︃ 1 1 Qab = . (4.10) 1 0\"},\"formula_index\":8,\"parent_page_sha256\":\"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1.pdf","asset_sha256":"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1","document_id":"arxiv:2608.07909","kind":"formula","page_number":12,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[238.63,564.7,260.77,21.11],\"bbox_pdf\":[238.63,206.19,499.4,227.3],\"fragments\":[{\"bbox_pdf\":[238.63,206.19,499.4,227.3],\"words\":[3724,3725,3726,3727,3728,3729,3730,3731,3732,3733,3734,3735,3736,3737,3738,3739,3740]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{d}{dr}\\\\left[\\\\lambda^{-1}\\\\left(\\\\dot{q}^{1}+\\\\dot{q}^{2}\\\\right)\\\\right]=%\\n\\\\lambda M^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"S4.Ex11\",\"source_text\":\"$\\\\frac{d}{dr}\\\\left[\\\\lambda^{-1}\\\\left(\\\\dot{q}^{1}+\\\\dot{q}^{2}\\\\right)\\\\right]=% \\\\lambda M^{2},$ (4.11)\",\"tag\":\"(4.11)\",\"text\":\"d [︁ (︁ )︁]︁ −1 1 2 2 λ q˙ + ˙q = λM , (4.11) dr\"},\"formula_index\":10,\"parent_page_sha256\":\"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1.pdf","asset_sha256":"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1","document_id":"arxiv:2608.07909","kind":"formula","page_number":12,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[256.99,605.71,242.41,21.11],\"bbox_pdf\":[256.99,165.18,499.4,186.29],\"fragments\":[{\"bbox_pdf\":[256.99,165.18,499.4,186.29],\"words\":[3742,3743,3744,3745,3746,3747,3748,3749,3750,3751,3752,3753,3754]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{d}{dr}\\\\left(\\\\lambda^{-1}\\\\dot{q}^{1}\\\\right)=\\\\lambda M^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"S4.Ex12\",\"source_text\":\"$\\\\frac{d}{dr}\\\\left(\\\\lambda^{-1}\\\\dot{q}^{1}\\\\right)=\\\\lambda M^{2},$ (4.12)\",\"tag\":\"(4.12)\",\"text\":\"d (︁ )︁ −1 1 2 λ q˙ = λM , (4.12) dr\"},\"formula_index\":12,\"parent_page_sha256\":\"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1.pdf","asset_sha256":"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1","document_id":"arxiv:2608.07909","kind":"formula","page_number":12,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[237.75,645.23,261.65,25.23],\"bbox_pdf\":[237.75,121.54,499.4,146.77],\"fragments\":[{\"bbox_pdf\":[237.75,121.54,499.4,146.77],\"words\":[3763,3764,3765,3766,3767,3768,3769,3770,3771,3772,3773,3774,3775,3776,3777,3778,3779,3780]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\lambda^{-2}\\\\left[\\\\left(\\\\dot{q}^{1}\\\\right)^{2}+2\\\\dot{q}^{1}\\\\dot{q}^{2}\\\\right]=%\\nM^{2}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"S4.Ex13\",\"source_text\":\"$\\\\lambda^{-2}\\\\left[\\\\left(\\\\dot{q}^{1}\\\\right)^{2}+2\\\\dot{q}^{1}\\\\dot{q}^{2}\\\\right]=% M^{2}.$ (4.13)\",\"tag\":\"(4.13)\",\"text\":\"[︂(︁ )︁2 ]︂ λ −2 q˙ 1 + 2 ˙q 1 q˙ 2 = M 2 . (4.13)\"},\"formula_index\":14,\"parent_page_sha256\":\"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1.pdf","asset_sha256":"26cee9fabcfaac103417c8cdccfb705dc3ffbd59e3f3121f058d3f335f82d5a1","document_id":"arxiv:2608.07909","kind":"formula","page_number":12,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,125.8,388.55,40.85],\"bbox_pdf\":[110.85,625.35,499.4,666.2],\"fragments\":[{\"bbox_pdf\":[110.85,625.35,499.4,666.2],\"words\":[3782,3783,3784,3785,3786,3787,3788,3789,3790,3791,3792,3793,3794,3795,3796,3797,3798,3799,3800,3801,3802,3803,3804,3805,3806,3807,3808,3809,3810,3811]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph116\",\"source_text\":\"Here, it is worthwhile to emphasize that these equations are consistent with the ones obtained directly from the original Einstein-Hilbert action [17]. In fact, Eqs. (4.11) and (4.12) together imply:\",\"tag\":null,\"text\":\"Here, it is worthwhile to emphasize that these equations are consistent with the ones obtained directly from the original Einstein-Hilbert action [17]. In fact, Eqs. (4.11) and (4.12) together imply:\"},{\"bbox\":[269.86,178.33,229.54,28.6],\"bbox_pdf\":[269.86,585.08,499.4,613.67],\"fragments\":[{\"bbox_pdf\":[269.86,585.08,499.4,613.67],\"words\":[3812,3813,3814,3815,3816,3818,3819,3820,3821]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{d\\\\left(\\\\lambda^{-1}\\\\dot{q}^{2}\\\\right)}{dr}=0.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S4.Ex14\",\"source_text\":\"$\\\\frac{d\\\\left(\\\\lambda^{-1}\\\\dot{q}^{2}\\\\right)}{dr}=0.$ (4.14)\",\"tag\":\"(4.14)\",\"text\":\"d (λ −1 q˙ 2 = 0. (4.14) dr\"},{\"bbox\":[110.85,216.02,388.55,41.2],\"bbox_pdf\":[110.85,534.78,499.4,575.98],\"fragments\":[{\"bbox_pdf\":[110.85,534.78,499.4,575.98],\"words\":[3822,3823,3824,3825,3826,3827,3828,3829,3830,3831,3832,3833,3834,3836,3837,3838,3839,3840,3841,3842,3843,3844,3845,3846,3847,3848,3849,3850,3851,3852,3853,3854,3855,3856,3857,3858,3859,3860]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph118\",\"source_text\":\"Recalling the definitions of the coordinates and that $\\\\lambda^{-1}=e^{\\\\alpha-\\\\beta+2\\\\gamma}$ , one can see that, up to a factor, Eq. (4.14) is equivalent to the field equation $R_{00}=0$ , which can be read from (4.5).\",\"tag\":null,\"text\":\"Recalling the definitions of the coordinates and that λ −1 = e α−β+2γ one can see that, up to a factor, Eq. (4.14) is equivalent to the field equation R00 = 0, which can be read from (4.5).\"},{\"bbox\":[128.41,259.71,329.32,11.96],\"bbox_pdf\":[128.41,520.34,457.74,532.29],\"fragments\":[{\"bbox_pdf\":[128.41,520.34,457.74,532.29],\"words\":[3861,3862,3863,3864,3865,3866,3867,3868,3869,3870,3871,3872]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph119\",\"source_text\":\"Also, by using Eqs. (4.13) and (4.14) in Eq. (4.11), one obtains:\",\"tag\":null,\"text\":\"Also, by using Eqs. (4.13) and (4.14) in Eq. (4.11), one obtains:\"},{\"bbox\":[222.62,289.66,276.78,39.3],\"bbox_pdf\":[222.62,463.04,499.4,502.34],\"fragments\":[{\"bbox_pdf\":[222.62,463.04,499.4,502.34],\"words\":[3873,3874,3875,3876,3877,3878,3879,3880,3881,3882,3883,3884,3885,3886,3887,3888,3889,3890,3891,3892,3893,3894]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{d}{dr}\\\\left[\\\\ln\\\\left(\\\\lambda^{-1}\\\\dot{q}^{1}\\\\right)-q^{1}-2q^{2}\\\\right]=0.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S4.Ex15\",\"source_text\":\"$\\\\frac{d}{dr}\\\\left[\\\\ln\\\\left(\\\\lambda^{-1}\\\\dot{q}^{1}\\\\right)-q^{1}-2q^{2}\\\\right]=0.$ (4.15)\",\"tag\":\"(4.15)\",\"text\":\"d [︁ (︁ )︁ ]︁ −1 1 1 2 ln λ q˙ − q − 2q = 0. (4.15) dr −1 1\"},{\"bbox\":[110.85,321.35,388.55,40.85],\"bbox_pdf\":[110.85,429.81,499.4,470.65],\"fragments\":[{\"bbox_pdf\":[110.85,429.81,499.4,470.65],\"words\":[3895,3896,3897,3898,3899,3900,3901,3902,3903,3904,3905,3906,3908,3909,3910,3911,3912,3913,3914,3915,3916,3917,3918,3919,3920,3921,3922,3923,3924,3925,3926,3927,3928,3930,3931,3932,3933,3934,3935,3936,3937,3938,3939,3940,3941,3942,3943,3944,3945,3946,3947,3948,3949]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph121\",\"source_text\":\"This is equivalent to $\\\\lambda^{-1}\\\\dot{q}^{1}=2Ke^{q^{1}+2q^{2}}$ , where $K$ is a constant. Returning to the original variables, this is the same as $d(e^{\\\\gamma})/dr=Ke^{\\\\alpha+\\\\beta}$ . We can use this intermediate result in $\\\\lambda^{-1}\\\\dot{q}^{2}=const.$ [cf. (4.14)] to show that this implies:\",\"tag\":null,\"text\":\"q 1 +2q 2 This is equivalent to λ q˙ = 2Ke where K is a constant. Returning to the original variables, this is the same as d(e γ )/dr = Ke α+β We can use this intermediate result in λ −1 q˙ 2 = const. [cf. (4.14)] to show that this implies:\"},{\"bbox\":[243.33,374.59,256.07,15.52],\"bbox_pdf\":[243.33,401.89,499.39,417.41],\"fragments\":[{\"bbox_pdf\":[243.33,401.89,499.39,417.41],\"words\":[3950,3951,3952,3953,3954,3955,3956,3957,3958,3959,3960,3961,3962,3963]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"e^{2\\\\alpha}=e^{2q^{2}}=c_{1}-c_{2}e^{-q^{1}},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S4.Ex16\",\"source_text\":\"$e^{2\\\\alpha}=e^{2q^{2}}=c_{1}-c_{2}e^{-q^{1}},$ (4.16)\",\"tag\":\"(4.16)\",\"text\":\"2α 2q 2 −q 1 e = e = c1 − c2e , (4.16)\"},{\"bbox\":[110.85,401.34,388.54,40.85],\"bbox_pdf\":[110.85,349.82,499.4,390.66],\"fragments\":[{\"bbox_pdf\":[110.85,349.82,499.4,390.66],\"words\":[3964,3965,3966,3967,3968,3969,3970,3971,3972,3973,3974,3975,3976,3977,3978,3979,3980,3981,3982,3983,3984,3985,3986,3987,3988,3989,3990,3991,3992,3993,3994,3995,3996,3997,3998,3999,4000,4001,4002,4003,4004,4005,4006,4007,4008,4009,4010,4011]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph123\",\"source_text\":\"where $c_{1}$ and $c_{2}$ are constants. Again: if one sets $e^{q^{1}}=e^{\\\\gamma}=r$ in the metric (4.1), one recovers the result $\\\\beta=-\\\\alpha$ , with $K=1$ . By asymptotic flatness, Eq. (4.16) is the familiar Schwarzschild factor $1-r_{s}/r=-g_{00}$ .\",\"tag\":null,\"text\":\"q 1 γ where c1 and c2 are constants. Again: if one sets e = e = r in the metric (4.1), one recovers the result β = −α, with K = 1. By asymptotic flatness, Eq. (4.16) is the familiar Schwarzschild factor 1 − rs/r = −g00.\"},{\"bbox\":[110.85,444.67,388.55,69.74],\"bbox_pdf\":[110.85,277.59,499.4,347.33],\"fragments\":[{\"bbox_pdf\":[110.85,277.59,499.4,347.33],\"words\":[4012,4013,4014,4015,4016,4017,4018,4019,4020,4021,4022,4023,4024,4025,4034,4026,4027,4028,4029,4035,4030,4031,4032,4033,4036,4037,4038,4039,4040,4041,4042,4043,4044,4045,4046,4047,4048,4049,4050,4051,4052,4053,4054,4055,4056,4057,4058,4059,4060,4061,4062,4063,4064,4065,4066,4067,4068,4069,4070,4071,4072,4073,4074,4075]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph124\",\"source_text\":\"We have completed the correspondence with classical results, reaffirming Birkhoff’s theorem [7]. Now we turn our attention again to the metric (4.10). Its eigenvalues are $\\\\varphi^{-}=(1-\\\\sqrt{5})/2$ and $\\\\varphi^{+}=(1+\\\\sqrt{5})/2$ . That is, the negative and positive Golden ratios have arisen (see the Appendix). By calculating the eigenvectors, one can see that the coordinate transformation\",\"tag\":null,\"text\":\"We have completed the correspondence with classical results, reaffirming Birkhoff’s theorem [7]. Now we √ turn our attention again √ to the metric (4.10). Its eigenvalues are φ − = (1 − 5)/2 and φ + = (1 + 5)/2. That is, the negative and positive Golden ratios have arisen (see the Appendix). By calculating the eigenvectors, one can see that the coordinate transformation\"},{\"bbox\":[225.3,516.79,274.1,50.81],\"bbox_pdf\":[225.3,224.4,499.4,275.21],\"fragments\":[{\"bbox_pdf\":[225.3,224.4,499.4,275.21],\"words\":[4076,4077,4078,4079,4080,4081,4082,4083,4084,4085,4086,4087,4088,4089,4090,4091,4092,4093,4094,4095,4096,4097,4098,4099,4100,4101,4102,4103,4104]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{array}[]{c}w^{1}=\\\\frac{1}{5^{1/4}}\\\\left(\\\\sqrt{-\\\\varphi^{-}}q^{1}-\\\\sqrt{%\\n\\\\varphi^{+}}q^{2}\\\\right),\\\\\\\\\\n\\\\\\\\\\nw^{2}=\\\\frac{1}{5^{1/4}}\\\\left(\\\\sqrt{\\\\varphi^{+}}q^{1}+\\\\sqrt{-\\\\varphi^{-}}q^{2}%\\n\\\\right),\\\\end{array}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S4.Ex17\",\"source_text\":\"$\\\\begin{array}[]{c}w^{1}=\\\\frac{1}{5^{1/4}}\\\\left(\\\\sqrt{-\\\\varphi^{-}}q^{1}-\\\\sqrt{% \\\\varphi^{+}}q^{2}\\\\right),\\\\\\\\ \\\\\\\\ w^{2}=\\\\frac{1}{5^{1/4}}\\\\left(\\\\sqrt{\\\\varphi^{+}}q^{1}+\\\\sqrt{-\\\\varphi^{-}}q^{2}% \\\\right),\\\\end{array}$ (4.17)\",\"tag\":\"(4.17)\",\"text\":\"(︁√ √ )︁ 1 1 −q1 +q2 w = 1/4 −φ − φ , 5 (︁√ √ )︁ (4.17) w 2 = 1 φ+q1 + −φ−q2 , 1/4 5\"},{\"bbox\":[110.85,581.26,351.98,14.1],\"bbox_pdf\":[110.85,196.64,462.84,210.74],\"fragments\":[{\"bbox_pdf\":[110.85,196.64,462.84,210.74],\"words\":[4105,4106,4107,4108,4109,4110,4111,4112,4113,4114,4115,4116,4118,4119,4120,4121]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"paragraph\",\"source_ref\":\"paragraph126\",\"source_text\":\"implies the equality $Q_{ab}dq^{a}dq^{b}=G_{ab}dw^{a}dw^{b}$ , where we have defined:\",\"tag\":null,\"text\":\"implies the equality Qabdq a dq b = Gabdw a dw b where we have defined:\"},{\"bbox\":[257.17,596.04,242.22,35.94],\"bbox_pdf\":[257.17,160.03,499.39,195.96],\"fragments\":[{\"bbox_pdf\":[257.17,160.03,499.39,195.96],\"words\":[4122,4123,4124,4125,4126,4127,4128,4129,4130,4131,4132,4133]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"G_{ab}=\\\\begin{pmatrix}\\\\varphi^{-}&0\\\\\\\\\\n0&\\\\varphi^{+}\\\\end{pmatrix}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"formula\",\"source_ref\":\"S4.Ex18\",\"source_text\":\"$G_{ab}=\\\\begin{pmatrix}\\\\varphi^{-}&0\\\\\\\\ 0&\\\\varphi^{+}\\\\end{pmatrix}.$ (4.18)\",\"tag\":\"(4.18)\",\"text\":\"(︃ )︃ φ − 0 Gab = + . (4.18) 0 φ\"},{\"bbox\":[110.85,645.85,388.54,28.19],\"bbox_pdf\":[110.85,117.96,499.4,146.15],\"fragments\":[{\"bbox_pdf\":[110.85,117.96,499.4,146.15],\"words\":[4134,4135,4136,4137,4138,4139,4140,4141,4142,4143,4144,4145,4146,4147,4149,4150,4151,4153,4154,4155,4156,4157,4158,4159,4160,4161,4162,4163,4165,4166,4167,4168,4169]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"paragraph\",\"source_ref\":\"paragraph128\",\"source_text\":\"Rescaling coordinates in the form $W^{1}=\\\\sqrt{-\\\\varphi^{-}}w^{1}$ , $W^{2}=\\\\sqrt{\\\\varphi^{+}}w^{2}$ , we obtain the usual (1+1)-Minkowski form $\\\\eta_{ab}dW^{a}dW^{b}$ , where $\\\\eta_{ab}=diag(-1,1)$ . However, the transformation (4.17) preserves the scale. Its inversion is:\",\"tag\":null,\"text\":\"√ √ 1 −w1 2 +w2 Rescaling coordinates in the form W = −φ W = φ we obtain the usual (1+1)-Minkowski form ηabdW a dW b where ηab = diag(−1, 1).\"},{\"bbox\":[299.27,690.18,11.71,11.96],\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"fragments\":[{\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"words\":[10387]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":13,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"13\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0012.pdf\",\"page_no\":13,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef.pdf","asset_sha256":"e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef","document_id":"arxiv:2608.07909","kind":"page","page_number":13,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[269.86,178.33,229.54,28.6],\"bbox_pdf\":[269.86,585.08,499.4,613.67],\"fragments\":[{\"bbox_pdf\":[269.86,585.08,499.4,613.67],\"words\":[3812,3813,3814,3815,3816,3818,3819,3820,3821]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{d\\\\left(\\\\lambda^{-1}\\\\dot{q}^{2}\\\\right)}{dr}=0.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S4.Ex14\",\"source_text\":\"$\\\\frac{d\\\\left(\\\\lambda^{-1}\\\\dot{q}^{2}\\\\right)}{dr}=0.$ (4.14)\",\"tag\":\"(4.14)\",\"text\":\"d (λ −1 q˙ 2 = 0. (4.14) dr\"},\"formula_index\":1,\"parent_page_sha256\":\"e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef.pdf","asset_sha256":"e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef","document_id":"arxiv:2608.07909","kind":"formula","page_number":13,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[222.62,289.66,276.78,39.3],\"bbox_pdf\":[222.62,463.04,499.4,502.34],\"fragments\":[{\"bbox_pdf\":[222.62,463.04,499.4,502.34],\"words\":[3873,3874,3875,3876,3877,3878,3879,3880,3881,3882,3883,3884,3885,3886,3887,3888,3889,3890,3891,3892,3893,3894]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{d}{dr}\\\\left[\\\\ln\\\\left(\\\\lambda^{-1}\\\\dot{q}^{1}\\\\right)-q^{1}-2q^{2}\\\\right]=0.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S4.Ex15\",\"source_text\":\"$\\\\frac{d}{dr}\\\\left[\\\\ln\\\\left(\\\\lambda^{-1}\\\\dot{q}^{1}\\\\right)-q^{1}-2q^{2}\\\\right]=0.$ (4.15)\",\"tag\":\"(4.15)\",\"text\":\"d [︁ (︁ )︁ ]︁ −1 1 1 2 ln λ q˙ − q − 2q = 0. (4.15) dr −1 1\"},\"formula_index\":4,\"parent_page_sha256\":\"e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef.pdf","asset_sha256":"e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef","document_id":"arxiv:2608.07909","kind":"formula","page_number":13,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[243.33,374.59,256.07,15.52],\"bbox_pdf\":[243.33,401.89,499.39,417.41],\"fragments\":[{\"bbox_pdf\":[243.33,401.89,499.39,417.41],\"words\":[3950,3951,3952,3953,3954,3955,3956,3957,3958,3959,3960,3961,3962,3963]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"e^{2\\\\alpha}=e^{2q^{2}}=c_{1}-c_{2}e^{-q^{1}},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S4.Ex16\",\"source_text\":\"$e^{2\\\\alpha}=e^{2q^{2}}=c_{1}-c_{2}e^{-q^{1}},$ (4.16)\",\"tag\":\"(4.16)\",\"text\":\"2α 2q 2 −q 1 e = e = c1 − c2e , (4.16)\"},\"formula_index\":6,\"parent_page_sha256\":\"e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef.pdf","asset_sha256":"e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef","document_id":"arxiv:2608.07909","kind":"formula","page_number":13,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[225.3,516.79,274.1,50.81],\"bbox_pdf\":[225.3,224.4,499.4,275.21],\"fragments\":[{\"bbox_pdf\":[225.3,224.4,499.4,275.21],\"words\":[4076,4077,4078,4079,4080,4081,4082,4083,4084,4085,4086,4087,4088,4089,4090,4091,4092,4093,4094,4095,4096,4097,4098,4099,4100,4101,4102,4103,4104]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{array}[]{c}w^{1}=\\\\frac{1}{5^{1/4}}\\\\left(\\\\sqrt{-\\\\varphi^{-}}q^{1}-\\\\sqrt{%\\n\\\\varphi^{+}}q^{2}\\\\right),\\\\\\\\\\n\\\\\\\\\\nw^{2}=\\\\frac{1}{5^{1/4}}\\\\left(\\\\sqrt{\\\\varphi^{+}}q^{1}+\\\\sqrt{-\\\\varphi^{-}}q^{2}%\\n\\\\right),\\\\end{array}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S4.Ex17\",\"source_text\":\"$\\\\begin{array}[]{c}w^{1}=\\\\frac{1}{5^{1/4}}\\\\left(\\\\sqrt{-\\\\varphi^{-}}q^{1}-\\\\sqrt{% \\\\varphi^{+}}q^{2}\\\\right),\\\\\\\\ \\\\\\\\ w^{2}=\\\\frac{1}{5^{1/4}}\\\\left(\\\\sqrt{\\\\varphi^{+}}q^{1}+\\\\sqrt{-\\\\varphi^{-}}q^{2}% \\\\right),\\\\end{array}$ (4.17)\",\"tag\":\"(4.17)\",\"text\":\"(︁√ √ )︁ 1 1 −q1 +q2 w = 1/4 −φ − φ , 5 (︁√ √ )︁ (4.17) w 2 = 1 φ+q1 + −φ−q2 , 1/4 5\"},\"formula_index\":9,\"parent_page_sha256\":\"e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef.pdf","asset_sha256":"e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef","document_id":"arxiv:2608.07909","kind":"formula","page_number":13,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[257.17,596.04,242.22,35.94],\"bbox_pdf\":[257.17,160.03,499.39,195.96],\"fragments\":[{\"bbox_pdf\":[257.17,160.03,499.39,195.96],\"words\":[4122,4123,4124,4125,4126,4127,4128,4129,4130,4131,4132,4133]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"G_{ab}=\\\\begin{pmatrix}\\\\varphi^{-}&0\\\\\\\\\\n0&\\\\varphi^{+}\\\\end{pmatrix}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"formula\",\"source_ref\":\"S4.Ex18\",\"source_text\":\"$G_{ab}=\\\\begin{pmatrix}\\\\varphi^{-}&0\\\\\\\\ 0&\\\\varphi^{+}\\\\end{pmatrix}.$ (4.18)\",\"tag\":\"(4.18)\",\"text\":\"(︃ )︃ φ − 0 Gab = + . (4.18) 0 φ\"},\"formula_index\":11,\"parent_page_sha256\":\"e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef.pdf","asset_sha256":"e66afc171891153b3a33633402c4ef206a9e159bac30e7e0c972a7d0f0784aef","document_id":"arxiv:2608.07909","kind":"formula","page_number":13,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,125.8,364.63,11.96],\"bbox_pdf\":[110.85,654.25,475.49,666.2],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[110.85,654.25,475.49,666.2],\"words\":[4170,4171,4172,4173,4174,4175,4176,4177,4178,4179]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph128\",\"source_text\":\"Rescaling coordinates in the form $W^{1}=\\\\sqrt{-\\\\varphi^{-}}w^{1}$ , $W^{2}=\\\\sqrt{\\\\varphi^{+}}w^{2}$ , we obtain the usual (1+1)-Minkowski form $\\\\eta_{ab}dW^{a}dW^{b}$ , where $\\\\eta_{ab}=diag(-1,1)$ . However, the transformation (4.17) preserves the scale. Its inversion is:\",\"tag\":null,\"text\":\"However, the transformation (4.17) preserves the scale. Its inversion is:\"},{\"bbox\":[219.21,147.9,280.19,31.97],\"bbox_pdf\":[219.21,612.13,499.4,644.1],\"fragments\":[{\"bbox_pdf\":[219.21,612.13,499.4,644.1],\"words\":[4180,4181,4182,4183,4184,4185,4186,4187,4188,4189,4190,4191,4192,4193,4194,4195,4196,4197,4198,4199,4200,4201,4202,4203,4204,4205,4206,4207,4208,4209,4210,4211,4212]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{array}[]{c}q^{1}=\\\\frac{1}{5^{1/4}}\\\\left(\\\\sqrt{-\\\\varphi^{-}}w^{1}+\\\\sqrt{%\\n\\\\varphi^{+}}w^{2}\\\\right),\\\\\\\\\\nq^{2}=\\\\frac{1}{5^{1/4}}\\\\left(-\\\\sqrt{\\\\varphi^{+}}w^{1}+\\\\sqrt{-\\\\varphi^{-}}w^{2}%\\n\\\\right).\\\\end{array}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S4.Ex19\",\"source_text\":\"$\\\\begin{array}[]{c}q^{1}=\\\\frac{1}{5^{1/4}}\\\\left(\\\\sqrt{-\\\\varphi^{-}}w^{1}+\\\\sqrt{% \\\\varphi^{+}}w^{2}\\\\right),\\\\\\\\ q^{2}=\\\\frac{1}{5^{1/4}}\\\\left(-\\\\sqrt{\\\\varphi^{+}}w^{1}+\\\\sqrt{-\\\\varphi^{-}}w^{2}% \\\\right).\\\\end{array}$ (4.19)\",\"tag\":\"(4.19)\",\"text\":\"(︁√ √ )︁ 1 1 −w1 +w2 q = 1/4 −φ + φ , 5 (︁ √ √ )︁ 2 1 +w1 −w2 (4.19) q = 1/4 − φ + −φ . 5\"},{\"bbox\":[110.85,186.71,388.55,84.54],\"bbox_pdf\":[110.85,520.75,499.4,605.29],\"fragments\":[{\"bbox_pdf\":[110.85,520.75,499.4,605.29],\"words\":[4213,4214,4215,4216,4217,4218,4219,4220,4221,4222,4223,4224,4225,4226,4227,4228,4229,4230,4231,4232,4233,4234,4235,4236,4237,4238,4239,4240,4241,4242,4243,4244,4245,4246,4247,4248,4249,4250,4251,4252,4253,4254,4255,4256,4257,4258,4259,4260,4261,4262,4263,4265,4266,4267,4268,4269,4270,4271,4272,4273,4274,4275,4276,4277,4278,4279,4280,4281,4282,4283,4284,4285,4286,4287,4288,4289,4290,4291,4292,4293,4294,4295,4296,4297,4298,4299,4300,4301,4302,4303]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph130\",\"source_text\":\"Equations (4.17) and (4.19) are SO(2) transformations. As $w^{1}$ and $w^{2}$ are orthogonal in the induced (1+1)-spacetime, we represent them in a Minkowski-like diagram. From Eq. (4.19), the $q^{2}$ and $q^{2}$ axes have slopes $\\\\Delta w^{1}/\\\\Delta w^{2}$ equal to $-\\\\varphi^{-}$ and $-\\\\varphi^{+}$ , respectively. Since $\\\\varphi^{+}\\\\varphi^{-}=-1$ , they are orthogonal in the diagram, while they are actually not orthogonal in the metric sense. The rotation is by an angle $\\\\arctan\\\\varphi^{+}\\\\approx 58.28{{}^{\\\\circ}}$ (see Fig. 4).\",\"tag\":null,\"text\":\"Equations (4.17) and (4.19) are SO(2) transformations. As w 1 and w 2 are orthogonal in the induced (1+1)-spacetime, we represent them in a Minkowskilike diagram. From Eq. (4.19), the q 2 and q 2 axes have slopes ∆w 1 /∆w 2 equal to −φ − and −φ + respectively. Since φ + φ − = −1, they are orthogonal in the diagram, while they are actually not orthogonal in the metric sense. The rotation is by an angle arctan φ + ≈ 58.28 ◦ (see Fig. 4).\"},{\"bbox\":[110.85,273.74,388.55,85.87],\"bbox_pdf\":[110.85,432.39,499.4,518.26],\"fragments\":[{\"bbox_pdf\":[110.85,432.39,499.4,518.26],\"words\":[4304,4305,4306,4307,4308,4309,4310,4311,4312,4313,4314,4315,4316,4317,4318,4319,4320,4321,4322,4323,4324,4325,4326,4327,4329,4330,4331,4332,4333,4334,4335,4336,4337,4338,4339,4340,4341,4342,4343,4344,4345,4347,4348,4349,4350,4351,4352,4353,4354,4355,4356,4357,4358,4359,4360,4361,4362,4363,4364,4365,4366,4367,4368,4369,4370,4371,4372,4373,4374,4375,4376,4377,4378,4379,4380,4381,4382,4383,4384,4385,4386,4387,4388,4389,4390,4391,4392,4393,4394,4395,4396,4397,4398,4399,4400,4401,4402,4403,4404,4405,4407,4408,4409,4410]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph131\",\"source_text\":\"From (4.9), (4.10) and (4.18) one has $G_{ab}dw^{a}dw^{b}=Q_{ab}dq^{a}dq^{b}=dq^{1}(dq^{1}+2dq^{2})$ . Setting it to zero yields the two null coordinates $q^{1}$ and $q^{1}+2q^{2}$ . Equivalently, $G_{ab}dw^{a}dw^{b}=-\\\\left(\\\\sqrt{-\\\\varphi^{-}}dw^{1}\\\\right)^{2}+\\\\left(\\\\sqrt{% \\\\varphi^{+}}dw^{2}\\\\right)^{2}$ admits a direct factorization (see the Appendix). In the diagram, lines $q^{1}=const.$ (as the $q^{2}$ orange axis) have slope of $-\\\\varphi^{+}$ with respect to the horizontal, while $q^{1}+2q^{2}=const.$ are lines with slope $\\\\varphi^{+}$ , as the grey line.\",\"tag\":null,\"text\":\"From (4.9), (4.10) and (4.18) one has Gabdw a dw b = Qabdq a dq b = dq 1 (dq 1 + 2dq 2 Setting it to zero yields the two null coordinates q 1 and q 1 + 2q 2 (︁√ )︁2 (︁√ )︁2 Equivalently, Gabdw a dw b = − −φ−dw1 + φ+dw2 admits a direct factorization (see the Appendix). In the diagram, lines q 1 = const. (as the q 2 orange axis) have slope of −φ + with respect to the horizontal, while q 1 + 2q 2 = const. are lines with slope φ + as the grey line.\"},{\"bbox\":[169.14,375.01,271.98,270.34],\"bbox_pdf\":[169.14,146.65,441.11,416.99],\"float_ref\":\"S4.F4\",\"fragments\":[{\"bbox_pdf\":[169.14,146.65,441.11,416.99],\"words\":[]}],\"html\":null,\"kind\":\"figure\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"figure\",\"source_ref\":\"S4.F4\",\"source_text\":\"\",\"tag\":null,\"text\":\"\"},{\"bbox\":[174.56,657.8,257.86,11.96],\"bbox_pdf\":[174.56,122.24,432.42,134.2],\"float_ref\":\"S4.F4\",\"fragments\":[{\"bbox_pdf\":[174.56,122.24,432.42,134.2],\"words\":[4411,4412,4413,4414,4415,4416,4417,4418,4419,4420,4421]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"caption\",\"source_ref\":\"caption133\",\"source_text\":\"Figure 4: SO(2) coordinate transformation from $w^{a}$ to $q^{a}$ .\",\"tag\":null,\"text\":\"Figure 4: SO(2) coordinate transformation from w a to q a\"},{\"bbox\":[299.27,690.18,11.71,11.96],\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"fragments\":[{\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"words\":[10388]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":6,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"14\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0013.pdf\",\"page_no\":14,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"bca170e6fec9dd72a72b8f41052c1b88e57cb32dc7e6dffa5e72d4c86e059eb2\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/bca170e6fec9dd72a72b8f41052c1b88e57cb32dc7e6dffa5e72d4c86e059eb2.pdf","asset_sha256":"bca170e6fec9dd72a72b8f41052c1b88e57cb32dc7e6dffa5e72d4c86e059eb2","document_id":"arxiv:2608.07909","kind":"page","page_number":14,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[219.21,147.9,280.19,31.97],\"bbox_pdf\":[219.21,612.13,499.4,644.1],\"fragments\":[{\"bbox_pdf\":[219.21,612.13,499.4,644.1],\"words\":[4180,4181,4182,4183,4184,4185,4186,4187,4188,4189,4190,4191,4192,4193,4194,4195,4196,4197,4198,4199,4200,4201,4202,4203,4204,4205,4206,4207,4208,4209,4210,4211,4212]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{array}[]{c}q^{1}=\\\\frac{1}{5^{1/4}}\\\\left(\\\\sqrt{-\\\\varphi^{-}}w^{1}+\\\\sqrt{%\\n\\\\varphi^{+}}w^{2}\\\\right),\\\\\\\\\\nq^{2}=\\\\frac{1}{5^{1/4}}\\\\left(-\\\\sqrt{\\\\varphi^{+}}w^{1}+\\\\sqrt{-\\\\varphi^{-}}w^{2}%\\n\\\\right).\\\\end{array}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S4.Ex19\",\"source_text\":\"$\\\\begin{array}[]{c}q^{1}=\\\\frac{1}{5^{1/4}}\\\\left(\\\\sqrt{-\\\\varphi^{-}}w^{1}+\\\\sqrt{% \\\\varphi^{+}}w^{2}\\\\right),\\\\\\\\ q^{2}=\\\\frac{1}{5^{1/4}}\\\\left(-\\\\sqrt{\\\\varphi^{+}}w^{1}+\\\\sqrt{-\\\\varphi^{-}}w^{2}% \\\\right).\\\\end{array}$ (4.19)\",\"tag\":\"(4.19)\",\"text\":\"(︁√ √ )︁ 1 1 −w1 +w2 q = 1/4 −φ + φ , 5 (︁ √ √ )︁ 2 1 +w1 −w2 (4.19) q = 1/4 − φ + −φ . 5\"},\"formula_index\":1,\"parent_page_sha256\":\"bca170e6fec9dd72a72b8f41052c1b88e57cb32dc7e6dffa5e72d4c86e059eb2\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/bca170e6fec9dd72a72b8f41052c1b88e57cb32dc7e6dffa5e72d4c86e059eb2.pdf","asset_sha256":"bca170e6fec9dd72a72b8f41052c1b88e57cb32dc7e6dffa5e72d4c86e059eb2","document_id":"arxiv:2608.07909","kind":"formula","page_number":14,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,125.8,357.47,11.96],\"bbox_pdf\":[110.85,654.25,468.32,666.2],\"fragments\":[{\"bbox_pdf\":[110.85,654.25,468.32,666.2],\"words\":[4423,4424,4425,4426,4427,4428,4429,4430,4431,4432]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph134\",\"source_text\":\"For completeness, now we work directly in the Golden representation.\",\"tag\":null,\"text\":\"For completeness, now we work directly in the Golden representation.\"},{\"bbox\":[110.85,159.93,381.43,17.22],\"bbox_pdf\":[110.85,614.86,492.29,632.07],\"fragments\":[{\"bbox_pdf\":[110.85,614.86,492.29,632.07],\"words\":[4433,4434,4435,4436,4437,4438]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"heading\",\"source_ref\":\"title135\",\"source_text\":\"5 Kinematics in the Golden Representation\",\"tag\":null,\"text\":\"5 Kinematics in the Golden Representation\"},{\"bbox\":[110.85,191.47,388.55,69.74],\"bbox_pdf\":[110.85,530.79,499.4,600.53],\"fragments\":[{\"bbox_pdf\":[110.85,530.79,499.4,600.53],\"words\":[4439,4440,4441,4442,4443,4444,4445,4446,4447,4448,4449,4450,4451,4452,4453,4454,4455,4456,4457,4458,4459,4460,4461,4462,4463,4464,4465,4466,4467,4468,4469,4470,4471,4472,4473,4474,4475,4476,4477,4478,4479,4480,4481,4482,4483,4484,4485,4486,4487,4488,4489,4490,4491,4492,4493,4494,4495]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph136\",\"source_text\":\"So far, we have shown that the dimensional reduction from the (1+3) dynamical Einstein-Hilbert action to a radial equivalent Lagrangian induces a (1+1)-spacetime. We can now show explicitly that the kinematics in this fiducial spacetime are equivalent to the curvature changes in the original spacetime. As we have shown, the Lagrangian (4.9) is the same as\",\"tag\":null,\"text\":\"So far, we have shown that the dimensional reduction from the (1+3) dynamical Einstein-Hilbert action to a radial equivalent Lagrangian induces a (1+1)-spacetime. We can now show explicitly that the kinematics in this fiducial spacetime are equivalent to the curvature changes in the original spacetime. As we have shown, the Lagrangian (4.9) is the same as\"},{\"bbox\":[239.5,270.97,259.89,28.24],\"bbox_pdf\":[239.5,492.79,499.4,521.03],\"fragments\":[{\"bbox_pdf\":[239.5,492.79,499.4,521.03],\"words\":[4496,4497,4498,4499,4500,4501,4502,4503,4504,4505,4506,4507,4508,4509,4510,4511,4512]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"L_{r}=\\\\frac{\\\\lambda^{-1}}{2}Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}+\\\\frac{\\\\lambda}{2}M^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S5.Ex1\",\"source_text\":\"$L_{r}=\\\\frac{\\\\lambda^{-1}}{2}Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}+\\\\frac{\\\\lambda}{2}M^{2},$ (5.1)\",\"tag\":\"(5.1)\",\"text\":\"λ −1 λ a b 2 Lr = Qab ˙q q˙ + M , (5.1) 2 2\"},{\"bbox\":[110.85,307.51,388.55,69.74],\"bbox_pdf\":[110.85,414.75,499.4,484.49],\"fragments\":[{\"bbox_pdf\":[110.85,414.75,499.4,484.49],\"words\":[4513,4514,4515,4516,4517,4518,4519,4520,4521,4522,4523,4524,4525,4526,4527,4528,4529,4530,4531,4532,4533,4534,4535,4536,4537,4538,4539,4540,4541,4542,4543,4544,4546,4547,4548,4549,4550,4551,4552,4553,4554,4555,4556,4558,4559,4560,4561,4562,4563,4564,4565,4566,4567,4568,4569,4570,4571,4572,4573,4575,4576,4577,4578,4579,4580,4581,4582,4583,4584,4585,4586,4587,4588,4590,4591,4592,4593,4594,4595,4596,4597,4598,4599,4600,4601,4602,4603,4604,4605,4606]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph138\",\"source_text\":\"where $Q_{ab}$ is the Q-Fibonacci metric in Eq. (4.10), while $\\\\lambda^{-1}=e^{-\\\\beta}e^{2q^{1}+q^{2}}$ and $M=e^{q^{1}+q^{2}}$ . Recall also the equality $Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}=G_{ab}\\\\dot{w}^{a}\\\\dot{w}^{b}$ , where we have the two equivalent forms (4.10) and (4.18). While $(w^{1},w^{2})$ are explicit orthogonal coordinates in this (1+1) version, $(q^{1},q^{2})$ are the ones with a direct translation in terms of both the Schwarzschild and the Tortoise coordinates.\",\"tag\":null,\"text\":\"−1 −β 2q 1 +q 2 where Qab is the Q-Fibonacci metric in Eq. (4.10), while λ = e e q 1 +q 2 a b a b and M = e Recall also the equality Qab ˙q q˙ = Gab ˙w w˙ where we have the two equivalent forms (4.10) and (4.18). While (w 1 , w 2 are explicit orthogonal coordinates in this (1+1) version, (q 1 , q 2 are the ones with a direct translation in terms of both the Schwarzschild and the Tortoise coordinates.\"},{\"bbox\":[110.85,379.74,388.55,113.08],\"bbox_pdf\":[110.85,299.18,499.4,412.26],\"fragments\":[{\"bbox_pdf\":[110.85,299.18,499.4,412.26],\"words\":[4607,4608,4609,4610,4611,4612,4613,4614,4615,4616,4617,4618,4619,4620,4621,4623,4624,4625,4626,4627,4628,4629,4630,4631,4632,4633,4634,4635,4636,4637,4638,4639,4640,4641,4642,4643,4644,4645,4646,4647,4648,4649,4650,4651,4652,4653,4654,4655,4656,4657,4658,4659,4660,4661,4662,4663,4664,4665,4666,4667,4668,4669,4670,4671,4672,4673,4674,4675,4676,4677,4678,4679,4680,4681,4682,4683,4684,4685,4686,4687,4688,4689,4690,4691,4692,4693,4694,4695,4696,4697,4698,4699,4700,4701,4702,4710,4703,4704,4705,4706,4707,4708,4709,4711,4712,4713,4714,4715,4716,4717,4718,4719,4720,4721,4722,4723,4724]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph139\",\"source_text\":\"Equation (4.13) is equivalent to $Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}=\\\\lambda^{2}M^{2}$ , which can also be obtained by varying the action into which (5.1) is inserted, with respect to $\\\\lambda$ . In this (1+1)-dimensional system, the gauge $\\\\lambda=M^{-1}$ would be the equivalent to taking the normalization $Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}=1$ for the tangent vector $dq^{a}/dr$ . However, the connection with the Schwarzschild geometry implies another norm, as we shall see. Nevertheless, this norm is spacelike, in agreement with all the previous comments about our fictitious particle system. With all this in mind, we substitute $\\\\lambda=M^{-1}\\\\sqrt{Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}}$ in (5.1), leading to:\",\"tag\":null,\"text\":\"Equation (4.13) is equivalent to Qab ˙q a q˙ b = λ 2 M 2 which can also be obtained by varying the action into which (5.1) is inserted, with respect to λ. In this (1+1)-dimensional system, the gauge λ = M −1 would be the equivalent to taking the normalization Qab ˙q a q˙ b = 1 for the tangent vector dq a /dr. How- ever, the connection with the Schwarzschild geometry implies another norm, as we shall see. Nevertheless, this norm is spacelike, in agreement with all the previous comments about our √︁ fictitious particle system. With all this in mind, we substitute λ = M −1 Qab ˙qa ˙qb in (5.1), leading to:\"},{\"bbox\":[257.98,494.37,241.42,24.9],\"bbox_pdf\":[257.98,272.73,499.4,297.63],\"fragments\":[{\"bbox_pdf\":[257.98,272.73,499.4,297.63],\"words\":[4725,4726,4727,4728,4729,4730,4731,4732]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"L_{r}=M\\\\sqrt{Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S5.Ex2\",\"source_text\":\"$L_{r}=M\\\\sqrt{Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}}.$ (5.2)\",\"tag\":\"(5.2)\",\"text\":\"√︁ Lr = M Qab ˙qa ˙qb. (5.2)\"},{\"bbox\":[110.85,528.22,388.55,40.85],\"bbox_pdf\":[110.85,222.93,499.4,263.78],\"fragments\":[{\"bbox_pdf\":[110.85,222.93,499.4,263.78],\"words\":[4733,4734,4735,4736,4737,4738,4739,4740,4741,4742,4743,4744,4745,4746,4747,4748,4749,4750,4751,4752,4753,4754,4755,4756,4757,4758,4759,4760,4761,4762,4763,4764,4769,4765,4766,4767,4768,4771]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph141\",\"source_text\":\"This is very similar to the relativistic point particle, but $M$ is not really constant. Since variations of (5.2) lead to the geodesics of this system, we define the Golden metric $\\\\tilde{\\\\Phi}$ , incorporating the conformal factor $M^{2}$ , as:\",\"tag\":null,\"text\":\"This is very similar to the relativistic point particle, but M is not really constant. Since variations of (5.2) lead to the geodesics of this system, we define the Golden metric Φ, M˜2 incorporating the conformal factor as:\"},{\"bbox\":[220.15,578.86,279.24,14.7],\"bbox_pdf\":[220.15,198.44,499.39,213.14],\"fragments\":[{\"bbox_pdf\":[220.15,198.44,499.39,213.14],\"words\":[4772,4773,4774,4775,4776,4777,4778,4779,4780,4781,4782,4783,4784,4785,4786,4787]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"d\\\\sigma^{2}=\\\\Phi_{ab}dq^{a}dq^{b}=M^{2}Q_{ab}dq^{a}dq^{b}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S5.Ex3\",\"source_text\":\"$d\\\\sigma^{2}=\\\\Phi_{ab}dq^{a}dq^{b}=M^{2}Q_{ab}dq^{a}dq^{b}.$ (5.3)\",\"tag\":\"(5.3)\",\"text\":\"dσ 2 = Φabdq a dq b = M 2 Qabdq a dq b . (5.3)\"},{\"bbox\":[110.85,602.51,388.54,55.29],\"bbox_pdf\":[110.85,134.2,499.4,189.49],\"fragments\":[{\"bbox_pdf\":[110.85,134.2,499.4,189.49],\"words\":[4788,4789,4790,4791,4792,4793,4794,4795,4796,4797,4798,4799,4800,4801,4802,4803,4804,4805,4806,4807,4808,4810,4811,4812,4813,4814,4815,4816,4817,4818,4819,4820,4821,4822,4823,4824,4825,4826,4827,4828]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph143\",\"source_text\":\"In terms of the rescaled coordinates defined below (4.18), this is explicitly conformal Minkowski, as $M^{2}\\\\eta_{ab}dW^{a}dW^{b}$ . The two-dimensional version of (4.2) for this metric is equivalent to a differential operator acting on $\\\\ln M$ [19]:\",\"tag\":null,\"text\":\"In terms of the rescaled coordinates defined below (4.18), this is explicitly conformal Minkowski, as M 2 ηabdW a dW b The two-dimensional version of (4.2) for this metric is equivalent to a differential operator acting on ln M [19]:\"},{\"bbox\":[211.67,659.34,287.73,15.86],\"bbox_pdf\":[211.67,116.8,499.4,132.66],\"fragments\":[{\"bbox_pdf\":[211.67,116.8,499.4,132.66],\"words\":[4829,4830,4831,4832,4833,4834,4835,4836,4837,4838,4839,4840,4841,4842,4843,4844]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\Gamma_{ab}^{c}=(\\\\delta_{b}^{c}\\\\partial_{a}+\\\\delta_{a}^{c}\\\\partial_{b}-Q^{cd}Q%\\n_{ab}\\\\partial_{d})\\\\ln M.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"S5.Ex4\",\"source_text\":\"$\\\\Gamma_{ab}^{c}=(\\\\delta_{b}^{c}\\\\partial_{a}+\\\\delta_{a}^{c}\\\\partial_{b}-Q^{cd}Q% _{ab}\\\\partial_{d})\\\\ln M.$ (5.4)\",\"tag\":\"(5.4)\",\"text\":\"Γ c c c cd Qab∂d) ln M. (5.4) ab = (δb∂a + δa∂b − Q\"},{\"bbox\":[299.27,690.18,11.71,11.96],\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"fragments\":[{\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"words\":[4845]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":11,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"15\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0014.pdf\",\"page_no\":15,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a.pdf","asset_sha256":"b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a","document_id":"arxiv:2608.07909","kind":"page","page_number":15,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[239.5,270.97,259.89,28.24],\"bbox_pdf\":[239.5,492.79,499.4,521.03],\"fragments\":[{\"bbox_pdf\":[239.5,492.79,499.4,521.03],\"words\":[4496,4497,4498,4499,4500,4501,4502,4503,4504,4505,4506,4507,4508,4509,4510,4511,4512]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"L_{r}=\\\\frac{\\\\lambda^{-1}}{2}Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}+\\\\frac{\\\\lambda}{2}M^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S5.Ex1\",\"source_text\":\"$L_{r}=\\\\frac{\\\\lambda^{-1}}{2}Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}+\\\\frac{\\\\lambda}{2}M^{2},$ (5.1)\",\"tag\":\"(5.1)\",\"text\":\"λ −1 λ a b 2 Lr = Qab ˙q q˙ + M , (5.1) 2 2\"},\"formula_index\":3,\"parent_page_sha256\":\"b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a.pdf","asset_sha256":"b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a","document_id":"arxiv:2608.07909","kind":"formula","page_number":15,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[257.98,494.37,241.42,24.9],\"bbox_pdf\":[257.98,272.73,499.4,297.63],\"fragments\":[{\"bbox_pdf\":[257.98,272.73,499.4,297.63],\"words\":[4725,4726,4727,4728,4729,4730,4731,4732]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"L_{r}=M\\\\sqrt{Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S5.Ex2\",\"source_text\":\"$L_{r}=M\\\\sqrt{Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}}.$ (5.2)\",\"tag\":\"(5.2)\",\"text\":\"√︁ Lr = M Qab ˙qa ˙qb. (5.2)\"},\"formula_index\":6,\"parent_page_sha256\":\"b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a.pdf","asset_sha256":"b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a","document_id":"arxiv:2608.07909","kind":"formula","page_number":15,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[220.15,578.86,279.24,14.7],\"bbox_pdf\":[220.15,198.44,499.39,213.14],\"fragments\":[{\"bbox_pdf\":[220.15,198.44,499.39,213.14],\"words\":[4772,4773,4774,4775,4776,4777,4778,4779,4780,4781,4782,4783,4784,4785,4786,4787]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"d\\\\sigma^{2}=\\\\Phi_{ab}dq^{a}dq^{b}=M^{2}Q_{ab}dq^{a}dq^{b}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S5.Ex3\",\"source_text\":\"$d\\\\sigma^{2}=\\\\Phi_{ab}dq^{a}dq^{b}=M^{2}Q_{ab}dq^{a}dq^{b}.$ (5.3)\",\"tag\":\"(5.3)\",\"text\":\"dσ 2 = Φabdq a dq b = M 2 Qabdq a dq b . (5.3)\"},\"formula_index\":8,\"parent_page_sha256\":\"b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a.pdf","asset_sha256":"b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a","document_id":"arxiv:2608.07909","kind":"formula","page_number":15,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[211.67,659.34,287.73,15.86],\"bbox_pdf\":[211.67,116.8,499.4,132.66],\"fragments\":[{\"bbox_pdf\":[211.67,116.8,499.4,132.66],\"words\":[4829,4830,4831,4832,4833,4834,4835,4836,4837,4838,4839,4840,4841,4842,4843,4844]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\Gamma_{ab}^{c}=(\\\\delta_{b}^{c}\\\\partial_{a}+\\\\delta_{a}^{c}\\\\partial_{b}-Q^{cd}Q%\\n_{ab}\\\\partial_{d})\\\\ln M.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"S5.Ex4\",\"source_text\":\"$\\\\Gamma_{ab}^{c}=(\\\\delta_{b}^{c}\\\\partial_{a}+\\\\delta_{a}^{c}\\\\partial_{b}-Q^{cd}Q% _{ab}\\\\partial_{d})\\\\ln M.$ (5.4)\",\"tag\":\"(5.4)\",\"text\":\"Γ c c c cd Qab∂d) ln M. (5.4) ab = (δb∂a + δa∂b − Q\"},\"formula_index\":10,\"parent_page_sha256\":\"b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a.pdf","asset_sha256":"b42e0c756308e4c78532232ea09ccbb07c2b17330c9ee742ad2300ca37410e5a","document_id":"arxiv:2608.07909","kind":"formula","page_number":15,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,122.78,388.54,43.87],\"bbox_pdf\":[110.85,625.35,499.39,669.22],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[110.85,625.35,499.39,669.22],\"words\":[5864,5865,5866,5867,5868,5869,5870,5871,5872,5873,5874,5875,5876,5877,5878,5879,5880,5881,5882,5883,5884,5885,5886,5887,5888,5889,5890,5891,5892,5893,5894,5895,5896,5897,5898,5899,5900,5901]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.981,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph167\",\"source_text\":\"Although the Golden representation lies in the kinematics induced by the Golden metric (5.3) and its connection with the four-dimensional Schwarzschild solution, a legitimate question is: what are the intrinsic geometric properties of this fictitious (1+1) spacetime? The short answer is: it is a flat spacetime. As we saw below Eq. (5.5), we have four non-trivial Christoffel symbols for the metric components in (5.3) $\\\\Phi_{ab}=e^{2(q^{1}+q^{2})}Q_{ab}$ , given by $\\\\Gamma_{11}^{1}=\\\\Gamma_{12}^{2}=1$ , $\\\\Gamma_{22}^{2}=2$ . Since they are constant, the Riemann tensor for this two-dimensional induced spacetime simplifies to $\\\\tilde{R}_{~{}~{}bcd}^{a}=\\\\Gamma_{ce}^{a}\\\\Gamma_{bd}^{e}-\\\\Gamma_{de}^{a}\\\\Gamma% _{bc}^{e}$ . By considering the symmetries of the tensor, it turns out that all components are equal to zero, showing that it is flat.\",\"tag\":null,\"text\":\"two-dimensional induced spacetime simplifies to ˜R a a e a e bcd = ΓceΓbd − ΓdeΓbc. By considering the symmetries of the tensor, it turns out that all components are equal to zero, showing that it is flat.\"},{\"bbox\":[110.85,169.14,388.54,40.85],\"bbox_pdf\":[110.85,582.02,499.4,622.86],\"fragments\":[{\"bbox_pdf\":[110.85,582.02,499.4,622.86],\"words\":[5902,5903,5904,5905,5906,5907,5908,5909,5910,5911,5912,5913,5914,5915,5916,5917,5918,5919,5920,5921,5922,5923,5924,5925,5926,5927,5928,5929,5930,5932,5933,5934,5935,5936,5937,5938,5939,5940,5941,5942,5943,5944,5945,5946,5947,5948,5949,5952]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph168\",\"source_text\":\"In fact, the explicit transformation to Minkowski form can be seen by writing explicitly the metric (5.3) in terms of the null coordinates discussed in the last part of Sect. 4, as $d\\\\sigma^{2}=e^{2(q^{1}+q^{2})}dq^{1}(dq^{1}+2dq^{2})$ .\",\"tag\":null,\"text\":\"In fact, the explicit transformation to Minkowski form can be seen by writing explicitly the metric (5.3) in terms of the null coordinates discussed 2 2(q 1 +q 2 1 1 2 in the last part of Sect. 4, as dσ = e dq (dq + 2dq 1\"},{\"bbox\":[110.85,212.47,388.55,141.97],\"bbox_pdf\":[110.85,437.56,499.4,579.53],\"fragments\":[{\"bbox_pdf\":[110.85,437.56,499.4,579.53],\"words\":[5951,5953,5954,5955,5956,5957,5959,5960,5961,5962,5963,5964,5965,5966,5967,5968,5969,5970,5971,5972,5973,5974,5975,5976,5977,5979,5980,5981,5982,5983,5984,5985,5986,5988,5989,5990,5991,5992,5993,5994,5995,5996,5997,5998,5999,6000,6001,6002,6003,6004,6005,6006,6007,6008,6009,6010,6011,6012,6013,6014,6015,6016,6017,6018,6019,6020,6022,6023,6024,6025,6026,6027,6028,6029,6030,6031,6032,6033,6034,6035,6036,6037,6038,6039,6040,6041,6042,6043,6044,6045,6046,6047,6048,6049,6050,6051,6052,6053,6054,6055,6056,6057,6058,6059,6060,6061,6062,6063,6064,6065,6066,6067,6068,6069,6070,6071,6072,6073,6074,6075,6076,6077,6078,6079,6080,6081,6082,6083,6084,6085,6086,6087,6088,6089,6090,6091,6092,6093,6094,6095,6096,6097,6098,6099,6100,6101,6102,6103,6104,6105,6106,6107,6108,6109,6110,6111]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph169\",\"source_text\":\"Since it can be factored in $dU=e^{q^{1}}dq^{1}$ and $dV:=-e^{(q^{1}+2q^{2})}(dq^{1}+2dq^{2})$ , we can make $d\\\\sigma^{2}=-dUdV$ . Rotating coordinates in the customary way $dV=dT-dX$ and $dU=dT+dX$ , yields the Minkowski form $d\\\\sigma^{2}=-dT^{2}+dX^{2}$ . The inversion of coordinates is $T=e^{q^{1}}(1-e^{2q^{2}})/2$ and $X=e^{q^{1}}(1+e^{2q^{2}})/2$ . By using (5.12) and $e^{q^{1}}=r$ , the Minkowski form of the Golden representation, under translation, is the same as $T=1/2$ and $X=r-1/2$ , which in fact can be rescaled and translated. The special geodesic solution (5.12) which represents the Schwarzschild spacetime, corresponds to a simple “simultaneity surface” in this Minkowski (1+1) representation, where the space dimension grows linearly with $r$ .\",\"tag\":null,\"text\":\"q 1 (q 1 +2q 2 1 2 Since it can be factored in dU = e dq and dV := −e (dq + 2dq we can make dσ 2 = −dU dV Rotating coordinates in the customary way dV = dT − dX and dU = dT + dX, yields the Minkowski form dσ 2 = 2 2 q 1 2q 2 −dT + dX The inversion of coordinates is T = e (1 − e )/2 and X = q 1 2q 2 q 1 e (1+e )/2. By using (5.12) and e = r, the Minkowski form of the Golden representation, under translation, is the same as T = 1/2 and X = r − 1/2, which in fact can be rescaled and translated. The special geodesic solution (5.12) which represents the Schwarzschild spacetime, corresponds to a simple “simultaneity surface” in this Minkowski (1+1) representation, where the space dimension grows linearly with r.\"},{\"bbox\":[110.85,356.93,388.54,26.4],\"bbox_pdf\":[110.85,408.67,499.39,435.07],\"fragments\":[{\"bbox_pdf\":[110.85,408.67,499.39,435.07],\"words\":[6112,6113,6114,6115,6116,6117,6118,6119,6120,6121,6122,6123,6124,6125,6126,6127,6128,6129,6130,6131,6132,6133,6134,6135,6136]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph170\",\"source_text\":\"We end this section by remarking that there is a third equivalent Lagrangian to (5.1) and (5.2) [17]. Consider the Legendre transform of (5.1):\",\"tag\":null,\"text\":\"We end this section by remarking that there is a third equivalent Lagrangian to (5.1) and (5.2) [17]. Consider the Legendre transform of (5.1):\"},{\"bbox\":[263.05,391.24,236.35,14.7],\"bbox_pdf\":[263.05,386.06,499.4,400.76],\"fragments\":[{\"bbox_pdf\":[263.05,386.06,499.4,400.76],\"words\":[6137,6138,6139,6140,6141,6142,6143,6144]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"H_{A}=\\\\dot{q}^{a}p_{a}-L_{r},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S5.Ex13\",\"source_text\":\"$H_{A}=\\\\dot{q}^{a}p_{a}-L_{r},$ (5.13)\",\"tag\":\"(5.13)\",\"text\":\"HA = ˙q a pa − Lr, (5.13)\"},{\"bbox\":[110.85,412.65,373.71,26.75],\"bbox_pdf\":[110.85,352.6,484.57,379.35],\"fragments\":[{\"bbox_pdf\":[110.85,352.6,484.57,379.35],\"words\":[6145,6146,6147,6148,6149,6150,6152,6153,6154,6155,6156,6157,6158,6159,6161,6162,6163,6164,6165,6167,6168,6170,6171,6172,6173,6175,6176,6177,6178,6179,6180,6181,6182,6183,6184,6185,6186]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph172\",\"source_text\":\"where $p_{a}=\\\\partial L_{r}/\\\\partial\\\\dot{q}^{a}$ . From (5.1), we obtain $p_{1}=\\\\lambda^{-1}\\\\left(\\\\dot{q}^{1}+\\\\dot{q}^{2}\\\\right)$ and $p_{2}=\\\\lambda^{-1}\\\\dot{q}^{1}$ . We use their inversions to eliminate the $\\\\dot{q}^{a}$ -dependence in (5.13):\",\"tag\":null,\"text\":\"where pa = ∂Lr/∂ ˙q a From (5.1), we obtain p1 = λ −1 ˙q 1 + ˙q 2 and p2 λ −1 q˙ 1 We use their inversions to eliminate the q˙ a -dependence in (5.13):\"},{\"bbox\":[228.12,444.5,271.28,29.84],\"bbox_pdf\":[228.12,317.66,499.4,347.5],\"fragments\":[{\"bbox_pdf\":[228.12,317.66,499.4,347.5],\"words\":[6187,6188,6189,6190,6191,6192,6193,6194,6195,6196,6197,6198,6199,6200,6201]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"H_{A}=\\\\frac{\\\\lambda}{2}\\\\left[\\\\left(2p_{1}-p_{2}\\\\right)p_{2}-M^{2}\\\\right].\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S5.Ex14\",\"source_text\":\"$H_{A}=\\\\frac{\\\\lambda}{2}\\\\left[\\\\left(2p_{1}-p_{2}\\\\right)p_{2}-M^{2}\\\\right].$ (5.14)\",\"tag\":\"(5.14)\",\"text\":\"λ [︁ ]︁ HA = (2p1 − p2) p2 − M 2 . (5.14) 2\"},{\"bbox\":[110.85,478.36,388.54,57.27],\"bbox_pdf\":[110.85,256.37,499.4,313.64],\"fragments\":[{\"bbox_pdf\":[110.85,256.37,499.4,313.64],\"words\":[6202,6203,6204,6205,6206,6207,6208,6210,6211,6212,6213,6214,6215,6216,6217,6218,6219,6220,6221,6222,6224,6225,6226,6227,6228,6229,6230,6231,6232,6234,6235,6236,6237,6238,6239,6240,6241,6242,6243,6244,6245,6246,6247,6248,6249,6250,6251,6252,6253,6254,6255,6256,6257,6258,6259,6260,6261,6262,6263,6264,6265,6266]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph174\",\"source_text\":\"Now, in terms of the $p^{a}$ , the Lagrangian (5.1) satisfies $2L_{r}=\\\\lambda\\\\left(2p_{1}-p_{2}\\\\right)p_{2}+\\\\lambda M^{2}$ . Then (5.14) is equivalent to $L_{r}=H_{A}-\\\\lambda M^{2}$ . Recalling that the inverse to (4.10) is $Q^{cd}=2\\\\delta_{1}^{(c}\\\\delta_{2}^{d)}-\\\\delta_{2}^{c}\\\\delta_{2}^{d}$ [above Eq. (5.5)], we can use (5.13) again to express the compact form for $L_{r}$ :\",\"tag\":null,\"text\":\"Now, in terms of the p a the Lagrangian (5.1) satisfies 2Lr = λ (2p1 − p2) p2+ λM 2 Then (5.14) is equivalent to Lr = HA−λM 2 Recalling that the inverse cd (c d) c d to (4.10) is Q = 2δ1 δ2 − δ2δ2 [above Eq. (5.5)], we can use (5.13) again to express the compact form for Lr:\"},{\"bbox\":[223.9,540.07,275.5,29.84],\"bbox_pdf\":[223.9,222.09,499.39,251.93],\"fragments\":[{\"bbox_pdf\":[223.9,222.09,499.39,251.93],\"words\":[6267,6268,6269,6270,6271,6272,6273,6274,6275,6276,6277,6278,6279,6280,6281,6282,6283]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"L_{r}=\\\\dot{q}^{a}p_{a}-\\\\frac{\\\\lambda}{2}\\\\left(p_{a}p_{b}Q^{ab}-M^{2}\\\\right).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S5.Ex15\",\"source_text\":\"$L_{r}=\\\\dot{q}^{a}p_{a}-\\\\frac{\\\\lambda}{2}\\\\left(p_{a}p_{b}Q^{ab}-M^{2}\\\\right).$ (5.15)\",\"tag\":\"(5.15)\",\"text\":\"λ (︁ )︁ a ab 2 Lr = ˙q pa − papbQ − M . (5.15) 2\"},{\"bbox\":[110.85,573.27,388.55,71.89],\"bbox_pdf\":[110.85,146.85,499.4,218.73],\"fragments\":[{\"bbox_pdf\":[110.85,146.85,499.4,218.73],\"words\":[6284,6285,6286,6287,6288,6289,6290,6291,6292,6293,6294,6295,6297,6298,6299,6300,6301,6302,6303,6304,6305,6306,6307,6308,6309,6310,6311,6312,6313,6314,6315,6317,6318,6319,6320,6321,6322,6323,6324,6325,6326,6327,6328,6329,6330,6331,6332,6333,6334,6335,6336,6337,6338,6339,6340,6342,6343,6344,6345,6346,6347,6348,6349,6350,6351,6352,6353,6354,6355,6356,6357,6358,6360,6361,6362,6363,6364,6365,6366]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph176\",\"source_text\":\"This form for the Lagrangian now depends on the momenta $p^{a}$ . Nevertheless, it allows for variations with respect to $\\\\lambda$ , which yields the constraint $p_{a}p_{b}Q^{ab}=M^{2}$ . By making the translations as before, in particular the identification $\\\\lambda\\\\left(2p_{1}-p_{2}\\\\right)p_{2}=\\\\lambda^{-1}Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}$ , this Hamiltonian constraint is the same as $Q_{ab}\\\\dot{q}^{a}\\\\dot{q}^{b}=\\\\lambda^{2}M^{2}$ , used for the developments of this Section.\",\"tag\":null,\"text\":\"This form for the Lagrangian now depends on the momenta p a Nevertheless, it allows for variations with respect to λ, which yields the constraint papbQ ab = M 2 By making the translations as before, in particular the identification λ (2p1 − p2) p2 = λ −1 Qab ˙q a q˙ b this Hamiltonian constraint is the same as Qab ˙q a q˙ b = λ 2 M 2 used for the developments of this Section.\"},{\"bbox\":[110.85,645.85,388.55,26.4],\"bbox_pdf\":[110.85,119.75,499.4,146.15],\"fragments\":[{\"bbox_pdf\":[110.85,119.75,499.4,146.15],\"words\":[6367,6368,6369,6370,6371,6372,6373,6374,6375,6376,6377,6378,6379,6380,6381,6382,6383,6384,6385]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"paragraph\",\"source_ref\":\"paragraph177\",\"source_text\":\"Now let us turn our attention to some visualizations of the Schwarzschild radial geometry, in the $q^{a}$ representation.\",\"tag\":null,\"text\":\"Now let us turn our attention to some visualizations of the Schwarzschild radial geometry, in the q a representation.\"},{\"bbox\":[299.27,690.18,11.71,11.96],\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"fragments\":[{\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"words\":[10391]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":11,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"18\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0017.pdf\",\"page_no\":18,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph167\",\"c_block\":0.98,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.98,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"710d964ade9d11a4e31f2e4ce7dc348e7c2efd9f50f54a4775923a6dd396634a\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/710d964ade9d11a4e31f2e4ce7dc348e7c2efd9f50f54a4775923a6dd396634a.pdf","asset_sha256":"710d964ade9d11a4e31f2e4ce7dc348e7c2efd9f50f54a4775923a6dd396634a","document_id":"arxiv:2608.07909","kind":"page","page_number":18,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[263.05,391.24,236.35,14.7],\"bbox_pdf\":[263.05,386.06,499.4,400.76],\"fragments\":[{\"bbox_pdf\":[263.05,386.06,499.4,400.76],\"words\":[6137,6138,6139,6140,6141,6142,6143,6144]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"H_{A}=\\\\dot{q}^{a}p_{a}-L_{r},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S5.Ex13\",\"source_text\":\"$H_{A}=\\\\dot{q}^{a}p_{a}-L_{r},$ (5.13)\",\"tag\":\"(5.13)\",\"text\":\"HA = ˙q a pa − Lr, (5.13)\"},\"formula_index\":4,\"parent_page_sha256\":\"710d964ade9d11a4e31f2e4ce7dc348e7c2efd9f50f54a4775923a6dd396634a\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/710d964ade9d11a4e31f2e4ce7dc348e7c2efd9f50f54a4775923a6dd396634a.pdf","asset_sha256":"710d964ade9d11a4e31f2e4ce7dc348e7c2efd9f50f54a4775923a6dd396634a","document_id":"arxiv:2608.07909","kind":"formula","page_number":18,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[228.12,444.5,271.28,29.84],\"bbox_pdf\":[228.12,317.66,499.4,347.5],\"fragments\":[{\"bbox_pdf\":[228.12,317.66,499.4,347.5],\"words\":[6187,6188,6189,6190,6191,6192,6193,6194,6195,6196,6197,6198,6199,6200,6201]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"H_{A}=\\\\frac{\\\\lambda}{2}\\\\left[\\\\left(2p_{1}-p_{2}\\\\right)p_{2}-M^{2}\\\\right].\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S5.Ex14\",\"source_text\":\"$H_{A}=\\\\frac{\\\\lambda}{2}\\\\left[\\\\left(2p_{1}-p_{2}\\\\right)p_{2}-M^{2}\\\\right].$ (5.14)\",\"tag\":\"(5.14)\",\"text\":\"λ [︁ ]︁ HA = (2p1 − p2) p2 − M 2 . (5.14) 2\"},\"formula_index\":6,\"parent_page_sha256\":\"710d964ade9d11a4e31f2e4ce7dc348e7c2efd9f50f54a4775923a6dd396634a\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/710d964ade9d11a4e31f2e4ce7dc348e7c2efd9f50f54a4775923a6dd396634a.pdf","asset_sha256":"710d964ade9d11a4e31f2e4ce7dc348e7c2efd9f50f54a4775923a6dd396634a","document_id":"arxiv:2608.07909","kind":"formula","page_number":18,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[223.9,540.07,275.5,29.84],\"bbox_pdf\":[223.9,222.09,499.39,251.93],\"fragments\":[{\"bbox_pdf\":[223.9,222.09,499.39,251.93],\"words\":[6267,6268,6269,6270,6271,6272,6273,6274,6275,6276,6277,6278,6279,6280,6281,6282,6283]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"L_{r}=\\\\dot{q}^{a}p_{a}-\\\\frac{\\\\lambda}{2}\\\\left(p_{a}p_{b}Q^{ab}-M^{2}\\\\right).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S5.Ex15\",\"source_text\":\"$L_{r}=\\\\dot{q}^{a}p_{a}-\\\\frac{\\\\lambda}{2}\\\\left(p_{a}p_{b}Q^{ab}-M^{2}\\\\right).$ (5.15)\",\"tag\":\"(5.15)\",\"text\":\"λ (︁ )︁ a ab 2 Lr = ˙q pa − papbQ − M . (5.15) 2\"},\"formula_index\":8,\"parent_page_sha256\":\"710d964ade9d11a4e31f2e4ce7dc348e7c2efd9f50f54a4775923a6dd396634a\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/710d964ade9d11a4e31f2e4ce7dc348e7c2efd9f50f54a4775923a6dd396634a.pdf","asset_sha256":"710d964ade9d11a4e31f2e4ce7dc348e7c2efd9f50f54a4775923a6dd396634a","document_id":"arxiv:2608.07909","kind":"formula","page_number":18,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,120.54,291.24,17.22],\"bbox_pdf\":[110.85,654.25,402.09,671.46],\"fragments\":[{\"bbox_pdf\":[110.85,654.25,402.09,671.46],\"words\":[6386,6387,6388,6389]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"heading\",\"source_ref\":\"title178\",\"source_text\":\"6 Visual Golden Representations\",\"tag\":null,\"text\":\"6 Visual Golden Representations\"},{\"bbox\":[110.85,152.08,389.11,127.06],\"bbox_pdf\":[110.85,512.86,499.96,639.92],\"fragments\":[{\"bbox_pdf\":[110.85,512.86,499.96,639.92],\"words\":[6390,6391,6392,6393,6394,6395,6396,6397,6398,6399,6400,6401,6402,6403,6404,6405,6417,6406,6407,6408,6409,6418,6410,6411,6412,6413,6414,6415,6416,6419,6420,6421,6422,6423,6424,6425,6426,6427,6428,6430,6431,6432,6433,6434,6435,6436,6437,6438,6439,6440,6441,6442,6443,6444,6445,6446,6447,6448,6449,6450,6452,6453,6454,6456,6457,6458,6459,6461,6462,6463,6464,6465,6466,6467,6468,6469,6470,6471,6472,6473,6474,6475,6476,6477,6478,6479,6480,6481,6482,6483,6484,6485,6486,6487,6488,6489,6490,6491,6492,6493,6494,6495,6496,6497,6498,6499,6500,6501,6502,6503,6504,6506,6507,6508,6509,6510,6511,6512,6513,6514,6516,6517,6518]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.974,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph179\",\"source_text\":\"The Golden representation of the Schwarzschild geometry in (2.3) can be understood in a similar way to the Flamm paraboloid in Fig. 1, defined by the functions $(\\\\rho,z)=\\\\left(r,\\\\sqrt{4r_{s}\\\\left(r-r_{s}\\\\right)}\\\\right)$ . In our present case, the Golden coordinates for the Schwarzschild spacetime are $\\\\left(q^{1},q^{2}\\\\right)=\\\\left(\\\\ln\\\\left(r/r_{s}\\\\right),\\\\ln\\\\sqrt{1-r_{s}/r}\\\\right)$ . In the (0+2) Flamm representation of Eq.(2.10), evaluating the interval from $r=r_{s}$ to $r\\\\rightarrow\\\\infty$ corresponds to $(\\\\rho,z)$ going from $(r_{s},0)$ to $(\\\\infty,\\\\infty)$ . In the (1+1) Golden representation, the corresponding limits are from $(q^{1},q^{2})\\\\rightarrow(0,-\\\\infty)$ to $(q^{1},q^{2})\\\\rightarrow(\\\\infty,0)$ .\",\"tag\":null,\"text\":\"The Golden representation of the Schwarzschild geometry in (2.3) can be understood in a similar (︂ way to the Flamm )︂ paraboloid in Fig. 1, defined by the √︁ functions (ρ, z) = r, 4rs (r − rs) In our present case, the Golden coordi- (︂ √︁ )︂ nates for the Schwarzschild spacetime are (q 1 , q 2 = ln (r/rs) ln 1 − rs/r In the (0+2) Flamm representation of Eq.(2.10), evaluating the interval from r = rs to r → ∞ corresponds to (ρ, z) going from (rs, 0) to (∞, ∞). In the (1+1) Golden representation, the corresponding limits are from (q 1 , q 2 → (0, −∞) to (q 1 , q 2 → (∞, 0).\"},{\"bbox\":[110.85,281.63,419.18,40.85],\"bbox_pdf\":[110.85,469.52,530.04,510.37],\"fragments\":[{\"bbox_pdf\":[110.85,469.52,530.04,510.37],\"words\":[6519,6520,6521,6522,6523,6524,6525,6526,6527,6528,6529,6530,6531,6532,6533,6534,6535,6536,6537,6538,6539,6540,6541,6542,6543,6544,6546,6547,6548,6549]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph180\",\"source_text\":\"Figure 5 shows the Golden representation of spatial sections of the Schwarzschild geometry. Note that in Fig. 4, this corresponds to rotating the coordinates $q^{a}$ , followed by an inversion.\",\"tag\":null,\"text\":\"Figure 5 shows the Golden representation of spatial sections of the Schwarzschild geometry. Note that in Fig. 4, this corresponds to rotating the coordinates q a followed by an inversion.\"},{\"bbox\":[149.71,337.32,310.84,231.74],\"bbox_pdf\":[149.71,222.93,460.55,454.68],\"float_ref\":\"S6.F5\",\"fragments\":[{\"bbox_pdf\":[149.71,222.93,460.55,454.68],\"words\":[]}],\"html\":null,\"kind\":\"figure\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"figure\",\"source_ref\":\"S6.F5\",\"source_text\":\"\",\"tag\":null,\"text\":\"\"},{\"bbox\":[110.85,579.03,388.54,23.91],\"bbox_pdf\":[110.85,189.06,499.39,212.97],\"float_ref\":\"S6.F5\",\"fragments\":[{\"bbox_pdf\":[110.85,189.06,499.39,212.97],\"words\":[6550,6551,6552,6553,6554,6555,6556,6557,6558,6559,6560,6561,6562,6563,6564,6565]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"caption\",\"source_ref\":\"caption182\",\"source_text\":\"Figure 5: The Golden curve of Eq. (5.13), showing some relevant points near the Golden radius.\",\"tag\":null,\"text\":\"Figure 5: The Golden curve of Eq. (5.13), showing some relevant points near the Golden radius.\"},{\"bbox\":[110.85,616.96,388.55,57.09],\"bbox_pdf\":[110.85,117.96,499.4,175.04],\"fragments\":[{\"bbox_pdf\":[110.85,117.96,499.4,175.04],\"words\":[6566,6567,6568,6569,6570,6571,6572,6573,6574,6575,6576,6577,6578,6579,6580,6581,6582,6583,6584,6585,6586,6587,6588,6589,6590,6591,6592,6593,6594,6595,6596,6597,6598,6599,6600,6601,6602,6603,6604,6605,6606,6607,6608,6609,6610,6611,6612,6613,6614,6615,6616,6617,6618,6619,6620,6621,6622,6623]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph183\",\"source_text\":\"Some important points are explicitly marked, such as the location where $r^{\\\\ast}=0$ ( $r\\\\approx 1.2786~{}r_{s}$ ), the photon sphere at $r=3r_{s}/2$ , the Golden radius ( $r=\\\\varphi^{+}\\\\approx 1.618r_{s}$ ), and the ISCO (Innermost Stable Circular Orbit) at $r=3r_{s}$ . Recall that in the Diamond representation in Fig. 3 we chose to change vertical ellipses from $r=\\\\varphi^{+}$ to $r^{\\\\ast}=0$ . In Fig. 5, this would represent a change in the curve from the points marked in yellow and purple.\",\"tag\":null,\"text\":\"Some important points are explicitly marked, such as the location where r ∗ = 0 (r ≈ 1.2786 rs), the photon sphere at r = 3rs/2, the Golden radius (r = φ + ≈ 1.618rs), and the ISCO (Innermost Stable Circular Orbit) at r = 3rs. Recall that in the Diamond representation in Fig. 3 we chose to\"},{\"bbox\":[299.27,690.18,11.71,11.96],\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"fragments\":[{\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"words\":[10392]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":6,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"19\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0018.pdf\",\"page_no\":19,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph179\",\"c_block\":0.97,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.995,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.97,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"62fde53757a7e957e7485505d7d55d758f65451f9ebbe6d86eaab87558ba99fd\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/62fde53757a7e957e7485505d7d55d758f65451f9ebbe6d86eaab87558ba99fd.pdf","asset_sha256":"62fde53757a7e957e7485505d7d55d758f65451f9ebbe6d86eaab87558ba99fd","document_id":"arxiv:2608.07909","kind":"page","page_number":19,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,125.8,388.54,40.85],\"bbox_pdf\":[110.85,625.35,499.39,666.2],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[110.85,625.35,499.39,666.2],\"words\":[7834,7835,7836,7837,7838,7839,7840,7841,7842,7843,7844,7845,7846,7847,7848,7849,7850,7851,7852,7853,7854,7855,7856,7857,7858,7859,7860,7861]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph201\",\"source_text\":\"We also highlighted a derivation of the Tortoise and Kruskal coordinates from the unified scheme of considering conformal (1+1) radial sections in the Schwarzschild geometry. The Tortoise coordinate (3.6) appears when considering the Cartesian form, while the Kruskal coordinates appear naturally by doing the analogous developments for a conformal flat Rindler transformation. We then presented the Diamond representation, the other main general possibility for these types of transformations [15].\",\"tag\":null,\"text\":\"doing the analogous developments for a conformal flat Rindler transformation. We then presented the Diamond representation, the other main general possibility for these types of transformations [15].\"},{\"bbox\":[110.85,169.14,388.55,84.18],\"bbox_pdf\":[110.85,538.68,499.4,622.86],\"fragments\":[{\"bbox_pdf\":[110.85,538.68,499.4,622.86],\"words\":[7862,7863,7864,7865,7866,7867,7868,7869,7870,7871,7872,7873,7874,7875,7876,7877,7878,7879,7880,7881,7882,7883,7884,7885,7886,7887,7888,7889,7890,7891,7892,7893,7894,7895,7896,7897,7898,7899,7900,7901,7902,7903,7904,7905,7906,7907,7908,7909,7910,7911,7912,7913,7914,7915,7916,7917,7918,7919,7920]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph202\",\"source_text\":\"In the last three sections we introduced the Golden representation, arising from transformations of the Einstein-Hilbert action. The radial Lagrangian results in a description similar to the kinematical Lagrangian of a relativistic particle [16][17]. In this article, we have deepened the significance of this dimensional reduction, clarified issues regarding the correct translation with Schwarzschild coordinates, and proposed useful visualizations.\",\"tag\":null,\"text\":\"In the last three sections we introduced the Golden representation, arising from transformations of the Einstein-Hilbert action. The radial Lagrangian results in a description similar to the kinematical Lagrangian of a relativistic particle [16][17]. In this article, we have deepened the significance of this dimensional reduction, clarified issues regarding the correct translation with Schwarzschild coordinates, and proposed useful visualizations.\"},{\"bbox\":[110.85,255.81,388.55,40.85],\"bbox_pdf\":[110.85,495.34,499.4,536.19],\"fragments\":[{\"bbox_pdf\":[110.85,495.34,499.4,536.19],\"words\":[7921,7922,7923,7924,7925,7926,7927,7928,7929,7930,7931,7932,7933,7934,7935,7936,7937,7938,7939,7940,7941,7942,7943,7944,7945,7946,7947,7948,7949,7950,7951,7952,7953,7954,7955,7956,7957,7958,7959,7960,7961]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph203\",\"source_text\":\"Let us make three final remarks about the representations we have explored. The first point is that the Golden radius $r=\\\\varphi^{+}r_{s}$ unites all of them in an interesting sense (see the Appendix). This radius corresponds to:\",\"tag\":null,\"text\":\"Let us make three final remarks about the representations we have explored. The first point is that the Golden radius r = φ + rs unites all of them in an interesting sense (see the Appendix). This radius corresponds to:\"},{\"bbox\":[140.13,312.1,293.06,11.96],\"bbox_pdf\":[140.13,467.94,433.19,479.9],\"fragments\":[{\"bbox_pdf\":[140.13,467.94,433.19,479.9],\"words\":[7962,7963,7966,7967,7968,7969,7970,7971,7972,7973,7974,7975,7976,7977,7978]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":1,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"list_item\",\"source_ref\":\"S7.I1.i1\",\"source_text\":\"• The location on the Flamm parabola $(\\\\varrho,z)=(\\\\varphi^{+},2\\\\sqrt{-\\\\varphi^{-}})r_{s}$ .\",\"tag\":null,\"text\":\"√ + The location on the Flamm parabola (ϱ, z) = (φ , 2 −φ\"},{\"bbox\":[140.12,336.51,359.28,26.4],\"bbox_pdf\":[140.12,429.09,499.4,455.49],\"fragments\":[{\"bbox_pdf\":[140.12,429.09,499.4,455.49],\"words\":[7980,7981,7982,7983,7984,7985,7986,7987,7988,7989,7990,7991,7992,7993,7994,7995,7996,7997,7998,7999,8000]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":1,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"list_item\",\"source_ref\":\"S7.I1.i2\",\"source_text\":\"• One of infinitely many radii that yield a horizontal degenerate ellipse in the Diamond representation (Fig. 3), given the $\\\\alpha$ chosen.\",\"tag\":null,\"text\":\"One of infinitely many radii that yield a horizontal degenerate ellipse in the Diamond representation (Fig. 3), given the α chosen.\"},{\"bbox\":[140.13,375.01,212.48,14.1],\"bbox_pdf\":[140.13,402.89,352.6,416.99],\"fragments\":[{\"bbox_pdf\":[140.13,402.89,352.6,416.99],\"words\":[8002,8003,8004,8005,8006,8007,8008,8009,8010,8011,8012,8013,8014]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":1,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"list_item\",\"source_ref\":\"S7.I1.i3\",\"source_text\":\"• A Tortoise coordinate $r^{\\\\ast}/r_{s}=\\\\varphi^{+}-\\\\ln\\\\varphi^{+}$ .\",\"tag\":null,\"text\":\"A Tortoise coordinate r ∗ /rs = φ + − ln φ +\"},{\"bbox\":[140.13,399.42,340.3,12.31],\"bbox_pdf\":[140.13,380.27,480.42,392.58],\"fragments\":[{\"bbox_pdf\":[140.13,380.27,480.42,392.58],\"words\":[8017,8018,8019,8020,8021,8022,8023,8024,8025,8026,8027,8028,8029,8030,8031,8032,8033]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":1,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"list_item\",\"source_ref\":\"S7.I1.i4\",\"source_text\":\"• The point $q^{1}=-q^{2}=\\\\ln\\\\varphi^{+}$ of the Golden representation (Fig. 5).\",\"tag\":null,\"text\":\"The point q 1 = −q 2 = ln φ + of the Golden representation (Fig. 5).\"},{\"bbox\":[140.13,424.18,358.0,11.96],\"bbox_pdf\":[140.13,355.87,498.12,367.82],\"fragments\":[{\"bbox_pdf\":[140.13,355.87,498.12,367.82],\"words\":[8034,8035,8036,8037,8038,8039,8041,8042,8043,8044,8045,8046,8047,8048,8049,8050,8051,8052]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":1,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"list_item\",\"source_ref\":\"S7.I1.i5\",\"source_text\":\"• A location $Z=-\\\\varphi^{-}e^{-i\\\\varphi^{-}\\\\ln\\\\varphi^{+}}$ in the complex representation (Fig. 6).\",\"tag\":null,\"text\":\"− −iφ − ln φ + A location Z = −φ e in the complex representation (Fig. 6).\"},{\"bbox\":[110.85,451.58,388.54,26.4],\"bbox_pdf\":[110.85,314.02,499.39,340.42],\"fragments\":[{\"bbox_pdf\":[110.85,314.02,499.39,340.42],\"words\":[8053,8054,8055,8056,8057,8058,8059,8060,8061,8062,8063,8064,8065,8066,8067,8068,8069]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph209\",\"source_text\":\"Furthermore, it is the radius where the proper distance (2.10) and the Tortoise coordinate (3.6) satisfy:\",\"tag\":null,\"text\":\"Furthermore, it is the radius where the proper distance (2.10) and the Tortoise coordinate (3.6) satisfy:\"},{\"bbox\":[253.32,485.23,246.09,23.43],\"bbox_pdf\":[253.32,283.34,499.4,306.77],\"fragments\":[{\"bbox_pdf\":[253.32,283.34,499.4,306.77],\"words\":[8070,8071,8072,8073,8074,8075,8076,8077,8078,8079,8080]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"3r^{\\\\ast}+2\\\\varrho=\\\\left(\\\\varphi^{+}\\\\right)^{4}r_{s}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S7.Ex1\",\"source_text\":\"$3r^{\\\\ast}+2\\\\varrho=\\\\left(\\\\varphi^{+}\\\\right)^{4}r_{s}.$ (7.1)\",\"tag\":\"(7.1)\",\"text\":\"(︁ )︁4 ∗ + 3r + 2ϱ = φ rs. (7.1)\"},{\"bbox\":[110.85,515.84,388.54,26.4],\"bbox_pdf\":[110.85,249.76,499.39,276.16],\"fragments\":[{\"bbox_pdf\":[110.85,249.76,499.39,276.16],\"words\":[8081,8082,8083,8084,8085,8086,8087,8088,8089,8090,8091,8092,8093,8094,8095,8096,8097,8098,8099,8100,8101,8102,8103]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"paragraph\",\"source_ref\":\"paragraph211\",\"source_text\":\"This equation beautifully links the Flamm and Tortoise representations, and mimics the purely numeric relation $3\\\\varphi^{+}+2=\\\\left(\\\\varphi^{+}\\\\right)^{4}$ .\",\"tag\":null,\"text\":\"This equation beautifully links the Flamm and Tortoise representations, and + + 4 mimics the purely numeric relation 3φ + 2 = (φ\"},{\"bbox\":[110.85,544.73,388.55,127.52],\"bbox_pdf\":[110.85,119.75,499.4,247.27],\"fragments\":[{\"bbox_pdf\":[110.85,119.75,499.4,247.27],\"words\":[8106,8107,8108,8109,8110,8111,8112,8113,8114,8115,8116,8117,8118,8119,8120,8121,8122,8123,8124,8125,8126,8127,8128,8129,8130,8131,8132,8133,8134,8135,8136,8137,8138,8139,8140,8141,8142,8143,8144,8145,8146,8147,8148,8149,8150,8151,8152,8153,8154,8155,8156,8157,8158,8159,8160,8163,8161,8162,8164,8165,8166,8167,8168,8169,8170,8171,8172,8173,8174,8175,8176,8177,8178,8179,8180,8181,8182,8183,8184,8185,8186,8187,8188,8189,8190,8191,8192,8193,8194,8195,8196,8197,8198,8199,8200,8201,8202,8203,8204,8205,8206,8207,8208,8209,8210,8212,8213,8214,8215,8216,8217,8218,8219,8220]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"paragraph\",\"source_ref\":\"paragraph212\",\"source_text\":\"A second point worth mentioning is the relation of the Golden representation to the others presented. For instance, the Golden representation retains a mixture of properties from the embedding and the Tortoise representations. Just as the Euclidean distance in the Flamm parabola allows one to read the integrated proper distance $\\\\varrho$ , the complex Golden representation (Fig. 6) allows one to directly read the redshift factor $\\\\sqrt{1-r_{s}/r}$ by measuring the distance to the origin (the event horizon). Also, the direct $q^{1}-q^{2}$ diagram of Fig. 5 implies that the event horizon has been pushed to $-\\\\infty$ in $q^{2}$ , just as it is in the Tortoise coordinate (3.6).\",\"tag\":null,\"text\":\"A second point worth mentioning is the relation of the Golden representation to the others presented. For instance, the Golden representation retains a mixture of properties from the embedding and the Tortoise representations. Just as the Euclidean distance in the Flamm parabola allows one to read the integrated proper distance ϱ, the complex Golden √︁ representation (Fig. 6) allows one to directly read the redshift factor 1 − rs/r by measuring the distance to the origin (the event horizon). Also, the direct q 1 − q 2 diagram of Fig. 5 implies that the event horizon has been pushed to −∞ in q 2 just as it is in the Tortoise coordinate (3.6).\"},{\"bbox\":[299.27,690.18,11.71,11.96],\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"fragments\":[{\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"words\":[10395]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":12,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"23\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0022.pdf\",\"page_no\":23,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":0.999,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.999,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"90203fb13d8845f0406d02e13d0bd16e0ce879f8c999aa5c7ea880f3e35ad3ae\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/90203fb13d8845f0406d02e13d0bd16e0ce879f8c999aa5c7ea880f3e35ad3ae.pdf","asset_sha256":"90203fb13d8845f0406d02e13d0bd16e0ce879f8c999aa5c7ea880f3e35ad3ae","document_id":"arxiv:2608.07909","kind":"page","page_number":23,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[253.32,485.23,246.09,23.43],\"bbox_pdf\":[253.32,283.34,499.4,306.77],\"fragments\":[{\"bbox_pdf\":[253.32,283.34,499.4,306.77],\"words\":[8070,8071,8072,8073,8074,8075,8076,8077,8078,8079,8080]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"3r^{\\\\ast}+2\\\\varrho=\\\\left(\\\\varphi^{+}\\\\right)^{4}r_{s}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S7.Ex1\",\"source_text\":\"$3r^{\\\\ast}+2\\\\varrho=\\\\left(\\\\varphi^{+}\\\\right)^{4}r_{s}.$ (7.1)\",\"tag\":\"(7.1)\",\"text\":\"(︁ )︁4 ∗ + 3r + 2ϱ = φ rs. (7.1)\"},\"formula_index\":9,\"parent_page_sha256\":\"90203fb13d8845f0406d02e13d0bd16e0ce879f8c999aa5c7ea880f3e35ad3ae\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/90203fb13d8845f0406d02e13d0bd16e0ce879f8c999aa5c7ea880f3e35ad3ae.pdf","asset_sha256":"90203fb13d8845f0406d02e13d0bd16e0ce879f8c999aa5c7ea880f3e35ad3ae","document_id":"arxiv:2608.07909","kind":"formula","page_number":23,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,125.44,388.54,26.75],\"bbox_pdf\":[110.85,639.8,499.4,666.56],\"fragments\":[{\"bbox_pdf\":[110.85,639.8,499.4,666.56],\"words\":[8221,8222,8223,8224,8225,8226,8227,8228,8229,8230,8231,8232,8233,8234,8235,8236,8237,8238,8239,8240,8241,8242]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph213\",\"source_text\":\"Furthermore, while the first null coordinate is $q^{1}=\\\\ln(r/r_{s})$ , the second one is, in the Schwarzschild interpretation (see the Appendix):\",\"tag\":null,\"text\":\"Furthermore, while the first null coordinate is q 1 = ln(r/rs), the second one is, in the Schwarzschild interpretation (see the Appendix):\"},{\"bbox\":[258.67,163.38,240.72,30.39],\"bbox_pdf\":[258.67,598.23,499.39,628.62],\"fragments\":[{\"bbox_pdf\":[258.67,598.23,499.39,628.62],\"words\":[8243,8244,8245,8246,8247,8248,8249,8250,8251,8252,8253,8254,8255]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"q^{1}+2q^{2}=\\\\frac{r^{\\\\ast}-r}{r_{s}}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S7.Ex2\",\"source_text\":\"$q^{1}+2q^{2}=\\\\frac{r^{\\\\ast}-r}{r_{s}}.$ (7.2)\",\"tag\":\"(7.2)\",\"text\":\"r ∗ − r q 1 + 2q 2 = . (7.2) rs\"},{\"bbox\":[110.85,196.51,388.55,58.12],\"bbox_pdf\":[110.85,537.37,499.4,595.49],\"fragments\":[{\"bbox_pdf\":[110.85,537.37,499.4,595.49],\"words\":[8256,8257,8258,8259,8260,8261,8262,8263,8264,8265,8266,8267,8268,8269,8270,8271,8272,8273,8274,8275,8276,8277,8278,8280,8281,8282,8283,8284,8285,8286,8287,8288,8289,8290,8291,8292,8293,8294,8295,8296,8297,8298,8299,8300,8301,8302,8303,8304,8305,8306,8307,8308]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.972,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph215\",\"source_text\":\"This measures how much the Tortoise advances past the Schwarzschild coordinate. Recalling from Eq. (3.9) that $\\\\rho^{2}=e^{\\\\frac{r^{\\\\ast}}{r_{s}}}$ , where $\\\\rho$ is the intermediate Rindler-type coordinate leading to the Kruskal transformations, exponentiating (7.2) yields the Kruskal connection $e^{q^{1}+2q^{2}}e^{\\\\frac{r}{r_{s}}}=\\\\rho^{2}$ .\",\"tag\":null,\"text\":\"This measures how much the Tortoise advances past the Schwarzschild coor- r ∗ dinate. Recalling from Eq. (3.9) that ρ 2 = ers where ρ is the intermediate Rindler-type coordinate leading to the Kruskal transformations, exponenti- 1 2 r q +2q rs = ρ2 ating (7.2) yields the Kruskal connection e e\"},{\"bbox\":[110.85,257.12,388.55,129.32],\"bbox_pdf\":[110.85,405.56,499.4,534.88],\"fragments\":[{\"bbox_pdf\":[110.85,405.56,499.4,534.88],\"words\":[8310,8311,8312,8313,8314,8315,8316,8317,8318,8319,8320,8321,8322,8323,8324,8325,8326,8327,8328,8329,8330,8331,8332,8333,8334,8335,8336,8337,8338,8339,8340,8341,8342,8343,8344,8345,8346,8347,8348,8349,8350,8351,8352,8353,8354,8355,8356,8357,8358,8359,8360,8361,8362,8363,8364,8365,8366,8367,8368,8369,8370,8371,8372,8373,8374,8375,8376,8377,8378,8379,8380,8381,8382,8384,8385,8386,8387,8388,8389,8390,8391,8392,8393,8394,8395,8396,8397,8398,8399,8400,8401,8402,8403,8404,8405,8406,8407,8408,8409,8410,8411,8412,8413,8414,8415,8416,8417,8418,8419,8420,8421,8422,8423,8424,8425,8426,8427,8428,8429,8430,8432,8433,8434,8435,8436,8437]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph216\",\"source_text\":\"There is also a relation to the Diamond representation, beyond just choosing a specific $\\\\alpha$ in Eq. (3.16). By taking any $\\\\alpha$ in Eqs. (3.13)-(3.16), one can see that the periodicity in the change of ellipses in Fig. 3 is driven by the argument $r^{\\\\ast}/\\\\alpha$ , while the oscillations in the spiral representations proposed at the end of Section 6 are driven by the angle $\\\\theta=-\\\\varphi^{-}q^{1}$ . However, while both make infinite oscillations when $r\\\\rightarrow\\\\infty$ , only the Diamond representation makes infinite oscillations when approaching the event horizon. The relation (6.2), together with (7.2) and the translations $q^{1}=\\\\ln(r/r_{s})$ and $e^{2q^{2}}=1-r_{s}/r=\\\\left|Z\\\\right|^{2}$ , allows us to obtain the relationship:\",\"tag\":null,\"text\":\"There is also a relation to the Diamond representation, beyond just choosing a specific α in Eq. (3.16). By taking any α in Eqs. (3.13)-(3.16), one can see that the periodicity in the change of ellipses in Fig. 3 is driven by the argument r ∗ /α, while the oscillations in the spiral representations proposed at the end of Section 6 are driven by the angle θ = −φ − q 1 However, while both make infinite oscillations when r → ∞, only the Diamond representation makes infinite oscillations when approaching the event horizon. The relation (6.2), together with (7.2) and the translations q 1 = ln(r/rs) and 2q 2 2 e = 1 − rs/r = |Z| allows us to obtain the relationship:\"},{\"bbox\":[223.75,388.96,275.64,33.43],\"bbox_pdf\":[223.75,369.61,499.39,403.04],\"fragments\":[{\"bbox_pdf\":[223.75,369.61,499.39,403.04],\"words\":[8438,8439,8440,8441,8442,8443,8444,8445,8446,8447,8448,8449,8450,8451,8452,8453,8454,8455,8456,8457,8458,8459]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{r^{\\\\ast}}{\\\\alpha}=\\\\frac{r_{s}}{\\\\alpha}\\\\left(\\\\varphi^{+}\\\\theta+\\\\ln\\\\left|Z%\\n\\\\right|^{2}+e^{\\\\varphi^{+}\\\\theta}\\\\right).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S7.Ex3\",\"source_text\":\"$\\\\frac{r^{\\\\ast}}{\\\\alpha}=\\\\frac{r_{s}}{\\\\alpha}\\\\left(\\\\varphi^{+}\\\\theta+\\\\ln\\\\left|Z% \\\\right|^{2}+e^{\\\\varphi^{+}\\\\theta}\\\\right).$ (7.3)\",\"tag\":\"(7.3)\",\"text\":\"r ∗ rs (︂ )︂ + 2 φ + θ = φ θ + ln |Z| + e . (7.3) α α\"},{\"bbox\":[110.85,429.16,388.54,57.09],\"bbox_pdf\":[110.85,305.75,499.4,362.84],\"fragments\":[{\"bbox_pdf\":[110.85,305.75,499.4,362.84],\"words\":[8460,8461,8462,8463,8464,8465,8466,8467,8468,8469,8470,8471,8472,8473,8474,8475,8476,8477,8478,8479,8480,8481,8482,8483,8484,8485,8486,8487,8488,8489,8490,8491,8492,8493,8494,8495,8496,8498,8499,8500,8501,8502,8503,8504,8505,8506,8507,8508,8509,8510,8511,8512,8513,8514,8515,8516,8517]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph218\",\"source_text\":\"This equation highlights two asymptotic connections. For $r\\\\rightarrow r_{s}$ , which has $\\\\theta\\\\rightarrow 0$ in the golden spirals, the logarithm dominates, and the near-horizon pulsations are governed by $r^{\\\\ast}/\\\\alpha\\\\cong(r_{s}/\\\\alpha)\\\\ln\\\\left|Z\\\\right|^{2}$ . On the other extreme, as $r\\\\rightarrow\\\\infty$ , the exponential dominates, yielding $r^{\\\\ast}/\\\\alpha\\\\cong(r_{s}/\\\\alpha)e^{\\\\varphi^{+}\\\\theta}$ .\",\"tag\":null,\"text\":\"This equation highlights two asymptotic connections. For r → rs, which has θ → 0 in the golden spirals, the logarithm dominates, and the near-horizon ∗ 2 pulsations are governed by r /α ∼= (rs/α) ln |Z| On the other extreme, as ∗ φ + θ r → ∞, the exponential dominates, yielding r /α ∼= (rs/α)e\"},{\"bbox\":[110.85,486.94,388.55,185.31],\"bbox_pdf\":[110.85,119.75,499.4,305.06],\"fragments\":[{\"bbox_pdf\":[110.85,119.75,499.4,305.06],\"words\":[8519,8520,8521,8522,8523,8524,8525,8526,8527,8528,8529,8530,8531,8532,8533,8534,8535,8536,8537,8538,8539,8540,8541,8542,8543,8544,8545,8546,8547,8548,8549,8550,8551,8552,8553,8554,8555,8556,8557,8558,8559,8560,8561,8562,8563,8564,8565,8566,8567,8568,8569,8570,8571,8572,8573,8574,8575,8576,8577,8578,8579,8580,8581,8582,8583,8584,8585,8586,8587,8588,8589,8590,8591,8592,8593,8594,8595,8596,8597,8599,8600,8601,8602,8603,8604,8605,8606,8607,8608,8609,8610,8611,8612,8613,8614,8615,8616,8617,8618,8619,8620,8621,8622,8623,8624,8625,8626,8627,8628,8629,8630,8631,8632,8633,8634,8635,8636,8637,8638,8639,8640,8641,8642,8643,8644,8645,8646,8647,8648,8649,8650,8651,8652,8653,8654,8655,8656,8657,8658,8659,8660,8661,8662,8663,8664,8665,8666,8667,8668,8669]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph219\",\"source_text\":\"The third and final point concerns other physical connections of the Golden representation and possible future work. For instance, notice that integrating $2(dr^{\\\\ast}-dr)$ yields what one would call a radial Shapiro time delay for a two-way light path, by setting the impact parameter to zero [1][38]. That is, by simply taking the difference of the null coordinate (7.2) at two distinct points, one obtains the time delay effect: $\\\\Delta t_{delay}=2r_{s}\\\\Delta(q^{1}+2q^{2})$ . Other physical aspects, such as thermodynamic aspects of the black hole, the use of Golden transformations or extensions as a toy arena to discuss implementations from a lower-dimensional manifold to four-dimensional or extradimensional ones, and the possible use of the phase space version of the Golden representation for deforming the two-dimensional flat version to obtain curved spacetimes, are left for future work (see Refs. [39]-[42] for interesting possibilities regarding these issues).\",\"tag\":null,\"text\":\"The third and final point concerns other physical connections of the Golden representation and possible future work. For instance, notice that integrating 2(dr ∗ − dr) yields what one would call a radial Shapiro time delay for a two-way light path, by setting the impact parameter to zero [1][38]. That is, by simply taking the difference of the null coordinate (7.2) at two distinct points, one obtains the time delay effect: ∆tdelay = 2rs∆(q 1 + 2q 2 Other physical aspects, such as thermodynamic aspects of the black hole, the use of Golden transformations or extensions as a toy arena to discuss implementations from a lower-dimensional manifold to four-dimensional or extradimensional ones, and the possible use of the phase space version of the Golden representation for deforming the two-dimensional flat version to obtain curved spacetimes, are left for future work (see Refs. [39]-[42] for interesting possibilities regarding these issues).\"},{\"bbox\":[299.27,690.18,11.71,11.96],\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"fragments\":[{\"bbox_pdf\":[299.27,89.86,310.98,101.82],\"words\":[10396]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":7,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"24\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.07909\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.07909v1-p0023.pdf\",\"page_no\":24,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2c0f62c33ddf5b64f40a957690a00fc95cf3929c3c493195f68ab1f4bb3d07a8\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph215\",\"c_block\":0.97,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.992,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.97,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"f7e7f6eb957cbc65ad6e09d3f7e1743424022ce1df0b465bde5583e94853a87b\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/f7e7f6eb957cbc65ad6e09d3f7e1743424022ce1df0b465bde5583e94853a87b.pdf","asset_sha256":"f7e7f6eb957cbc65ad6e09d3f7e1743424022ce1df0b465bde5583e94853a87b","document_id":"arxiv:2608.07909","kind":"page","page_number":24,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[258.67,163.38,240.72,30.39],\"bbox_pdf\":[258.67,598.23,499.39,628.62],\"fragments\":[{\"bbox_pdf\":[258.67,598.23,499.39,628.62],\"words\":[8243,8244,8245,8246,8247,8248,8249,8250,8251,8252,8253,8254,8255]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"q^{1}+2q^{2}=\\\\frac{r^{\\\\ast}-r}{r_{s}}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S7.Ex2\",\"source_text\":\"$q^{1}+2q^{2}=\\\\frac{r^{\\\\ast}-r}{r_{s}}.$ (7.2)\",\"tag\":\"(7.2)\",\"text\":\"r ∗ − r q 1 + 2q 2 = . (7.2) rs\"},\"formula_index\":1,\"parent_page_sha256\":\"f7e7f6eb957cbc65ad6e09d3f7e1743424022ce1df0b465bde5583e94853a87b\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/f7e7f6eb957cbc65ad6e09d3f7e1743424022ce1df0b465bde5583e94853a87b.pdf","asset_sha256":"f7e7f6eb957cbc65ad6e09d3f7e1743424022ce1df0b465bde5583e94853a87b","document_id":"arxiv:2608.07909","kind":"formula","page_number":24,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[223.75,388.96,275.64,33.43],\"bbox_pdf\":[223.75,369.61,499.39,403.04],\"fragments\":[{\"bbox_pdf\":[223.75,369.61,499.39,403.04],\"words\":[8438,8439,8440,8441,8442,8443,8444,8445,8446,8447,8448,8449,8450,8451,8452,8453,8454,8455,8456,8457,8458,8459]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{r^{\\\\ast}}{\\\\alpha}=\\\\frac{r_{s}}{\\\\alpha}\\\\left(\\\\varphi^{+}\\\\theta+\\\\ln\\\\left|Z%\\n\\\\right|^{2}+e^{\\\\varphi^{+}\\\\theta}\\\\right).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S7.Ex3\",\"source_text\":\"$\\\\frac{r^{\\\\ast}}{\\\\alpha}=\\\\frac{r_{s}}{\\\\alpha}\\\\left(\\\\varphi^{+}\\\\theta+\\\\ln\\\\left|Z% \\\\right|^{2}+e^{\\\\varphi^{+}\\\\theta}\\\\right).$ (7.3)\",\"tag\":\"(7.3)\",\"text\":\"r ∗ rs (︂ )︂ + 2 φ + θ = φ θ + ln |Z| + e . (7.3) α α\"},\"formula_index\":4,\"parent_page_sha256\":\"f7e7f6eb957cbc65ad6e09d3f7e1743424022ce1df0b465bde5583e94853a87b\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/f7e7f6eb957cbc65ad6e09d3f7e1743424022ce1df0b465bde5583e94853a87b.pdf","asset_sha256":"f7e7f6eb957cbc65ad6e09d3f7e1743424022ce1df0b465bde5583e94853a87b","document_id":"arxiv:2608.07909","kind":"formula","page_number":24,"paper_attribution_json":"{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,120.54,151.83,17.22],\"bbox_pdf\":[110.85,654.25,262.69,671.46],\"fragments\":[{\"bbox_pdf\":[110.85,654.25,262.69,671.46],\"words\":[8670]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"heading\",\"source_ref\":\"title220\",\"source_text\":\"Acknowledgments\",\"tag\":null,\"text\":\"Acknowledgments\"},{\"bbox\":[110.85,152.08,388.55,55.29],\"bbox_pdf\":[110.85,584.62,499.4,639.92],\"fragments\":[{\"bbox_pdf\":[110.85,584.62,499.4,639.92],\"words\":[8671,8672,8673,8674,8675,8676,8677,8678,8679,8680,8681,8682,8683,8684,8685,8686,8687,8688,8689,8690,8691,8692,8693,8694,8695,8696,8697,8698,8699,8700,8701,8702,8703,8704,8705,8706,8707,8708,8709]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph221\",\"source_text\":\"The author thanks J. A. Nieto and A. Meza for useful comments on this subject. He also recognizes the Secretaría de Ciencia, Humanidades, Tecnología e Innovación (SECIHTI) for support through the Sistema Nacional de Investigadoras e Investigadores (SNII).\",\"tag\":null,\"text\":\"The author thanks J. A. Nieto and A. Meza for useful comments on this subject. He also recognizes the Secretar´ıa de Ciencia, Humanidades, Tec- nolog´ıa e Innovaci´on (SECIHTI) for support through the Sistema Nacional de Investigadoras e Investigadores (SNII).\"},{\"bbox\":[110.85,230.09,371.83,17.22],\"bbox_pdf\":[110.85,544.69,482.68,561.91],\"fragments\":[{\"bbox_pdf\":[110.85,544.69,482.68,561.91],\"words\":[8710,8711,8712,8713,8714,8715]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"heading\",\"source_ref\":\"title222\",\"source_text\":\"Appendix: Golden ratio, and Golden radius\",\"tag\":null,\"text\":\"Appendix: Golden ratio, and Golden radius\"},{\"bbox\":[110.85,261.64,388.55,26.4],\"bbox_pdf\":[110.85,503.96,499.4,530.36],\"fragments\":[{\"bbox_pdf\":[110.85,503.96,499.4,530.36],\"words\":[8716,8717,8718,8719,8720,8721,8722,8723,8724,8725,8726,8727,8728,8729,8730,8731,8733,8734,8735,8736,8737]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph223\",\"source_text\":\"The golden ratio arises from the standard continuous division condition $\\\\varphi=(\\\\varphi-1)^{-1}$ , yielding the defining relation [43][44]:\",\"tag\":null,\"text\":\"The golden ratio arises from the standard continuous division condition φ = (φ − 1) −1 yielding the defining relation [43][44]:\"},{\"bbox\":[275.46,301.53,223.94,12.91],\"bbox_pdf\":[275.46,477.56,499.39,490.47],\"fragments\":[{\"bbox_pdf\":[275.46,477.56,499.39,490.47],\"words\":[8738,8739,8740,8741,8742,8743,8744]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\varphi^{2}=\\\\varphi+1.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"Ax1.Ex1\",\"source_text\":\"$\\\\varphi^{2}=\\\\varphi+1.$ (A.1)\",\"tag\":\"(A.1)\",\"text\":\"φ 2 = φ + 1. (A.1)\"},{\"bbox\":[110.85,328.53,388.55,58.81],\"bbox_pdf\":[110.85,404.66,499.4,463.47],\"fragments\":[{\"bbox_pdf\":[110.85,404.66,499.4,463.47],\"words\":[8745,8746,8747,8748,8749,8750,8751,8752,8753,8754,8755,8756,8757,8758,8759,8760,8761,8762,8763,8764,8765,8766,8767,8768,8769,8770,8771,8772,8773,8774,8775,8776,8777,8778,8779,8780,8781,8782,8783,8784,8785,8786,8787,8788,8789,8790,8791,8792,8793,8794,8795,8796,8797,8798,8799,8800,8801,8802,8803,8804,8805,8806,8807,8808,8809,8810,8811,8812,8813,8814,8815,8816,8817,8818]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.971,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph225\",\"source_text\":\"Solving the quadratic leads to two values: $\\\\varphi^{+}=\\\\left(1+\\\\sqrt{5}\\\\right)/2$ and $\\\\varphi^{-}=\\\\left(1-\\\\sqrt{5}\\\\right)/2$ . From this, we have $\\\\varphi^{+}\\\\varphi^{-}=-1$ , $\\\\varphi^{+}+\\\\varphi^{-}=1$ , and $\\\\varphi^{+}-\\\\varphi^{-}=\\\\sqrt{5}$ . By successive multiplications of (A.1) by $\\\\varphi$ , one can note that the following relation holds for integers $n\\\\geq 0$ :\",\"tag\":null,\"text\":\"(︁ √ )︁ Solving the quadratic leads to two values: φ + = 1 + 5 /2 and φ − = (︁ √ )︁ 1 − 5 /2. From this, we have φ + φ − = −1, φ + + φ − = 1, and φ + − φ − = √ 5. By successive multiplications of (A.1) by φ, one can note that the following relation holds for integers n ≥ 0:\"},{\"bbox\":[229.58,393.12,269.82,21.64],\"bbox_pdf\":[229.58,377.24,499.4,398.88],\"fragments\":[{\"bbox_pdf\":[229.58,377.24,499.4,398.88],\"words\":[8819,8820,8821,8822,8823,8824,8825,8826,8827,8828,8829,8830,8831,8832,8833,8834]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\left(\\\\varphi^{\\\\pm}\\\\right)^{n+2}=\\\\left(\\\\varphi^{\\\\pm}\\\\right)^{n+1}+\\\\left(%\\n\\\\varphi^{\\\\pm}\\\\right)^{n}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"Ax1.Ex2\",\"source_text\":\"$\\\\left(\\\\varphi^{\\\\pm}\\\\right)^{n+2}=\\\\left(\\\\varphi^{\\\\pm}\\\\right)^{n+1}+\\\\left(% \\\\varphi^{\\\\pm}\\\\right)^{n}.$ (A.2)\",\"tag\":\"(A.2)\",\"text\":\"(︁ )︁n+2 (︁ )︁n+1 (︁ )︁n ± ± ± φ = φ + φ . (A.2)\"},{\"bbox\":[110.85,431.08,392.82,55.29],\"bbox_pdf\":[110.85,305.63,503.68,360.92],\"fragments\":[{\"bbox_pdf\":[110.85,305.63,503.68,360.92],\"words\":[8835,8836,8837,8838,8839,8840,8841,8842,8843,8844,8845,8846,8847,8848,8849,8850,8851,8853,8854,8855,8856,8857,8858,8860,8861,8862,8863,8864,8865,8866,8867,8869,8870,8871,8872,8873,8874,8875,8876,8877,8878,8879,8880,8881,8882,8883,8884,8885,8886,8887,8888,8889,8890,8891,8892,8893,8894,8895,8896,8897,8898,8899,8900,8901,8902,8903,8906,8907,8908,8909,8910,8911,8912,8913,8914,8915,8916,8917]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.939,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph227\",\"source_text\":\"Thus, $\\\\left(\\\\varphi^{\\\\pm}\\\\right)^{2}=\\\\left(3\\\\pm\\\\sqrt{5}\\\\right)/2$ , $\\\\left(\\\\varphi^{\\\\pm}\\\\right)^{3}=\\\\left(4\\\\pm 2\\\\sqrt{5}\\\\right)/2$ and $\\\\left(\\\\varphi^{\\\\pm}\\\\right)^{4}=\\\\left(7\\\\pm 3\\\\sqrt{5}\\\\right)/2$ . The Fibonacci sequence, starting with $F_{0}=0$ , has elements $(0,1,1,2,3,5,8...)$ . By starting with (A.1) and calculating integer powers of $\\\\left(\\\\varphi^{\\\\pm}\\\\right)^{n}$ , one can convert them to a linear form in $\\\\varphi^{\\\\pm}$ :\",\"tag\":null,\"text\":\"2 (︁ √ )︁ 3 (︁ √ )︁ 4 (︁ √ )︁ ± ± ± Thus, (φ = 3 ± 5 /2, (φ = 4 ± 2 5 /2 and (φ = 7 ± 3 5 /2. The Fibonacci sequence, starting with F0 = 0, has elements (0, 1, 1, 2, 3, 5, 8...). ± n By starting with (A.1) and calculating integer powers of (φ one can convert them to a linear form in φ ±\"},{\"bbox\":[246.74,491.13,252.65,23.43],\"bbox_pdf\":[246.74,277.44,499.39,300.87],\"fragments\":[{\"bbox_pdf\":[246.74,277.44,499.39,300.87],\"words\":[8919,8920,8921,8922,8923,8924,8925,8926,8927,8928]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\left(\\\\varphi^{\\\\pm}\\\\right)^{n}=F_{n}\\\\varphi^{\\\\pm}+F_{n-1}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"Ax1.Ex3\",\"source_text\":\"$\\\\left(\\\\varphi^{\\\\pm}\\\\right)^{n}=F_{n}\\\\varphi^{\\\\pm}+F_{n-1}.$ (A.3)\",\"tag\":\"(A.3)\",\"text\":\"(︁ )︁n ± ± φ = Fnφ + Fn−1. (A.3)\"},{\"bbox\":[110.85,527.22,388.54,40.85],\"bbox_pdf\":[110.85,223.94,499.4,264.78],\"fragments\":[{\"bbox_pdf\":[110.85,223.94,499.4,264.78],\"words\":[8929,8930,8931,8932,8933,8934,8935,8936,8937,8938,8939,8940,8941,8942,8943,8944,8945,8946,8947,8948,8949,8950,8951,8952,8953,8954,8955,8956,8957,8958,8959,8960,8961,8962,8963,8964,8965,8967,8968]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.913,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph229\",\"source_text\":\"In the relations $\\\\sqrt{\\\\varphi^{+}}+\\\\sqrt{-\\\\varphi^{-}}$ and $\\\\sqrt{\\\\varphi^{+}}-\\\\sqrt{-\\\\varphi^{-}}$ , we can factor out $\\\\sqrt{\\\\varphi^{+}}$ and $\\\\sqrt{-\\\\varphi^{-}}$ , respectively. By considering again that $\\\\varphi^{+}\\\\varphi^{-}=-1$ and $\\\\varphi^{+}=1-\\\\varphi^{-}$ , one obtains:\",\"tag\":null,\"text\":\"√ √ √ √ √ In the relations φ++ −φ− and φ+− −φ−, we can factor out φ+ and √ −φ−, respectively. By considering again that φ+ φ − = −1 and φ + = 1−φ − one obtains:\"},{\"bbox\":[236.33,559.76,263.06,22.74],\"bbox_pdf\":[236.33,209.49,499.39,232.24],\"fragments\":[{\"bbox_pdf\":[236.33,209.49,499.39,232.24],\"words\":[8969,8970,8971,8972,8973,8974,8975,8976,8977,8978]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\sqrt{\\\\varphi^{+}}+\\\\sqrt{-\\\\varphi^{-}}=\\\\varphi^{+}\\\\sqrt{\\\\varphi^{+}},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"Ax1.Ex4\",\"source_text\":\"$\\\\sqrt{\\\\varphi^{+}}+\\\\sqrt{-\\\\varphi^{-}}=\\\\varphi^{+}\\\\sqrt{\\\\varphi^{+}},$ (A.4)\",\"tag\":\"(A.4)\",\"text\":\"√︁ √︁ √︁ φ+ + −φ− = φ+ φ+, (A.4)\"},{\"bbox\":[110.85,591.47,18.86,11.96],\"bbox_pdf\":[110.85,188.57,129.71,200.53],\"fragments\":[{\"bbox_pdf\":[110.85,188.57,129.71,200.53],\"words\":[8979]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"paragraph\",\"source_ref\":\"paragraph231\",\"source_text\":\"and\",\"tag\":null,\"text\":\"and\"},{\"bbox\":[226.93,609.58,272.46,22.74],\"bbox_pdf\":[226.93,159.68,499.39,182.42],\"fragments\":[{\"bbox_pdf\":[226.93,159.68,499.39,182.42],\"words\":[8980,8981,8982,8983,8984,8985,8986,8987,8988,8989]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\sqrt{\\\\varphi^{+}}-\\\\sqrt{-\\\\varphi^{-}}=-\\\\varphi^{-}\\\\sqrt{-\\\\varphi^{-}}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"Ax1.Ex5\",\"source_text\":\"$\\\\sqrt{\\\\varphi^{+}}-\\\\sqrt{-\\\\varphi^{-}}=-\\\\varphi^{-}\\\\sqrt{-\\\\varphi^{-}}.$ (A.5)\",\"tag\":\"(A.5)\",\"text\":\"√︁ √︁ √︁ φ+ − −φ− = −φ− −φ−. 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{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.07909\",\"authors\":\"Edgar Alejandro Le\\\\'on\",\"title\":\"Mapping Schwarzschild Spacetime: From Kruskal to Diamond and Golden Representations\",\"url\":\"https://arxiv.org/abs/2608.07909v1\",\"version\":1},\"blocks\":[{\"bbox\":[110.85,120.54,90.63,17.22],\"bbox_pdf\":[110.85,654.25,201.48,671.46],\"fragments\":[{\"bbox_pdf\":[110.85,654.25,201.48,671.46],\"words\":[9513]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"heading\",\"source_ref\":\"title246\",\"source_text\":\"References\",\"tag\":null,\"text\":\"References\"},{\"bbox\":[116.71,152.08,382.69,26.4],\"bbox_pdf\":[116.71,613.52,499.4,639.92],\"fragments\":[{\"bbox_pdf\":[116.71,613.52,499.4,639.92],\"words\":[9514,9515,9516,9517,9518,9519,9520,9521,9522,9523,9524,9525,9526,9527,9528,9529,9530,9531,9532,9533,9534]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"reference\",\"source_ref\":\"bib.bib1\",\"source_text\":\"C. 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{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[90.0,90.05,432.0,38.01],\"bbox_pdf\":[90.0,663.94,522.0,701.95],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[90.0,663.94,522.0,701.95],\"words\":[429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,450,451,452,453,454,455,456,457,458,459,460,461,462,463]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph6\",\"source_text\":\"Wall-pressure fluctuations beneath a turbulent boundary layer drive flow-induced noise and structural loading, and their frequency spectrum is the central input to rough-wall noise prediction (Blake, 1970; Goody, 2004). On a smooth wall the inner-scaled integrated variance grows with Reynolds number (Farabee and Casarella, 1991; Bull, 1996), and can be represented logarithmically by $\\\\overline{p^{\\\\prime 2}}^{+}=A\\\\ln\\\\delta^{+}+\\\\mathrm{const}$ , where $\\\\delta^{+}\\\\equiv\\\\delta u_{\\\\tau}/\\\\nu$ is the friction Reynolds number built from the boundary-layer thickness $\\\\delta$ , the friction velocity $u_{\\\\tau}$ and the kinematic viscosity $\\\\nu$ , and superscript $+$ denotes viscous scaling throughout, with $A\\\\approx 2.3$ from fits to the data of Schlatter and Örlü (2010); Lee and Moser (2015) (see §5) and the analyses of Panton et al. (2017); Massey et al. (2026c). This growth is the wall imprint of pressure sources distributed across the logarithmic layer, whose active range widens as $\\\\ln\\\\delta^{+}$ . In a source-term reading the fluctuating kinematic pressure $\\\\pi^{\\\\prime}$ satisfies the Poisson equation\",\"tag\":null,\"text\":\"is the wall imprint of pressure sources distributed across the logarithmic layer, whose active range widens as ln 𝛿 + In a source-term reading the fluctuating kinematic pressure 𝜋 ′ satisfies the Poisson equation\"},{\"bbox\":[142.45,132.08,379.55,26.61],\"bbox_pdf\":[142.45,633.31,522.0,659.92],\"fragments\":[{\"bbox_pdf\":[142.45,633.31,522.0,659.92],\"words\":[464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\partial_{i}\\\\partial_{i}\\\\pi^{\\\\prime}=\\\\mathcal{L}+\\\\mathcal{Q},\\\\qquad\\\\mathcal{L}%\\n=-2(\\\\partial_{j}\\\\bar{U}_{i})(\\\\partial_{i}u^{\\\\prime}_{j}),\\\\qquad\\\\mathcal{Q}=-%\\n\\\\partial_{i}\\\\partial_{j}\\\\!\\\\left(u^{\\\\prime}_{i}u^{\\\\prime}_{j}-\\\\overline{u^{%\\n\\\\prime}_{i}u^{\\\\prime}_{j}}\\\\right),\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S1.E1\",\"source_text\":\"$\\\\partial_{i}\\\\partial_{i}\\\\pi^{\\\\prime}=\\\\mathcal{L}+\\\\mathcal{Q},\\\\qquad\\\\mathcal{L}% =-2(\\\\partial_{j}\\\\bar{U}_{i})(\\\\partial_{i}u^{\\\\prime}_{j}),\\\\qquad\\\\mathcal{Q}=-% \\\\partial_{i}\\\\partial_{j}\\\\!\\\\left(u^{\\\\prime}_{i}u^{\\\\prime}_{j}-\\\\overline{u^{% \\\\prime}_{i}u^{\\\\prime}_{j}}\\\\right),$ (1)\",\"tag\":\"(1)\",\"text\":\"(︂ )︂ 𝜕𝑖𝜕𝑖𝜋 ′ = L + Q, L = −2(𝜕𝑗𝑈𝑖) (𝜕𝑖𝑢¯′ Q = −𝜕𝑖𝜕𝑗 𝑢 ′ ′ ′ ′ , (1) 𝑗), 𝑖𝑢 𝑗 − 𝑢𝑖𝑢 𝑗\"},{\"bbox\":[90.0,169.88,432.0,94.42],\"bbox_pdf\":[90.0,527.7,522.0,622.12],\"fragments\":[{\"bbox_pdf\":[90.0,527.7,522.0,622.12],\"words\":[492,493,497,494,495,496,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph8\",\"source_text\":\"with $\\\\bar{U}_{i}$ the mean velocity, $u^{\\\\prime}_{i}$ the velocity fluctuation, $\\\\partial_{i}\\\\equiv\\\\partial/\\\\partial x_{i}$ and an overbar the time average: the variance separates into a linear source $\\\\mathcal{L}$ (the product of mean shear and fluctuating strain; the ‘rapid’ source of the pressure-strain literature) and a nonlinear source $\\\\mathcal{Q}$ quadratic in the fluctuations (‘slow’) (Kim, 1989; Chang et al., 1999; Anantharamu and Mahesh, 2020). In the smooth-wall two-component model of Massey et al. (2026c) the $\\\\ln\\\\delta^{+}$ growth is carried by the nonlinear source over the inertial range (Massey et al., 2026b), while the linear source sets a near-wall offset (Massey et al., 2026a).\",\"tag\":null,\"text\":\"with 𝑈𝑖 𝑢¯′ the mean velocity, 𝑖 the velocity fluctuation, 𝜕𝑖 ≡ 𝜕/𝜕𝑥𝑖 and an overbar the time average: the variance separates into a linear source L (the product of mean shear and fluctuating strain; the ‘rapid’ source of the pressure-strain literature) and a nonlinear source Q quadratic in the fluctuations (‘slow’) [Kim, 1989, Chang et al., 1999, Anantharamu and Mahesh, 2020]. In the smooth-wall two-component model of Massey et al. [2026c] the ln 𝛿 + growth is carried by the nonlinear source over the inertial range [Massey et al., 2026b], while the linear source sets a near-wall offset [Massey et al., 2026a].\"},{\"bbox\":[90.0,266.94,432.01,227.7],\"bbox_pdf\":[90.0,297.36,522.01,525.06],\"fragments\":[{\"bbox_pdf\":[90.0,297.36,522.01,525.06],\"words\":[595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.996,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph9\",\"source_text\":\"Over a rough wall the picture must change where the roughness changes the flow. Once fully rough ( $k_{s}^{+}\\\\gg 1$ , $k_{s}/\\\\delta\\\\ll 1$ , with $k_{s}$ the equivalent sandgrain roughness) the viscous length is buried within the roughness sublayer and outer-layer similarity holds above it (Townsend, 1976; Raupach et al., 1991; Jiménez, 2004; Schultz and Flack, 2007; Squire et al., 2016): the mean shear keeps its log-layer form $\\\\mathrm{d}U^{+}/\\\\mathrm{d}y_{d}^{+}=1/(\\\\kappa y_{d}^{+})$ , with $\\\\kappa$ the von Kármán constant and $y_{d}$ the wall-normal coordinate displaced by the zero-plane height (§2), shifted only by the roughness function. Rough-wall pressure spectra have been measured and modelled extensively. Blake (1970) introduced roughness inner scaling for the high-frequency range; Meyers et al. (2015) identified a three-region structure with a $k_{s}$ -scaled intermediate band; and empirical Goody-type models (Goody, 2004; Catlett et al., 2022; Joseph et al., 2022; Fritsch et al., 2023) extend the smooth-wall spectrum to rough walls, differing chiefly in which roughness parameter they carry. A parallel theoretical line treats the rough-wall spectrum at low (acoustic and sub-convective) wavenumbers through enhanced source strengths and redistributed source near-fields (Howe, 1988; Glegg and Devenport, 2009); it anticipates the source-density-times-Green-weighting partition used below but targets the scattering-dominated far field rather than the energetic convective peak, and we do not pursue it (cf. the sublayer-over-scattering assessment of Totten et al. 2026). What is missing is a simple account of what sets the growth of the variance once the wall is fully rough.\",\"tag\":null,\"text\":\"Over a rough wall the picture must change where the roughness changes the flow. Once fully rough (𝑘 + 𝑠 ≫ 1, 𝑘𝑠/𝛿 ≪ 1, with 𝑘𝑠 the equivalent sandgrain roughness) the viscous length is buried within the roughness sublayer and outer-layer similarity holds above it [Townsend, 1976, Raupach et al., 1991, Jim´enez, 2004, Schultz and Flack, 2007, Squire et al., 2016]: the mean shear keeps its loglayer form d𝑈 + /d𝑦 + = 1/(𝜅𝑦 + with 𝜅 the von K´arm´an constant and 𝑦𝑑 the wall-normal coordinate 𝑑 𝑑 displaced by the zero-plane height (§2), shifted only by the roughness function. Rough-wall pressure spectra have been measured and modelled extensively. Blake [1970] introduced roughness inner scaling for the high-frequency range; Meyers et al. [2015] identified a three-region structure with a 𝑘𝑠-scaled intermediate band; and empirical Goody-type models [Goody, 2004, Catlett et al., 2022, Joseph et al., 2022, Fritsch et al., 2023] extend the smooth-wall spectrum to rough walls, differing chiefly in which roughness parameter they carry. A parallel theoretical line treats the rough-wall spectrum at low (acoustic and sub-convective) wavenumbers through enhanced source strengths and redistributed source near-fields [Howe, 1988, Glegg and Devenport, 2009]; it anticipates the sourcedensity-times-Green-weighting partition used below but targets the scattering-dominated far field rather than the energetic convective peak, and we do not pursue it (cf. the sublayer-over-scattering assessment of Totten et al. 2026). What is missing is a simple account of what sets the growth of the variance once the wall is fully rough.\"},{\"bbox\":[90.0,497.28,432.01,200.6],\"bbox_pdf\":[90.0,94.12,522.01,294.72],\"fragments\":[{\"bbox_pdf\":[90.0,94.12,522.01,294.72],\"words\":[847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph10\",\"source_text\":\"This work proposes that account and makes two contributions. First, we carry the source-term interpretation of the wall-pressure spectrum to a fully rough wall, following its linear and nonlinear sources separately and asking how each rescales on its natural length: the linear source becomes roughness-local and scales on $k_{s}$ , while the nonlinear source acts over a logarithmic wall-normal range now cut off below by the roughness sublayer rather than by the viscous scale. Second, and as a direct corollary, the Reynolds-number growth of the integrated variance must count the geometric range $\\\\eta_{s}\\\\equiv\\\\ln(\\\\delta/k_{s})$ in place of $\\\\ln\\\\delta^{+}$ . The reasoning is elementary: through the transitional band the viscous length $\\\\nu/u_{\\\\tau}$ and the roughness $k_{s}$ compete for the lower cutoff of the active source range, but once fully rough the viscous length is buried in the sublayer, $k_{s}^{+}$ loses independent content, and $\\\\delta/k_{s}$ inherits the counting role of $\\\\delta^{+}$ . This is the same outer-layer similarity that collapses the mean velocity profile, applied now to a range-counting quantity — a consequence of standard fully rough phenomenology, not a rival scaling. The same similarity then fixes the rate: any canopy influence is set by $k_{s}$ , so it decays in $y_{d}/k_{s}$ and integrates over the logarithmic measure to a constant, leaving the growth coefficient at its smooth-wall value and confining roughness to an additive offset. Two premises are inherited and stated plainly at the outset: attributing the smooth-wall slope $A\\\\approx 2.3$ to the nonlinear source over the log layer is imported from the smooth-wall analysis of Massey et al. (2026b), and the split of the spectrum into a linear and a nonlinear part is an interpretation, not something the wall pressure measures directly.\",\"tag\":null,\"text\":\"This work proposes that account and makes two contributions. First, we carry the source-term interpretation of the wall-pressure spectrum to a fully rough wall, following its linear and nonlinear sources separately and asking how each rescales on its natural length: the linear source becomes roughness-local and scales on 𝑘𝑠, while the nonlinear source acts over a logarithmic wall-normal range now cut off below by the roughness sublayer rather than by the viscous scale. Second, and as a direct corollary, the Reynolds-number growth of the integrated variance must count the geometric range 𝜂𝑠 ≡ ln(𝛿/𝑘𝑠) in place of ln 𝛿 + The reasoning is elementary: through the transitional band the viscous length 𝜈/𝑢𝜏 and the roughness 𝑘𝑠 compete for the lower cutoff of the active source range, but once fully rough the viscous length is buried in the sublayer, 𝑘 + 𝑠 loses independent content, and 𝛿/𝑘𝑠 inherits the counting role of 𝛿 + This is the same outer-layer similarity that collapses the mean velocity profile, applied now to a range-counting quantity a consequence of standard fully rough phenomenology, not a rival scaling. The same similarity then fixes the rate: any canopy influence is set by 𝑘𝑠, so it decays in 𝑦𝑑/𝑘𝑠 and integrates over the logarithmic measure to a constant, leaving the growth coefficient at its smooth-wall value and confining roughness to an additive offset. Two premises are inherited and stated plainly at the outset: attributing the smooth-wall slope 𝐴 ≈ 2.3 to\"},{\"bbox\":[303.27,720.98,5.45,10.91],\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"fragments\":[{\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"words\":[9381]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":5,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"2\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0001.pdf\",\"page_no\":2,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph9\",\"c_block\":0.996,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.998,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.996,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"6a4e6d6539101fd47df831f8f4a6e5d0c948bc01a1c644c371ece8335ae3c8d2\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6a4e6d6539101fd47df831f8f4a6e5d0c948bc01a1c644c371ece8335ae3c8d2.pdf","asset_sha256":"6a4e6d6539101fd47df831f8f4a6e5d0c948bc01a1c644c371ece8335ae3c8d2","document_id":"arxiv:2608.16729","kind":"page","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[142.45,132.08,379.55,26.61],\"bbox_pdf\":[142.45,633.31,522.0,659.92],\"fragments\":[{\"bbox_pdf\":[142.45,633.31,522.0,659.92],\"words\":[464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\partial_{i}\\\\partial_{i}\\\\pi^{\\\\prime}=\\\\mathcal{L}+\\\\mathcal{Q},\\\\qquad\\\\mathcal{L}%\\n=-2(\\\\partial_{j}\\\\bar{U}_{i})(\\\\partial_{i}u^{\\\\prime}_{j}),\\\\qquad\\\\mathcal{Q}=-%\\n\\\\partial_{i}\\\\partial_{j}\\\\!\\\\left(u^{\\\\prime}_{i}u^{\\\\prime}_{j}-\\\\overline{u^{%\\n\\\\prime}_{i}u^{\\\\prime}_{j}}\\\\right),\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S1.E1\",\"source_text\":\"$\\\\partial_{i}\\\\partial_{i}\\\\pi^{\\\\prime}=\\\\mathcal{L}+\\\\mathcal{Q},\\\\qquad\\\\mathcal{L}% =-2(\\\\partial_{j}\\\\bar{U}_{i})(\\\\partial_{i}u^{\\\\prime}_{j}),\\\\qquad\\\\mathcal{Q}=-% \\\\partial_{i}\\\\partial_{j}\\\\!\\\\left(u^{\\\\prime}_{i}u^{\\\\prime}_{j}-\\\\overline{u^{% \\\\prime}_{i}u^{\\\\prime}_{j}}\\\\right),$ (1)\",\"tag\":\"(1)\",\"text\":\"(︂ )︂ 𝜕𝑖𝜕𝑖𝜋 ′ = L + Q, L = −2(𝜕𝑗𝑈𝑖) (𝜕𝑖𝑢¯′ Q = −𝜕𝑖𝜕𝑗 𝑢 ′ ′ ′ ′ , (1) 𝑗), 𝑖𝑢 𝑗 − 𝑢𝑖𝑢 𝑗\"},\"formula_index\":1,\"parent_page_sha256\":\"6a4e6d6539101fd47df831f8f4a6e5d0c948bc01a1c644c371ece8335ae3c8d2\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6a4e6d6539101fd47df831f8f4a6e5d0c948bc01a1c644c371ece8335ae3c8d2.pdf","asset_sha256":"6a4e6d6539101fd47df831f8f4a6e5d0c948bc01a1c644c371ece8335ae3c8d2","document_id":"arxiv:2608.16729","kind":"formula","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[90.0,90.05,432.0,38.01],\"bbox_pdf\":[90.0,663.94,522.0,701.95],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[90.0,663.94,522.0,701.95],\"words\":[1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph10\",\"source_text\":\"This work proposes that account and makes two contributions. First, we carry the source-term interpretation of the wall-pressure spectrum to a fully rough wall, following its linear and nonlinear sources separately and asking how each rescales on its natural length: the linear source becomes roughness-local and scales on $k_{s}$ , while the nonlinear source acts over a logarithmic wall-normal range now cut off below by the roughness sublayer rather than by the viscous scale. Second, and as a direct corollary, the Reynolds-number growth of the integrated variance must count the geometric range $\\\\eta_{s}\\\\equiv\\\\ln(\\\\delta/k_{s})$ in place of $\\\\ln\\\\delta^{+}$ . The reasoning is elementary: through the transitional band the viscous length $\\\\nu/u_{\\\\tau}$ and the roughness $k_{s}$ compete for the lower cutoff of the active source range, but once fully rough the viscous length is buried in the sublayer, $k_{s}^{+}$ loses independent content, and $\\\\delta/k_{s}$ inherits the counting role of $\\\\delta^{+}$ . This is the same outer-layer similarity that collapses the mean velocity profile, applied now to a range-counting quantity — a consequence of standard fully rough phenomenology, not a rival scaling. The same similarity then fixes the rate: any canopy influence is set by $k_{s}$ , so it decays in $y_{d}/k_{s}$ and integrates over the logarithmic measure to a constant, leaving the growth coefficient at its smooth-wall value and confining roughness to an additive offset. Two premises are inherited and stated plainly at the outset: attributing the smooth-wall slope $A\\\\approx 2.3$ to the nonlinear source over the log layer is imported from the smooth-wall analysis of Massey et al. (2026b), and the split of the spectrum into a linear and a nonlinear part is an interpretation, not something the wall pressure measures directly.\",\"tag\":null,\"text\":\"the nonlinear source over the log layer is imported from the smooth-wall analysis of Massey et al. [2026b], and the split of the spectrum into a linear and a nonlinear part is an interpretation, not something the wall pressure measures directly.\"},{\"bbox\":[90.0,130.7,432.01,119.3],\"bbox_pdf\":[90.0,542.0,522.01,661.3],\"fragments\":[{\"bbox_pdf\":[90.0,542.0,522.01,661.3],\"words\":[1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.992,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph11\",\"source_text\":\"The wall-pressure spectrum can be read as the wall imprint of these two mechanisms. Over a rough wall the mean shear is largest where the canopy lifts the velocity onto its log profile, so $\\\\mathcal{L}$ is concentrated in the roughness sublayer, scales on the roughness height, and produces a spectral feature at roughness-set frequencies. The growth-relevant amplitude of $\\\\mathcal{Q}$ lives over the inertial region above the sublayer, and the $y^{-1}$ decay there supplies the energetic outer peak at frequencies set by $\\\\delta$ (Massey et al., 2026c, b). The viscous (Stokes) pressure also contributes through the treatment of the boundary condition, but is subdominant, as discussed in §2. Figure 1 sketches this two-population structure and how the Reynolds-number growth attaches to the nonlinear population alone.\",\"tag\":null,\"text\":\"The wall-pressure spectrum can be read as the wall imprint of these two mechanisms. Over a rough wall the mean shear is largest where the canopy lifts the velocity onto its log profile, so L is concentrated in the roughness sublayer, scales on the roughness height, and produces a spectral feature at roughness-set frequencies. The growth-relevant amplitude of Q lives over the inertial region above the sublayer, and the 𝑦 −1 decay there supplies the energetic outer peak at frequencies set by 𝛿 [Massey et al., 2026c,b]. The viscous (Stokes) pressure also contributes through the treatment of the boundary condition, but is subdominant, as discussed in §2. Figure 1 sketches this two-population structure and how the Reynolds-number growth attaches to the nonlinear population alone.\"},{\"bbox\":[90.0,270.42,168.74,14.35],\"bbox_pdf\":[90.0,507.24,258.74,521.58],\"fragments\":[{\"bbox_pdf\":[90.0,507.24,258.74,521.58],\"words\":[1258,1259,1260]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"heading\",\"source_ref\":\"title12\",\"source_text\":\"2 Source-term framework\",\"tag\":null,\"text\":\"2 Source-term framework\"},{\"bbox\":[90.0,298.69,432.0,105.75],\"bbox_pdf\":[90.0,387.55,522.0,493.31],\"fragments\":[{\"bbox_pdf\":[90.0,387.55,522.0,493.31],\"words\":[1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.99,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph13\",\"source_text\":\"In smooth-wall canonical flows, source-term analyses associate the near-wall $O(1)$ offset of the wall-pressure variance primarily with the linear pressure source $\\\\mathcal{L}$ and the inertial-layer contribution with the nonlinear source $\\\\mathcal{Q}$ (Kim, 1989; Chang et al., 1999; Panton et al., 2017; Anantharamu and Mahesh, 2020; Massey et al., 2026b, a); Chang et al. (1999) computed exactly this Green-weighted linear/nonlinear partition in channel flow. Here we posit how each source rescales in a fully rough turbulent boundary layer when roughness is represented by the equivalent sandgrain roughness $k_{s}$ , and use that scaling to construct a closed-form two-lognormal premultiplied spectrum with predicted break frequencies.\",\"tag\":null,\"text\":\"In smooth-wall canonical flows, source-term analyses associate the near-wall 𝑂 (1) offset of the wallpressure variance primarily with the linear pressure source L and the inertial-layer contribution with the nonlinear source Q [Kim, 1989, Chang et al., 1999, Panton et al., 2017, Anantharamu and Mahesh, 2020, Massey et al., 2026b,a]; Chang et al. [1999] computed exactly this Green-weighted linear/nonlinear partition in channel flow. Here we posit how each source rescales in a fully rough turbulent boundary layer when roughness is represented by the equivalent sandgrain roughness 𝑘𝑠, and use that scaling to construct a closed-form two-lognormal premultiplied spectrum with predicted break frequencies.\"},{\"bbox\":[90.0,406.07,432.01,98.16],\"bbox_pdf\":[90.0,287.78,522.01,385.93],\"fragments\":[{\"bbox_pdf\":[90.0,287.78,522.01,385.93],\"words\":[1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474,1475,1476,1477,1478,1479,1480,1481,1482,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph14\",\"source_text\":\"Let the roughness geometry be fixed, let $k_{s}^{+}\\\\gg 1$ , and assume $k_{s}/\\\\delta\\\\ll 1$ . The displaced wall-normal coordinate is $y_{d}=y-d$ , where $d$ is the displacement height, with $x_{2}\\\\equiv y_{d}$ in the sources of (1). Above the roughness sublayer the mean field reduces to $\\\\bar{U}_{i}=U(y_{d})\\\\,\\\\delta_{i1}$ , so $\\\\mathcal{L}=-2(dU/dy_{d})\\\\,\\\\partial_{x}v^{\\\\prime}$ ; using incompressibility the nonlinear source admits the local form $\\\\mathcal{Q}=-(\\\\partial_{j}u^{\\\\prime}_{k})(\\\\partial_{k}u^{\\\\prime}_{j})=\\\\tfrac{1% }{2}\\\\omega_{i}\\\\omega_{i}-s_{ij}s_{ij}$ , the difference between fluctuating enstrophy and strain magnitude squared, with $\\\\omega_{i}$ the fluctuating vorticity and $s_{ij}$ the fluctuating strain rate. The two forms of $\\\\mathcal{Q}$ differ by the non-fluctuating term $\\\\partial_{i}\\\\partial_{j}\\\\overline{u^{\\\\prime}_{i}u^{\\\\prime}_{j}}$ , which contributes only to the mean pressure.\",\"tag\":null,\"text\":\"Let the roughness geometry be fixed, let 𝑘 + ≫ 1, and assume 𝑘𝑠/𝛿 ≪ 1. The displaced 𝑠 wall-normal coordinate is 𝑦𝑑 = 𝑦 − 𝑑, where 𝑑 is the displacement height, with 𝑥2 ≡ 𝑦𝑑 in the sources of (1). Above the roughness sublayer the mean field reduces to 𝑈𝑖¯ = 𝑈 (𝑦𝑑) 𝛿𝑖1, so L = −2(𝑑𝑈/𝑑𝑦𝑑) 𝜕𝑥𝑣 ′ using incompressibility the nonlinear source admits the local form Q = −(𝜕𝑗𝑢 ′ ) (𝜕𝑘𝑢 ′ = 1 𝑘 𝑗) magnitude squared, with 𝜔𝑖 the fluctuating vorticity and 𝑠𝑖 𝑗 the fluctuating strain rate. The two 2𝜔𝑖𝜔𝑖 − 𝑠𝑖 𝑗𝑠𝑖 𝑗, the difference between fluctuating enstrophy and strain forms of Q differ by the non-fluctuating term 𝜕𝑖𝜕𝑗𝑢 ′ ′ 𝑖𝑢 𝑗, which contributes only to the mean pressure.\"},{\"bbox\":[90.0,503.22,432.0,24.46],\"bbox_pdf\":[90.0,264.32,522.0,288.78],\"fragments\":[{\"bbox_pdf\":[90.0,264.32,522.0,288.78],\"words\":[1495,1496,1497,1498,1499,1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512,1513,1514,1515]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph15\",\"source_text\":\"For a wall-parallel wavenumber vector $\\\\bm{k}=(k_{x},k_{z})$ with magnitude $k=|\\\\bm{k}|$ , the wall pressure is the Green projection\",\"tag\":null,\"text\":\"For a wall-parallel wavenumber vector k = (𝑘𝑥, 𝑘𝑧) with magnitude 𝑘 = |k|, the wall pressure is the Green projection\"},{\"bbox\":[125.03,538.56,396.97,27.93],\"bbox_pdf\":[125.03,225.52,522.0,253.44],\"fragments\":[{\"bbox_pdf\":[125.03,225.52,522.0,253.44],\"words\":[1516,1517,1518,1519,1520,1521,1522,1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534,1535,1536,1537,1538,1539,1540,1541,1542,1543,1544,1545,1546,1547,1548,1549,1550]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\widehat{p}^{\\\\prime}(\\\\bm{k},t)=\\\\int_{0}^{\\\\delta}G(k,y_{d})\\\\left[\\\\widehat{%\\n\\\\mathcal{L}}(\\\\bm{k},y_{d},t)+\\\\widehat{\\\\mathcal{Q}}(\\\\bm{k},y_{d},t)\\\\right]%\\n\\\\mathrm{d}y_{d},\\\\qquad G(k,y_{d})\\\\sim-\\\\frac{e^{-ky_{d}}}{k},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S2.E2\",\"source_text\":\"$\\\\widehat{p}^{\\\\prime}(\\\\bm{k},t)=\\\\int_{0}^{\\\\delta}G(k,y_{d})\\\\left[\\\\widehat{% \\\\mathcal{L}}(\\\\bm{k},y_{d},t)+\\\\widehat{\\\\mathcal{Q}}(\\\\bm{k},y_{d},t)\\\\right]% \\\\mathrm{d}y_{d},\\\\qquad G(k,y_{d})\\\\sim-\\\\frac{e^{-ky_{d}}}{k},$ (2)\",\"tag\":\"(2)\",\"text\":\"∫ 𝛿 [︂ ]︂ −𝑘 𝑦𝑑 ′ ˆ︁ 𝑒 𝑝ˆ︁ (k, 𝑡) = 𝐺 (𝑘, 𝑦𝑑) L (k, 𝑦𝑑, 𝑡) + ˆ︁Q (k, 𝑦𝑑, 𝑡) d𝑦𝑑, 𝐺 (𝑘, 𝑦𝑑) ∼ − , (2) 0 𝑘\"},{\"bbox\":[90.0,572.69,432.0,129.31],\"bbox_pdf\":[90.0,90.0,522.0,219.31],\"fragments\":[{\"bbox_pdf\":[90.0,90.0,522.0,219.31],\"words\":[1551,1552,1553,1554,1555,1556,1557,1558,1559,1560,1561,1562,1563,1564,1565,1566,1567,1568,1569,1570,1571,1572,1573,1574,1575,1576,1577,1578,1579,1580,1581,1582,1583,1584,1585,1586,1587,1588,1589,1590,1591,1592,1593,1594,1595,1597,1598,1599,1600,1602,1603,1604,1605,1606,1607,1608,1609,1610,1611,1612,1613,1614,1615,1616,1618,1619,1620,1621,1622,1623,1624,1625,1627,1628,1629,1630,1631,1632,1633,1634,1635,1636,1637,1638,1639,1640,1641,1642,1643,1644,1645,1646,1647,1648,1650,1651,1652,1653,1654,1655,1656,1657,1658,1659,1660,1661,1662,1663,1664,1665,1666,1667,1668,1669,1670,1671,1672,1673,1674,1675,1676,1677,1678,1679,1680,1681,1682,1683,1684,1685,1686,1687,1688,1689,1690,1691,1692,1693,1694,1695,1696,1697,1698,1699,1700,1701,1702,1703,1704,1705,1706,1707,1708,1709,1710,1711,1712,1713,1714,1715,1716,1717,1718,1719,1720,1721,1722,1723]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.993,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph17\",\"source_text\":\"in the single-wall or large- $k\\\\delta$ limit. Throughout, $\\\\widehat{p}^{\\\\prime}=\\\\pi^{\\\\prime}_{y=0}$ is the wall trace of the pressure fluctuation, and superscript $+$ denotes viscous scaling ( $\\\\overline{p^{\\\\prime 2}}^{+}\\\\equiv\\\\overline{p^{\\\\prime 2}}/u_{\\\\tau}^{4}$ for the kinematic variable, and likewise the premultiplied spectrum ${f\\\\phi_{pp}}^{+}$ , with $\\\\phi_{pp}(f)$ the single-point frequency spectrum of the wall pressure at frequency $f$ , normalised so that $\\\\overline{p^{\\\\prime 2}}=\\\\int_{0}^{\\\\infty}\\\\phi_{pp}\\\\,\\\\mathrm{d}f$ ). The wall signal is a Green-weighted integral of wall-normal source densities; its variance involves $|G|^{2}\\\\sim e^{-2ky_{d}}/k^{2}$ , so sources at height $y_{d}$ contribute to the wall with a factor that decays exponentially in $ky_{d}$ . What is used below is this wall-normal attenuation factor $e^{-ky_{d}}$ , the dependence on $ky_{d}$ that selects which sources reach the wall; we do not rely on its accuracy as a $k\\\\delta\\\\to\\\\infty$ approximation at the energetic peak, where $k\\\\delta\\\\sim O(1)$ . This elliptic attenuation is the central mechanism by which the wall-normal location of each source, and any wallward redistribution imposed by the canopy, enters the wall-projected variance.\",\"tag\":null,\"text\":\"in the single-wall or large-𝑘𝛿 limit. Throughout, 𝑝ˆ︁ ′ = 𝜋 ′ is the wall trace of the pressure 𝑦=0 + fluctuation, and superscript + denotes viscous scaling (𝑝′2 ≡ 𝑝′2/𝑢4 𝜏 for the kinematic variable, and likewise the premultiplied spectrum 𝑓 𝜙𝑝 𝑝 + with 𝜙𝑝 𝑝( 𝑓 the single-point frequency spectrum ∫ ∞ ′2 = of the wall pressure at frequency 𝑓 normalised so that 𝑝 𝜙𝑝 𝑝 d 𝑓 The wall signal is a 0 Green-weighted integral of wall-normal source densities; its variance involves |𝐺 | 2 ∼ 𝑒 −2𝑘 𝑦𝑑/𝑘2 so sources at height 𝑦𝑑 contribute to the wall with a factor that decays exponentially in 𝑘 𝑦𝑑. What is used below is this wall-normal attenuation factor 𝑒 −𝑘 𝑦𝑑, the dependence on 𝑘 𝑦𝑑 that selects which sources reach the wall; we do not rely on its accuracy as a 𝑘𝛿 → ∞ approximation at the energetic peak, where 𝑘𝛿 ∼ 𝑂 (1). This elliptic attenuation is the central mechanism by which the\"},{\"bbox\":[303.27,720.98,5.45,10.91],\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"fragments\":[{\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"words\":[9382]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":8,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"3\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0002.pdf\",\"page_no\":3,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph13\",\"c_block\":0.99,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.997,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.99,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"d4e89cb23f95ad8d0f64502186bbbbd887f0c9728f1de1b04322a815b3dcced1\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/d4e89cb23f95ad8d0f64502186bbbbd887f0c9728f1de1b04322a815b3dcced1.pdf","asset_sha256":"d4e89cb23f95ad8d0f64502186bbbbd887f0c9728f1de1b04322a815b3dcced1","document_id":"arxiv:2608.16729","kind":"page","page_number":3,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[125.03,538.56,396.97,27.93],\"bbox_pdf\":[125.03,225.52,522.0,253.44],\"fragments\":[{\"bbox_pdf\":[125.03,225.52,522.0,253.44],\"words\":[1516,1517,1518,1519,1520,1521,1522,1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534,1535,1536,1537,1538,1539,1540,1541,1542,1543,1544,1545,1546,1547,1548,1549,1550]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\widehat{p}^{\\\\prime}(\\\\bm{k},t)=\\\\int_{0}^{\\\\delta}G(k,y_{d})\\\\left[\\\\widehat{%\\n\\\\mathcal{L}}(\\\\bm{k},y_{d},t)+\\\\widehat{\\\\mathcal{Q}}(\\\\bm{k},y_{d},t)\\\\right]%\\n\\\\mathrm{d}y_{d},\\\\qquad G(k,y_{d})\\\\sim-\\\\frac{e^{-ky_{d}}}{k},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S2.E2\",\"source_text\":\"$\\\\widehat{p}^{\\\\prime}(\\\\bm{k},t)=\\\\int_{0}^{\\\\delta}G(k,y_{d})\\\\left[\\\\widehat{% \\\\mathcal{L}}(\\\\bm{k},y_{d},t)+\\\\widehat{\\\\mathcal{Q}}(\\\\bm{k},y_{d},t)\\\\right]% \\\\mathrm{d}y_{d},\\\\qquad G(k,y_{d})\\\\sim-\\\\frac{e^{-ky_{d}}}{k},$ (2)\",\"tag\":\"(2)\",\"text\":\"∫ 𝛿 [︂ ]︂ −𝑘 𝑦𝑑 ′ ˆ︁ 𝑒 𝑝ˆ︁ (k, 𝑡) = 𝐺 (𝑘, 𝑦𝑑) L (k, 𝑦𝑑, 𝑡) + ˆ︁Q (k, 𝑦𝑑, 𝑡) d𝑦𝑑, 𝐺 (𝑘, 𝑦𝑑) ∼ − , (2) 0 𝑘\"},\"formula_index\":6,\"parent_page_sha256\":\"d4e89cb23f95ad8d0f64502186bbbbd887f0c9728f1de1b04322a815b3dcced1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/d4e89cb23f95ad8d0f64502186bbbbd887f0c9728f1de1b04322a815b3dcced1.pdf","asset_sha256":"d4e89cb23f95ad8d0f64502186bbbbd887f0c9728f1de1b04322a815b3dcced1","document_id":"arxiv:2608.16729","kind":"formula","page_number":3,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[90.0,86.61,246.51,14.35],\"bbox_pdf\":[90.0,691.04,336.51,705.39],\"fragments\":[{\"bbox_pdf\":[90.0,691.04,336.51,705.39],\"words\":[2173,2174,2175,2176,2177,2178,2179,2180]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"heading\",\"source_ref\":\"title31\",\"source_text\":\"3 Linear source over a fully rough wall\",\"tag\":null,\"text\":\"3 Linear source over a fully rough wall\"},{\"bbox\":[90.0,114.89,432.0,65.11],\"bbox_pdf\":[90.0,612.0,522.0,677.11],\"fragments\":[{\"bbox_pdf\":[90.0,612.0,522.0,677.11],\"words\":[2181,2182,2183,2184,2185,2186,2187,2188,2189,2190,2191,2192,2193,2194,2195,2196,2197,2198,2199,2200,2201,2202,2204,2205,2206,2207,2208,2209,2210,2211,2212,2213,2214,2215,2216,2217,2218,2219,2220,2221,2222,2223,2224,2225,2226,2227,2228,2229,2230,2231,2232,2233,2234,2235,2236,2237,2238,2239,2240]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph32\",\"source_text\":\"Above the roughness sublayer the displaced mean velocity keeps the smooth-wall logarithmic form, shifted down only by the roughness function $\\\\Delta U^{+}$ . This persistence of the log law above the roughness is Townsend’s outer-layer similarity hypothesis (Townsend, 1976), well supported for the mean flow across and below the present roughness range (Schultz and Flack, 2007; Mehdi et al., 2013),\",\"tag\":null,\"text\":\"Above the roughness sublayer the displaced mean velocity keeps the smooth-wall logarithmic form, shifted down only by the roughness function Δ𝑈 + This persistence of the log law above the roughness is Townsend’s outer-layer similarity hypothesis [Townsend, 1976], well supported for the mean flow across and below the present roughness range [Schultz and Flack, 2007, Mehdi et al., 2013],\"},{\"bbox\":[173.73,181.07,348.27,15.79],\"bbox_pdf\":[173.73,595.14,522.0,610.93],\"fragments\":[{\"bbox_pdf\":[173.73,595.14,522.0,610.93],\"words\":[2241,2242,2243,2244,2245,2246,2247,2248,2249,2250,2251,2252,2253,2254,2255,2256,2257,2258,2259,2260,2261,2262,2263,2264,2265,2266,2267,2268,2269,2270]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"U^{+}(y_{d}^{+})\\\\approx\\\\kappa^{-1}\\\\ln y_{d}^{+}+B-\\\\Delta U^{+},\\\\qquad\\\\Delta U^%\\n{+}\\\\approx\\\\kappa^{-1}\\\\ln k_{s}^{+}+B_{s},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S3.E3\",\"source_text\":\"$U^{+}(y_{d}^{+})\\\\approx\\\\kappa^{-1}\\\\ln y_{d}^{+}+B-\\\\Delta U^{+},\\\\qquad\\\\Delta U^% {+}\\\\approx\\\\kappa^{-1}\\\\ln k_{s}^{+}+B_{s},$ (3)\",\"tag\":\"(3)\",\"text\":\"𝑈 + (𝑦 + −1 ln 𝑦 + + , Δ𝑈 + ≈ 𝜅 −1 ln 𝑘 + (3) 𝑠 + 𝐵𝑠, 𝑑) ≈ 𝜅 𝑑 + 𝐵 − Δ𝑈\"},{\"bbox\":[90.0,201.64,432.0,40.67],\"bbox_pdf\":[90.0,549.7,522.0,590.36],\"fragments\":[{\"bbox_pdf\":[90.0,549.7,522.0,590.36],\"words\":[2271,2272,2273,2274,2275,2276,2277,2278,2279,2280,2281,2282,2283,2284,2285,2286,2287,2288,2289,2290,2291,2292,2293,2294,2295,2296,2297,2298,2299,2300,2301,2302,2303,2304,2305,2306,2307,2308,2309,2310,2311,2312,2313,2314,2315,2316,2317,2318,2319,2320,2321,2322,2323,2324]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph34\",\"source_text\":\"the roughness function $\\\\Delta U^{+}$ being the downward shift of the log law with roughness, in the form first formulated by Hama (1954). Equivalently $U^{+}(y_{d})\\\\approx\\\\kappa^{-1}\\\\ln(y_{d}/k_{s})+B_{r}$ , with $B$ the smooth-wall log-law intercept, $B_{s}$ the fully rough intercept constant, and $B_{r}\\\\equiv B-B_{s}$ . In either representation,\",\"tag\":null,\"text\":\"the roughness function Δ𝑈 + being the downward shift of the log law with roughness, in the form first formulated by Hama [1954]. Equivalently 𝑈 + (𝑦𝑑) ≈ 𝜅 −1 ln(𝑦𝑑/𝑘𝑠) + 𝐵𝑟, with 𝐵 the smooth-wall log-law intercept, 𝐵𝑠 the fully rough intercept constant, and 𝐵𝑟 ≡ 𝐵 − 𝐵𝑠. In either representation,\"},{\"bbox\":[278.63,250.99,243.37,30.77],\"bbox_pdf\":[278.63,510.23,522.0,541.01],\"fragments\":[{\"bbox_pdf\":[278.63,510.23,522.0,541.01],\"words\":[2325,2326,2327,2328,2329,2330,2331,2332,2333,2334,2335,2336]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{\\\\partial U^{+}}{\\\\partial y_{d}^{+}}\\\\approx\\\\frac{1}{\\\\kappa y_{d}^{+}},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S3.E4\",\"source_text\":\"$\\\\frac{\\\\partial U^{+}}{\\\\partial y_{d}^{+}}\\\\approx\\\\frac{1}{\\\\kappa y_{d}^{+}},$ (4)\",\"tag\":\"(4)\",\"text\":\"𝜕𝑈 + 1 + ≈ + , (4) 𝜕𝑦 𝑑 𝜅𝑦 𝑑\"},{\"bbox\":[90.0,290.46,432.01,105.75],\"bbox_pdf\":[90.0,395.79,522.01,501.54],\"fragments\":[{\"bbox_pdf\":[90.0,395.79,522.01,501.54],\"words\":[2337,2338,2339,2340,2341,2342,2343,2344,2345,2346,2347,2348,2349,2350,2351,2352,2353,2354,2355,2356,2357,2358,2359,2360,2361,2363,2364,2365,2366,2367,2368,2369,2370,2371,2372,2373,2374,2375,2376,2377,2378,2379,2380,2381,2382,2383,2384,2385,2386,2387,2388,2389,2390,2391,2393,2394,2395,2396,2397,2398,2399,2400,2401,2402,2403,2404,2405,2406,2407,2408,2409,2410,2411,2412,2413,2414,2415,2416,2417,2418,2419,2420,2421,2422,2423,2424,2425,2426,2427,2428,2429,2430,2431,2432,2433,2434,2435,2436,2437,2438,2439,2440,2441,2442,2443,2444,2445,2446,2447,2448,2449,2450,2451,2452,2453,2454,2455,2456,2457,2458]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph36\",\"source_text\":\"so that the log-region mean shear has the same functional form as on the smooth wall, shifted only by the additive roughness function $\\\\Delta U^{+}$ . Because $\\\\Delta U^{+}$ enters $U$ but not $dU/dy_{d}$ , it does not by itself introduce any new $\\\\delta^{+}$ dependence into the log-layer part of $\\\\mathcal{L}=-2(dU/dy_{d})\\\\,\\\\partial_{x}v^{\\\\prime}$ . A leading change in the $\\\\delta^{+}$ scaling of the linear contribution would therefore have to come from altered wall-parallel spectral weights, altered convection, or altered Green-projection into the one-dimensional frequency spectrum, not from the displaced logarithmic mean shear alone. In the absence of those spectral changes, the linear-source contribution to the inner-scaled wall-pressure variance over the rough wall (subscript $r$ ) reduces to a constant,\",\"tag\":null,\"text\":\"so that the log-region mean shear has the same functional form as on the smooth wall, shifted only by the additive roughness function Δ𝑈 + Because Δ𝑈 + enters 𝑈 but not 𝑑𝑈/𝑑𝑦𝑑, it does not by itself introduce any new 𝛿 + dependence into the log-layer part of L = −2(𝑑𝑈/𝑑𝑦𝑑) 𝜕𝑥𝑣 ′ A leading change in the 𝛿 + scaling of the linear contribution would therefore have to come from altered wallparallel spectral weights, altered convection, or altered Green-projection into the one-dimensional frequency spectrum, not from the displaced logarithmic mean shear alone. In the absence of those spectral changes, the linear-source contribution to the inner-scaled wall-pressure variance over the rough wall (subscript 𝑟) reduces to a constant,\"},{\"bbox\":[261.08,408.24,260.93,15.79],\"bbox_pdf\":[261.08,367.96,522.01,383.76],\"fragments\":[{\"bbox_pdf\":[261.08,367.96,522.01,383.76],\"words\":[2459,2460,2461,2462,2463,2464,2465,2466,2467,2468,2469,2470]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\langle p_{\\\\mathcal{L}}^{\\\\prime 2}\\\\rangle_{r}^{+}\\\\approx B_{\\\\mathcal{L}}^{r}+O%\\n(1)\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S3.E5\",\"source_text\":\"$\\\\langle p_{\\\\mathcal{L}}^{\\\\prime 2}\\\\rangle_{r}^{+}\\\\approx B_{\\\\mathcal{L}}^{r}+O% (1)$ (5)\",\"tag\":\"(5)\",\"text\":\"⟨𝑝 ′2 + ≈ 𝐵 𝑟 (5) L⟩𝑟 L + 𝑂 (1)\"},{\"bbox\":[90.0,433.3,432.01,27.12],\"bbox_pdf\":[90.0,331.59,522.01,358.7],\"fragments\":[{\"bbox_pdf\":[90.0,331.59,522.01,358.7],\"words\":[2471,2472,2473,2474,2475,2476,2477,2478,2479,2480,2481,2482,2483,2484,2485,2486,2487,2488,2489,2490,2491,2492,2493,2494,2495,2496,2497,2498,2499,2500,2501]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph38\",\"source_text\":\"as $\\\\delta^{+}\\\\to\\\\infty$ at fixed fully rough state. The constant $B_{\\\\mathcal{L}}^{r}$ is roughness dependent but is not expected to grow logarithmically with $\\\\delta^{+}$ or with $\\\\delta/k_{s}$ .\",\"tag\":null,\"text\":\"as 𝛿 + → ∞ at fixed fully rough state. The constant 𝐵 𝑟 is roughness dependent but is not expected L to grow logarithmically with 𝛿 + or with 𝛿/𝑘𝑠.\"},{\"bbox\":[89.62,461.42,432.39,146.4],\"bbox_pdf\":[89.62,184.18,522.01,330.58],\"fragments\":[{\"bbox_pdf\":[89.62,184.18,522.01,330.58],\"words\":[2502,2503,2504,2505,2506,2507,2508,2509,2510,2511,2512,2513,2514,2515,2516,2517,2518,2519,2520,2521,2522,2523,2524,2525,2526,2527,2528,2529,2530,2531,2532,2533,2534,2535,2536,2537,2538,2539,2540,2541,2542,2543,2544,2545,2546,2547,2548,2549,2550,2551,2552,2553,2554,2555,2556,2557,2558,2559,2560,2561,2562,2563,2564,2565,2566,2567,2568,2569,2570,2571,2572,2573,2574,2575,2576,2577,2578,2579,2580,2581,2582,2583,2584,2585,2586,2587,2588,2589,2590,2591,2592,2593,2594,2595,2596,2597,2598,2599,2600,2601,2602,2603,2604,2605,2606,2607,2608,2609,2610,2611,2612,2613,2614,2615,2616,2617,2618,2619,2620,2621,2622,2624,2625,2627,2628,2629,2630,2631,2632,2633,2634,2635,2636,2637,2638,2639,2640,2641,2642,2643,2644,2645,2646,2647,2648,2649,2650,2651,2652,2653,2654,2655,2656,2657,2658]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.993,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph39\",\"source_text\":\"The important linear modification is local. Within the roughness sublayer the mean field is three-dimensional (crest-scale gradients, sheltering, separation/reattachment) and the linear source couples those to fluctuations, giving a roughness-coherent contribution that replaces the smooth-wall buffer-layer feature. In the reduced model this is parameterised by $k_{s}$ : a roughness-scaled period $T^{s}\\\\equiv TU_{e}/k_{s}$ , with $T=1/f$ the period and $U_{e}$ the boundary-layer edge velocity, locates the peak and its amplitude $B_{\\\\mathcal{L}}^{r}$ is a single fitted constant. This $k_{s}$ -scaled axis follows from the work of Blake (1970) who first proposed roughness inner scaling $\\\\omega k_{s}/u_{\\\\tau}$ for the high-frequency rough-wall spectrum. Geometry sensitivity (element shape, solidity, sheltering, the dispersive source of §2, the canopy-to-wall projection) is absorbed into $B_{\\\\mathcal{L}}^{r}$ , $T_{b\\\\mathcal{L}}^{s,r}$ , $\\\\sigma_{\\\\mathcal{L}}^{r}$ for the Virginia Tech (VT) geometry of §7; the assumption is that these constants, whatever their values, do not grow with $\\\\delta/k_{s}$ at fixed roughness state.\",\"tag\":null,\"text\":\"The important linear modification is local. Within the roughness sublayer the mean field is three-dimensional (crest-scale gradients, sheltering, separation/reattachment) and the linear source couples those to fluctuations, giving a roughness-coherent contribution that replaces the smooth-wall buffer-layer feature. In the reduced model this is parameterised by 𝑘𝑠: a roughness-scaled period 𝑇 𝑠 ≡ 𝑇𝑈𝑒/𝑘𝑠, with 𝑇 = 1/ 𝑓 the period and 𝑈𝑒 the boundary-layer edge velocity, locates the peak and its amplitude 𝐵 𝑟 is a single fitted constant. This 𝑘𝑠-scaled axis follows from the work of L Blake [1970] who first proposed roughness inner scaling 𝜔𝑘𝑠/𝑢𝜏 for the high-frequency rough-wall spectrum. Geometry sensitivity (element shape, solidity, sheltering, the dispersive source of §2, the canopy-to-wall projection) is absorbed into 𝐵 𝑟 𝑇 𝑠,𝑟 𝜎 𝑟 for the Virginia Tech (VT) geometry of L 𝑏 L L §7; the assumption is that these constants, whatever their values, do not grow with 𝛿/𝑘𝑠 at fixed roughness state.\"},{\"bbox\":[90.0,628.59,266.22,14.35],\"bbox_pdf\":[90.0,149.07,356.22,163.41],\"fragments\":[{\"bbox_pdf\":[90.0,149.07,356.22,163.41],\"words\":[2659,2660,2661,2662,2663,2664,2665,2666]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"heading\",\"source_ref\":\"title40\",\"source_text\":\"4 Nonlinear source over a fully rough wall\",\"tag\":null,\"text\":\"4 Nonlinear source over a fully rough wall\"},{\"bbox\":[90.0,656.86,432.0,24.46],\"bbox_pdf\":[90.0,110.68,522.0,135.14],\"fragments\":[{\"bbox_pdf\":[90.0,110.68,522.0,135.14],\"words\":[2667,2668,2669,2670,2671,2672,2673,2674,2675,2676,2677,2678,2679,2680,2681,2682,2683,2684,2685,2686,2687,2688,2689,2690,2691,2692,2693,2694,2695,2696]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.98,\"verified\":true},\"order\":10,\"role\":\"paragraph\",\"source_ref\":\"paragraph41\",\"source_text\":\"The nonlinear source $\\\\mathcal{Q}$ is the dominant inertial-layer pressure mechanism in the smooth-wall source-term interpretation (Massey et al., 2026b, a). Over a fully rough wall the active wall-normal range of $\\\\mathcal{Q}$ is cut off below by the roughness sublayer rather than by the viscous scale, so the natural log-range parameter is\",\"tag\":null,\"text\":\"The nonlinear source Q is the dominant inertial-layer pressure mechanism in the smooth-wall sourceterm interpretation [Massey et al., 2026b,a]. Over a fully rough wall the active wall-normal range of\"},{\"bbox\":[303.27,720.98,5.45,10.91],\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"fragments\":[{\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"words\":[9384]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":11,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"5\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0004.pdf\",\"page_no\":5,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph41\",\"c_block\":0.98,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.997,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.98,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"f4e579aa273038604b864876089d97d32acac4acbe3c1a59cb4d4f9383c1ea08\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/f4e579aa273038604b864876089d97d32acac4acbe3c1a59cb4d4f9383c1ea08.pdf","asset_sha256":"f4e579aa273038604b864876089d97d32acac4acbe3c1a59cb4d4f9383c1ea08","document_id":"arxiv:2608.16729","kind":"page","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[173.73,181.07,348.27,15.79],\"bbox_pdf\":[173.73,595.14,522.0,610.93],\"fragments\":[{\"bbox_pdf\":[173.73,595.14,522.0,610.93],\"words\":[2241,2242,2243,2244,2245,2246,2247,2248,2249,2250,2251,2252,2253,2254,2255,2256,2257,2258,2259,2260,2261,2262,2263,2264,2265,2266,2267,2268,2269,2270]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"U^{+}(y_{d}^{+})\\\\approx\\\\kappa^{-1}\\\\ln y_{d}^{+}+B-\\\\Delta U^{+},\\\\qquad\\\\Delta U^%\\n{+}\\\\approx\\\\kappa^{-1}\\\\ln k_{s}^{+}+B_{s},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S3.E3\",\"source_text\":\"$U^{+}(y_{d}^{+})\\\\approx\\\\kappa^{-1}\\\\ln y_{d}^{+}+B-\\\\Delta U^{+},\\\\qquad\\\\Delta U^% {+}\\\\approx\\\\kappa^{-1}\\\\ln k_{s}^{+}+B_{s},$ (3)\",\"tag\":\"(3)\",\"text\":\"𝑈 + (𝑦 + −1 ln 𝑦 + + , Δ𝑈 + ≈ 𝜅 −1 ln 𝑘 + (3) 𝑠 + 𝐵𝑠, 𝑑) ≈ 𝜅 𝑑 + 𝐵 − Δ𝑈\"},\"formula_index\":2,\"parent_page_sha256\":\"f4e579aa273038604b864876089d97d32acac4acbe3c1a59cb4d4f9383c1ea08\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/f4e579aa273038604b864876089d97d32acac4acbe3c1a59cb4d4f9383c1ea08.pdf","asset_sha256":"f4e579aa273038604b864876089d97d32acac4acbe3c1a59cb4d4f9383c1ea08","document_id":"arxiv:2608.16729","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[278.63,250.99,243.37,30.77],\"bbox_pdf\":[278.63,510.23,522.0,541.01],\"fragments\":[{\"bbox_pdf\":[278.63,510.23,522.0,541.01],\"words\":[2325,2326,2327,2328,2329,2330,2331,2332,2333,2334,2335,2336]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{\\\\partial U^{+}}{\\\\partial y_{d}^{+}}\\\\approx\\\\frac{1}{\\\\kappa y_{d}^{+}},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S3.E4\",\"source_text\":\"$\\\\frac{\\\\partial U^{+}}{\\\\partial y_{d}^{+}}\\\\approx\\\\frac{1}{\\\\kappa y_{d}^{+}},$ (4)\",\"tag\":\"(4)\",\"text\":\"𝜕𝑈 + 1 + ≈ + , (4) 𝜕𝑦 𝑑 𝜅𝑦 𝑑\"},\"formula_index\":4,\"parent_page_sha256\":\"f4e579aa273038604b864876089d97d32acac4acbe3c1a59cb4d4f9383c1ea08\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/f4e579aa273038604b864876089d97d32acac4acbe3c1a59cb4d4f9383c1ea08.pdf","asset_sha256":"f4e579aa273038604b864876089d97d32acac4acbe3c1a59cb4d4f9383c1ea08","document_id":"arxiv:2608.16729","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[261.08,408.24,260.93,15.79],\"bbox_pdf\":[261.08,367.96,522.01,383.76],\"fragments\":[{\"bbox_pdf\":[261.08,367.96,522.01,383.76],\"words\":[2459,2460,2461,2462,2463,2464,2465,2466,2467,2468,2469,2470]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\langle p_{\\\\mathcal{L}}^{\\\\prime 2}\\\\rangle_{r}^{+}\\\\approx B_{\\\\mathcal{L}}^{r}+O%\\n(1)\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S3.E5\",\"source_text\":\"$\\\\langle p_{\\\\mathcal{L}}^{\\\\prime 2}\\\\rangle_{r}^{+}\\\\approx B_{\\\\mathcal{L}}^{r}+O% (1)$ (5)\",\"tag\":\"(5)\",\"text\":\"⟨𝑝 ′2 + ≈ 𝐵 𝑟 (5) L⟩𝑟 L + 𝑂 (1)\"},\"formula_index\":6,\"parent_page_sha256\":\"f4e579aa273038604b864876089d97d32acac4acbe3c1a59cb4d4f9383c1ea08\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/f4e579aa273038604b864876089d97d32acac4acbe3c1a59cb4d4f9383c1ea08.pdf","asset_sha256":"f4e579aa273038604b864876089d97d32acac4acbe3c1a59cb4d4f9383c1ea08","document_id":"arxiv:2608.16729","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[90.0,90.05,432.0,24.46],\"bbox_pdf\":[90.0,677.49,522.0,701.95],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[90.0,677.49,522.0,701.95],\"words\":[2697,2698,2699,2700,2701,2702,2703,2704,2705,2706,2707,2708,2709,2710,2711,2712,2713,2714,2715,2716,2717]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.98,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph41\",\"source_text\":\"The nonlinear source $\\\\mathcal{Q}$ is the dominant inertial-layer pressure mechanism in the smooth-wall source-term interpretation (Massey et al., 2026b, a). Over a fully rough wall the active wall-normal range of $\\\\mathcal{Q}$ is cut off below by the roughness sublayer rather than by the viscous scale, so the natural log-range parameter is\",\"tag\":null,\"text\":\"Q is cut off below by the roughness sublayer rather than by the viscous scale, so the natural log-range parameter is\"},{\"bbox\":[276.34,107.57,245.66,35.42],\"bbox_pdf\":[276.34,649.0,522.0,684.43],\"fragments\":[{\"bbox_pdf\":[276.34,649.0,522.0,684.43],\"words\":[2718,2719,2720,2721,2722,2723,2724,2725,2726]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\eta_{s}\\\\equiv\\\\ln\\\\!\\\\left(\\\\frac{\\\\delta}{k_{s}}\\\\right),\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S4.E6\",\"source_text\":\"$\\\\eta_{s}\\\\equiv\\\\ln\\\\!\\\\left(\\\\frac{\\\\delta}{k_{s}}\\\\right),$ (6)\",\"tag\":\"(6)\",\"text\":\"(︃ )︃ 𝛿 𝜂𝑠 ≡ ln , (6) 𝑘𝑠\"},{\"bbox\":[90.0,146.25,432.0,94.86],\"bbox_pdf\":[90.0,550.89,522.0,645.75],\"fragments\":[{\"bbox_pdf\":[90.0,550.89,522.0,645.75],\"words\":[2727,2728,2729,2730,2731,2732,2733,2734,2735,2736,2737,2738,2739,2740,2741,2742,2743,2744,2745,2746,2747,2748,2749,2750,2751,2752,2753,2754,2755,2756,2757,2758,2759,2760,2761,2762,2763,2764,2765,2766,2767,2768,2769,2770,2771,2772,2773,2774,2775,2776,2777,2778,2779,2780,2781,2782,2783,2784,2785,2786,2787,2788,2789,2790,2791,2792,2793,2794,2795,2796,2797,2798,2799,2800,2801,2802,2803,2804,2805,2806,2807,2808,2809,2810,2811,2812,2813,2814,2815,2816,2817,2818,2819,2820,2821,2822,2823,2824,2825,2826,2827,2828,2829,2830,2831,2832,2833,2834,2835,2836,2837,2838,2839,2840,2841]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph43\",\"source_text\":\"which sets the outer extent of the fully rough logarithmic source region in place of $\\\\ln\\\\delta^{+}$ on the smooth wall. This is the pressure-variance counterpart of the rough-wall outer-layer similarity measured for the velocity statistics (Squire et al., 2016): above the sublayer the mean shear and energetic turbulence take their smooth-wall log form, so the nonlinear source density the variance integrates does too, now over $[k_{s},\\\\delta]$ rather than $[\\\\nu/u_{\\\\tau},\\\\delta]$ . At leading order in the fully rough regime, $k_{s}^{+}$ does not enter $\\\\mathcal{Q}$ as an independent dynamical parameter: the viscous length is hidden inside the roughness sublayer and the active range depends on $\\\\delta$ and $k_{s}$ only through their ratio.\",\"tag\":null,\"text\":\"which sets the outer extent of the fully rough logarithmic source region in place of ln 𝛿 + on the smooth wall. This is the pressure-variance counterpart of the rough-wall outer-layer similarity measured for the velocity statistics [Squire et al., 2016]: above the sublayer the mean shear and energetic turbulence take their smooth-wall log form, so the nonlinear source density the variance integrates does too, now over [𝑘𝑠, 𝛿] rather than [𝜈/𝑢𝜏, 𝛿]. At leading order in the fully rough regime, 𝑘 + 𝑠 does not enter Q as an independent dynamical parameter: the viscous length is hidden inside the roughness sublayer and the active range depends on 𝛿 and 𝑘𝑠 only through their ratio.\"},{\"bbox\":[90.0,242.12,432.0,78.66],\"bbox_pdf\":[90.0,471.23,522.0,549.88],\"fragments\":[{\"bbox_pdf\":[90.0,471.23,522.0,549.88],\"words\":[2842,2843,2844,2845,2846,2847,2848,2849,2850,2851,2852,2853,2854,2855,2856,2857,2858,2859,2860,2861,2862,2863,2864,2865,2866,2867,2868,2869,2870,2871,2872,2873,2874,2875,2876,2877,2878,2879,2880,2881,2882,2883,2884,2885,2886,2887,2888,2889,2890,2891,2892,2893,2894,2895,2896,2897,2898,2899,2900,2901,2902,2903,2904,2905,2906,2907,2908,2909,2910,2911,2912,2913,2914,2915,2916,2917,2918,2919,2920,2921,2922,2923,2924,2925,2926,2927,2928,2929,2930,2931,2932,2933,2934,2935,2936,2937,2938,2939]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph44\",\"source_text\":\"A second effect is entirely about the wall-normal position of the sources, all of which sit at $y_{d}>0$ above the displacement plane. The elliptic kernel attenuates each source by $e^{-ky_{d}}$ , so a source deeper in the layer (larger $y_{d}$ ) reaches the wall more weakly. “Wallward” therefore means toward smaller $y_{d}$ : if the canopy holds the active nonlinear source closer to the wall than its smooth-wall counterpart sits, by an effective amount $\\\\Delta_{\\\\mathcal{Q}}(y_{d})>0$ , that source is less attenuated and its wall imprint rises. Collect that departure in a single local gain,\",\"tag\":null,\"text\":\"A second effect is entirely about the wall-normal position of the sources, all of which sit at 𝑦𝑑 > 0 above the displacement plane. The elliptic kernel attenuates each source by 𝑒 −𝑘 𝑦𝑑, so a source deeper in the layer (larger 𝑦𝑑) reaches the wall more weakly. “Wallward” therefore means toward smaller 𝑦𝑑: if the canopy holds the active nonlinear source closer to the wall than its smoothwall counterpart sits, by an effective amount ΔQ (𝑦𝑑) > 0, that source is less attenuated and its wall imprint rises. Collect that departure in a single local gain,\"},{\"bbox\":[162.89,331.38,359.11,12.56],\"bbox_pdf\":[162.89,448.06,522.0,460.62],\"fragments\":[{\"bbox_pdf\":[162.89,448.06,522.0,460.62],\"words\":[2940,2941,2942,2943,2944,2946,2947,2948,2949,2950,2951,2952,2953]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\chi_{G}(y_{d})=\\\\exp\\\\!\\\\left[2k_{*}(y_{d})\\\\,\\\\Delta_{\\\\mathcal{Q}}(y_{d})\\\\right],%\\n\\\\qquad\\\\chi_{G}\\\\to 1\\\\quad\\\\text{as }y_{d}/k_{s}\\\\to\\\\infty,\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S4.E7\",\"source_text\":\"$\\\\chi_{G}(y_{d})=\\\\exp\\\\!\\\\left[2k_{*}(y_{d})\\\\,\\\\Delta_{\\\\mathcal{Q}}(y_{d})\\\\right],% \\\\qquad\\\\chi_{G}\\\\to 1\\\\quad\\\\text{as }y_{d}/k_{s}\\\\to\\\\infty,$ (7)\",\"tag\":\"(7)\",\"text\":\"𝜒𝐺(𝑦𝑑) = exp[2𝑘∗(𝑦𝑑) ΔQ (𝑦𝑑)] 𝜒𝐺 → 1 as 𝑦𝑑/𝑘𝑠 → ∞, (7)\"},{\"bbox\":[90.0,352.93,432.0,66.74],\"bbox_pdf\":[90.0,372.33,522.0,439.07],\"fragments\":[{\"bbox_pdf\":[90.0,372.33,522.0,439.07],\"words\":[2954,2955,2956,2957,2958,2959,2960,2961,2962,2963,2964,2965,2966,2967,2968,2969,2970,2971,2972,2973,2974,2975,2976,2977,2978,2979,2980,2981,2982,2983,2984,2985,2986,2987,2988,2989,2990,2991,2992,2993,2994,2995,2996,2997,2998,2999,3000,3001,3002,3003,3004,3005,3006,3007,3008,3009,3010,3011,3012,3013,3014,3015,3016,3017,3018,3019,3020,3021,3022,3023,3024,3025,3026,3027,3028,3029,3030,3031,3032,3033,3034,3035,3036,3037,3038,3039,3040,3041,3042,3043,3044]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph46\",\"source_text\":\"so that $\\\\chi_{G}\\\\equiv 1$ for the smooth wall and $\\\\chi_{G}>1$ for a wallward shift, with $k_{*}(y_{d})$ the wall-parallel wavenumber most relevant to the source at $y_{d}$ . Nothing in the construction fixes the sign of the shift: outward redistribution ( $\\\\Delta_{\\\\mathcal{Q}}<0$ , $\\\\chi_{G}<1$ ) is equally admissible. The limit in (7) is not an extra assumption. The canopy has one length, $k_{s}$ , and above the sublayer outer-layer similarity leaves no other, so $\\\\chi_{G}-1$ is a function of $y_{d}/k_{s}$ alone and must vanish as that ratio grows.\",\"tag\":null,\"text\":\"so that 𝜒𝐺 ≡ 1 for the smooth wall and 𝜒𝐺 > 1 for a wallward shift, with 𝑘∗(𝑦𝑑) the wall-parallel wavenumber most relevant to the source at 𝑦𝑑. Nothing in the construction fixes the sign of the shift: outward redistribution (ΔQ < 0, 𝜒𝐺 < 1) is equally admissible. The limit in (7) is not an extra assumption. The canopy has one length, 𝑘𝑠, and above the sublayer outer-layer similarity leaves no other, so 𝜒𝐺 − 1 is a function of 𝑦𝑑/𝑘𝑠 alone and must vanish as that ratio grows.\"},{\"bbox\":[106.94,420.68,383.81,24.46],\"bbox_pdf\":[106.94,346.87,490.75,371.32],\"fragments\":[{\"bbox_pdf\":[106.94,346.87,490.75,371.32],\"words\":[3045,3046,3047,3048,3049,3050,3051,3052,3053,3054,3055,3056,3057,3063,3064,3065,3066,3067,3068]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph47\",\"source_text\":\"Splitting the wall-projected nonlinear variance at the roughness height and writing $\\\\chi_{G}=1+(\\\\chi_{G}-1)$ in the outer part gives\",\"tag\":null,\"text\":\"Splitting the wall-projected nonlinear variance at the roughness height and writing 𝜒𝐺 = 1) in the outer part gives\"},{\"bbox\":[126.98,447.59,395.03,62.59],\"bbox_pdf\":[126.98,281.82,522.0,344.41],\"fragments\":[{\"bbox_pdf\":[126.98,281.82,522.0,344.41],\"words\":[3069,3070,3071,3072,3073,3074,3075,3076,3077,3078,3079,3080,3081,3082,3083,3084,3085,3086,3087,3088,3089,3090,3091,3092,3093,3094,3095,3096,3097,3098,3099,3100,3101,3102,3103,3104,3105,3106,3107,3108,3109,3110,3111,3112,3113,3114]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\langle p_{\\\\mathcal{Q}}^{\\\\prime 2}\\\\rangle_{r}^{+}\\\\sim\\\\underbrace{\\\\int_{y_{d,0}%\\n}^{k_{s}}a_{\\\\mathcal{Q}}^{\\\\mathrm{src},r}(y_{d})\\\\,\\\\chi_{G}(y_{d})\\\\,\\\\mathrm{d}%\\n\\\\ln y_{d}}_{\\\\displaystyle\\\\equiv\\\\,B_{\\\\mathcal{Q},i}^{r}}\\\\;+\\\\;A_{\\\\mathcal{Q},%\\n\\\\eta}^{\\\\mathrm{base}}\\\\left[\\\\,\\\\eta_{s}+\\\\int_{1}^{\\\\delta/k_{s}}\\\\bigl{(}\\\\chi_{G}-%\\n1\\\\bigr{)}\\\\,\\\\mathrm{d}\\\\ln\\\\frac{y_{d}}{k_{s}}\\\\,\\\\right],\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"S4.E8\",\"source_text\":\"$\\\\langle p_{\\\\mathcal{Q}}^{\\\\prime 2}\\\\rangle_{r}^{+}\\\\sim\\\\underbrace{\\\\int_{y_{d,0}% }^{k_{s}}a_{\\\\mathcal{Q}}^{\\\\mathrm{src},r}(y_{d})\\\\,\\\\chi_{G}(y_{d})\\\\,\\\\mathrm{d}% \\\\ln y_{d}}_{\\\\displaystyle\\\\equiv\\\\,B_{\\\\mathcal{Q},i}^{r}}\\\\;+\\\\;A_{\\\\mathcal{Q},% \\\\eta}^{\\\\mathrm{base}}\\\\left[\\\\,\\\\eta_{s}+\\\\int_{1}^{\\\\delta/k_{s}}\\\\bigl{(}\\\\chi_{G}-% 1\\\\bigr{)}\\\\,\\\\mathrm{d}\\\\ln\\\\frac{y_{d}}{k_{s}}\\\\,\\\\right],$ (8)\",\"tag\":\"(8)\",\"text\":\"∫ 𝑘𝑠 [︃ ∫ 𝛿/𝑘𝑠 ]︃ ′2 + src,𝑟 base (︁ )︁ 𝑦𝑑 ⟨𝑝 Q⟩𝑟 ∼ 𝑎 (𝑦𝑑) 𝜒𝐺(𝑦𝑑) d ln 𝑦𝑑 + 𝐴 Q, 𝜂 𝜂𝑠 + 𝜒𝐺 − 1 d ln , (8) Q 𝑦𝑑,0 1 𝑘𝑠 ⏞ ⏟⏟ ⏞ ≡ 𝐵 𝑟 Q,𝑖\"},{\"bbox\":[90.0,518.0,432.0,93.83],\"bbox_pdf\":[90.0,180.18,522.0,274.0],\"fragments\":[{\"bbox_pdf\":[90.0,180.18,522.0,274.0],\"words\":[3115,3116,3117,3118,3119,3120,3121,3122,3123,3124,3125,3126,3127,3128,3129,3130,3131,3132,3133,3134,3135,3136,3137,3139,3140,3141,3142,3143,3144,3145,3146,3147,3148,3149,3150,3151,3152,3153,3154,3155,3156,3157,3158,3159,3160,3161,3162,3163,3164,3165,3166,3167,3168,3169,3170,3171,3172,3173,3174,3175,3176,3177,3178,3179,3180,3181,3182,3184,3185,3186,3187,3188,3189,3190,3191,3192,3193,3194,3195,3196,3197,3198,3199,3200,3201,3202,3203,3204,3205,3206,3207,3208,3209,3210,3211,3212,3213,3214,3215,3216,3217,3218,3219,3220,3221]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph49\",\"source_text\":\"with $y_{d,0}$ a finite inner floor at the element scale that, with the rest of the sublayer contribution, is absorbed into $B_{\\\\mathcal{Q},i}^{r}$ . The first term is the inner-scaled invariant part of $\\\\mathcal{Q}$ , generated within the roughness sublayer (crest-scale turbulence, separation and reattachment, the canopy-coherent motions of rough-channel DNS; Bhaganagar et al. 2007). In the second term $a_{\\\\mathcal{Q}}^{\\\\mathrm{src},r}$ has been set to $A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}$ , its universal log-layer value: outer-layer similarity supplies the source density above the sublayer (the attached-eddy/uniform-momentum-zone baseline, supported on rough walls by the outer-layer UMZ similarity of Zhang et al. 2026), and roughness does not renormalise it.\",\"tag\":null,\"text\":\"with 𝑦𝑑,0 a finite inner floor at the element scale that, with the rest of the sublayer contribution, is absorbed into 𝐵 𝑟 The first term is the inner-scaled invariant part of Q, generated within Q,𝑖 the roughness sublayer (crest-scale turbulence, separation and reattachment, the canopy-coherent motions of rough-channel DNS; Bhaganagar et al. 2007). In the second term 𝑎 src,𝑟 has been set Q to 𝐴 base its universal log-layer value: outer-layer similarity supplies the source density above the Q, 𝜂 sublayer (the attached-eddy/uniform-momentum-zone baseline, supported on rough walls by the outer-layer UMZ similarity of Zhang et al. 2026), and roughness does not renormalise it.\"},{\"bbox\":[106.94,614.46,293.61,12.55],\"bbox_pdf\":[106.94,164.99,400.55,177.54],\"fragments\":[{\"bbox_pdf\":[106.94,164.99,400.55,177.54],\"words\":[3222,3223,3224,3225,3226,3227,3228,3229,3230,3231,3232]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph50\",\"source_text\":\"Because $\\\\chi_{G}-1$ decays in ${y_{d}/k_{s}}$ , the remaining integral converges,\",\"tag\":null,\"text\":\"Because 𝜒𝐺 − 1 decays in 𝑦𝑑/𝑘𝑠, the remaining integral converges,\"},{\"bbox\":[140.31,628.38,381.69,35.38],\"bbox_pdf\":[140.31,128.23,522.0,163.62],\"fragments\":[{\"bbox_pdf\":[140.31,128.23,522.0,163.62],\"words\":[3233,3234,3235,3236,3237,3238,3239,3240,3241,3242,3243,3244,3245,3246,3247,3248,3249,3250,3251,3252,3253,3254,3255,3256,3257,3258,3259,3260,3261,3262,3263]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\int_{1}^{\\\\delta/k_{s}}\\\\bigl{(}\\\\chi_{G}-1\\\\bigr{)}\\\\,\\\\mathrm{d}\\\\ln\\\\frac{y_{d}}{k%\\n_{s}}\\\\;\\\\longrightarrow\\\\;\\\\mathcal{I}_{\\\\chi}\\\\equiv\\\\int_{1}^{\\\\infty}\\\\bigl{(}\\\\chi_%\\n{G}-1\\\\bigr{)}\\\\,\\\\mathrm{d}\\\\ln\\\\frac{y_{d}}{k_{s}}\\\\qquad(\\\\delta/k_{s}\\\\to\\\\infty),\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"S4.E9\",\"source_text\":\"$\\\\int_{1}^{\\\\delta/k_{s}}\\\\bigl{(}\\\\chi_{G}-1\\\\bigr{)}\\\\,\\\\mathrm{d}\\\\ln\\\\frac{y_{d}}{k% _{s}}\\\\;\\\\longrightarrow\\\\;\\\\mathcal{I}_{\\\\chi}\\\\equiv\\\\int_{1}^{\\\\infty}\\\\bigl{(}\\\\chi_% {G}-1\\\\bigr{)}\\\\,\\\\mathrm{d}\\\\ln\\\\frac{y_{d}}{k_{s}}\\\\qquad(\\\\delta/k_{s}\\\\to\\\\infty),$ (9)\",\"tag\":\"(9)\",\"text\":\"∫ 𝛿/𝑘𝑠 (︁ )︁ 𝑦𝑑 ∫ ∞ (︁ )︁ 𝑦𝑑 𝜒𝐺 − 1 d ln −→ I𝜒 ≡ 𝜒𝐺 − 1 d ln (𝛿/𝑘𝑠 → ∞), (9) 1 𝑘𝑠 1 𝑘𝑠\"},{\"bbox\":[90.0,669.56,331.78,12.55],\"bbox_pdf\":[90.0,109.9,421.78,122.44],\"fragments\":[{\"bbox_pdf\":[90.0,109.9,421.78,122.44],\"words\":[3264,3265,3266,3267,3268,3269,3270,3271,3272,3273,3274,3275,3276]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"paragraph\",\"source_ref\":\"paragraph52\",\"source_text\":\"to a number set by the roughness geometry and independent of $\\\\delta/k_{s}$ . Hence\",\"tag\":null,\"text\":\"to a number set by the roughness geometry and independent of 𝛿/𝑘𝑠. Hence\"},{\"bbox\":[183.03,689.52,338.97,16.14],\"bbox_pdf\":[183.03,86.33,522.0,102.48],\"fragments\":[{\"bbox_pdf\":[183.03,86.33,522.0,102.48],\"words\":[3277,3278,3279,3280,3281,3282,3283,3284,3285,3286,3287,3288,3289,3290,3291,3292,3293,3294,3295,3296,3297,3298,3299,3300,3301]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\langle p_{\\\\mathcal{Q}}^{\\\\prime 2}\\\\rangle_{r}^{+}\\\\approx A_{\\\\mathcal{Q},\\\\eta}^%\\n{\\\\mathrm{base}}\\\\,\\\\eta_{s}+B_{\\\\mathcal{Q},G}^{r},\\\\qquad B_{\\\\mathcal{Q},G}^{r}%\\n\\\\equiv B_{\\\\mathcal{Q},i}^{r}+A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\mathcal{I}%\\n_{\\\\chi},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"S4.E10\",\"source_text\":\"$\\\\langle p_{\\\\mathcal{Q}}^{\\\\prime 2}\\\\rangle_{r}^{+}\\\\approx A_{\\\\mathcal{Q},\\\\eta}^% {\\\\mathrm{base}}\\\\,\\\\eta_{s}+B_{\\\\mathcal{Q},G}^{r},\\\\qquad B_{\\\\mathcal{Q},G}^{r}% \\\\equiv B_{\\\\mathcal{Q},i}^{r}+A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\mathcal{I}% _{\\\\chi},$ (10)\",\"tag\":\"(10)\",\"text\":\"⟨𝑝 ′2 + ≈ 𝐴 base 𝑟 𝐵 𝑟 𝑟 base (10) Q⟩𝑟 Q, 𝜂 𝜂𝑠 + 𝐵Q,𝐺, Q,𝐺 ≡ 𝐵Q,𝑖 + 𝐴Q, 𝜂 I𝜒,\"},{\"bbox\":[303.27,720.98,5.45,10.91],\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"fragments\":[{\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"words\":[9385]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":13,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"6\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0005.pdf\",\"page_no\":6,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph41\",\"c_block\":0.98,\"c_lines\":1.0,\"c_pdf\":0.999,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.98,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a.pdf","asset_sha256":"4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a","document_id":"arxiv:2608.16729","kind":"page","page_number":6,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[276.34,107.57,245.66,35.42],\"bbox_pdf\":[276.34,649.0,522.0,684.43],\"fragments\":[{\"bbox_pdf\":[276.34,649.0,522.0,684.43],\"words\":[2718,2719,2720,2721,2722,2723,2724,2725,2726]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\eta_{s}\\\\equiv\\\\ln\\\\!\\\\left(\\\\frac{\\\\delta}{k_{s}}\\\\right),\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S4.E6\",\"source_text\":\"$\\\\eta_{s}\\\\equiv\\\\ln\\\\!\\\\left(\\\\frac{\\\\delta}{k_{s}}\\\\right),$ (6)\",\"tag\":\"(6)\",\"text\":\"(︃ )︃ 𝛿 𝜂𝑠 ≡ ln , (6) 𝑘𝑠\"},\"formula_index\":1,\"parent_page_sha256\":\"4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a.pdf","asset_sha256":"4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a","document_id":"arxiv:2608.16729","kind":"formula","page_number":6,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[162.89,331.38,359.11,12.56],\"bbox_pdf\":[162.89,448.06,522.0,460.62],\"fragments\":[{\"bbox_pdf\":[162.89,448.06,522.0,460.62],\"words\":[2940,2941,2942,2943,2944,2946,2947,2948,2949,2950,2951,2952,2953]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\chi_{G}(y_{d})=\\\\exp\\\\!\\\\left[2k_{*}(y_{d})\\\\,\\\\Delta_{\\\\mathcal{Q}}(y_{d})\\\\right],%\\n\\\\qquad\\\\chi_{G}\\\\to 1\\\\quad\\\\text{as }y_{d}/k_{s}\\\\to\\\\infty,\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S4.E7\",\"source_text\":\"$\\\\chi_{G}(y_{d})=\\\\exp\\\\!\\\\left[2k_{*}(y_{d})\\\\,\\\\Delta_{\\\\mathcal{Q}}(y_{d})\\\\right],% \\\\qquad\\\\chi_{G}\\\\to 1\\\\quad\\\\text{as }y_{d}/k_{s}\\\\to\\\\infty,$ (7)\",\"tag\":\"(7)\",\"text\":\"𝜒𝐺(𝑦𝑑) = exp[2𝑘∗(𝑦𝑑) ΔQ (𝑦𝑑)] 𝜒𝐺 → 1 as 𝑦𝑑/𝑘𝑠 → ∞, (7)\"},\"formula_index\":4,\"parent_page_sha256\":\"4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a.pdf","asset_sha256":"4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a","document_id":"arxiv:2608.16729","kind":"formula","page_number":6,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[126.98,447.59,395.03,62.59],\"bbox_pdf\":[126.98,281.82,522.0,344.41],\"fragments\":[{\"bbox_pdf\":[126.98,281.82,522.0,344.41],\"words\":[3069,3070,3071,3072,3073,3074,3075,3076,3077,3078,3079,3080,3081,3082,3083,3084,3085,3086,3087,3088,3089,3090,3091,3092,3093,3094,3095,3096,3097,3098,3099,3100,3101,3102,3103,3104,3105,3106,3107,3108,3109,3110,3111,3112,3113,3114]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\langle p_{\\\\mathcal{Q}}^{\\\\prime 2}\\\\rangle_{r}^{+}\\\\sim\\\\underbrace{\\\\int_{y_{d,0}%\\n}^{k_{s}}a_{\\\\mathcal{Q}}^{\\\\mathrm{src},r}(y_{d})\\\\,\\\\chi_{G}(y_{d})\\\\,\\\\mathrm{d}%\\n\\\\ln y_{d}}_{\\\\displaystyle\\\\equiv\\\\,B_{\\\\mathcal{Q},i}^{r}}\\\\;+\\\\;A_{\\\\mathcal{Q},%\\n\\\\eta}^{\\\\mathrm{base}}\\\\left[\\\\,\\\\eta_{s}+\\\\int_{1}^{\\\\delta/k_{s}}\\\\bigl{(}\\\\chi_{G}-%\\n1\\\\bigr{)}\\\\,\\\\mathrm{d}\\\\ln\\\\frac{y_{d}}{k_{s}}\\\\,\\\\right],\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"S4.E8\",\"source_text\":\"$\\\\langle p_{\\\\mathcal{Q}}^{\\\\prime 2}\\\\rangle_{r}^{+}\\\\sim\\\\underbrace{\\\\int_{y_{d,0}% }^{k_{s}}a_{\\\\mathcal{Q}}^{\\\\mathrm{src},r}(y_{d})\\\\,\\\\chi_{G}(y_{d})\\\\,\\\\mathrm{d}% \\\\ln y_{d}}_{\\\\displaystyle\\\\equiv\\\\,B_{\\\\mathcal{Q},i}^{r}}\\\\;+\\\\;A_{\\\\mathcal{Q},% \\\\eta}^{\\\\mathrm{base}}\\\\left[\\\\,\\\\eta_{s}+\\\\int_{1}^{\\\\delta/k_{s}}\\\\bigl{(}\\\\chi_{G}-% 1\\\\bigr{)}\\\\,\\\\mathrm{d}\\\\ln\\\\frac{y_{d}}{k_{s}}\\\\,\\\\right],$ (8)\",\"tag\":\"(8)\",\"text\":\"∫ 𝑘𝑠 [︃ ∫ 𝛿/𝑘𝑠 ]︃ ′2 + src,𝑟 base (︁ )︁ 𝑦𝑑 ⟨𝑝 Q⟩𝑟 ∼ 𝑎 (𝑦𝑑) 𝜒𝐺(𝑦𝑑) d ln 𝑦𝑑 + 𝐴 Q, 𝜂 𝜂𝑠 + 𝜒𝐺 − 1 d ln , (8) Q 𝑦𝑑,0 1 𝑘𝑠 ⏞ ⏟⏟ ⏞ ≡ 𝐵 𝑟 Q,𝑖\"},\"formula_index\":7,\"parent_page_sha256\":\"4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a.pdf","asset_sha256":"4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a","document_id":"arxiv:2608.16729","kind":"formula","page_number":6,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[140.31,628.38,381.69,35.38],\"bbox_pdf\":[140.31,128.23,522.0,163.62],\"fragments\":[{\"bbox_pdf\":[140.31,128.23,522.0,163.62],\"words\":[3233,3234,3235,3236,3237,3238,3239,3240,3241,3242,3243,3244,3245,3246,3247,3248,3249,3250,3251,3252,3253,3254,3255,3256,3257,3258,3259,3260,3261,3262,3263]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\int_{1}^{\\\\delta/k_{s}}\\\\bigl{(}\\\\chi_{G}-1\\\\bigr{)}\\\\,\\\\mathrm{d}\\\\ln\\\\frac{y_{d}}{k%\\n_{s}}\\\\;\\\\longrightarrow\\\\;\\\\mathcal{I}_{\\\\chi}\\\\equiv\\\\int_{1}^{\\\\infty}\\\\bigl{(}\\\\chi_%\\n{G}-1\\\\bigr{)}\\\\,\\\\mathrm{d}\\\\ln\\\\frac{y_{d}}{k_{s}}\\\\qquad(\\\\delta/k_{s}\\\\to\\\\infty),\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"S4.E9\",\"source_text\":\"$\\\\int_{1}^{\\\\delta/k_{s}}\\\\bigl{(}\\\\chi_{G}-1\\\\bigr{)}\\\\,\\\\mathrm{d}\\\\ln\\\\frac{y_{d}}{k% _{s}}\\\\;\\\\longrightarrow\\\\;\\\\mathcal{I}_{\\\\chi}\\\\equiv\\\\int_{1}^{\\\\infty}\\\\bigl{(}\\\\chi_% {G}-1\\\\bigr{)}\\\\,\\\\mathrm{d}\\\\ln\\\\frac{y_{d}}{k_{s}}\\\\qquad(\\\\delta/k_{s}\\\\to\\\\infty),$ (9)\",\"tag\":\"(9)\",\"text\":\"∫ 𝛿/𝑘𝑠 (︁ )︁ 𝑦𝑑 ∫ ∞ (︁ )︁ 𝑦𝑑 𝜒𝐺 − 1 d ln −→ I𝜒 ≡ 𝜒𝐺 − 1 d ln (𝛿/𝑘𝑠 → ∞), (9) 1 𝑘𝑠 1 𝑘𝑠\"},\"formula_index\":10,\"parent_page_sha256\":\"4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a.pdf","asset_sha256":"4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a","document_id":"arxiv:2608.16729","kind":"formula","page_number":6,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[183.03,689.52,338.97,16.14],\"bbox_pdf\":[183.03,86.33,522.0,102.48],\"fragments\":[{\"bbox_pdf\":[183.03,86.33,522.0,102.48],\"words\":[3277,3278,3279,3280,3281,3282,3283,3284,3285,3286,3287,3288,3289,3290,3291,3292,3293,3294,3295,3296,3297,3298,3299,3300,3301]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\langle p_{\\\\mathcal{Q}}^{\\\\prime 2}\\\\rangle_{r}^{+}\\\\approx A_{\\\\mathcal{Q},\\\\eta}^%\\n{\\\\mathrm{base}}\\\\,\\\\eta_{s}+B_{\\\\mathcal{Q},G}^{r},\\\\qquad B_{\\\\mathcal{Q},G}^{r}%\\n\\\\equiv B_{\\\\mathcal{Q},i}^{r}+A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\mathcal{I}%\\n_{\\\\chi},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"S4.E10\",\"source_text\":\"$\\\\langle p_{\\\\mathcal{Q}}^{\\\\prime 2}\\\\rangle_{r}^{+}\\\\approx A_{\\\\mathcal{Q},\\\\eta}^% {\\\\mathrm{base}}\\\\,\\\\eta_{s}+B_{\\\\mathcal{Q},G}^{r},\\\\qquad B_{\\\\mathcal{Q},G}^{r}% \\\\equiv B_{\\\\mathcal{Q},i}^{r}+A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\mathcal{I}% _{\\\\chi},$ (10)\",\"tag\":\"(10)\",\"text\":\"⟨𝑝 ′2 + ≈ 𝐴 base 𝑟 𝐵 𝑟 𝑟 base (10) Q⟩𝑟 Q, 𝜂 𝜂𝑠 + 𝐵Q,𝐺, Q,𝐺 ≡ 𝐵Q,𝑖 + 𝐴Q, 𝜂 I𝜒,\"},\"formula_index\":12,\"parent_page_sha256\":\"4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a.pdf","asset_sha256":"4c8ab7b06e3949477c5d1dc563a43fdac825e0bf3f7d8defd34d0f9fc355fa8a","document_id":"arxiv:2608.16729","kind":"formula","page_number":6,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[90.0,90.05,432.0,24.46],\"bbox_pdf\":[90.0,677.49,522.0,701.95],\"fragments\":[{\"bbox_pdf\":[90.0,677.49,522.0,701.95],\"words\":[3302,3303,3304,3305,3306,3307,3308,3309,3310,3311,3312,3313,3314,3315,3316,3317]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph54\",\"source_text\":\"the smooth-wall coefficient on the rough-wall range, with every roughness-set effect in a single non-growing offset.\",\"tag\":null,\"text\":\"the smooth-wall coefficient on the rough-wall range, with every roughness-set effect in a single non-growing offset.\"},{\"bbox\":[89.88,117.15,432.13,78.66],\"bbox_pdf\":[89.88,596.2,522.01,674.85],\"fragments\":[{\"bbox_pdf\":[89.88,596.2,522.01,674.85],\"words\":[3318,3319,3320,3321,3322,3323,3324,3325,3326,3327,3328,3329,3330,3331,3332,3333,3334,3335,3336,3337,3338,3339,3340,3341,3342,3343,3344,3345,3346,3347,3348,3349,3350,3351,3352,3353,3354,3355,3356,3357,3358,3359,3360,3361,3362,3363,3364,3365,3366,3367,3368,3369,3370,3371,3372,3373,3374,3375,3377,3378,3379,3380,3381,3382,3383,3384,3385,3386,3387,3388,3389,3390,3391,3392,3393,3394,3395,3396,3397,3398,3399,3400,3401]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.988,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph55\",\"source_text\":\"This subsumes the loose ends of §2 under one principle. The dispersive linear source, the $O(kk_{s})$ canopy correction to the half-space kernel, the sublayer-generated nonlinear turbulence and any Green-weighted redistribution of the log-layer source are all set by the roughness and all die away with $y_{d}/k_{s}$ . Each therefore contributes a convergent, $\\\\eta_{s}$ -independent amount to $B_{\\\\mathcal{Q},G}^{r}$ , and none of them can alter the coefficient of $\\\\eta_{s}$ . The wall spectrum constrains only their sum; separating them requires wall-normal source statistics (§9).\",\"tag\":null,\"text\":\"This subsumes the loose ends of §2 under one principle. The dispersive linear source, the 𝑂 (𝑘 𝑘𝑠) canopy correction to the half-space kernel, the sublayer-generated nonlinear turbulence and any Green-weighted redistribution of the log-layer source are all set by the roughness and all die away with 𝑦𝑑/𝑘𝑠. Each therefore contributes a convergent, 𝜂𝑠-independent amount to 𝐵 𝑟 and Q,𝐺 none of them can alter the coefficient of 𝜂𝑠. The wall spectrum constrains only their sum; separating them requires wall-normal source statistics (§9).\"},{\"bbox\":[90.0,198.44,432.0,92.21],\"bbox_pdf\":[90.0,501.35,522.0,593.56],\"fragments\":[{\"bbox_pdf\":[90.0,501.35,522.0,593.56],\"words\":[3402,3403,3404,3405,3406,3407,3408,3409,3410,3411,3412,3413,3414,3415,3416,3417,3418,3419,3420,3421,3422,3423,3424,3425,3426,3427,3429,3430,3431,3432,3433,3434,3435,3436,3437,3438,3439,3440,3441,3442,3443,3444,3445,3446,3447,3448,3449,3450,3451,3452,3453,3454,3455,3456,3457,3458,3459,3460,3461,3462,3463,3464,3465,3466,3467,3468,3469,3470,3471,3472,3473,3474,3475,3476,3477,3478,3479,3480,3481,3482,3483,3484,3485,3486,3487,3488,3489,3490,3491,3492,3493,3494,3495,3496,3497,3498,3499,3500,3501,3502,3503,3504,3505,3506,3507,3508,3509,3510,3511,3512,3513,3514,3515,3516,3517,3518]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph56\",\"source_text\":\"Framed as a falsifiable test, a multiplicative gain on the growth rate would require a canopy influence that persists undiminished to the top of the layer — $\\\\Delta_{\\\\mathcal{Q}}\\\\propto y_{d}$ all the way to $y_{d}\\\\sim\\\\delta$ , so that roughness of height $k_{s}\\\\sim\\\\delta/30$ displaces the mid-layer source by $O(\\\\delta)$ . That is what outer-layer similarity forbids, and it is the same similarity that produced $\\\\eta_{s}$ . The growth rate is therefore not a free parameter of the model but a prediction, and a measured rate departing from $A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}$ is evidence against outer-layer similarity for the pressure source—or against attributing the whole smooth-wall rate to $\\\\mathcal{Q}$ (§5)—rather than a canopy gain to be absorbed.\",\"tag\":null,\"text\":\"Framed as a falsifiable test, a multiplicative gain on the growth rate would require a canopy influence that persists undiminished to the top of the layer ΔQ ∝ 𝑦𝑑 all the way to 𝑦𝑑 ∼ 𝛿, so that roughness of height 𝑘𝑠 ∼ 𝛿/30 displaces the mid-layer source by 𝑂 (𝛿). That is what outer-layer similarity forbids, and it is the same similarity that produced 𝜂𝑠. The growth rate is therefore not a free parameter of the model but a prediction, and a measured rate departing from 𝐴 base is evidence Q, 𝜂 against outer-layer similarity for the pressure source—or against attributing the whole smooth-wall rate to Q (§5)—rather than a canopy gain to be absorbed.\"},{\"bbox\":[90.0,310.7,225.36,14.35],\"bbox_pdf\":[90.0,466.95,315.36,481.3],\"fragments\":[{\"bbox_pdf\":[90.0,466.95,315.36,481.3],\"words\":[3519,3520,3521,3522]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"heading\",\"source_ref\":\"title57\",\"source_text\":\"5 Two-lognormal rough-wall model\",\"tag\":null,\"text\":\"5 Two-lognormal rough-wall model\"},{\"bbox\":[90.0,337.96,432.0,25.48],\"bbox_pdf\":[90.0,428.56,522.0,454.04],\"fragments\":[{\"bbox_pdf\":[90.0,428.56,522.0,454.04],\"words\":[3523,3524,3525,3526,3527,3528,3529,3530,3531,3532,3533,3534,3536,3537,3538,3539,3540,3541,3542,3543,3544,3545,3546,3547]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph58\",\"source_text\":\"The smooth-wall two-component model writes the premultiplied spectrum (at sufficient $\\\\delta^{+}$ ) as an inner population plus an outer population (Massey et al., 2026c),\",\"tag\":null,\"text\":\"The smooth-wall two-component model writes the premultiplied spectrum (at sufficient 𝛿 + as an inner population plus an outer population [Massey et al., 2026c],\"},{\"bbox\":[135.1,370.82,386.9,27.11],\"bbox_pdf\":[135.1,394.07,522.0,421.18],\"fragments\":[{\"bbox_pdf\":[135.1,394.07,522.0,421.18],\"words\":[3548,3549,3550,3551,3552,3553,3554,3555,3556,3557,3558,3559,3560,3561,3562,3563,3564,3565,3567,3568,3569,3570,3571,3572,3574,3575,3576,3577,3578,3579,3580,3581,3582]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"{{f\\\\phi_{pp}}}^{+}=g_{1}(T^{+})+g_{2}(T^{o};\\\\delta^{+}),\\\\qquad T=1/f,\\\\qquad T^%\\n{+}\\\\equiv\\\\frac{Tu_{\\\\tau}^{2}}{\\\\nu},\\\\qquad T^{o}=\\\\frac{TU_{e}}{\\\\delta}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S5.E11\",\"source_text\":\"${{f\\\\phi_{pp}}}^{+}=g_{1}(T^{+})+g_{2}(T^{o};\\\\delta^{+}),\\\\qquad T=1/f,\\\\qquad T^% {+}\\\\equiv\\\\frac{Tu_{\\\\tau}^{2}}{\\\\nu},\\\\qquad T^{o}=\\\\frac{TU_{e}}{\\\\delta}.$ (11)\",\"tag\":\"(11)\",\"text\":\"𝑇𝑢 2 𝑇𝑈𝑒 + + 𝑜 + + 𝜏 𝑜 𝑓 𝜙𝑝 𝑝 = 𝑔1(𝑇 ) + 𝑔2(𝑇 𝛿 ), 𝑇 = 1/ 𝑓 𝑇 ≡ , 𝑇 = . (11) 𝜈 𝛿\"},{\"bbox\":[90.0,402.34,297.75,10.91],\"bbox_pdf\":[90.0,378.75,387.75,389.66],\"fragments\":[{\"bbox_pdf\":[90.0,378.75,387.75,389.66],\"words\":[3583,3584,3585,3586,3587,3588,3589,3590,3591,3592,3593,3594]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph60\",\"source_text\":\"For fully rough walls, splitting the spectrum by its two sources gives\",\"tag\":null,\"text\":\"For fully rough walls, splitting the spectrum by its two sources gives\"},{\"bbox\":[139.68,412.92,382.32,35.42],\"bbox_pdf\":[139.68,343.65,522.0,379.08],\"fragments\":[{\"bbox_pdf\":[139.68,343.65,522.0,379.08],\"words\":[3595,3596,3597,3598,3599,3600,3601,3602,3603,3604,3605,3606,3607,3608,3609,3610,3612,3613,3614,3615,3616,3618,3619,3620,3621,3622,3623,3624,3625,3626,3627,3628,3629]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"{f\\\\phi_{pp,r}}^{+}=g_{\\\\mathcal{L}}^{r}(T^{s};\\\\mathcal{R}_{s})+g_{\\\\mathcal{Q}}^%\\n{r}(T^{o};\\\\eta_{s},\\\\mathcal{R}_{s}),\\\\qquad T^{s}\\\\equiv\\\\frac{TU_{e}}{k_{s}},%\\n\\\\qquad\\\\eta_{s}\\\\equiv\\\\ln\\\\!\\\\left(\\\\frac{\\\\delta}{k_{s}}\\\\right),\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"S5.E12\",\"source_text\":\"${f\\\\phi_{pp,r}}^{+}=g_{\\\\mathcal{L}}^{r}(T^{s};\\\\mathcal{R}_{s})+g_{\\\\mathcal{Q}}^% {r}(T^{o};\\\\eta_{s},\\\\mathcal{R}_{s}),\\\\qquad T^{s}\\\\equiv\\\\frac{TU_{e}}{k_{s}},% \\\\qquad\\\\eta_{s}\\\\equiv\\\\ln\\\\!\\\\left(\\\\frac{\\\\delta}{k_{s}}\\\\right),$ (12)\",\"tag\":\"(12)\",\"text\":\"(︃ )︃ + 𝑟 𝑠 𝑟 𝑜 𝑠 𝑇𝑈𝑒 𝛿 𝑓 𝜙𝑝 𝑝,𝑟 = 𝑔 L(𝑇 R𝑠) + 𝑔 Q (𝑇 𝜂𝑠, R𝑠), 𝑇 ≡ , 𝜂𝑠 ≡ ln , (12) 𝑘𝑠 𝑘𝑠\"},{\"bbox\":[90.0,454.64,432.0,93.23],\"bbox_pdf\":[90.0,244.14,522.0,337.36],\"fragments\":[{\"bbox_pdf\":[90.0,244.14,522.0,337.36],\"words\":[3630,3631,3632,3633,3634,3635,3636,3637,3638,3639,3640,3641,3642,3643,3644,3645,3646,3647,3648,3649,3650,3651,3652,3653,3654,3655,3656,3657,3658,3659,3660,3661,3662,3663,3664,3665,3666,3667,3668,3669,3670,3671,3672,3673,3674,3675,3676,3677,3678,3679,3680,3681,3682,3683,3684,3685,3686,3687,3688,3689,3690,3691,3692,3693,3694,3695,3696,3697,3698,3699,3700,3701,3702,3703,3704,3705,3706,3707,3708,3709,3710,3711,3712,3713,3714,3715,3716,3717,3718,3719]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.978,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph62\",\"source_text\":\"where $\\\\mathcal{R}_{s}$ denotes the fixed fully rough hydrodynamic state (absorbing all geometry-set constants).1 Only two lognormal populations are retained, and $\\\\eta_{s}$ is the sole carrier of Reynolds-number growth at fixed roughness state. The closest spectral-model antecedent is the three-region rough-wall structure of Meyers et al. (2015), built on the same VT-lineage measurements; the present model differs in deriving its two populations from the two Poisson source terms, carrying all Reynolds-number growth in the outer area through $\\\\eta_{s}$ , and retaining no viscous-scaled range once fully rough. Define the lognormal-in- $T$ density\",\"tag\":null,\"text\":\"where R𝑠 denotes the fixed fully rough hydrodynamic state (absorbing all geometry-set constants). 1 Only two lognormal populations are retained, and 𝜂𝑠 is the sole carrier of Reynolds-number growth at fixed roughness state. The closest spectral-model antecedent is the three-region rough-wall structure of Meyers et al. [2015], built on the same VT-lineage measurements; the present model differs in deriving its two populations from the two Poisson source terms, carrying all Reynolds-number growth in the outer area through 𝜂𝑠, and retaining no viscous-scaled range once fully rough. Define the lognormal-in-𝑇 density\"},{\"bbox\":[200.37,547.54,321.63,39.21],\"bbox_pdf\":[200.37,205.25,522.0,244.46],\"fragments\":[{\"bbox_pdf\":[200.37,205.25,522.0,244.46],\"words\":[3720,3721,3722,3723,3724,3725,3726,3727,3728,3729,3730,3731,3732,3733,3734,3735,3736,3737,3738,3739]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathcal{N}(T;T_{b},\\\\sigma,A)=\\\\frac{A}{\\\\sqrt{2\\\\pi}\\\\sigma}\\\\exp\\\\!\\\\left[-\\\\frac{1}%\\n{2}\\\\!\\\\left(\\\\frac{\\\\ln(T/T_{b})}{\\\\sigma}\\\\right)^{\\\\!2}\\\\right],\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S5.E13\",\"source_text\":\"$\\\\mathcal{N}(T;T_{b},\\\\sigma,A)=\\\\frac{A}{\\\\sqrt{2\\\\pi}\\\\sigma}\\\\exp\\\\!\\\\left[-\\\\frac{1}% {2}\\\\!\\\\left(\\\\frac{\\\\ln(T/T_{b})}{\\\\sigma}\\\\right)^{\\\\!2}\\\\right],$ (13)\",\"tag\":\"(13)\",\"text\":\"[︄ (︃ )︃2]︄ 𝐴 1 ln(𝑇/𝑇𝑏) N (𝑇; 𝑇𝑏, 𝜎, 𝐴) = √ exp − , (13) 2𝜋𝜎 2 𝜎\"},{\"bbox\":[90.0,597.46,337.35,10.91],\"bbox_pdf\":[90.0,183.63,427.35,194.54],\"fragments\":[{\"bbox_pdf\":[90.0,183.63,427.35,194.54],\"words\":[3740,3741,3742,3743,3744,3745,3746,3747,3748,3749,3750,3751,3752,3753,3754,3755,3756,3757]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"paragraph\",\"source_ref\":\"paragraph65\",\"source_text\":\"so that $\\\\int_{0}^{\\\\infty}\\\\mathcal{N}\\\\,\\\\mathrm{d}\\\\ln f=A$ over an infinite logarithmic frequency domain. Then\",\"tag\":null,\"text\":\"∫ ∞ so that N d ln 𝑓 = 𝐴 over an infinite logarithmic frequency domain. Then 0\"},{\"bbox\":[226.52,610.15,295.48,50.91],\"bbox_pdf\":[226.52,130.95,522.0,181.85],\"fragments\":[{\"bbox_pdf\":[226.52,130.95,522.0,181.85],\"words\":[3758,3759,3760,3761,3762,3763,3764,3765,3766,3767,3769,3770,3771,3772,3773,3774,3775,3776,3777,3778,3779,3780,3781,3782,3783,3784,3785,3786,3787,3788,3789,3790,3791,3793,3794,3795,3796,3797,3798,3799,3800,3801,3802,3803,3804,3805,3806]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle g_{\\\\mathcal{L}}^{r}(T^{s}) \\\\displaystyle=\\\\mathcal{N}\\\\!\\\\left(T^{s};T_{b\\\\mathcal{L}}^{s,r},\\\\sigma_{\\\\mathcal%\\n{L}}^{r},B_{\\\\mathcal{L}}^{r}\\\\right), \\\\\\\\ \\\\displaystyle g_{\\\\mathcal{Q}}^{r}(T^{o}) \\\\displaystyle=\\\\mathcal{N}\\\\!\\\\left(T^{o};T_{b\\\\mathcal{Q}}^{o,r},\\\\sigma_{\\\\mathcal%\\n{Q}}^{r},A_{\\\\mathcal{Q},\\\\mathrm{obs}}^{r}\\\\right), \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"formula\",\"source_ref\":\"S9.EGx1\",\"source_text\":\"$\\\\displaystyle g_{\\\\mathcal{L}}^{r}(T^{s})$ $\\\\displaystyle=\\\\mathcal{N}\\\\!\\\\left(T^{s};T_{b\\\\mathcal{L}}^{s,r},\\\\sigma_{\\\\mathcal% {L}}^{r},B_{\\\\mathcal{L}}^{r}\\\\right),$ (14) $\\\\displaystyle g_{\\\\mathcal{Q}}^{r}(T^{o})$ $\\\\displaystyle=\\\\mathcal{N}\\\\!\\\\left(T^{o};T_{b\\\\mathcal{Q}}^{o,r},\\\\sigma_{\\\\mathcal% {Q}}^{r},A_{\\\\mathcal{Q},\\\\mathrm{obs}}^{r}\\\\right),$ (15)\",\"tag\":\"(14) (15)\",\"text\":\"(︂ )︂ 𝑔 𝑟 𝑠 ) = N 𝑇 𝑠 𝑇 𝑠,𝑟 , 𝜎 𝑟 𝑟 , (14) L(𝑇 𝑏 L L, 𝐵L (︂ )︂ 𝑔 𝑟 𝑜 ) = N 𝑇 𝑜 𝑇 𝑜,𝑟 , 𝜎 𝑟 𝑟 , (15) Q (𝑇 𝑏 Q Q, 𝐴Q,obs\"},{\"bbox\":[90.0,671.12,432.0,30.88],\"bbox_pdf\":[90.0,90.0,522.0,120.88],\"fragments\":[{\"bbox_pdf\":[90.0,90.0,522.0,120.88],\"words\":[3807,3808,3809,3810,3811,3812,3813,3814,3815,3816,3817,3818,3819,3820,3821,3822,3823,3824,3825,3826,3827,3828,3829,3830,3831,3832,3833,3834,3835,3836,3837,3838,3839,3840,3841,3842,3843,3844,3845,3846,3847,3848,3849,3850,3851,3852,3853]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.976,\"verified\":true},\"order\":12,\"role\":\"footnote\",\"source_ref\":\"footnote63\",\"source_text\":\"11Both populations use the measured edge velocity $U_{e}$ , which keeps both breaks on one directly measurable axis; a canopy velocity $U_{d}<U_{e}$ for the roughness-local feature would rescale $T_{b\\\\mathcal{L}}^{s,r}$ by the constant $U_{e}/U_{d}$ and is absorbed into the fitted constant (cf. (18)).\",\"tag\":null,\"text\":\"1 Both populations use the measured edge velocity 𝑈𝑒, which keeps both breaks on one directly measurable axis; a canopy velocity 𝑈𝑑 < 𝑈𝑒 for the roughness-local feature would rescale 𝑇 𝑠,𝑟 by the constant 𝑈𝑒/𝑈𝑑 and is absorbed into 𝑏 L the fitted constant (cf. (18)).\"},{\"bbox\":[303.27,720.98,5.45,10.91],\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"fragments\":[{\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"words\":[9386]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":13,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"7\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0006.pdf\",\"page_no\":7,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"footnote63\",\"c_block\":0.973,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.986,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.973,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e.pdf","asset_sha256":"ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e","document_id":"arxiv:2608.16729","kind":"page","page_number":7,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[135.1,370.82,386.9,27.11],\"bbox_pdf\":[135.1,394.07,522.0,421.18],\"fragments\":[{\"bbox_pdf\":[135.1,394.07,522.0,421.18],\"words\":[3548,3549,3550,3551,3552,3553,3554,3555,3556,3557,3558,3559,3560,3561,3562,3563,3564,3565,3567,3568,3569,3570,3571,3572,3574,3575,3576,3577,3578,3579,3580,3581,3582]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"{{f\\\\phi_{pp}}}^{+}=g_{1}(T^{+})+g_{2}(T^{o};\\\\delta^{+}),\\\\qquad T=1/f,\\\\qquad T^%\\n{+}\\\\equiv\\\\frac{Tu_{\\\\tau}^{2}}{\\\\nu},\\\\qquad T^{o}=\\\\frac{TU_{e}}{\\\\delta}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S5.E11\",\"source_text\":\"${{f\\\\phi_{pp}}}^{+}=g_{1}(T^{+})+g_{2}(T^{o};\\\\delta^{+}),\\\\qquad T=1/f,\\\\qquad T^% {+}\\\\equiv\\\\frac{Tu_{\\\\tau}^{2}}{\\\\nu},\\\\qquad T^{o}=\\\\frac{TU_{e}}{\\\\delta}.$ (11)\",\"tag\":\"(11)\",\"text\":\"𝑇𝑢 2 𝑇𝑈𝑒 + + 𝑜 + + 𝜏 𝑜 𝑓 𝜙𝑝 𝑝 = 𝑔1(𝑇 ) + 𝑔2(𝑇 𝛿 ), 𝑇 = 1/ 𝑓 𝑇 ≡ , 𝑇 = . (11) 𝜈 𝛿\"},\"formula_index\":5,\"parent_page_sha256\":\"ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e.pdf","asset_sha256":"ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e","document_id":"arxiv:2608.16729","kind":"formula","page_number":7,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[139.68,412.92,382.32,35.42],\"bbox_pdf\":[139.68,343.65,522.0,379.08],\"fragments\":[{\"bbox_pdf\":[139.68,343.65,522.0,379.08],\"words\":[3595,3596,3597,3598,3599,3600,3601,3602,3603,3604,3605,3606,3607,3608,3609,3610,3612,3613,3614,3615,3616,3618,3619,3620,3621,3622,3623,3624,3625,3626,3627,3628,3629]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"{f\\\\phi_{pp,r}}^{+}=g_{\\\\mathcal{L}}^{r}(T^{s};\\\\mathcal{R}_{s})+g_{\\\\mathcal{Q}}^%\\n{r}(T^{o};\\\\eta_{s},\\\\mathcal{R}_{s}),\\\\qquad T^{s}\\\\equiv\\\\frac{TU_{e}}{k_{s}},%\\n\\\\qquad\\\\eta_{s}\\\\equiv\\\\ln\\\\!\\\\left(\\\\frac{\\\\delta}{k_{s}}\\\\right),\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"S5.E12\",\"source_text\":\"${f\\\\phi_{pp,r}}^{+}=g_{\\\\mathcal{L}}^{r}(T^{s};\\\\mathcal{R}_{s})+g_{\\\\mathcal{Q}}^% {r}(T^{o};\\\\eta_{s},\\\\mathcal{R}_{s}),\\\\qquad T^{s}\\\\equiv\\\\frac{TU_{e}}{k_{s}},% \\\\qquad\\\\eta_{s}\\\\equiv\\\\ln\\\\!\\\\left(\\\\frac{\\\\delta}{k_{s}}\\\\right),$ (12)\",\"tag\":\"(12)\",\"text\":\"(︃ )︃ + 𝑟 𝑠 𝑟 𝑜 𝑠 𝑇𝑈𝑒 𝛿 𝑓 𝜙𝑝 𝑝,𝑟 = 𝑔 L(𝑇 R𝑠) + 𝑔 Q (𝑇 𝜂𝑠, R𝑠), 𝑇 ≡ , 𝜂𝑠 ≡ ln , (12) 𝑘𝑠 𝑘𝑠\"},\"formula_index\":7,\"parent_page_sha256\":\"ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e.pdf","asset_sha256":"ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e","document_id":"arxiv:2608.16729","kind":"formula","page_number":7,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[200.37,547.54,321.63,39.21],\"bbox_pdf\":[200.37,205.25,522.0,244.46],\"fragments\":[{\"bbox_pdf\":[200.37,205.25,522.0,244.46],\"words\":[3720,3721,3722,3723,3724,3725,3726,3727,3728,3729,3730,3731,3732,3733,3734,3735,3736,3737,3738,3739]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathcal{N}(T;T_{b},\\\\sigma,A)=\\\\frac{A}{\\\\sqrt{2\\\\pi}\\\\sigma}\\\\exp\\\\!\\\\left[-\\\\frac{1}%\\n{2}\\\\!\\\\left(\\\\frac{\\\\ln(T/T_{b})}{\\\\sigma}\\\\right)^{\\\\!2}\\\\right],\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S5.E13\",\"source_text\":\"$\\\\mathcal{N}(T;T_{b},\\\\sigma,A)=\\\\frac{A}{\\\\sqrt{2\\\\pi}\\\\sigma}\\\\exp\\\\!\\\\left[-\\\\frac{1}% {2}\\\\!\\\\left(\\\\frac{\\\\ln(T/T_{b})}{\\\\sigma}\\\\right)^{\\\\!2}\\\\right],$ (13)\",\"tag\":\"(13)\",\"text\":\"[︄ (︃ )︃2]︄ 𝐴 1 ln(𝑇/𝑇𝑏) N (𝑇; 𝑇𝑏, 𝜎, 𝐴) = √ exp − , (13) 2𝜋𝜎 2 𝜎\"},\"formula_index\":9,\"parent_page_sha256\":\"ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e.pdf","asset_sha256":"ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e","document_id":"arxiv:2608.16729","kind":"formula","page_number":7,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[226.52,610.15,295.48,50.91],\"bbox_pdf\":[226.52,130.95,522.0,181.85],\"fragments\":[{\"bbox_pdf\":[226.52,130.95,522.0,181.85],\"words\":[3758,3759,3760,3761,3762,3763,3764,3765,3766,3767,3769,3770,3771,3772,3773,3774,3775,3776,3777,3778,3779,3780,3781,3782,3783,3784,3785,3786,3787,3788,3789,3790,3791,3793,3794,3795,3796,3797,3798,3799,3800,3801,3802,3803,3804,3805,3806]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle g_{\\\\mathcal{L}}^{r}(T^{s}) \\\\displaystyle=\\\\mathcal{N}\\\\!\\\\left(T^{s};T_{b\\\\mathcal{L}}^{s,r},\\\\sigma_{\\\\mathcal%\\n{L}}^{r},B_{\\\\mathcal{L}}^{r}\\\\right), \\\\\\\\ \\\\displaystyle g_{\\\\mathcal{Q}}^{r}(T^{o}) \\\\displaystyle=\\\\mathcal{N}\\\\!\\\\left(T^{o};T_{b\\\\mathcal{Q}}^{o,r},\\\\sigma_{\\\\mathcal%\\n{Q}}^{r},A_{\\\\mathcal{Q},\\\\mathrm{obs}}^{r}\\\\right), \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"formula\",\"source_ref\":\"S9.EGx1\",\"source_text\":\"$\\\\displaystyle g_{\\\\mathcal{L}}^{r}(T^{s})$ $\\\\displaystyle=\\\\mathcal{N}\\\\!\\\\left(T^{s};T_{b\\\\mathcal{L}}^{s,r},\\\\sigma_{\\\\mathcal% {L}}^{r},B_{\\\\mathcal{L}}^{r}\\\\right),$ (14) $\\\\displaystyle g_{\\\\mathcal{Q}}^{r}(T^{o})$ $\\\\displaystyle=\\\\mathcal{N}\\\\!\\\\left(T^{o};T_{b\\\\mathcal{Q}}^{o,r},\\\\sigma_{\\\\mathcal% {Q}}^{r},A_{\\\\mathcal{Q},\\\\mathrm{obs}}^{r}\\\\right),$ (15)\",\"tag\":\"(14) (15)\",\"text\":\"(︂ )︂ 𝑔 𝑟 𝑠 ) = N 𝑇 𝑠 𝑇 𝑠,𝑟 , 𝜎 𝑟 𝑟 , (14) L(𝑇 𝑏 L L, 𝐵L (︂ )︂ 𝑔 𝑟 𝑜 ) = N 𝑇 𝑜 𝑇 𝑜,𝑟 , 𝜎 𝑟 𝑟 , (15) Q (𝑇 𝑏 Q Q, 𝐴Q,obs\"},\"formula_index\":11,\"parent_page_sha256\":\"ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e.pdf","asset_sha256":"ff85973d6519ef15c2adc6c305db6d3e3eb7132aa00229dbcf8cb2c3064dc89e","document_id":"arxiv:2608.16729","kind":"formula","page_number":7,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[90.0,90.05,152.06,10.91],\"bbox_pdf\":[90.0,691.04,242.06,701.95],\"fragments\":[{\"bbox_pdf\":[90.0,691.04,242.06,701.95],\"words\":[3854,3855,3856,3857,3858,3859,3860]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph67\",\"source_text\":\"with the nonlinear area set by (10),\",\"tag\":null,\"text\":\"with the nonlinear area set by (10),\"},{\"bbox\":[210.83,112.99,311.18,16.14],\"bbox_pdf\":[210.83,662.87,522.01,679.01],\"fragments\":[{\"bbox_pdf\":[210.83,662.87,522.01,679.01],\"words\":[3861,3862,3863,3864,3865,3866,3867,3868,3869,3870,3871,3872,3873,3874,3875,3876,3877,3878]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"A_{\\\\mathcal{Q},\\\\mathrm{obs}}^{r}=A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\eta_{s%\\n}+B_{\\\\mathcal{Q},G}^{r},\\\\qquad A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\approx 2.3,\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S5.E16\",\"source_text\":\"$A_{\\\\mathcal{Q},\\\\mathrm{obs}}^{r}=A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\eta_{s% }+B_{\\\\mathcal{Q},G}^{r},\\\\qquad A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\approx 2.3,$ (16)\",\"tag\":\"(16)\",\"text\":\"𝐴 𝑟 base 𝑟 𝐴 base (16) Q,obs = 𝐴Q, 𝜂 𝜂𝑠 + 𝐵Q,𝐺, Q, 𝜂 ≈ 2.3,\"},{\"bbox\":[90.0,139.07,432.0,73.69],\"bbox_pdf\":[90.0,579.25,522.0,652.93],\"fragments\":[{\"bbox_pdf\":[90.0,579.25,522.0,652.93],\"words\":[3879,3880,3881,3882,3883,3884,3885,3886,3887,3888,3890,3891,3892,3893,3894,3895,3896,3897,3898,3899,3900,3901,3902,3903,3904,3905,3906,3907,3908,3909,3910,3911,3912,3913,3914,3915,3916,3917,3918,3919,3920,3921,3922,3923,3924,3925,3926,3927,3928,3929,3930,3931,3932,3933,3934,3935,3936,3937,3938,3939,3940,3941,3942,3943,3944,3945,3946,3947,3948,3949,3950,3951,3952,3953,3954,3955,3956,3957,3958,3959,3960,3961]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph69\",\"source_text\":\"with the intercept collecting every non-growing nonlinear contribution of §4 — the inner-scaled sublayer offset and the convergent canopy Green term. Here $A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\equiv a_{\\\\mathcal{Q}}^{\\\\mathrm{src},r}$ is the bare log-range coefficient set by the universal log-layer nonlinear-source density. We take $A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}=A_{\\\\mathcal{Q}}^{\\\\mathrm{smooth}}\\\\approx 2.3$ , the natural-log slope of $\\\\overline{p^{\\\\prime 2}}^{+}$ against $\\\\ln\\\\delta^{+}$ in the smooth-wall data of Schlatter and Örlü (2010); Lee and Moser (2015) (fitted slopes $2.42$ and $2.24$ ).\",\"tag\":null,\"text\":\"with the intercept collecting every non-growing nonlinear contribution of §4 the inner-scaled sublayer offset and the convergent canopy Green term. Here 𝐴 base ≡ 𝑎 src,𝑟 is the bare log-range Q, 𝜂 Q coefficient set by the universal log-layer nonlinear-source density. We take 𝐴 base = 𝐴 smooth ≈ 2.3, Q, 𝜂 Q + ′2 + the natural-log slope of 𝑝 against ln 𝛿 in the smooth-wall data of Schlatter and ¨Orl¨u [2010], Lee and Moser [2015] (fitted slopes 2.42 and 2.24).\"},{\"bbox\":[90.0,214.37,432.0,109.47],\"bbox_pdf\":[90.0,468.16,522.0,577.63],\"fragments\":[{\"bbox_pdf\":[90.0,468.16,522.0,577.63],\"words\":[3962,3963,3964,3965,3966,3967,3968,3969,3970,3971,3972,3973,3974,3975,3976,3977,3978,3979,3981,3982,3983,3984,3985,3986,3987,3988,3989,3990,3991,3992,3993,3994,3995,3996,3997,3998,3999,4000,4001,4002,4003,4004,4005,4006,4007,4008,4009,4010,4011,4012,4013,4014,4015,4016,4017,4018,4019,4020,4021,4022,4023,4024,4025,4026,4027,4028,4029,4030,4031,4032,4033,4034,4035,4036,4037,4038,4039,4040,4041,4042,4043,4044,4045,4046,4047,4048,4049,4050,4051,4052,4053,4054,4055,4056,4057,4058,4059,4060,4061,4062,4063,4064,4065,4066,4067,4068,4069,4070,4071,4072,4073,4074,4075,4076,4077,4078,4079,4080,4081,4082,4083,4084,4085,4086]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph70\",\"source_text\":\"Attributing the entire smooth slope to $\\\\mathcal{Q}$ takes the linear source to carry no $\\\\ln\\\\delta^{+}$ growth — the partition Massey et al. (2026a) measure on the smooth wall, against the classical attribution of the overlap-range spectrum, and hence part of the growth, to the mean-shear (linear) source (Bradshaw, 1967; Panton and Linebarger, 1974). The rate prediction is less sensitive to this partition than it may appear: the log-layer mean shear keeps its smooth-wall form above the sublayer (§3), so any log-layer linear growth would count on $\\\\eta_{s}$ alongside $\\\\mathcal{Q}$ and leave the total coefficient at $\\\\approx 2.3$ ; the partition fixes only which source carries the growth, an attribution a wall-pressure sweep cannot resolve (§4). No $k_{s}^{+}$ -dependent term appears in (16).\",\"tag\":null,\"text\":\"Attributing the entire smooth slope to Q takes the linear source to carry no ln 𝛿 + growth the partition Massey et al. [2026a] measure on the smooth wall, against the classical attribution of the overlap-range spectrum, and hence part of the growth, to the mean-shear (linear) source [Bradshaw, 1967, Panton and Linebarger, 1974]. The rate prediction is less sensitive to this partition than it may appear: the log-layer mean shear keeps its smooth-wall form above the sublayer (§3), so any log-layer linear growth would count on 𝜂𝑠 alongside Q and leave the total coefficient at ≈ 2.3; the partition fixes only which source carries the growth, an attribution a wall-pressure sweep cannot resolve (§4). No 𝑘 + 𝑠 -dependent term appears in (16).\"},{\"bbox\":[90.0,341.91,132.19,14.35],\"bbox_pdf\":[90.0,435.74,222.19,450.09],\"fragments\":[{\"bbox_pdf\":[90.0,435.74,222.19,450.09],\"words\":[4087,4088,4089]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"heading\",\"source_ref\":\"title71\",\"source_text\":\"6 Break frequencies\",\"tag\":null,\"text\":\"6 Break frequencies\"},{\"bbox\":[90.0,370.19,432.0,38.01],\"bbox_pdf\":[90.0,383.8,522.0,421.81],\"fragments\":[{\"bbox_pdf\":[90.0,383.8,522.0,421.81],\"words\":[4090,4091,4092,4093,4094,4095,4096,4097,4098,4099,4100,4101,4102,4103,4104,4105,4106,4107,4108,4109,4110,4111,4112,4113,4114,4115,4116,4117,4118,4119,4120,4121,4122,4123,4124,4125,4126,4127,4128,4129,4130,4131,4132,4133,4134,4135,4136,4137]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph72\",\"source_text\":\"The break frequency of each population follows from where its source sits and how fast that source is carried past the wall. For the roughness-local linear population the break sits on the roughness period axis $T^{s}$ and, at fixed roughness state, should not drift with $\\\\delta^{+}$ :\",\"tag\":null,\"text\":\"The break frequency of each population follows from where its source sits and how fast that source is carried past the wall. For the roughness-local linear population the break sits on the roughness period axis 𝑇 𝑠 and, at fixed roughness state, should not drift with 𝛿 +\"},{\"bbox\":[202.5,418.01,319.5,33.34],\"bbox_pdf\":[202.5,340.65,522.0,373.99],\"fragments\":[{\"bbox_pdf\":[202.5,340.65,522.0,373.99],\"words\":[4139,4140,4141,4142,4143,4144,4145,4146,4147,4148,4149,4150,4151,4152,4153,4154,4155,4156,4157,4158,4159,4160,4161,4162,4163,4164,4165,4166,4167,4168,4169,4170,4171,4172,4173,4174,4175,4176]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"f_{b\\\\mathcal{L}}^{r}\\\\sim\\\\frac{U_{c,\\\\mathcal{L}}^{r}}{C_{b\\\\mathcal{L}}^{r}k_{s}%\\n},\\\\qquad T_{b\\\\mathcal{L}}^{s,r}\\\\equiv\\\\frac{U_{e}}{f_{b\\\\mathcal{L}}^{r}k_{s}}%\\n\\\\sim C_{b\\\\mathcal{L}}^{r}\\\\frac{U_{e}}{U_{c,\\\\mathcal{L}}^{r}},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S6.E17\",\"source_text\":\"$f_{b\\\\mathcal{L}}^{r}\\\\sim\\\\frac{U_{c,\\\\mathcal{L}}^{r}}{C_{b\\\\mathcal{L}}^{r}k_{s}% },\\\\qquad T_{b\\\\mathcal{L}}^{s,r}\\\\equiv\\\\frac{U_{e}}{f_{b\\\\mathcal{L}}^{r}k_{s}}% \\\\sim C_{b\\\\mathcal{L}}^{r}\\\\frac{U_{e}}{U_{c,\\\\mathcal{L}}^{r}},$ (17)\",\"tag\":\"(17)\",\"text\":\"𝑈 𝑟 𝑟 𝑐, L 𝑠,𝑟 𝑈𝑒 𝑟 𝑈𝑒 𝑓 𝑏 L ∼ 𝑟 , 𝑇 𝑏 L ≡ 𝑟 ∼ 𝐶 𝑏 L 𝑟 , (17) 𝐶 𝑏 L 𝑘𝑠 𝑓 𝑏 L 𝑘𝑠 𝑈 𝑐, L\"},{\"bbox\":[90.0,460.25,432.0,52.58],\"bbox_pdf\":[90.0,279.18,522.0,331.75],\"fragments\":[{\"bbox_pdf\":[90.0,279.18,522.0,331.75],\"words\":[4177,4178,4179,4180,4181,4182,4183,4184,4185,4186,4187,4188,4189,4190,4191,4192,4193,4194,4195,4196,4197,4198,4199,4200,4201,4202,4203,4204,4205,4206,4207,4208,4209,4210,4211,4212,4213,4214,4215,4216,4217,4218,4219,4220,4221,4222,4223,4224,4225,4227,4228,4229,4230,4231,4232,4233,4234,4235,4236,4237,4238,4239]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph74\",\"source_text\":\"with $U_{c,\\\\mathcal{L}}^{r}$ the convection velocity of the roughness-local motions and $C_{b\\\\mathcal{L}}^{r}=O(1)$ . What sets the numerical value of the break is the physical mechanism behind the feature, and two natural candidates give predictions that differ by about a factor of five — a gap wide enough that the fit of §7 can tell them apart.\",\"tag\":null,\"text\":\"with 𝑈 𝑟 the convection velocity of the roughness-local motions and 𝐶 𝑟 = 𝑂 (1). What sets 𝑐, L 𝑏 L the numerical value of the break is the physical mechanism behind the feature, and two natural candidates give predictions that differ by about a factor of five a gap wide enough that the fit of §7 can tell them apart.\"},{\"bbox\":[90.0,515.46,432.0,78.66],\"bbox_pdf\":[90.0,197.88,522.0,276.54],\"fragments\":[{\"bbox_pdf\":[90.0,197.88,522.0,276.54],\"words\":[4240,4241,4242,4243,4244,4245,4246,4247,4248,4249,4250,4251,4252,4253,4254,4255,4256,4257,4258,4259,4260,4261,4262,4263,4264,4265,4266,4267,4268,4269,4270,4271,4272,4273,4274,4275,4276,4277,4278,4279,4280,4281,4282,4283,4284,4285,4286,4287,4288,4289,4290,4291,4292,4293,4294,4295,4296,4297,4298,4299,4300,4301,4302,4303,4304,4305,4306,4307,4308,4309,4310,4311,4312,4313,4314,4315,4316,4317,4318,4319,4320,4321,4322,4323,4324,4325,4326,4327,4328,4329,4330,4331]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph75\",\"source_text\":\"The first candidate is unsteady vortex shedding from the roughness elements. The VT bed is staggered circular cylinders of diameter $d_{c}=3.14$ mm (Totten et al., 2026), and a cylinder in cross-flow sheds a Kármán vortex street at the Strouhal frequency $f_{s}=St_{d_{c}}\\\\,U/d_{c}$ , with $St_{d_{c}}\\\\approx 0.2$ over a broad Reynolds-number range. The approach speed the elements feel is not the edge velocity but the reduced canopy velocity $U_{d}\\\\approx(0.4$ – $0.5)U_{e}$ they sit in low in the layer, so $f_{s}=St_{d_{c}}\\\\,U_{d}/d_{c}$ and, on the roughness period axis,\",\"tag\":null,\"text\":\"The first candidate is unsteady vortex shedding from the roughness elements. The VT bed is staggered circular cylinders of diameter 𝑑𝑐 = 3.14 mm [Totten et al., 2026], and a cylinder in cross-flow sheds a K´arm´an vortex street at the Strouhal frequency 𝑓𝑠 = 𝑆𝑡𝑑𝑐 𝑈/𝑑𝑐, with 𝑆𝑡𝑑𝑐 ≈ 0.2 over a broad Reynolds-number range. The approach speed the elements feel is not the edge velocity but the reduced canopy velocity 𝑈𝑑 ≈ (0.4–0.5)𝑈𝑒 they sit in low in the layer, so 𝑓𝑠 = 𝑆𝑡𝑑𝑐 𝑈𝑑/𝑑𝑐 and, on the roughness period axis,\"},{\"bbox\":[243.99,604.91,278.01,28.61],\"bbox_pdf\":[243.99,158.48,522.0,187.09],\"fragments\":[{\"bbox_pdf\":[243.99,158.48,522.0,187.09],\"words\":[4332,4333,4334,4335,4336,4337,4338,4339,4340,4341,4342,4343,4344,4345,4346,4347]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"T_{b\\\\mathcal{L}}^{s,r}=\\\\frac{U_{e}}{f_{s}k_{s}}=\\\\frac{1}{St_{d_{c}}}\\\\,\\\\frac{U_%\\n{e}}{U_{d}}\\\\,\\\\frac{d_{c}}{k_{s}},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S6.E18\",\"source_text\":\"$T_{b\\\\mathcal{L}}^{s,r}=\\\\frac{U_{e}}{f_{s}k_{s}}=\\\\frac{1}{St_{d_{c}}}\\\\,\\\\frac{U_% {e}}{U_{d}}\\\\,\\\\frac{d_{c}}{k_{s}},$ (18)\",\"tag\":\"(18)\",\"text\":\"𝑠,𝑟 𝑈𝑒 1 𝑈𝑒 𝑑𝑐 𝑇 𝑏 L = = , (18) 𝑓𝑠𝑘𝑠 𝑆𝑡𝑑𝑐 𝑈𝑑 𝑘𝑠\"},{\"bbox\":[90.0,641.51,432.0,52.58],\"bbox_pdf\":[90.0,97.91,522.0,150.49],\"fragments\":[{\"bbox_pdf\":[90.0,97.91,522.0,150.49],\"words\":[4348,4349,4350,4351,4352,4353,4354,4355,4356,4357,4358,4359,4360,4361,4363,4364,4365,4366,4367,4368,4369,4370,4371,4372,4373,4374,4375,4376,4377,4378,4379,4380,4381,4382,4383,4384,4385,4386,4387,4388,4389,4390,4391,4392,4393,4394,4395,4396,4397,4398,4399,4400,4401,4402,4404,4405,4406,4407,4408]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.978,\"verified\":true},\"order\":10,\"role\":\"paragraph\",\"source_ref\":\"paragraph77\",\"source_text\":\"fixed by the geometric ratio $d_{c}/k_{s}$ and the convection deficit, not by $\\\\delta^{+}$ . With $St_{d_{c}}\\\\approx 0.2$ , $U_{d}/U_{e}\\\\approx 0.4$ – $0.5$ and $d_{c}/k_{s}\\\\approx 0.95$ , (18) predicts $T_{b\\\\mathcal{L}}^{s,r}\\\\approx 10$ – $12$ . Shedding is a slow dynamical instability, so it would place the feature at long period, well to the low-frequency (large- $T^{s}$ ) side of the roughness axis.\",\"tag\":null,\"text\":\"fixed by the geometric ratio 𝑑𝑐/𝑘𝑠 and the convection deficit, not by 𝛿 + With 𝑆𝑡𝑑𝑐 ≈ 0.2, 𝑈𝑑/𝑈𝑒 ≈ 0.4–0.5 and 𝑑𝑐/𝑘𝑠 ≈ 0.95, (18) predicts 𝑇 𝑠,𝑟 ≈ 10–12. Shedding is a slow dynamical instability, so 𝑏 L it would place the feature at long period, well to the low-frequency (large-𝑇 𝑠 side of the roughness axis.\"},{\"bbox\":[303.27,720.98,5.45,10.91],\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"fragments\":[{\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"words\":[9387]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":11,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"8\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0007.pdf\",\"page_no\":8,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph77\",\"c_block\":0.975,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.997,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.975,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"69fb66d29559c858e6a8afb89f7bc3df55853ad9785c91e8de65153b2f6c4f27\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/69fb66d29559c858e6a8afb89f7bc3df55853ad9785c91e8de65153b2f6c4f27.pdf","asset_sha256":"69fb66d29559c858e6a8afb89f7bc3df55853ad9785c91e8de65153b2f6c4f27","document_id":"arxiv:2608.16729","kind":"page","page_number":8,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[210.83,112.99,311.18,16.14],\"bbox_pdf\":[210.83,662.87,522.01,679.01],\"fragments\":[{\"bbox_pdf\":[210.83,662.87,522.01,679.01],\"words\":[3861,3862,3863,3864,3865,3866,3867,3868,3869,3870,3871,3872,3873,3874,3875,3876,3877,3878]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"A_{\\\\mathcal{Q},\\\\mathrm{obs}}^{r}=A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\eta_{s%\\n}+B_{\\\\mathcal{Q},G}^{r},\\\\qquad A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\approx 2.3,\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S5.E16\",\"source_text\":\"$A_{\\\\mathcal{Q},\\\\mathrm{obs}}^{r}=A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\eta_{s% }+B_{\\\\mathcal{Q},G}^{r},\\\\qquad A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\approx 2.3,$ (16)\",\"tag\":\"(16)\",\"text\":\"𝐴 𝑟 base 𝑟 𝐴 base (16) Q,obs = 𝐴Q, 𝜂 𝜂𝑠 + 𝐵Q,𝐺, Q, 𝜂 ≈ 2.3,\"},\"formula_index\":1,\"parent_page_sha256\":\"69fb66d29559c858e6a8afb89f7bc3df55853ad9785c91e8de65153b2f6c4f27\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/69fb66d29559c858e6a8afb89f7bc3df55853ad9785c91e8de65153b2f6c4f27.pdf","asset_sha256":"69fb66d29559c858e6a8afb89f7bc3df55853ad9785c91e8de65153b2f6c4f27","document_id":"arxiv:2608.16729","kind":"formula","page_number":8,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[202.5,418.01,319.5,33.34],\"bbox_pdf\":[202.5,340.65,522.0,373.99],\"fragments\":[{\"bbox_pdf\":[202.5,340.65,522.0,373.99],\"words\":[4139,4140,4141,4142,4143,4144,4145,4146,4147,4148,4149,4150,4151,4152,4153,4154,4155,4156,4157,4158,4159,4160,4161,4162,4163,4164,4165,4166,4167,4168,4169,4170,4171,4172,4173,4174,4175,4176]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"f_{b\\\\mathcal{L}}^{r}\\\\sim\\\\frac{U_{c,\\\\mathcal{L}}^{r}}{C_{b\\\\mathcal{L}}^{r}k_{s}%\\n},\\\\qquad T_{b\\\\mathcal{L}}^{s,r}\\\\equiv\\\\frac{U_{e}}{f_{b\\\\mathcal{L}}^{r}k_{s}}%\\n\\\\sim C_{b\\\\mathcal{L}}^{r}\\\\frac{U_{e}}{U_{c,\\\\mathcal{L}}^{r}},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S6.E17\",\"source_text\":\"$f_{b\\\\mathcal{L}}^{r}\\\\sim\\\\frac{U_{c,\\\\mathcal{L}}^{r}}{C_{b\\\\mathcal{L}}^{r}k_{s}% },\\\\qquad T_{b\\\\mathcal{L}}^{s,r}\\\\equiv\\\\frac{U_{e}}{f_{b\\\\mathcal{L}}^{r}k_{s}}% \\\\sim C_{b\\\\mathcal{L}}^{r}\\\\frac{U_{e}}{U_{c,\\\\mathcal{L}}^{r}},$ (17)\",\"tag\":\"(17)\",\"text\":\"𝑈 𝑟 𝑟 𝑐, L 𝑠,𝑟 𝑈𝑒 𝑟 𝑈𝑒 𝑓 𝑏 L ∼ 𝑟 , 𝑇 𝑏 L ≡ 𝑟 ∼ 𝐶 𝑏 L 𝑟 , (17) 𝐶 𝑏 L 𝑘𝑠 𝑓 𝑏 L 𝑘𝑠 𝑈 𝑐, L\"},\"formula_index\":6,\"parent_page_sha256\":\"69fb66d29559c858e6a8afb89f7bc3df55853ad9785c91e8de65153b2f6c4f27\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/69fb66d29559c858e6a8afb89f7bc3df55853ad9785c91e8de65153b2f6c4f27.pdf","asset_sha256":"69fb66d29559c858e6a8afb89f7bc3df55853ad9785c91e8de65153b2f6c4f27","document_id":"arxiv:2608.16729","kind":"formula","page_number":8,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[243.99,604.91,278.01,28.61],\"bbox_pdf\":[243.99,158.48,522.0,187.09],\"fragments\":[{\"bbox_pdf\":[243.99,158.48,522.0,187.09],\"words\":[4332,4333,4334,4335,4336,4337,4338,4339,4340,4341,4342,4343,4344,4345,4346,4347]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"T_{b\\\\mathcal{L}}^{s,r}=\\\\frac{U_{e}}{f_{s}k_{s}}=\\\\frac{1}{St_{d_{c}}}\\\\,\\\\frac{U_%\\n{e}}{U_{d}}\\\\,\\\\frac{d_{c}}{k_{s}},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S6.E18\",\"source_text\":\"$T_{b\\\\mathcal{L}}^{s,r}=\\\\frac{U_{e}}{f_{s}k_{s}}=\\\\frac{1}{St_{d_{c}}}\\\\,\\\\frac{U_% {e}}{U_{d}}\\\\,\\\\frac{d_{c}}{k_{s}},$ (18)\",\"tag\":\"(18)\",\"text\":\"𝑠,𝑟 𝑈𝑒 1 𝑈𝑒 𝑑𝑐 𝑇 𝑏 L = = , (18) 𝑓𝑠𝑘𝑠 𝑆𝑡𝑑𝑐 𝑈𝑑 𝑘𝑠\"},\"formula_index\":9,\"parent_page_sha256\":\"69fb66d29559c858e6a8afb89f7bc3df55853ad9785c91e8de65153b2f6c4f27\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/69fb66d29559c858e6a8afb89f7bc3df55853ad9785c91e8de65153b2f6c4f27.pdf","asset_sha256":"69fb66d29559c858e6a8afb89f7bc3df55853ad9785c91e8de65153b2f6c4f27","document_id":"arxiv:2608.16729","kind":"formula","page_number":8,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[89.62,90.05,432.39,92.2],\"bbox_pdf\":[89.62,609.75,522.01,701.95],\"fragments\":[{\"bbox_pdf\":[89.62,609.75,522.01,701.95],\"words\":[4409,4410,4411,4412,4413,4414,4415,4416,4417,4418,4419,4420,4421,4422,4423,4424,4425,4426,4427,4428,4429,4430,4431,4432,4433,4434,4435,4436,4437,4438,4439,4440,4441,4442,4443,4444,4445,4446,4447,4448,4449,4450,4451,4452,4453,4454,4455,4456,4457,4458,4459,4460,4461,4462,4463,4464,4465,4466,4467,4469,4470,4471,4472,4473,4474,4475,4476,4477,4478,4479,4480,4481,4482,4483,4484,4485,4486,4487,4488,4489,4490,4491,4492,4493,4494,4495,4496,4498,4499,4500,4501,4502,4503,4504,4505,4506,4507,4508,4509,4510,4511,4512,4513,4514,4515,4516,4517,4518,4519,4520,4521,4522,4523,4524,4525]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph78\",\"source_text\":\"The second candidate is simpler: an element-scale structure convected past the wall at the roughness convection velocity, with no instability frequency of its own. Using the measured $U_{c}\\\\approx 0.49\\\\,U_{e}$ (Totten et al., 2026) and a streamwise wavelength of order the element size, $\\\\lambda_{x}\\\\sim k_{s}$ , this feature falls at $T_{b\\\\mathcal{L}}^{s,r}=U_{e}\\\\lambda_{x}/(U_{c}k_{s})\\\\sim U_{e}/U_{c}\\\\approx 2$ — of order unity on the roughness axis, a factor of about five below the shedding value. The two mechanisms are thus cleanly separated on $T^{s}$ : shedding near $T^{s}\\\\approx 10$ – $12$ , convected element-scale structure near $T^{s}\\\\sim 2$ . Which of the two the data select is left to the fit of §7.\",\"tag\":null,\"text\":\"The second candidate is simpler: an element-scale structure convected past the wall at the roughness convection velocity, with no instability frequency of its own. Using the measured 𝑈𝑐 ≈ 0.49 𝑈𝑒 [Totten et al., 2026] and a streamwise wavelength of order the element size, 𝜆𝑥 ∼ 𝑘𝑠, this feature falls at 𝑇 𝑠,𝑟 = 𝑈𝑒𝜆𝑥/(𝑈𝑐𝑘𝑠) ∼ 𝑈𝑒/𝑈𝑐 ≈ 2 of order unity on the roughness axis, a 𝑏 L factor of about five below the shedding value. The two mechanisms are thus cleanly separated on 𝑇 𝑠 shedding near 𝑇 𝑠 ≈ 10–12, convected element-scale structure near 𝑇 𝑠 ∼ 2. Which of the two the data select is left to the fit of §7.\"},{\"bbox\":[106.94,184.89,401.53,12.7],\"bbox_pdf\":[106.94,594.41,508.46,607.11],\"fragments\":[{\"bbox_pdf\":[106.94,594.41,508.46,607.11],\"words\":[4526,4527,4528,4529,4530,4531,4532,4533,4534,4535,4536,4537,4538,4539,4540]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph79\",\"source_text\":\"For the nonlinear population, let the wall-effective active source height be $y_{\\\\mathcal{Q},\\\\mathrm{eff}}\\\\gtrsim k_{s}$ . Then\",\"tag\":null,\"text\":\"For the nonlinear population, let the wall-effective active source height be 𝑦Q,eff ≳ 𝑘𝑠. Then\"},{\"bbox\":[235.56,204.17,286.44,30.51],\"bbox_pdf\":[235.56,557.33,522.0,587.83],\"fragments\":[{\"bbox_pdf\":[235.56,557.33,522.0,587.83],\"words\":[4541,4542,4543,4544,4545,4546,4547,4548,4549,4550,4551,4552,4553,4554,4555,4556,4557,4558,4559]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"T_{b\\\\mathcal{Q}}^{o,r}\\\\sim C_{b\\\\mathcal{Q}}^{r}\\\\,\\\\frac{U_{e}}{U_{c,\\\\mathcal{Q}%\\n}^{r}(y_{\\\\mathcal{Q},\\\\mathrm{eff}})}\\\\frac{y_{\\\\mathcal{Q},\\\\mathrm{eff}}}{\\\\delta}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S6.E19\",\"source_text\":\"$T_{b\\\\mathcal{Q}}^{o,r}\\\\sim C_{b\\\\mathcal{Q}}^{r}\\\\,\\\\frac{U_{e}}{U_{c,\\\\mathcal{Q}% }^{r}(y_{\\\\mathcal{Q},\\\\mathrm{eff}})}\\\\frac{y_{\\\\mathcal{Q},\\\\mathrm{eff}}}{\\\\delta}.$ (19)\",\"tag\":\"(19)\",\"text\":\"𝑜,𝑟 𝑟 𝑈𝑒 𝑦Q,eff 𝑇 ∼ 𝐶 𝑏 Q 𝑟 . (19) 𝑏 Q 𝑈 (𝑦Q,eff) 𝛿 𝑐, Q\"},{\"bbox\":[89.62,241.31,432.38,54.21],\"bbox_pdf\":[89.62,496.48,522.0,550.69],\"fragments\":[{\"bbox_pdf\":[89.62,496.48,522.0,550.69],\"words\":[4560,4561,4562,4563,4564,4565,4566,4567,4568,4569,4570,4571,4572,4573,4574,4575,4576,4577,4578,4579,4580,4581,4582,4583,4584,4585,4586,4587,4588,4589,4590,4591,4592,4593,4594,4595,4596,4597,4598,4599,4600,4601,4602,4603,4604,4605,4606,4607,4608,4609,4610,4611,4612,4613,4614,4615,4616,4617,4618,4619,4620,4621,4622,4623,4624,4625,4626,4627,4628,4629,4630,4631]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph81\",\"source_text\":\"Here $U_{c,\\\\mathcal{Q}}^{r}(y_{\\\\mathcal{Q},\\\\mathrm{eff}})$ is the convection velocity of the energetic nonlinear motions at that height and $C_{b\\\\mathcal{Q}}^{r}$ an $O(1)$ prefactor. At fixed roughness state an increase in $\\\\delta/k_{s}$ should grow the area of $g_{\\\\mathcal{Q}}^{r}$ through (16) without moving the break; a break shift in $T^{o}$ would instead signal a change in $y_{\\\\mathcal{Q},\\\\mathrm{eff}}/\\\\delta$ , $U_{c,\\\\mathcal{Q}}^{r}/U_{e}$ , or the Green-weighted source distribution.\",\"tag\":null,\"text\":\"Here 𝑈 𝑟 (𝑦Q,eff) is the convection velocity of the energetic nonlinear motions at that height and 𝑐, Q 𝐶 𝑟 an 𝑂 (1) prefactor. At fixed roughness state an increase in 𝛿/𝑘𝑠 should grow the area of 𝑔 𝑟 𝑏 Q Q through (16) without moving the break; a break shift in 𝑇 𝑜 would instead signal a change in 𝑦Q,eff/𝛿, 𝑈 𝑟 /𝑈𝑒, or the Green-weighted source distribution. 𝑐, Q\"},{\"bbox\":[89.88,296.53,432.12,92.2],\"bbox_pdf\":[89.88,403.27,522.0,495.47],\"fragments\":[{\"bbox_pdf\":[89.88,403.27,522.0,495.47],\"words\":[4632,4633,4634,4635,4636,4637,4638,4639,4640,4641,4642,4643,4644,4645,4646,4647,4648,4649,4650,4651,4652,4653,4654,4655,4656,4658,4659,4660,4661,4662,4663,4664,4665,4666,4667,4668,4669,4670,4671,4672,4673,4674,4675,4676,4677,4678,4679,4680,4681,4682,4683,4684,4685,4686,4687,4688,4689,4690,4691,4692,4693,4694,4695,4696,4697,4698,4699,4700,4701,4702,4703,4704,4705,4706,4707,4708,4709,4710,4711,4712,4713,4714,4715,4716,4717,4718,4719,4720,4721,4722,4723,4724,4725,4726,4727,4728,4729,4730,4731,4732,4733,4734,4735,4736,4737,4738,4739,4740,4741,4742,4743,4744,4745,4746,4747,4748,4749,4750,4751,4752,4753,4754,4755,4756,4757,4758,4759,4760,4761,4762,4763,4764,4765,4767,4768,4769]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.977,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph82\",\"source_text\":\"The prefactor is not pinned by the construction, and the inferred source height depends on it inversely ( $y_{\\\\mathcal{Q},\\\\mathrm{eff}}/\\\\delta=T_{b\\\\mathcal{Q}}^{o,r}(U_{c,\\\\mathcal{Q}}^{% r}/U_{e})/C_{b\\\\mathcal{Q}}^{r}$ ); the elliptic kernel constrains the admissible combinations. At the energetic peak $T^{o}\\\\approx 1$ with $U_{c}\\\\approx 0.5\\\\,U_{e}$ the convected wavelength is $\\\\lambda_{x}\\\\approx 0.5\\\\,\\\\delta$ , and a source at $y_{d}=0.5\\\\,\\\\delta$ ( $C_{b\\\\mathcal{Q}}^{r}=1$ ) has $ky_{d}\\\\approx 2\\\\pi$ and negligible kernel weight $e^{-4\\\\pi}$ at the wall; an $O(1)$ kernel ( $ky_{d}\\\\lesssim 1$ ) instead puts the source in the log layer, $y_{\\\\mathcal{Q},\\\\mathrm{eff}}/\\\\delta\\\\lesssim 0.1$ , with $C_{b\\\\mathcal{Q}}^{r}=O(2\\\\pi)$ absorbing the aspect ratio of wall-attached structures. The wall spectrum alone does not distinguish the two readings; § 7 reports both.\",\"tag\":null,\"text\":\"The prefactor is not pinned by the construction, and the inferred source height depends on it inversely (𝑦Q,eff/𝛿 = 𝑇 𝑜,𝑟 (𝑈 𝑟 /𝑈𝑒)/𝐶 𝑟 the elliptic kernel constrains the admissible combina- 𝑏 Q 𝑐, Q 𝑏 Q tions. At the energetic peak 𝑇 𝑜 ≈ 1 with 𝑈𝑐 ≈ 0.5 𝑈𝑒 the convected wavelength is 𝜆𝑥 ≈ 0.5 𝛿, and a source at 𝑦𝑑 = 0.5 𝛿 (𝐶 𝑟 = 1) has 𝑘 𝑦𝑑 ≈ 2𝜋 and negligible kernel weight 𝑒 −4 𝜋 at the wall; an 𝑏 Q 𝑂 (1) kernel (𝑘 𝑦𝑑 ≲ 1) instead puts the source in the log layer, 𝑦Q,eff/𝛿 ≲ 0.1, with 𝐶 𝑟 = 𝑂 (2𝜋) 𝑏 Q absorbing the aspect ratio of wall-attached structures. The wall spectrum alone does not distinguish the two readings; 7 reports both.\"},{\"bbox\":[90.0,409.13,201.08,14.35],\"bbox_pdf\":[90.0,368.52,291.08,382.87],\"fragments\":[{\"bbox_pdf\":[90.0,368.52,291.08,382.87],\"words\":[4770,4771,4772,4773,4774,4775,4776]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"heading\",\"source_ref\":\"title83\",\"source_text\":\"7 Model results at the VT cases\",\"tag\":null,\"text\":\"7 Model results at the VT cases\"},{\"bbox\":[89.62,437.41,432.39,202.23],\"bbox_pdf\":[89.62,152.36,522.01,354.59],\"fragments\":[{\"bbox_pdf\":[89.62,152.36,522.01,354.59],\"words\":[4777,4778,4779,4780,4781,4782,4783,4784,4785,4786,4787,4788,4789,4790,4791,4792,4793,4794,4795,4796,4797,4798,4799,4800,4801,4802,4803,4804,4805,4806,4807,4808,4809,4810,4811,4812,4813,4814,4815,4816,4817,4818,4819,4820,4821,4822,4823,4824,4825,4826,4827,4828,4829,4830,4831,4832,4833,4834,4835,4836,4837,4838,4839,4840,4841,4842,4843,4844,4845,4846,4847,4848,4849,4850,4851,4852,4853,4854,4855,4856,4857,4858,4859,4860,4861,4862,4863,4864,4865,4866,4867,4868,4869,4870,4871,4872,4873,4874,4875,4876,4877,4878,4879,4880,4881,4882,4883,4884,4885,4886,4887,4888,4890,4891,4892,4893,4894,4896,4897,4898,4899,4900,4901,4902,4903,4904,4905,4906,4907,4908,4909,4910,4911,4912,4913,4914,4915,4916,4917,4918,4919,4920,4921,4922,4923,4924,4925,4926,4927,4928,4929,4930,4931,4932,4933,4934,4935,4936,4937,4938,4939,4940,4941,4942,4943,4944,4945,4946,4947,4948,4949,4950,4951,4952,4953,4954,4955,4956,4957,4958,4959,4960,4961,4962,4963,4964,4965,4966,4967,4968,4970,4971,4972,4973,4974,4975,4976,4977,4978,4979,4980,4981,4982,4983,4984,4985,4986,4987,4988,4989,4990,4991,4992,4993,4994,4995,4996,4997,4998,4999,5000,5001,5002,5003,5004,5005,5006,5007,5008,5009,5010,5011,5012,5013,5014,5015]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph84\",\"source_text\":\"The model of §5 is evaluated at fourteen rough-wall cases from the Virginia Tech Stability Wind Tunnel rough-wall campaign (Vishwanathan et al., 2023; Fritsch et al., 2023): a single cylinder-roughness plate (staggered cylinders, $d_{c}=3.14$ mm, $k_{g}=2$ mm, pitch $6.93$ mm; Totten et al. 2026) at nominally zero pressure gradient (the tunnel’s NACA 0012 at zero angle of attack), sampled at seven streamwise stations $x=1.25$ – $4.91$ m in each of two unit-Reynolds-number families (nominal $U_{e}=35$ and $65$ m/s; measured $U_{e}\\\\approx 32$ – $36$ and $57$ – $65$ across stations): a spatially developing fully rough boundary layer measured along its growth. The measured flow state ( $Re_{\\\\tau}\\\\equiv\\\\delta^{+}$ , $U_{e}$ , $\\\\delta$ , $u_{\\\\tau}$ , $C_{f}$ ) is supplied for each case; the roughness is represented by the equivalent sandgrain height $k_{s}=3.3$ mm, the value implied by the dataset’s own roughness-scaled spectral axes and some $3\\\\%$ above the $k_{s}\\\\approx 1.6k_{g}$ quoted by Totten et al. (2026), whose sub-convective measurements on the same roughness lineage also supply the convection velocity used below. Every case is fully rough, $k_{s}^{+}\\\\in[361,702]$ , with $\\\\delta/k_{s}\\\\in[18,42]$ — low enough that Townsend similarity is an applied premise (Jiménez, 2004), though one the boundary-layer evidence supports down to this range (Schultz and Flack, 2007; Squire et al., 2016). The model treats all fourteen as canonical fully rough states parameterised by $(\\\\delta^{+},k_{s})$ alone.\",\"tag\":null,\"text\":\"The model of §5 is evaluated at fourteen rough-wall cases from the Virginia Tech Stability Wind Tunnel rough-wall campaign [Vishwanathan et al., 2023, Fritsch et al., 2023]: a single cylinderroughness plate (staggered cylinders, 𝑑𝑐 = 3.14 mm, 𝑘𝑔 = 2 mm, pitch 6.93 mm; Totten et al. 2026) at nominally zero pressure gradient (the tunnel’s NACA 0012 at zero angle of attack), sampled at seven streamwise stations 𝑥 = 1.25–4.91 m in each of two unit-Reynolds-number families (nominal 𝑈𝑒 = 35 and 65 m/s; measured 𝑈𝑒 ≈ 32–36 and 57–65 across stations): a spatially developing fully rough boundary layer measured along its growth. The measured flow state (𝑅𝑒𝜏 ≡ 𝛿 + 𝑈𝑒, 𝛿, 𝑢𝜏, 𝐶 𝑓 is supplied for each case; the roughness is represented by the equivalent sandgrain height 𝑘𝑠 = 3.3 mm, the value implied by the dataset’s own roughness-scaled spectral axes and some 3% above the 𝑘𝑠 ≈ 1.6𝑘𝑔 quoted by Totten et al. [2026], whose sub-convective measurements on the same roughness lineage also supply the convection velocity used below. Every case is fully rough, 𝑘 + 𝑠 ∈ [361, 702], with 𝛿/𝑘𝑠 ∈ [18, 42] low enough that Townsend similarity is an applied premise [Jim´enez, 2004], though one the boundary-layer evidence supports down to this range [Schultz and Flack, 2007, Squire et al., 2016]. The model treats all fourteen as canonical fully rough states parameterised by (𝛿 + , 𝑘𝑠) alone.\"},{\"bbox\":[90.0,655.58,86.02,11.96],\"bbox_pdf\":[90.0,124.46,176.02,136.42],\"fragments\":[{\"bbox_pdf\":[90.0,124.46,176.02,136.42],\"words\":[5016,5017]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":2,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"heading\",\"source_ref\":\"title85\",\"source_text\":\"7.1 Calibration\",\"tag\":null,\"text\":\"7.1 Calibration\"},{\"bbox\":[90.0,677.54,432.0,24.46],\"bbox_pdf\":[90.0,90.0,522.0,114.46],\"fragments\":[{\"bbox_pdf\":[90.0,90.0,522.0,114.46],\"words\":[5018,5019,5020,5021,5022,5023,5024,5025,5026,5027,5028,5029,5030,5031,5032,5033,5034,5035,5036,5037,5038,5039,5040,5041,5042,5043,5044,5045,5046,5047,5048,5049,5050,5051]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph86\",\"source_text\":\"The six parameters are estimated in two separate stages. It is worth being explicit that all fourteen cases are used to calibrate: with a single roughness geometry there is no independent set to validate against, so the model is calibrated rather than cross-validated, and its honesty rests on an error bar that reflects how few independent measurements the data really contain (below), not on a train/test split the data cannot support.\",\"tag\":null,\"text\":\"The six parameters are estimated in two separate stages. It is worth being explicit that all fourteen cases are used to calibrate: with a single roughness geometry there is no independent set to validate\"},{\"bbox\":[303.27,720.98,5.45,10.91],\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"fragments\":[{\"bbox_pdf\":[303.27,60.11,308.73,71.02],\"words\":[9388]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":9,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"9\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0008.pdf\",\"page_no\":9,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph82\",\"c_block\":0.974,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.994,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.974,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"c44eb9dd8b49c980d4470029c69afec0aa1b37570cfb4895261937500323bcba\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/c44eb9dd8b49c980d4470029c69afec0aa1b37570cfb4895261937500323bcba.pdf","asset_sha256":"c44eb9dd8b49c980d4470029c69afec0aa1b37570cfb4895261937500323bcba","document_id":"arxiv:2608.16729","kind":"page","page_number":9,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[235.56,204.17,286.44,30.51],\"bbox_pdf\":[235.56,557.33,522.0,587.83],\"fragments\":[{\"bbox_pdf\":[235.56,557.33,522.0,587.83],\"words\":[4541,4542,4543,4544,4545,4546,4547,4548,4549,4550,4551,4552,4553,4554,4555,4556,4557,4558,4559]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"T_{b\\\\mathcal{Q}}^{o,r}\\\\sim C_{b\\\\mathcal{Q}}^{r}\\\\,\\\\frac{U_{e}}{U_{c,\\\\mathcal{Q}%\\n}^{r}(y_{\\\\mathcal{Q},\\\\mathrm{eff}})}\\\\frac{y_{\\\\mathcal{Q},\\\\mathrm{eff}}}{\\\\delta}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S6.E19\",\"source_text\":\"$T_{b\\\\mathcal{Q}}^{o,r}\\\\sim C_{b\\\\mathcal{Q}}^{r}\\\\,\\\\frac{U_{e}}{U_{c,\\\\mathcal{Q}% }^{r}(y_{\\\\mathcal{Q},\\\\mathrm{eff}})}\\\\frac{y_{\\\\mathcal{Q},\\\\mathrm{eff}}}{\\\\delta}.$ (19)\",\"tag\":\"(19)\",\"text\":\"𝑜,𝑟 𝑟 𝑈𝑒 𝑦Q,eff 𝑇 ∼ 𝐶 𝑏 Q 𝑟 . (19) 𝑏 Q 𝑈 (𝑦Q,eff) 𝛿 𝑐, Q\"},\"formula_index\":2,\"parent_page_sha256\":\"c44eb9dd8b49c980d4470029c69afec0aa1b37570cfb4895261937500323bcba\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/c44eb9dd8b49c980d4470029c69afec0aa1b37570cfb4895261937500323bcba.pdf","asset_sha256":"c44eb9dd8b49c980d4470029c69afec0aa1b37570cfb4895261937500323bcba","document_id":"arxiv:2608.16729","kind":"formula","page_number":9,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[90.0,90.05,432.0,38.01],\"bbox_pdf\":[90.0,663.94,522.0,701.95],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[90.0,663.94,522.0,701.95],\"words\":[5052,5053,5054,5055,5056,5057,5058,5059,5060,5061,5062,5063,5064,5065,5066,5067,5068,5069,5070,5071,5072,5073,5074,5075,5076,5077,5078,5079,5080,5081,5082,5083,5084,5085,5086,5087,5088]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph86\",\"source_text\":\"The six parameters are estimated in two separate stages. It is worth being explicit that all fourteen cases are used to calibrate: with a single roughness geometry there is no independent set to validate against, so the model is calibrated rather than cross-validated, and its honesty rests on an error bar that reflects how few independent measurements the data really contain (below), not on a train/test split the data cannot support.\",\"tag\":null,\"text\":\"against, so the model is calibrated rather than cross-validated, and its honesty rests on an error bar that reflects how few independent measurements the data really contain (below), not on a train/test split the data cannot support.\"},{\"bbox\":[89.82,128.64,432.18,102.06],\"bbox_pdf\":[89.82,561.3,522.01,663.36],\"fragments\":[{\"bbox_pdf\":[89.82,561.3,522.01,663.36],\"words\":[5089,5090,5091,5092,5093,5094,5095,5096,5097,5098,5099,5100,5101,5102,5103,5104,5105,5106,5107,5109,5110,5111,5112,5113,5114,5115,5116,5117,5118,5119,5120,5121,5122,5123,5124,5125,5126,5127,5128,5129,5130,5131,5132,5133,5134,5135,5136,5137,5138,5139,5140,5141,5142,5143,5144,5145,5146,5147,5148,5149,5150,5151,5152,5153,5154,5155,5156,5157,5158,5159,5160,5161,5162,5163,5164,5165,5166,5167,5168,5169,5170,5171,5172,5173,5174,5175,5176,5177,5178,5179,5180,5181,5182,5183,5184,5185,5186,5187,5188,5189,5190,5191,5192,5193,5194,5195,5196,5197,5198,5199,5200,5201,5202,5203,5204,5205,5206,5207,5208,5209,5210,5211,5212,5213,5214,5215]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.989,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph87\",\"source_text\":\"Stage A (shape) fits the four shape parameters $(T_{b\\\\mathcal{L}}^{s,r},\\\\sigma_{\\\\mathcal{L}}^{r},T_{b\\\\mathcal{Q}}^{o,r},\\\\sigma% _{\\\\mathcal{Q}}^{r})$ and the linear area $B_{\\\\mathcal{L}}^{r}$ by bounded least squares to the area-normalised spectra ${f\\\\phi_{pp}}^{+}/\\\\overline{p^{\\\\prime 2}}^{+}$ ( $2221$ spectral points across the fourteen cases), where $\\\\overline{p^{\\\\prime 2}}^{+}=\\\\int{f\\\\phi_{pp}}^{+}\\\\,\\\\mathrm{d}\\\\ln f$ is each case’s measured integrated variance. Dividing each spectrum by its own variance removes the $\\\\pm 40\\\\%$ amplitude scatter that the campaign’s $\\\\pm 20\\\\%$ $C_{f}$ uncertainty imposes, leaving only the shape; the non-growing linear population enters as a constant absolute area. Stage B (growth) then tests the predicted coefficient against the fourteen measured variances and calibrates the single remaining offset,\",\"tag\":null,\"text\":\"Stage A (shape) fits the four shape parameters (𝑇 𝑠,𝑟 , 𝜎 𝑟 , 𝑇 𝑜,𝑟 , 𝜎 𝑟 and the linear area 𝐵 𝑟 by 𝑏 L L 𝑏 Q Q L + + ′2 bounded least squares to the area-normalised spectra 𝑓 𝜙𝑝 𝑝 /𝑝 (2221 spectral points across the + ∫ ′2 + fourteen cases), where 𝑝 = 𝑓 𝜙𝑝 𝑝 d ln 𝑓 is each case’s measured integrated variance. Dividing each spectrum by its own variance removes the ±40% amplitude scatter that the campaign’s ±20% 𝐶 𝑓 uncertainty imposes, leaving only the shape; the non-growing linear population enters as a constant absolute area. Stage B (growth) then tests the predicted coefficient against the fourteen measured variances and calibrates the single remaining offset,\"},{\"bbox\":[212.09,246.6,309.91,55.0],\"bbox_pdf\":[212.09,490.4,522.0,545.4],\"fragments\":[{\"bbox_pdf\":[212.09,490.4,522.0,545.4],\"words\":[5216,5217,5218,5219,5220,5221,5222,5223,5224,5225,5226,5227,5228,5229,5230,5231,5232,5233,5234,5235,5236,5237,5238,5239,5240,5241,5242,5243,5244,5245,5246,5247,5248,5249,5250,5251]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\overline{p^{\\\\prime 2}}^{+} \\\\displaystyle=V_{0}+A_{\\\\mathcal{Q},\\\\eta}^{r}\\\\,\\\\eta_{s} \\\\displaystyle\\\\quad\\\\text{(test, $A_{\\\\mathcal{Q},\\\\eta}^{r}$ free)}, \\\\\\\\ \\\\displaystyle\\\\overline{p^{\\\\prime 2}}^{+} \\\\displaystyle=V_{0}+A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\eta_{s} \\\\displaystyle\\\\quad\\\\text{(calibration)}, \\\\\\\\ \\\\displaystyle B_{\\\\mathcal{Q},G}^{r} \\\\displaystyle=V_{0}-B_{\\\\mathcal{L}}^{r}, \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S7.E20\",\"source_text\":\"$\\\\displaystyle\\\\overline{p^{\\\\prime 2}}^{+}$ $\\\\displaystyle=V_{0}+A_{\\\\mathcal{Q},\\\\eta}^{r}\\\\,\\\\eta_{s}$ $\\\\displaystyle\\\\quad\\\\text{(test, $A_{\\\\mathcal{Q},\\\\eta}^{r}$ free)},$ (20) $\\\\displaystyle\\\\overline{p^{\\\\prime 2}}^{+}$ $\\\\displaystyle=V_{0}+A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\eta_{s}$ $\\\\displaystyle\\\\quad\\\\text{(calibration)},$ $\\\\displaystyle B_{\\\\mathcal{Q},G}^{r}$ $\\\\displaystyle=V_{0}-B_{\\\\mathcal{L}}^{r},$\",\"tag\":null,\"text\":\"+ 𝑝′2 = 𝑉0 + 𝐴 𝑟 (test, 𝐴 𝑟 Q, 𝜂 𝜂𝑠 Q, 𝜂 free), + 𝑝′2 = 𝑉0 + 𝐴 base (calibration), (20) Q, 𝜂 𝜂𝑠 𝐵 𝑟 𝑟 Q,𝐺 = 𝑉0 − 𝐵L,\"},{\"bbox\":[90.0,311.51,432.0,24.46],\"bbox_pdf\":[90.0,456.03,522.0,480.49],\"fragments\":[{\"bbox_pdf\":[90.0,456.03,522.0,480.49],\"words\":[5252,5253,5254,5255,5256,5257,5258,5259,5260,5261,5262,5263,5264,5265,5266,5267,5268,5269,5270,5271,5272,5273,5274,5275,5276,5277,5278,5279,5280]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph89\",\"source_text\":\"with $V_{0}$ the intercept and each case’s $\\\\eta_{s}=\\\\ln(\\\\delta/k_{s})$ from its measured $\\\\delta$ ; the slope reads as the nonlinear growth rate because the linear population does not grow.\",\"tag\":null,\"text\":\"with 𝑉0 the intercept and each case’s 𝜂𝑠 = ln(𝛿/𝑘𝑠) from its measured 𝛿; the slope reads as the nonlinear growth rate because the linear population does not grow.\"},{\"bbox\":[90.0,338.61,432.01,146.4],\"bbox_pdf\":[90.0,306.99,522.01,453.39],\"fragments\":[{\"bbox_pdf\":[90.0,306.99,522.01,453.39],\"words\":[5281,5282,5283,5284,5285,5286,5287,5288,5289,5290,5291,5292,5293,5294,5295,5296,5297,5298,5299,5300,5301,5302,5303,5304,5305,5306,5307,5308,5309,5310,5311,5312,5313,5314,5315,5316,5317,5318,5319,5320,5321,5322,5323,5324,5325,5326,5327,5328,5329,5330,5331,5332,5333,5334,5335,5336,5337,5338,5339,5340,5341,5342,5343,5344,5345,5346,5347,5348,5349,5350,5351,5352,5353,5354,5355,5356,5357,5358,5359,5360,5361,5362,5363,5364,5365,5366,5367,5368,5369,5370,5371,5372,5373,5374,5375,5376,5377,5378,5379,5380,5381,5382,5383,5384,5385,5386,5387,5388,5390,5391,5392,5393,5394,5395,5396,5397,5398,5399,5400,5401,5402,5403,5404,5405,5407,5408,5409,5410,5411,5412,5413,5414,5415,5416,5417,5418,5419,5420,5421,5422,5423,5424,5425,5426,5427,5428,5429,5430,5431,5432,5433,5434,5435,5436,5437,5438,5439,5440,5441,5442]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph90\",\"source_text\":\"Keeping the two stages apart matters because they rest on very different amounts of information. The shape is pinned by $2221$ spectral samples freed of amplitude noise; the growth test and the offset rest on a single fourteen-point regression whose $\\\\eta_{s}$ range is generated by streamwise development, so it inherits whatever non-equilibrium history and residual tunnel gradient the developing layer carries (Vishwanathan et al., 2023) and is an estimate, not a controlled measurement. Uncertainties for both stages come from a case-level bootstrap: the fourteen cases are resampled with replacement, both stages refitted on each of $1000$ replicates, and the central $68\\\\%$ spread reported (table 1). Resampling whole cases — not individual spectral points, which are strongly correlated within a case and would manufacture false precision — keeps the effective sample size at the fourteen cases (arguably the two speed families) that vary independently. This is why the shape intervals come out at a few percent while the free-slope test interval is wide.\",\"tag\":null,\"text\":\"Keeping the two stages apart matters because they rest on very different amounts of information. The shape is pinned by 2221 spectral samples freed of amplitude noise; the growth test and the offset rest on a single fourteen-point regression whose 𝜂𝑠 range is generated by streamwise development, so it inherits whatever non-equilibrium history and residual tunnel gradient the developing layer carries [Vishwanathan et al., 2023] and is an estimate, not a controlled measurement. Uncertainties for both stages come from a case-level bootstrap: the fourteen cases are resampled with replacement, both stages refitted on each of 1000 replicates, and the central 68% spread reported (table 1). Resampling whole cases not individual spectral points, which are strongly correlated within a case and would manufacture false precision keeps the effective sample size at the fourteen cases (arguably the two speed families) that vary independently. This is why the shape intervals come out at a few percent while the free-slope test interval is wide.\"},{\"bbox\":[90.0,502.96,73.38,11.96],\"bbox_pdf\":[90.0,277.09,163.38,289.04],\"fragments\":[{\"bbox_pdf\":[90.0,277.09,163.38,289.04],\"words\":[5443,5444]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":2,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"heading\",\"source_ref\":\"title91\",\"source_text\":\"7.2 Outcome\",\"tag\":null,\"text\":\"7.2 Outcome\"},{\"bbox\":[90.0,524.92,432.0,177.72],\"bbox_pdf\":[90.0,89.37,522.0,267.08],\"fragments\":[{\"bbox_pdf\":[90.0,89.37,522.0,267.08],\"words\":[5445,5446,5447,5448,5449,5450,5451,5452,5453,5454,5455,5456,5457,5458,5459,5460,5461,5462,5463,5464,5465,5466,5467,5468,5469,5470,5471,5472,5473,5474,5475,5476,5477,5478,5479,5480,5481,5482,5483,5484,5485,5486,5487,5488,5489,5490,5491,5492,5493,5494,5495,5496,5497,5498,5499,5500,5501,5502,5503,5504,5505,5506,5507,5508,5509,5510,5511,5512,5513,5514,5515,5516,5517,5518,5519,5520,5521,5522,5523,5524,5525,5526,5527,5528,5529,5530,5531,5532,5533,5534,5535,5536,5537,5538,5539,5540,5541,5542,5543,5544,5545,5546,5547,5548,5549,5550,5551,5552,5553,5554,5555,5556,5557,5558,5559,5560,5561,5562,5563,5564,5565,5566,5567,5568,5569,5570,5571,5572,5573,5574,5575,5576,5577,5578,5579,5580,5581,5582,5583,5584,5585,5586,5587,5588,5589,5590,5591,5592,5593,5594,5595,5597,5598,5599,5600,5601,5602,5603,5604,5605,5606,5607,5608,5609,5610,5611,5612,5613,5614,5615,5616,5617,5618,5619,5620,5621,5622,5623,5624,5625,5626,5627,5628,5629,5630,5631,5633,5634,5635,5636,5637,5638,5639,5640,5641,5642,5643,5644,5645,5646,5647,5648,5649,5650,5651,5652,5653,5654,5655,5656,5657,5658,5659,5661,5662,5663,5664,5665,5666,5667,5668,5669,5670,5671,5672,5673,5674,5675,5676,5677,5678,5679,5680,5681,5682,5683,5684,5685,5686,5687,5688,5689,5690,5691,5692,5693,5694]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph92\",\"source_text\":\"The shape is tightly constrained; the free-slope test is not. Stage A collapses the fourteen area-normalised spectra onto a common shape at an r.m.s. residual of $0.02$ , with bootstrap intervals a few percent of the central values (figure 2a, table 1). Stage B returns a free slope $A_{\\\\mathcal{Q},\\\\eta}^{r}\\\\approx 3.8$ , but the variances scatter about the regression with $R^{2}\\\\approx 0.19$ : the bootstrap $68\\\\%$ interval is $[1.5,\\\\,6.1]$ and the $95\\\\%$ interval $[-1.0,\\\\,9.5]$ does not exclude zero slope (figure 2b). The predicted $A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}=2.3$ sits comfortably inside it, and imposing it costs almost nothing: the residual sum of squares rises by $3.6\\\\%$ while one parameter is removed. The VT data are therefore consistent with the predicted rate but cannot yet discriminate it from a rate half or twice as large. Nor can they prefer the $\\\\eta_{s}$ basis over $\\\\ln\\\\delta^{+}$ : regressed on either, the fourteen variances scatter about the trend with the same $R^{2}\\\\approx 0.19$ , so the campaign does not favour one basis on fit quality. The two are not the same regressor — because $k_{s}^{+}$ varies across the campaign they are only moderately correlated ( $\\\\mathrm{corr}\\\\approx 0.66$ ), spanning $\\\\Delta\\\\eta_{s}\\\\approx 0.84$ against $\\\\Delta\\\\ln\\\\delta^{+}\\\\approx 1.20$ — and what separates them is exactly the family offset $\\\\Delta\\\\ln k_{s}^{+}\\\\approx 0.53$ between the two speed families at matched $\\\\eta_{s}$ ; the choice of $\\\\eta_{s}$ therefore rests on the outer-similarity argument of §4, not on a data preference. Separating them requires varying $k_{s}^{+}$ at matched $\\\\delta/k_{s}$ , and the two speed families are exactly that comparison, taken up below. The point estimate there lies nearer the $\\\\ln\\\\delta^{+}$ prediction than zero, though neither is excluded.\",\"tag\":null,\"text\":\"The shape is tightly constrained; the free-slope test is not. Stage A collapses the fourteen areanormalised spectra onto a common shape at an r.m.s. residual of 0.02, with bootstrap intervals a few percent of the central values (figure 2a, table 1). Stage B returns a free slope 𝐴 𝑟 ≈ 3.8, but the Q, 𝜂 variances scatter about the regression with 𝑅 2 ≈ 0.19: the bootstrap 68% interval is [1.5, 6.1] and the 95% interval [−1.0, 9.5] does not exclude zero slope (figure 2b). The predicted 𝐴 base = 2.3 sits Q, 𝜂 comfortably inside it, and imposing it costs almost nothing: the residual sum of squares rises by 3.6% while one parameter is removed. The VT data are therefore consistent with the predicted rate but cannot yet discriminate it from a rate half or twice as large. Nor can they prefer the 𝜂𝑠 basis over ln 𝛿 + regressed on either, the fourteen variances scatter about the trend with the same 𝑅 2 ≈ 0.19, so the campaign does not favour one basis on fit quality. The two are not the same regressor because 𝑘 + 𝑠 varies across the campaign they are only moderately correlated (corr ≈ 0.66), spanning Δ𝜂𝑠 ≈ 0.84 against Δ ln 𝛿 + ≈ 1.20 and what separates them is exactly the family offset Δ ln 𝑘 + 𝑠 ≈ 0.53 between the two speed families at matched 𝜂𝑠; the choice of 𝜂𝑠 therefore rests on the outer-similarity argument\"},{\"bbox\":[300.55,720.98,10.91,10.91],\"bbox_pdf\":[300.55,60.11,311.46,71.02],\"fragments\":[{\"bbox_pdf\":[300.55,60.11,311.46,71.02],\"words\":[9389]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":7,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"10\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0009.pdf\",\"page_no\":10,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph87\",\"c_block\":0.988,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.999,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.988,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"c219880cf0e0f7c93988ffb2ef1f337ddbbaabdf964d4602fb71be8e774d3967\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/c219880cf0e0f7c93988ffb2ef1f337ddbbaabdf964d4602fb71be8e774d3967.pdf","asset_sha256":"c219880cf0e0f7c93988ffb2ef1f337ddbbaabdf964d4602fb71be8e774d3967","document_id":"arxiv:2608.16729","kind":"page","page_number":10,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[212.09,246.6,309.91,55.0],\"bbox_pdf\":[212.09,490.4,522.0,545.4],\"fragments\":[{\"bbox_pdf\":[212.09,490.4,522.0,545.4],\"words\":[5216,5217,5218,5219,5220,5221,5222,5223,5224,5225,5226,5227,5228,5229,5230,5231,5232,5233,5234,5235,5236,5237,5238,5239,5240,5241,5242,5243,5244,5245,5246,5247,5248,5249,5250,5251]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\overline{p^{\\\\prime 2}}^{+} \\\\displaystyle=V_{0}+A_{\\\\mathcal{Q},\\\\eta}^{r}\\\\,\\\\eta_{s} \\\\displaystyle\\\\quad\\\\text{(test, $A_{\\\\mathcal{Q},\\\\eta}^{r}$ free)}, \\\\\\\\ \\\\displaystyle\\\\overline{p^{\\\\prime 2}}^{+} \\\\displaystyle=V_{0}+A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\eta_{s} \\\\displaystyle\\\\quad\\\\text{(calibration)}, \\\\\\\\ \\\\displaystyle B_{\\\\mathcal{Q},G}^{r} \\\\displaystyle=V_{0}-B_{\\\\mathcal{L}}^{r}, \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S7.E20\",\"source_text\":\"$\\\\displaystyle\\\\overline{p^{\\\\prime 2}}^{+}$ $\\\\displaystyle=V_{0}+A_{\\\\mathcal{Q},\\\\eta}^{r}\\\\,\\\\eta_{s}$ $\\\\displaystyle\\\\quad\\\\text{(test, $A_{\\\\mathcal{Q},\\\\eta}^{r}$ free)},$ (20) $\\\\displaystyle\\\\overline{p^{\\\\prime 2}}^{+}$ $\\\\displaystyle=V_{0}+A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\eta_{s}$ $\\\\displaystyle\\\\quad\\\\text{(calibration)},$ $\\\\displaystyle B_{\\\\mathcal{Q},G}^{r}$ $\\\\displaystyle=V_{0}-B_{\\\\mathcal{L}}^{r},$\",\"tag\":null,\"text\":\"+ 𝑝′2 = 𝑉0 + 𝐴 𝑟 (test, 𝐴 𝑟 Q, 𝜂 𝜂𝑠 Q, 𝜂 free), + 𝑝′2 = 𝑉0 + 𝐴 base (calibration), (20) Q, 𝜂 𝜂𝑠 𝐵 𝑟 𝑟 Q,𝐺 = 𝑉0 − 𝐵L,\"},\"formula_index\":2,\"parent_page_sha256\":\"c219880cf0e0f7c93988ffb2ef1f337ddbbaabdf964d4602fb71be8e774d3967\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/c219880cf0e0f7c93988ffb2ef1f337ddbbaabdf964d4602fb71be8e774d3967.pdf","asset_sha256":"c219880cf0e0f7c93988ffb2ef1f337ddbbaabdf964d4602fb71be8e774d3967","document_id":"arxiv:2608.16729","kind":"formula","page_number":10,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[90.0,89.03,432.0,39.03],\"bbox_pdf\":[90.0,663.94,522.0,702.97],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[90.0,663.94,522.0,702.97],\"words\":[5695,5696,5697,5698,5699,5700,5701,5702,5703,5704,5705,5706,5707,5708,5709,5710,5711,5712,5713,5714,5715,5716,5717,5718,5719,5720,5721,5722,5723,5724,5725,5726,5727,5728,5729,5730,5731,5732,5733,5734,5735,5736,5737,5738,5739,5740]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph92\",\"source_text\":\"The shape is tightly constrained; the free-slope test is not. Stage A collapses the fourteen area-normalised spectra onto a common shape at an r.m.s. residual of $0.02$ , with bootstrap intervals a few percent of the central values (figure 2a, table 1). Stage B returns a free slope $A_{\\\\mathcal{Q},\\\\eta}^{r}\\\\approx 3.8$ , but the variances scatter about the regression with $R^{2}\\\\approx 0.19$ : the bootstrap $68\\\\%$ interval is $[1.5,\\\\,6.1]$ and the $95\\\\%$ interval $[-1.0,\\\\,9.5]$ does not exclude zero slope (figure 2b). The predicted $A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}=2.3$ sits comfortably inside it, and imposing it costs almost nothing: the residual sum of squares rises by $3.6\\\\%$ while one parameter is removed. The VT data are therefore consistent with the predicted rate but cannot yet discriminate it from a rate half or twice as large. Nor can they prefer the $\\\\eta_{s}$ basis over $\\\\ln\\\\delta^{+}$ : regressed on either, the fourteen variances scatter about the trend with the same $R^{2}\\\\approx 0.19$ , so the campaign does not favour one basis on fit quality. The two are not the same regressor — because $k_{s}^{+}$ varies across the campaign they are only moderately correlated ( $\\\\mathrm{corr}\\\\approx 0.66$ ), spanning $\\\\Delta\\\\eta_{s}\\\\approx 0.84$ against $\\\\Delta\\\\ln\\\\delta^{+}\\\\approx 1.20$ — and what separates them is exactly the family offset $\\\\Delta\\\\ln k_{s}^{+}\\\\approx 0.53$ between the two speed families at matched $\\\\eta_{s}$ ; the choice of $\\\\eta_{s}$ therefore rests on the outer-similarity argument of §4, not on a data preference. Separating them requires varying $k_{s}^{+}$ at matched $\\\\delta/k_{s}$ , and the two speed families are exactly that comparison, taken up below. The point estimate there lies nearer the $\\\\ln\\\\delta^{+}$ prediction than zero, though neither is excluded.\",\"tag\":null,\"text\":\"of §4, not on a data preference. Separating them requires varying 𝑘 + 𝑠 at matched 𝛿/𝑘𝑠, and the two speed families are exactly that comparison, taken up below. The point estimate there lies nearer the ln 𝛿 + prediction than zero, though neither is excluded.\"},{\"bbox\":[90.0,130.7,432.0,80.29],\"bbox_pdf\":[90.0,581.01,522.0,661.3],\"fragments\":[{\"bbox_pdf\":[90.0,581.01,522.0,661.3],\"words\":[5741,5742,5743,5744,5745,5746,5747,5748,5749,5750,5751,5752,5753,5754,5755,5756,5757,5758,5759,5760,5761,5762,5763,5764,5765,5767,5768,5769,5770,5771,5772,5773,5774,5775,5776,5777,5779,5780,5781,5782,5783,5784,5785,5786,5787,5788,5789,5790,5791,5792,5793,5794,5795,5796,5797,5798,5799,5800,5801,5802,5803,5804,5805,5806,5807,5808,5809,5810,5811,5812,5813,5814,5815,5816,5817,5818,5819,5820,5821,5822,5823,5824,5825,5826,5827,5828,5829,5830,5831,5832,5833,5834,5835,5836,5837,5838,5839,5840,5841,5842,5843,5844,5845,5846,5847,5848]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph93\",\"source_text\":\"The rate is, moreover, hostage to single cases at the ends of so short a range (figure 2b). The low pair near $\\\\eta_{s}\\\\approx 3.5$ — the $35$ and $65$ m/s stations at $x=3.69$ m — sits below the trend; removing the $35$ m/s member alone steepens the fitted slope from $3.8$ to $4.6$ and lifts $R^{2}$ from $0.19$ to $0.31$ . Removing instead the single highest point at the other end ( $65$ m/s, $x=4.91$ m) flattens the slope to $1.7$ . Both bracket the predicted $2.3$ , and no one deletion makes the rate robust; over fourteen points spanning only $\\\\Delta\\\\eta_{s}\\\\approx 0.84$ none should.\",\"tag\":null,\"text\":\"The rate is, moreover, hostage to single cases at the ends of so short a range (figure 2b). The low pair near 𝜂𝑠 ≈ 3.5 the 35 and 65 m/s stations at 𝑥 = 3.69 m sits below the trend; removing the 35 m/s member alone steepens the fitted slope from 3.8 to 4.6 and lifts 𝑅 2 from 0.19 to 0.31. Removing instead the single highest point at the other end (65 m/s, 𝑥 = 4.91 m) flattens the slope to 1.7. Both bracket the predicted 2.3, and no one deletion makes the rate robust; over fourteen points spanning only Δ𝜂𝑠 ≈ 0.84 none should.\"},{\"bbox\":[90.0,211.99,432.0,51.56],\"bbox_pdf\":[90.0,528.45,522.0,580.01],\"fragments\":[{\"bbox_pdf\":[90.0,528.45,522.0,580.01],\"words\":[5849,5850,5851,5852,5853,5854,5855,5856,5857,5858,5859,5860,5861,5862,5863,5864,5865,5866,5867,5868,5869,5870,5871,5872,5873,5874,5875,5876,5877,5878,5879,5880,5881,5882,5883,5884,5885,5886,5887,5888,5889,5890,5891,5892,5894,5895,5896,5897,5898,5899,5900,5901,5902,5903,5904,5905,5906,5907,5908,5909,5910,5911,5912,5913,5914,5915,5916,5917,5918,5919,5920,5921]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph94\",\"source_text\":\"A common-mode $C_{f}$ bias is degenerate with the prediction, since scaling all variances by $c$ tilts the test slope to $2.3c$ . Closing the gap from $2.3$ to $3.8$ requires $c=1.65$ , that is $u_{\\\\tau}$ overestimated by $13\\\\%$ and $C_{f}$ by $28\\\\%$ — beyond the campaign’s quoted $\\\\pm 20\\\\%$ , so calibration bias at the stated level accounts for much but not all of the difference, the remainder lying well inside $[1.5,\\\\,6.1]$ .\",\"tag\":null,\"text\":\"A common-mode 𝐶 𝑓 bias is degenerate with the prediction, since scaling all variances by 𝑐 tilts the test slope to 2.3𝑐. Closing the gap from 2.3 to 3.8 requires 𝑐 = 1.65, that is 𝑢𝜏 overestimated by 13% and 𝐶 𝑓 by 28% beyond the campaign’s quoted ±20%, so calibration bias at the stated level accounts for much but not all of the difference, the remainder lying well inside [1.5, 6.1].\"},{\"bbox\":[89.62,266.19,432.39,139.48],\"bbox_pdf\":[89.62,386.33,522.01,525.81],\"fragments\":[{\"bbox_pdf\":[89.62,386.33,522.01,525.81],\"words\":[5922,5923,5924,5925,5926,5927,5928,5929,5930,5931,5932,5933,5934,5935,5936,5937,5938,5939,5940,5941,5942,5943,5944,5945,5946,5947,5948,5949,5950,5951,5952,5953,5954,5955,5956,5957,5958,5959,5960,5961,5962,5963,5964,5965,5966,5967,5968,5969,5970,5971,5972,5973,5974,5975,5976,5977,5978,5979,5980,5981,5982,5983,5984,5985,5986,5987,5988,5989,5990,5991,5992,5993,5994,5995,5996,5997,5998,5999,6000,6001,6002,6003,6004,6005,6006,6007,6008,6009,6010,6011,6012,6013,6014,6015,6016,6017,6018,6019,6020,6021,6022,6023,6024,6025,6027,6028,6029,6030,6031,6032,6033,6034,6035,6036,6037,6038,6039,6040,6041,6042,6043,6044,6045,6046,6047,6048,6049,6050,6051,6052,6053,6054,6055,6056,6057,6058,6059,6060,6061,6062,6063,6064,6065,6066,6067,6068,6069,6070,6071,6072,6073,6074,6075,6076,6077,6078,6079,6080,6081,6082,6083,6084,6085,6086,6087,6088,6089,6090,6091,6092,6093,6094,6095,6096,6097,6098,6099,6100,6101,6102,6103,6104,6105,6106,6107,6108,6109,6110,6111,6112,6113,6114,6115,6116,6117,6118]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph95\",\"source_text\":\"What the free-slope test leaves loose the calibration pins: at the predicted slope the offset is well determined, $B_{\\\\mathcal{Q},G}^{r}\\\\approx 6.4\\\\pm 0.6$ ( $68\\\\%$ ). Dividing it by the range coefficient converts it into the currency of the log range, $B_{\\\\mathcal{Q},G}^{r}/A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}=\\\\mathcal{I}_{\\\\chi}+% B_{\\\\mathcal{Q},i}^{r}/A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\approx 2.8$ : the sublayer and canopy together add as much inner-scaled variance as a further factor $e^{2.8}\\\\approx 16$ in $\\\\delta/k_{s}$ would. That is comparable to the whole range these layers possess ( $\\\\eta_{s}\\\\approx 2.9$ – $3.7$ ), so the non-growing offset supplies over $40\\\\%$ of their variance — consistent in sign with the enhanced near-wall pressure activity of rough-wall DNS (Bhaganagar et al., 2007). The outer break $T_{b\\\\mathcal{Q}}^{o,r}\\\\approx 0.98$ is likewise tight: with the measured $U_{c}\\\\approx 0.49\\\\,U_{e}$ (Totten et al., 2026) it fixes the convected wavelength $\\\\lambda_{x}\\\\approx 0.5\\\\,\\\\delta$ , consistent with the smooth-wall outer population of Massey et al. (2026c), and via (19) places the effective source height at $y_{\\\\mathcal{Q},\\\\mathrm{eff}}/\\\\delta\\\\approx 0.5$ if $C_{b\\\\mathcal{Q}}^{r}=1$ or $\\\\lesssim 0.1$ under the kernel-consistent $C_{b\\\\mathcal{Q}}^{r}=O(2\\\\pi)$ .\",\"tag\":null,\"text\":\"What the free-slope test leaves loose the calibration pins: at the predicted slope the offset is well determined, 𝐵 𝑟 ≈ 6.4 ± 0.6 (68%). Dividing it by the range coefficient converts it into the Q,𝐺 currency of the log range, 𝐵 𝑟 /𝐴 base = I𝜒 + 𝐵 𝑟 /𝐴 base ≈ 2.8: the sublayer and canopy together Q,𝐺 Q, 𝜂 Q,𝑖 Q, 𝜂 add as much inner-scaled variance as a further factor 𝑒 2.8 ≈ 16 in 𝛿/𝑘𝑠 would. That is comparable to the whole range these layers possess (𝜂𝑠 ≈ 2.9–3.7), so the non-growing offset supplies over 40% of their variance consistent in sign with the enhanced near-wall pressure activity of rough-wall DNS [Bhaganagar et al., 2007]. The outer break 𝑇 𝑜,𝑟 ≈ 0.98 is likewise tight: with the measured 𝑏 Q 𝑈𝑐 ≈ 0.49 𝑈𝑒 [Totten et al., 2026] it fixes the convected wavelength 𝜆𝑥 ≈ 0.5 𝛿, consistent with the smooth-wall outer population of Massey et al. [2026c], and via (19) places the effective source height at 𝑦Q,eff/𝛿 ≈ 0.5 if 𝐶 𝑟 = 1 or ≲ 0.1 under the kernel-consistent 𝐶 𝑟 = 𝑂 (2𝜋). 𝑏 Q 𝑏 Q 𝑠,𝑟\"},{\"bbox\":[90.0,406.52,432.0,105.75],\"bbox_pdf\":[90.0,279.73,522.0,385.48],\"fragments\":[{\"bbox_pdf\":[90.0,279.73,522.0,385.48],\"words\":[6119,6120,6121,6122,6123,6124,6125,6126,6127,6128,6129,6130,6131,6132,6133,6134,6135,6136,6137,6138,6139,6140,6141,6142,6143,6144,6145,6146,6147,6148,6149,6150,6151,6152,6153,6154,6155,6156,6157,6158,6159,6160,6161,6162,6163,6164,6165,6166,6167,6168,6169,6170,6171,6172,6173,6174,6175,6176,6177,6178,6179,6180,6181,6182,6183,6184,6185,6186,6187,6188,6189,6190,6191,6192,6193,6194,6195,6196,6197,6198,6199,6200,6201,6202,6203,6204,6205,6206,6207,6208,6209,6210,6211,6212,6213,6214,6215,6216,6217,6218,6219,6220,6221,6222,6223,6224,6225,6226,6227,6228,6229,6230,6231,6232,6233,6234,6235,6236,6237,6238,6239,6240,6241,6242,6243,6244,6245,6246,6247,6248,6249,6250,6251,6252,6253,6254]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph96\",\"source_text\":\"The fitted linear break settles the mechanism left open in §6. It falls at $T_{b\\\\mathcal{L}}^{s,r}\\\\approx 1.87$ (table 1), on the convected element-scale prediction $U_{e}/U_{c}\\\\approx 2$ and a factor of five to six below the shedding value $10$ – $12$ of (18): the roughness-local feature is element-scale structure convected past the wall, not vortex shedding from the cylinders. The axis relation $T^{s}=(\\\\delta/k_{s})\\\\,T^{o}$ also reassigns a feature seen near $St\\\\equiv fk_{s}/U_{e}\\\\approx 0.02$ in the VT spectra, read tentatively in earlier work as a canopy ‘breathing’ mode: it maps to $T^{o}\\\\approx 1.2$ – $2.8$ across these $\\\\delta/k_{s}$ , i.e. onto the energetic nonlinear population at $T^{o}\\\\sim 1$ rather than a roughness-set mode, so the genuinely roughness-local content is the $g_{\\\\mathcal{L}}^{r}$ feature at $T^{s}\\\\sim 2$ .\",\"tag\":null,\"text\":\"The fitted linear break settles the mechanism left open in §6. It falls at 𝑇 𝑏 L ≈ 1.87 (table 1), on the convected element-scale prediction 𝑈𝑒/𝑈𝑐 ≈ 2 and a factor of five to six below the shedding value 10–12 of (18): the roughness-local feature is element-scale structure convected past the wall, not vortex shedding from the cylinders. The axis relation 𝑇 𝑠 = (𝛿/𝑘𝑠) 𝑇 𝑜 also reassigns a feature seen near 𝑆𝑡 ≡ 𝑓 𝑘𝑠/𝑈𝑒 ≈ 0.02 in the VT spectra, read tentatively in earlier work as a canopy ‘breathing’ mode: it maps to 𝑇 𝑜 ≈ 1.2–2.8 across these 𝛿/𝑘𝑠, i.e. onto the energetic nonlinear population at 𝑇 𝑜 ∼ 1 rather than a roughness-set mode, so the genuinely roughness-local content is the 𝑔 𝑟 feature at 𝑇 𝑠 ∼ 2. L\"},{\"bbox\":[90.0,514.91,432.01,82.59],\"bbox_pdf\":[90.0,194.5,522.01,277.09],\"fragments\":[{\"bbox_pdf\":[90.0,194.5,522.01,277.09],\"words\":[6255,6256,6257,6258,6259,6260,6261,6262,6263,6264,6265,6266,6267,6268,6269,6270,6271,6272,6273,6274,6275,6276,6277,6278,6279,6280,6281,6282,6283,6284,6285,6286,6287,6288,6289,6290,6291,6292,6293,6294,6295,6296,6297,6298,6299,6300,6301,6302,6303,6304,6305,6306,6307,6308,6309,6310,6311,6312,6313,6314,6315,6316,6317,6318,6319,6320,6321,6322,6323,6324,6325,6326,6327,6328,6329,6330,6331,6332,6333,6334,6335,6336,6337,6338,6339,6340,6341,6342,6343,6344,6345,6346,6347,6348,6349,6350,6351,6352,6353,6354,6355,6356,6357,6358,6359,6360,6361,6362,6363,6364]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph101\",\"source_text\":\"Figure 3 overlays the calibrated model on the VT spectra in roughness-scaled and outer-scaled coordinates, evaluated at the measured per-case flow states. The energetic peak collapses on the outer axis at $T^{o}\\\\sim 1$ , the roughness-local feature sits near $T^{s}\\\\sim 2$ , and the predicted peak and variance rise across the sweep ( $\\\\max{f\\\\phi_{pp}}^{+}\\\\approx 4.0\\\\to 4.5$ and $\\\\overline{p^{\\\\prime 2}}^{+}\\\\approx 13.8\\\\to 15.4$ from $Re_{\\\\tau}\\\\approx 7400$ to $24600$ ), inside the VT envelopes. The r.m.s. residual of the assembled model is $0.50$ on the dimensional ${f\\\\phi_{pp}}^{+}$ (peaks $\\\\sim 4$ – $7$ ); the $0.02$ of Stage A is on the unit-area shape.\",\"tag\":null,\"text\":\"Figure 3 overlays the calibrated model on the VT spectra in roughness-scaled and outer-scaled coordinates, evaluated at the measured per-case flow states. The energetic peak collapses on the outer axis at 𝑇 𝑜 ∼ 1, the roughness-local feature sits near 𝑇 𝑠 ∼ 2, and the predicted peak and variance rise + + ′2 across the sweep (max 𝑓 𝜙𝑝 𝑝 ≈ 4.0 → 4.5 and 𝑝 ≈ 13.8 → 15.4 from 𝑅𝑒𝜏 ≈ 7400 to 24600), inside the VT envelopes. The r.m.s. residual of the assembled model is 0.50 on the dimensional 𝑓 𝜙𝑝 𝑝 + (peaks ∼ 4–7); the 0.02 of Stage A is on the unit-area shape.\"},{\"bbox\":[90.0,597.48,432.01,94.48],\"bbox_pdf\":[90.0,100.04,522.01,194.52],\"fragments\":[{\"bbox_pdf\":[90.0,100.04,522.01,194.52],\"words\":[6365,6366,6367,6368,6369,6370,6371,6372,6373,6374,6375,6376,6377,6378,6379,6380,6381,6382,6383,6384,6385,6386,6387,6388,6389,6390,6391,6392,6393,6394,6395,6396,6397,6398,6399,6400,6401,6402,6403,6404,6405,6406,6407,6408,6409,6410,6411,6412,6413,6414,6415,6416,6417,6418,6419,6420,6421,6422,6423,6424,6425,6426,6427,6428,6429,6430,6431,6432,6433,6434,6436,6437,6438,6439,6440,6441,6442,6443,6444,6445,6446,6447,6448,6449,6450,6451,6452,6453,6454,6455,6456,6457,6458,6459,6460,6461,6462,6463,6464,6465,6466,6467,6468,6469,6470,6471,6472,6473,6474,6475,6476,6477,6478,6479,6481,6482,6483,6484,6485,6486,6487,6488,6489,6490,6491,6492,6493,6494,6495,6496,6497,6498,6499,6500,6501,6502,6503,6504,6505,6506,6507,6508,6509,6510,6511,6512]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.971,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph102\",\"source_text\":\"The two speed families overlap in $\\\\eta_{s}$ at distinctly different $k_{s}^{+}$ (family means $\\\\approx 393$ and $669$ ) and so directly probe whether $k_{s}^{+}$ drops out once fully rough. The probe is inconclusive. The families return different fitted slopes ( $A_{\\\\mathcal{Q},\\\\eta}^{r}\\\\approx 2.75$ at $35$ m/s, $5.07$ at $65$ m/s), but a common-slope fit with a family offset gives $+1.5\\\\pm 1.2$ at matched $\\\\eta_{s}$ — within $1.3$ standard errors of zero (the $\\\\eta_{s}$ basis) and within a third of a standard error of the $A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\Delta\\\\ln k_{s}^{+}\\\\approx 1.2$ the $\\\\ln\\\\delta^{+}$ basis predicts ( $\\\\Delta\\\\ln k_{s}^{+}\\\\approx 0.53$ between families) — and the split hangs on the same single case as above: the high-variance $65$ m/s point at $x=4.91$ m, whose removal drops the $65$ m/s family slope to $0.8$ . Genuine residual $k_{s}^{+}$ sensitivity, non-equilibrium development effects, a family-level $C_{f}$ bias ( $\\\\sim 5\\\\%$ suffices) and experimental scatter are not separable from the VT data alone (§9).\",\"tag\":null,\"text\":\"The two speed families overlap in 𝜂𝑠 at distinctly different 𝑘 + (family means ≈ 393 and 669) 𝑠 and so directly probe whether 𝑘 + drops out once fully rough. The probe is inconclusive. The 𝑠 families return different fitted slopes (𝐴 𝑟 ≈ 2.75 at 35 m/s, 5.07 at 65 m/s), but a common-slope Q, 𝜂 fit with a family offset gives +1.5 ± 1.2 at matched 𝜂𝑠 within 1.3 standard errors of zero (the 𝜂𝑠 basis) and within a third of a standard error of the 𝐴 base Δ ln 𝑘 + ≈ 1.2 the ln 𝛿 + basis predicts 𝑠 Q, 𝜂 (Δ ln 𝑘 + ≈ 0.53 between families) and the split hangs on the same single case as above: the 𝑠 high-variance 65 m/s point at 𝑥 = 4.91 m, whose removal drops the 65 m/s family slope to 0.8.\"},{\"bbox\":[300.54,720.98,10.91,10.91],\"bbox_pdf\":[300.54,60.11,311.45,71.02],\"fragments\":[{\"bbox_pdf\":[300.54,60.11,311.45,71.02],\"words\":[9390]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":7,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"11\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0010.pdf\",\"page_no\":11,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph102\",\"c_block\":0.983,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.995,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.983,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"1ed5e01e4d774ce18782b75ea94ddaec6f32da102488b2d88313c99b59951531\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/1ed5e01e4d774ce18782b75ea94ddaec6f32da102488b2d88313c99b59951531.pdf","asset_sha256":"1ed5e01e4d774ce18782b75ea94ddaec6f32da102488b2d88313c99b59951531","document_id":"arxiv:2608.16729","kind":"page","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[90.0,233.5,432.02,211.09],\"bbox_pdf\":[90.0,347.41,522.02,558.5],\"float_ref\":\"S7.F2\",\"fragments\":[{\"bbox_pdf\":[90.0,347.41,522.02,558.5],\"words\":[]}],\"html\":null,\"kind\":\"figure\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"figure\",\"source_ref\":\"S7.F2\",\"source_text\":\"\",\"tag\":null,\"text\":\"\"},{\"bbox\":[90.0,457.2,432.0,98.92],\"bbox_pdf\":[90.0,235.88,522.0,334.8],\"float_ref\":\"S7.F2\",\"fragments\":[{\"bbox_pdf\":[90.0,235.88,522.0,334.8],\"words\":[6513,6514,6515,6516,6517,6518,6519,6520,6521,6522,6523,6524,6525,6526,6527,6528,6529,6530,6531,6532,6533,6534,6535,6537,6538,6539,6540,6541,6542,6543,6544,6545,6546,6547,6548,6549,6550,6551,6552,6553,6554,6555,6556,6557,6558,6559,6560,6561,6562,6563,6564,6565,6566,6567,6568,6569,6570,6571,6572,6573,6574,6575,6576,6577,6578,6579,6580,6581,6582,6584,6585,6586,6587,6588,6589,6590,6591,6592,6593,6594,6595,6596,6597,6598,6599,6600,6601,6602,6603,6604,6605,6606,6607,6608,6609,6610,6611,6612,6613,6614,6615,6616,6617,6618,6619,6620,6621,6622,6623,6624,6625,6626,6627,6628,6629,6630,6631,6632,6633]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.989,\"verified\":true},\"order\":1,\"role\":\"caption\",\"source_ref\":\"caption100\",\"source_text\":\"Figure 2: The decoupled calibration. (a) The fourteen area-normalised premultiplied spectra ${f\\\\phi_{pp}}^{+}/\\\\overline{p^{\\\\prime 2}}^{+}$ on the outer axis $T^{o}$ , coloured by $Re_{\\\\tau}$ , with the fitted energetic lognormal $g_{\\\\mathcal{Q}}^{r}$ overlaid (heavy black, at the mean energetic area fraction $\\\\approx 0.95$ ); the energetic peak collapses at $T^{o}\\\\sim 1$ and the roughness-local population $g_{\\\\mathcal{L}}^{r}$ appears as the small shoulder at low $T^{o}$ . (b) Measured integrated variance $\\\\overline{p^{\\\\prime 2}}^{+}$ versus $\\\\eta_{s}=\\\\ln(\\\\delta/k_{s})$ for the two speed families, with the predicted slope $A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}=2.3$ at the fitted offset (20) (heavy black), the free-slope least-squares fit (dashed) and its bootstrap $68\\\\%$ band (grey), and the family fits (thin).\",\"tag\":null,\"text\":\"Figure 2: The decoupled calibration. (a) The fourteen area-normalised premultiplied spectra + 𝑓 𝜙𝑝 𝑝 + /𝑝′2 on the outer axis 𝑇 𝑜 coloured by 𝑅𝑒𝜏, with the fitted energetic lognormal 𝑔 𝑟 overlaid Q (heavy black, at the mean energetic area fraction ≈ 0.95); the energetic peak collapses at 𝑇 𝑜 ∼ 1 and the roughness-local population 𝑔 𝑟 appears as the small shoulder at low 𝑇 𝑜 (b) Measured integrated L + ′2 base variance 𝑝 versus 𝜂𝑠 = ln(𝛿/𝑘𝑠) for the two speed families, with the predicted slope 𝐴 = 2.3 Q, 𝜂 at the fitted offset (20) (heavy black), the free-slope least-squares fit (dashed) and its bootstrap 68% band (grey), and the family fits (thin).\"},{\"bbox\":[300.54,720.98,10.91,10.91],\"bbox_pdf\":[300.54,60.11,311.45,71.02],\"fragments\":[{\"bbox_pdf\":[300.54,60.11,311.45,71.02],\"words\":[9391]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":2,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"12\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0011.pdf\",\"page_no\":12,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"caption100\",\"c_block\":0.988,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.982,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.982,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"79372cfeda1be974d03101d6ab6da70185d0cae58355a50abd75837a35fd57ab\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/79372cfeda1be974d03101d6ab6da70185d0cae58355a50abd75837a35fd57ab.pdf","asset_sha256":"79372cfeda1be974d03101d6ab6da70185d0cae58355a50abd75837a35fd57ab","document_id":"arxiv:2608.16729","kind":"page","page_number":12,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[135.63,88.8,340.74,140.42],\"bbox_pdf\":[135.63,562.77,476.37,703.2],\"float_ref\":\"S7.T1\",\"fragments\":[{\"bbox_pdf\":[135.63,562.77,476.37,703.2],\"words\":[6634,6635,6636,6637,6638,6639,6640,6641,6642,6643,6644,6645,6646,6647,6648,6649,6650,6651,6652,6653,6654,6655,6656,6657,6658,6659,6660,6661,6662,6663,6664,6665,6666,6667,6668,6669,6670,6671,6672,6673,6674,6675,6676,6677,6678,6679,6680,6681,6682,6683,6684,6685,6686,6687,6688,6689,6690,6691,6692,6693,6694,6695,6696,6697,6698,6699,6700,6701,6702,6703,6704,6705,6706,6707,6708,6709,6710,6711,6713,6714,6715,6716,6717,6718,6719,6720,6721,6722,6723,6724,6725,6726,6727,6728,6729,6730,6731,6732,6733,6734,6735,6736,6737,6738,6739,6740]}],\"html\":\"<table><tr><td>Parameter</td><td>Symbol</td><td>Value</td><td>68% interval</td><td>Source</td></tr><tr><td colspan=\\\"5\\\">Stage A: spectral shape (area-normalised spectra)</td></tr><tr><td>Linear break (roughness scaling)</td><td>Tb⁢ℒs,r</td><td>1.87</td><td>[1.81, 1.96]</td><td>fitted</td></tr><tr><td>Linear width</td><td>σℒr</td><td>0.50</td><td>[0.48, 0.52]</td><td>fitted</td></tr><tr><td>Linear area</td><td>Bℒr</td><td>0.70</td><td>[0.67, 0.78]</td><td>fitted</td></tr><tr><td>Nonlinear break (outer scaling)</td><td>Tb⁢𝒬o,r</td><td>0.98</td><td>[0.94, 1.04]</td><td>fitted</td></tr><tr><td>Nonlinear width</td><td>σ𝒬r</td><td>1.297</td><td>[1.280, 1.304]</td><td>fitted</td></tr><tr><td colspan=\\\"5\\\">Stage B: variance growth (14-point regression on ηs )</td></tr><tr><td>Fully rough range coefficient</td><td>A𝒬,ηbase</td><td>2.30</td><td>—</td><td>predicted</td></tr><tr><td>free-slope fit (test)</td><td>A𝒬,ηr</td><td>3.8</td><td>[1.5, 6.1]</td><td>regression slope</td></tr><tr><td>Nonlinear baseline</td><td>B𝒬,Gr</td><td>6.4</td><td>[5.7, 6.9]</td><td>V0−Bℒr</td></tr></table>\",\"kind\":\"table\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"table\",\"source_ref\":\"tab97\",\"source_text\":\"Parameter Symbol Value 68% interval Source Stage A: spectral shape (area-normalised spectra) Linear break (roughness scaling) Tb⁢ℒs,r 1.87 [1.81, 1.96] fitted Linear width σℒr 0.50 [0.48, 0.52] fitted Linear area Bℒr 0.70 [0.67, 0.78] fitted Nonlinear break (outer scaling) Tb⁢𝒬o,r 0.98 [0.94, 1.04] fitted Nonlinear width σ𝒬r 1.297 [1.280, 1.304] fitted Stage B: variance growth (14-point regression on ηs ) Fully rough range coefficient A𝒬,ηbase 2.30 — predicted free-slope fit (test) A𝒬,ηr 3.8 [1.5, 6.1] regression slope Nonlinear baseline B𝒬,Gr 6.4 [5.7, 6.9] V0−Bℒr\",\"tag\":null,\"text\":\"Parameter Symbol Value 68% interval Source Stage A: spectral shape (area-normalised spectra) Linear break (roughness scaling) Tb⁢ℒs,r 1.87 [1.81, 1.96] fitted Linear width σℒr 0.50 [0.48, 0.52] fitted Linear area Bℒr 0.70 [0.67, 0.78] fitted Nonlinear break (outer scaling) Tb⁢𝒬o,r 0.98 [0.94, 1.04] fitted Nonlinear width σ𝒬r 1.297 [1.280, 1.304] fitted Stage B: variance growth (14-point regression on ηs ) Fully rough range coefficient A𝒬,ηbase 2.30 — predicted free-slope fit (test) A𝒬,ηr 3.8 [1.5, 6.1] regression slope Nonlinear baseline B𝒬,Gr 6.4 [5.7, 6.9] V0−Bℒr\"},{\"bbox\":[90.0,238.23,432.0,65.11],\"bbox_pdf\":[90.0,488.66,522.0,553.77],\"float_ref\":\"S7.T1\",\"fragments\":[{\"bbox_pdf\":[90.0,488.66,522.0,553.77],\"words\":[6741,6742,6743,6744,6745,6746,6747,6748,6749,6750,6751,6752,6753,6754,6755,6756,6757,6758,6759,6760,6761,6762,6763,6764,6765,6766,6767,6768,6769,6770,6771,6772,6773,6774,6775,6776,6777,6778,6779,6780,6781,6782,6783,6784,6785,6786,6787,6788,6789,6790,6791,6792,6793,6794,6795,6796,6797,6798,6799,6800,6801,6802,6803,6804,6805,6806,6807,6808,6809,6810,6811,6812,6813,6814,6815,6816,6817,6818,6819,6820,6821]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"caption\",\"source_ref\":\"caption98\",\"source_text\":\"Table 1: Two-lognormal rough-wall parameters from the two stages of §7. The range coefficient is predicted, not fitted; the free-slope fit is shown as the test of it. Intervals are central $68\\\\%$ ranges from a case-level bootstrap ( $1000$ replicates, whole cases) and quantify sampling scatter only: a common-mode $C_{f}$ calibration bias ( $\\\\pm 20\\\\%$ ) would rescale the free-slope test and the offset by up to $\\\\approx\\\\pm 40\\\\%$ without registering here. Shape parameters and $B_{\\\\mathcal{Q},G}^{r}$ are specific to the VT geometry.\",\"tag\":null,\"text\":\"Table 1: Two-lognormal rough-wall parameters from the two stages of §7. The range coefficient is predicted, not fitted; the free-slope fit is shown as the test of it. Intervals are central 68% ranges from a case-level bootstrap (1000 replicates, whole cases) and quantify sampling scatter only: a common-mode 𝐶 𝑓 calibration bias (±20%) would rescale the free-slope test and the offset by up to ≈ ±40% without registering here. Shape parameters and 𝐵 𝑟 are specific to the VT geometry. Q,𝐺\"},{\"bbox\":[90.0,328.58,432.0,24.46],\"bbox_pdf\":[90.0,438.96,522.0,463.42],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[90.0,438.96,522.0,463.42],\"words\":[6822,6823,6824,6825,6826,6827,6828,6829,6830,6831,6832,6833,6834,6835,6836,6837,6838,6839,6840,6841,6842,6843,6844,6845,6846,6847,6848,6849,6850]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.971,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph102\",\"source_text\":\"The two speed families overlap in $\\\\eta_{s}$ at distinctly different $k_{s}^{+}$ (family means $\\\\approx 393$ and $669$ ) and so directly probe whether $k_{s}^{+}$ drops out once fully rough. The probe is inconclusive. The families return different fitted slopes ( $A_{\\\\mathcal{Q},\\\\eta}^{r}\\\\approx 2.75$ at $35$ m/s, $5.07$ at $65$ m/s), but a common-slope fit with a family offset gives $+1.5\\\\pm 1.2$ at matched $\\\\eta_{s}$ — within $1.3$ standard errors of zero (the $\\\\eta_{s}$ basis) and within a third of a standard error of the $A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\,\\\\Delta\\\\ln k_{s}^{+}\\\\approx 1.2$ the $\\\\ln\\\\delta^{+}$ basis predicts ( $\\\\Delta\\\\ln k_{s}^{+}\\\\approx 0.53$ between families) — and the split hangs on the same single case as above: the high-variance $65$ m/s point at $x=4.91$ m, whose removal drops the $65$ m/s family slope to $0.8$ . Genuine residual $k_{s}^{+}$ sensitivity, non-equilibrium development effects, a family-level $C_{f}$ bias ( $\\\\sim 5\\\\%$ suffices) and experimental scatter are not separable from the VT data alone (§9).\",\"tag\":null,\"text\":\"Genuine residual 𝑘 + 𝑠 sensitivity, non-equilibrium development effects, a family-level 𝐶 𝑓 bias (∼ 5% suffices) and experimental scatter are not separable from the VT data alone (§9).\"},{\"bbox\":[90.0,373.81,189.73,14.35],\"bbox_pdf\":[90.0,403.84,279.73,418.19],\"fragments\":[{\"bbox_pdf\":[90.0,403.84,279.73,418.19],\"words\":[6851,6852,6853]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"heading\",\"source_ref\":\"title105\",\"source_text\":\"8 Smooth-to-rough transition\",\"tag\":null,\"text\":\"8 Smooth-to-rough transition\"},{\"bbox\":[90.0,402.09,432.0,146.4],\"bbox_pdf\":[90.0,243.51,522.0,389.91],\"fragments\":[{\"bbox_pdf\":[90.0,243.51,522.0,389.91],\"words\":[6854,6855,6856,6857,6858,6859,6860,6861,6862,6863,6864,6865,6866,6867,6868,6869,6870,6871,6872,6873,6874,6875,6876,6877,6878,6879,6880,6881,6882,6883,6884,6885,6886,6887,6888,6889,6890,6891,6892,6893,6894,6895,6896,6897,6898,6899,6900,6901,6902,6903,6904,6905,6906,6907,6908,6909,6910,6911,6912,6913,6914,6915,6916,6917,6918,6919,6920,6921,6922,6923,6924,6925,6926,6927,6928,6929,6930,6931,6932,6933,6934,6935,6936,6937,6938,6939,6940,6941,6942,6943,6944,6945,6946,6947,6948,6949,6950,6951,6952,6953,6954,6955,6956,6957,6958,6959,6960,6961,6962,6963,6964,6965,6966,6967,6968,6969,6970,6971,6972,6973,6974,6976,6977,6978,6979,6980,6981,6982,6983,6984,6985,6986,6987,6988,6989,6990,6991,6992,6993,6994,6995,6996,6997,6998,6999,7000,7001,7002,7003,7004,7005,7006,7007,7008,7009,7010,7012,7013,7014,7015,7016,7017,7018,7019,7020,7021,7022,7023,7024,7025,7026,7027,7028,7029,7030,7031,7032,7033,7034,7035,7036,7037,7038]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.987,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph106\",\"source_text\":\"The same two-source construction sketches how the spectrum reorganises as the wall is taken from hydraulically smooth, through the transitional band ( $5\\\\lesssim k_{s}^{+}\\\\lesssim 70$ ), into the fully rough regime; only the two asymptotic limits are fixed by the framework, the band between them requiring an interpolating ansatz it does not derive. In the smooth limit the inner population sits at the viscous peak $T^{+}\\\\sim 20$ and the outer population grows as $\\\\ln\\\\delta^{+}$ at the rate $A_{\\\\mathcal{Q}}^{\\\\mathrm{smooth}}\\\\approx 2.3$ ; in the fully rough limit the viscous inner peak is replaced by the roughness-local linear population $g_{\\\\mathcal{L}}^{r}(T^{s})$ , the inner-scaled invariant nonlinear part contributes the offset $B_{\\\\mathcal{Q},i}^{r}$ , and the outer population grows instead on $\\\\eta_{s}$ . Through the band the viscous peak is hollowed out while a roughness-local feature fills the same viscous decade, and the outer peak, fixed in $T^{o}$ , only changes amplitude; a peak that appears to move in viscous units across roughness states is therefore an artefact of the rescaling, not of a moving source.\",\"tag\":null,\"text\":\"The same two-source construction sketches how the spectrum reorganises as the wall is taken from hydraulically smooth, through the transitional band (5 ≲ 𝑘 + ≲ 70), into the fully rough regime; 𝑠 only the two asymptotic limits are fixed by the framework, the band between them requiring an interpolating ansatz it does not derive. In the smooth limit the inner population sits at the viscous peak 𝑇 + ∼ 20 and the outer population grows as ln 𝛿 + at the rate 𝐴 smooth ≈ 2.3; in the fully rough Q limit the viscous inner peak is replaced by the roughness-local linear population 𝑔 𝑟 (𝑇 𝑠 ), the inner- L scaled invariant nonlinear part contributes the offset 𝐵 𝑟 and the outer population grows instead Q,𝑖 on 𝜂𝑠. Through the band the viscous peak is hollowed out while a roughness-local feature fills the same viscous decade, and the outer peak, fixed in 𝑇 𝑜 only changes amplitude; a peak that appears to move in viscous units across roughness states is therefore an artefact of the rescaling, not of a moving source.\"},{\"bbox\":[90.0,550.11,432.01,123.21],\"bbox_pdf\":[90.0,118.68,522.01,241.89],\"fragments\":[{\"bbox_pdf\":[90.0,118.68,522.01,241.89],\"words\":[7039,7040,7041,7042,7043,7044,7045,7046,7047,7048,7049,7050,7051,7052,7053,7054,7055,7056,7057,7058,7059,7060,7061,7062,7063,7064,7065,7066,7067,7068,7069,7070,7071,7072,7073,7074,7075,7076,7077,7078,7079,7080,7081,7082,7083,7084,7085,7086,7087,7088,7089,7090,7091,7092,7093,7094,7095,7096,7097,7098,7099,7100,7101,7102,7103,7104,7105,7106,7107,7108,7109,7110,7111,7112,7113,7114,7115,7116,7117,7119,7120,7121,7122,7123,7124,7125,7126,7127,7128,7129,7130,7131,7132,7133,7134,7135,7136,7137,7138,7139,7140,7141,7142,7143,7144,7145,7146,7147,7148,7149,7150,7151,7152,7153,7154,7155,7157,7158,7159,7160,7161,7162,7163,7164,7165,7166,7168,7169,7170,7171,7172,7173,7174,7175,7176,7177,7178,7179,7180,7181,7182,7183,7184,7185,7186,7187,7188,7189,7190,7191,7192,7193,7194,7195,7196,7197,7198,7199,7200,7201,7202,7203,7204,7205,7206,7207,7208,7209,7210,7211,7212,7213,7214,7215,7216,7217,7218,7219,7220,7221,7222,7223,7224,7225,7226,7227]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph107\",\"source_text\":\"This yields a structural prediction sharper than any fitted number (figure 4). At fixed $k_{s}^{+}$ the inner-scaled variance grows linearly in $\\\\ln\\\\delta^{+}$ at the coefficient $2.3$ throughout, because $\\\\eta_{s}=\\\\ln\\\\delta^{+}-\\\\ln k_{s}^{+}$ and $k_{s}^{+}$ is fixed: roughness enters the growth not through the rate but through a vertical offset. The transitional band is traversed not along these curves but across them, by changing $k_{s}^{+}$ : the offset rises as $A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\ln k_{s}^{+}$ while the lower cutoff of the active range is still $\\\\nu/u_{\\\\tau}$ , and saturates at a geometry-set constant once $k_{s}$ takes over. It is that crossover — not any change of rate at fixed $k_{s}^{+}$ — whose width and shape are properties of the chosen blend rather than of the derivation. A $\\\\delta^{+}$ sweep at fixed $k_{s}^{+}$ in the fully rough regime, at controlled pressure gradient, would test the rate directly and measure the offset $B_{\\\\mathcal{Q},G}^{r}$ that the present narrow $\\\\eta_{s}$ range holds only through an extrapolation to $\\\\delta=k_{s}$ .\",\"tag\":null,\"text\":\"This yields a structural prediction sharper than any fitted number (figure 4). At fixed 𝑘 + 𝑠 the innerscaled variance grows linearly in ln 𝛿 + at the coefficient 2.3 throughout, because 𝜂𝑠 = ln 𝛿 + − ln 𝑘 + 𝑠 and 𝑘 + 𝑠 is fixed: roughness enters the growth not through the rate but through a vertical offset. The transitional band is traversed not along these curves but across them, by changing 𝑘 + 𝑠 the offset rises as 𝐴 base ln 𝑘 + 𝑠 while the lower cutoff of the active range is still 𝜈/𝑢𝜏, and saturates at a geometry-set Q, 𝜂 constant once 𝑘𝑠 takes over. It is that crossover not any change of rate at fixed 𝑘 + 𝑠 whose width and shape are properties of the chosen blend rather than of the derivation. A 𝛿 + sweep at fixed 𝑘 + 𝑠 in the fully rough regime, at controlled pressure gradient, would test the rate directly and measure the offset 𝐵 𝑟 that the present narrow 𝜂𝑠 range holds only through an extrapolation to 𝛿 = 𝑘𝑠. Q,𝐺\"},{\"bbox\":[300.54,720.98,10.91,10.91],\"bbox_pdf\":[300.54,60.11,311.45,71.02],\"fragments\":[{\"bbox_pdf\":[300.54,60.11,311.45,71.02],\"words\":[9392]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":6,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"13\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0012.pdf\",\"page_no\":13,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph102\",\"c_block\":0.983,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.993,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.983,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"401cd8b1c02b17b05d5410b95ae34c98dbf626a0e5900acd178a24ddfd4b10df\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/401cd8b1c02b17b05d5410b95ae34c98dbf626a0e5900acd178a24ddfd4b10df.pdf","asset_sha256":"401cd8b1c02b17b05d5410b95ae34c98dbf626a0e5900acd178a24ddfd4b10df","document_id":"arxiv:2608.16729","kind":"page","page_number":13,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[111.6,167.1,388.81,389.03],\"bbox_pdf\":[111.6,235.87,500.41,624.9],\"float_ref\":\"S7.F3\",\"fragments\":[{\"bbox_pdf\":[111.6,235.87,500.41,624.9],\"words\":[]}],\"html\":null,\"kind\":\"figure\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"figure\",\"source_ref\":\"S7.F3\",\"source_text\":\"\",\"tag\":null,\"text\":\"\"},{\"bbox\":[90.0,568.73,432.0,51.56],\"bbox_pdf\":[90.0,171.71,522.0,223.27],\"float_ref\":\"S7.F3\",\"fragments\":[{\"bbox_pdf\":[90.0,171.71,522.0,223.27],\"words\":[7228,7229,7230,7231,7232,7233,7234,7235,7236,7237,7238,7239,7240,7241,7242,7243,7244,7245,7246,7247,7248,7249,7250,7251,7252,7253,7254,7255,7256,7257,7258,7259,7260,7261,7262,7263,7264,7265,7266,7267,7268,7269,7270,7271,7272,7273,7274,7275,7276,7277,7278,7279,7280,7281,7282,7283,7284,7285,7286,7287,7288,7289]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"caption\",\"source_ref\":\"caption104\",\"source_text\":\"Figure 3: Calibrated two-lognormal model (heavy coloured curves) overlaid on the VT premultiplied spectra (thin coloured lines), in roughness-scaled ( $TU_{e}/k_{s}$ ) and outer-scaled ( $T^{o}=TU_{e}/\\\\delta$ ) coordinates, at the parameter values of table 1 and each case’s measured flow state. Dashed and dotted grey curves show $g_{\\\\mathcal{L}}^{r}$ and $g_{\\\\mathcal{Q}}^{r}$ individually for a representative case in each row.\",\"tag\":null,\"text\":\"Figure 3: Calibrated two-lognormal model (heavy coloured curves) overlaid on the VT premultiplied spectra (thin coloured lines), in roughness-scaled (𝑇𝑈𝑒/𝑘𝑠) and outer-scaled (𝑇 𝑜 = 𝑇𝑈𝑒/𝛿) coordinates, at the parameter values of table 1 and each case’s measured flow state. Dashed and dotted grey curves show 𝑔 𝑟 and 𝑔 𝑟 individually for a representative case in each row. L Q\"},{\"bbox\":[300.54,720.98,10.91,10.91],\"bbox_pdf\":[300.54,60.11,311.45,71.02],\"fragments\":[{\"bbox_pdf\":[300.54,60.11,311.45,71.02],\"words\":[9393]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":2,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"14\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0013.pdf\",\"page_no\":14,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"1a8f6a292ec36698a1cc44bf206409e049dfb33bbaa87bb56c802871ff2bb89a\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/1a8f6a292ec36698a1cc44bf206409e049dfb33bbaa87bb56c802871ff2bb89a.pdf","asset_sha256":"1a8f6a292ec36698a1cc44bf206409e049dfb33bbaa87bb56c802871ff2bb89a","document_id":"arxiv:2608.16729","kind":"page","page_number":14,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[90.0,220.34,432.01,210.96],\"bbox_pdf\":[90.0,360.7,522.01,571.66],\"float_ref\":\"S8.F4\",\"fragments\":[{\"bbox_pdf\":[90.0,360.7,522.01,571.66],\"words\":[]}],\"html\":null,\"kind\":\"figure\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"figure\",\"source_ref\":\"S8.F4\",\"source_text\":\"\",\"tag\":null,\"text\":\"\"},{\"bbox\":[90.0,442.88,432.01,126.42],\"bbox_pdf\":[90.0,222.7,522.01,349.12],\"float_ref\":\"S8.F4\",\"fragments\":[{\"bbox_pdf\":[90.0,222.7,522.01,349.12],\"words\":[7290,7291,7292,7293,7294,7295,7296,7297,7298,7299,7300,7301,7302,7303,7304,7305,7306,7307,7308,7309,7310,7311,7312,7313,7314,7315,7316,7317,7318,7320,7321,7322,7323,7324,7325,7326,7327,7328,7329,7330,7331,7332,7333,7334,7335,7336,7337,7338,7339,7340,7341,7342,7343,7344,7345,7346,7347,7348,7349,7350,7351,7352,7353,7354,7355,7356,7357,7358,7359,7360,7361,7362,7363,7364,7366,7367,7368,7369,7370,7371,7372,7373,7374,7375,7376,7377,7378,7379,7380,7381,7382,7383,7384,7385,7386,7387,7388,7389,7390,7391,7392,7394,7395,7396,7397,7398,7399,7400,7401,7402,7403,7404,7405,7406,7407,7408,7409,7410,7411,7412,7413,7414,7415,7416,7417,7418,7419,7420,7421,7422,7423,7424,7425,7426,7427,7428,7429,7430,7431,7432,7433,7434,7435,7436,7437,7438,7439,7440,7441,7442,7443,7444,7445,7446,7447,7448,7449,7450,7451,7452,7453,7454,7455,7456,7457,7458,7459,7460,7461,7462,7463,7464,7465,7466,7467,7468,7469,7470,7471,7472,7473,7474,7475,7476,7477,7478,7479]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.993,\"verified\":true},\"order\":1,\"role\":\"caption\",\"source_ref\":\"caption109\",\"source_text\":\"Figure 4: Effect of $k_{s}^{+}$ on the logarithmic growth of the wall-pressure variance, from a smooth-to-rough blend of the present model at fixed $C_{f}$ . (a) Integrated variance $\\\\overline{p^{\\\\prime 2}}^{+}$ versus $\\\\delta^{+}$ at six fixed $k_{s}^{+}$ from hydraulically smooth ( $k_{s}^{+}=0.5$ ) to fully rough ( $k_{s}^{+}=1000$ ), drawn only where $\\\\delta/k_{s}\\\\geq 10$ ; the curves are parallel (grey guide, $2.3$ per $\\\\ln\\\\delta^{+}$ ), roughness entering as a vertical offset alone. (b) That offset, $\\\\overline{p^{\\\\prime 2}}^{+}-A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\eta_{s}$ , at fixed $k_{s}^{+}$ : it rises as $A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\ln k_{s}^{+}$ while the lower cutoff of the active range is still $\\\\nu/u_{\\\\tau}$ (blue dashed), and saturates at the geometry-set $B_{\\\\mathcal{L}}^{r}+B_{\\\\mathcal{Q},G}^{r}$ (red dashed) once $k_{s}$ takes over, i.e. once $k_{s}^{+}\\\\gtrsim O(100)$ , so the VT cases (red band) sit on the plateau. The grey band marks where the cutoff migrates from $\\\\nu/u_{\\\\tau}$ to $k_{s}$ . Only the two limits are fixed by the framework; the width and shape of the crossover are the blend’s.\",\"tag\":null,\"text\":\"Figure 4: Effect of 𝑘 + 𝑠 on the logarithmic growth of the wall-pressure variance, from a smooth-to- + rough blend of the present model at fixed 𝐶 𝑓 (a) Integrated variance 𝑝′2 versus 𝛿 + at six fixed 𝑘 + 𝑠 from hydraulically smooth (𝑘 + + 𝑠 = 0.5) to fully rough (𝑘𝑠 = 1000), drawn only where 𝛿/𝑘𝑠 ≥ 10; the curves are parallel (grey guide, 2.3 per ln 𝛿 + roughness entering as a vertical offset alone. (b) That + offset, 𝑝′2 − 𝐴 base 𝜂𝑠, at fixed 𝑘 + base ln 𝑘 + Q, 𝜂 𝑠 it rises as 𝐴Q, 𝜂 𝑠 while the lower cutoff of the active range is still 𝜈/𝑢𝜏 (blue dashed), and saturates at the geometry-set 𝐵 𝑟 + 𝐵 𝑟 (red dashed) once 𝑘𝑠 takes L Q,𝐺 over, i.e. once 𝑘 + ≳ 𝑂 (100), so the VT cases (red band) sit on the plateau. The grey band marks 𝑠 where the cutoff migrates from 𝜈/𝑢𝜏 to 𝑘𝑠. Only the two limits are fixed by the framework; the width and shape of the crossover are the blend’s.\"},{\"bbox\":[300.54,720.98,10.91,10.91],\"bbox_pdf\":[300.54,60.11,311.45,71.02],\"fragments\":[{\"bbox_pdf\":[300.54,60.11,311.45,71.02],\"words\":[9394]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":2,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"15\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0014.pdf\",\"page_no\":15,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"caption109\",\"c_block\":0.992,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.996,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.992,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"ed02954b81c23a6fc85b8b4cf58799686f365c852bf9701c73dde969ff263822\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/ed02954b81c23a6fc85b8b4cf58799686f365c852bf9701c73dde969ff263822.pdf","asset_sha256":"ed02954b81c23a6fc85b8b4cf58799686f365c852bf9701c73dde969ff263822","document_id":"arxiv:2608.16729","kind":"page","page_number":15,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1},\"blocks\":[{\"bbox\":[90.0,86.61,186.54,14.35],\"bbox_pdf\":[90.0,691.04,276.54,705.39],\"fragments\":[{\"bbox_pdf\":[90.0,691.04,276.54,705.39],\"words\":[7480,7481,7482,7483]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"heading\",\"source_ref\":\"title110\",\"source_text\":\"9 Discussion and conclusions\",\"tag\":null,\"text\":\"9 Discussion and conclusions\"},{\"bbox\":[90.0,114.89,432.0,132.85],\"bbox_pdf\":[90.0,544.26,522.0,677.11],\"fragments\":[{\"bbox_pdf\":[90.0,544.26,522.0,677.11],\"words\":[7484,7485,7486,7487,7488,7489,7490,7491,7492,7493,7494,7495,7496,7497,7498,7499,7500,7501,7502,7503,7504,7505,7506,7507,7508,7509,7510,7511,7512,7513,7514,7515,7516,7517,7518,7519,7520,7521,7522,7523,7524,7525,7527,7528,7529,7530,7531,7532,7533,7534,7535,7536,7537,7539,7540,7541,7542,7543,7544,7545,7546,7547,7548,7549,7550,7551,7552,7553,7554,7555,7556,7557,7558,7559,7560,7561,7562,7563,7564,7565,7566,7567,7568,7569,7570,7571,7572,7573,7574,7575,7576,7577,7578,7579,7580,7581,7582,7583,7584,7585,7586,7587,7588,7589,7590,7591,7592,7593,7594,7595,7596,7597,7598,7599,7600,7601,7602,7603,7604,7605,7606,7607,7608,7609,7610,7611,7612,7613,7614,7615,7616,7617,7618,7619,7620,7621,7622,7623,7624,7625,7626,7627,7628,7629,7630,7631,7632,7633,7634,7635,7636,7637,7638,7639,7640,7641,7642,7643]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph111\",\"source_text\":\"What the fourteen cases establish robustly is spectral, and it belongs at the front. The energetic population collapses at $T^{o}\\\\sim 1$ and its shape is Reynolds-independent at fixed geometry; the integrated variance is consistent with growth in $\\\\eta_{s}=\\\\ln(\\\\delta/k_{s})$ — a positive trend at the $68\\\\%$ but not the $95\\\\%$ level — rather than demonstrating it. Behind that collapse is a simple picture: the rough-wall pressure spectrum is the sum of two overlapping log-normal distributions, each modelling a separate contribution to the total energy. One is roughness-local, from the mean-shear (linear) source, and sits at high frequency on the roughness period $T^{s}=TU_{e}/k_{s}$ ; the other is the energetic contribution from the nonlinear source, at the outer period $T^{o}=TU_{e}/\\\\delta\\\\sim 1$ , and it alone grows with Reynolds number. Splitting the total variance into these two contributions is what lets the shape be pinned tightly while the growth rate is predicted rather than fitted.\",\"tag\":null,\"text\":\"What the fourteen cases establish robustly is spectral, and it belongs at the front. The energetic population collapses at 𝑇 𝑜 ∼ 1 and its shape is Reynolds-independent at fixed geometry; the integrated variance is consistent with growth in 𝜂𝑠 = ln(𝛿/𝑘𝑠) a positive trend at the 68% but not the 95% level rather than demonstrating it. Behind that collapse is a simple picture: the rough-wall pressure spectrum is the sum of two overlapping log-normal distributions, each modelling a separate contribution to the total energy. One is roughness-local, from the mean-shear (linear) source, and sits at high frequency on the roughness period 𝑇 𝑠 = 𝑇𝑈𝑒/𝑘𝑠; the other is the energetic contribution from the nonlinear source, at the outer period 𝑇 𝑜 = 𝑇𝑈𝑒/𝛿 ∼ 1, and it alone grows with Reynolds number. Splitting the total variance into these two contributions is what lets the shape be pinned tightly while the growth rate is predicted rather than fitted.\"},{\"bbox\":[90.0,250.38,432.01,105.75],\"bbox_pdf\":[90.0,435.87,522.01,541.62],\"fragments\":[{\"bbox_pdf\":[90.0,435.87,522.01,541.62],\"words\":[7644,7645,7646,7647,7648,7649,7650,7651,7652,7653,7654,7655,7656,7657,7658,7659,7660,7661,7662,7663,7664,7665,7666,7667,7668,7669,7670,7671,7672,7673,7674,7675,7676,7677,7678,7679,7680,7681,7682,7683,7684,7685,7686,7687,7688,7689,7690,7691,7692,7693,7694,7695,7696,7697,7698,7699,7700,7701,7702,7703,7704,7705,7706,7707,7708,7709,7710,7711,7712,7713,7714,7715,7716,7717,7718,7719,7720,7721,7722,7723,7724,7725,7726,7727,7728,7729,7730,7731,7732,7733,7734,7735,7736,7737,7738,7739,7740,7741,7742,7743,7744,7745,7746,7747,7748,7749,7750,7751,7752,7753,7754,7755,7756,7757,7758,7759,7760,7761,7762,7763,7764,7765,7766,7767,7768,7769,7770,7771]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph112\",\"source_text\":\"Each source scales on its own natural length. The linear source $\\\\mathcal{L}$ is roughness-local and contributes a non-growing $O(1)$ offset; the nonlinear source $\\\\mathcal{Q}$ acts over a logarithmic wall-normal range cut off below by the roughness sublayer ( $\\\\sim k_{s}$ ) and above by $\\\\delta$ , so its area carries all the Reynolds-number growth through the single term $\\\\eta_{s}$ , with no $k_{s}^{+}$ dependence at leading order. The fitted outer break $T_{b\\\\mathcal{Q}}^{o,r}\\\\approx 0.98$ , with the measured $U_{c}\\\\approx 0.49\\\\,U_{e}$ (Totten et al., 2026), gives a convected wavelength $\\\\lambda_{x}\\\\approx 0.5\\\\,\\\\delta$ , matching the smooth-wall outer population of Massey et al. (2026c); kernel consistency favours a log-layer source ( $y_{\\\\mathcal{Q},\\\\mathrm{eff}}/\\\\delta\\\\lesssim 0.1$ ) over a mid-layer one, whose contribution the elliptic weight $e^{-4\\\\pi}$ would suppress.\",\"tag\":null,\"text\":\"Each source scales on its own natural length. The linear source L is roughness-local and contributes a non-growing 𝑂 (1) offset; the nonlinear source Q acts over a logarithmic wall-normal range cut off below by the roughness sublayer (∼ 𝑘𝑠) and above by 𝛿, so its area carries all the Reynolds-number growth through the single term 𝜂𝑠, with no 𝑘 + dependence at leading order. 𝑠 The fitted outer break 𝑇 𝑜,𝑟 ≈ 0.98, with the measured 𝑈𝑐 ≈ 0.49 𝑈𝑒 [Totten et al., 2026], gives 𝑏 Q a convected wavelength 𝜆𝑥 ≈ 0.5 𝛿, matching the smooth-wall outer population of Massey et al. [2026c]; kernel consistency favours a log-layer source (𝑦Q,eff/𝛿 ≲ 0.1) over a mid-layer one, whose contribution the elliptic weight 𝑒 −4 𝜋 would suppress.\"},{\"bbox\":[90.0,358.77,432.01,24.46],\"bbox_pdf\":[90.0,408.77,522.01,433.23],\"fragments\":[{\"bbox_pdf\":[90.0,408.77,522.01,433.23],\"words\":[7772,7773,7774,7775,7776,7777,7778,7779,7780,7781,7782,7783,7784,7785,7786,7787,7788,7789,7790,7791]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph113\",\"source_text\":\"The construction makes four predictions, each with a clear experimental test; the offset carries the calibration uncertainty of table 1.\",\"tag\":null,\"text\":\"The construction makes four predictions, each with a clear experimental test; the offset carries the calibration uncertainty of table 1.\"},{\"bbox\":[103.64,394.84,418.37,106.8],\"bbox_pdf\":[103.64,290.36,522.0,397.16],\"fragments\":[{\"bbox_pdf\":[103.64,290.36,522.0,397.16],\"words\":[7792,7793,7794,7795,7796,7797,7798,7799,7800,7801,7802,7803,7804,7805,7806,7807,7808,7809,7810,7811,7812,7813,7814,7815,7816,7817,7818,7819,7820,7821,7822,7823,7824,7825,7826,7827,7828,7829,7830,7831,7833,7834,7835,7836,7837,7838,7839,7840,7841,7842,7843,7844,7845,7846,7847,7848,7849,7850,7851,7852,7853,7854,7855,7856,7857,7858,7859,7860,7861,7862,7863,7864,7865,7866,7867,7868,7869,7870,7871,7872,7873,7874,7875,7876,7877,7878,7879,7880,7881,7882,7883,7884,7885,7886,7887,7888,7889,7890,7891,7892,7893,7894,7895,7896,7897,7898,7899,7900,7901,7902,7903,7904,7905,7906,7907,7908,7909,7910,7911,7912]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":1,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"list_item\",\"source_ref\":\"S9.I1.i1\",\"source_text\":\"1. The growth rate is universal. Once fully rough, the inner-scaled variance grows on $\\\\eta_{s}$ at the smooth-wall rate $A_{\\\\mathcal{Q},\\\\eta}^{\\\\mathrm{base}}\\\\approx 2.3$ , with no free parameter: every canopy effect is confined to the non-growing offset $B_{\\\\mathcal{Q},G}^{r}$ , because anything set by $k_{s}$ decays in $y_{d}/k_{s}$ and integrates over $\\\\mathrm{d}\\\\ln y_{d}$ to a finite constant (§4). Test: a $\\\\delta^{+}$ sweep at fixed $k_{s}^{+}$ over decades, at controlled pressure gradient, measures the slope directly. A rate significantly above $2.3$ would falsify outer-layer similarity for the pressure source, not calibrate a gain. On the VT data the free slope is $3.8$ with $68\\\\%$ interval $[1.5,\\\\,6.1]$ , consistent with the prediction and unable to discriminate it.\",\"tag\":null,\"text\":\"1. The growth rate is universal. Once fully rough, the inner-scaled variance grows on 𝜂𝑠 at the smooth-wall rate 𝐴 base ≈ 2.3, with no free parameter: every canopy effect is confined to the Q, 𝜂 non-growing offset 𝐵 𝑟 because anything set by 𝑘𝑠 decays in 𝑦𝑑/𝑘𝑠 and integrates over Q,𝐺 d ln 𝑦𝑑 to a finite constant (§4). Test: a 𝛿 + sweep at fixed 𝑘 + over decades, at controlled 𝑠 pressure gradient, measures the slope directly. A rate significantly above 2.3 would falsify outer-layer similarity for the pressure source, not calibrate a gain. On the VT data the free slope is 3.8 with 68% interval [1.5, 6.1], consistent with the prediction and unable to discriminate it.\"},{\"bbox\":[103.64,512.23,418.37,52.58],\"bbox_pdf\":[103.64,227.19,522.0,279.77],\"fragments\":[{\"bbox_pdf\":[103.64,227.19,522.0,279.77],\"words\":[7913,7914,7915,7916,7917,7918,7919,7920,7921,7922,7923,7924,7925,7926,7927,7928,7929,7930,7931,7932,7933,7934,7935,7936,7937,7938,7939,7940,7941,7942,7943,7944,7945,7946,7947,7948,7949,7950,7951,7952,7953,7954,7955,7956,7957,7958,7959,7960,7961,7962,7963,7964,7965,7966,7967,7968,7969,7970,7971,7972,7973,7974]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":1,\"match\":{\"src_placed\":0.983,\"verified\":true},\"order\":5,\"role\":\"list_item\",\"source_ref\":\"S9.I1.i2\",\"source_text\":\"2. $k_{s}^{+}$ drops out once fully rough. At fixed geometry all growth is carried by $\\\\delta/k_{s}$ , not by the roughness Reynolds number. Test: vary $k_{s}^{+}$ (change roughness size or speed) at matched $\\\\delta/k_{s}$ and look for a change in the spectrum; there should be none. The two VT speed families are the only present instance and are inconclusive.\",\"tag\":null,\"text\":\"2. 𝑘 + drops out once fully rough. At fixed geometry all growth is carried by 𝛿/𝑘𝑠, not by the 𝑠 roughness Reynolds number. Test: vary 𝑘 + 𝑠 (change roughness size or speed) at matched 𝛿/𝑘𝑠 and look for a change in the spectrum; there should be none. The two VT speed families are the only present instance and are inconclusive.\"},{\"bbox\":[103.64,575.39,418.37,52.58],\"bbox_pdf\":[103.64,164.03,522.0,216.61],\"fragments\":[{\"bbox_pdf\":[103.64,164.03,522.0,216.61],\"words\":[7975,7976,7977,7978,7979,7980,7981,7982,7983,7984,7985,7986,7987,7988,7989,7990,7991,7992,7993,7994,7995,7996,7997,7998,7999,8000,8001,8002,8003,8004,8005,8006,8007,8008,8010,8011,8012,8013,8014,8015,8016,8017,8018,8019,8020,8021,8022,8024,8025,8026,8027,8028,8029,8030,8031,8032,8033,8034,8035,8036,8037,8038,8039,8040,8041,8042,8043,8044,8045,8046,8047,8048,8049,8050,8051]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":1,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"list_item\",\"source_ref\":\"S9.I1.i3\",\"source_text\":\"3. The energetic break is fixed at $T^{o}\\\\sim 1$ . It does not move with $\\\\delta/k_{s}$ ; its apparent drift on the roughness axis is only the rescaling $T_{b\\\\mathcal{Q}}^{s,r}=(\\\\delta/k_{s})\\\\,T_{b\\\\mathcal{Q}}^{o,r}$ . Test: track the peak period across a $\\\\delta/k_{s}$ range — it should stay fixed in outer units and move in exact proportion to $\\\\delta/k_{s}$ in roughness units. The VT collapse at $T^{o}\\\\sim 1$ already supports this.\",\"tag\":null,\"text\":\"3. The energetic break is fixed at 𝑇 𝑜 ∼ 1. It does not move with 𝛿/𝑘𝑠; its apparent drift on the roughness axis is only the rescaling 𝑇 𝑠,𝑟 = (𝛿/𝑘𝑠) 𝑇 𝑜,𝑟 Test: track the peak period across 𝑏 Q 𝑏 Q a 𝛿/𝑘𝑠 range it should stay fixed in outer units and move in exact proportion to 𝛿/𝑘𝑠 in roughness units. The VT collapse at 𝑇 𝑜 ∼ 1 already supports this.\"},{\"bbox\":[103.64,638.55,418.36,57.77],\"bbox_pdf\":[103.64,95.68,522.0,153.45],\"fragments\":[{\"bbox_pdf\":[103.64,95.68,522.0,153.45],\"words\":[8052,8053,8054,8055,8056,8057,8058,8059,8060,8061,8062,8063,8064,8065,8066,8067,8068,8069,8070,8071,8072,8073,8074,8075,8076,8077,8078,8079,8080,8081,8082,8083,8084,8085,8086,8087,8088,8089,8090,8091,8092,8093,8094,8095,8096,8097,8098,8099,8100,8101,8102,8103,8104,8105,8106,8107,8108,8109]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":1,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"list_item\",\"source_ref\":\"S9.I1.i4\",\"source_text\":\"4. Across geometries the growth lines are parallel. Geometry enters $B_{\\\\mathcal{Q},G}^{r}$ and not the slope, a falsifiable consequence of confining canopy physics to the offset. Test: plot $\\\\overline{p^{\\\\prime 2}}^{+}$ against $\\\\eta_{s}$ for different roughness types; the lines should be offset but not tilted. Existing rough-wall campaigns (Blake, 1970; Meyers et al., 2015) may already permit this.\",\"tag\":null,\"text\":\"4. Across geometries the growth lines are parallel. Geometry enters 𝐵 𝑟 and not the slope, a Q,𝐺 + falsifiable consequence of confining canopy physics to the offset. Test: plot 𝑝′2 against 𝜂𝑠 for different roughness types; the lines should be offset but not tilted. Existing rough-wall campaigns [Blake, 1970, Meyers et al., 2015] may already permit this.\"},{\"bbox\":[300.54,720.98,10.91,10.91],\"bbox_pdf\":[300.54,60.11,311.45,71.02],\"fragments\":[{\"bbox_pdf\":[300.54,60.11,311.45,71.02],\"words\":[9395]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":8,\"role\":\"page_footer\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"16\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.16729\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.16729v1-p0015.pdf\",\"page_no\":16,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5fe8ffddab5be85eb53966d1be97a9ddd6175a58962de06818d14d7bea6d92f0\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"S9.I1.i2\",\"c_block\":0.981,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.981,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"1c3044407073be1a3c4a466e281c05254f9c56a01df87cefbd87d7ca5fbc9f3e\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/1c3044407073be1a3c4a466e281c05254f9c56a01df87cefbd87d7ca5fbc9f3e.pdf","asset_sha256":"1c3044407073be1a3c4a466e281c05254f9c56a01df87cefbd87d7ca5fbc9f3e","document_id":"arxiv:2608.16729","kind":"page","page_number":16,"paper_attribution_json":"{\"arxiv_id\":\"2608.16729\",\"authors\":\"J. M. O. Massey, A. J. Smits and B. J. McKeon\",\"title\":\"A source-term interpretation of turbulent rough wall-pressure spectra\",\"url\":\"https://arxiv.org/abs/2608.16729v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.20432\",\"authors\":\"Shane Caldwell\",\"title\":\"ProofJudge: Tool-Grounded LLM Evaluation of Formal Proof Quality in Mathlib\",\"url\":\"https://arxiv.org/abs/2608.20432v1\",\"version\":1},\"blocks\":[{\"bbox\":[100.49,78.92,302.43,7.97],\"bbox_pdf\":[100.49,705.11,402.92,713.08],\"fragments\":[{\"bbox_pdf\":[100.49,705.11,402.92,713.08],\"words\":[1566,1567,1568,1569,1570,1571,1572,1573,1574,1575]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":0,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"ProofJudge: Tool-Grounded LLM Evaluation of Formal Proof Quality in Mathlib\"},{\"bbox\":[100.49,106.42,411.02,33.87],\"bbox_pdf\":[100.49,651.71,511.51,685.58],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[100.49,651.71,511.51,685.58],\"words\":[429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph11\",\"source_text\":\"We introduce ProofJudge, an Agentic Judge [7] system that evaluates formal proof quality beyond compilation. The Lean kernel provides an objective correctness anchor, while the judge evaluates softer quality dimensions—for example, statement quality—via a rubric based on Mathlib’s review norms. ProofJudge’s tool access (querying Mathlib via bash) enables the judge to ground its assessments in the actual state of the library as a human reviewer does.\",\"tag\":null,\"text\":\"evaluates softer quality dimensions—for example, statement quality—via a rubric based on Mathlib’s review norms. ProofJudge’s tool access (querying Mathlib via bash) enables the judge to ground its assessments in the actual state of the library as a human reviewer does.\"},{\"bbox\":[100.49,157.07,114.99,11.96],\"bbox_pdf\":[100.49,622.97,215.47,634.93],\"fragments\":[{\"bbox_pdf\":[100.49,622.97,215.47,634.93],\"words\":[469,470,471]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":2,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"heading\",\"source_ref\":\"title12\",\"source_text\":\"2.1 Rubric Design\",\"tag\":null,\"text\":\"2.1 Rubric Design\"},{\"bbox\":[100.49,177.76,411.02,21.92],\"bbox_pdf\":[100.49,592.33,511.51,614.24],\"fragments\":[{\"bbox_pdf\":[100.49,592.33,511.51,614.24],\"words\":[472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph13\",\"source_text\":\"Recent work applies LLM judges to the semantic correctness of PRs[6]; we instead focus on whether a proof meets the qualitative standards for library inclusion.\",\"tag\":null,\"text\":\"Recent work applies LLM judges to the semantic correctness of PRs[6]; we instead focus on whether a proof meets the qualitative standards for library inclusion.\"},{\"bbox\":[100.49,201.83,411.03,45.83],\"bbox_pdf\":[100.49,544.35,511.51,590.17],\"fragments\":[{\"bbox_pdf\":[100.49,544.35,511.51,590.17],\"words\":[497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph14\",\"source_text\":\"The rubric asks the judge agent to decompose the verdict: the model scores five dimensions independently on a 1–10 scale. Those are: library leverage, automation fit, structural clarity, statement quality and Mathlib conventions. The harness takes those numbers and weights them towards a final score.\",\"tag\":null,\"text\":\"The rubric asks the judge agent to decompose the verdict: the model scores five dimensions independently on a 1–10 scale. Those are: library leverage, automation fit, structural clarity, statement quality and Mathlib conventions. The harness takes those numbers and weights them towards a final score.\"},{\"bbox\":[100.49,264.44,156.96,11.96],\"bbox_pdf\":[100.49,515.6,257.45,527.56],\"fragments\":[{\"bbox_pdf\":[100.49,515.6,257.45,527.56],\"words\":[542,543,544]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":2,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"heading\",\"source_ref\":\"title15\",\"source_text\":\"2.2 Dataset Construction\",\"tag\":null,\"text\":\"2.2 Dataset Construction\"},{\"bbox\":[100.49,285.12,411.03,69.74],\"bbox_pdf\":[100.49,437.14,511.52,506.88],\"fragments\":[{\"bbox_pdf\":[100.49,437.14,511.52,506.88],\"words\":[545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph16\",\"source_text\":\"We evaluate ProofJudge on an initial dataset of 218 declaration pairs drawn from Mathlib pull requests. To develop the dataset, we used claude-sonnet-5 to review past PRs and determine which had significant differences between the earliest and final revision that did not come down to linting or involve a new declaration. The dataset, which we have open-sourced, includes a dev split of 123 declaration pairs that was used to tune the rubric, along with the 218-pair test split for the eval itself.\",\"tag\":null,\"text\":\"We evaluate ProofJudge on an initial dataset of 218 declaration pairs drawn from Mathlib pull requests. To develop the dataset, we used claude-sonnet-5 to review past PRs and determine which had significant differences between the earliest and final revision that did not come down to linting or involve a new declaration. The dataset, which we have open-sourced, includes a dev split of 123 declaration pairs that was used to tune the rubric, along with the 218-pair test split for the eval itself.\"},{\"bbox\":[100.49,374.31,76.07,14.35],\"bbox_pdf\":[100.49,403.34,176.56,417.69],\"fragments\":[{\"bbox_pdf\":[100.49,403.34,176.56,417.69],\"words\":[627,628]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"heading\",\"source_ref\":\"title17\",\"source_text\":\"3 Results\",\"tag\":null,\"text\":\"3 Results\"},{\"bbox\":[100.49,400.82,411.03,81.69],\"bbox_pdf\":[100.49,309.49,511.52,391.18],\"fragments\":[{\"bbox_pdf\":[100.49,309.49,511.52,391.18],\"words\":[629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph18\",\"source_text\":\"For each pair, the judge scores both the pre-revision and post-revision proof independently, and we measure whether the judge’s preference aligns with the reviewer’s. By alignment, we refer to the post-revision proof (that was accepted into the library) receiving a higher score than the pre-revision proof (that was rejected). The judge considers each PR independently, and is unaware of the score it provided the other revision or that any other revision is being scored at all. The judge agent is allowed to use twenty tool calls before it is forced to make a determination to limit inference costs.\",\"tag\":null,\"text\":\"For each pair, the judge scores both the pre-revision and post-revision proof independently, and we measure whether the judge’s preference aligns with the reviewer’s. By alignment, we refer to the post-revision proof (that was accepted into the library) receiving a higher score than the pre-revision proof (that was rejected). The judge considers each PR independently, and is unaware of the score it provided the other revision or that any other revision is being scored at all. The judge agent is allowed to use twenty tool calls before it is forced to make a determination to limit inference costs.\"},{\"bbox\":[100.49,484.66,411.02,21.92],\"bbox_pdf\":[100.49,285.42,511.51,307.34],\"fragments\":[{\"bbox_pdf\":[100.49,285.42,511.51,307.34],\"words\":[727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph19\",\"source_text\":\"We evaluate six judges—three open-weight, three closed—on the 218-pair test split with three replicates each. We compare against a 50% random-chance baseline.\",\"tag\":null,\"text\":\"We evaluate six judges—three open-weight, three closed—on the 218-pair test split with three replicates each. We compare against a 50% random-chance baseline.\"},{\"bbox\":[167.28,520.1,277.45,79.72],\"bbox_pdf\":[167.28,192.18,444.72,271.9],\"float_ref\":\"S3.T1\",\"fragments\":[{\"bbox_pdf\":[167.28,192.18,444.72,271.9],\"words\":[749,750,751,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785]}],\"html\":\"<table><tr><th>Judge</th><th>Weights</th><th>Alignment % (95% CI)</th><th>USD/pair</th></tr><tr><td>claude-sonnet-5</td><td>closed</td><td>80.8 (73.6–88.2)</td><td>1.392</td></tr><tr><td>gemini-3.7-flash</td><td>closed</td><td>75.2 (69.3–81.2)</td><td>0.186</td></tr><tr><td>muse-glimmer-30b</td><td>open</td><td>70.2 (63.4–77.3)</td><td>0.140</td></tr><tr><td>inkling-small</td><td>open</td><td>69.3 (60.2–77.8)</td><td>0.126</td></tr><tr><td>gpt-5.4-mini</td><td>closed</td><td>68.7 (60.6–77.1)</td><td>0.136</td></tr><tr><td>deepseek-v4-flash</td><td>open</td><td>63.5 (56.0–71.6)</td><td>0.029</td></tr></table>\",\"kind\":\"table\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"table\",\"source_ref\":\"tab20\",\"source_text\":\"Judge Weights Alignment % (95% CI) USD/pair claude-sonnet-5 closed 80.8 (73.6–88.2) 1.392 gemini-3.7-flash closed 75.2 (69.3–81.2) 0.186 muse-glimmer-30b open 70.2 (63.4–77.3) 0.140 inkling-small open 69.3 (60.2–77.8) 0.126 gpt-5.4-mini closed 68.7 (60.6–77.1) 0.136 deepseek-v4-flash open 63.5 (56.0–71.6) 0.029\",\"tag\":null,\"text\":\"Judge Weights Alignment % (95% CI) USD/pair claude-sonnet-5 closed 80.8 (73.6–88.2) 1.392 gemini-3.7-flash closed 75.2 (69.3–81.2) 0.186 muse-glimmer-30b open 70.2 (63.4–77.3) 0.140 inkling-small open 69.3 (60.2–77.8) 0.126 gpt-5.4-mini closed 68.7 (60.6–77.1) 0.136 deepseek-v4-flash open 63.5 (56.0–71.6) 0.029\"},{\"bbox\":[100.49,608.55,411.03,21.92],\"bbox_pdf\":[100.49,161.53,511.51,183.45],\"float_ref\":\"S3.T1\",\"fragments\":[{\"bbox_pdf\":[100.49,161.53,511.51,183.45],\"words\":[786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"caption\",\"source_ref\":\"caption21\",\"source_text\":\"Table 1: Six judges on the 218-pair test split, three replicates each. Intervals are a clustered bootstrap over declarations and replicates. Cost is published list rates times token counts.\",\"tag\":null,\"text\":\"Table 1: Six judges on the 218-pair test split, three replicates each. Intervals are a clustered bootstrap over declarations and replicates. Cost is published list rates times token counts.\"},{\"bbox\":[100.49,641.75,411.03,45.83],\"bbox_pdf\":[100.49,104.42,511.51,150.25],\"fragments\":[{\"bbox_pdf\":[100.49,104.42,511.51,150.25],\"words\":[815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"paragraph\",\"source_ref\":\"paragraph24\",\"source_text\":\"Every judge recovers the reviewers’ preference far above chance, from claude-sonnet-5 at 80.8% of declarations down to deepseek-v4-flash at 63.5%, each at $p<10^{-5}$ by sign test. Whatever reviewers apply when they send a proof back is legible to a model reading the two revisions.\",\"tag\":null,\"text\":\"Every judge recovers the reviewers’ preference far above chance, from claude-sonnet-5 at 80.8% of declarations down to deepseek-v4-flash at 63.5%, each at p < 10 −5 by sign test. Whatever reviewers apply when they send a proof back is legible to a model reading the two revisions.\"},{\"bbox\":[100.49,703.52,4.98,9.96],\"bbox_pdf\":[100.49,78.52,105.47,88.48],\"fragments\":[{\"bbox_pdf\":[100.49,78.52,105.47,88.48],\"words\":[862]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":13,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"2\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.20432\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.20432v1-p0001.pdf\",\"page_no\":2,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5c23513f2812266bdae6041c4dfd34ae96488acb747df05a17f2139b3cd95573\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"c6d253085ed97643047a6184a1426d2baeb150108c1dad78cd8e1c3e28c2ccb5\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/c6d253085ed97643047a6184a1426d2baeb150108c1dad78cd8e1c3e28c2ccb5.pdf","asset_sha256":"c6d253085ed97643047a6184a1426d2baeb150108c1dad78cd8e1c3e28c2ccb5","document_id":"arxiv:2608.20432","kind":"page","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.20432\",\"authors\":\"Shane Caldwell\",\"title\":\"ProofJudge: Tool-Grounded LLM Evaluation of Formal Proof Quality in Mathlib\",\"url\":\"https://arxiv.org/abs/2608.20432v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.20432\",\"authors\":\"Shane Caldwell\",\"title\":\"ProofJudge: Tool-Grounded LLM Evaluation of Formal Proof Quality in Mathlib\",\"url\":\"https://arxiv.org/abs/2608.20432v1\",\"version\":1},\"blocks\":[{\"bbox\":[100.49,78.92,302.43,7.97],\"bbox_pdf\":[100.49,705.11,402.92,713.08],\"fragments\":[{\"bbox_pdf\":[100.49,705.11,402.92,713.08],\"words\":[1576,1577,1578,1579,1580,1581,1582,1583,1584,1585]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":0,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"ProofJudge: Tool-Grounded LLM Evaluation of Formal Proof Quality in Mathlib\"},{\"bbox\":[100.49,36.0,411.02,351.61],\"bbox_pdf\":[100.49,404.39,511.51,756.0],\"float_ref\":\"S3.F1\",\"fragments\":[{\"bbox_pdf\":[100.49,404.39,511.51,756.0],\"words\":[]}],\"html\":null,\"kind\":\"figure\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"figure\",\"source_ref\":\"S3.F1\",\"source_text\":\"\",\"tag\":null,\"text\":\"\"},{\"bbox\":[100.49,387.61,411.02,21.92],\"bbox_pdf\":[100.49,382.47,511.51,404.39],\"float_ref\":\"S3.F1\",\"fragments\":[{\"bbox_pdf\":[100.49,382.47,511.51,404.39],\"words\":[916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"caption\",\"source_ref\":\"caption23\",\"source_text\":\"Figure 1: Alignment against cost per declaration pair, with the intervals of Table 1. Open-weight judges are drawn hollow.\",\"tag\":null,\"text\":\"Figure 1: Alignment against cost per declaration pair, with the intervals of Table 1. Openweight judges are drawn hollow.\"},{\"bbox\":[100.49,428.23,411.03,21.92],\"bbox_pdf\":[100.49,341.85,511.52,363.77],\"fragments\":[{\"bbox_pdf\":[100.49,341.85,511.52,363.77],\"words\":[936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph25\",\"source_text\":\"Reviewing Mathlib PRs this way is not limited to frontier models. Two open-weight judges, muse-glimmer-30b and inkling-small, sit at 70.2% and 69.3%.\",\"tag\":null,\"text\":\"Reviewing Mathlib PRs this way is not limited to frontier models. Two open-weight judges, muse-glimmer-30b and inkling-small, sit at 70.2% and 69.3%.\"},{\"bbox\":[100.49,456.04,411.02,45.83],\"bbox_pdf\":[100.49,290.13,511.51,335.96],\"fragments\":[{\"bbox_pdf\":[100.49,290.13,511.51,335.96],\"words\":[958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph26\",\"source_text\":\"However, the judges are noisy. Re-running one over the same declarations flips between a fifth and nearly half of its verdicts, and a single run’s interval hides that badly: the same ablation returns $p=0.34$ on one replicate and $p=0.006$ on another. A single-run version of this judge benchmark will report differences that replication does not support.\",\"tag\":null,\"text\":\"However, the judges are noisy. Re-running one over the same declarations flips between a fifth and nearly half of its verdicts, and a single run’s interval hides that badly: the same ablation returns p = 0.34 on one replicate and p = 0.006 on another. A single-run version of this judge benchmark will report differences that replication does not support.\"},{\"bbox\":[100.49,538.48,156.25,11.96],\"bbox_pdf\":[100.49,241.56,256.74,253.52],\"fragments\":[{\"bbox_pdf\":[100.49,241.56,256.74,253.52],\"words\":[1018,1019,1020,1021]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":2,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"heading\",\"source_ref\":\"title27\",\"source_text\":\"3.1 An Example Grading\",\"tag\":null,\"text\":\"3.1 An Example Grading\"},{\"bbox\":[100.49,566.12,411.03,21.92],\"bbox_pdf\":[100.49,203.96,511.52,225.88],\"fragments\":[{\"bbox_pdf\":[100.49,203.96,511.52,225.88],\"words\":[1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph28\",\"source_text\":\"To help illustrate the eval, we provide a representative example of a PR that was evaluated correctly during the ProofJudge evaluation.\",\"tag\":null,\"text\":\"To help illustrate the eval, we provide a representative example of a PR that was evaluated correctly during the ProofJudge evaluation.\"},{\"bbox\":[100.49,593.93,411.03,93.65],\"bbox_pdf\":[100.49,104.42,511.52,198.07],\"fragments\":[{\"bbox_pdf\":[100.49,104.42,511.52,198.07],\"words\":[1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph29\",\"source_text\":\"In PR 11640, the initial proof of Set.restrictPreimage_isClosedMap reconstructed a result that Mathlib already provided. The merged revision replaces it in one line, with H.restrictPreimage s. Scoring the initial variant, the judge searched the repository for the existing declaration, quoted the signature it found, and marked the initial PR down on the dimension of library leverage. Every judge reduced the score of the initial PR in every rollout. Note that while this would not be possible with regular expressions or linting, an agent with tool access can search the library, understand the semantic meaning of what it finds, and act as a reviewer would.\",\"tag\":null,\"text\":\"In PR 11640, the initial proof of Set.restrictPreimage_isClosedMap reconstructed a result that Mathlib already provided. The merged revision replaces it in one line, with H. restrictPreimage s. Scoring the initial variant, the judge searched the repository for the existing declaration, quoted the signature it found, and marked the initial PR down on the dimension of library leverage. Every judge reduced the score of the initial PR in every rollout. Note that while this would not be possible with regular expressions or linting, an agent with tool access can search the library, understand the semantic meaning of what it finds, and act as a reviewer would.\"},{\"bbox\":[506.53,703.52,4.98,9.96],\"bbox_pdf\":[506.53,78.52,511.51,88.48],\"fragments\":[{\"bbox_pdf\":[506.53,78.52,511.51,88.48],\"words\":[1586]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":8,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"3\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.20432\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.20432v1-p0002.pdf\",\"page_no\":3,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5c23513f2812266bdae6041c4dfd34ae96488acb747df05a17f2139b3cd95573\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.117,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"0a09ad771dc225c821065977919949bf347d6248bd19f46cbdd1a192bb7b5423\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/0a09ad771dc225c821065977919949bf347d6248bd19f46cbdd1a192bb7b5423.pdf","asset_sha256":"0a09ad771dc225c821065977919949bf347d6248bd19f46cbdd1a192bb7b5423","document_id":"arxiv:2608.20432","kind":"page","page_number":3,"paper_attribution_json":"{\"arxiv_id\":\"2608.20432\",\"authors\":\"Shane Caldwell\",\"title\":\"ProofJudge: Tool-Grounded LLM Evaluation of Formal Proof Quality in Mathlib\",\"url\":\"https://arxiv.org/abs/2608.20432v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.20432\",\"authors\":\"Shane Caldwell\",\"title\":\"ProofJudge: Tool-Grounded LLM Evaluation of Formal Proof Quality in Mathlib\",\"url\":\"https://arxiv.org/abs/2608.20432v1\",\"version\":1},\"blocks\":[{\"bbox\":[100.49,78.92,302.43,7.97],\"bbox_pdf\":[100.49,705.11,402.92,713.08],\"fragments\":[{\"bbox_pdf\":[100.49,705.11,402.92,713.08],\"words\":[1587,1588,1589,1590,1591,1592,1593,1594,1595,1596]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":0,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"ProofJudge: Tool-Grounded LLM Evaluation of Formal Proof Quality in Mathlib\"},{\"bbox\":[100.49,102.03,101.41,14.35],\"bbox_pdf\":[100.49,675.62,201.9,689.97],\"fragments\":[{\"bbox_pdf\":[100.49,675.62,201.9,689.97],\"words\":[1148,1149]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"heading\",\"source_ref\":\"title30\",\"source_text\":\"4 Conclusion\",\"tag\":null,\"text\":\"4 Conclusion\"},{\"bbox\":[100.49,128.24,411.03,81.69],\"bbox_pdf\":[100.49,582.07,511.51,663.76],\"fragments\":[{\"bbox_pdf\":[100.49,582.07,511.51,663.76],\"words\":[1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph31\",\"source_text\":\"Correctness is not the only property of formal proof that matters. It is important to mathematicians that proofs be understandable, and that contributions to mathematical libraries enable future work and remain accessible to readers. While there is work to be done, the alignment scores from modern models evaluated with ProofJudge suggest that these qualities are not illegible to language model agents. With that established, this signal should be used to relieve maintainer burden where possible, and create agents capable of writing Lean 4 PRs that maintainers would welcome.\",\"tag\":null,\"text\":\"Correctness is not the only property of formal proof that matters. It is important to mathematicians that proofs be understandable, and that contributions to mathematical libraries enable future work and remain accessible to readers. While there is work to be done, the alignment scores from modern models evaluated with ProofJudge suggest that these qualities are not illegible to language model agents. With that established, this signal should be used to relieve maintainer burden where possible, and create agents capable of writing Lean 4 PRs that maintainers would welcome.\"},{\"bbox\":[100.49,225.86,106.35,11.96],\"bbox_pdf\":[100.49,554.18,206.84,566.14],\"fragments\":[{\"bbox_pdf\":[100.49,554.18,206.84,566.14],\"words\":[1240,1241,1242]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":2,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"heading\",\"source_ref\":\"title32\",\"source_text\":\"4.1 Future Work\",\"tag\":null,\"text\":\"4.1 Future Work\"},{\"bbox\":[100.49,246.24,411.03,57.78],\"bbox_pdf\":[100.49,487.97,511.52,545.76],\"fragments\":[{\"bbox_pdf\":[100.49,487.97,511.52,545.76],\"words\":[1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph33\",\"source_text\":\"Future work should focus on making judges that perform at the same level as claude-sonnet-5 but cheaper and open-source to help reduce undue burden on Mathlib maintainers. In addition to a cost focus, a less noisy judge would give a model a signal to iterate a PR against, improving quality before a human reviews it. Reinforcement learning using the ProofJudge results as a reward may create models that write better proofs before review.\",\"tag\":null,\"text\":\"Future work should focus on making judges that perform at the same level as claude-sonnet-5 but cheaper and open-source to help reduce undue burden on Mathlib maintainers. In addition to a cost focus, a less noisy judge would give a model a signal to iterate a PR against, improving quality before a human reviews it. Reinforcement learning using the ProofJudge results as a reward may create models that write better proofs before review.\"},{\"bbox\":[100.49,306.02,411.02,33.87],\"bbox_pdf\":[100.49,452.11,511.51,485.98],\"fragments\":[{\"bbox_pdf\":[100.49,452.11,511.51,485.98],\"words\":[1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph34\",\"source_text\":\"Two directions for reducing judge noise are left to future work: splitting the rubric into five single-dimension judges so each makes more focused use of tools, and treating the tool-call budget itself as a tunable parameter.\",\"tag\":null,\"text\":\"Two directions for reducing judge noise are left to future work: splitting the rubric into five single-dimension judges so each makes more focused use of tools, and treating the tool-call budget itself as a tunable parameter.\"},{\"bbox\":[100.49,356.99,75.52,14.35],\"bbox_pdf\":[100.49,420.66,176.01,435.01],\"fragments\":[{\"bbox_pdf\":[100.49,420.66,176.01,435.01],\"words\":[1352]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"heading\",\"source_ref\":\"title35\",\"source_text\":\"References\",\"tag\":null,\"text\":\"References\"},{\"bbox\":[100.49,382.21,236.59,8.97],\"bbox_pdf\":[100.49,400.82,337.08,409.79],\"fragments\":[{\"bbox_pdf\":[100.49,400.82,337.08,409.79],\"words\":[1353,1354,1355,1356,1357,1358,1359,1360,1361,1362]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"reference\",\"source_ref\":\"bib.bib1\",\"source_text\":\"Jeremy Avigad. Mathematicians in the age of ai, 2026.\",\"tag\":null,\"text\":\"[1] Jeremy Avigad. Mathematicians in the age of ai, 2026.\"},{\"bbox\":[100.49,395.16,411.03,52.8],\"bbox_pdf\":[100.49,344.04,511.52,396.84],\"fragments\":[{\"bbox_pdf\":[100.49,344.04,511.52,396.84],\"words\":[1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"reference\",\"source_ref\":\"bib.bib2\",\"source_text\":\"Nathan Lambert, Jacob Morrison, Valentina Pyatkin, Shengyi Huang, Hamish Ivison, Faeze Brahman, Lester James V. Miranda, Alisa Liu, Nouha Dziri, Shane Lyu, Yuling Gu, Saumya Malik, Victoria Graf, Jena D. Hwang, Jiangjiang Yang, Ronan Le Bras, Oyvind Tafjord, Chris Wilhelm, Luca Soldaini, Noah A. Smith, Yizhong Wang, Pradeep Dasigi, and Hannaneh Hajishirzi. Tulu 3: Pushing frontiers in open language model post-training, 2025.\",\"tag\":null,\"text\":\"[2] Nathan Lambert, Jacob Morrison, Valentina Pyatkin, Shengyi Huang, Hamish Ivison, Faeze Brahman, Lester James V. Miranda, Alisa Liu, Nouha Dziri, Shane Lyu, Yuling Gu, Saumya Malik, Victoria Graf, Jena D. Hwang, Jiangjiang Yang, Ronan Le Bras, Oyvind Tafjord, Chris Wilhelm, Luca Soldaini, Noah A. Smith, Yizhong Wang, Pradeep Dasigi, and Hannaneh Hajishirzi. Tulu 3: Pushing frontiers in open language model post-training, 2025.\"},{\"bbox\":[100.49,451.95,411.03,41.84],\"bbox_pdf\":[100.49,298.21,511.52,340.05],\"fragments\":[{\"bbox_pdf\":[100.49,298.21,511.52,340.05],\"words\":[1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474,1475,1476,1477,1478]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"reference\",\"source_ref\":\"bib.bib3\",\"source_text\":\"Z. Z. Ren, Zhihong Shao, Junxiao Song, Huajian Xin, Haocheng Wang, Wanjia Zhao, Liyue Zhang, Zhe Fu, Qihao Zhu, Dejian Yang, Z. F. Wu, Zhibin Gou, Shirong Ma, Hongxuan Tang, Yuxuan Liu, Wenjun Gao, Daya Guo, and Chong Ruan. Deepseek-prover-v2: Advancing formal mathematical reasoning via reinforcement learning for subgoal decomposition, 2025.\",\"tag\":null,\"text\":\"[3] Z. Z. Ren, Zhihong Shao, Junxiao Song, Huajian Xin, Haocheng Wang, Wanjia Zhao, Liyue Zhang, Zhe Fu, Qihao Zhu, Dejian Yang, Z. F. Wu, Zhibin Gou, Shirong Ma, Hongxuan Tang, Yuxuan Liu, Wenjun Gao, Daya Guo, and Chong Ruan. Deepseek-prover-v2: Advancing formal mathematical reasoning via reinforcement learning for subgoal decomposition, 2025.\"},{\"bbox\":[100.49,497.78,411.02,19.93],\"bbox_pdf\":[100.49,274.3,511.51,294.22],\"fragments\":[{\"bbox_pdf\":[100.49,274.3,511.51,294.22],\"words\":[1479,1480,1481,1482,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494,1495,1496,1497,1498,1499]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"reference\",\"source_ref\":\"bib.bib4\",\"source_text\":\"Peiyang Song, Kaiyu Yang, and Anima Anandkumar. Lean copilot: Large language models as copilots for theorem proving in lean, 2025.\",\"tag\":null,\"text\":\"[4] Peiyang Song, Kaiyu Yang, and Anima Anandkumar. Lean copilot: Large language models as copilots for theorem proving in lean, 2025.\"},{\"bbox\":[100.49,521.69,411.03,19.93],\"bbox_pdf\":[100.49,250.39,511.51,270.31],\"fragments\":[{\"bbox_pdf\":[100.49,250.39,511.51,270.31],\"words\":[1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512,1513,1514,1515,1516,1517,1518,1519,1520,1521,1522]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"reference\",\"source_ref\":\"bib.bib5\",\"source_text\":\"Floris van Doorn, Gabriel Ebner, and Robert Y. Lewis. Maintaining a Library of Formal Mathematics, page 251–267. Springer International Publishing, 2020.\",\"tag\":null,\"text\":\"[5] Floris van Doorn, Gabriel Ebner, and Robert Y. Lewis. Maintaining a Library of Formal Mathematics, page 251–267. Springer International Publishing, 2020.\"},{\"bbox\":[100.49,545.6,411.03,19.93],\"bbox_pdf\":[100.49,226.48,511.51,246.4],\"fragments\":[{\"bbox_pdf\":[100.49,226.48,511.51,246.4],\"words\":[1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534,1535,1536,1537,1538,1539,1540,1541,1542,1543,1544]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"reference\",\"source_ref\":\"bib.bib6\",\"source_text\":\"Huajian Xin, Luming Li, Xiaoran Jin, Jacques Fleuriot, and Wenda Li. Ape-bench: Evaluating automated proof engineering for formal math libraries, 2026.\",\"tag\":null,\"text\":\"[6] Huajian Xin, Luming Li, Xiaoran Jin, Jacques Fleuriot, and Wenda Li. Ape-bench: Evaluating automated proof engineering for formal math libraries, 2026.\"},{\"bbox\":[100.49,569.51,411.02,19.93],\"bbox_pdf\":[100.49,202.57,511.51,222.49],\"fragments\":[{\"bbox_pdf\":[100.49,202.57,511.51,222.49],\"words\":[1545,1546,1547,1548,1549,1550,1551,1552,1553,1554,1555,1556,1557,1558,1559,1560,1561,1562,1563,1564]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":13,\"role\":\"reference\",\"source_ref\":\"bib.bib7\",\"source_text\":\"Runyang You, Hongru Cai, Caiqi Zhang, Qiancheng Xu, Meng Liu, Tiezheng Yu, Yongqi Li, and Wenjie Li. Agent-as-a-judge, 2026.\",\"tag\":null,\"text\":\"[7] Runyang You, Hongru Cai, Caiqi Zhang, Qiancheng Xu, Meng Liu, Tiezheng Yu, Yongqi Li, and Wenjie Li. Agent-as-a-judge, 2026.\"},{\"bbox\":[100.49,703.52,4.98,9.96],\"bbox_pdf\":[100.49,78.52,105.47,88.48],\"fragments\":[{\"bbox_pdf\":[100.49,78.52,105.47,88.48],\"words\":[1565]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":14,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"4\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.20432\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.20432v1-p0003.pdf\",\"page_no\":4,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"5c23513f2812266bdae6041c4dfd34ae96488acb747df05a17f2139b3cd95573\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"9e46abd08f5db1eb1fb107cffaaf68af16657d375fd5d91b5f41a502b37e759c\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/9e46abd08f5db1eb1fb107cffaaf68af16657d375fd5d91b5f41a502b37e759c.pdf","asset_sha256":"9e46abd08f5db1eb1fb107cffaaf68af16657d375fd5d91b5f41a502b37e759c","document_id":"arxiv:2608.20432","kind":"page","page_number":4,"paper_attribution_json":"{\"arxiv_id\":\"2608.20432\",\"authors\":\"Shane Caldwell\",\"title\":\"ProofJudge: Tool-Grounded LLM Evaluation of Formal Proof Quality in Mathlib\",\"url\":\"https://arxiv.org/abs/2608.20432v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1},\"blocks\":[{\"bbox\":[133.77,124.8,194.79,11.46],\"bbox_pdf\":[133.77,655.74,328.56,667.2],\"fragments\":[{\"bbox_pdf\":[133.77,655.74,328.56,667.2],\"words\":[246,247,248,249,250,251,252]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph16\",\"source_text\":\"Both subspaces are invariant under $\\\\mathcal{L}_{A}$ since\",\"tag\":null,\"text\":\"Both subspaces are invariant under LA since\"},{\"bbox\":[260.73,144.7,82.65,13.48],\"bbox_pdf\":[260.73,633.82,343.38,647.3],\"fragments\":[{\"bbox_pdf\":[260.73,633.82,343.38,647.3],\"words\":[253,254,255,256,257]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"{\\\\mathcal{L}_{A}(X)}^{\\\\top}=\\\\mathcal{L}_{A}(X^{\\\\top}).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S1.Ex4\",\"source_text\":\"${\\\\mathcal{L}_{A}(X)}^{\\\\top}=\\\\mathcal{L}_{A}(X^{\\\\top}).$\",\"tag\":null,\"text\":\"⊤ ⊤ LA(X) = LA(X\"},{\"bbox\":[133.77,168.64,343.71,33.87],\"bbox_pdf\":[133.77,589.49,477.48,623.36],\"fragments\":[{\"bbox_pdf\":[133.77,589.49,477.48,623.36],\"words\":[259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph18\",\"source_text\":\"Relative to this orthogonal decomposition, $\\\\mathcal{L}_{A}$ is therefore block diagonal and its operator norm is the maximum of the norms of these blocks (see also [CZQ09, Eq. (5)]):\",\"tag\":null,\"text\":\"Relative to this orthogonal decomposition, LA is therefore block diagonal and its operator norm is the maximum of the norms of these blocks (see also [CZQ09, Eq. (5)]):\"},{\"bbox\":[231.69,204.5,245.79,12.45],\"bbox_pdf\":[231.69,575.04,477.48,587.5],\"fragments\":[{\"bbox_pdf\":[231.69,575.04,477.48,587.5],\"words\":[287,288,289,290,291,293]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\|\\\\mathcal{L}_{A}\\\\|=\\\\max\\\\left\\\\{\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{S}_{n}}\\\\|,\\\\|%\\n\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{n}}\\\\|\\\\right\\\\}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S1.E2\",\"source_text\":\"$\\\\|\\\\mathcal{L}_{A}\\\\|=\\\\max\\\\left\\\\{\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{S}_{n}}\\\\|,\\\\|% \\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{n}}\\\\|\\\\right\\\\}.$ (2)\",\"tag\":\"(2)\",\"text\":\"∥LA∥ = max {∥LA|Sn∥, ∥LA|Kn∥} (2)\"},{\"bbox\":[133.77,222.44,343.71,21.92],\"bbox_pdf\":[133.77,547.65,477.48,569.56],\"fragments\":[{\"bbox_pdf\":[133.77,547.65,477.48,569.56],\"words\":[294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph20\",\"source_text\":\"Thus a maximizer in (1) can always be chosen in one of these two subspaces. The question is whether it can always be chosen symmetric.\",\"tag\":null,\"text\":\"Thus a maximizer in (1) can always be chosen in one of these two subspaces. The question is whether it can always be chosen symmetric.\"},{\"bbox\":[133.77,253.69,315.31,10.59],\"bbox_pdf\":[133.77,527.72,449.07,538.31],\"fragments\":[{\"bbox_pdf\":[133.77,527.72,449.07,538.31],\"words\":[319,320,321,322,323,324,325,326,327,328]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph21\",\"source_text\":\"Conjecture 1 (Symmetric-maximizer conjecture). For every $A\\\\in\\\\mathbb{R}^{n\\\\times n}$ ,\",\"tag\":null,\"text\":\"Conjecture 1 (Symmetric-maximizer conjecture). For every A ∈ R n×n\"},{\"bbox\":[261.47,276.23,88.3,12.45],\"bbox_pdf\":[261.47,503.31,349.77,515.77],\"fragments\":[{\"bbox_pdf\":[261.47,503.31,349.77,515.77],\"words\":[330,331,332]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{n}}\\\\|\\\\leq\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{S}_{n}}\\\\|.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S1.Ex5\",\"source_text\":\"$\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{n}}\\\\|\\\\leq\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{S}_{n}}\\\\|.$\",\"tag\":null,\"text\":\"∥LA|Kn∥ ≤ ∥LA|Sn∥.\"},{\"bbox\":[133.77,298.15,300.9,9.96],\"bbox_pdf\":[133.77,483.88,434.67,493.85],\"fragments\":[{\"bbox_pdf\":[133.77,483.88,434.67,493.85],\"words\":[333,334,335,336,337,338,339,340,341,342,343]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph23\",\"source_text\":\"Equivalently, the maximum in (1) is attained at a symmetric matrix.\",\"tag\":null,\"text\":\"Equivalently, the maximum in (1) is attained at a symmetric matrix.\"},{\"bbox\":[133.77,318.08,343.72,33.87],\"bbox_pdf\":[133.77,440.05,477.48,473.92],\"fragments\":[{\"bbox_pdf\":[133.77,440.05,477.48,473.92],\"words\":[344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,363,364,365,366,367,368,369,370]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph24\",\"source_text\":\"The statement originated as Theorem 9 of Byers and Nash [BN87], up to replacing $A$ by $A^{\\\\top}$ . Their main motivation concerned the smallest singular value,\",\"tag\":null,\"text\":\"The statement originated as Theorem 9 of Byers and Nash [BN87], up to replacing A by A ⊤ Their main motivation concerned the smallest singular value,\"},{\"bbox\":[225.0,352.75,252.48,25.66],\"bbox_pdf\":[225.0,413.59,477.48,439.25],\"fragments\":[{\"bbox_pdf\":[225.0,413.59,477.48,439.25],\"words\":[371,372,373,374,375,376,377,378,380,381,382,383,384,385]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\operatorname{sep}(A,-A^{\\\\top})=\\\\min_{X\\\\neq 0}\\\\frac{\\\\|AX+XA^{\\\\top}\\\\|_{F}}{\\\\|X%\\n\\\\|_{F}},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S1.E3\",\"source_text\":\"$\\\\operatorname{sep}(A,-A^{\\\\top})=\\\\min_{X\\\\neq 0}\\\\frac{\\\\|AX+XA^{\\\\top}\\\\|_{F}}{\\\\|X% \\\\|_{F}},$ (3)\",\"tag\":\"(3)\",\"text\":\"∥AX + XA ⊤ ∥F ⊤ sep(A, −A = min X̸=0 ∥X∥F , (3)\"},{\"bbox\":[133.77,383.34,343.71,33.87],\"bbox_pdf\":[133.77,374.79,477.48,408.66],\"fragments\":[{\"bbox_pdf\":[133.77,374.79,477.48,408.66],\"words\":[386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"paragraph\",\"source_ref\":\"paragraph26\",\"source_text\":\"which is closely related to the conditioning of the Lyapunov equation. They proved, in particular, that a symmetric minimizer exists when $A$ is Hurwitz stable, while the analogous statement fails for general $A$ .\",\"tag\":null,\"text\":\"which is closely related to the conditioning of the Lyapunov equation. They proved, in particular, that a symmetric minimizer exists when A is Hurwitz stable, while the analogous statement fails for general A.\"},{\"bbox\":[133.77,419.2,343.72,81.69],\"bbox_pdf\":[133.77,291.11,477.48,372.8],\"fragments\":[{\"bbox_pdf\":[133.77,291.11,477.48,372.8],\"words\":[419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,494,495,496,497,498,499,500,501]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"paragraph\",\"source_ref\":\"paragraph27\",\"source_text\":\"In 2015, Chen and Tian identified an error in the proof of the maximization result and formulated it as a conjecture; they proved it for $n\\\\leq 5$ [CT15]. The problem also appears in [CHE01, CZQ09]. Feng, Lam, Yang, and Li proved the conjecture for entrywise nonnegative, entrywise nonpositive, and tridiagonal matrices [FLY+15]. As of 2021, the problem was described as open for every $n\\\\geq 6$ [KT21]. For $A,B\\\\in\\\\mathbb{R}^{n\\\\times n}$ , the generalized continuous-time Lyapunov operator is defined by\",\"tag\":null,\"text\":\"In 2015, Chen and Tian identified an error in the proof of the maximization result and formulated it as a conjecture; they proved it for n ≤ 5 [CT15]. The problem also appears in [Che01, CZQ09]. Feng, Lam, Yang, and Li proved the conjecture for entrywise nonnegative, entrywise nonpositive, and tridiagonal matrices [FLYL15]. As of 2021, the problem was described as open for every n ≥ 6 [KT21]. For A, B ∈ R n×n the generalized continuous-time Lyapunov operator is defined by\"},{\"bbox\":[206.12,511.73,199.01,12.58],\"bbox_pdf\":[206.12,267.69,405.13,280.27],\"fragments\":[{\"bbox_pdf\":[206.12,267.69,405.13,280.27],\"words\":[502,503,504,505,506,507,508,509,510,511,512,513,514]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathcal{L}_{A,B}(X)=AXB^{\\\\top}+BXA^{\\\\top},\\\\qquad X\\\\in\\\\mathbb{R}^{n\\\\times n}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"S1.Ex6\",\"source_text\":\"$\\\\mathcal{L}_{A,B}(X)=AXB^{\\\\top}+BXA^{\\\\top},\\\\qquad X\\\\in\\\\mathbb{R}^{n\\\\times n}.$\",\"tag\":null,\"text\":\"LA,B(X) = AXB ⊤ + BXA ⊤ , X ∈ R n×n .\"},{\"bbox\":[133.77,534.77,343.71,23.41],\"bbox_pdf\":[133.77,233.82,477.48,257.23],\"fragments\":[{\"bbox_pdf\":[133.77,233.82,477.48,257.23],\"words\":[515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":13,\"role\":\"paragraph\",\"source_ref\":\"paragraph29\",\"source_text\":\"Chen and Tian proved that the corresponding symmetric-maximizer statement for $\\\\mathcal{L}_{A,B}$ holds for $n\\\\leq 3$ and gave a counterexample of order four [CT16].\",\"tag\":null,\"text\":\"Chen and Tian proved that the corresponding symmetric-maximizer statement for LA,B holds for n ≤ 3 and gave a counterexample of order four [CT16].\"},{\"bbox\":[133.77,558.68,343.71,33.87],\"bbox_pdf\":[133.77,199.45,477.48,233.32],\"fragments\":[{\"bbox_pdf\":[133.77,199.45,477.48,233.32],\"words\":[539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"paragraph\",\"source_ref\":\"paragraph30\",\"source_text\":\"We disprove Conjecture 1 in every order $n\\\\geq 7$ by giving an exact certificate in order seven and then applying a direct-sum construction for any $n>7$ . This leaves order six as the only unresolved dimension.\",\"tag\":null,\"text\":\"We disprove Conjecture 1 in every order n ≥ 7 by giving an exact certificate in order seven and then applying a direct-sum construction for any n > 7. This leaves order six as the only unresolved dimension.\"},{\"bbox\":[303.13,692.67,4.98,9.96],\"bbox_pdf\":[303.13,89.37,308.11,99.33],\"fragments\":[{\"bbox_pdf\":[303.13,89.37,308.11,99.33],\"words\":[1971]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":15,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"2\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.20875\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.20875v1-p0001.pdf\",\"page_no\":2,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"b5918b771dce42fb0e3e6bcec792c7a8006d4a7f624bdd95f20a1f34641c02ff\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230.pdf","asset_sha256":"7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230","document_id":"arxiv:2608.20875","kind":"page","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[260.73,144.7,82.65,13.48],\"bbox_pdf\":[260.73,633.82,343.38,647.3],\"fragments\":[{\"bbox_pdf\":[260.73,633.82,343.38,647.3],\"words\":[253,254,255,256,257]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"{\\\\mathcal{L}_{A}(X)}^{\\\\top}=\\\\mathcal{L}_{A}(X^{\\\\top}).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S1.Ex4\",\"source_text\":\"${\\\\mathcal{L}_{A}(X)}^{\\\\top}=\\\\mathcal{L}_{A}(X^{\\\\top}).$\",\"tag\":null,\"text\":\"⊤ ⊤ LA(X) = LA(X\"},\"formula_index\":1,\"parent_page_sha256\":\"7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230.pdf","asset_sha256":"7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230","document_id":"arxiv:2608.20875","kind":"formula","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[231.69,204.5,245.79,12.45],\"bbox_pdf\":[231.69,575.04,477.48,587.5],\"fragments\":[{\"bbox_pdf\":[231.69,575.04,477.48,587.5],\"words\":[287,288,289,290,291,293]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\|\\\\mathcal{L}_{A}\\\\|=\\\\max\\\\left\\\\{\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{S}_{n}}\\\\|,\\\\|%\\n\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{n}}\\\\|\\\\right\\\\}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S1.E2\",\"source_text\":\"$\\\\|\\\\mathcal{L}_{A}\\\\|=\\\\max\\\\left\\\\{\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{S}_{n}}\\\\|,\\\\|% \\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{n}}\\\\|\\\\right\\\\}.$ (2)\",\"tag\":\"(2)\",\"text\":\"∥LA∥ = max {∥LA|Sn∥, ∥LA|Kn∥} (2)\"},\"formula_index\":3,\"parent_page_sha256\":\"7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230.pdf","asset_sha256":"7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230","document_id":"arxiv:2608.20875","kind":"formula","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[261.47,276.23,88.3,12.45],\"bbox_pdf\":[261.47,503.31,349.77,515.77],\"fragments\":[{\"bbox_pdf\":[261.47,503.31,349.77,515.77],\"words\":[330,331,332]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{n}}\\\\|\\\\leq\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{S}_{n}}\\\\|.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S1.Ex5\",\"source_text\":\"$\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{n}}\\\\|\\\\leq\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{S}_{n}}\\\\|.$\",\"tag\":null,\"text\":\"∥LA|Kn∥ ≤ ∥LA|Sn∥.\"},\"formula_index\":6,\"parent_page_sha256\":\"7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230.pdf","asset_sha256":"7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230","document_id":"arxiv:2608.20875","kind":"formula","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[225.0,352.75,252.48,25.66],\"bbox_pdf\":[225.0,413.59,477.48,439.25],\"fragments\":[{\"bbox_pdf\":[225.0,413.59,477.48,439.25],\"words\":[371,372,373,374,375,376,377,378,380,381,382,383,384,385]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\operatorname{sep}(A,-A^{\\\\top})=\\\\min_{X\\\\neq 0}\\\\frac{\\\\|AX+XA^{\\\\top}\\\\|_{F}}{\\\\|X%\\n\\\\|_{F}},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S1.E3\",\"source_text\":\"$\\\\operatorname{sep}(A,-A^{\\\\top})=\\\\min_{X\\\\neq 0}\\\\frac{\\\\|AX+XA^{\\\\top}\\\\|_{F}}{\\\\|X% \\\\|_{F}},$ (3)\",\"tag\":\"(3)\",\"text\":\"∥AX + XA ⊤ ∥F ⊤ sep(A, −A = min X̸=0 ∥X∥F , (3)\"},\"formula_index\":9,\"parent_page_sha256\":\"7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230.pdf","asset_sha256":"7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230","document_id":"arxiv:2608.20875","kind":"formula","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[206.12,511.73,199.01,12.58],\"bbox_pdf\":[206.12,267.69,405.13,280.27],\"fragments\":[{\"bbox_pdf\":[206.12,267.69,405.13,280.27],\"words\":[502,503,504,505,506,507,508,509,510,511,512,513,514]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathcal{L}_{A,B}(X)=AXB^{\\\\top}+BXA^{\\\\top},\\\\qquad X\\\\in\\\\mathbb{R}^{n\\\\times n}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"S1.Ex6\",\"source_text\":\"$\\\\mathcal{L}_{A,B}(X)=AXB^{\\\\top}+BXA^{\\\\top},\\\\qquad X\\\\in\\\\mathbb{R}^{n\\\\times n}.$\",\"tag\":null,\"text\":\"LA,B(X) = AXB ⊤ + BXA ⊤ , X ∈ R n×n .\"},\"formula_index\":12,\"parent_page_sha256\":\"7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230.pdf","asset_sha256":"7b26232ce2ba8eefd7b58e59d853e84cf8add4b30d30ed005cd0a64d85cff230","document_id":"arxiv:2608.20875","kind":"formula","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1},\"blocks\":[{\"bbox\":[133.77,124.8,343.71,21.92],\"bbox_pdf\":[133.77,645.28,477.48,667.2],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[133.77,645.28,477.48,667.2],\"words\":[1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.964,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph66\",\"source_text\":\"Proof. For $m=n-7$ , let $\\\\widehat{A}=A\\\\oplus 0_{m}$ , where $0_{m}$ denotes the $m\\\\times m$ zero matrix. Writing a symmetric or skew-symmetric matrix conformally with this block decomposition gives orthogonal decompositions into the $7\\\\times 7$ part, the off-diagonal part, and the $m\\\\times m$ part. On the off-diagonal part the Lyapunov operator acts as $Y\\\\mapsto AY$ , while it vanishes on the $m\\\\times m$ part. Hence\",\"tag\":null,\"text\":\"off-diagonal part, and the m × m part. On the off-diagonal part the Lyapunov operator acts as Y ↦→ AY while it vanishes on the m × m part. Hence\"},{\"bbox\":[224.61,150.05,252.88,45.43],\"bbox_pdf\":[224.61,596.52,477.48,641.95],\"fragments\":[{\"bbox_pdf\":[224.61,596.52,477.48,641.95],\"words\":[1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\left\\\\|\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{S}_{n}}\\\\right\\\\|^{2} \\\\displaystyle=\\\\max\\\\left\\\\{\\\\left\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{S}_{7}}\\\\right\\\\|^{2},%\\n\\\\|A\\\\|_{2}^{2}\\\\right\\\\}, \\\\\\\\ \\\\displaystyle\\\\left\\\\|\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{K}_{n}}\\\\right\\\\|^{2} \\\\displaystyle=\\\\max\\\\left\\\\{\\\\left\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{7}}\\\\right\\\\|^{2},%\\n\\\\|A\\\\|_{2}^{2}\\\\right\\\\}. \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S5.EGx1\",\"source_text\":\"$\\\\displaystyle\\\\left\\\\|\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{S}_{n}}\\\\right\\\\|^{2}$ $\\\\displaystyle=\\\\max\\\\left\\\\{\\\\left\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{S}_{7}}\\\\right\\\\|^{2},% \\\\|A\\\\|_{2}^{2}\\\\right\\\\},$ (10) $\\\\displaystyle\\\\left\\\\|\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{K}_{n}}\\\\right\\\\|^{2}$ $\\\\displaystyle=\\\\max\\\\left\\\\{\\\\left\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{7}}\\\\right\\\\|^{2},% \\\\|A\\\\|_{2}^{2}\\\\right\\\\}.$ (11)\",\"tag\":\"(10) (11)\",\"text\":\"⃦ ⃦ {︂ }︂ 2 2 2 ⃦L ˆ︁ ⃦ = max ∥LA|S7∥ , ∥A∥ , (10) A|Sn 2 ⃦ ⃦ {︂ }︂ 2 2 2 ⃦L ˆ︁ ⃦ = max ∥LA|K7∥ , ∥A∥ . (11) A|Kn 2\"},{\"bbox\":[133.77,207.38,343.71,21.92],\"bbox_pdf\":[133.77,562.7,477.48,584.62],\"fragments\":[{\"bbox_pdf\":[133.77,562.7,477.48,584.62],\"words\":[1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph68\",\"source_text\":\"After independent row and column permutations, the nonzero part of $A$ is the orthogonal direct sum of\",\"tag\":null,\"text\":\"After independent row and column permutations, the nonzero part of A is the orthogonal direct sum of\"},{\"bbox\":[178.56,232.63,254.13,41.84],\"bbox_pdf\":[178.56,517.53,432.69,559.37],\"fragments\":[{\"bbox_pdf\":[178.56,517.53,432.69,559.37],\"words\":[1274,1275,1276,1277,1278,1279,1280,1281,1282,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"A_{1}=\\\\begin{pmatrix}6&-5&-13\\\\\\\\\\n-14&-18&0\\\\\\\\\\n12&-12&11\\\\end{pmatrix},\\\\qquad A_{2}=\\\\begin{pmatrix}-6&0&-14\\\\\\\\\\n6&-18&0\\\\end{pmatrix}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S3.Ex15\",\"source_text\":\"$A_{1}=\\\\begin{pmatrix}6&-5&-13\\\\\\\\ -14&-18&0\\\\\\\\ 12&-12&11\\\\end{pmatrix},\\\\qquad A_{2}=\\\\begin{pmatrix}-6&0&-14\\\\\\\\ 6&-18&0\\\\end{pmatrix}.$\",\"tag\":null,\"text\":\"⎛ ⎞ 6 −5 −13 (︃ )︃ ⎝ ⎠ −6 0 −14 A1 = −14 −18 0 A2 = . 6 −18 0 12 −12 11\"},{\"bbox\":[133.77,285.98,41.84,9.96],\"bbox_pdf\":[133.77,496.06,175.61,506.02],\"fragments\":[{\"bbox_pdf\":[133.77,496.06,175.61,506.02],\"words\":[1301]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph70\",\"source_text\":\"Therefore\",\"tag\":null,\"text\":\"Therefore\"},{\"bbox\":[210.99,306.77,266.48,26.03],\"bbox_pdf\":[210.99,459.2,477.47,485.23],\"fragments\":[{\"bbox_pdf\":[210.99,459.2,477.47,485.23],\"words\":[1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1315,1317,1318,1319,1320,1321,1322,1323,1324]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\|A\\\\|_{2}^{2} \\\\displaystyle=\\\\max\\\\{\\\\|A_{1}\\\\|_{2}^{2},\\\\|A_{2}\\\\|_{2}^{2}\\\\}\\\\leq\\\\max\\\\{\\\\|A_{1}\\\\|_{%\\nF}^{2},\\\\|A_{2}\\\\|_{F}^{2}\\\\} \\\\\\\\ \\\\displaystyle=\\\\max\\\\{1159,592\\\\}<1196. \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S5.EGx2\",\"source_text\":\"$\\\\displaystyle\\\\|A\\\\|_{2}^{2}$ $\\\\displaystyle=\\\\max\\\\{\\\\|A_{1}\\\\|_{2}^{2},\\\\|A_{2}\\\\|_{2}^{2}\\\\}\\\\leq\\\\max\\\\{\\\\|A_{1}\\\\|_{% F}^{2},\\\\|A_{2}\\\\|_{F}^{2}\\\\}$ $\\\\displaystyle=\\\\max\\\\{1159,592\\\\}<1196.$ (12)\",\"tag\":\"(12)\",\"text\":\"2 2 2 2 2 2 = max{∥A1∥2, ∥A2∥2} ≤ max{∥A1∥F ∥A2∥F = max{1159, 592} < 1196. < 1196. (12)\"},{\"bbox\":[133.77,344.76,150.74,9.96],\"bbox_pdf\":[133.77,437.28,284.51,447.24],\"fragments\":[{\"bbox_pdf\":[133.77,437.28,284.51,447.24],\"words\":[1325,1326,1327,1328,1329]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph72\",\"source_text\":\"Equations (5) and (10)–(12) imply\",\"tag\":null,\"text\":\"Equations (5) and (10)–(12) imply\"},{\"bbox\":[238.46,360.24,134.32,21.8],\"bbox_pdf\":[238.46,409.96,372.78,431.76],\"fragments\":[{\"bbox_pdf\":[238.46,409.96,372.78,431.76],\"words\":[1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\left\\\\|\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{S}_{n}}\\\\right\\\\|^{2}<1196<\\\\left\\\\|%\\n\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{K}_{n}}\\\\right\\\\|^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"S3.Ex17\",\"source_text\":\"$\\\\left\\\\|\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{S}_{n}}\\\\right\\\\|^{2}<1196<\\\\left\\\\|% \\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{K}_{n}}\\\\right\\\\|^{2},$\",\"tag\":null,\"text\":\"⃦ ⃦2 ⃦ ⃦2 ⃦L ˆ︁ ⃦ < 1196 < ⃦L ˆ︁ ⃦ , A|Sn A|Kn\"},{\"bbox\":[133.77,390.62,102.48,9.96],\"bbox_pdf\":[133.77,391.41,236.25,401.38],\"fragments\":[{\"bbox_pdf\":[133.77,391.41,236.25,401.38],\"words\":[1346,1347,1348,1349]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph74\",\"source_text\":\"which proves the claim. ∎\",\"tag\":null,\"text\":\"which proves the claim.\"},{\"bbox\":[133.77,419.18,343.72,32.28],\"bbox_pdf\":[133.77,340.54,477.49,372.82],\"fragments\":[{\"bbox_pdf\":[133.77,340.54,477.49,372.82],\"words\":[1350,1351,1352,1353,1354,1355,1356,1357]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"heading\",\"source_ref\":\"title75\",\"source_text\":\"4 Computational discovery and use of artificial intelligence\",\"tag\":null,\"text\":\"4 Computational discovery and use of artificial intelligence\"},{\"bbox\":[133.77,463.32,343.71,69.74],\"bbox_pdf\":[133.77,258.94,477.48,328.68],\"fragments\":[{\"bbox_pdf\":[133.77,258.94,477.48,328.68],\"words\":[1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"paragraph\",\"source_ref\":\"paragraph76\",\"source_text\":\"The computational discovery used OpenAI’s gpt-5.6-sol at high reasoning effort. The model was given the conjecture and was allowed to design the numerical test itself, rather than being supplied with an objective function or an optimization method. Its initial approach led directly to the desired counterexample: exploit the orthogonal splitting into symmetric and skew-symmetric matrices and numerically maximize the gap\",\"tag\":null,\"text\":\"The computational discovery used OpenAI’s gpt-5.6-sol at high reasoning effort. The model was given the conjecture and was allowed to design the numerical test itself, rather than being supplied with an objective function or an optimization method. Its initial approach led directly to the desired counterexample: exploit the orthogonal splitting into symmetric and skew-symmetric matrices and numerically maximize the gap\"},{\"bbox\":[245.22,545.02,120.81,12.45],\"bbox_pdf\":[245.22,234.53,366.03,246.98],\"fragments\":[{\"bbox_pdf\":[245.22,234.53,366.03,246.98],\"words\":[1420,1421,1422,1423,1424]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"g(A):=\\\\left\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{n}}\\\\right\\\\|-\\\\left\\\\|\\\\mathcal{L}_{A}|%\\n_{\\\\mathbb{S}_{n}}\\\\right\\\\|\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"formula\",\"source_ref\":\"S4.Ex18\",\"source_text\":\"$g(A):=\\\\left\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{n}}\\\\right\\\\|-\\\\left\\\\|\\\\mathcal{L}_{A}|% _{\\\\mathbb{S}_{n}}\\\\right\\\\|$\",\"tag\":null,\"text\":\"g(A) := ∥LA|Kn∥ − ∥LA|Sn∥\"},{\"bbox\":[133.77,566.93,343.71,45.83],\"bbox_pdf\":[133.77,179.24,477.48,225.07],\"fragments\":[{\"bbox_pdf\":[133.77,179.24,477.48,225.07],\"words\":[1425,1426,1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"paragraph\",\"source_ref\":\"paragraph78\",\"source_text\":\"on the Frobenius unit sphere $\\\\|A\\\\|_{F}=1$ . For each trial matrix, the two restricted norms were evaluated as largest singular values in orthonormal coordinate bases. The corresponding singular vectors supplied a gradient, which was projected onto the tangent space of the sphere before the next optimization step.\",\"tag\":null,\"text\":\"on the Frobenius unit sphere ∥A∥F = 1. For each trial matrix, the two restricted norms were evaluated as largest singular values in orthonormal coordinate bases. The corresponding singular vectors supplied a gradient, which was projected onto the tangent space of the sphere before the next optimization step.\"},{\"bbox\":[133.77,614.75,343.71,57.78],\"bbox_pdf\":[133.77,119.46,477.48,177.25],\"fragments\":[{\"bbox_pdf\":[133.77,119.46,477.48,177.25],\"words\":[1473,1474,1475,1476,1477,1478,1479,1480,1481,1482,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494,1495,1496,1497,1498,1499,1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512,1513,1514,1515,1516,1517,1518,1519,1520,1521,1522,1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":13,\"role\":\"paragraph\",\"source_ref\":\"paragraph79\",\"source_text\":\"The first counterexample obtained in the completed search had order nine and was found with Adam [KB15] from nine Gaussian random starts. The moment parameters had the standard values $\\\\beta_{1}=0.9$ and $\\\\beta_{2}=0.999$ . The problem-specific stepsize was initially $0.02$ for $800$ iterations and $0.006$ during refinement, with a quadratic decay to $15\\\\%$ of its initial value. All these values were chosen by the model. The matrix was normalized back to the Frobenius unit sphere after every step. Manual follow-up prompting then asked the same model to reduce the dimension and to seek matrices with fewer nonzero entries and smaller coefficients. Repeated numerical searches and exact checks led to the sparse order-seven integer matrix in (4).\",\"tag\":null,\"text\":\"The first counterexample obtained in the completed search had order nine and was found with Adam [KB15] from nine Gaussian random starts. The moment parameters had the standard values β1 = 0.9 and β2 = 0.999. The problem-specific stepsize was initially 0.02 for 800 iterations and 0.006 during refinement, with a quadratic decay to 15% of its initial value. All these values\"},{\"bbox\":[303.13,692.67,4.98,9.96],\"bbox_pdf\":[303.13,89.36,308.11,99.33],\"fragments\":[{\"bbox_pdf\":[303.13,89.36,308.11,99.33],\"words\":[1974]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":14,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"5\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.20875\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.20875v1-p0004.pdf\",\"page_no\":5,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"b5918b771dce42fb0e3e6bcec792c7a8006d4a7f624bdd95f20a1f34641c02ff\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph66\",\"c_block\":0.979,\"c_lines\":1.0,\"c_pdf\":0.999,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.979,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc.pdf","asset_sha256":"318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc","document_id":"arxiv:2608.20875","kind":"page","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[224.61,150.05,252.88,45.43],\"bbox_pdf\":[224.61,596.52,477.48,641.95],\"fragments\":[{\"bbox_pdf\":[224.61,596.52,477.48,641.95],\"words\":[1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\left\\\\|\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{S}_{n}}\\\\right\\\\|^{2} \\\\displaystyle=\\\\max\\\\left\\\\{\\\\left\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{S}_{7}}\\\\right\\\\|^{2},%\\n\\\\|A\\\\|_{2}^{2}\\\\right\\\\}, \\\\\\\\ \\\\displaystyle\\\\left\\\\|\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{K}_{n}}\\\\right\\\\|^{2} \\\\displaystyle=\\\\max\\\\left\\\\{\\\\left\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{7}}\\\\right\\\\|^{2},%\\n\\\\|A\\\\|_{2}^{2}\\\\right\\\\}. \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S5.EGx1\",\"source_text\":\"$\\\\displaystyle\\\\left\\\\|\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{S}_{n}}\\\\right\\\\|^{2}$ $\\\\displaystyle=\\\\max\\\\left\\\\{\\\\left\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{S}_{7}}\\\\right\\\\|^{2},% \\\\|A\\\\|_{2}^{2}\\\\right\\\\},$ (10) $\\\\displaystyle\\\\left\\\\|\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{K}_{n}}\\\\right\\\\|^{2}$ $\\\\displaystyle=\\\\max\\\\left\\\\{\\\\left\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{7}}\\\\right\\\\|^{2},% \\\\|A\\\\|_{2}^{2}\\\\right\\\\}.$ (11)\",\"tag\":\"(10) (11)\",\"text\":\"⃦ ⃦ {︂ }︂ 2 2 2 ⃦L ˆ︁ ⃦ = max ∥LA|S7∥ , ∥A∥ , (10) A|Sn 2 ⃦ ⃦ {︂ }︂ 2 2 2 ⃦L ˆ︁ ⃦ = max ∥LA|K7∥ , ∥A∥ . (11) A|Kn 2\"},\"formula_index\":1,\"parent_page_sha256\":\"318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc.pdf","asset_sha256":"318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc","document_id":"arxiv:2608.20875","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[178.56,232.63,254.13,41.84],\"bbox_pdf\":[178.56,517.53,432.69,559.37],\"fragments\":[{\"bbox_pdf\":[178.56,517.53,432.69,559.37],\"words\":[1274,1275,1276,1277,1278,1279,1280,1281,1282,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"A_{1}=\\\\begin{pmatrix}6&-5&-13\\\\\\\\\\n-14&-18&0\\\\\\\\\\n12&-12&11\\\\end{pmatrix},\\\\qquad A_{2}=\\\\begin{pmatrix}-6&0&-14\\\\\\\\\\n6&-18&0\\\\end{pmatrix}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S3.Ex15\",\"source_text\":\"$A_{1}=\\\\begin{pmatrix}6&-5&-13\\\\\\\\ -14&-18&0\\\\\\\\ 12&-12&11\\\\end{pmatrix},\\\\qquad A_{2}=\\\\begin{pmatrix}-6&0&-14\\\\\\\\ 6&-18&0\\\\end{pmatrix}.$\",\"tag\":null,\"text\":\"⎛ ⎞ 6 −5 −13 (︃ )︃ ⎝ ⎠ −6 0 −14 A1 = −14 −18 0 A2 = . 6 −18 0 12 −12 11\"},\"formula_index\":3,\"parent_page_sha256\":\"318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc.pdf","asset_sha256":"318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc","document_id":"arxiv:2608.20875","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[210.99,306.77,266.48,26.03],\"bbox_pdf\":[210.99,459.2,477.47,485.23],\"fragments\":[{\"bbox_pdf\":[210.99,459.2,477.47,485.23],\"words\":[1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1315,1317,1318,1319,1320,1321,1322,1323,1324]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\|A\\\\|_{2}^{2} \\\\displaystyle=\\\\max\\\\{\\\\|A_{1}\\\\|_{2}^{2},\\\\|A_{2}\\\\|_{2}^{2}\\\\}\\\\leq\\\\max\\\\{\\\\|A_{1}\\\\|_{%\\nF}^{2},\\\\|A_{2}\\\\|_{F}^{2}\\\\} \\\\\\\\ \\\\displaystyle=\\\\max\\\\{1159,592\\\\}<1196. \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S5.EGx2\",\"source_text\":\"$\\\\displaystyle\\\\|A\\\\|_{2}^{2}$ $\\\\displaystyle=\\\\max\\\\{\\\\|A_{1}\\\\|_{2}^{2},\\\\|A_{2}\\\\|_{2}^{2}\\\\}\\\\leq\\\\max\\\\{\\\\|A_{1}\\\\|_{% F}^{2},\\\\|A_{2}\\\\|_{F}^{2}\\\\}$ $\\\\displaystyle=\\\\max\\\\{1159,592\\\\}<1196.$ (12)\",\"tag\":\"(12)\",\"text\":\"2 2 2 2 2 2 = max{∥A1∥2, ∥A2∥2} ≤ max{∥A1∥F ∥A2∥F = max{1159, 592} < 1196. < 1196. (12)\"},\"formula_index\":5,\"parent_page_sha256\":\"318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc.pdf","asset_sha256":"318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc","document_id":"arxiv:2608.20875","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[238.46,360.24,134.32,21.8],\"bbox_pdf\":[238.46,409.96,372.78,431.76],\"fragments\":[{\"bbox_pdf\":[238.46,409.96,372.78,431.76],\"words\":[1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\left\\\\|\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{S}_{n}}\\\\right\\\\|^{2}<1196<\\\\left\\\\|%\\n\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{K}_{n}}\\\\right\\\\|^{2},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"S3.Ex17\",\"source_text\":\"$\\\\left\\\\|\\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{S}_{n}}\\\\right\\\\|^{2}<1196<\\\\left\\\\|% \\\\mathcal{L}_{\\\\widehat{A}}|_{\\\\mathbb{K}_{n}}\\\\right\\\\|^{2},$\",\"tag\":null,\"text\":\"⃦ ⃦2 ⃦ ⃦2 ⃦L ˆ︁ ⃦ < 1196 < ⃦L ˆ︁ ⃦ , A|Sn A|Kn\"},\"formula_index\":7,\"parent_page_sha256\":\"318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc.pdf","asset_sha256":"318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc","document_id":"arxiv:2608.20875","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[245.22,545.02,120.81,12.45],\"bbox_pdf\":[245.22,234.53,366.03,246.98],\"fragments\":[{\"bbox_pdf\":[245.22,234.53,366.03,246.98],\"words\":[1420,1421,1422,1423,1424]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"g(A):=\\\\left\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{n}}\\\\right\\\\|-\\\\left\\\\|\\\\mathcal{L}_{A}|%\\n_{\\\\mathbb{S}_{n}}\\\\right\\\\|\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"formula\",\"source_ref\":\"S4.Ex18\",\"source_text\":\"$g(A):=\\\\left\\\\|\\\\mathcal{L}_{A}|_{\\\\mathbb{K}_{n}}\\\\right\\\\|-\\\\left\\\\|\\\\mathcal{L}_{A}|% _{\\\\mathbb{S}_{n}}\\\\right\\\\|$\",\"tag\":null,\"text\":\"g(A) := ∥LA|Kn∥ − ∥LA|Sn∥\"},\"formula_index\":11,\"parent_page_sha256\":\"318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc.pdf","asset_sha256":"318b19861147fb42f9115ab9ef8a3085c07ba99bc2cd015f20c1dfb07930b1bc","document_id":"arxiv:2608.20875","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.20875\",\"authors\":\"Daniel Kressner and Bart Vandereycken\",\"title\":\"A counterexample to the symmetric-maximizer conjecture for Lyapunov operators\",\"url\":\"https://arxiv.org/abs/2608.20875v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2},\"blocks\":[{\"bbox\":[189.0,73.12,270.71,7.97],\"bbox_pdf\":[189.0,710.91,459.71,718.88],\"fragments\":[{\"bbox_pdf\":[189.0,710.91,459.71,718.88],\"words\":[11464,11465,11466,11467,11468,11469,11470]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":0,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"SHINSUKE IWAO, FUMIHIKO NAKAMURA, AND YUSHI NAKANO\"},{\"bbox\":[110.16,94.09,428.4,54.78],\"bbox_pdf\":[110.16,643.12,538.56,697.91],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[110.16,643.12,538.56,697.91],\"words\":[435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,499]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph13\",\"source_text\":\"The same symbolic viewpoint also covers the tent map, namely the piecewise linear map with slope $2$ on the left half of the interval and slope $-2$ on the right half. We rank the first $N$ orbit points $x,\\\\;f(x),\\\\;f^{2}(x),\\\\ldots,f^{N-1}(x)$ and consider the LIS of the resulting order pattern. Equivalently, by Schensted insertion, we study the first row of the Young diagram generated by this orbit. The resulting sequence is not i.i.d., and its induced order pattern is not a uniform random permutation. Thus, the problem does not fall under either the classical i.i.d. setting or the Plancherel framework. Our main result shows that the expected LIS nevertheless has the same first-order asymptotics as in the Ulam–Hammersley problem: if $\\\\lambda_{1}^{(N)}(x)$ denotes the LIS of the first $N$ orbit points $\\\\{f^{n}(x)\\\\}_{n=0}^{N-1}$ , then\",\"tag\":null,\"text\":\"pattern is not a uniform random permutation. Thus, the problem does not fall under either the classical i.i.d. setting or the Plancherel framework. Our main result shows that the expected LIS nevertheless has the same first-order asymptotics as in the Ulam–Hammersley (N n N −1 problem: if λ (x) denotes the LIS of the first N orbit points {f (x)} 1 n=0 then\"},{\"bbox\":[281.31,158.17,86.1,30.01],\"bbox_pdf\":[281.31,603.83,367.41,633.83],\"fragments\":[{\"bbox_pdf\":[281.31,603.83,367.41,633.83],\"words\":[500,502,504,505,506,507,508,509,510]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\lim_{N\\\\to\\\\infty}\\\\frac{\\\\mathbb{E}[\\\\lambda_{1}^{(N)}]}{\\\\sqrt{N}}=2.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S1.Ex2\",\"source_text\":\"$\\\\lim_{N\\\\to\\\\infty}\\\\frac{\\\\mathbb{E}[\\\\lambda_{1}^{(N)}]}{\\\\sqrt{N}}=2.$\",\"tag\":null,\"text\":\"(N E[λ1 lim √ = 2. N →∞ N\"},{\"bbox\":[110.16,197.36,428.4,23.86],\"bbox_pdf\":[110.16,570.78,538.56,594.64],\"fragments\":[{\"bbox_pdf\":[110.16,570.78,538.56,594.64],\"words\":[511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph15\",\"source_text\":\"Here and throughout the paper, probabilities and expectations of random variables determined by the initial point $x$ are taken with respect to Lebesgue measure on $[0,1)$ .\",\"tag\":null,\"text\":\"Here and throughout the paper, probabilities and expectations of random variables determined by the initial point x are taken with respect to Lebesgue measure on [0, 1).\"},{\"bbox\":[110.16,223.26,428.4,140.42],\"bbox_pdf\":[110.16,428.31,538.56,568.74],\"fragments\":[{\"bbox_pdf\":[110.16,428.31,538.56,568.74],\"words\":[539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph16\",\"source_text\":\"A natural first guess is that such a result should follow from the strong statistical properties of expanding maps. For many expanding or hyperbolic systems, exponential decay of correlations, often combined with martingale or reverse-martingale decompositions of Gordin–Liverani type, provides a standard route to central limit theorems and related limit laws [6, 8, 12, 13]. However, this mechanism does not seem to control the LIS in any direct way. The LIS is neither an additive observable nor a local statistic depending on a fixed number of orbit coordinates. A maximal increasing subsequence may choose many possible sets of times, and even the comparison of two orbit points can require long, overlapping tails of the binary expansion of the same initial point. Thus an i.i.d.-type approximation obtained only from exponential mixing is not by itself suited to the non-local optimization problem considered here.\",\"tag\":null,\"text\":\"A natural first guess is that such a result should follow from the strong statistical properties of expanding maps. For many expanding or hyperbolic systems, exponential decay of correlations, often combined with martingale or reverse-martingale decompositions of Gordin–Liverani type, provides a standard route to central limit theorems and related limit laws [6, 8, 12, 13]. However, this mechanism does not seem to control the LIS in any direct way. The LIS is neither an additive observable nor a local statistic depending on a fixed number of orbit coordinates. A maximal increasing subsequence may choose many possible sets of times, and even the comparison of two orbit points can require long, overlapping tails of the binary expansion of the same initial point. Thus an i.i.d.-type approximation obtained only from exponential mixing is not by itself suited to the non-local optimization problem considered here.\"},{\"bbox\":[110.16,365.73,428.4,36.81],\"bbox_pdf\":[110.16,389.46,538.56,426.27],\"fragments\":[{\"bbox_pdf\":[110.16,389.46,538.56,426.27],\"words\":[683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph17\",\"source_text\":\"The main technical idea is to use the dyadic symbolic structure more deeply than what is captured by abstract mixing estimates. In the symbolic coordinate used in the proof, an initial point is written as\",\"tag\":null,\"text\":\"The main technical idea is to use the dyadic symbolic structure more deeply than what is captured by abstract mixing estimates. In the symbolic coordinate used in the proof, an initial point is written as\"},{\"bbox\":[285.0,410.34,47.6,12.55],\"bbox_pdf\":[285.0,369.12,332.6,381.66],\"fragments\":[{\"bbox_pdf\":[285.0,369.12,332.6,381.66],\"words\":[718,719,720]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"x=0.b_{1}b_{2}b_{3}\\\\cdots,\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S1.Ex3\",\"source_text\":\"$x=0.b_{1}b_{2}b_{3}\\\\cdots,$\",\"tag\":null,\"text\":\"x = 0.b1b2b3\"},{\"bbox\":[110.16,431.31,428.4,153.37],\"bbox_pdf\":[110.16,207.32,538.56,360.69],\"fragments\":[{\"bbox_pdf\":[110.16,207.32,538.56,360.69],\"words\":[725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph19\",\"source_text\":\"and the dynamics is the left shift; under the uniform initial distribution, the digits $b_{i}$ are independent Bernoulli variables. The difficulty is that comparisons of $f^{i}(x)$ and $f^{j}(x)$ involve overlapping tails of this same binary sequence. For the lower bound, we cut the binary expansion into blocks, search for prescribed finite words that force selected orbit points into specified dyadic intervals, and use the remaining tail digits as independent labels. After collision events are excluded, the relative order of the selected orbit points agrees in distribution with that of an independent uniform sequence; together with the order-isomorphism statement for i.i.d. samples, this reduces the selected subsequence to the uniform permutation model. At the lower-bound level, the precise reduction is the combination of Proposition 2.3 with Proposition 3.7, followed by the uniform-permutation estimate in Theorem 3.8; see Section 3.3. The upper-bound version appears in Section 4.2, where the comparison is applied after thinning and leads to the estimate (28).\",\"tag\":null,\"text\":\"and the dynamics is the left shift; under the uniform initial distribution, the digits bi are independent Bernoulli variables. The difficulty is that comparisons of f i (x) and f j (x) involve overlapping tails of this same binary sequence. For the lower bound, we cut the binary expansion into blocks, search for prescribed finite words that force selected orbit points into specified dyadic intervals, and use the remaining tail digits as independent labels. After collision events are excluded, the relative order of the selected orbit points agrees in distribution with that of an independent uniform sequence; together with the order-isomorphism statement for i.i.d. samples, this reduces the selected subsequence to the uniform permutation model. At the lower-bound level, the precise reduction is the combination of Proposition 2.3 with Proposition 3.7, followed by the uniform-permutation estimate in Theorem 3.8; see Section 3.3. The upper-bound version appears in Section 4.2, where the comparison is applied after thinning and leads to the estimate (28).\"},{\"bbox\":[110.16,586.72,428.4,23.86],\"bbox_pdf\":[110.16,181.42,538.56,205.28],\"fragments\":[{\"bbox_pdf\":[110.16,181.42,538.56,205.28],\"words\":[887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph20\",\"source_text\":\"For a different perspective based on the full distributions of order patterns generated by interval maps, see Abrams et al. [1].\",\"tag\":null,\"text\":\"For a different perspective based on the full distributions of order patterns generated by interval maps, see Abrams et al. [1].\"},{\"bbox\":[110.16,612.63,428.4,36.81],\"bbox_pdf\":[110.16,142.56,538.56,179.37],\"fragments\":[{\"bbox_pdf\":[110.16,142.56,538.56,179.37],\"words\":[908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph21\",\"source_text\":\"The paper is organized as follows. Section 2 states the main results, introduces the pattern construction and the class of admissible orders. Sections 3 and 4 prove the lower and upper bounds, respectively.\",\"tag\":null,\"text\":\"The paper is organized as follows. Section 2 states the main results, introduces the pattern construction and the class of admissible orders. Sections 3 and 4 prove the lower and upper bounds, respectively.\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.24575\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.24575v2-p0001.pdf\",\"page_no\":2,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"b2d46808ff2e296dea9c8edc44ee5a098a7d8aa0\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"9da159a0600a0a7ff1719755312c6159bf47af137eaa2cb5edfec454220b158b\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"a58617fd6e94d905fa00779842c61dfe6a13f109b33bc02fa6b015dd56f31457\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/a58617fd6e94d905fa00779842c61dfe6a13f109b33bc02fa6b015dd56f31457.pdf","asset_sha256":"a58617fd6e94d905fa00779842c61dfe6a13f109b33bc02fa6b015dd56f31457","document_id":"arxiv:2608.24575","kind":"page","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[281.31,158.17,86.1,30.01],\"bbox_pdf\":[281.31,603.83,367.41,633.83],\"fragments\":[{\"bbox_pdf\":[281.31,603.83,367.41,633.83],\"words\":[500,502,504,505,506,507,508,509,510]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\lim_{N\\\\to\\\\infty}\\\\frac{\\\\mathbb{E}[\\\\lambda_{1}^{(N)}]}{\\\\sqrt{N}}=2.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S1.Ex2\",\"source_text\":\"$\\\\lim_{N\\\\to\\\\infty}\\\\frac{\\\\mathbb{E}[\\\\lambda_{1}^{(N)}]}{\\\\sqrt{N}}=2.$\",\"tag\":null,\"text\":\"(N E[λ1 lim √ = 2. N →∞ N\"},\"formula_index\":2,\"parent_page_sha256\":\"a58617fd6e94d905fa00779842c61dfe6a13f109b33bc02fa6b015dd56f31457\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/a58617fd6e94d905fa00779842c61dfe6a13f109b33bc02fa6b015dd56f31457.pdf","asset_sha256":"a58617fd6e94d905fa00779842c61dfe6a13f109b33bc02fa6b015dd56f31457","document_id":"arxiv:2608.24575","kind":"formula","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[285.0,410.34,47.6,12.55],\"bbox_pdf\":[285.0,369.12,332.6,381.66],\"fragments\":[{\"bbox_pdf\":[285.0,369.12,332.6,381.66],\"words\":[718,719,720]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"x=0.b_{1}b_{2}b_{3}\\\\cdots,\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S1.Ex3\",\"source_text\":\"$x=0.b_{1}b_{2}b_{3}\\\\cdots,$\",\"tag\":null,\"text\":\"x = 0.b1b2b3\"},\"formula_index\":6,\"parent_page_sha256\":\"a58617fd6e94d905fa00779842c61dfe6a13f109b33bc02fa6b015dd56f31457\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/a58617fd6e94d905fa00779842c61dfe6a13f109b33bc02fa6b015dd56f31457.pdf","asset_sha256":"a58617fd6e94d905fa00779842c61dfe6a13f109b33bc02fa6b015dd56f31457","document_id":"arxiv:2608.24575","kind":"formula","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2},\"blocks\":[{\"bbox\":[121.13,73.12,333.02,7.97],\"bbox_pdf\":[121.13,710.91,454.15,718.88],\"fragments\":[{\"bbox_pdf\":[121.13,710.91,454.15,718.88],\"words\":[11479,11480,11481,11482,11483,11484,11485]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":0,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"LONGEST INCREASING SUBSEQUENCES OF DYADIC-TYPE CHAOTIC ORBITS\"},{\"bbox\":[73.44,94.09,428.4,25.5],\"bbox_pdf\":[73.44,672.41,501.84,697.91],\"fragments\":[{\"bbox_pdf\":[73.44,672.41,501.84,697.91],\"words\":[1922,1923,1924,1925,1926,1927,1928,1929,1930,1931,1932,1933,1934,1935,1936,1937,1938,1939,1940,1942,1943]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph65\",\"source_text\":\"where $wa$ is the concatenation of $w$ and $a$ . The map $\\\\sigma_{w}(a)$ is well-defined by (5). Note that $\\\\sigma_{w}(0)\\\\neq\\\\sigma_{w}(1)$ .\",\"tag\":null,\"text\":\"where wa is the concatenation of w and a. The map σw(a) is well-defined by (5). Note that σw(0) = σw(1).\"},{\"bbox\":[85.4,120.0,248.76,12.55],\"bbox_pdf\":[85.4,659.46,334.16,672.0],\"fragments\":[{\"bbox_pdf\":[85.4,659.46,334.16,672.0],\"words\":[1944,1945,1946,1947,1948,1950,1951,1952,1953,1957,1958,1962]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph66\",\"source_text\":\"We define the map $\\\\tau:\\\\Omega\\\\to\\\\Omega$ ; $c_{1}c_{2}\\\\ldots\\\\mapsto d_{1}d_{2}\\\\dots$ by\",\"tag\":null,\"text\":\"We define the map τ Ω → Ω; c1c2 ↦→ d1d2 by\"},{\"bbox\":[214.09,138.4,147.11,13.76],\"bbox_pdf\":[214.09,639.84,361.19,653.6],\"fragments\":[{\"bbox_pdf\":[214.09,639.84,361.19,653.6],\"words\":[1963,1964,1965,1966,1967,1968]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"d_{j}:=\\\\sigma_{c_{1}\\\\cdots c_{j-1}}(c_{j}),\\\\qquad(j\\\\geq 1).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S2.Ex12\",\"source_text\":\"$d_{j}:=\\\\sigma_{c_{1}\\\\cdots c_{j-1}}(c_{j}),\\\\qquad(j\\\\geq 1).$\",\"tag\":null,\"text\":\"dj := σc1···cj−1(cj), (j ≥ 1).\"},{\"bbox\":[73.44,156.8,428.4,38.59],\"bbox_pdf\":[73.44,596.61,501.84,635.2],\"fragments\":[{\"bbox_pdf\":[73.44,596.61,501.84,635.2],\"words\":[1969,1970,1971,1972,1973,1974,1978,1979,1980,1981,1982,1983,1987,1988,1989,1990,1991,1992,1993,1997,1998,1999,2003,2004,2005,2006,2007,2008,2009,2010,2011,2012,2013,2014,2015,2016,2017,2018,2019,2020,2021,2022,2023,2024,2025,2029,2030,2031,2032,2033,2034,2035,2036,2040,2041,2042]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph68\",\"source_text\":\"Since the first $\\\\ell$ digits $d_{1}d_{2}\\\\dots d_{\\\\ell}$ are determined only by $c_{1}c_{2}\\\\dots c_{\\\\ell}$ , we have a one-to-one correspondence $c_{1}c_{2}\\\\dots c_{\\\\ell}\\\\leftrightarrow d_{1}d_{2}\\\\dots d_{\\\\ell}$ for any $\\\\ell\\\\geq 1$ . This implies that $\\\\tau$ is a bijection of $\\\\Omega$ . Moreover, for any finite words $c=c_{1}c_{2}\\\\dots c_{\\\\ell}$ and $d=\\\\tau(c)=d_{1}d_{2}\\\\dots d_{\\\\ell}$ , we have\",\"tag\":null,\"text\":\"Since the first ℓ digits d1d2 dℓ are determined only by c1c2 cℓ, we have a one-to-one correspondence c1c2 cℓ ↔ d1d2 dℓ for any ℓ ≥ 1. This implies that τ is a bijection of Ω. Moreover, for any finite words c = c1c2 cℓ and d = τ (c) = d1d2 dℓ, we have\"},{\"bbox\":[192.28,200.36,190.73,12.47],\"bbox_pdf\":[192.28,579.17,383.0,591.64],\"fragments\":[{\"bbox_pdf\":[192.28,579.17,383.0,591.64],\"words\":[2043,2044,2045,2046,2047,2048,2049,2050,2051,2052,2053,2054]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\tau([c])=[d]\\\\quad\\\\text{and}\\\\quad\\\\nu([c])=\\\\nu([d])=2^{-\\\\ell},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S2.Ex13\",\"source_text\":\"$\\\\tau([c])=[d]\\\\quad\\\\text{and}\\\\quad\\\\nu([c])=\\\\nu([d])=2^{-\\\\ell},$\",\"tag\":null,\"text\":\"τ ([c]) = [d] and ν([c]) = ν([d]) = 2 −ℓ ,\"},{\"bbox\":[73.44,220.32,428.4,23.86],\"bbox_pdf\":[73.44,547.81,501.84,571.68],\"fragments\":[{\"bbox_pdf\":[73.44,547.81,501.84,571.68],\"words\":[2055,2056,2057,2058,2059,2060,2061,2062,2063,2064,2065,2066,2067,2068,2069,2070,2071,2072,2073,2074,2075,2076,2077,2078,2079,2080,2081,2082]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.966,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph70\",\"source_text\":\"which implies that $\\\\tau$ preserves the product measure of all cylinder sets. Since the cylinder sets generate the Borel $\\\\sigma$ -algebra of $\\\\Omega$ , we conclude that $\\\\tau$ preserves $\\\\nu$ .\",\"tag\":null,\"text\":\"which implies that τ preserves the product measure of all cylinder sets. Since the cylinder sets generate the Borel σ-algebra of Ω, we conclude that τ preserves ν.\"},{\"bbox\":[73.44,245.21,428.4,26.52],\"bbox_pdf\":[73.44,520.27,501.84,546.79],\"fragments\":[{\"bbox_pdf\":[73.44,520.27,501.84,546.79],\"words\":[2083,2084,2085,2086,2087,2088,2090,2091,2092,2093,2094,2095,2096,2097,2098,2099,2100,2101,2102,2103,2104,2105,2106,2107,2108,2110,2111,2112,2113,2114,2115,2116,2117,2118,2119,2120,2121,2122]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph71\",\"source_text\":\"If $c,c^{\\\\prime}\\\\in\\\\Omega\\\\setminus\\\\mathfrak{z}$ first differ at position $j$ , their images $d=\\\\tau(c)$ and $d^{\\\\prime}=\\\\tau(c^{\\\\prime})$ also first differ at position $j$ . By the definition of $\\\\sigma_{w}$ , we have\",\"tag\":null,\"text\":\"If c, c ′ ∈ Ω z first differ at position j, their images d = τ (c) and d ′ = τ (c ′ also first differ at position j. By the definition of σw, we have\"},{\"bbox\":[131.9,275.12,303.71,32.63],\"bbox_pdf\":[131.9,484.26,435.61,516.88],\"fragments\":[{\"bbox_pdf\":[131.9,484.26,435.61,516.88],\"words\":[2123,2124,2125,2126,2128,2129,2130,2131,2132,2133,2137,2138,2139,2143,2144,2145,2146,2147,2148,2149,2150]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\beta(c)\\\\prec\\\\beta(c^{\\\\prime}) \\\\displaystyle\\\\iff C_{c_{1}c_{2}\\\\dots c_{j}}\\\\prec C_{c_{1}c_{2}\\\\dots\\\\overline{c%\\n_{j}}} \\\\\\\\ \\\\displaystyle\\\\iff d_{1}d_{2}\\\\dots d_{j}<_{\\\\mathrm{lex}}d_{1}d_{2}\\\\dots%\\n\\\\overline{d_{j}}\\\\iff\\\\tau(c)<_{\\\\mathrm{lex}}\\\\tau(c^{\\\\prime}). \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S2.Ex14\",\"source_text\":\"$\\\\displaystyle\\\\beta(c)\\\\prec\\\\beta(c^{\\\\prime})$ $\\\\displaystyle\\\\iff C_{c_{1}c_{2}\\\\dots c_{j}}\\\\prec C_{c_{1}c_{2}\\\\dots\\\\overline{c% _{j}}}$ $\\\\displaystyle\\\\iff d_{1}d_{2}\\\\dots d_{j}<_{\\\\mathrm{lex}}d_{1}d_{2}\\\\dots% \\\\overline{d_{j}}\\\\iff\\\\tau(c)<_{\\\\mathrm{lex}}\\\\tau(c^{\\\\prime}).$\",\"tag\":null,\"text\":\"β(c) ≺ β(c ′ ⇐⇒ Cc1c2...cj ≺ Cc1c2...cj ⇐⇒ d1d2 dj <lex d1d2 dj ⇐⇒ τ (c) <lex τ (c ′\"},{\"bbox\":[73.44,313.96,428.39,24.88],\"bbox_pdf\":[73.44,453.16,501.84,478.04],\"fragments\":[{\"bbox_pdf\":[73.44,453.16,501.84,478.04],\"words\":[2152,2153,2154,2155,2157,2158,2159,2160,2161,2163,2164,2165,2166,2167,2168,2169,2170,2171,2172,2173,2174,2175,2176,2178,2179,2180,2181,2182,2183,2184,2185,2186,2187,2188,2190,2192,2193,2194,2196,2198,2199,2200,2201]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph73\",\"source_text\":\"Let $A:=(\\\\Omega\\\\setminus\\\\mathfrak{z})\\\\cap\\\\tau^{-1}(\\\\Omega\\\\setminus\\\\mathfrak{z})$ be a maximal subset of $\\\\Omega$ on which both of $\\\\beta$ and $\\\\beta\\\\circ\\\\tau$ are injective. Set $B:=\\\\tau(A).$ Then, both $(\\\\Omega\\\\setminus\\\\mathfrak{z})\\\\setminus A$ and $(\\\\Omega\\\\setminus\\\\mathfrak{z})\\\\setminus B$ are countable. Define\",\"tag\":null,\"text\":\"Let A := (Ω z) ∩ τ −1 (Ω z) be a maximal subset of Ω on which both of β and β τ are injective. Set B := τ (A). Then, both (Ω z) A and (Ω z) B are countable. Define\"},{\"bbox\":[185.97,346.34,203.34,12.55],\"bbox_pdf\":[185.97,433.12,389.31,445.66],\"fragments\":[{\"bbox_pdf\":[185.97,433.12,389.31,445.66],\"words\":[2202,2203,2204,2205,2207,2208,2209,2210,2211,2213]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"Z_{1}:=[0,1)\\\\setminus\\\\beta(A),\\\\qquad Z_{2}:=[0,1)\\\\setminus\\\\beta(B).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"S2.Ex15\",\"source_text\":\"$Z_{1}:=[0,1)\\\\setminus\\\\beta(A),\\\\qquad Z_{2}:=[0,1)\\\\setminus\\\\beta(B).$\",\"tag\":null,\"text\":\"Z1 := [0, 1) β(A), Z2 := [0, 1) β(B).\"},{\"bbox\":[73.44,364.74,343.54,12.55],\"bbox_pdf\":[73.44,414.71,416.99,427.26],\"fragments\":[{\"bbox_pdf\":[73.44,414.71,416.99,427.26],\"words\":[2214,2215,2216,2217,2218,2219,2220,2221,2222,2223,2224]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"paragraph\",\"source_ref\":\"paragraph75\",\"source_text\":\"Then $Z_{1}$ and $Z_{2}$ are countable, and hence Lebesgue-null. Finally, define\",\"tag\":null,\"text\":\"Then Z1 and Z2 are countable, and hence Lebesgue-null. Finally, define\"},{\"bbox\":[214.34,375.73,146.61,22.44],\"bbox_pdf\":[214.34,393.83,360.94,416.27],\"fragments\":[{\"bbox_pdf\":[214.34,393.83,360.94,416.27],\"words\":[2225,2226,2227,2228,2229,2230,2231,2233,2235,2237,2238,2239]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\iota:=\\\\beta\\\\circ\\\\tau\\\\circ\\\\bigl{(}\\\\beta|_{A}\\\\bigr{)}^{-1}:Z_{1}^{c}\\\\to Z_{2}^{%\\nc}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"S2.Ex16\",\"source_text\":\"$\\\\iota:=\\\\beta\\\\circ\\\\tau\\\\circ\\\\bigl{(}\\\\beta|_{A}\\\\bigr{)}^{-1}:Z_{1}^{c}\\\\to Z_{2}^{% c}.$\",\"tag\":null,\"text\":\"(︁ )︁−1 c c ι := β τ β|A Z1 → Z2.\"},{\"bbox\":[73.44,402.97,343.46,10.91],\"bbox_pdf\":[73.44,378.12,416.9,389.03],\"fragments\":[{\"bbox_pdf\":[73.44,378.12,416.9,389.03],\"words\":[2240,2241,2242,2243,2244,2245,2246,2247,2248,2249,2250,2251]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":13,\"role\":\"paragraph\",\"source_ref\":\"paragraph77\",\"source_text\":\"This map is a bijection, and the preceding order comparison shows that\",\"tag\":null,\"text\":\"This map is a bijection, and the preceding order comparison shows that\"},{\"bbox\":[223.32,421.37,128.64,10.91],\"bbox_pdf\":[223.32,359.72,351.96,370.63],\"fragments\":[{\"bbox_pdf\":[223.32,359.72,351.96,370.63],\"words\":[2252,2253,2254,2255,2256,2257,2258]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"x\\\\prec y\\\\quad\\\\Longleftrightarrow\\\\quad\\\\iota(x)<\\\\iota(y).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"S2.Ex17\",\"source_text\":\"$x\\\\prec y\\\\quad\\\\Longleftrightarrow\\\\quad\\\\iota(x)<\\\\iota(y).$\",\"tag\":null,\"text\":\"x ≺ y ⇐⇒ ι(x) < ι(y).\"},{\"bbox\":[73.44,439.78,409.0,10.91],\"bbox_pdf\":[73.44,341.31,482.44,352.22],\"fragments\":[{\"bbox_pdf\":[73.44,341.31,482.44,352.22],\"words\":[2259,2260,2261,2262,2263,2264,2265,2266,2267,2268,2269,2270,2271]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":15,\"role\":\"paragraph\",\"source_ref\":\"paragraph79\",\"source_text\":\"Moreover, since $\\\\beta$ and $\\\\tau$ are measure-preserving, $\\\\iota$ also preserves the Lebesgue measure. ∎\",\"tag\":null,\"text\":\"Moreover, since β and τ are measure-preserving, ι also preserves the Lebesgue measure.\"},{\"bbox\":[85.4,459.92,410.22,10.91],\"bbox_pdf\":[85.4,321.17,495.61,332.08],\"fragments\":[{\"bbox_pdf\":[85.4,321.17,495.61,332.08],\"words\":[2273,2274,2275,2276,2277,2278,2279,2280,2281,2282,2283,2284,2285,2286,2287]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":16,\"role\":\"paragraph\",\"source_ref\":\"paragraph80\",\"source_text\":\"By Lemma 2.2, we obtain the following statement for i.i.d. random variables on $[0,1)$ .\",\"tag\":null,\"text\":\"By Lemma 2.2, we obtain the following statement for i.i.d. random variables on [0, 1).\"},{\"bbox\":[73.44,479.87,428.4,39.51],\"bbox_pdf\":[73.44,272.62,501.84,312.13],\"fragments\":[{\"bbox_pdf\":[73.44,272.62,501.84,312.13],\"words\":[2288,2289,2290,2291,2292,2293,2294,2295,2296,2297,2298,2302,2303,2304,2305,2306,2307,2308,2309,2310,2311,2312,2313,2314,2315,2316,2317,2318,2319,2320,2321,2322,2323,2324,2325,2326,2327,2328,2329,2330,2331,2332,2333,2334,2335,2336,2337,2338,2339,2340,2341,2342,2343,2344]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":17,\"role\":\"paragraph\",\"source_ref\":\"paragraph81\",\"source_text\":\"Proposition 2.3. For an i.i.d. uniform sequence $a=(a_{1},a_{2},\\\\dots,a_{n})$ taking values in $[0,1)$ , let $U_{n}(a)$ denote the length of the LIS taken with respect to the usual order, and let $U_{n}^{\\\\succ}(a)$ denote the length of the LIS taken with respect to the order $\\\\succ$ . Then, we have\",\"tag\":null,\"text\":\"Proposition 2.3. For an i.i.d. uniform sequence a = (a1, a2, ,an) taking values in [0, 1), let Un(a) denote the length of the LIS taken with respect to the usual order, and let U ≻ n (a) denote the length of the LIS taken with respect to the order ≻. Then, we have\"},{\"bbox\":[264.45,524.51,42.85,16.85],\"bbox_pdf\":[264.45,250.64,307.3,267.49],\"fragments\":[{\"bbox_pdf\":[264.45,250.64,307.3,267.49],\"words\":[2345,2346,2347,2348,2349]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"U_{n}\\\\stackrel{{\\\\scriptstyle d}}{{=}}U_{n}^{\\\\succ}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":18,\"role\":\"formula\",\"source_ref\":\"S2.Ex18\",\"source_text\":\"$U_{n}\\\\stackrel{{\\\\scriptstyle d}}{{=}}U_{n}^{\\\\succ}.$\",\"tag\":null,\"text\":\"d ≻ Un = Un\"},{\"bbox\":[73.44,547.71,428.4,39.21],\"bbox_pdf\":[73.44,205.08,501.84,244.29],\"fragments\":[{\"bbox_pdf\":[73.44,205.08,501.84,244.29],\"words\":[2351,2352,2353,2354,2355,2356,2357,2358,2359,2360,2361,2362,2363,2364,2365,2366,2367,2368,2372,2373,2374,2375,2376,2377,2378,2379,2380,2381,2382,2383,2384,2385,2386,2387,2388,2389,2390,2391,2393,2394,2398,2399,2400]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":19,\"role\":\"paragraph\",\"source_ref\":\"paragraph83\",\"source_text\":\"Let $x\\\\in[0,1)$ be a uniformly distributed random variable, and define the sequence $a_{1},a_{2},a_{3},\\\\dots\\\\in[0,1)$ by setting $a_{1}=x$ and $a_{n+1}:=f(a_{n})$ . For the first $N$ terms $a^{(N)}=(a_{1},\\\\dots,a_{N})$ , we let\",\"tag\":null,\"text\":\"Let x ∈ [0, 1) be a uniformly distributed random variable, and define the sequence a1, a2, a3, ∈ [0, 1) by setting a1 = x and an+1 := f (an). For the first N terms a (N = (a1, ,aN), we let\"},{\"bbox\":[227.77,587.88,118.39,16.2],\"bbox_pdf\":[227.77,187.92,346.16,204.12],\"fragments\":[{\"bbox_pdf\":[227.77,187.92,346.16,204.12],\"words\":[2401,2403,2405,2406,2407,2409,2410]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\pi_{\\\\succ}^{(N)}:=\\\\mathrm{Pat}_{\\\\succ}(a^{(N)})\\\\in S_{N}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":20,\"role\":\"formula\",\"source_ref\":\"S2.Ex19\",\"source_text\":\"$\\\\pi_{\\\\succ}^{(N)}:=\\\\mathrm{Pat}_{\\\\succ}(a^{(N)})\\\\in S_{N}$\",\"tag\":null,\"text\":\"(N (N π≻ := Pat≻(a ∈ SN\"},{\"bbox\":[73.44,609.58,428.4,39.86],\"bbox_pdf\":[73.44,142.56,501.84,182.42],\"fragments\":[{\"bbox_pdf\":[73.44,142.56,501.84,182.42],\"words\":[2411,2413,2414,2415,2416,2417,2418,2419,2420,2421,2422,2423,2424,2425,2426,2427,2428,2429,2430,2432,2434,2435,2436,2437,2438,2439,2440,2441,2443,2444,2445,2446,2447,2448,2449,2450,2451,2452]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":21,\"role\":\"paragraph\",\"source_ref\":\"paragraph85\",\"source_text\":\"be the sequence with respect to the order $\\\\succ$ (see (1)). Let $\\\\lambda_{\\\\succ}^{(N)}$ be the Young diagram obtained by applying Schensted’s row-insertion algorithm to $\\\\pi_{\\\\succ}^{(N)}$ , and let $\\\\lambda_{\\\\succ,1}^{(N)}$ denote the length of its first row.\",\"tag\":null,\"text\":\"(N be the sequence with respect to the order ≻ (see (1)). Let λ≻ be the Young diagram (N (N obtained by applying Schensted’s row-insertion algorithm to π≻ and let λ≻,1 denote the length of its first row.\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.24575\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.24575v2-p0004.pdf\",\"page_no\":5,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"b2d46808ff2e296dea9c8edc44ee5a098a7d8aa0\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"9da159a0600a0a7ff1719755312c6159bf47af137eaa2cb5edfec454220b158b\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph70\",\"c_block\":0.958,\"c_lines\":1.0,\"c_pdf\":0.999,\"c_region\":1.0,\"c_src\":0.994,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.958,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b.pdf","asset_sha256":"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b","document_id":"arxiv:2608.24575","kind":"page","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[214.09,138.4,147.11,13.76],\"bbox_pdf\":[214.09,639.84,361.19,653.6],\"fragments\":[{\"bbox_pdf\":[214.09,639.84,361.19,653.6],\"words\":[1963,1964,1965,1966,1967,1968]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"d_{j}:=\\\\sigma_{c_{1}\\\\cdots c_{j-1}}(c_{j}),\\\\qquad(j\\\\geq 1).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S2.Ex12\",\"source_text\":\"$d_{j}:=\\\\sigma_{c_{1}\\\\cdots c_{j-1}}(c_{j}),\\\\qquad(j\\\\geq 1).$\",\"tag\":null,\"text\":\"dj := σc1···cj−1(cj), (j ≥ 1).\"},\"formula_index\":3,\"parent_page_sha256\":\"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b.pdf","asset_sha256":"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b","document_id":"arxiv:2608.24575","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[192.28,200.36,190.73,12.47],\"bbox_pdf\":[192.28,579.17,383.0,591.64],\"fragments\":[{\"bbox_pdf\":[192.28,579.17,383.0,591.64],\"words\":[2043,2044,2045,2046,2047,2048,2049,2050,2051,2052,2053,2054]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\tau([c])=[d]\\\\quad\\\\text{and}\\\\quad\\\\nu([c])=\\\\nu([d])=2^{-\\\\ell},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S2.Ex13\",\"source_text\":\"$\\\\tau([c])=[d]\\\\quad\\\\text{and}\\\\quad\\\\nu([c])=\\\\nu([d])=2^{-\\\\ell},$\",\"tag\":null,\"text\":\"τ ([c]) = [d] and ν([c]) = ν([d]) = 2 −ℓ ,\"},\"formula_index\":5,\"parent_page_sha256\":\"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b.pdf","asset_sha256":"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b","document_id":"arxiv:2608.24575","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[131.9,275.12,303.71,32.63],\"bbox_pdf\":[131.9,484.26,435.61,516.88],\"fragments\":[{\"bbox_pdf\":[131.9,484.26,435.61,516.88],\"words\":[2123,2124,2125,2126,2128,2129,2130,2131,2132,2133,2137,2138,2139,2143,2144,2145,2146,2147,2148,2149,2150]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\beta(c)\\\\prec\\\\beta(c^{\\\\prime}) \\\\displaystyle\\\\iff C_{c_{1}c_{2}\\\\dots c_{j}}\\\\prec C_{c_{1}c_{2}\\\\dots\\\\overline{c%\\n_{j}}} \\\\\\\\ \\\\displaystyle\\\\iff d_{1}d_{2}\\\\dots d_{j}<_{\\\\mathrm{lex}}d_{1}d_{2}\\\\dots%\\n\\\\overline{d_{j}}\\\\iff\\\\tau(c)<_{\\\\mathrm{lex}}\\\\tau(c^{\\\\prime}). \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S2.Ex14\",\"source_text\":\"$\\\\displaystyle\\\\beta(c)\\\\prec\\\\beta(c^{\\\\prime})$ $\\\\displaystyle\\\\iff C_{c_{1}c_{2}\\\\dots c_{j}}\\\\prec C_{c_{1}c_{2}\\\\dots\\\\overline{c% _{j}}}$ $\\\\displaystyle\\\\iff d_{1}d_{2}\\\\dots d_{j}<_{\\\\mathrm{lex}}d_{1}d_{2}\\\\dots% \\\\overline{d_{j}}\\\\iff\\\\tau(c)<_{\\\\mathrm{lex}}\\\\tau(c^{\\\\prime}).$\",\"tag\":null,\"text\":\"β(c) ≺ β(c ′ ⇐⇒ Cc1c2...cj ≺ Cc1c2...cj ⇐⇒ d1d2 dj <lex d1d2 dj ⇐⇒ τ (c) <lex τ (c ′\"},\"formula_index\":8,\"parent_page_sha256\":\"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b.pdf","asset_sha256":"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b","document_id":"arxiv:2608.24575","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[185.97,346.34,203.34,12.55],\"bbox_pdf\":[185.97,433.12,389.31,445.66],\"fragments\":[{\"bbox_pdf\":[185.97,433.12,389.31,445.66],\"words\":[2202,2203,2204,2205,2207,2208,2209,2210,2211,2213]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"Z_{1}:=[0,1)\\\\setminus\\\\beta(A),\\\\qquad Z_{2}:=[0,1)\\\\setminus\\\\beta(B).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"S2.Ex15\",\"source_text\":\"$Z_{1}:=[0,1)\\\\setminus\\\\beta(A),\\\\qquad Z_{2}:=[0,1)\\\\setminus\\\\beta(B).$\",\"tag\":null,\"text\":\"Z1 := [0, 1) β(A), Z2 := [0, 1) β(B).\"},\"formula_index\":10,\"parent_page_sha256\":\"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b.pdf","asset_sha256":"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b","document_id":"arxiv:2608.24575","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[214.34,375.73,146.61,22.44],\"bbox_pdf\":[214.34,393.83,360.94,416.27],\"fragments\":[{\"bbox_pdf\":[214.34,393.83,360.94,416.27],\"words\":[2225,2226,2227,2228,2229,2230,2231,2233,2235,2237,2238,2239]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\iota:=\\\\beta\\\\circ\\\\tau\\\\circ\\\\bigl{(}\\\\beta|_{A}\\\\bigr{)}^{-1}:Z_{1}^{c}\\\\to Z_{2}^{%\\nc}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"S2.Ex16\",\"source_text\":\"$\\\\iota:=\\\\beta\\\\circ\\\\tau\\\\circ\\\\bigl{(}\\\\beta|_{A}\\\\bigr{)}^{-1}:Z_{1}^{c}\\\\to Z_{2}^{% c}.$\",\"tag\":null,\"text\":\"(︁ )︁−1 c c ι := β τ β|A Z1 → Z2.\"},\"formula_index\":12,\"parent_page_sha256\":\"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b.pdf","asset_sha256":"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b","document_id":"arxiv:2608.24575","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[223.32,421.37,128.64,10.91],\"bbox_pdf\":[223.32,359.72,351.96,370.63],\"fragments\":[{\"bbox_pdf\":[223.32,359.72,351.96,370.63],\"words\":[2252,2253,2254,2255,2256,2257,2258]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"x\\\\prec y\\\\quad\\\\Longleftrightarrow\\\\quad\\\\iota(x)<\\\\iota(y).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"S2.Ex17\",\"source_text\":\"$x\\\\prec y\\\\quad\\\\Longleftrightarrow\\\\quad\\\\iota(x)<\\\\iota(y).$\",\"tag\":null,\"text\":\"x ≺ y ⇐⇒ ι(x) < ι(y).\"},\"formula_index\":14,\"parent_page_sha256\":\"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b.pdf","asset_sha256":"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b","document_id":"arxiv:2608.24575","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[264.45,524.51,42.85,16.85],\"bbox_pdf\":[264.45,250.64,307.3,267.49],\"fragments\":[{\"bbox_pdf\":[264.45,250.64,307.3,267.49],\"words\":[2345,2346,2347,2348,2349]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"U_{n}\\\\stackrel{{\\\\scriptstyle d}}{{=}}U_{n}^{\\\\succ}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":18,\"role\":\"formula\",\"source_ref\":\"S2.Ex18\",\"source_text\":\"$U_{n}\\\\stackrel{{\\\\scriptstyle d}}{{=}}U_{n}^{\\\\succ}.$\",\"tag\":null,\"text\":\"d ≻ Un = Un\"},\"formula_index\":18,\"parent_page_sha256\":\"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b.pdf","asset_sha256":"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b","document_id":"arxiv:2608.24575","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[227.77,587.88,118.39,16.2],\"bbox_pdf\":[227.77,187.92,346.16,204.12],\"fragments\":[{\"bbox_pdf\":[227.77,187.92,346.16,204.12],\"words\":[2401,2403,2405,2406,2407,2409,2410]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\pi_{\\\\succ}^{(N)}:=\\\\mathrm{Pat}_{\\\\succ}(a^{(N)})\\\\in S_{N}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":20,\"role\":\"formula\",\"source_ref\":\"S2.Ex19\",\"source_text\":\"$\\\\pi_{\\\\succ}^{(N)}:=\\\\mathrm{Pat}_{\\\\succ}(a^{(N)})\\\\in S_{N}$\",\"tag\":null,\"text\":\"(N (N π≻ := Pat≻(a ∈ SN\"},\"formula_index\":20,\"parent_page_sha256\":\"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b.pdf","asset_sha256":"368fa46617be3363a1f63c0cebc00fdb1e07e81e7cea44c17edd7903b9d4de8b","document_id":"arxiv:2608.24575","kind":"formula","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2},\"blocks\":[{\"bbox\":[110.16,73.12,8.47,7.97],\"bbox_pdf\":[110.16,710.91,118.63,718.88],\"fragments\":[{\"bbox_pdf\":[110.16,710.91,118.63,718.88],\"words\":[6510]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":0,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"16\"},{\"bbox\":[189.01,73.12,270.71,7.97],\"bbox_pdf\":[189.01,710.91,459.71,718.88],\"fragments\":[{\"bbox_pdf\":[189.01,710.91,459.71,718.88],\"words\":[6511,6512,6513,6514,6515,6516,6517]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":1,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"SHINSUKE IWAO, FUMIHIKO NAKAMURA, AND YUSHI NAKANO\"},{\"bbox\":[122.12,94.09,146.61,10.91],\"bbox_pdf\":[122.12,687.0,268.72,697.91],\"fragments\":[{\"bbox_pdf\":[122.12,687.0,268.72,697.91],\"words\":[6518,6519,6520,6521,6522,6523]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph283\",\"source_text\":\"If we choose the parameters as\",\"tag\":null,\"text\":\"If we choose the parameters as\"},{\"bbox\":[136.45,109.23,402.11,25.77],\"bbox_pdf\":[136.45,657.0,538.56,682.77],\"fragments\":[{\"bbox_pdf\":[136.45,657.0,538.56,682.77],\"words\":[6524,6525,6526,6527,6528,6529,6530,6531,6532,6533,6534,6535,6536,6537,6538,6539,6540,6541,6542,6543,6544,6545,6546,6547,6548,6549,6550,6551]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"m\\\\sim\\\\frac{2}{5}\\\\log_{2}N,\\\\quad P\\\\sim\\\\frac{1}{4}\\\\log_{2}N,\\\\quad S\\\\sim N^{1/10}%\\n,\\\\quad t\\\\sim 100\\\\log_{2}N,\\\\quad B=S+t,\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S3.E22\",\"source_text\":\"$m\\\\sim\\\\frac{2}{5}\\\\log_{2}N,\\\\quad P\\\\sim\\\\frac{1}{4}\\\\log_{2}N,\\\\quad S\\\\sim N^{1/10}% ,\\\\quad t\\\\sim 100\\\\log_{2}N,\\\\quad B=S+t,$ (22)\",\"tag\":\"(22)\",\"text\":\"2 1 1/10 m ∼ log2 N, P ∼ log2 N, S ∼ N , t ∼ 100 log2 N, B = S + t, (22) 5 4\"},{\"bbox\":[110.16,137.66,37.58,10.91],\"bbox_pdf\":[110.16,643.43,147.74,654.34],\"fragments\":[{\"bbox_pdf\":[110.16,643.43,147.74,654.34],\"words\":[6552,6553]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph285\",\"source_text\":\"we have\",\"tag\":null,\"text\":\"we have\"},{\"bbox\":[221.3,146.57,205.62,25.77],\"bbox_pdf\":[221.3,619.66,426.92,645.43],\"fragments\":[{\"bbox_pdf\":[221.3,619.66,426.92,645.43],\"words\":[6554,6555,6556,6557,6558,6559,6560,6561,6562,6563,6564,6565,6566,6567,6568,6569,6570,6571,6572]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"2^{P}\\\\gg S\\\\gg m,\\\\quad\\\\frac{T}{B}\\\\sim N^{1/2},\\\\quad\\\\frac{NS}{4^{m}B}\\\\sim N^{1/5}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S3.Ex85\",\"source_text\":\"$2^{P}\\\\gg S\\\\gg m,\\\\quad\\\\frac{T}{B}\\\\sim N^{1/2},\\\\quad\\\\frac{NS}{4^{m}B}\\\\sim N^{1/5}$\",\"tag\":null,\"text\":\"P T 1/2 N S 1/5 2 ≫ S ≫ m, ∼ N , ∼ N mB B 4\"},{\"bbox\":[110.16,173.33,405.53,10.91],\"bbox_pdf\":[110.16,607.76,515.69,618.67],\"fragments\":[{\"bbox_pdf\":[110.16,607.76,515.69,618.67],\"words\":[6573,6574,6575,6576,6577,6578,6579,6580,6581,6582,6583,6584,6585,6586,6587,6588,6590]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph287\",\"source_text\":\"as $N\\\\to\\\\infty$ . Thus Lemma 3.5 applies uniformly to every $r$ satisfying $\\\\chi(r)\\\\geq P$ . From\",\"tag\":null,\"text\":\"as N → ∞. Thus Lemma 3.5 applies uniformly to every r satisfying χ(r) ≥ P From\"},{\"bbox\":[137.21,184.47,370.07,33.77],\"bbox_pdf\":[137.21,573.76,507.28,607.53],\"fragments\":[{\"bbox_pdf\":[137.21,573.76,507.28,607.53],\"words\":[6591,6592,6593,6594,6595,6596,6597,6598,6599,6600,6601,6602,6603,6604,6605,6606,6607,6608,6609,6610,6611,6612,6613,6614,6615,6617,6618,6619,6621,6622,6623]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{2^{P}}{2^{m}}\\\\to 0,\\\\quad\\\\frac{S}{B}\\\\to 1,\\\\quad 2^{m-t}\\\\left(\\\\frac{N}{2^{%\\nm}B}\\\\right)^{3}=o(\\\\sqrt{N}),\\\\quad 2^{m}n(\\\\epsilon)^{2}=O(N^{2/5})=o(\\\\sqrt{N})\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"S3.Ex86\",\"source_text\":\"$\\\\frac{2^{P}}{2^{m}}\\\\to 0,\\\\quad\\\\frac{S}{B}\\\\to 1,\\\\quad 2^{m-t}\\\\left(\\\\frac{N}{2^{% m}B}\\\\right)^{3}=o(\\\\sqrt{N}),\\\\quad 2^{m}n(\\\\epsilon)^{2}=O(N^{2/5})=o(\\\\sqrt{N})$\",\"tag\":null,\"text\":\"(︃ )︃3 2 P S N √ √ → 0, → 1, 2 m−t = o( N ), 2 m n(ϵ) 2 = O(N 2/5 = o( N 2m B 2mB\"},{\"bbox\":[110.16,224.38,93.64,10.91],\"bbox_pdf\":[110.16,556.71,203.8,567.62],\"fragments\":[{\"bbox_pdf\":[110.16,556.71,203.8,567.62],\"words\":[6624,6625,6626,6627]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph289\",\"source_text\":\"and (21), we obtain\",\"tag\":null,\"text\":\"and (21), we obtain\"},{\"bbox\":[270.55,229.43,103.4,20.68],\"bbox_pdf\":[270.55,541.89,373.95,562.57],\"fragments\":[{\"bbox_pdf\":[270.55,541.89,373.95,562.57],\"words\":[6628,6629,6630,6631,6632,6633,6634,6635,6636]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathcal{E}\\\\geq(2-\\\\epsilon-o(1))\\\\sqrt{N},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S3.Ex87\",\"source_text\":\"$\\\\mathcal{E}\\\\geq(2-\\\\epsilon-o(1))\\\\sqrt{N},$\",\"tag\":null,\"text\":\"√ E ≥ (2 − ϵ − o(1)) N\"},{\"bbox\":[110.16,254.88,220.67,10.91],\"bbox_pdf\":[110.16,526.21,330.83,537.12],\"fragments\":[{\"bbox_pdf\":[110.16,526.21,330.83,537.12],\"words\":[6638,6639,6640,6641,6642,6643,6644,6645]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"paragraph\",\"source_ref\":\"paragraph291\",\"source_text\":\"which proves the desired lower bound by (20).\",\"tag\":null,\"text\":\"which proves the desired lower bound by (20).\"},{\"bbox\":[281.75,278.2,85.22,10.91],\"bbox_pdf\":[281.75,502.89,366.97,513.8],\"fragments\":[{\"bbox_pdf\":[281.75,502.89,366.97,513.8],\"words\":[6646,6647,6648]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"heading\",\"source_ref\":\"title292\",\"source_text\":\"4. Upper bound\",\"tag\":null,\"text\":\"4. Upper bound\"},{\"bbox\":[110.16,297.63,428.4,23.86],\"bbox_pdf\":[110.16,470.51,538.56,494.37],\"fragments\":[{\"bbox_pdf\":[110.16,470.51,538.56,494.37],\"words\":[6649,6650,6651,6652,6653,6654,6655,6656,6657,6658,6659,6660,6661,6662,6663,6664,6665,6666,6667,6668,6669,6670,6671,6672]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"paragraph\",\"source_ref\":\"paragraph293\",\"source_text\":\"To obtain an upper bound, we need a more careful argument. We will define a complete prefix set $\\\\mathcal{W}$ in a more refined way.\",\"tag\":null,\"text\":\"To obtain an upper bound, we need a more careful argument. We will define a complete prefix set W in a more refined way.\"},{\"bbox\":[110.16,330.93,214.55,10.91],\"bbox_pdf\":[110.16,450.16,324.71,461.07],\"fragments\":[{\"bbox_pdf\":[110.16,450.16,324.71,461.07],\"words\":[6673,6674,6675,6676,6677,6678,6679]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":2,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":13,\"role\":\"heading\",\"source_ref\":\"title294\",\"source_text\":\"4.1. Definition of the complete prefix set $\\\\mathcal{W}(P,Q)$\",\"tag\":null,\"text\":\"4.1. Definition of the complete prefix set\"},{\"bbox\":[110.16,330.93,428.4,36.81],\"bbox_pdf\":[110.16,424.26,538.56,461.07],\"fragments\":[{\"bbox_pdf\":[110.16,424.26,538.56,461.07],\"words\":[6680,6681,6682,6683,6684,6685,6686,6687,6688,6689,6690,6691,6692,6693,6694,6695,6696,6697,6698,6702,6703,6704,6705,6706,6707,6708,6709,6710,6711,6712,6713,6714,6715,6716,6717,6718,6719,6720,6721,6722]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"paragraph\",\"source_ref\":\"paragraph295\",\"source_text\":\"Fix natural numbers $1<P<Q$ , which will serve as thresholds. A binary word $b_{1}b_{2}\\\\ldots b_{\\\\ell}$ of length $\\\\ell\\\\geq P$ is said to have a short period if it has a period smaller than $P$ .\",\"tag\":null,\"text\":\"W(P, Q). Fix natural numbers 1 < P < Q, which will serve as thresholds. A binary word b1b2 bℓ of length ℓ ≥ P is said to have a short period if it has a period smaller than P\"},{\"bbox\":[122.12,369.78,256.07,10.91],\"bbox_pdf\":[122.12,411.31,378.19,422.22],\"fragments\":[{\"bbox_pdf\":[122.12,411.31,378.19,422.22],\"words\":[6724,6725,6726,6727,6728,6729,6730,6731,6732,6733,6734,6735]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":15,\"role\":\"paragraph\",\"source_ref\":\"paragraph296\",\"source_text\":\"For a binary word $w$ , let $\\\\mathord{\\\\ell}(w)$ denote the length of $w$ .\",\"tag\":null,\"text\":\"For a binary word w, let ℓ(w) denote the length of w.\"},{\"bbox\":[110.16,389.47,428.4,23.86],\"bbox_pdf\":[110.16,378.67,538.56,402.53],\"fragments\":[{\"bbox_pdf\":[110.16,378.67,538.56,402.53],\"words\":[6736,6737,6738,6739,6740,6741,6742,6743,6744,6745,6746,6747,6748,6749,6750,6751,6752,6753,6754]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":16,\"role\":\"paragraph\",\"source_ref\":\"paragraph297\",\"source_text\":\"Definition 4.1. Let $\\\\mathcal{W}(P,Q)$ denote the set of binary sequences $w$ satisfying one of the following properties (i)–(iii):\",\"tag\":null,\"text\":\"Definition 4.1. Let W(P, Q) denote the set of binary sequences w satisfying one of the following properties (i)–(iii):\"},{\"bbox\":[129.99,417.87,195.92,10.91],\"bbox_pdf\":[129.99,363.22,325.92,374.13],\"fragments\":[{\"bbox_pdf\":[129.99,363.22,325.92,374.13],\"words\":[6755,6756,6757,6758,6760,6761,6762,6763,6764,6765]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":1,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":17,\"role\":\"list_item\",\"source_ref\":\"S4.I1.ix1\",\"source_text\":\"(i) $\\\\mathord{\\\\ell}(w)=P$ , and $w$ has no short period.\",\"tag\":null,\"text\":\"(i) ℓ(w) = P and w has no short period.\"},{\"bbox\":[126.96,430.82,411.6,23.86],\"bbox_pdf\":[126.96,337.32,538.56,361.18],\"fragments\":[{\"bbox_pdf\":[126.96,337.32,538.56,361.18],\"words\":[6766,6767,6768,6769,6770,6771,6772,6773,6774,6775,6776,6777,6778,6779,6780,6781,6782,6783,6784,6785,6786,6787,6788,6789]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":1,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":18,\"role\":\"list_item\",\"source_ref\":\"S4.I1.ix2\",\"source_text\":\"(ii) $P<\\\\mathord{\\\\ell}(w)\\\\leq Q$ , $w$ has no short period, but the subsequence obtained by deleting its last digit has a short period.\",\"tag\":null,\"text\":\"(ii) P < ℓ(w) ≤ Q, w has no short period, but the subsequence obtained by deleting its last digit has a short period.\"},{\"bbox\":[123.93,456.72,195.96,10.91],\"bbox_pdf\":[123.93,324.37,319.89,335.28],\"fragments\":[{\"bbox_pdf\":[123.93,324.37,319.89,335.28],\"words\":[6790,6791,6792,6793,6794,6795,6796,6797,6798,6799]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":1,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":19,\"role\":\"list_item\",\"source_ref\":\"S4.I1.ix3\",\"source_text\":\"(iii) $\\\\mathord{\\\\ell}(w)=Q$ , and $w$ has a short period.\",\"tag\":null,\"text\":\"(iii) ℓ(w) = Q, and w has a short period.\"},{\"bbox\":[110.16,472.17,428.4,49.76],\"bbox_pdf\":[110.16,270.07,538.56,319.83],\"fragments\":[{\"bbox_pdf\":[110.16,270.07,538.56,319.83],\"words\":[6800,6801,6802,6803,6804,6805,6806,6807,6808,6809,6810,6811,6812,6813,6814,6815,6816,6817,6818,6819,6820,6821,6822,6823,6824,6825,6826,6827,6828,6829,6830,6831,6832,6833,6834,6835,6836,6837,6838,6839,6840,6841,6842,6843,6844,6845,6846,6847,6848,6849,6850,6851,6852]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":20,\"role\":\"paragraph\",\"source_ref\":\"paragraph301\",\"source_text\":\"We write $\\\\mathcal{W}_{\\\\mathrm{regular}}(P,Q)\\\\subset\\\\mathcal{W}(P,Q)$ for the subset consisting of those $w$ satisfying (i) or (ii), and $\\\\mathcal{W}_{\\\\mathrm{exceptional}}(P,Q)\\\\subset\\\\mathcal{W}(P,Q)$ for the subset consisting of those $w$ satisfying (iii). For a binary code $w\\\\in\\\\mathcal{W}(P,Q)$ , denote by $C_{w}\\\\subset[0,1)$ the cylinder defined in (4).\",\"tag\":null,\"text\":\"We write Wregular(P, Q) ⊂ W(P, Q) for the subset consisting of those w satisfying (i) or (ii), and Wexceptional(P, Q) ⊂ W(P, Q) for the subset consisting of those w satisfying (iii). For a binary code w ∈ W(P, Q), denote by Cw ⊂ [0, 1) the cylinder defined in (4).\"},{\"bbox\":[110.16,530.71,428.4,23.86],\"bbox_pdf\":[110.16,237.43,538.56,261.29],\"fragments\":[{\"bbox_pdf\":[110.16,237.43,538.56,261.29],\"words\":[6853,6854,6855,6856,6857,6858,6859,6860,6861,6862,6863,6864,6865,6866,6867,6868,6869,6870,6871,6872,6873,6874,6875,6876,6877,6878,6879,6880]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":21,\"role\":\"paragraph\",\"source_ref\":\"paragraph302\",\"source_text\":\"Lemma 4.2. Every element of $[0,1)$ belongs to exactly one cylinder $C_{w}$ with $w\\\\in\\\\mathcal{W}(P,Q)$ . In other words, $\\\\mathcal{W}(P,Q)$ is a complete prefix set.\",\"tag\":null,\"text\":\"Lemma 4.2. Every element of [0, 1) belongs to exactly one cylinder Cw with w ∈ W(P, Q). In other words, W(P, Q) is a complete prefix set.\"},{\"bbox\":[110.16,563.35,427.9,23.86],\"bbox_pdf\":[110.16,204.79,538.06,228.65],\"fragments\":[{\"bbox_pdf\":[110.16,204.79,538.06,228.65],\"words\":[6881,6882,6883,6884,6885,6886,6887,6888,6889,6890,6894,6895,6896,6897,6898,6899,6900,6901,6905,6906,6907,6908,6909,6910,6911]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":22,\"role\":\"paragraph\",\"source_ref\":\"paragraph303\",\"source_text\":\"Proof. For $x\\\\in[0,1)$ , let $x=0.c_{1}c_{2}c_{3}\\\\dots$ be the binary expansion and $w_{m}(x):=c_{1}c_{2}\\\\dots c_{m}$ be its first $m$ digits. Define\",\"tag\":null,\"text\":\"Proof. For x ∈ [0, 1), let x = 0.c1c2c3 be the binary expansion and wm(x) := c1c2 cm be its first m digits. Define\"},{\"bbox\":[184.93,592.68,274.02,14.11],\"bbox_pdf\":[184.93,185.21,458.94,199.32],\"fragments\":[{\"bbox_pdf\":[184.93,185.21,458.94,199.32],\"words\":[6912,6913,6914,6915,6916,6917,6918,6920,6921,6922,6923,6924,6925,6926]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"m^{\\\\ast}:=\\\\min\\\\left(\\\\{m\\\\geq P:w_{m}(x)\\\\text{ has no short period}\\\\}\\\\cup\\\\{%\\n\\\\infty\\\\}\\\\right).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":23,\"role\":\"formula\",\"source_ref\":\"S4.Ex88\",\"source_text\":\"$m^{\\\\ast}:=\\\\min\\\\left(\\\\{m\\\\geq P:w_{m}(x)\\\\text{ has no short period}\\\\}\\\\cup\\\\{% \\\\infty\\\\}\\\\right).$\",\"tag\":null,\"text\":\"m ∗ := min ({m ≥ P wm(x) has no short period} ∪ {∞})\"},{\"bbox\":[110.16,612.19,428.4,37.25],\"bbox_pdf\":[110.16,142.56,538.56,179.81],\"fragments\":[{\"bbox_pdf\":[110.16,142.56,538.56,179.81],\"words\":[6928,6929,6930,6931,6932,6933,6934,6935,6936,6937,6938,6939,6940,6941,6942,6943,6944,6945,6946,6947,6949,6950,6951,6952,6953,6954,6955,6956,6957,6958,6959,6960,6961,6962,6963,6964,6965,6966,6967,6968,6969,6970,6971,6972,6973,6974,6975,6976,6977,6978,6979,6980,6981,6982,6983,6984,6985,6986,6987,6988,6989]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":24,\"role\":\"paragraph\",\"source_ref\":\"paragraph305\",\"source_text\":\"Put $w:=w_{\\\\min\\\\{m^{\\\\ast},Q\\\\}}(x)$ . Then, we have $w\\\\in\\\\mathcal{W}(P,Q)$ and $x\\\\in C_{w}$ . If $m^{\\\\ast}=P$ , then $w$ satisfies (i); if $P<m^{\\\\ast}\\\\leq Q$ , then $w$ satisfies (ii); if $m^{\\\\ast}>Q$ , then $w$ satisfies (iii). To show the uniqueness, note that if a binary word has no short period, its extensions also have no short period. Therefore, if $w$ is an element of $\\\\mathcal{W}(P,Q)$ , any extension of $w$ is not an element of $\\\\mathcal{W}(P,Q)$ . Then, from Lemma 2.1, we find that $C_{w}\\\\cap C_{w^{\\\\prime}}\\\\neq\\\\emptyset$ for some $w,w^{\\\\prime}\\\\in\\\\mathcal{W}(P,Q)$ implies $w=w^{\\\\prime}$ .\",\"tag\":null,\"text\":\"∗ Put w := wmin{m∗,Q}(x). Then, we have w ∈ W(P, Q) and x ∈ Cw. If m = P then w satisfies (i); if P < m ∗ ≤ Q, then w satisfies (ii); if m ∗ > Q, then w satisfies (iii). To show the uniqueness, note that if a binary word has no short period, its extensions also\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.24575\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.24575v2-p0015.pdf\",\"page_no\":16,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"b2d46808ff2e296dea9c8edc44ee5a098a7d8aa0\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"9da159a0600a0a7ff1719755312c6159bf47af137eaa2cb5edfec454220b158b\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1.pdf","asset_sha256":"6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1","document_id":"arxiv:2608.24575","kind":"page","page_number":16,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[136.45,109.23,402.11,25.77],\"bbox_pdf\":[136.45,657.0,538.56,682.77],\"fragments\":[{\"bbox_pdf\":[136.45,657.0,538.56,682.77],\"words\":[6524,6525,6526,6527,6528,6529,6530,6531,6532,6533,6534,6535,6536,6537,6538,6539,6540,6541,6542,6543,6544,6545,6546,6547,6548,6549,6550,6551]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"m\\\\sim\\\\frac{2}{5}\\\\log_{2}N,\\\\quad P\\\\sim\\\\frac{1}{4}\\\\log_{2}N,\\\\quad S\\\\sim N^{1/10}%\\n,\\\\quad t\\\\sim 100\\\\log_{2}N,\\\\quad B=S+t,\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S3.E22\",\"source_text\":\"$m\\\\sim\\\\frac{2}{5}\\\\log_{2}N,\\\\quad P\\\\sim\\\\frac{1}{4}\\\\log_{2}N,\\\\quad S\\\\sim N^{1/10}% ,\\\\quad t\\\\sim 100\\\\log_{2}N,\\\\quad B=S+t,$ (22)\",\"tag\":\"(22)\",\"text\":\"2 1 1/10 m ∼ log2 N, P ∼ log2 N, S ∼ N , t ∼ 100 log2 N, B = S + t, (22) 5 4\"},\"formula_index\":3,\"parent_page_sha256\":\"6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1.pdf","asset_sha256":"6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1","document_id":"arxiv:2608.24575","kind":"formula","page_number":16,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[221.3,146.57,205.62,25.77],\"bbox_pdf\":[221.3,619.66,426.92,645.43],\"fragments\":[{\"bbox_pdf\":[221.3,619.66,426.92,645.43],\"words\":[6554,6555,6556,6557,6558,6559,6560,6561,6562,6563,6564,6565,6566,6567,6568,6569,6570,6571,6572]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"2^{P}\\\\gg S\\\\gg m,\\\\quad\\\\frac{T}{B}\\\\sim N^{1/2},\\\\quad\\\\frac{NS}{4^{m}B}\\\\sim N^{1/5}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S3.Ex85\",\"source_text\":\"$2^{P}\\\\gg S\\\\gg m,\\\\quad\\\\frac{T}{B}\\\\sim N^{1/2},\\\\quad\\\\frac{NS}{4^{m}B}\\\\sim N^{1/5}$\",\"tag\":null,\"text\":\"P T 1/2 N S 1/5 2 ≫ S ≫ m, ∼ N , ∼ N mB B 4\"},\"formula_index\":5,\"parent_page_sha256\":\"6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1.pdf","asset_sha256":"6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1","document_id":"arxiv:2608.24575","kind":"formula","page_number":16,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[137.21,184.47,370.07,33.77],\"bbox_pdf\":[137.21,573.76,507.28,607.53],\"fragments\":[{\"bbox_pdf\":[137.21,573.76,507.28,607.53],\"words\":[6591,6592,6593,6594,6595,6596,6597,6598,6599,6600,6601,6602,6603,6604,6605,6606,6607,6608,6609,6610,6611,6612,6613,6614,6615,6617,6618,6619,6621,6622,6623]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\frac{2^{P}}{2^{m}}\\\\to 0,\\\\quad\\\\frac{S}{B}\\\\to 1,\\\\quad 2^{m-t}\\\\left(\\\\frac{N}{2^{%\\nm}B}\\\\right)^{3}=o(\\\\sqrt{N}),\\\\quad 2^{m}n(\\\\epsilon)^{2}=O(N^{2/5})=o(\\\\sqrt{N})\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"S3.Ex86\",\"source_text\":\"$\\\\frac{2^{P}}{2^{m}}\\\\to 0,\\\\quad\\\\frac{S}{B}\\\\to 1,\\\\quad 2^{m-t}\\\\left(\\\\frac{N}{2^{% m}B}\\\\right)^{3}=o(\\\\sqrt{N}),\\\\quad 2^{m}n(\\\\epsilon)^{2}=O(N^{2/5})=o(\\\\sqrt{N})$\",\"tag\":null,\"text\":\"(︃ )︃3 2 P S N √ √ → 0, → 1, 2 m−t = o( N ), 2 m n(ϵ) 2 = O(N 2/5 = o( N 2m B 2mB\"},\"formula_index\":7,\"parent_page_sha256\":\"6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1.pdf","asset_sha256":"6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1","document_id":"arxiv:2608.24575","kind":"formula","page_number":16,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[270.55,229.43,103.4,20.68],\"bbox_pdf\":[270.55,541.89,373.95,562.57],\"fragments\":[{\"bbox_pdf\":[270.55,541.89,373.95,562.57],\"words\":[6628,6629,6630,6631,6632,6633,6634,6635,6636]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathcal{E}\\\\geq(2-\\\\epsilon-o(1))\\\\sqrt{N},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S3.Ex87\",\"source_text\":\"$\\\\mathcal{E}\\\\geq(2-\\\\epsilon-o(1))\\\\sqrt{N},$\",\"tag\":null,\"text\":\"√ E ≥ (2 − ϵ − o(1)) N\"},\"formula_index\":9,\"parent_page_sha256\":\"6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1.pdf","asset_sha256":"6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1","document_id":"arxiv:2608.24575","kind":"formula","page_number":16,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[184.93,592.68,274.02,14.11],\"bbox_pdf\":[184.93,185.21,458.94,199.32],\"fragments\":[{\"bbox_pdf\":[184.93,185.21,458.94,199.32],\"words\":[6912,6913,6914,6915,6916,6917,6918,6920,6921,6922,6923,6924,6925,6926]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"m^{\\\\ast}:=\\\\min\\\\left(\\\\{m\\\\geq P:w_{m}(x)\\\\text{ has no short period}\\\\}\\\\cup\\\\{%\\n\\\\infty\\\\}\\\\right).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":23,\"role\":\"formula\",\"source_ref\":\"S4.Ex88\",\"source_text\":\"$m^{\\\\ast}:=\\\\min\\\\left(\\\\{m\\\\geq P:w_{m}(x)\\\\text{ has no short period}\\\\}\\\\cup\\\\{% \\\\infty\\\\}\\\\right).$\",\"tag\":null,\"text\":\"m ∗ := min ({m ≥ P wm(x) has no short period} ∪ {∞})\"},\"formula_index\":23,\"parent_page_sha256\":\"6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1.pdf","asset_sha256":"6d57c334b6360283bf64fb54bc7d236961f416d3f2fa479378f7e2751f6d37b1","document_id":"arxiv:2608.24575","kind":"formula","page_number":16,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2},\"blocks\":[{\"bbox\":[121.13,73.12,333.02,7.97],\"bbox_pdf\":[121.13,710.91,454.15,718.88],\"fragments\":[{\"bbox_pdf\":[121.13,710.91,454.15,718.88],\"words\":[11569,11570,11571,11572,11573,11574,11575]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":0,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"LONGEST INCREASING SUBSEQUENCES OF DYADIC-TYPE CHAOTIC ORBITS\"},{\"bbox\":[493.37,73.12,8.47,7.97],\"bbox_pdf\":[493.37,710.91,501.84,718.88],\"fragments\":[{\"bbox_pdf\":[493.37,710.91,501.84,718.88],\"words\":[11576]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":1,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"23\"},{\"bbox\":[73.44,94.09,409.0,10.91],\"bbox_pdf\":[73.44,687.0,482.44,697.91],\"fragments\":[{\"bbox_pdf\":[73.44,687.0,482.44,697.91],\"words\":[9541,9542,9543,9544,9545,9546,9547,9548,9549,9550,9554,9555,9556,9557,9558,9559]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph411\",\"source_text\":\"By summing up these inequalities for $i=0,1,\\\\dots,\\\\mathord{\\\\ell}(w)-1$ , we obtain the desired result. ∎\",\"tag\":null,\"text\":\"By summing up these inequalities for i = 0, 1, ,ℓ(w)−1, we obtain the desired result.\"},{\"bbox\":[85.4,112.2,394.17,14.63],\"bbox_pdf\":[85.4,665.18,479.57,679.8],\"fragments\":[{\"bbox_pdf\":[85.4,665.18,479.57,679.8],\"words\":[9561,9562,9563,9564,9565,9566,9567,9568,9569,9570,9571,9572,9573,9574,9575,9576,9577,9578]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph412\",\"source_text\":\"Next, we consider $M_{w}-M_{w}^{\\\\prime}$ , which is the number of elements of the difference set\",\"tag\":null,\"text\":\"Next, we consider Mw − M ′ w, which is the number of elements of the difference set\"},{\"bbox\":[187.54,130.85,200.2,16.86],\"bbox_pdf\":[187.54,644.29,387.74,661.15],\"fragments\":[{\"bbox_pdf\":[187.54,644.29,387.74,661.15],\"words\":[9579,9580,9584,9587,9588,9592,9593,9594]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\{a_{\\\\tau_{1}},a_{\\\\tau_{2}},\\\\dots,a_{\\\\tau_{M_{w}}}\\\\}\\\\setminus\\\\{a_{\\\\sigma_{1}},%\\na_{\\\\sigma_{2}},\\\\dots,a_{\\\\sigma_{M^{\\\\prime}_{w}}}\\\\}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S4.Ex124\",\"source_text\":\"$\\\\{a_{\\\\tau_{1}},a_{\\\\tau_{2}},\\\\dots,a_{\\\\tau_{M_{w}}}\\\\}\\\\setminus\\\\{a_{\\\\sigma_{1}},% a_{\\\\sigma_{2}},\\\\dots,a_{\\\\sigma_{M^{\\\\prime}_{w}}}\\\\}.$\",\"tag\":null,\"text\":\"{aτ1, aτ2, ,aτMw {aσ1, aσ2, ,aσ }. M′w\"},{\"bbox\":[73.44,150.66,428.39,26.8],\"bbox_pdf\":[73.44,614.54,501.83,641.34],\"fragments\":[{\"bbox_pdf\":[73.44,614.54,501.83,641.34],\"words\":[9595,9596,9597,9598,9599,9600,9601,9602,9603,9604,9605,9606,9607,9611,9612,9613,9614,9615,9616,9617,9618,9619,9620,9621,9622,9623,9627,9628,9629,9630]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph414\",\"source_text\":\"Define the random variable $V_{i}$ , which means “ $a_{i}$ is contained in $\\\\{a_{\\\\tau_{1}},a_{\\\\tau_{2}},\\\\dots,a_{\\\\tau_{M_{w}}}\\\\}$ , but has a possibility not to be contained in $\\\\{a_{\\\\sigma_{1}},a_{\\\\sigma_{2}},\\\\dots,a_{\\\\sigma_{M^{\\\\prime}_{w}}}\\\\}$ ,” by\",\"tag\":null,\"text\":\"Define the random variable Vi, which means “ai is contained in {aτ1, aτ2, ,aτMw }, but has a possibility not to be contained in {aσ1, aσ2, ,aσ },” by M′w\"},{\"bbox\":[147.28,175.87,354.56,31.2],\"bbox_pdf\":[147.28,584.93,501.84,616.13],\"fragments\":[{\"bbox_pdf\":[147.28,584.93,501.84,616.13],\"words\":[9631,9632,9633,9634,9635,9636,9637,9638,9642,9643,9644,9645,9646,9647,9648,9649,9650,9651,9652,9653]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"V_{i}:=\\\\begin{cases}1&\\\\mathrm{hit}(i)\\\\wedge(\\\\exists d\\\\in\\\\{1,2,\\\\dots,D-1\\\\}\\\\ \\\\ %\\n\\\\text{s.t.}\\\\ \\\\ \\\\mathrm{hit}(i+d)),\\\\\\\\\\n0&(\\\\text{otherwise}).\\\\end{cases}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S4.E32\",\"source_text\":\"$V_{i}:=\\\\begin{cases}1&\\\\mathrm{hit}(i)\\\\wedge(\\\\exists d\\\\in\\\\{1,2,\\\\dots,D-1\\\\}\\\\ \\\\ % \\\\text{s.t.}\\\\ \\\\ \\\\mathrm{hit}(i+d)),\\\\\\\\ 0&(\\\\text{otherwise}).\\\\end{cases}$ (32)\",\"tag\":\"(32)\",\"text\":\"{︄ 1 hit(i) ∧ (∃d ∈ {1, 2, ,D − 1} s.t. hit(i + d)), Vi := (32) 0 (otherwise).\"},{\"bbox\":[73.44,221.41,138.48,10.91],\"bbox_pdf\":[73.44,559.68,211.93,570.59],\"fragments\":[{\"bbox_pdf\":[73.44,559.68,211.93,570.59],\"words\":[9654,9655,9656,9657,9658]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph416\",\"source_text\":\"Then, we have the inequality\",\"tag\":null,\"text\":\"Then, we have the inequality\"},{\"bbox\":[241.15,233.84,260.69,34.69],\"bbox_pdf\":[241.15,523.46,501.84,558.16],\"fragments\":[{\"bbox_pdf\":[241.15,523.46,501.84,558.16],\"words\":[9659,9660,9661,9662,9663,9664,9665,9666,9667,9668,9669]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"M_{w}-M_{w}^{\\\\prime}\\\\leq\\\\sum_{i=u}^{v-1}V_{i}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S4.E33\",\"source_text\":\"$M_{w}-M_{w}^{\\\\prime}\\\\leq\\\\sum_{i=u}^{v-1}V_{i}.$ (33)\",\"tag\":\"(33)\",\"text\":\"∑︂ v−1 Mw − M ′ Vi. (33) w ≤ i=u\"},{\"bbox\":[85.4,270.31,92.3,10.91],\"bbox_pdf\":[85.4,510.78,177.7,521.69],\"fragments\":[{\"bbox_pdf\":[85.4,510.78,177.7,521.69],\"words\":[9670,9671,9672,9673]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph418\",\"source_text\":\"From (32), we have\",\"tag\":null,\"text\":\"From (32), we have\"},{\"bbox\":[207.93,283.66,293.91,34.96],\"bbox_pdf\":[207.93,473.38,501.84,508.34],\"fragments\":[{\"bbox_pdf\":[207.93,473.38,501.84,508.34],\"words\":[9674,9675,9676,9677,9678,9679,9680,9681,9682,9683,9684,9685,9686,9687,9688]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathbb{E}[V_{i}]\\\\leq\\\\sum_{d=1}^{D-1}\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge%\\n\\\\mathrm{hit}(i+d)\\\\bigr{)}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"S4.E34\",\"source_text\":\"$\\\\mathbb{E}[V_{i}]\\\\leq\\\\sum_{d=1}^{D-1}\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge% \\\\mathrm{hit}(i+d)\\\\bigr{)}.$ (34)\",\"tag\":\"(34)\",\"text\":\"D−1 ∑︂ (︁ )︁ E[Vi] ≤ P hit(i) ∧ hit(i + d) . (34) d=1\"},{\"bbox\":[73.44,320.39,428.4,23.86],\"bbox_pdf\":[73.44,447.75,501.84,471.61],\"fragments\":[{\"bbox_pdf\":[73.44,447.75,501.84,471.61],\"words\":[9689,9690,9691,9692,9693,9694,9695,9696,9697,9698,9699,9700,9701,9702,9703,9704,9705,9706,9707,9708,9709,9710,9711,9712,9713]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"paragraph\",\"source_ref\":\"paragraph420\",\"source_text\":\"Let $C\\\\leq\\\\mathord{\\\\ell}(w)$ denote the period of $w$ . For $1\\\\leq d<C$ , the two events $\\\\mathrm{hit}(i)$ and $\\\\mathrm{hit}(i+d)$ cannot occur simultaneously:\",\"tag\":null,\"text\":\"Let C ≤ ℓ(w) denote the period of w. For 1 ≤ d <C, the two events hit(i) and hit(i + d) cannot occur simultaneously:\"},{\"bbox\":[226.65,342.14,121.98,19.75],\"bbox_pdf\":[226.65,430.12,348.63,449.86],\"fragments\":[{\"bbox_pdf\":[226.65,430.12,348.63,449.86],\"words\":[9714,9715,9716,9717,9718,9719,9720,9721,9722,9723]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge\\\\mathrm{hit}(i+d)\\\\bigr{)}=0.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"S4.Ex125\",\"source_text\":\"$\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge\\\\mathrm{hit}(i+d)\\\\bigr{)}=0.$\",\"tag\":null,\"text\":\"(︁ )︁ P hit(i) ∧ hit(i + d) = 0.\"},{\"bbox\":[73.44,368.65,428.4,23.86],\"bbox_pdf\":[73.44,399.49,501.84,423.35],\"fragments\":[{\"bbox_pdf\":[73.44,399.49,501.84,423.35],\"words\":[9724,9725,9726,9727,9728,9729,9730,9731,9732,9733,9734,9735,9736,9737,9738,9739,9740,9741,9742,9743,9744,9745,9746,9747]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":13,\"role\":\"paragraph\",\"source_ref\":\"paragraph422\",\"source_text\":\"For $C\\\\leq d<\\\\mathord{\\\\ell}(w)$ , if the two events $\\\\mathrm{hit}(i)$ and $\\\\mathrm{hit}(i+d)$ occur simultaneously, they prescribe the digits on the union\",\"tag\":null,\"text\":\"For C ≤ d < ℓ(w), if the two events hit(i) and hit(i + d) occur simultaneously, they prescribe the digits on the union\"},{\"bbox\":[128.23,399.24,318.82,10.91],\"bbox_pdf\":[128.23,381.85,447.05,392.76],\"fragments\":[{\"bbox_pdf\":[128.23,381.85,447.05,392.76],\"words\":[9748,9749,9750,9751,9755,9756,9757,9758,9759,9760,9761,9762,9763,9764,9765,9766,9767,9768,9772,9773,9774,9775,9776,9777,9778]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\{i,i+1,\\\\dots,i+\\\\mathord{\\\\ell}(w)-1\\\\}\\\\cup\\\\{i+d,i+d+1,\\\\dots,i+d+\\\\mathord{\\\\ell}(%\\nw)-1\\\\},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"S4.Ex126\",\"source_text\":\"$\\\\{i,i+1,\\\\dots,i+\\\\mathord{\\\\ell}(w)-1\\\\}\\\\cup\\\\{i+d,i+d+1,\\\\dots,i+d+\\\\mathord{\\\\ell}(% w)-1\\\\},$\",\"tag\":null,\"text\":\"{i, i + 1, ,i + ℓ(w) − 1} ∪ {i + d, i + d + 1, ,i + d + ℓ(w) − 1},\"},{\"bbox\":[73.44,416.87,256.12,10.91],\"bbox_pdf\":[73.44,364.22,329.56,375.13],\"fragments\":[{\"bbox_pdf\":[73.44,364.22,329.56,375.13],\"words\":[9779,9780,9781,9782,9783,9784,9785,9786,9787]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":15,\"role\":\"paragraph\",\"source_ref\":\"paragraph424\",\"source_text\":\"which contains $\\\\mathord{\\\\ell}(w)+d$ numbers. Therefore, we have\",\"tag\":null,\"text\":\"which contains ℓ(w) + d numbers. Therefore, we have\"},{\"bbox\":[206.17,426.92,159.41,19.75],\"bbox_pdf\":[206.17,345.34,365.58,365.08],\"fragments\":[{\"bbox_pdf\":[206.17,345.34,365.58,365.08],\"words\":[9788,9789,9790,9791,9792,9793,9794,9795,9796,9797,9798]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge\\\\mathrm{hit}(i+d)\\\\bigr{)}\\\\leq 2^{-(%\\n\\\\mathord{\\\\ell}(w)+d)}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":16,\"role\":\"formula\",\"source_ref\":\"S4.Ex127\",\"source_text\":\"$\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge\\\\mathrm{hit}(i+d)\\\\bigr{)}\\\\leq 2^{-(% \\\\mathord{\\\\ell}(w)+d)}.$\",\"tag\":null,\"text\":\"(︁ )︁ −(ℓ(w)+d) P hit(i) ∧ hit(i + d) ≤ 2\"},{\"bbox\":[73.44,453.43,378.68,10.91],\"bbox_pdf\":[73.44,327.66,452.12,338.57],\"fragments\":[{\"bbox_pdf\":[73.44,327.66,452.12,338.57],\"words\":[9800,9801,9802,9803,9804,9805,9806,9807,9808,9809,9810,9811,9812,9813,9814,9815,9816]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":17,\"role\":\"paragraph\",\"source_ref\":\"paragraph426\",\"source_text\":\"For $\\\\ell(w)\\\\leq d<D$ , the two events involve disjoint sets of $\\\\ell(w)$ digits, and hence\",\"tag\":null,\"text\":\"For ℓ(w) ≤ d < D, the two events involve disjoint sets of ℓ(w) digits, and hence\"},{\"bbox\":[212.82,463.48,146.11,19.75],\"bbox_pdf\":[212.82,308.77,358.93,328.52],\"fragments\":[{\"bbox_pdf\":[212.82,308.77,358.93,328.52],\"words\":[9817,9818,9819,9820,9821,9822,9823,9824,9825,9826,9827]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge\\\\mathrm{hit}(i+d)\\\\bigr{)}=2^{-2\\\\mathord%\\n{\\\\ell}(w)}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":18,\"role\":\"formula\",\"source_ref\":\"S4.Ex128\",\"source_text\":\"$\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge\\\\mathrm{hit}(i+d)\\\\bigr{)}=2^{-2\\\\mathord% {\\\\ell}(w)}.$\",\"tag\":null,\"text\":\"(︁ )︁ −2ℓ(w) P hit(i) ∧ hit(i + d) = 2\"},{\"bbox\":[85.4,490.0,159.18,10.91],\"bbox_pdf\":[85.4,291.1,244.58,302.0],\"fragments\":[{\"bbox_pdf\":[85.4,291.1,244.58,302.0],\"words\":[9829,9830,9831,9832,9833]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":19,\"role\":\"paragraph\",\"source_ref\":\"paragraph428\",\"source_text\":\"Consequently, from (34), we have\",\"tag\":null,\"text\":\"Consequently, from (34), we have\"},{\"bbox\":[137.4,508.41,299.29,66.1],\"bbox_pdf\":[137.4,217.5,436.69,283.59],\"fragments\":[{\"bbox_pdf\":[137.4,217.5,436.69,283.59],\"words\":[9834,9835,9836,9837,9838,9839,9840,9841,9842,9843,9844,9845,9846,9847,9848,9849,9850,9851,9852,9853,9854,9855,9856,9857,9858,9859,9860,9861,9862,9863,9864,9865,9866,9867,9868,9880]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\mathbb{E}[V_{i}] \\\\displaystyle\\\\leq\\\\sum_{d=C}^{\\\\mathord{\\\\ell}(w)-1}2^{-(\\\\mathord{\\\\ell}(w)+d)}+%\\n\\\\sum_{d=\\\\mathord{\\\\ell}(w)}^{D-1}2^{-2\\\\mathord{\\\\ell}(w)}<\\\\frac{2}{2^{\\\\mathord{%\\n\\\\ell}(w)+C}}+\\\\frac{D-\\\\mathord{\\\\ell}(w)}{4^{\\\\mathord{\\\\ell}(w)}} \\\\\\\\ \\\\displaystyle=\\\\frac{2}{2^{\\\\mathord{\\\\ell}(w)+C}}+\\\\frac{t}{4^{\\\\mathord{\\\\ell}(w)}%\\n}\\\\leq\\\\frac{t+2}{2^{\\\\mathord{\\\\ell}(w)+C}}. \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":20,\"role\":\"formula\",\"source_ref\":\"S4.Ex129\",\"source_text\":\"$\\\\displaystyle\\\\mathbb{E}[V_{i}]$ $\\\\displaystyle\\\\leq\\\\sum_{d=C}^{\\\\mathord{\\\\ell}(w)-1}2^{-(\\\\mathord{\\\\ell}(w)+d)}+% \\\\sum_{d=\\\\mathord{\\\\ell}(w)}^{D-1}2^{-2\\\\mathord{\\\\ell}(w)}<\\\\frac{2}{2^{\\\\mathord{% \\\\ell}(w)+C}}+\\\\frac{D-\\\\mathord{\\\\ell}(w)}{4^{\\\\mathord{\\\\ell}(w)}}$ $\\\\displaystyle=\\\\frac{2}{2^{\\\\mathord{\\\\ell}(w)+C}}+\\\\frac{t}{4^{\\\\mathord{\\\\ell}(w)}% }\\\\leq\\\\frac{t+2}{2^{\\\\mathord{\\\\ell}(w)+C}}.$\",\"tag\":null,\"text\":\"ℓ(w)−1 ∑︂ D−1∑︂ 2 D − ℓ(w) E[Vi] ≤ 2 −(ℓ(w)+d) + 2 −2ℓ(w) < + 2ℓ(w)+C 4ℓ(w) d=C d=ℓ(w) 2 t t + 2 = + ≤ . ℓ(w)+C ℓ(w) ℓ(w)+C 2 4 2 ℓ(w).\"},{\"bbox\":[73.44,577.45,235.89,10.91],\"bbox_pdf\":[73.44,203.64,309.33,214.55],\"fragments\":[{\"bbox_pdf\":[73.44,203.64,309.33,214.55],\"words\":[9869,9870,9871,9872,9873,9874,9875,9876,9877,9878,9879]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":21,\"role\":\"paragraph\",\"source_ref\":\"paragraph430\",\"source_text\":\"The first sum is understood to be zero when $C=\\\\mathord{\\\\ell}(w)$ .\",\"tag\":null,\"text\":\"The first sum is understood to be zero when C =\"},{\"bbox\":[85.4,590.41,139.18,10.91],\"bbox_pdf\":[85.4,190.69,224.58,201.59],\"fragments\":[{\"bbox_pdf\":[85.4,190.69,224.58,201.59],\"words\":[9881,9882,9883,9884,9885,9886]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":22,\"role\":\"paragraph\",\"source_ref\":\"paragraph431\",\"source_text\":\"It now follows from (33) that\",\"tag\":null,\"text\":\"It now follows from (33) that\"},{\"bbox\":[200.47,607.98,301.37,34.69],\"bbox_pdf\":[200.47,149.33,501.84,184.02],\"fragments\":[{\"bbox_pdf\":[200.47,149.33,501.84,184.02],\"words\":[9887,9888,9889,9890,9891,9892,9893,9894,9895,9896,9897,9898,9899,9900,9901,9902,9903]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathbb{E}[M_{w}-M_{w}^{\\\\prime}]\\\\leq\\\\sum_{i=u}^{v-1}\\\\mathbb{E}[V_{i}]\\\\leq\\\\frac%\\n{(t+2)n}{2^{\\\\mathord{\\\\ell}(w)+C}}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":23,\"role\":\"formula\",\"source_ref\":\"S4.E35\",\"source_text\":\"$\\\\mathbb{E}[M_{w}-M_{w}^{\\\\prime}]\\\\leq\\\\sum_{i=u}^{v-1}\\\\mathbb{E}[V_{i}]\\\\leq\\\\frac% {(t+2)n}{2^{\\\\mathord{\\\\ell}(w)+C}}.$ (35)\",\"tag\":\"(35)\",\"text\":\"∑︂ v−1 (t + 2)n E[Mw − M ′ E[Vi] ≤ . (35) w] ≤ 2ℓ(w)+C i=u\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.24575\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.24575v2-p0022.pdf\",\"page_no\":23,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"b2d46808ff2e296dea9c8edc44ee5a098a7d8aa0\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"9da159a0600a0a7ff1719755312c6159bf47af137eaa2cb5edfec454220b158b\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d.pdf","asset_sha256":"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d","document_id":"arxiv:2608.24575","kind":"page","page_number":23,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
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{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[147.28,175.87,354.56,31.2],\"bbox_pdf\":[147.28,584.93,501.84,616.13],\"fragments\":[{\"bbox_pdf\":[147.28,584.93,501.84,616.13],\"words\":[9631,9632,9633,9634,9635,9636,9637,9638,9642,9643,9644,9645,9646,9647,9648,9649,9650,9651,9652,9653]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"V_{i}:=\\\\begin{cases}1&\\\\mathrm{hit}(i)\\\\wedge(\\\\exists d\\\\in\\\\{1,2,\\\\dots,D-1\\\\}\\\\ \\\\ %\\n\\\\text{s.t.}\\\\ \\\\ \\\\mathrm{hit}(i+d)),\\\\\\\\\\n0&(\\\\text{otherwise}).\\\\end{cases}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S4.E32\",\"source_text\":\"$V_{i}:=\\\\begin{cases}1&\\\\mathrm{hit}(i)\\\\wedge(\\\\exists d\\\\in\\\\{1,2,\\\\dots,D-1\\\\}\\\\ \\\\ % \\\\text{s.t.}\\\\ \\\\ \\\\mathrm{hit}(i+d)),\\\\\\\\ 0&(\\\\text{otherwise}).\\\\end{cases}$ (32)\",\"tag\":\"(32)\",\"text\":\"{︄ 1 hit(i) ∧ (∃d ∈ {1, 2, ,D − 1} s.t. hit(i + d)), Vi := (32) 0 (otherwise).\"},\"formula_index\":6,\"parent_page_sha256\":\"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d.pdf","asset_sha256":"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d","document_id":"arxiv:2608.24575","kind":"formula","page_number":23,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[241.15,233.84,260.69,34.69],\"bbox_pdf\":[241.15,523.46,501.84,558.16],\"fragments\":[{\"bbox_pdf\":[241.15,523.46,501.84,558.16],\"words\":[9659,9660,9661,9662,9663,9664,9665,9666,9667,9668,9669]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"M_{w}-M_{w}^{\\\\prime}\\\\leq\\\\sum_{i=u}^{v-1}V_{i}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S4.E33\",\"source_text\":\"$M_{w}-M_{w}^{\\\\prime}\\\\leq\\\\sum_{i=u}^{v-1}V_{i}.$ (33)\",\"tag\":\"(33)\",\"text\":\"∑︂ v−1 Mw − M ′ Vi. (33) w ≤ i=u\"},\"formula_index\":8,\"parent_page_sha256\":\"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d.pdf","asset_sha256":"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d","document_id":"arxiv:2608.24575","kind":"formula","page_number":23,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[207.93,283.66,293.91,34.96],\"bbox_pdf\":[207.93,473.38,501.84,508.34],\"fragments\":[{\"bbox_pdf\":[207.93,473.38,501.84,508.34],\"words\":[9674,9675,9676,9677,9678,9679,9680,9681,9682,9683,9684,9685,9686,9687,9688]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathbb{E}[V_{i}]\\\\leq\\\\sum_{d=1}^{D-1}\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge%\\n\\\\mathrm{hit}(i+d)\\\\bigr{)}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"S4.E34\",\"source_text\":\"$\\\\mathbb{E}[V_{i}]\\\\leq\\\\sum_{d=1}^{D-1}\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge% \\\\mathrm{hit}(i+d)\\\\bigr{)}.$ (34)\",\"tag\":\"(34)\",\"text\":\"D−1 ∑︂ (︁ )︁ E[Vi] ≤ P hit(i) ∧ hit(i + d) . (34) d=1\"},\"formula_index\":10,\"parent_page_sha256\":\"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d.pdf","asset_sha256":"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d","document_id":"arxiv:2608.24575","kind":"formula","page_number":23,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
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{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[206.17,426.92,159.41,19.75],\"bbox_pdf\":[206.17,345.34,365.58,365.08],\"fragments\":[{\"bbox_pdf\":[206.17,345.34,365.58,365.08],\"words\":[9788,9789,9790,9791,9792,9793,9794,9795,9796,9797,9798]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge\\\\mathrm{hit}(i+d)\\\\bigr{)}\\\\leq 2^{-(%\\n\\\\mathord{\\\\ell}(w)+d)}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":16,\"role\":\"formula\",\"source_ref\":\"S4.Ex127\",\"source_text\":\"$\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge\\\\mathrm{hit}(i+d)\\\\bigr{)}\\\\leq 2^{-(% \\\\mathord{\\\\ell}(w)+d)}.$\",\"tag\":null,\"text\":\"(︁ )︁ −(ℓ(w)+d) P hit(i) ∧ hit(i + d) ≤ 2\"},\"formula_index\":16,\"parent_page_sha256\":\"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d.pdf","asset_sha256":"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d","document_id":"arxiv:2608.24575","kind":"formula","page_number":23,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[212.82,463.48,146.11,19.75],\"bbox_pdf\":[212.82,308.77,358.93,328.52],\"fragments\":[{\"bbox_pdf\":[212.82,308.77,358.93,328.52],\"words\":[9817,9818,9819,9820,9821,9822,9823,9824,9825,9826,9827]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge\\\\mathrm{hit}(i+d)\\\\bigr{)}=2^{-2\\\\mathord%\\n{\\\\ell}(w)}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":18,\"role\":\"formula\",\"source_ref\":\"S4.Ex128\",\"source_text\":\"$\\\\mathbb{P}\\\\bigl{(}\\\\mathrm{hit}(i)\\\\wedge\\\\mathrm{hit}(i+d)\\\\bigr{)}=2^{-2\\\\mathord% {\\\\ell}(w)}.$\",\"tag\":null,\"text\":\"(︁ )︁ −2ℓ(w) P hit(i) ∧ hit(i + d) = 2\"},\"formula_index\":18,\"parent_page_sha256\":\"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d.pdf","asset_sha256":"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d","document_id":"arxiv:2608.24575","kind":"formula","page_number":23,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[137.4,508.41,299.29,66.1],\"bbox_pdf\":[137.4,217.5,436.69,283.59],\"fragments\":[{\"bbox_pdf\":[137.4,217.5,436.69,283.59],\"words\":[9834,9835,9836,9837,9838,9839,9840,9841,9842,9843,9844,9845,9846,9847,9848,9849,9850,9851,9852,9853,9854,9855,9856,9857,9858,9859,9860,9861,9862,9863,9864,9865,9866,9867,9868,9880]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\mathbb{E}[V_{i}] \\\\displaystyle\\\\leq\\\\sum_{d=C}^{\\\\mathord{\\\\ell}(w)-1}2^{-(\\\\mathord{\\\\ell}(w)+d)}+%\\n\\\\sum_{d=\\\\mathord{\\\\ell}(w)}^{D-1}2^{-2\\\\mathord{\\\\ell}(w)}<\\\\frac{2}{2^{\\\\mathord{%\\n\\\\ell}(w)+C}}+\\\\frac{D-\\\\mathord{\\\\ell}(w)}{4^{\\\\mathord{\\\\ell}(w)}} \\\\\\\\ \\\\displaystyle=\\\\frac{2}{2^{\\\\mathord{\\\\ell}(w)+C}}+\\\\frac{t}{4^{\\\\mathord{\\\\ell}(w)}%\\n}\\\\leq\\\\frac{t+2}{2^{\\\\mathord{\\\\ell}(w)+C}}. \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":20,\"role\":\"formula\",\"source_ref\":\"S4.Ex129\",\"source_text\":\"$\\\\displaystyle\\\\mathbb{E}[V_{i}]$ $\\\\displaystyle\\\\leq\\\\sum_{d=C}^{\\\\mathord{\\\\ell}(w)-1}2^{-(\\\\mathord{\\\\ell}(w)+d)}+% \\\\sum_{d=\\\\mathord{\\\\ell}(w)}^{D-1}2^{-2\\\\mathord{\\\\ell}(w)}<\\\\frac{2}{2^{\\\\mathord{% \\\\ell}(w)+C}}+\\\\frac{D-\\\\mathord{\\\\ell}(w)}{4^{\\\\mathord{\\\\ell}(w)}}$ $\\\\displaystyle=\\\\frac{2}{2^{\\\\mathord{\\\\ell}(w)+C}}+\\\\frac{t}{4^{\\\\mathord{\\\\ell}(w)}% }\\\\leq\\\\frac{t+2}{2^{\\\\mathord{\\\\ell}(w)+C}}.$\",\"tag\":null,\"text\":\"ℓ(w)−1 ∑︂ D−1∑︂ 2 D − ℓ(w) E[Vi] ≤ 2 −(ℓ(w)+d) + 2 −2ℓ(w) < + 2ℓ(w)+C 4ℓ(w) d=C d=ℓ(w) 2 t t + 2 = + ≤ . ℓ(w)+C ℓ(w) ℓ(w)+C 2 4 2 ℓ(w).\"},\"formula_index\":20,\"parent_page_sha256\":\"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d.pdf","asset_sha256":"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d","document_id":"arxiv:2608.24575","kind":"formula","page_number":23,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[200.47,607.98,301.37,34.69],\"bbox_pdf\":[200.47,149.33,501.84,184.02],\"fragments\":[{\"bbox_pdf\":[200.47,149.33,501.84,184.02],\"words\":[9887,9888,9889,9890,9891,9892,9893,9894,9895,9896,9897,9898,9899,9900,9901,9902,9903]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathbb{E}[M_{w}-M_{w}^{\\\\prime}]\\\\leq\\\\sum_{i=u}^{v-1}\\\\mathbb{E}[V_{i}]\\\\leq\\\\frac%\\n{(t+2)n}{2^{\\\\mathord{\\\\ell}(w)+C}}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":23,\"role\":\"formula\",\"source_ref\":\"S4.E35\",\"source_text\":\"$\\\\mathbb{E}[M_{w}-M_{w}^{\\\\prime}]\\\\leq\\\\sum_{i=u}^{v-1}\\\\mathbb{E}[V_{i}]\\\\leq\\\\frac% {(t+2)n}{2^{\\\\mathord{\\\\ell}(w)+C}}.$ (35)\",\"tag\":\"(35)\",\"text\":\"∑︂ v−1 (t + 2)n E[Mw − M ′ E[Vi] ≤ . (35) w] ≤ 2ℓ(w)+C i=u\"},\"formula_index\":23,\"parent_page_sha256\":\"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d.pdf","asset_sha256":"6becf5ba75dc941e412336b6c69bc22aa0e708970f7ea5297226c9c86fe6786d","document_id":"arxiv:2608.24575","kind":"formula","page_number":23,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2},\"blocks\":[{\"bbox\":[110.16,73.12,8.47,7.97],\"bbox_pdf\":[110.16,710.91,118.63,718.88],\"fragments\":[{\"bbox_pdf\":[110.16,710.91,118.63,718.88],\"words\":[10623]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":0,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"26\"},{\"bbox\":[189.01,73.12,270.71,7.97],\"bbox_pdf\":[189.01,710.91,459.71,718.88],\"fragments\":[{\"bbox_pdf\":[189.01,710.91,459.71,718.88],\"words\":[10624,10625,10626,10627,10628,10629,10630]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":1,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"SHINSUKE IWAO, FUMIHIKO NAKAMURA, AND YUSHI NAKANO\"},{\"bbox\":[110.16,94.09,128.18,10.91],\"bbox_pdf\":[110.16,687.0,238.34,697.91],\"fragments\":[{\"bbox_pdf\":[110.16,687.0,238.34,697.91],\"words\":[10631,10632,10633,10634,10635]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph471\",\"source_text\":\"Then, we have the identity\",\"tag\":null,\"text\":\"Then, we have the identity\"},{\"bbox\":[288.06,101.69,250.5,34.54],\"bbox_pdf\":[288.06,655.77,538.56,690.31],\"fragments\":[{\"bbox_pdf\":[288.06,655.77,538.56,690.31],\"words\":[10636,10637,10639,10640,10641,10642,10643,10644]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\lambda_{\\\\succ,1}^{(N)}=\\\\sum_{w\\\\in\\\\mathcal{W}}L^{\\\\ast}_{w}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S4.E37\",\"source_text\":\"$\\\\lambda_{\\\\succ,1}^{(N)}=\\\\sum_{w\\\\in\\\\mathcal{W}}L^{\\\\ast}_{w}$ (37)\",\"tag\":\"(37)\",\"text\":\"∑︂ (N ∗ λ≻,1 = Lw (37) w∈W\"},{\"bbox\":[122.12,141.48,357.28,10.91],\"bbox_pdf\":[122.12,639.61,479.39,650.52],\"fragments\":[{\"bbox_pdf\":[122.12,639.61,479.39,650.52],\"words\":[10645,10646,10647,10648,10649,10650,10651,10652,10653,10654,10655,10656,10657,10658]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph473\",\"source_text\":\"Under the event on which all of (i)–(iii) in Proposition 4.16 occur, we have\",\"tag\":null,\"text\":\"Under the event on which all of (i)–(iii) in Proposition 4.16 occur, we have\"},{\"bbox\":[115.68,154.22,417.36,121.39],\"bbox_pdf\":[115.68,516.4,533.04,637.78],\"fragments\":[{\"bbox_pdf\":[115.68,516.4,533.04,637.78],\"words\":[10659,10660,10662,10663,10664,10665,10666,10667,10668,10669,10670,10671,10672,10673,10674,10675,10676,10677,10679,10680,10681,10682,10683,10684,10685,10686,10687,10688,10689,10690,10691,10692,10693,10694,10695,10696,10697,10698,10699,10700,10701,10702,10703,10704,10705,10706,10707,10708,10709,10710,10711,10712,10713,10714,10715,10716,10717,10718,10719,10720,10721,10722,10723,10724,10725,10726,10727,10728,10729,10730,10731,10733,10734]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\lambda_{\\\\succ,1}^{(N)} \\\\displaystyle=\\\\sum_{w\\\\in\\\\mathcal{W}}L^{\\\\ast}_{w} \\\\\\\\ \\\\displaystyle\\\\leq\\\\sum_{w\\\\in\\\\mathcal{W}}(W^{\\\\ast}_{w}+M^{\\\\ast}_{w}-{M^{\\\\prime}_%\\n{w}}^{\\\\ast})\\\\qquad(\\\\text{Lemma \\\\ref{lemma:combin_W_L}}) \\\\\\\\ \\\\displaystyle\\\\leq\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\left((2+\\\\epsilon)\\\\sqrt{{M_{w}^{\\\\prime}%\\n}^{\\\\ast}}+N^{1/10}+(1+\\\\delta)\\\\frac{(t+2)\\\\#J^{\\\\ast}_{w}}{4^{P}}+g\\\\right)\\\\qquad(%\\n\\\\text{Prop.~{}\\\\ref{prop:ideal} (i),(iii)}) \\\\\\\\ \\\\displaystyle=(2+\\\\epsilon)\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\sqrt{{M_{w}^{\\\\prime}}^{\\\\ast}}%\\n+(1+\\\\delta)\\\\frac{(t+2)N}{4^{P}}+R(g+N^{1/10}). \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"Ax1.EGx3\",\"source_text\":\"$\\\\displaystyle\\\\lambda_{\\\\succ,1}^{(N)}$ $\\\\displaystyle=\\\\sum_{w\\\\in\\\\mathcal{W}}L^{\\\\ast}_{w}$ $\\\\displaystyle\\\\leq\\\\sum_{w\\\\in\\\\mathcal{W}}(W^{\\\\ast}_{w}+M^{\\\\ast}_{w}-{M^{\\\\prime}_% {w}}^{\\\\ast})\\\\qquad(\\\\text{Lemma \\\\ref{lemma:combin_W_L}})$ $\\\\displaystyle\\\\leq\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\left((2+\\\\epsilon)\\\\sqrt{{M_{w}^{\\\\prime}% }^{\\\\ast}}+N^{1/10}+(1+\\\\delta)\\\\frac{(t+2)\\\\#J^{\\\\ast}_{w}}{4^{P}}+g\\\\right)\\\\qquad(% \\\\text{Prop.~{}\\\\ref{prop:ideal} (i),(iii)})$ $\\\\displaystyle=(2+\\\\epsilon)\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\sqrt{{M_{w}^{\\\\prime}}^{\\\\ast}}% +(1+\\\\delta)\\\\frac{(t+2)N}{4^{P}}+R(g+N^{1/10}).$\",\"tag\":null,\"text\":\"∑︂ (N ∗ λ≻,1 = Lw w∈W ∑︂ ∗ ∗ ′ ∗ ≤ (Ww + Mw − Mw (Lemma 4.9) w∈W (︃ )︃ ∑︂ √︁ ∗ ∗ 1/10 (t + 2)#Jw ≤ (2 + ϵ) M′ + N + (1 + δ) + g (Prop. 4.16 (i),(iii)) w 4P w∈W ∑︂ √︁ (t + 2)N ′ ∗ 1/10 = (2 + ϵ) Mw + (1 + δ) + R(g + N 4P w∈W\"},{\"bbox\":[122.12,295.78,244.91,10.91],\"bbox_pdf\":[122.12,485.31,367.02,496.22],\"fragments\":[{\"bbox_pdf\":[122.12,485.31,367.02,496.22],\"words\":[10735,10736,10737,10738,10739,10740,10741]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph475\",\"source_text\":\"Here, by the Cauchy–Schwarz inequality, we obtain\",\"tag\":null,\"text\":\"Here, by the Cauchy–Schwarz inequality, we obtain\"},{\"bbox\":[176.87,335.4,294.98,152.41],\"bbox_pdf\":[176.87,304.19,471.85,456.6],\"fragments\":[{\"bbox_pdf\":[176.87,304.19,471.85,456.6],\"words\":[10748,10749,10750,10751,10752,10753,10754,10755,10756,10757,10758,10759,10760,10761,10762,10763,10764,10765,10766,10767,10768,10769,10770,10771,10772,10773,10774,10775,10776,10777,10778,10779,10780,10781,10782,10783,10784,10785,10786,10787,10789,10790,10791,10792,10793,10794,10795,10796,10797,10798,10799,10800,10801,10802,10803]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\sqrt{{M_{w}^{\\\\prime}}^{\\\\ast}} \\\\displaystyle\\\\leq\\\\sqrt{\\\\left(\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\frac{1}{2^{\\\\mathord{\\\\ell}(%\\nw)}}\\\\right)\\\\left(\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\ell(w)}{M_{w}^{\\\\prime}}^{\\\\ast}%\\n\\\\right)} \\\\\\\\ \\\\displaystyle=\\\\sqrt{\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\mathord{\\\\ell}(w)}{M_{w}^{\\\\prime}%\\n}^{\\\\ast}}\\\\qquad(\\\\text{Lemma \\\\ref{lemma:2_minusL}}) \\\\\\\\ \\\\displaystyle\\\\leq\\\\sqrt{\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\mathord{\\\\ell}(w)}M^{\\\\ast}_{w}} \\\\\\\\ \\\\displaystyle\\\\leq\\\\sqrt{(1+\\\\delta)N+(\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\mathord{\\\\ell}(w)%\\n})f}\\\\qquad(\\\\text{Prop.~{}\\\\ref{prop:ideal} (ii)}) \\\\\\\\ \\\\displaystyle\\\\leq\\\\sqrt{(1+\\\\delta)N+2^{P+Q+3}f}\\\\qquad(\\\\text{Lemma \\\\ref{lemma:%\\nsum_of_powers_of_2}}). \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"Ax1.EGx4\",\"source_text\":\"$\\\\displaystyle\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\sqrt{{M_{w}^{\\\\prime}}^{\\\\ast}}$ $\\\\displaystyle\\\\leq\\\\sqrt{\\\\left(\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\frac{1}{2^{\\\\mathord{\\\\ell}(% w)}}\\\\right)\\\\left(\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\ell(w)}{M_{w}^{\\\\prime}}^{\\\\ast}% \\\\right)}$ $\\\\displaystyle=\\\\sqrt{\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\mathord{\\\\ell}(w)}{M_{w}^{\\\\prime}% }^{\\\\ast}}\\\\qquad(\\\\text{Lemma \\\\ref{lemma:2_minusL}})$ $\\\\displaystyle\\\\leq\\\\sqrt{\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\mathord{\\\\ell}(w)}M^{\\\\ast}_{w}}$ $\\\\displaystyle\\\\leq\\\\sqrt{(1+\\\\delta)N+(\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\mathord{\\\\ell}(w)% })f}\\\\qquad(\\\\text{Prop.~{}\\\\ref{prop:ideal} (ii)})$ $\\\\displaystyle\\\\leq\\\\sqrt{(1+\\\\delta)N+2^{P+Q+3}f}\\\\qquad(\\\\text{Lemma \\\\ref{lemma:% sum_of_powers_of_2}}).$\",\"tag\":null,\"text\":\"∑︂ √︁ ⃓ ∑︂ 1 ∑︂ M′ ∗ ≤ ⎷ 2ℓ(w)M′ ∗ w 2ℓ(w) w w∈W w∈W w∈W √︄ ∑︂ = 2ℓ(w)M′ ∗ (Lemma 4.8) w w∈W √︄ ∑︂ ≤ 2ℓ(w)M∗ w w∈W √︄ ∑︂ ≤ (1 + δ)N + 2ℓ(w))f (Prop. 4.16 (ii)) w∈W √︂ ≤ (1 + δ)N + 2P +Q+3f (Lemma 4.7).\"},{\"bbox\":[110.16,498.98,204.3,10.91],\"bbox_pdf\":[110.16,282.11,314.46,293.02],\"fragments\":[{\"bbox_pdf\":[110.16,282.11,314.46,293.02],\"words\":[10804,10805,10806,10807,10808,10809,10810]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph477\",\"source_text\":\"From these two inequalities, it follows that\",\"tag\":null,\"text\":\"From these two inequalities, it follows that\"},{\"bbox\":[300.6,518.67,47.52,16.84],\"bbox_pdf\":[300.6,256.49,348.12,273.33],\"fragments\":[{\"bbox_pdf\":[300.6,256.49,348.12,273.33],\"words\":[10811,10813,10814,10815]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\lambda_{\\\\succ,1}^{(N)}\\\\leq\\\\mathcal{F},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S4.Ex153\",\"source_text\":\"$\\\\lambda_{\\\\succ,1}^{(N)}\\\\leq\\\\mathcal{F},$\",\"tag\":null,\"text\":\"(N λ≻,1 ≤ F,\"},{\"bbox\":[110.16,542.38,27.91,10.91],\"bbox_pdf\":[110.16,238.71,138.07,249.62],\"fragments\":[{\"bbox_pdf\":[110.16,238.71,138.07,249.62],\"words\":[10816]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"paragraph\",\"source_ref\":\"paragraph479\",\"source_text\":\"where\",\"tag\":null,\"text\":\"where\"},{\"bbox\":[155.87,553.25,382.69,31.73],\"bbox_pdf\":[155.87,207.02,538.56,238.75],\"fragments\":[{\"bbox_pdf\":[155.87,207.02,538.56,238.75],\"words\":[10817,10818,10819,10820,10821,10822,10823,10824,10825,10826,10827,10828,10829,10830,10831,10832,10833,10834,10835,10836,10837,10838,10839,10840,10841,10843,10844]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathcal{F}:=(2+\\\\epsilon)\\\\sqrt{(1+\\\\delta)N+2^{P+Q+3}f}+(1+\\\\delta)\\\\frac{(t+2)N}%\\n{4^{P}}+R(g+N^{1/10}).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"formula\",\"source_ref\":\"S4.E38\",\"source_text\":\"$\\\\mathcal{F}:=(2+\\\\epsilon)\\\\sqrt{(1+\\\\delta)N+2^{P+Q+3}f}+(1+\\\\delta)\\\\frac{(t+2)N}% {4^{P}}+R(g+N^{1/10}).$ (38)\",\"tag\":\"(38)\",\"text\":\"√︂ (t + 2)N 1/10 P +Q+3f + (1 + δ) F := (2 + ϵ) (1 + δ)N + 2 + R(g + N (38) 4P\"},{\"bbox\":[122.12,590.29,248.09,10.91],\"bbox_pdf\":[122.12,190.8,370.2,201.71],\"fragments\":[{\"bbox_pdf\":[122.12,190.8,370.2,201.71],\"words\":[10845,10846,10847,10848,10849,10850,10851,10852]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"paragraph\",\"source_ref\":\"paragraph481\",\"source_text\":\"In other words, we obtain the following proposition.\",\"tag\":null,\"text\":\"In other words, we obtain the following proposition.\"},{\"bbox\":[110.16,611.5,264.36,10.91],\"bbox_pdf\":[110.16,169.59,374.53,180.5],\"fragments\":[{\"bbox_pdf\":[110.16,169.59,374.53,180.5],\"words\":[10853,10854,10855,10856,10857,10858,10859,10860]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":13,\"role\":\"paragraph\",\"source_ref\":\"paragraph482\",\"source_text\":\"Proposition 4.17. Under the inequality (36), we have\",\"tag\":null,\"text\":\"Proposition 4.17. Under the inequality (36), we have\"},{\"bbox\":[277.91,623.31,92.89,26.14],\"bbox_pdf\":[277.91,142.54,370.8,168.69],\"fragments\":[{\"bbox_pdf\":[277.91,142.54,370.8,168.69],\"words\":[10861,10862,10863,10865,10866,10867,10868,10869,10870]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathbb{P}\\\\left(\\\\lambda_{\\\\succ,1}^{(N)}>\\\\mathcal{F}\\\\right)<\\\\mathcal{H}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"S4.Ex154\",\"source_text\":\"$\\\\mathbb{P}\\\\left(\\\\lambda_{\\\\succ,1}^{(N)}>\\\\mathcal{F}\\\\right)<\\\\mathcal{H}.$\",\"tag\":null,\"text\":\"(︂ )︂ (N P λ≻,1 > F < H.\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.24575\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.24575v2-p0025.pdf\",\"page_no\":26,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"b2d46808ff2e296dea9c8edc44ee5a098a7d8aa0\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"9da159a0600a0a7ff1719755312c6159bf47af137eaa2cb5edfec454220b158b\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed.pdf","asset_sha256":"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed","document_id":"arxiv:2608.24575","kind":"page","page_number":26,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[288.06,101.69,250.5,34.54],\"bbox_pdf\":[288.06,655.77,538.56,690.31],\"fragments\":[{\"bbox_pdf\":[288.06,655.77,538.56,690.31],\"words\":[10636,10637,10639,10640,10641,10642,10643,10644]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\lambda_{\\\\succ,1}^{(N)}=\\\\sum_{w\\\\in\\\\mathcal{W}}L^{\\\\ast}_{w}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S4.E37\",\"source_text\":\"$\\\\lambda_{\\\\succ,1}^{(N)}=\\\\sum_{w\\\\in\\\\mathcal{W}}L^{\\\\ast}_{w}$ (37)\",\"tag\":\"(37)\",\"text\":\"∑︂ (N ∗ λ≻,1 = Lw (37) w∈W\"},\"formula_index\":3,\"parent_page_sha256\":\"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed.pdf","asset_sha256":"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed","document_id":"arxiv:2608.24575","kind":"formula","page_number":26,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[115.68,154.22,417.36,121.39],\"bbox_pdf\":[115.68,516.4,533.04,637.78],\"fragments\":[{\"bbox_pdf\":[115.68,516.4,533.04,637.78],\"words\":[10659,10660,10662,10663,10664,10665,10666,10667,10668,10669,10670,10671,10672,10673,10674,10675,10676,10677,10679,10680,10681,10682,10683,10684,10685,10686,10687,10688,10689,10690,10691,10692,10693,10694,10695,10696,10697,10698,10699,10700,10701,10702,10703,10704,10705,10706,10707,10708,10709,10710,10711,10712,10713,10714,10715,10716,10717,10718,10719,10720,10721,10722,10723,10724,10725,10726,10727,10728,10729,10730,10731,10733,10734]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\lambda_{\\\\succ,1}^{(N)} \\\\displaystyle=\\\\sum_{w\\\\in\\\\mathcal{W}}L^{\\\\ast}_{w} \\\\\\\\ \\\\displaystyle\\\\leq\\\\sum_{w\\\\in\\\\mathcal{W}}(W^{\\\\ast}_{w}+M^{\\\\ast}_{w}-{M^{\\\\prime}_%\\n{w}}^{\\\\ast})\\\\qquad(\\\\text{Lemma \\\\ref{lemma:combin_W_L}}) \\\\\\\\ \\\\displaystyle\\\\leq\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\left((2+\\\\epsilon)\\\\sqrt{{M_{w}^{\\\\prime}%\\n}^{\\\\ast}}+N^{1/10}+(1+\\\\delta)\\\\frac{(t+2)\\\\#J^{\\\\ast}_{w}}{4^{P}}+g\\\\right)\\\\qquad(%\\n\\\\text{Prop.~{}\\\\ref{prop:ideal} (i),(iii)}) \\\\\\\\ \\\\displaystyle=(2+\\\\epsilon)\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\sqrt{{M_{w}^{\\\\prime}}^{\\\\ast}}%\\n+(1+\\\\delta)\\\\frac{(t+2)N}{4^{P}}+R(g+N^{1/10}). \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"Ax1.EGx3\",\"source_text\":\"$\\\\displaystyle\\\\lambda_{\\\\succ,1}^{(N)}$ $\\\\displaystyle=\\\\sum_{w\\\\in\\\\mathcal{W}}L^{\\\\ast}_{w}$ $\\\\displaystyle\\\\leq\\\\sum_{w\\\\in\\\\mathcal{W}}(W^{\\\\ast}_{w}+M^{\\\\ast}_{w}-{M^{\\\\prime}_% {w}}^{\\\\ast})\\\\qquad(\\\\text{Lemma \\\\ref{lemma:combin_W_L}})$ $\\\\displaystyle\\\\leq\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\left((2+\\\\epsilon)\\\\sqrt{{M_{w}^{\\\\prime}% }^{\\\\ast}}+N^{1/10}+(1+\\\\delta)\\\\frac{(t+2)\\\\#J^{\\\\ast}_{w}}{4^{P}}+g\\\\right)\\\\qquad(% \\\\text{Prop.~{}\\\\ref{prop:ideal} (i),(iii)})$ $\\\\displaystyle=(2+\\\\epsilon)\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\sqrt{{M_{w}^{\\\\prime}}^{\\\\ast}}% +(1+\\\\delta)\\\\frac{(t+2)N}{4^{P}}+R(g+N^{1/10}).$\",\"tag\":null,\"text\":\"∑︂ (N ∗ λ≻,1 = Lw w∈W ∑︂ ∗ ∗ ′ ∗ ≤ (Ww + Mw − Mw (Lemma 4.9) w∈W (︃ )︃ ∑︂ √︁ ∗ ∗ 1/10 (t + 2)#Jw ≤ (2 + ϵ) M′ + N + (1 + δ) + g (Prop. 4.16 (i),(iii)) w 4P w∈W ∑︂ √︁ (t + 2)N ′ ∗ 1/10 = (2 + ϵ) Mw + (1 + δ) + R(g + N 4P w∈W\"},\"formula_index\":5,\"parent_page_sha256\":\"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed.pdf","asset_sha256":"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed","document_id":"arxiv:2608.24575","kind":"formula","page_number":26,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[176.87,335.4,294.98,152.41],\"bbox_pdf\":[176.87,304.19,471.85,456.6],\"fragments\":[{\"bbox_pdf\":[176.87,304.19,471.85,456.6],\"words\":[10748,10749,10750,10751,10752,10753,10754,10755,10756,10757,10758,10759,10760,10761,10762,10763,10764,10765,10766,10767,10768,10769,10770,10771,10772,10773,10774,10775,10776,10777,10778,10779,10780,10781,10782,10783,10784,10785,10786,10787,10789,10790,10791,10792,10793,10794,10795,10796,10797,10798,10799,10800,10801,10802,10803]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\sqrt{{M_{w}^{\\\\prime}}^{\\\\ast}} \\\\displaystyle\\\\leq\\\\sqrt{\\\\left(\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\frac{1}{2^{\\\\mathord{\\\\ell}(%\\nw)}}\\\\right)\\\\left(\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\ell(w)}{M_{w}^{\\\\prime}}^{\\\\ast}%\\n\\\\right)} \\\\\\\\ \\\\displaystyle=\\\\sqrt{\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\mathord{\\\\ell}(w)}{M_{w}^{\\\\prime}%\\n}^{\\\\ast}}\\\\qquad(\\\\text{Lemma \\\\ref{lemma:2_minusL}}) \\\\\\\\ \\\\displaystyle\\\\leq\\\\sqrt{\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\mathord{\\\\ell}(w)}M^{\\\\ast}_{w}} \\\\\\\\ \\\\displaystyle\\\\leq\\\\sqrt{(1+\\\\delta)N+(\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\mathord{\\\\ell}(w)%\\n})f}\\\\qquad(\\\\text{Prop.~{}\\\\ref{prop:ideal} (ii)}) \\\\\\\\ \\\\displaystyle\\\\leq\\\\sqrt{(1+\\\\delta)N+2^{P+Q+3}f}\\\\qquad(\\\\text{Lemma \\\\ref{lemma:%\\nsum_of_powers_of_2}}). \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"formula\",\"source_ref\":\"Ax1.EGx4\",\"source_text\":\"$\\\\displaystyle\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\sqrt{{M_{w}^{\\\\prime}}^{\\\\ast}}$ $\\\\displaystyle\\\\leq\\\\sqrt{\\\\left(\\\\sum_{w\\\\in\\\\mathcal{W}}\\\\frac{1}{2^{\\\\mathord{\\\\ell}(% w)}}\\\\right)\\\\left(\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\ell(w)}{M_{w}^{\\\\prime}}^{\\\\ast}% \\\\right)}$ $\\\\displaystyle=\\\\sqrt{\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\mathord{\\\\ell}(w)}{M_{w}^{\\\\prime}% }^{\\\\ast}}\\\\qquad(\\\\text{Lemma \\\\ref{lemma:2_minusL}})$ $\\\\displaystyle\\\\leq\\\\sqrt{\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\mathord{\\\\ell}(w)}M^{\\\\ast}_{w}}$ $\\\\displaystyle\\\\leq\\\\sqrt{(1+\\\\delta)N+(\\\\sum_{w\\\\in\\\\mathcal{W}}2^{\\\\mathord{\\\\ell}(w)% })f}\\\\qquad(\\\\text{Prop.~{}\\\\ref{prop:ideal} (ii)})$ $\\\\displaystyle\\\\leq\\\\sqrt{(1+\\\\delta)N+2^{P+Q+3}f}\\\\qquad(\\\\text{Lemma \\\\ref{lemma:% sum_of_powers_of_2}}).$\",\"tag\":null,\"text\":\"∑︂ √︁ ⃓ ∑︂ 1 ∑︂ M′ ∗ ≤ ⎷ 2ℓ(w)M′ ∗ w 2ℓ(w) w w∈W w∈W w∈W √︄ ∑︂ = 2ℓ(w)M′ ∗ (Lemma 4.8) w w∈W √︄ ∑︂ ≤ 2ℓ(w)M∗ w w∈W √︄ ∑︂ ≤ (1 + δ)N + 2ℓ(w))f (Prop. 4.16 (ii)) w∈W √︂ ≤ (1 + δ)N + 2P +Q+3f (Lemma 4.7).\"},\"formula_index\":7,\"parent_page_sha256\":\"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed.pdf","asset_sha256":"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed","document_id":"arxiv:2608.24575","kind":"formula","page_number":26,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[300.6,518.67,47.52,16.84],\"bbox_pdf\":[300.6,256.49,348.12,273.33],\"fragments\":[{\"bbox_pdf\":[300.6,256.49,348.12,273.33],\"words\":[10811,10813,10814,10815]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\lambda_{\\\\succ,1}^{(N)}\\\\leq\\\\mathcal{F},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"S4.Ex153\",\"source_text\":\"$\\\\lambda_{\\\\succ,1}^{(N)}\\\\leq\\\\mathcal{F},$\",\"tag\":null,\"text\":\"(N λ≻,1 ≤ F,\"},\"formula_index\":9,\"parent_page_sha256\":\"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed.pdf","asset_sha256":"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed","document_id":"arxiv:2608.24575","kind":"formula","page_number":26,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[155.87,553.25,382.69,31.73],\"bbox_pdf\":[155.87,207.02,538.56,238.75],\"fragments\":[{\"bbox_pdf\":[155.87,207.02,538.56,238.75],\"words\":[10817,10818,10819,10820,10821,10822,10823,10824,10825,10826,10827,10828,10829,10830,10831,10832,10833,10834,10835,10836,10837,10838,10839,10840,10841,10843,10844]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathcal{F}:=(2+\\\\epsilon)\\\\sqrt{(1+\\\\delta)N+2^{P+Q+3}f}+(1+\\\\delta)\\\\frac{(t+2)N}%\\n{4^{P}}+R(g+N^{1/10}).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"formula\",\"source_ref\":\"S4.E38\",\"source_text\":\"$\\\\mathcal{F}:=(2+\\\\epsilon)\\\\sqrt{(1+\\\\delta)N+2^{P+Q+3}f}+(1+\\\\delta)\\\\frac{(t+2)N}% {4^{P}}+R(g+N^{1/10}).$ (38)\",\"tag\":\"(38)\",\"text\":\"√︂ (t + 2)N 1/10 P +Q+3f + (1 + δ) F := (2 + ϵ) (1 + δ)N + 2 + R(g + N (38) 4P\"},\"formula_index\":11,\"parent_page_sha256\":\"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed.pdf","asset_sha256":"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed","document_id":"arxiv:2608.24575","kind":"formula","page_number":26,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[277.91,623.31,92.89,26.14],\"bbox_pdf\":[277.91,142.54,370.8,168.69],\"fragments\":[{\"bbox_pdf\":[277.91,142.54,370.8,168.69],\"words\":[10861,10862,10863,10865,10866,10867,10868,10869,10870]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathbb{P}\\\\left(\\\\lambda_{\\\\succ,1}^{(N)}>\\\\mathcal{F}\\\\right)<\\\\mathcal{H}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"S4.Ex154\",\"source_text\":\"$\\\\mathbb{P}\\\\left(\\\\lambda_{\\\\succ,1}^{(N)}>\\\\mathcal{F}\\\\right)<\\\\mathcal{H}.$\",\"tag\":null,\"text\":\"(︂ )︂ (N P λ≻,1 > F < H.\"},\"formula_index\":14,\"parent_page_sha256\":\"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed.pdf","asset_sha256":"b812da6ca04774f50982ff746e72e60944058ce017dfb1e2f2bd1ed2f4f78fed","document_id":"arxiv:2608.24575","kind":"formula","page_number":26,"paper_attribution_json":"{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.24575\",\"authors\":\"Shinsuke Iwao, Fumihiko Nakamura, and Yushi Nakano\",\"title\":\"Longest increasing subsequences of dyadic-type chaotic orbits\",\"url\":\"https://arxiv.org/abs/2608.24575v2\",\"version\":2},\"blocks\":[{\"bbox\":[121.13,73.12,333.02,7.97],\"bbox_pdf\":[121.13,710.91,454.15,718.88],\"fragments\":[{\"bbox_pdf\":[121.13,710.91,454.15,718.88],\"words\":[11593,11594,11595,11596,11597,11598,11599]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":0,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"LONGEST INCREASING SUBSEQUENCES OF DYADIC-TYPE CHAOTIC ORBITS\"},{\"bbox\":[493.37,73.12,8.47,7.97],\"bbox_pdf\":[493.37,710.91,501.84,718.88],\"fragments\":[{\"bbox_pdf\":[493.37,710.91,501.84,718.88],\"words\":[11600]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":1,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"27\"},{\"bbox\":[73.44,95.86,428.4,24.16],\"bbox_pdf\":[73.44,671.99,501.84,696.14],\"fragments\":[{\"bbox_pdf\":[73.44,671.99,501.84,696.14],\"words\":[10871,10873,10874,10875,10876,10877,10878,10879,10880,10881,10882,10883,10884,10885,10886,10887,10888,10889,10890,10891,10892]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph484\",\"source_text\":\"Finally, we complete the estimation of $\\\\mathbb{E}[\\\\lambda_{\\\\succ,1}^{(N)}]$ . For a sufficiently small constant $\\\\theta>0$ , we choose the parameters as\",\"tag\":null,\"text\":\"(N Finally, we complete the estimation of E[λ≻,1]. For a sufficiently small constant θ > 0, we choose the parameters as\"},{\"bbox\":[153.5,117.43,268.28,53.99],\"bbox_pdf\":[153.5,620.58,421.78,674.57],\"fragments\":[{\"bbox_pdf\":[153.5,620.58,421.78,674.57],\"words\":[10893,10894,10895,10896,10897,10898,10899,10900,10901,10902,10903,10904,10905,10906,10907,10908,10909,10910,10911,10912,10913,10914,10915,10916,10917,10918,10919,10920,10921,10922,10923,10924,10925,10926,10927,10928,10929,10930,10931]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{gathered}\\\\delta=\\\\frac{\\\\epsilon}{10},\\\\quad P\\\\sim\\\\left(\\\\frac{1}{4}+\\\\theta%\\n\\\\right)\\\\log_{2}N,\\\\quad Q\\\\sim\\\\left(\\\\frac{3}{4}-2\\\\theta\\\\right)\\\\log_{2}N,\\\\\\\\\\nt\\\\sim 10\\\\log_{2}N,\\\\quad f\\\\sim N^{\\\\theta/2},\\\\quad g\\\\sim N^{1/10}\\\\end{gathered}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"formula\",\"source_ref\":\"S4.Ex155\",\"source_text\":\"$\\\\begin{gathered}\\\\delta=\\\\frac{\\\\epsilon}{10},\\\\quad P\\\\sim\\\\left(\\\\frac{1}{4}+\\\\theta% \\\\right)\\\\log_{2}N,\\\\quad Q\\\\sim\\\\left(\\\\frac{3}{4}-2\\\\theta\\\\right)\\\\log_{2}N,\\\\\\\\ t\\\\sim 10\\\\log_{2}N,\\\\quad f\\\\sim N^{\\\\theta/2},\\\\quad g\\\\sim N^{1/10}\\\\end{gathered}$\",\"tag\":null,\"text\":\"(︃ )︃ (︃ )︃ ϵ 1 3 δ = , P ∼ + θ log2 N, Q ∼ − 2θ log2 N, 10 4 4 t ∼ 10 log2 N, f ∼ N θ/2 , g ∼ N 1/10\"},{\"bbox\":[73.44,176.23,367.08,10.91],\"bbox_pdf\":[73.44,604.86,440.52,615.77],\"fragments\":[{\"bbox_pdf\":[73.44,604.86,440.52,615.77],\"words\":[10932,10933,10934,10935,10936,10937,10938,10939,10940,10941,10942,10943,10944,10945]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph486\",\"source_text\":\"as $N\\\\to\\\\infty$ . Then the inequality (36) is satisfied, so Proposition 4.17 applies.\",\"tag\":null,\"text\":\"as N → ∞. Then the inequality (36) is satisfied, so Proposition 4.17 applies.\"},{\"bbox\":[85.4,189.18,134.03,10.91],\"bbox_pdf\":[85.4,591.91,219.43,602.82],\"fragments\":[{\"bbox_pdf\":[85.4,591.91,219.43,602.82],\"words\":[10946,10947,10948,10949,10950]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph487\",\"source_text\":\"By Proposition 4.6, we have\",\"tag\":null,\"text\":\"By Proposition 4.6, we have\"},{\"bbox\":[181.22,205.99,212.85,12.48],\"bbox_pdf\":[181.22,573.53,394.07,586.01],\"fragments\":[{\"bbox_pdf\":[181.22,573.53,394.07,586.01],\"words\":[10951,10952,10953,10954,10955,10956,10957,10958,10959,10960,10961,10962,10963,10964,10965,10966]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"R<2^{P+1}(Q-P+1)<N^{1/3}\\\\qquad(N\\\\to\\\\infty).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"S4.Ex156\",\"source_text\":\"$R<2^{P+1}(Q-P+1)<N^{1/3}\\\\qquad(N\\\\to\\\\infty).$\",\"tag\":null,\"text\":\"R < 2 P +1 (Q − P + 1) < N 1/3 (N → ∞).\"},{\"bbox\":[73.44,224.97,428.4,24.88],\"bbox_pdf\":[73.44,542.15,501.84,567.03],\"fragments\":[{\"bbox_pdf\":[73.44,542.15,501.84,567.03],\"words\":[10967,10968,10969,10970,10971,10972,10973,10974,10975,10976,10977,10978,10979,10980,10981,10982,10983,10985,10986,10987,10988,10989,10990]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph489\",\"source_text\":\"From this, we see that each term appearing in $\\\\mathcal{F}$ is of order at most $N^{1/2}$ , and that its leading term is\",\"tag\":null,\"text\":\"From this, we see that each term appearing in F is of order at most N 1/2 and that its leading term is\"},{\"bbox\":[190.01,243.17,195.27,20.77],\"bbox_pdf\":[190.01,528.06,385.27,548.83],\"fragments\":[{\"bbox_pdf\":[190.01,528.06,385.27,548.83],\"words\":[10991,10992,10993,10994,10995,10996,10997,10998,10999,11000,11001,11002]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathcal{F}\\\\sim(2+\\\\epsilon)\\\\sqrt{(1+\\\\epsilon/10)N}\\\\qquad(N\\\\to\\\\infty).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S4.Ex157\",\"source_text\":\"$\\\\mathcal{F}\\\\sim(2+\\\\epsilon)\\\\sqrt{(1+\\\\epsilon/10)N}\\\\qquad(N\\\\to\\\\infty).$\",\"tag\":null,\"text\":\"√︁ F ∼ (2 + ϵ) (1 + ϵ/10)N (N → ∞).\"},{\"bbox\":[73.44,268.49,237.69,10.91],\"bbox_pdf\":[73.44,512.6,311.13,523.51],\"fragments\":[{\"bbox_pdf\":[73.44,512.6,311.13,523.51],\"words\":[11003,11004,11005,11006,11007,11008,11009,11010,11011,11012,11013]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph491\",\"source_text\":\"On the other hand, since the leading term of $\\\\mathcal{H}$ is\",\"tag\":null,\"text\":\"On the other hand, since the leading term of H is\"},{\"bbox\":[253.79,276.82,67.7,33.77],\"bbox_pdf\":[253.79,481.41,321.49,515.18],\"fragments\":[{\"bbox_pdf\":[253.79,481.41,321.49,515.18],\"words\":[11014,11015,11016,11017,11018,11019,11020,11021,11022]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"RN^{2}\\\\binom{N}{2}2^{-t},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"formula\",\"source_ref\":\"S4.Ex158\",\"source_text\":\"$RN^{2}\\\\binom{N}{2}2^{-t},$\",\"tag\":null,\"text\":\"(︃ )︃ 2 N −t RN 2 , 2\"},{\"bbox\":[73.44,315.91,66.73,10.91],\"bbox_pdf\":[73.44,465.18,140.17,476.09],\"fragments\":[{\"bbox_pdf\":[73.44,465.18,140.17,476.09],\"words\":[11023,11024,11025]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"paragraph\",\"source_ref\":\"paragraph493\",\"source_text\":\"it follows that\",\"tag\":null,\"text\":\"it follows that\"},{\"bbox\":[256.67,327.54,54.17,12.47],\"bbox_pdf\":[256.67,451.99,310.84,464.46],\"fragments\":[{\"bbox_pdf\":[256.67,451.99,310.84,464.46],\"words\":[11026,11027,11028,11029]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathcal{H}=o(N^{-1}).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"S4.Ex159\",\"source_text\":\"$\\\\mathcal{H}=o(N^{-1}).$\",\"tag\":null,\"text\":\"H = o(N −1\"},{\"bbox\":[85.4,347.06,207.1,14.03],\"bbox_pdf\":[85.4,430.91,292.49,444.94],\"fragments\":[{\"bbox_pdf\":[85.4,430.91,292.49,444.94],\"words\":[11031,11033,11034,11035,11036,11038,11039,11040,11041]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":13,\"role\":\"paragraph\",\"source_ref\":\"paragraph495\",\"source_text\":\"As $\\\\lambda_{\\\\succ,1}^{(N)}\\\\leq N$ , Proposition 4.17 implies that\",\"tag\":null,\"text\":\"(N As λ≻,1 ≤ N Proposition 4.17 implies that\"},{\"bbox\":[178.25,359.37,211.0,23.89],\"bbox_pdf\":[178.25,408.73,389.25,432.63],\"fragments\":[{\"bbox_pdf\":[178.25,408.73,389.25,432.63],\"words\":[11042,11043,11045,11046,11047,11048,11049,11050,11051,11052,11053,11054,11055,11057]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathbb{E}[\\\\lambda_{\\\\succ,1}^{(N)}]\\\\leq(2+\\\\epsilon)\\\\sqrt{(1+\\\\epsilon/10)N}+N%\\n\\\\cdot o(N^{-1}).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"S4.Ex160\",\"source_text\":\"$\\\\mathbb{E}[\\\\lambda_{\\\\succ,1}^{(N)}]\\\\leq(2+\\\\epsilon)\\\\sqrt{(1+\\\\epsilon/10)N}+N% \\\\cdot o(N^{-1}).$\",\"tag\":null,\"text\":\"√︁ (N −1 E[λ≻,1] ≤ (2 + ϵ) (1 + ϵ/10)N + N o(N\"},{\"bbox\":[73.44,387.25,48.85,10.91],\"bbox_pdf\":[73.44,393.84,122.29,404.75],\"fragments\":[{\"bbox_pdf\":[73.44,393.84,122.29,404.75],\"words\":[11059]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":15,\"role\":\"paragraph\",\"source_ref\":\"paragraph497\",\"source_text\":\"Therefore,\",\"tag\":null,\"text\":\"Therefore,\"},{\"bbox\":[195.13,391.05,185.03,33.77],\"bbox_pdf\":[195.13,367.18,380.16,400.95],\"fragments\":[{\"bbox_pdf\":[195.13,367.18,380.16,400.95],\"words\":[11060,11061,11062,11063,11064,11066,11067,11068,11069,11070,11071,11072,11073,11074,11075]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathbb{E}[\\\\lambda_{\\\\succ,1}^{(N)}]\\\\leq\\\\left(2+\\\\frac{6}{5}\\\\epsilon\\\\right)\\\\sqrt%\\n{N}\\\\qquad(N\\\\to\\\\infty)\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":16,\"role\":\"formula\",\"source_ref\":\"S4.Ex161\",\"source_text\":\"$\\\\mathbb{E}[\\\\lambda_{\\\\succ,1}^{(N)}]\\\\leq\\\\left(2+\\\\frac{6}{5}\\\\epsilon\\\\right)\\\\sqrt% {N}\\\\qquad(N\\\\to\\\\infty)$\",\"tag\":null,\"text\":\"(︃ )︃ 6 √ (N E[λ≻,1] ≤ 2 + ϵ N (N → ∞) 5\"},{\"bbox\":[73.44,428.37,346.72,16.03],\"bbox_pdf\":[73.44,347.6,420.16,363.63],\"fragments\":[{\"bbox_pdf\":[73.44,347.6,420.16,363.63],\"words\":[11076,11077,11078,11079,11080,11081,11082,11083,11084,11085,11086,11087,11088]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":17,\"role\":\"paragraph\",\"source_ref\":\"paragraph499\",\"source_text\":\"follows. Finally, by replacing $\\\\frac{6}{5}\\\\epsilon$ with $\\\\epsilon$ , we obtain the desired inequality.\",\"tag\":null,\"text\":\"follows. Finally, by replacing 6 5ϵ with ϵ, we obtain the desired inequality.\"},{\"bbox\":[233.45,452.44,108.37,10.91],\"bbox_pdf\":[233.45,328.65,341.82,339.56],\"fragments\":[{\"bbox_pdf\":[233.45,328.65,341.82,339.56],\"words\":[11089]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":18,\"role\":\"heading\",\"source_ref\":\"title500\",\"source_text\":\"Acknowledgements\",\"tag\":null,\"text\":\"Acknowledgements\"},{\"bbox\":[73.44,471.86,428.4,49.76],\"bbox_pdf\":[73.44,270.37,501.84,320.14],\"fragments\":[{\"bbox_pdf\":[73.44,270.37,501.84,320.14],\"words\":[11090,11091,11092,11093,11094,11095,11096,11097,11098,11099,11100,11101,11102,11103,11104,11105,11106,11107,11108,11109,11110,11111,11112,11113,11114,11115,11116,11117,11118,11119,11120,11121,11122,11123,11124,11125,11126,11127,11128,11129]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":19,\"role\":\"paragraph\",\"source_ref\":\"paragraph501\",\"source_text\":\"S. Iwao was supported by JSPS KAKENHI Grant Numbers 22K03239 and 23K03056. F. Nakamura was supported by JSPS KAKENHI Grant Numbers 23K25785 and 24K16942. Y. Nakano was supported by JSPS KAKENHI Grant Number 23K03188 and JST PRESTO Grant Number JPMJPR25K8.\",\"tag\":null,\"text\":\"S. Iwao was supported by JSPS KAKENHI Grant Numbers 22K03239 and 23K03056. F. Nakamura was supported by JSPS KAKENHI Grant Numbers 23K25785 and 24K16942. Y. Nakano was supported by JSPS KAKENHI Grant Number 23K03188 and JST PRESTO Grant Number JPMJPR25K8.\"},{\"bbox\":[255.83,533.42,63.61,10.91],\"bbox_pdf\":[255.83,247.67,319.45,258.58],\"fragments\":[{\"bbox_pdf\":[255.83,247.67,319.45,258.58],\"words\":[11130]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":20,\"role\":\"heading\",\"source_ref\":\"title502\",\"source_text\":\"References\",\"tag\":null,\"text\":\"References\"},{\"bbox\":[78.05,552.8,423.79,19.93],\"bbox_pdf\":[78.05,219.27,501.84,239.2],\"fragments\":[{\"bbox_pdf\":[78.05,219.27,501.84,239.2],\"words\":[11131,11132,11133,11134,11135,11136,11137,11138,11139,11140,11141,11142,11143,11144,11145,11146,11147,11148,11149,11150,11151,11152,11153,11154,11155,11156,11157]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":21,\"role\":\"reference\",\"source_ref\":\"bib.bib1\",\"source_text\":\"A. Abrams, E. Babson, H. Landau, Z. Landau, and J. Pommersheim, Distributions of order patterns of interval maps, Combin. Probab. Comput. 22 (2013), no. 3, 319–341.\",\"tag\":null,\"text\":\"[1] A. Abrams, E. Babson, H. Landau, Z. Landau, and J. Pommersheim, Distributions of order patterns of interval maps, Combin. Probab. Comput. 22 (2013), no. 3, 319–341.\"},{\"bbox\":[78.05,574.72,423.79,30.88],\"bbox_pdf\":[78.05,186.4,501.84,217.28],\"fragments\":[{\"bbox_pdf\":[78.05,186.4,501.84,217.28],\"words\":[11158,11159,11160,11161,11162,11163,11164,11165,11166,11167,11168,11169,11170,11171,11172,11173,11174,11175,11176,11177,11178,11179,11180,11181,11182,11183,11184,11185,11186,11187,11188,11189]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":22,\"role\":\"reference\",\"source_ref\":\"bib.bib2\",\"source_text\":\"D. Arroyo and A. Gonzalo. Application of Gray codes to the study of the theory of symbolic dynamics of unimodal maps, Communications in Nonlinear Science and Numerical Simulation 19(7) (2014), 2345-2353.\",\"tag\":null,\"text\":\"[2] D. Arroyo and A. Gonzalo. Application of Gray codes to the study of the theory of symbolic dynamics of unimodal maps, Communications in Nonlinear Science and Numerical Simulation 19(7) (2014), 2345-2353.\"},{\"bbox\":[78.05,607.6,423.8,19.93],\"bbox_pdf\":[78.05,164.48,501.84,184.4],\"fragments\":[{\"bbox_pdf\":[78.05,164.48,501.84,184.4],\"words\":[11190,11191,11192,11193,11194,11195,11196,11197,11198,11199,11200,11201,11202,11203,11204,11205,11206,11207,11208,11209,11210,11211,11212,11213,11214,11215,11216,11217,11218,11219,11220,11221]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":23,\"role\":\"reference\",\"source_ref\":\"bib.bib3\",\"source_text\":\"J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), no. 4, 1119–1178.\",\"tag\":null,\"text\":\"[3] J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), no. 4, 1119–1178.\"},{\"bbox\":[78.05,629.51,423.79,19.93],\"bbox_pdf\":[78.05,142.56,501.84,162.49],\"fragments\":[{\"bbox_pdf\":[78.05,142.56,501.84,162.49],\"words\":[11222,11223,11224,11225,11226,11227,11228,11229,11230,11231,11232,11233,11234,11235,11236,11237,11238,11239,11240,11241]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":24,\"role\":\"reference\",\"source_ref\":\"bib.bib4\",\"source_text\":\"S. Boucheron, G. Lugosi, and P. Massart, Concentration Inequalities: A Nonasymptotic Theory of Independence, Oxford University Press, Oxford, 2013.\",\"tag\":null,\"text\":\"[4] S. Boucheron, G. Lugosi, and P. 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{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.24622\",\"authors\":\"Davide Rovelli, Philipp Berdesinski, Rodrigo Otoni, Patrick Eugster\",\"title\":\"Scalable datacenter replication with mostly-synchronous consensus on hardware\",\"url\":\"https://arxiv.org/abs/2608.24622v1\",\"version\":1},\"blocks\":[{\"bbox\":[54.0,46.5,265.17,7.97],\"bbox_pdf\":[54.0,737.53,319.17,745.5],\"fragments\":[{\"bbox_pdf\":[54.0,737.53,319.17,745.5],\"words\":[18134,18135,18136,18137,18138,18139,18140,18141]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":0,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"Scalable datacenter replication with mostly-synchronous consensus on hardware\"},{\"bbox\":[54.0,72.09,241.73,129.42],\"bbox_pdf\":[54.0,590.49,295.73,719.91],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[54.0,590.49,295.73,719.91],\"words\":[3002,3003,3004,3005,3006,3007,3008,3009,3010,3011,3012,3013,3014,3015,3016,3017,3018,3019,3020,3021,3022,3023,3024,3025,3026,3027,3028,3029,3030,3031,3032,3033,3034,3035,3036,3037,3038,3039,3040,3041,3042,3043,3044,3045,3046,3047,3048,3049,3050,3051,3052,3053,3054,3055,3056,3057,3058,3059,3060,3061,3062,3063,3064,3065,3066,3067,3068,3069,3070,3071,3072,3073,3074,3075,3076,3077,3078,3079,3080,3081,3082,3083,3084,3085,3086,3087,3088,3089,3090,3091,3092,3093,3094,3095,3096,3097,3098,3099,3100]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph57\",\"source_text\":\"Fig. 2 outlines the architecture of scarHW in two variants: vanilla (2(a)), containing one replication unit (green box), and pipelined (2(b)), aggregating multiple units. Every smartNIC running scarHW is a replica executing the same logic. The brain of the system is the consensus engine module which implements the POPUC algorithm (cf. Alg. 1). In short, the module takes arbitrary values from one or more applications as inputs, proposes them with a unique id for the current consensus instance, and outputs consensus decisions. The consensus engine operates in synchronous rounds in which all replicas disseminate messages to each other. In a failure-free execution, all replicas propose values simultaneously and, after two rounds, output an ordered array of all proposed values as the consensus decision. This key feature of scarHW enables load distribution, which we will discuss in later sections. The engine keeps track of packets it receives in every round and flags a remote replica as faulty if its message is not received within the expected timeframe, i.e., it observes an omission failure. To avoid false positives, replicas can send empty heartbeat values in absence of pending proposals. Otherwise, replicas can choose to not send a heartbeat on purpose to scale up/down, or allowing re-configuration during “quiet” periods with few requests. In the presence of up to $t$ failures, a decision can take between 2 and $t$ rounds, and replicas can decide in different rounds. When a replica observes more than $t$ failures, e.g., due to a network outage, the consensus engine self-declares itself faulty and quits execution.\",\"tag\":null,\"text\":\"not received within the expected timeframe, i.e., it observes an omission failure. To avoid false positives, replicas can send empty heartbeat values in absence of pending proposals. Otherwise, replicas can choose to not send a heartbeat on purpose to scale up/down, or allowing re-configuration during “quiet” periods with few requests. In the presence of up to 𝑡 failures, a decision can take between 2 and 𝑡 rounds, and replicas can decide in different rounds. When a replica observes more than 𝑡 failures, e.g., due to a network outage, the consensus engine self-declares itself faulty and quits execution.\"},{\"bbox\":[53.63,203.71,242.1,284.73],\"bbox_pdf\":[53.63,303.56,295.73,588.29],\"fragments\":[{\"bbox_pdf\":[53.63,303.56,295.73,588.29],\"words\":[3101,3102,3103,3104,3105,3106,3107,3108,3109,3110,3111,3112,3113,3114,3115,3116,3117,3118,3119,3120,3121,3122,3123,3124,3125,3126,3127,3128,3129,3130,3131,3132,3133,3134,3135,3136,3137,3138,3139,3140,3141,3142,3143,3144,3145,3146,3147,3148,3149,3150,3151,3152,3153,3154,3155,3156,3157,3158,3159,3160,3161,3162,3163,3164,3165,3166,3167,3168,3169,3170,3171,3172,3173,3174,3175,3176,3177,3178,3179,3180,3181,3182,3183,3184,3185,3186,3187,3188,3189,3190,3191,3192,3193,3194,3195,3196,3197,3198,3199,3200,3201,3202,3203,3204,3205,3206,3207,3208,3209,3210,3211,3212,3213,3214,3215,3216,3217,3218,3219,3220,3221,3222,3223,3224,3225,3226,3227,3228,3229,3230,3231,3232,3233,3234,3235,3236,3237,3238,3239,3240,3241,3242,3243,3244,3245,3246,3247,3248,3249,3250,3251,3252,3253,3254,3255,3256,3257,3258,3259,3260,3261,3262,3263,3264,3265,3266,3267,3268,3269,3270,3271,3272,3273,3274,3275,3276,3278,3279,3280,3281,3282,3283,3284,3285,3286,3287,3288,3289,3290,3291,3292,3293,3294,3295,3296,3297,3298,3299,3300,3301,3302,3303]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.966,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph58\",\"source_text\":\"To avoid costly retransmissions, which would compromise our algorithm’s performance, scarHW replicas communicate via UDP over Ethernet through the UDP/IP4 and MAC modules. The communication between the host and the consensus engine involves two memory-mapped buffers separated for input proposals and consensus decisions using memory-to-stream (MM2S) interfaces which bridge the stream nature of the UDP/IPv4 layer and the memory-oriented structure of host-FPGA transfers. Host processes interact with scarHW via scarHW-server: a thin software layer that can handle client requests, seamlessly integrate with applications, and configure hardware settings. The packet broadcaster turns the payload supplied by the consensus engine module into a suitable stream for the UDP/IP4 layer, while the parser performs the opposite function. The bootstrapper module is of particular relevance since it allows the synchronization of the replicas before the start of the first consensus instance by adequately triggering the round timer after an initial message exchange. After this bootstrap sequence, the consensus engine is normally set to run back-to-back consensus instances in sequence in order to achieve SMR (cf. § 5.1) while maintaining synchronization. The consensus engine can adjust the round timer dynamically to correct timer drift, in a similar fashion as clock synchronization methods (IEEE Instrumentation and Measurement Society, 2020).\",\"tag\":null,\"text\":\"To avoid costly retransmissions, which would compromise our algorithm’s performance, scarHW replicas communicate via UDP over Ethernet through the UDP/IP4 and MAC modules. The communication between the host and the consensus engine involves two memory-mapped buffers separated for input proposals and consensus decisions using memoryto-stream (MM2S) interfaces which bridge the stream nature of the UDP/IPv4 layer and the memory-oriented structure of host-FPGA transfers. Host processes interact with scarHW via scarHW-server: a thin software layer that can handle client requests, seamlessly integrate with applications, and configure hardware settings. The packet broadcaster turns the payload supplied by the consensus engine module into a suitable stream for the UDP/IP4 layer, while the parser performs the opposite function. The bootstrapper module is of particular relevance since it allows the synchronization of the replicas before the start of the first consensus instance by adequately triggering the round timer after an initial message exchange. After this bootstrap sequence, the consensus engine is normally set to run back-to-back consensus instances in sequence in order to achieve SMR (cf. 5.1) while maintaining synchronization. The consensus engine can adjust the round timer dynamically to correct timer drift, in a similar fashion as clock synchronization methods [32].\"},{\"bbox\":[53.75,490.28,242.13,201.39],\"bbox_pdf\":[53.75,100.32,295.88,301.72],\"fragments\":[{\"bbox_pdf\":[53.75,100.32,295.88,301.72],\"words\":[3304,3305,3306,3307,3308,3309,3310,3311,3312,3313,3314,3315,3316,3317,3318,3319,3320,3321,3322,3323,3324,3325,3326,3327,3328,3329,3330,3331,3332,3333,3334,3335,3336,3337,3338,3339,3340,3341,3342,3343,3344,3345,3346,3347,3348,3349,3350,3351,3352,3353,3354,3355,3356,3357,3358,3359,3360,3361,3362,3363,3364,3365,3366,3367,3368,3369,3370,3371,3372,3373,3374,3375,3376,3377,3378,3379,3380,3381,3382,3383,3384,3385,3386,3387,3388,3389,3390,3391,3392,3393,3394,3395,3396,3397,3398,3399,3400,3401,3402,3403,3404,3405,3406,3407,3408,3409,3410,3411,3412,3413,3414,3415,3416,3417,3418,3419,3420,3421,3422,3423,3424,3425,3426,3427,3428,3429,3430,3431,3432,3433,3434,3435,3436,3437,3438,3439,3440,3441,3442,3443,3444,3445,3446,3447]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph59\",\"source_text\":\"The replication throughput of scarHW vanilla is limited by the round time, i.e., worst-case latency of the setup plus a safety margin, since replicas have to wait for the decision from a consensus instance before starting the next one. scarHW pipelined uses multiple consensus engines, timers and packet handling modules in parallel (grouped into a green module in Fig. 2(a)) to maximize decision rate. Individual green modules in different scarHWs have matching ids and form a replication group to avoid overlapping protocol states. Dispatcher and aggregator modules provide basic functionalities to respectively parallelize and serialize consensus decisions. Serialization sorts parallel decisions by green module ids to ensure deterministic order across different replicas. Dispatcher and aggregator settings allow a single application to use different replication groups to increase throughput and also multiple applications to use different replication groups as unrelated SMR instances.\",\"tag\":null,\"text\":\"The replication throughput of scarHW vanilla is limited by the round time, i.e., worst-case latency of the setup plus a safety margin, since replicas have to wait for the decision from a consensus instance before starting the next one. scarHW pipelined uses multiple consensus engines, timers and packet handling modules in parallel (grouped into a green module in Fig. 2a) to maximize decision rate. Individual green modules in different scarHWs have matching ids and form a replication group to avoid overlapping protocol states. Dispatcher and aggregator modules provide basic functionalities to respectively parallelize and serialize consensus decisions. Serialization sorts parallel decisions by green module ids to ensure deterministic order across different replicas. Dispatcher and aggregator settings allow a single application to use different replication groups to increase throughput and also multiple applications to use different replication groups as unrelated SMR instances.\"},{\"bbox\":[317.95,36.0,241.73,236.51],\"bbox_pdf\":[317.95,519.49,559.68,756.0],\"float_ref\":\"S3.F3\",\"fragments\":[{\"bbox_pdf\":[317.95,519.49,559.68,756.0],\"words\":[]}],\"html\":null,\"kind\":\"figure\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"figure\",\"source_ref\":\"S3.F3\",\"source_text\":\"\",\"tag\":null,\"text\":\"\"},{\"bbox\":[317.95,272.51,241.73,45.63],\"bbox_pdf\":[317.95,473.86,559.68,519.49],\"float_ref\":\"S3.F3\",\"fragments\":[{\"bbox_pdf\":[317.95,473.86,559.68,519.49],\"words\":[3465,3466,3467,3468,3469,3470,3471,3472,3473,3474,3475,3476,3477,3478,3479,3480,3481,3482,3483,3484,3485,3486,3487,3488,3489,3490,3491,3492,3493]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"caption\",\"source_ref\":\"caption63\",\"source_text\":\"Figure 3. Possible deployments for replication clusters using scarHW. Classical leader-based consensus solutions allow only one clients-service interaction (left) while scarHW enables two additional configurations (center and right).\",\"tag\":null,\"text\":\"Figure 3. Possible deployments for replication clusters using scarHW. Classical leader-based consensus solutions allow only one clients-service interaction (left) while scarHW enables two additional configurations (center and right).\"},{\"bbox\":[317.95,415.71,110.2,9.96],\"bbox_pdf\":[317.95,366.33,428.15,376.29],\"fragments\":[{\"bbox_pdf\":[317.95,366.33,428.15,376.29],\"words\":[3494,3495,3496]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":2,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"heading\",\"source_ref\":\"title60\",\"source_text\":\"3.2. Deployment modes\",\"tag\":null,\"text\":\"3.2 Deployment modes\"},{\"bbox\":[317.59,430.65,242.1,272.99],\"bbox_pdf\":[317.59,88.37,559.69,361.35],\"fragments\":[{\"bbox_pdf\":[317.59,88.37,559.69,361.35],\"words\":[3497,3498,3499,3500,3501,3502,3503,3504,3505,3506,3507,3508,3509,3510,3511,3512,3513,3514,3515,3516,3517,3518,3519,3520,3521,3522,3523,3524,3525,3526,3527,3528,3529,3530,3531,3532,3533,3534,3535,3536,3537,3538,3539,3540,3541,3542,3543,3544,3545,3546,3547,3548,3549,3550,3551,3552,3553,3554,3555,3556,3557,3558,3559,3560,3561,3562,3563,3564,3565,3566,3567,3568,3569,3570,3571,3572,3573,3574,3575,3576,3577,3578,3579,3580,3581,3582,3583,3584,3585,3586,3587,3588,3589,3590,3591,3592,3593,3594,3595,3596,3597,3598,3599,3600,3601,3602,3603,3604,3605,3606,3607,3608,3609,3610,3611,3612,3613,3614,3615,3616,3617,3618,3619,3620,3621,3622,3623,3624,3625,3626,3627,3628,3629,3630,3631,3632,3633,3634,3635,3636,3637,3638,3639,3640,3641,3642,3643,3644,3645,3646,3647,3648,3649,3650,3651,3652,3653,3654,3655,3656,3657,3658,3659,3660,3661,3662,3663,3664,3665,3666,3667,3668,3669,3670,3671,3672,3673,3674,3675,3676,3677,3678,3679,3680,3681]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.911,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph64\",\"source_text\":\"Consistently-replicated services running on top of scarHW can leverage the power of load distribution since all replicas can propose values simultaneously. This allows programmers to choose among the different deployment options outlined in Fig. 3. In scarHW leader (scarHW-L) mode, all clients forward requests to one node which in turn forwards requests to the replication cluster. This option is the least beneficial and is included only to compare scarHW against other leader-based approaches. scarHW load-balanced (scarHW-B) mode mitigates the bottleneck by using a load balancer to dispatch the requests to all remote replicas, allowing for custom load shaping strategies. Lastly, scarHW microservice (scarHW-u) is tailored to microservices architectures where client processes are co-located with application server processes. Here, clients interact with respective local instances of scarHW-server to safely replicate requests before sending them to application processes. The separation between scarHW processes, assuming mostly synchronous behavior thanks to deployment on programmable hardware, and (asynchronous) application processes naturally fits modern sidecar (Zhu et al., 2023; Sahu et al., 2025) microservice architectures. scarHW-u also allows seamless integration with popular container orchestration services such as Kubernetes (The Kubernetes Authors, 2024) and docker-swarm (Docker Inc., 2024).\",\"tag\":null,\"text\":\"Consistently-replicated services running on top of scarHW can leverage the power of load distribution since all replicas can propose values simultaneously. This allows programmers to choose among the different deployment options outlined in Fig. 3. In scarHW leader (scarHW-L) mode, all clients forward requests to one node which in turn forwards requests to the replication cluster. This option is the least beneficial and is included only to compare scarHW against other leader-based approaches. scarHW load-balanced (scarHW-B) mode mitigates the bottleneck by using a load balancer to dispatch the requests to all remote replicas, allowing for custom load shaping strategies. Lastly, scarHW microservice (scarHW-u) is tailored to microservices architectures where client processes are co-located with application server processes. Here, clients interact with respective local instances of scarHW-server to safely replicate requests before sending them to application processes. The separation between scarHW processes, assuming mostly synchronous behavior thanks to deployment on programmable hardware, and (asynchronous) application processes naturally fits modern sidecar [67, 84] microservice architectures. scarHW-u also allows seamless integration with popular container orchestration services such as Kubernetes [75] and\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.24622\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.24622v1-p0004.pdf\",\"page_no\":5,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"b2d46808ff2e296dea9c8edc44ee5a098a7d8aa0\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"2ff1c46330db7b29ed29a460b546feb56c8a5213b3311870bbaf98c8757fb93c\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph64\",\"c_block\":0.911,\"c_lines\":1.0,\"c_pdf\":0.997,\"c_region\":1.0,\"c_src\":0.952,\"figure_text_share\":0.033,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.911,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"37fc9e5648d22f378472413ec932060473ae691a2f3e13341a820e13d4c544ea\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/37fc9e5648d22f378472413ec932060473ae691a2f3e13341a820e13d4c544ea.pdf","asset_sha256":"37fc9e5648d22f378472413ec932060473ae691a2f3e13341a820e13d4c544ea","document_id":"arxiv:2608.24622","kind":"page","page_number":5,"paper_attribution_json":"{\"arxiv_id\":\"2608.24622\",\"authors\":\"Davide Rovelli, Philipp Berdesinski, Rodrigo Otoni, Patrick Eugster\",\"title\":\"Scalable datacenter replication with mostly-synchronous consensus on hardware\",\"url\":\"https://arxiv.org/abs/2608.24622v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.28229\",\"authors\":\"Vincenzo Collura, Karim Tit, Eleonora Giunchiglia, Mike Papadakis, Maxime Cordy\",\"title\":\"Stay Within Your Bounds: Distance-Guided Decoding for Guaranteed Context-Free Grammar Compliance\",\"url\":\"https://arxiv.org/abs/2608.28229v1\",\"version\":1},\"blocks\":[{\"bbox\":[70.87,70.94,218.45,65.08],\"bbox_pdf\":[70.87,705.87,289.32,770.95],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[70.87,705.87,289.32,770.95],\"words\":[1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph23\",\"source_text\":\"These methods primarily enforce local prefix feasibility: at each step, the decoder checks whether a token keeps the current prefix extendable to at least one valid completion. This provides prefix safety, but not necessarily completed-output validity under a finite token budget. If decoding stops because the maximum number of tokens is reached before an accepting configuration, the returned output may still be syntactically invalid, even though every generated prefix was locally feasible (Ugare et al., 2025). Thus, local masking answers whether a prefix can still be completed, but not whether it is close enough to acceptance within the remaining budget.\",\"tag\":null,\"text\":\"every generated prefix was locally feasible (Ugare et al., 2025). Thus, local masking answers whether a prefix can still be completed, but not whether it is close enough to acceptance within the remaining budget.\"},{\"bbox\":[70.87,147.52,71.2,11.08],\"bbox_pdf\":[70.87,683.29,142.07,694.37],\"fragments\":[{\"bbox_pdf\":[70.87,683.29,142.07,694.37],\"words\":[1152,1153]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":4,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"heading\",\"source_ref\":\"title24\",\"source_text\":\"Global control.\",\"tag\":null,\"text\":\"Global control.\"},{\"bbox\":[70.51,147.52,220.46,390.46],\"bbox_pdf\":[70.51,303.91,290.96,694.37],\"fragments\":[{\"bbox_pdf\":[70.51,303.91,290.96,694.37],\"words\":[1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph25\",\"source_text\":\"A complementary line of work studies controlled generation through global objectives, automata-based dynamic programming, or probabilistic inference. GeLaTo formulates tractable control for autoregressive generation with automata and dynamic programming (Zhang et al., 2023). Ctrl-G extends this view to adaptable logical control (Zhang et al., 2024). SEM-CTRL steers generation toward semantic constraints (Albinhassan et al., 2026). ABS enforces constraints that can be represented as a deterministic finite automaton (DFA) using the distance from the accepting states (Collura et al., 2025). Following a similar approach, TruncProof addresses finite-budget constrained decoding using LL(1) parsing and remaining-token-cost estimates (Kato and Tarashima, 2026). Importantly, LL(1) grammars form a strict subset of deterministic CFGs (DCFGs), which are themselves a strict subset of general CFGs. In contrast, our method, SWYB, operates on generic PDAs and therefore supports general CFG constraints. Sequential Monte Carlo and adaptive weighted rejection sampling provide sampling-based control mechanisms for syntactic, semantic, or black-box constraints (Loula et al., 2025; Lipkin et al., 2025). These methods show the value of lookahead, but global control is much harder for CFGs than for regular constraints because pushdown automata have unbounded stacks.\",\"tag\":null,\"text\":\"A complementary line of work studies controlled generation through global objectives, automata-based dynamic programming, or probabilistic inference. GeLaTo formulates tractable control for autoregressive generation with automata and dynamic programming (Zhang et al., 2023). Ctrl-G extends this view to adaptable logical control (Zhang et al., 2024). SEM-CTRL steers generation toward semantic constraints (Albinhassan et al., 2026). ABS enforces constraints that can be represented as a deterministic finite automaton (DFA) using the distance from the accepting states (Collura et al., 2025). Following a similar approach, TruncProof addresses finite-budget constrained decoding using LL(1) parsing and remaining-token-cost estimates (Kato and Tarashima, 2026). Importantly, LL(1) grammars form a strict subset of deterministic CFGs (DCFGs), which are themselves a strict subset of general CFGs. In contrast, our method, SWYB, operates on generic PDAs and therefore supports general CFG constraints. Sequential Monte Carlo and adaptive weighted rejection sampling provide sampling-based control mechanisms for syntactic, semantic, or black-box constraints (Loula et al., 2025; Lipkin et al., 2025). These methods show the value of lookahead, but global control is much harder for CFGs than for regular constraints because pushdown automata have unbounded stacks.\"},{\"bbox\":[70.87,549.48,102.78,11.08],\"bbox_pdf\":[70.87,281.33,173.64,292.41],\"fragments\":[{\"bbox_pdf\":[70.87,281.33,173.64,292.41],\"words\":[1344,1345]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":4,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"heading\",\"source_ref\":\"title26\",\"source_text\":\"Reachability analysis.\",\"tag\":null,\"text\":\"Reachability analysis.\"},{\"bbox\":[70.05,549.48,220.9,173.67],\"bbox_pdf\":[70.05,118.74,290.95,292.41],\"fragments\":[{\"bbox_pdf\":[70.05,118.74,290.95,292.41],\"words\":[1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426,1427,1428]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph27\",\"source_text\":\"The formal-methods literature provides symbolic tools for infinite-state pushdown systems. Reachability for pushdown automata can be represented using finite stack summaries and saturation procedures (Bouajjani et al., 1997). Weighted pushdown systems extend this idea by attaching weights to transitions and computing path quantities, with applications to interprocedural dataflow analysis (Reps et al., 2003). SWYB adapts this perspective to constrained decoding by computing bounded pushdown summaries and upper-bound distances to acceptance, then using them to guide token pruning and re-ranking.\",\"tag\":null,\"text\":\"The formal-methods literature provides symbolic tools for infinite-state pushdown systems. Reachability for pushdown automata can be represented using finite stack summaries and saturation procedures (Bouajjani et al., 1997). Weighted pushdown systems extend this idea by attaching weights to transitions and computing path quantities, with applications to interprocedural dataflow analysis (Reps et al., 2003). SWYB adapts this perspective to constrained decoding by computing bounded pushdown summaries and upper-bound distances to acceptance, then using them to guide token pruning and re-ranking.\"},{\"bbox\":[70.87,734.61,117.24,11.13],\"bbox_pdf\":[70.87,96.15,188.1,107.28],\"fragments\":[{\"bbox_pdf\":[70.87,96.15,188.1,107.28],\"words\":[1429,1430]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":4,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"heading\",\"source_ref\":\"title28\",\"source_text\":\"Program constraints.\",\"tag\":null,\"text\":\"Program constraints.\"},{\"bbox\":[70.87,734.61,218.27,38.23],\"bbox_pdf\":[70.87,69.06,289.14,107.28],\"fragments\":[{\"bbox_pdf\":[70.87,69.06,289.14,107.28],\"words\":[1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.986,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph29\",\"source_text\":\"Several methods specialize in constrained decoding for code or programming-language structure. Type-constrained decoding restricts generation using type information (Mündler et al., 2025), while correctness-guaranteed code generation studies decoding under stronger program-level specifications (Li et al., 2025). CFG-constrained decoding has also been extended to diffusion language models (mündler2025constraineddecodingdiffusionllms). These approaches are complementary to ours: they enrich the constraint language or model class, whereas SWYB adds sound distance-guided lookahead to CFG/PDA decoding.\",\"tag\":null,\"text\":\"Several methods specialize in constrained decoding for code or programming-language structure. Type-\"},{\"bbox\":[306.14,73.29,218.27,247.36],\"bbox_pdf\":[306.14,521.25,524.41,768.6],\"float_ref\":\"S2.T1\",\"fragments\":[{\"bbox_pdf\":[306.14,521.25,524.41,768.6],\"words\":[1443,1444,1445,1446,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474,1475,1476,1477,1478,1479,1480,1481,1482,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494,1495,1496,1497,1498,1499,1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512,1513,1514,1515,1516,1517,1518,1519,1520,1521,1522,1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534,1535,1536,1537,1538,1539,1540,1541,1542,1543,1544,1545,1546,1547,1548,1549,1550,1551,1552,1553,1554,1555,1556,1557,1558,1559,1560,1561,1562,1563,1564,1565,1566,1567,1568]}],\"html\":\"<table><tr><th>Method</th><th>Constraint</th><th>Prefix</th><th>Budget</th></tr><tr><th>Guidance (Lundberg and et. al., 2024)</th><td>CFG</td><td>partial</td><td>no</td></tr><tr><th>Outlines (Willard and Louf, 2023)</th><td>CFG</td><td>yes</td><td>no</td></tr><tr><th>DOMINO (Beurer-Kellner et al., 2024)</th><td>CFG</td><td>yes</td><td>no</td></tr><tr><th>SynCode (Ugare et al., 2025)</th><td>CFG</td><td>yes</td><td>no</td></tr><tr><th>XGrammar (Dong et al., 2025)</th><td>CFG</td><td>yes</td><td>no</td></tr><tr><th>GreatGramma (Park et al., 2025)</th><td>CFG</td><td>yes</td><td>no</td></tr><tr><th>Pre3 (Chen et al., 2025)</th><td>DCFG</td><td>yes</td><td>no</td></tr><tr><th>GeLaTo (Zhang et al., 2023)</th><td>REG</td><td>yes</td><td>yes</td></tr><tr><th>Ctrl-G (Zhang et al., 2024)</th><td>REG</td><td>yes</td><td>yes</td></tr><tr><th>ABS (Collura et al., 2025)</th><td>REG</td><td>yes</td><td>yes</td></tr><tr><th>GenLM.control (Loula et al., 2025; Lipkin et al., 2025)</th><td>CFG/BB</td><td>yes</td><td>yes</td></tr><tr><th>SEM-CTRL (Albinhassan et al., 2026)</th><td>sem.</td><td>no</td><td>partial</td></tr><tr><th>Type-constr. (Mündler et al., 2025)</th><td>program</td><td>yes</td><td>no</td></tr><tr><th>TruncProof (Kato and Tarashima, 2026)</th><td>LL(1)</td><td>yes</td><td>yes</td></tr><tr><th>SWYB</th><td>CFG</td><td>yes</td><td>yes</td></tr></table>\",\"kind\":\"table\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"table\",\"source_ref\":\"tab30\",\"source_text\":\"Method Constraint Prefix Budget Guidance (Lundberg and et. al., 2024) CFG partial no Outlines (Willard and Louf, 2023) CFG yes no DOMINO (Beurer-Kellner et al., 2024) CFG yes no SynCode (Ugare et al., 2025) CFG yes no XGrammar (Dong et al., 2025) CFG yes no GreatGramma (Park et al., 2025) CFG yes no Pre3 (Chen et al., 2025) DCFG yes no GeLaTo (Zhang et al., 2023) REG yes yes Ctrl-G (Zhang et al., 2024) REG yes yes ABS (Collura et al., 2025) REG yes yes GenLM.control (Loula et al., 2025; Lipkin et al., 2025) CFG/BB yes yes SEM-CTRL (Albinhassan et al., 2026) sem. no partial Type-constr. (Mündler et al., 2025) program yes no TruncProof (Kato and Tarashima, 2026) LL(1) yes yes SWYB CFG yes yes\",\"tag\":null,\"text\":\"Method Constraint Prefix Budget Guidance (Lundberg and et. al., 2024) CFG partial no Outlines (Willard and Louf, 2023) CFG yes no DOMINO (Beurer-Kellner et al., 2024) CFG yes no SynCode (Ugare et al., 2025) CFG yes no XGrammar (Dong et al., 2025) CFG yes no GreatGramma (Park et al., 2025) CFG yes no Pre3 (Chen et al., 2025) DCFG yes no GeLaTo (Zhang et al., 2023) REG yes yes Ctrl-G (Zhang et al., 2024) REG yes yes ABS (Collura et al., 2025) REG yes yes GenLM.control (Loula et al., 2025; Lipkin et al., 2025) CFG/BB yes yes SEM-CTRL (Albinhassan et al., 2026) sem. no partial Type-constr. (Mündler et al., 2025) program yes no TruncProof (Kato and Tarashima, 2026) LL(1) yes yes SWYB CFG yes yes\"},{\"bbox\":[305.78,334.82,220.28,69.91],\"bbox_pdf\":[305.78,437.17,526.06,507.07],\"float_ref\":\"S2.T1\",\"fragments\":[{\"bbox_pdf\":[305.78,437.17,526.06,507.07],\"words\":[1569,1570,1571,1572,1573,1574,1575,1576,1577,1578,1579,1580,1581,1582,1583,1584,1585,1586,1587,1588,1589,1590,1591,1592,1593,1594,1595,1596,1597,1598,1599,1600,1601,1602,1603,1604,1605,1606,1607,1608,1609]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"caption\",\"source_ref\":\"caption31\",\"source_text\":\"Table 1: Positioning of SWYB relative to constrained and controlled decoding methods. \\\"Prefix\\\" indicates whether generation is kept within the prefix closure of the constraint language. \\\"Budget\\\" indicates whether decoding uses a remaining-horizon, distance-to-acceptance, or related lookahead signal.\",\"tag\":null,\"text\":\"Table 1: Positioning of SWYB relative to constrained and controlled decoding methods. \\\"Prefix\\\" indicates whether generation is kept within the prefix closure of the constraint language. \\\"Budget\\\" indicates whether decoding uses a remaining-horizon, distanceto-acceptance, or related lookahead signal.\"},{\"bbox\":[306.14,433.43,220.08,146.62],\"bbox_pdf\":[306.14,261.84,526.22,408.46],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[306.14,261.84,526.22,408.46],\"words\":[1610,1611,1612,1613,1614,1615,1616,1617,1618,1619,1620,1621,1622,1623,1624,1625,1626,1627,1628,1629,1630,1631,1632,1633,1634,1635,1636,1637,1638,1639,1640,1641,1642,1643,1644,1645,1646,1647,1648,1649,1650,1651,1652,1653,1654,1655,1656,1657,1658,1659,1660,1661,1662,1663,1664,1665,1666,1667,1668,1669,1670,1671,1672]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.986,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph29\",\"source_text\":\"Several methods specialize in constrained decoding for code or programming-language structure. Type-constrained decoding restricts generation using type information (Mündler et al., 2025), while correctness-guaranteed code generation studies decoding under stronger program-level specifications (Li et al., 2025). CFG-constrained decoding has also been extended to diffusion language models (mündler2025constraineddecodingdiffusionllms). These approaches are complementary to ours: they enrich the constraint language or model class, whereas SWYB adds sound distance-guided lookahead to CFG/PDA decoding.\",\"tag\":null,\"text\":\"constrained decoding restricts generation using type information (Mündler et al., 2025), while correctness-guaranteed code generation studies decoding under stronger program-level specifications (Li et al., 2025). CFG-constrained decoding has also been extended to diffusion language models (Mündler et al., 2026). These approaches are complementary to ours: they enrich the constraint language or model class, whereas SWYB adds sound distance-guided lookahead to CFG/PDA decoding.\"},{\"bbox\":[306.14,599.28,129.15,10.96],\"bbox_pdf\":[306.14,231.65,435.29,242.61],\"fragments\":[{\"bbox_pdf\":[306.14,231.65,435.29,242.61],\"words\":[1673,1674,1675]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":4,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"heading\",\"source_ref\":\"title32\",\"source_text\":\"Budget saturation problem.\",\"tag\":null,\"text\":\"Budget saturation problem.\"},{\"bbox\":[306.14,599.28,220.08,173.55],\"bbox_pdf\":[306.14,69.06,526.22,242.61],\"fragments\":[{\"bbox_pdf\":[306.14,69.06,526.22,242.61],\"words\":[1676,1677,1678,1679,1680,1681,1682,1683,1684,1685,1686,1687,1688,1689,1690,1691,1692,1693,1694,1695,1696,1697,1698,1699,1700,1701,1702,1703,1704,1705,1706,1707,1708,1709,1710,1711,1712,1713,1714,1715,1716,1717,1718,1719,1720,1721,1722,1723,1724,1725,1726,1727,1728,1729,1730,1731,1732,1733,1734,1735,1736,1737,1738,1739,1740,1741,1742,1743,1744,1745,1746,1747,1748,1749,1750,1751,1752,1753,1754,1755,1756,1757,1758,1759,1760,1761]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"paragraph\",\"source_ref\":\"paragraph33\",\"source_text\":\"Local constrained decoding methods enforce constraints based primarily on the current decoding state, but lack an explicit notion of distance to an accepting configuration. Consequently, locally valid actions may lead to increasingly complex intermediate structures without providing guidance on when to stop expanding and start closing them. This myopia can cause performance to saturate as the generation budget increases: providing additional tokens does not necessarily bring the decoder closer to satisfying the global constraints, as can be observed empirically in Appendix C.\",\"tag\":null,\"text\":\"Local constrained decoding methods enforce constraints based primarily on the current decoding state, but lack an explicit notion of distance to an accepting configuration. Consequently, locally valid actions may lead to increasingly complex intermediate structures without providing guidance on when to stop expanding and start closing them. This myopia can cause performance to saturate as the generation budget increases: providing additional tokens does not necessarily bring the decoder closer to satisfying the global constraints, as can be observed empirically in Appendix C.\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.28229\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.28229v1-p0002.pdf\",\"page_no\":3,\"page_size\":[595.28,841.89],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"9a7bba731e1c794e37ef2f3c98e6e3bbb6f41e98f53e8b0023321431e17b8c66\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph29\",\"c_block\":0.986,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.995,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.986,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"e9e67205617c354afae9d24187a0819b9a9165d63697ea12a5c5001cc94870d3\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/e9e67205617c354afae9d24187a0819b9a9165d63697ea12a5c5001cc94870d3.pdf","asset_sha256":"e9e67205617c354afae9d24187a0819b9a9165d63697ea12a5c5001cc94870d3","document_id":"arxiv:2608.28229","kind":"page","page_number":3,"paper_attribution_json":"{\"arxiv_id\":\"2608.28229\",\"authors\":\"Vincenzo Collura, Karim Tit, Eleonora Giunchiglia, Mike Papadakis, Maxime Cordy\",\"title\":\"Stay Within Your Bounds: Distance-Guided Decoding for Guaranteed Context-Free Grammar Compliance\",\"url\":\"https://arxiv.org/abs/2608.28229v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.28229\",\"authors\":\"Vincenzo Collura, Karim Tit, Eleonora Giunchiglia, Mike Papadakis, Maxime Cordy\",\"title\":\"Stay Within Your Bounds: Distance-Guided Decoding for Guaranteed Context-Free Grammar Compliance\",\"url\":\"https://arxiv.org/abs/2608.28229v1\",\"version\":1},\"blocks\":[{\"bbox\":[70.87,69.87,210.78,25.9],\"bbox_pdf\":[70.87,746.12,281.65,772.02],\"fragments\":[{\"bbox_pdf\":[70.87,746.12,281.65,772.02],\"words\":[1762,1763,1764,1765,1766,1767,1768,1769]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"heading\",\"source_ref\":\"title34\",\"source_text\":\"3 CFG-Constrained Decoding as a PDA Reachability Problem\",\"tag\":null,\"text\":\"3 CFG-Constrained Decoding as a PDA Reachability Problem\"},{\"bbox\":[70.35,106.16,220.59,105.97],\"bbox_pdf\":[70.35,629.76,290.94,735.73],\"fragments\":[{\"bbox_pdf\":[70.35,629.76,290.94,735.73],\"words\":[1770,1771,1772,1773,1774,1775,1776,1777,1778,1779,1780,1781,1782,1783,1784,1785,1786,1787,1788,1789,1790,1791,1792,1793,1794,1795,1796,1797,1798,1799,1800,1801,1802,1803,1804,1805,1806,1807,1808,1809,1810,1811,1812,1813,1814,1815,1816,1817,1818,1819,1820,1821,1822,1823]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph35\",\"source_text\":\"We consider LLM decoding under a context-free grammar (CFG) constraint. Since CFG and pushdown automata (PDA) define the same class of languages, any CFG can be represented by an equivalent PDA, and conversely, any PDA recognizes a context-free language (Hopcroft and Ullman, 1979). We therefore formulate the constraint through a PDA\",\"tag\":null,\"text\":\"We consider LLM decoding under a context-free grammar (CFG) constraint. Since CFG and pushdown automata (PDA) define the same class of languages, any CFG can be represented by an equivalent PDA, and conversely, any PDA recognizes a context-free language (Hopcroft and Ullman, 1979). We therefore formulate the constraint through a PDA\"},{\"bbox\":[118.01,226.45,171.86,12.55],\"bbox_pdf\":[118.01,602.9,289.87,615.44],\"fragments\":[{\"bbox_pdf\":[118.01,602.9,289.87,615.44],\"words\":[1824,1825,1826,1827,1828,1829,1830,1831,1832,1834]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathcal{P}=(Q,\\\\Sigma,\\\\Gamma,\\\\Delta,q_{0},z_{0},F),\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S3.E1\",\"source_text\":\"$\\\\mathcal{P}=(Q,\\\\Sigma,\\\\Gamma,\\\\Delta,q_{0},z_{0},F),$ (1)\",\"tag\":\"(1)\",\"text\":\"P = (Q, Σ, Γ, ∆, q0, z0, F (1)\"},{\"bbox\":[70.47,251.46,220.03,105.97],\"bbox_pdf\":[70.47,484.46,290.5,590.43],\"fragments\":[{\"bbox_pdf\":[70.47,484.46,290.5,590.43],\"words\":[1835,1836,1837,1838,1839,1840,1841,1842,1843,1844,1845,1846,1847,1848,1849,1850,1851,1852,1853,1854,1855,1856,1857,1858,1859,1860,1861,1862,1863,1864,1865,1866,1867,1868,1869,1870,1871,1872,1873,1874,1875,1876,1877,1878,1879,1880,1881,1882,1883,1884,1885,1886,1887,1888,1889,1890,1891,1892,1893,1894,1895,1896,1897,1898,1899,1900,1901,1902,1903,1904,1905,1906,1907,1908,1909,1911,1912,1913,1914,1915,1916,1917,1918,1919,1920,1921,1922,1923,1924]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph37\",\"source_text\":\"where $Q$ is a finite set of control states, $\\\\Sigma$ is the input alphabet whose elements correspond to CFG terminals, $\\\\Gamma$ is the stack alphabet, $\\\\Delta\\\\subseteq Q\\\\times(\\\\Sigma\\\\cup\\\\{\\\\varepsilon\\\\})\\\\times\\\\Gamma\\\\times Q\\\\times% \\\\Gamma^{*}$ is the transition relation, $q_{0}\\\\in Q$ is the initial state, $z_{0}\\\\in\\\\Gamma$ is the initial stack symbol, and $F\\\\subseteq Q$ is the set of accepting states. A PDA configuration is a pair $(q,\\\\gamma)\\\\in Q\\\\times\\\\Gamma^{*}$ , where $\\\\gamma$ is the stack, written with its top on the left. We write\",\"tag\":null,\"text\":\"where Q is a finite set of control states, Σ is the input alphabet whose elements correspond to CFG terminals, Γ is the stack alphabet, ∆ ⊆ Q × (Σ ∪ {ε})×Γ×Q×Γ ∗ is the transition relation, q0 ∈ Q is the initial state, z0 ∈ Γ is the initial stack symbol, and F ⊆ Q is the set of accepting states. A PDA configuration is a pair (q, γ) ∈ Q × Γ ∗ where γ is the stack, written with its top on the left. We write\"},{\"bbox\":[143.78,370.19,146.08,12.47],\"bbox_pdf\":[143.78,459.23,289.86,471.7],\"fragments\":[{\"bbox_pdf\":[143.78,459.23,289.86,471.7],\"words\":[1925,1926,1927,1928,1929,1930,1931,1932,1933,1935]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"(q,\\\\gamma)\\\\xrightarrow{a}(q^{\\\\prime},\\\\gamma^{\\\\prime})\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"formula\",\"source_ref\":\"S3.E2\",\"source_text\":\"$(q,\\\\gamma)\\\\xrightarrow{a}(q^{\\\\prime},\\\\gamma^{\\\\prime})$ (2)\",\"tag\":\"(2)\",\"text\":\"a ′ ′ (q, γ) −→ (q , γ (2)\"},{\"bbox\":[70.47,396.99,218.66,38.01],\"bbox_pdf\":[70.47,406.9,289.14,444.9],\"fragments\":[{\"bbox_pdf\":[70.47,406.9,289.14,444.9],\"words\":[1936,1937,1938,1939,1940,1941,1942,1943,1944,1945,1946,1947,1948,1949,1950,1951,1952,1953,1954,1955,1956,1958,1959,1960,1961,1962,1963,1964,1965,1966,1967,1968]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph39\",\"source_text\":\"when the PDA can consume $a\\\\in\\\\Sigma\\\\cup\\\\{\\\\varepsilon\\\\}$ and move from $(q,\\\\gamma)$ to $(q^{\\\\prime},\\\\gamma^{\\\\prime})$ . The language on $\\\\Sigma$ accepted by $\\\\mathcal{P}$ is denoted as $\\\\mathcal{L}(\\\\mathcal{P})$ .\",\"tag\":null,\"text\":\"when the PDA can consume a ∈ Σ ∪ {ε} and move from (q, γ) to (q ′ , γ ′ The language on Σ accepted by P is denoted as L(P).\"},{\"bbox\":[70.87,445.72,126.55,11.02],\"bbox_pdf\":[70.87,385.15,197.42,396.17],\"fragments\":[{\"bbox_pdf\":[70.87,385.15,197.42,396.17],\"words\":[1969,1970]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":4,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"heading\",\"source_ref\":\"title40\",\"source_text\":\"Token-level configurations.\",\"tag\":null,\"text\":\"Token-level configurations.\"},{\"bbox\":[69.59,445.72,221.36,146.51],\"bbox_pdf\":[69.59,249.66,290.95,396.17],\"fragments\":[{\"bbox_pdf\":[69.59,249.66,290.95,396.17],\"words\":[1971,1972,1973,1974,1975,1976,1977,1978,1980,1981,1982,1983,1984,1985,1986,1987,1988,1989,1990,1991,1992,1993,1994,1995,1996,1997,1998,1999,2000,2001,2002,2003,2004,2005,2006,2007,2008,2009,2010,2011,2012,2013,2014,2015,2016,2017,2018,2019,2020,2021,2022,2023,2024,2025,2026,2028,2029,2030,2031,2032,2033,2034,2035,2036,2037,2038,2039,2040,2041,2042,2043,2044,2045,2046,2047,2048,2049,2050,2051,2052,2053,2054,2055,2056,2057,2058,2060,2061,2062,2063,2064,2065,2066,2067]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph41\",\"source_text\":\"LLMs generate tokens from a vocabulary $V$ , whereas the PDA consumes terminals from $\\\\Sigma$ . A token may represent a full terminal, several terminals, a strict prefix of a terminal, or a string crossing terminal boundaries. We therefore track an extended configuration $(q,\\\\gamma,\\\\rho)\\\\in Q\\\\times\\\\Gamma^{*}\\\\times\\\\Sigma^{*}$ , where $\\\\rho$ is the open prefix of a terminal currently being matched; $\\\\rho=\\\\varepsilon$ means that no terminal is open. Let $\\\\mathcal{E}=Q\\\\times\\\\Gamma^{*}\\\\times\\\\Sigma^{*}$ . Token consumption is represented as a transition function\",\"tag\":null,\"text\":\"LLMs generate tokens from a vocabulary V whereas the PDA consumes terminals from Σ. A token may represent a full terminal, several terminals, a strict prefix of a terminal, or a string crossing terminal boundaries. We therefore track an extended configuration (q, γ, ρ) ∈ Q × Γ ∗ × Σ ∗ where ρ is the open prefix of a terminal currently being matched; ρ = ε means that no terminal is open. Let E = Q × Γ ∗ × Σ ∗ Token consumption is represented as a transition function\"},{\"bbox\":[123.62,604.99,166.24,12.47],\"bbox_pdf\":[123.62,224.43,289.86,236.9],\"fragments\":[{\"bbox_pdf\":[123.62,224.43,289.86,236.9],\"words\":[2068,2070,2071,2072,2073,2074,2075,2076,2077]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\mathrm{Consume}:\\\\mathcal{E}\\\\times V\\\\to 2^{\\\\mathcal{E}},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"formula\",\"source_ref\":\"S3.E3\",\"source_text\":\"$\\\\mathrm{Consume}:\\\\mathcal{E}\\\\times V\\\\to 2^{\\\\mathcal{E}},$ (3)\",\"tag\":\"(3)\",\"text\":\"Consume E × V → 2 E , (3)\"},{\"bbox\":[70.47,631.78,218.67,51.56],\"bbox_pdf\":[70.47,158.55,289.14,210.11],\"fragments\":[{\"bbox_pdf\":[70.47,158.55,289.14,210.11],\"words\":[2078,2079,2080,2081,2082,2083,2084,2085,2086,2087,2088,2089,2090,2091,2092,2093,2094,2095,2096,2097,2098,2099,2100,2101,2102]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"paragraph\",\"source_ref\":\"paragraph43\",\"source_text\":\"where $\\\\mathrm{Consume}((q,\\\\gamma,\\\\rho),t)$ contains all extended configurations reachable by matching the decoded token string against CFG terminals and applying the induced PDA transitions.\",\"tag\":null,\"text\":\"where Consume((q, γ, ρ), t) contains all extended configurations reachable by matching the decoded token string against CFG terminals and applying the induced PDA transitions.\"},{\"bbox\":[70.87,693.96,157.16,11.13],\"bbox_pdf\":[70.87,136.8,228.03,147.93],\"fragments\":[{\"bbox_pdf\":[70.87,136.8,228.03,147.93],\"words\":[2103,2104,2105,2106]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":4,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"heading\",\"source_ref\":\"title44\",\"source_text\":\"Reachability and token distance.\",\"tag\":null,\"text\":\"Reachability and token distance.\"},{\"bbox\":[70.47,693.96,220.57,78.87],\"bbox_pdf\":[70.47,69.06,291.04,147.93],\"fragments\":[{\"bbox_pdf\":[70.47,69.06,291.04,147.93],\"words\":[2107,2108,2109,2110,2111,2112,2113,2114,2115,2116,2117,2118,2119,2120,2121,2122,2123,2124,2125,2126,2127,2128,2129,2130,2131,2132,2133,2134,2135,2136,2137,2138,2139,2140,2141,2142,2143,2144,2145,2146,2148,2149,2150,2151,2152,2153,2154]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"paragraph\",\"source_ref\":\"paragraph45\",\"source_text\":\"For a stack bound $H$ , let $\\\\mathcal{S}_{H}\\\\subseteq Q\\\\times\\\\Gamma^{\\\\leq H}$ be a bounded pushdown summary. Reachability in this summary follows the symbolic view of pushdown systems, in which predecessor sets can be computed by $\\\\mathrm{Pre}^{*}$ -style saturation procedures (Bouajjani et al., 1997). We write $\\\\mathrm{Reach}_{H}(q,\\\\gamma)$ when $(q,\\\\gamma)$ can reach an accepting configuration within $\\\\mathcal{S}_{H}$ .\",\"tag\":null,\"text\":\"For a stack bound H, let SH ⊆ Q × Γ ≤H be a bounded pushdown summary. Reachability in this summary follows the symbolic view of pushdown systems, in which predecessor sets can be computed by Pre ∗ style saturation procedures (Bouajjani et al., 1997).\"},{\"bbox\":[305.63,70.7,218.78,24.68],\"bbox_pdf\":[305.63,746.52,524.41,771.19],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[305.63,746.52,524.41,771.19],\"words\":[2155,2156,2157,2158,2159,2160,2161,2162,2163,2164,2165,2166,2167,2168]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"paragraph\",\"source_ref\":\"paragraph45\",\"source_text\":\"For a stack bound $H$ , let $\\\\mathcal{S}_{H}\\\\subseteq Q\\\\times\\\\Gamma^{\\\\leq H}$ be a bounded pushdown summary. Reachability in this summary follows the symbolic view of pushdown systems, in which predecessor sets can be computed by $\\\\mathrm{Pre}^{*}$ -style saturation procedures (Bouajjani et al., 1997). We write $\\\\mathrm{Reach}_{H}(q,\\\\gamma)$ when $(q,\\\\gamma)$ can reach an accepting configuration within $\\\\mathcal{S}_{H}$ .\",\"tag\":null,\"text\":\"We write ReachH(q, γ) when (q, γ) can reach an accepting configuration within SH.\"},{\"bbox\":[306.14,97.84,220.08,200.77],\"bbox_pdf\":[306.14,543.28,526.23,744.05],\"fragments\":[{\"bbox_pdf\":[306.14,543.28,526.23,744.05],\"words\":[2169,2170,2171,2172,2173,2174,2175,2176,2177,2178,2179,2180,2181,2182,2183,2184,2185,2186,2187,2188,2189,2190,2191,2192,2193,2194,2195,2196,2197,2198,2199,2200,2201,2202,2203,2204,2205,2206,2207,2208,2209,2210,2211,2212,2213,2214,2215,2216,2217,2218,2219,2220,2221,2222,2223,2224,2225,2226,2227,2228,2229,2230,2231,2232,2233,2234,2235,2236,2237,2238,2239,2240,2241,2242,2243,2244,2245,2246,2247,2248,2249,2250,2251,2252,2253,2254,2255,2256,2257,2258,2259,2260,2261,2262,2263,2264,2265,2266,2267,2268,2269,2270,2271,2272,2273,2274,2275,2276,2277,2278,2279,2280,2281,2282,2283]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":13,\"role\":\"paragraph\",\"source_ref\":\"paragraph46\",\"source_text\":\"To associate summarized configurations with a token-level distance estimate, we turn the PDA into a weighted pushdown system. For each PDA transition consuming a terminal $a$ , we assign the weight $w(a)=$ the minimum number of LLM tokens required to generate $a$ . If a terminal is specified by a regular expression $R$ , we first identify a shortest string $s_{R}\\\\in L(R)$ and compute the minimum tokenization length of $s_{R}$ using the tokenizer vocabulary. Epsilon transitions are assigned weight zero. Under a stack-height bound $H$ , we then apply the standard weighted $\\\\mathrm{Pre}^{*}$ algorithm to compute the minimum accumulated token cost of reaching an accepting configuration. We denote the resulting quantity by\",\"tag\":null,\"text\":\"To associate summarized configurations with a token-level distance estimate, we turn the PDA into a weighted pushdown system. For each PDA transition consuming a terminal a, we assign the weight w(a) = the minimum number of LLM tokens required to generate a. If a terminal is specified by a regular expression R, we first identify a shortest string sR ∈ L(R) and compute the minimum tokenization length of sR using the tokenizer vocabulary. Epsilon transitions are assigned weight zero. Under a stack-height bound H, we then apply the standard weighted Pre ∗ algorithm to compute the minimum accumulated token cost of reaching an accepting configuration. We denote the resulting quantity by\"},{\"bbox\":[365.61,308.19,159.52,12.6],\"bbox_pdf\":[365.61,521.1,525.14,533.7],\"fragments\":[{\"bbox_pdf\":[365.61,521.1,525.14,533.7],\"words\":[2284,2286,2287,2288,2289,2290,2291]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"d_{H}:\\\\mathcal{S}_{H}\\\\to\\\\mathbb{N}\\\\cup\\\\{\\\\infty\\\\}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"S3.E4\",\"source_text\":\"$d_{H}:\\\\mathcal{S}_{H}\\\\to\\\\mathbb{N}\\\\cup\\\\{\\\\infty\\\\}.$ (4)\",\"tag\":\"(4)\",\"text\":\"dH SH → N ∪ {∞}. (4)\"},{\"bbox\":[305.75,328.68,220.57,65.11],\"bbox_pdf\":[305.75,448.1,526.32,513.21],\"fragments\":[{\"bbox_pdf\":[305.75,448.1,526.32,513.21],\"words\":[2292,2293,2294,2295,2296,2297,2298,2299,2300,2301,2302,2303,2304,2305,2306,2307,2308,2309,2310,2311,2312,2313,2314,2315,2316,2317,2318,2319,2320,2321,2322,2323,2324,2325,2326,2327,2328,2329,2330,2331]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":15,\"role\":\"paragraph\",\"source_ref\":\"paragraph48\",\"source_text\":\"In the exact weighted bounded system, $d_{H}(q,\\\\gamma)$ coincides with the true minimum token cost $d_{H}^{*}(q,\\\\gamma)$ . More generally, when conservative approximations are used in the construction of the terminal weights, we require the estimate to satisfy\",\"tag\":null,\"text\":\"In the exact weighted bounded system, dH(q, γ) coincides with the true minimum token cost d ∗ H(q, γ). More generally, when conservative approximations are used in the construction of the terminal weights, we require the estimate to satisfy\"},{\"bbox\":[367.89,401.81,157.25,15.17],\"bbox_pdf\":[367.89,424.91,525.15,440.08],\"fragments\":[{\"bbox_pdf\":[367.89,424.91,525.15,440.08],\"words\":[2332,2333,2334,2335,2336,2337,2338,2339]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"d_{H}(q,\\\\gamma)\\\\geq d_{H}^{*}(q,\\\\gamma).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":16,\"role\":\"formula\",\"source_ref\":\"S3.E5\",\"source_text\":\"$d_{H}(q,\\\\gamma)\\\\geq d_{H}^{*}(q,\\\\gamma).$ (5)\",\"tag\":\"(5)\",\"text\":\"dH(q, γ) ≥ d ∗ (5) H(q, γ).\"},{\"bbox\":[305.8,423.64,220.51,65.32],\"bbox_pdf\":[305.8,352.92,526.32,418.25],\"fragments\":[{\"bbox_pdf\":[305.8,352.92,526.32,418.25],\"words\":[2340,2341,2342,2343,2344,2345,2346,2347,2348,2349,2350,2351,2352,2353,2354,2355,2356,2357,2358,2359,2360,2361,2362,2363,2364,2365,2366,2367,2368,2369,2370,2371,2372,2373,2374,2375,2376,2377,2378]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":17,\"role\":\"paragraph\",\"source_ref\":\"paragraph50\",\"source_text\":\"Thus, if the remaining token budget is $K$ and $d_{H}(q,\\\\gamma)>K$ , no accepting continuation of length at most $K$ exists within the bounded summary. The resulting estimate is therefore used for budget-aware pruning in Equation (5).\",\"tag\":null,\"text\":\"Thus, if the remaining token budget is K and dH(q, γ) > K, no accepting continuation of length at most K exists within the bounded summary. The resulting estimate is therefore used for budgetaware pruning in Equation (5).\"},{\"bbox\":[306.14,498.09,48.29,11.13],\"bbox_pdf\":[306.14,332.67,354.43,343.8],\"fragments\":[{\"bbox_pdf\":[306.14,332.67,354.43,343.8],\"words\":[2379]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":4,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":18,\"role\":\"heading\",\"source_ref\":\"title51\",\"source_text\":\"Objective.\",\"tag\":null,\"text\":\"Objective.\"},{\"bbox\":[306.14,498.09,220.08,51.77],\"bbox_pdf\":[306.14,292.02,526.22,343.8],\"fragments\":[{\"bbox_pdf\":[306.14,292.02,526.22,343.8],\"words\":[2380,2381,2382,2383,2384,2385,2386,2387,2388,2389,2390,2391,2392,2393,2394,2395,2396,2397,2398,2399,2400,2401,2402,2403]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":19,\"role\":\"paragraph\",\"source_ref\":\"paragraph52\",\"source_text\":\"We formulate CFG-constrained decoding as the problem of producing the highest-probability sequence accepted by the PDA within a finite token budget $T$ :\",\"tag\":null,\"text\":\"We formulate CFG-constrained decoding as the problem of producing the highestprobability sequence accepted by the PDA within a finite token budget T\"},{\"bbox\":[335.18,559.27,189.96,39.47],\"bbox_pdf\":[335.18,243.15,525.14,282.62],\"fragments\":[{\"bbox_pdf\":[335.18,243.15,525.14,282.62],\"words\":[2405,2406,2407,2408,2409,2410,2411,2412,2413,2414,2415,2416,2417,2418,2419]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"y^{\\\\star}\\\\in\\\\arg\\\\max_{\\\\begin{subarray}{c}y\\\\in\\\\mathcal{L}(\\\\mathcal{P})\\\\\\\\\\n|y|\\\\leq T\\\\end{subarray}}\\\\sum_{i=1}^{|y|}\\\\log p_{\\\\theta}(y_{i}\\\\mid y_{<i}).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":20,\"role\":\"formula\",\"source_ref\":\"S3.E6\",\"source_text\":\"$y^{\\\\star}\\\\in\\\\arg\\\\max_{\\\\begin{subarray}{c}y\\\\in\\\\mathcal{L}(\\\\mathcal{P})\\\\\\\\ |y|\\\\leq T\\\\end{subarray}}\\\\sum_{i=1}^{|y|}\\\\log p_{\\\\theta}(y_{i}\\\\mid y_{<i}).$ (6)\",\"tag\":\"(6)\",\"text\":\"∑︂ |y| y ⋆ ∈ arg max log pθ(yi | y<i). (6) y∈L(P) |y|≤T i=1\"},{\"bbox\":[306.14,606.46,220.08,91.99],\"bbox_pdf\":[306.14,143.44,526.23,235.43],\"fragments\":[{\"bbox_pdf\":[306.14,143.44,526.23,235.43],\"words\":[2420,2421,2422,2423,2424,2425,2426,2427,2428,2429,2430,2431,2432,2433,2434,2435,2436,2437,2438,2439,2440,2441,2442,2443,2444,2445,2446,2447,2448,2449,2450,2451,2452,2453,2454,2455,2456,2457,2458,2459,2460,2461,2462,2463]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":21,\"role\":\"paragraph\",\"source_ref\":\"paragraph54\",\"source_text\":\"Exact optimization of this sequence-level objective is intractable in general: for autoregressive models, exact maximum-a-posteriori decoding is NP-hard, even under simple unary constraints (Pachet and Roy, 2026). Accordingly, SWYB treats Eq. (6) as an approximate search objective, guided by reachability and distance estimates.\",\"tag\":null,\"text\":\"Exact optimization of this sequence-level objective is intractable in general: for autoregressive models, exact maximum-a-posteriori decoding is NP-hard, even under simple unary constraints (Pachet and Roy, 2026). Accordingly, SWYB treats Eq. (6) as an approximate search objective, guided by reachability and distance estimates.\"},{\"bbox\":[306.14,709.6,108.37,11.96],\"bbox_pdf\":[306.14,120.33,414.52,132.29],\"fragments\":[{\"bbox_pdf\":[306.14,120.33,414.52,132.29],\"words\":[2464,2465,2466]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":22,\"role\":\"heading\",\"source_ref\":\"title55\",\"source_text\":\"4 Proposed Method\",\"tag\":null,\"text\":\"4 Proposed Method\"},{\"bbox\":[306.14,730.9,69.16,10.91],\"bbox_pdf\":[306.14,100.08,375.31,110.99],\"fragments\":[{\"bbox_pdf\":[306.14,100.08,375.31,110.99],\"words\":[2467,2468]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":2,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":23,\"role\":\"heading\",\"source_ref\":\"title56\",\"source_text\":\"4.1 Overview\",\"tag\":null,\"text\":\"4.1 Overview\"},{\"bbox\":[306.14,748.16,220.18,24.68],\"bbox_pdf\":[306.14,69.06,526.32,93.73],\"fragments\":[{\"bbox_pdf\":[306.14,69.06,526.32,93.73],\"words\":[2469,2470,2471,2472,2473,2474,2475,2476,2477,2478,2479,2480,2481,2482,2483]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":24,\"role\":\"paragraph\",\"source_ref\":\"paragraph57\",\"source_text\":\"SWYB is a two-phase decoder for LLM generation under CFG constraints represented as PDAs. Offline, it builds the bounded summary $\\\\mathcal{S}_{H}$ with reachability labels $\\\\mathrm{Reach}_{H}$ and token-distance estimates $d_{H}$ . Online, our beam search, with a width of $M$ , maintains extended configurations $(q,\\\\gamma,\\\\rho)$ and uses these labels to validate, prune, and rank token candidates. Algorithm 1 describes the online decoding loop, Figure 1 summarizes the full pipeline, and a practical example is given in Appendix D. Each beam stores a prefix $y$ , a score $\\\\ell$ , and an extended configuration $(q,\\\\gamma,\\\\rho)$ . Given a total token budget $T$ , at step $k$ , with the remaining horizon $K=T-k$ , the decoder builds a candidate set, executes candidates through the PDA, filters out unreachable or over-budget successors, and keeps the top $M$ scored extensions. Here $\\\\mathrm{Accept}(q,\\\\gamma,\\\\rho)$ means that the PDA acceptance condition holds and no terminal is open, i.e., $\\\\rho=\\\\varepsilon$ .\",\"tag\":null,\"text\":\"SWYB is a two-phase decoder for LLM generation under CFG constraints represented as PDAs.\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.28229\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.28229v1-p0003.pdf\",\"page_no\":4,\"page_size\":[595.28,841.89],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"9a7bba731e1c794e37ef2f3c98e6e3bbb6f41e98f53e8b0023321431e17b8c66\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"29abcb1007eeb83f3fb64614134a7a1dae4f0485eecd601e3f4693b12810984f\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/29abcb1007eeb83f3fb64614134a7a1dae4f0485eecd601e3f4693b12810984f.pdf","asset_sha256":"29abcb1007eeb83f3fb64614134a7a1dae4f0485eecd601e3f4693b12810984f","document_id":"arxiv:2608.28229","kind":"page","page_number":4,"paper_attribution_json":"{\"arxiv_id\":\"2608.28229\",\"authors\":\"Vincenzo Collura, Karim Tit, Eleonora Giunchiglia, Mike Papadakis, Maxime Cordy\",\"title\":\"Stay Within Your Bounds: Distance-Guided Decoding for Guaranteed Context-Free Grammar Compliance\",\"url\":\"https://arxiv.org/abs/2608.28229v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
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{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[365.61,308.19,159.52,12.6],\"bbox_pdf\":[365.61,521.1,525.14,533.7],\"fragments\":[{\"bbox_pdf\":[365.61,521.1,525.14,533.7],\"words\":[2284,2286,2287,2288,2289,2290,2291]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"d_{H}:\\\\mathcal{S}_{H}\\\\to\\\\mathbb{N}\\\\cup\\\\{\\\\infty\\\\}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"S3.E4\",\"source_text\":\"$d_{H}:\\\\mathcal{S}_{H}\\\\to\\\\mathbb{N}\\\\cup\\\\{\\\\infty\\\\}.$ (4)\",\"tag\":\"(4)\",\"text\":\"dH SH → N ∪ {∞}. (4)\"},\"formula_index\":14,\"parent_page_sha256\":\"29abcb1007eeb83f3fb64614134a7a1dae4f0485eecd601e3f4693b12810984f\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/29abcb1007eeb83f3fb64614134a7a1dae4f0485eecd601e3f4693b12810984f.pdf","asset_sha256":"29abcb1007eeb83f3fb64614134a7a1dae4f0485eecd601e3f4693b12810984f","document_id":"arxiv:2608.28229","kind":"formula","page_number":4,"paper_attribution_json":"{\"arxiv_id\":\"2608.28229\",\"authors\":\"Vincenzo Collura, Karim Tit, Eleonora Giunchiglia, Mike Papadakis, Maxime Cordy\",\"title\":\"Stay Within Your Bounds: Distance-Guided Decoding for Guaranteed Context-Free Grammar Compliance\",\"url\":\"https://arxiv.org/abs/2608.28229v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
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{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.28229\",\"authors\":\"Vincenzo Collura, Karim Tit, Eleonora Giunchiglia, Mike Papadakis, Maxime Cordy\",\"title\":\"Stay Within Your Bounds: Distance-Guided Decoding for Guaranteed Context-Free Grammar Compliance\",\"url\":\"https://arxiv.org/abs/2608.28229v1\",\"version\":1},\"blocks\":[{\"bbox\":[70.87,70.7,70.55,11.13],\"bbox_pdf\":[70.87,760.07,141.41,771.19],\"fragments\":[{\"bbox_pdf\":[70.87,760.07,141.41,771.19],\"words\":[5248,5249]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":4,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"heading\",\"source_ref\":\"title123\",\"source_text\":\"Main findings.\",\"tag\":null,\"text\":\"Main findings.\"},{\"bbox\":[70.47,70.7,220.48,146.62],\"bbox_pdf\":[70.47,624.57,290.95,771.19],\"fragments\":[{\"bbox_pdf\":[70.47,624.57,290.95,771.19],\"words\":[5250,5251,5252,5253,5254,5255,5256,5257,5258,5259,5260,5261,5262,5263,5264,5265,5266,5267,5268,5269,5270,5271,5272,5273,5274,5275,5276,5277,5278,5279,5280,5281,5282,5283,5284,5285,5286,5287,5288,5289,5290,5291,5292,5293,5294,5295,5296,5297,5298,5299,5300,5301,5302,5303,5304,5305,5306,5307,5308,5309,5310,5311,5312,5313,5314,5315,5316,5317,5318,5319,5320,5321,5322,5323,5324]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"paragraph\",\"source_ref\":\"paragraph124\",\"source_text\":\"Across all three domains and base models, SWYB is the only method that achieves $100\\\\%$ syntactic correctness in every setting. On JSON, this also corresponds to $100\\\\%$ schema validity across all models. On SQL and LTL, SWYB improves task-specific accuracy while preserving syntactic validity. The cost is a moderate decoding-time overhead relative to lightweight local masking methods, but it is comparable to that of repeated sample-and-verify, which achieves lower performance in all domains.\",\"tag\":null,\"text\":\"Across all three domains and base models, SWYB is the only method that achieves 100% syntactic correctness in every setting. On JSON, this also corresponds to 100% schema validity across all models. On SQL and LTL, SWYB improves task-specific accuracy while preserving syntactic validity. The cost is a moderate decoding-time overhead relative to lightweight local masking methods, but it is comparable to that of repeated sample-and-verify, which achieves lower performance in all domains.\"},{\"bbox\":[70.87,229.33,77.23,10.77],\"bbox_pdf\":[70.87,601.79,148.1,612.56],\"fragments\":[{\"bbox_pdf\":[70.87,601.79,148.1,612.56],\"words\":[5325,5326]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":4,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"heading\",\"source_ref\":\"title125\",\"source_text\":\"Ablation studies.\",\"tag\":null,\"text\":\"Ablation studies.\"},{\"bbox\":[70.87,229.33,220.08,200.46],\"bbox_pdf\":[70.87,412.1,290.95,612.56],\"fragments\":[{\"bbox_pdf\":[70.87,412.1,290.95,612.56],\"words\":[5327,5328,5329,5330,5331,5332,5333,5334,5335,5336,5337,5338,5339,5340,5341,5342,5343,5344,5345,5346,5347,5348,5349,5350,5351,5352,5353,5354,5355,5356,5357,5358,5359,5360,5361,5362,5363,5364,5365,5366,5367,5368,5369,5370,5371,5372,5373,5374,5375,5376,5377,5378,5379,5380,5381,5382,5383,5384,5385,5386,5387,5388,5389,5390,5391,5392,5393,5394,5395,5396,5397,5398,5399,5400,5401,5402,5403,5404,5405,5406,5407,5408,5409,5410,5411,5412,5413,5414,5415,5416,5417,5418,5419,5420,5421,5422,5423,5424,5425,5426,5427,5428,5429]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph126\",\"source_text\":\"We assess the contributions of the two main components of SWYB, Beam Search and Distance-Guided Scoring (DGS), through ablations on SQL and LTL, where task-quality metrics are available (Appendix B). Across all three models, syntactic correctness remains at $100\\\\%$ in every configuration, confirming that syntax guarantees are independent of these components. Beam Search provides the largest gains, substantially improving execution accuracy on SQL and semantic accuracy on LTL while reducing perplexity. DGS adds smaller but consistent improvements in task accuracy with negligible decoding overhead. Overall, combining Beam Search and DGS yields the best trade-off between efficiency and quality.\",\"tag\":null,\"text\":\"We assess the contributions of the two main components of SWYB, Beam Search and Distance-Guided Scoring (DGS), through ablations on SQL and LTL, where task-quality metrics are available (Appendix B). Across all three models, syntactic correctness remains at 100% in every configuration, confirming that syntax guarantees are independent of these components. Beam Search provides the largest gains, substantially improving execution accuracy on SQL and semantic accuracy on LTL while reducing perplexity. DGS adds smaller but consistent improvements in task accuracy with negligible decoding overhead. Overall, combining Beam Search and DGS yields the best trade-off between efficiency and quality.\"},{\"bbox\":[70.87,443.14,107.88,10.91],\"bbox_pdf\":[70.87,387.84,178.75,398.75],\"fragments\":[{\"bbox_pdf\":[70.87,387.84,178.75,398.75],\"words\":[5430,5431,5432]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":2,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"heading\",\"source_ref\":\"title127\",\"source_text\":\"5.1 JSON Generation\",\"tag\":null,\"text\":\"5.1 JSON Generation\"},{\"bbox\":[70.47,461.47,220.57,200.82],\"bbox_pdf\":[70.47,179.61,291.04,380.42],\"fragments\":[{\"bbox_pdf\":[70.47,179.61,291.04,380.42],\"words\":[5433,5434,5435,5436,5437,5438,5439,5440,5441,5442,5443,5444,5445,5446,5447,5448,5449,5450,5451,5452,5453,5454,5455,5456,5457,5458,5459,5460,5461,5462,5463,5464,5465,5466,5467,5468,5469,5470,5471,5472,5473,5474,5475,5476,5477,5478,5479,5480,5481,5482,5483,5484,5485,5486,5487,5488,5489,5490,5491,5492,5493,5494,5495,5496,5497,5498,5499,5500,5501,5502,5503,5504,5505,5506,5507,5508,5509,5510,5511,5512,5513,5514,5515,5516,5517,5518,5519,5520,5521,5522,5523,5524,5525,5526,5527,5528,5529,5530,5531,5532,5533,5534,5535,5536,5537,5538,5539,5540,5541]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph128\",\"source_text\":\"Table 2 reports results on json-mode-eval. SWYB is the only method that achieves perfect syntactic and schema validity across all three base models. Other constrained decoders generally improve over the Base LM on the two Llama models, but their gains are less consistent on Qwen2.5-7B, where the unconstrained model is already strong. SWYB maintains 100% validity in all settings, at a moderate runtime cost: it is slower than lightweight masking-based methods, but faster than Sample-Verify and competitive with GenLM at higher particle counts. Its perplexity remains close to the best baseline, suggesting that the guarantee does not come at the cost of substantially lower model likelihood.\",\"tag\":null,\"text\":\"Table 2 reports results on json-mode-eval. SWYB is the only method that achieves perfect syntactic and schema validity across all three base models. Other constrained decoders generally improve over the Base LM on the two Llama models, but their gains are less consistent on Qwen2.5-7B, where the unconstrained model is already strong. SWYB maintains 100% validity in all settings, at a moderate runtime cost: it is slower than lightweight masking-based methods, but faster than SampleVerify and competitive with GenLM at higher particle counts. Its perplexity remains close to the best baseline, suggesting that the guarantee does not come at the cost of substantially lower model likelihood.\"},{\"bbox\":[70.87,675.63,101.83,10.91],\"bbox_pdf\":[70.87,155.35,172.69,166.26],\"fragments\":[{\"bbox_pdf\":[70.87,155.35,172.69,166.26],\"words\":[5542,5543,5544]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":2,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"heading\",\"source_ref\":\"title129\",\"source_text\":\"5.2 SQL Generation\",\"tag\":null,\"text\":\"5.2 SQL Generation\"},{\"bbox\":[70.53,694.27,220.41,78.57],\"bbox_pdf\":[70.53,69.06,290.94,147.62],\"fragments\":[{\"bbox_pdf\":[70.53,69.06,290.94,147.62],\"words\":[5545,5546,5547,5548,5549,5550,5551,5552,5553,5554,5555,5556,5557,5558,5559,5560,5561,5562,5563,5564,5565,5566,5567,5568,5569,5570,5571,5572,5573,5574,5575,5576,5577,5578,5579,5580,5581,5582,5583,5584,5585,5586,5587,5588,5589]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph130\",\"source_text\":\"Table 3 reports results on Spider. SWYB achieves perfect syntactic correctness across all three base models and the highest execution accuracy in every setting. Sample-Verify also reaches perfect or near-perfect syntax but does not improve execution accuracy to the same extent, showing that syntactic filtering alone is insufficient for this task. SynCode variants underperform the Base LM in execution accuracy, while GenLM is competitive on the Llama models but degrades on Qwen2.5-7B-Instruct. In contrast, SWYB consistently improves both syntactic correctness and execution accuracy across architectures. Its runtime is comparable to Sample-Verify and GenLM on the Llama models, though it is slower on Qwen, and its perplexity remains the best or second-best in all settings. Overall, SWYB provides the strongest trade-off between syntactic guarantees, execution accuracy, and decoding cost.\",\"tag\":null,\"text\":\"Table 3 reports results on Spider. SWYB achieves perfect syntactic correctness across all three base models and the highest execution accuracy in every setting. Sample-Verify also reaches perfect or near-perfect syntax but does not improve execution accuracy to the same extent, showing that syntactic\"},{\"bbox\":[305.75,71.13,220.48,159.73],\"bbox_pdf\":[305.75,611.02,526.23,770.76],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[305.75,611.02,526.23,770.76],\"words\":[5590,5591,5592,5593,5594,5595,5596,5597,5598,5599,5600,5601,5602,5603,5604,5605,5606,5607,5608,5609,5610,5611,5612,5613,5614,5615,5616,5617,5618,5619,5620,5621,5622,5623,5624,5625,5626,5627,5628,5629,5630,5631,5632,5633,5634,5635,5636,5637,5638,5639,5640,5641,5642,5643,5644,5645,5646,5647,5648,5649,5650,5651,5652,5653,5654,5655,5656,5657,5658,5659,5660,5661,5662,5663,5664,5665,5666,5667,5668,5669,5670,5671,5672,5673,5674,5675,5676]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph130\",\"source_text\":\"Table 3 reports results on Spider. SWYB achieves perfect syntactic correctness across all three base models and the highest execution accuracy in every setting. Sample-Verify also reaches perfect or near-perfect syntax but does not improve execution accuracy to the same extent, showing that syntactic filtering alone is insufficient for this task. SynCode variants underperform the Base LM in execution accuracy, while GenLM is competitive on the Llama models but degrades on Qwen2.5-7B-Instruct. In contrast, SWYB consistently improves both syntactic correctness and execution accuracy across architectures. Its runtime is comparable to Sample-Verify and GenLM on the Llama models, though it is slower on Qwen, and its perplexity remains the best or second-best in all settings. Overall, SWYB provides the strongest trade-off between syntactic guarantees, execution accuracy, and decoding cost.\",\"tag\":null,\"text\":\"filtering alone is insufficient for this task. SynCode variants underperform the Base LM in execution accuracy, while GenLM is competitive on the Llama models but degrades on Qwen2.5-7B-Instruct. In contrast, SWYB consistently improves both syntactic correctness and execution accuracy across architectures. Its runtime is comparable to SampleVerify and GenLM on the Llama models, though it is slower on Qwen, and its perplexity remains the best or second-best in all settings. Overall, SWYB provides the strongest trade-off between syntactic guarantees, execution accuracy, and decoding cost.\"},{\"bbox\":[306.14,248.51,100.82,10.91],\"bbox_pdf\":[306.14,582.47,406.96,593.38],\"fragments\":[{\"bbox_pdf\":[306.14,582.47,406.96,593.38],\"words\":[5677,5678,5679]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":2,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"heading\",\"source_ref\":\"title131\",\"source_text\":\"5.3 LTL Generation\",\"tag\":null,\"text\":\"5.3 LTL Generation\"},{\"bbox\":[305.8,269.64,220.42,159.84],\"bbox_pdf\":[305.8,412.41,526.23,572.26],\"fragments\":[{\"bbox_pdf\":[305.8,412.41,526.23,572.26],\"words\":[5680,5681,5682,5683,5684,5685,5686,5687,5688,5689,5690,5691,5692,5693,5694,5695,5696,5697,5698,5699,5700,5701,5702,5703,5704,5705,5706,5707,5708,5709,5710,5711,5712,5713,5714,5715,5716,5717,5718,5719,5720,5721,5722,5723,5724,5725,5726,5727,5728,5729,5730,5731,5732,5733,5734,5735,5736,5737,5738,5739,5740,5741,5742,5743,5744,5745,5746,5747,5748,5749,5750,5751,5752,5753,5754,5755,5756,5757,5758,5759,5760,5761,5762,5763,5764,5765,5766]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"paragraph\",\"source_ref\":\"paragraph132\",\"source_text\":\"Table 4 reports results on the LTL dataset. SWYB consistently achieves $100\\\\%$ syntactic correctness and the highest task accuracy across all three models. SynCode improves task accuracy over the Base LM in every setting, but provides smaller gains and does not consistently guarantee syntactic correctness, reaching $100\\\\%$ only for Qwen2.5-7B. In particular, for Llama-3.1-8B, its syntactic correctness is slightly below that of Sample-Verify. Overall, SWYB provides the strongest combination of syntax guarantees and task accuracy, at the cost of a moderate decoding-time overhead.\",\"tag\":null,\"text\":\"Table 4 reports results on the LTL dataset. SWYB consistently achieves 100% syntactic correctness and the highest task accuracy across all three models. SynCode improves task accuracy over the Base LM in every setting, but provides smaller gains and does not consistently guarantee syntactic correctness, reaching 100% only for Qwen2.5-7B. In particular, for Llama-3.1-8B, its syntactic correctness is slightly below that of Sample-Verify. Overall, SWYB provides the strongest combination of syntax guarantees and task accuracy, at the cost of a moderate decoding-time overhead.\"},{\"bbox\":[306.14,447.46,75.07,11.96],\"bbox_pdf\":[306.14,382.48,381.21,394.43],\"fragments\":[{\"bbox_pdf\":[306.14,382.48,381.21,394.43],\"words\":[5767,5768]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"heading\",\"source_ref\":\"title133\",\"source_text\":\"6 Conclusion\",\"tag\":null,\"text\":\"6 Conclusion\"},{\"bbox\":[305.63,473.61,218.79,38.23],\"bbox_pdf\":[305.63,330.05,524.41,368.28],\"fragments\":[{\"bbox_pdf\":[305.63,330.05,524.41,368.28],\"words\":[5769,5770,5771,5772,5773,5774,5775,5776,5777,5778,5779,5780,5781,5782,5783,5784,5785]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"paragraph\",\"source_ref\":\"paragraph134\",\"source_text\":\"We have introduced SWYB, a distance-guided decoding framework for guaranteed context-free grammar compliance in large language models.\",\"tag\":null,\"text\":\"We have introduced SWYB, a distance-guided decoding framework for guaranteed context-free grammar compliance in large language models.\"},{\"bbox\":[306.14,516.4,220.08,132.71],\"bbox_pdf\":[306.14,192.78,526.23,325.49],\"fragments\":[{\"bbox_pdf\":[306.14,192.78,526.23,325.49],\"words\":[5786,5787,5788,5789,5790,5791,5792,5793,5794,5795,5796,5797,5798,5799,5800,5801,5802,5803,5804,5805,5806,5807,5808,5809,5810,5811,5812,5813,5814,5815,5816,5817,5818,5819,5820,5821,5822,5823,5824,5825,5826,5827,5828,5829,5830,5831,5832,5833,5834,5835,5836,5837,5838,5839,5840,5841,5842,5843,5844,5845,5846,5847]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":13,\"role\":\"paragraph\",\"source_ref\":\"paragraph135\",\"source_text\":\"Unlike prior grammar-constrained decoders that rely solely on local prefix feasibility, SWYB augments token-level validation with bounded pushdown summaries and upper-bound distance estimates to acceptance. This enables budget-aware pruning and reranking: configurations whose estimated completion cost exceeds the remaining token horizon are discarded; conversely, our beam search softly promotes tokens that make measurable structural progress toward acceptance.\",\"tag\":null,\"text\":\"Unlike prior grammar-constrained decoders that rely solely on local prefix feasibility, SWYB augments token-level validation with bounded pushdown summaries and upper-bound distance estimates to acceptance. This enables budget-aware pruning and reranking: configurations whose estimated completion cost exceeds the remaining token horizon are discarded; conversely, our beam search softly promotes tokens that make measurable structural progress toward acceptance.\"},{\"bbox\":[305.75,653.31,220.47,119.52],\"bbox_pdf\":[305.75,69.06,526.22,188.58],\"fragments\":[{\"bbox_pdf\":[305.75,69.06,526.22,188.58],\"words\":[5848,5849,5850,5851,5852,5853,5854,5855,5856,5857,5858,5859,5860,5861,5862,5863,5864,5865,5866,5867,5868,5869,5870,5871,5872,5873,5874,5875,5876,5877,5878,5879,5880,5881,5882,5883,5884,5885,5886,5887,5888,5889,5890,5891,5892,5893,5894,5895,5896,5897,5898,5899,5900,5901]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"paragraph\",\"source_ref\":\"paragraph136\",\"source_text\":\"Experiments on JSON, SQL, and LTL generation demonstrate that SWYB consistently achieves perfect syntactic validity across three diverse base models while also improving task-specific accuracy over existing methods, including Sample-Verify, SynCode, XGrammar, and GenLM. The method maintains competitive perplexity and decoding time, offering a favorable trade-off between guarantee and efficiency.\",\"tag\":null,\"text\":\"Experiments on JSON, SQL, and LTL generation demonstrate that SWYB consistently achieves perfect syntactic validity across three diverse base models while also improving task-specific accuracy over existing methods, including SampleVerify, SynCode, XGrammar, and GenLM. The method maintains competitive perplexity and decoding time, offering a favorable trade-off between guarantee and efficiency.\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.28229\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.28229v1-p0008.pdf\",\"page_no\":9,\"page_size\":[595.28,841.89],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"9a7bba731e1c794e37ef2f3c98e6e3bbb6f41e98f53e8b0023321431e17b8c66\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"777bf5a9ed004b6936a87a02be6a1d10b4ce29fbba8280513b8cac1260258035\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/777bf5a9ed004b6936a87a02be6a1d10b4ce29fbba8280513b8cac1260258035.pdf","asset_sha256":"777bf5a9ed004b6936a87a02be6a1d10b4ce29fbba8280513b8cac1260258035","document_id":"arxiv:2608.28229","kind":"page","page_number":9,"paper_attribution_json":"{\"arxiv_id\":\"2608.28229\",\"authors\":\"Vincenzo Collura, Karim Tit, Eleonora Giunchiglia, Mike Papadakis, Maxime Cordy\",\"title\":\"Stay Within Your Bounds: Distance-Guided Decoding for Guaranteed Context-Free Grammar Compliance\",\"url\":\"https://arxiv.org/abs/2608.28229v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.28764\",\"authors\":\"M. Flores, A. Kahrs, F. Rinderknecht, and Arturo C. Mart\\\\'i\",\"title\":\"Stable levitation of ring magnets: A low-cost, 3D-printed experiment on the magnetic force-distance law\",\"url\":\"https://arxiv.org/abs/2608.28764v2\",\"version\":2},\"blocks\":[{\"bbox\":[54.0,52.07,83.3,9.96],\"bbox_pdf\":[54.0,729.96,137.3,739.93],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[54.0,729.96,137.3,739.93],\"words\":[752,753,754,755,756]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"paragraph\",\"source_ref\":\"paragraph16\",\"source_text\":\"In static equilibrium, the net force vector on the upper magnet must vanish. The magnitude of the magnetic force therefore balances exactly the total weight of the levitating system, which comprises the mass of the upper magnet ( $m_{\\\\text{magnet}}$ ) plus any additional mass ( $m_{\\\\text{add}}$ ) placed on top of it,\",\"tag\":null,\"text\":\"placed on top of it,\"},{\"bbox\":[116.07,74.14,183.01,11.46],\"bbox_pdf\":[116.07,706.41,299.08,717.86],\"fragments\":[{\"bbox_pdf\":[116.07,706.41,299.08,717.86],\"words\":[757,758,759,760,761,762,763]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"F_{\\\\text{mag}}=\\\\left(m_{\\\\text{magnet}}+m_{\\\\text{add}}\\\\right)g,\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S2.E2\",\"source_text\":\"$F_{\\\\text{mag}}=\\\\left(m_{\\\\text{magnet}}+m_{\\\\text{add}}\\\\right)g,$ (2)\",\"tag\":\"(2)\",\"text\":\"Fmag = (mmagnet + madd) g, (2)\"},{\"bbox\":[54.0,96.2,245.08,32.88],\"bbox_pdf\":[54.0,662.92,299.08,695.8],\"fragments\":[{\"bbox_pdf\":[54.0,662.92,299.08,695.8],\"words\":[764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"paragraph\",\"source_ref\":\"paragraph18\",\"source_text\":\"where $g$ is the gravitational acceleration. Combining Eqs. (1) and (2) shows that the equilibrium separation $h$ is inversely proportional to the applied weight.\",\"tag\":null,\"text\":\"where g is the gravitational acceleration. Combining Eqs. (1) and (2) shows that the equilibrium separation h is inversely proportional to the applied weight.\"},{\"bbox\":[54.0,130.89,245.08,78.7],\"bbox_pdf\":[54.0,582.4,299.08,661.11],\"fragments\":[{\"bbox_pdf\":[54.0,582.4,299.08,661.11],\"words\":[788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph19\",\"source_text\":\"Consequently, the model predicts a linear relationship between the inverse separation $h^{-1}$ and the magnitude of the equilibrium magnetic force. This prediction allows students not only to verify the physical law but also to determine experimentally an intrinsic parameter of the magnet, namely its effective current $I$ or its magnetic dipole moment $\\\\mu$ .\",\"tag\":null,\"text\":\"Consequently, the model predicts a linear relationship between the inverse separation h −1 and the magnitude of the equilibrium magnetic force. This prediction allows students not only to verify the physical law but also to determine experimentally an intrinsic parameter of the magnet, namely its effective current I or its magnetic dipole moment µ.\"},{\"bbox\":[54.0,211.41,245.08,78.7],\"bbox_pdf\":[54.0,501.88,299.08,580.59],\"fragments\":[{\"bbox_pdf\":[54.0,501.88,299.08,580.59],\"words\":[842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"paragraph\",\"source_ref\":\"paragraph20\",\"source_text\":\"Alternatively, the axial magnetic field $B(z)$ generated by a single ring magnet at a distance $z$ along its symmetry axis can be modeled by considering the magnet as an equivalent thin current-loop. Under this representation, the on-axis magnetic field strength can be written as a function of the magnetic dipole moment $\\\\mu$ and a purely geometric factor $g(z)$ according to [2]:\",\"tag\":null,\"text\":\"Alternatively, the axial magnetic field B(z) generated by a single ring magnet at a distance z along its symmetry axis can be modeled by considering the magnet as an equivalent thin current-loop. Under this representation, the on-axis magnetic field strength can be written as a function of the magnetic dipole moment µ and a purely geometric factor g(z) according to [2]:\"},{\"bbox\":[93.77,294.64,205.31,31.02],\"bbox_pdf\":[93.77,466.33,299.08,497.36],\"fragments\":[{\"bbox_pdf\":[93.77,466.33,299.08,497.36],\"words\":[903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"B(z)=\\\\mu\\\\,g(z)=\\\\mu\\\\left[\\\\frac{\\\\mu_{0}}{2\\\\pi}\\\\frac{1}{(z^{2}+R^{2})^{3/2}}%\\n\\\\right].\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S2.E3\",\"source_text\":\"$B(z)=\\\\mu\\\\,g(z)=\\\\mu\\\\left[\\\\frac{\\\\mu_{0}}{2\\\\pi}\\\\frac{1}{(z^{2}+R^{2})^{3/2}}% \\\\right].$ (3)\",\"tag\":\"(3)\",\"text\":\"[︃ ]︃ µ0 1 B(z) = µ g(z) = µ . (3) 2π (z2 + R2)3/2\"},{\"bbox\":[54.0,336.71,245.09,32.88],\"bbox_pdf\":[54.0,422.41,299.09,455.29],\"fragments\":[{\"bbox_pdf\":[54.0,422.41,299.09,455.29],\"words\":[919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"paragraph\",\"source_ref\":\"paragraph22\",\"source_text\":\"This alternative expression provides an independent, first-principles-based method to characterize the magnet’s parameters through a fit.\",\"tag\":null,\"text\":\"This alternative expression provides an independent, first-principles-based method to characterize the magnet’s parameters through a fit.\"},{\"bbox\":[93.65,395.47,165.79,8.97],\"bbox_pdf\":[93.65,387.57,259.43,396.53],\"fragments\":[{\"bbox_pdf\":[93.65,387.57,259.43,396.53],\"words\":[1048,1049,1050]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":1,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"heading\",\"source_ref\":\"title23\",\"source_text\":\"III Experimental method\",\"tag\":null,\"text\":\"III. EXPERIMENTAL METHOD\"},{\"bbox\":[54.0,420.42,245.09,113.08],\"bbox_pdf\":[54.0,258.5,299.09,371.58],\"fragments\":[{\"bbox_pdf\":[54.0,258.5,299.09,371.58],\"words\":[1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph24\",\"source_text\":\"The experimental setup was designed for simplicity and reproducibility. To counteract the natural tendency of the upper magnet to flip and stick laterally to the lower one—as dictated by the Earnshaw instability—we designed a coaxial cylindrical guide. The apparatus utilizes two identical ferrite ring magnets with an internal diameter of $17~{}\\\\text{mm}$ , an external diameter of $32~{}\\\\text{mm}$ , and an axial thickness of $6~{}\\\\text{mm}$ . The conceptual principle of the assembly and its transition to a stable solution are illustrated in Fig. 1.\",\"tag\":null,\"text\":\"The experimental setup was designed for simplicity and reproducibility. To counteract the natural tendency of the upper magnet to flip and stick laterally to the lower one—as dictated by the Earnshaw instability—we designed a coaxial cylindrical guide. The apparatus utilizes two identical ferrite ring magnets with an internal diameter of 17 mm, an external diameter of 32 mm, and an axial thickness of 6 mm. The conceptual principle of the assembly and its transition to a stable solution are illustrated in Fig. 1.\"},{\"bbox\":[100.33,548.04,152.43,117.99],\"bbox_pdf\":[100.33,125.98,252.75,243.96],\"float_ref\":\"S3.F1\",\"fragments\":[{\"bbox_pdf\":[100.33,125.98,252.75,243.96],\"words\":[]}],\"html\":null,\"kind\":\"figure\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"figure\",\"source_ref\":\"S3.F1\",\"source_text\":\"\",\"tag\":null,\"text\":\"\"},{\"bbox\":[54.0,677.48,245.08,40.35],\"bbox_pdf\":[54.0,74.17,299.08,114.52],\"float_ref\":\"S3.F1\",\"fragments\":[{\"bbox_pdf\":[54.0,74.17,299.08,114.52],\"words\":[1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.974,\"verified\":true},\"order\":10,\"role\":\"caption\",\"source_ref\":\"caption26\",\"source_text\":\"Figure 1: Illustration of magnetic levitation. On the left, two ring-shaped magnets face each other, one above the other. On the right, the same arrangement is shown with a cylindrical guide that provides stability to the levitating magnet.\",\"tag\":null,\"text\":\"FIG. 1. Illustration of magnetic levitation. On the left, two ring-shaped magnets face each other, one above the other. On the right, the same arrangement is shown with a cylindrical guide that provides stability to the levitating magnet.\"},{\"bbox\":[317.01,52.07,245.08,101.62],\"bbox_pdf\":[317.01,638.31,562.1,739.93],\"fragments\":[{\"bbox_pdf\":[317.01,638.31,562.1,739.93],\"words\":[936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":11,\"role\":\"paragraph\",\"source_ref\":\"paragraph27\",\"source_text\":\"To realize the cylindrical guide while minimizing friction and ensuring perfect concentric alignment, we modeled and 3D-printed a coupled male–female device using ABS (acrylonitrile butadiene styrene), a non-magnetic material that does not interfere with the field lines. The geometric details of the printed parts are shown in Fig. 2. To facilitate reproduction of the setup in other educational settings, the 3D models are provided as STL files in a public, open-access repository (see Sec. VI).\",\"tag\":null,\"text\":\"To realize the cylindrical guide while minimizing friction and ensuring perfect concentric alignment, we modeled and 3D-printed a coupled male–female device using ABS (acrylonitrile butadiene styrene), a non-magnetic material that does not interfere with the field lines. The geometric details of the printed parts are shown in Fig. 2. To facilitate reproduction of the setup in other educational settings, the 3D models are provided as STL files in a public, open-access repository (see Sec. VI).\"},{\"bbox\":[353.18,169.11,172.75,138.16],\"bbox_pdf\":[353.18,484.73,525.93,622.89],\"float_ref\":\"S3.F2\",\"fragments\":[{\"bbox_pdf\":[353.18,484.73,525.93,622.89],\"words\":[]}],\"html\":null,\"kind\":\"figure\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"figure\",\"source_ref\":\"S3.F2\",\"source_text\":\"\",\"tag\":null,\"text\":\"\"},{\"bbox\":[317.01,318.73,245.08,19.43],\"bbox_pdf\":[317.01,453.84,562.1,473.27],\"float_ref\":\"S3.F2\",\"fragments\":[{\"bbox_pdf\":[317.01,453.84,562.1,473.27],\"words\":[1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.955,\"verified\":true},\"order\":13,\"role\":\"caption\",\"source_ref\":\"caption29\",\"source_text\":\"Figure 2: Experimental design produced for 3D printing. On the left is the “female” part and on the right the “male” part.\",\"tag\":null,\"text\":\"FIG. 2. Experimental design produced for 3D printing. On the left is the “female” part and on the right the “male” part.\"},{\"bbox\":[317.01,353.82,245.08,21.42],\"bbox_pdf\":[317.01,416.76,562.09,438.18],\"fragments\":[{\"bbox_pdf\":[317.01,416.76,562.09,438.18],\"words\":[1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"paragraph\",\"source_ref\":\"paragraph30\",\"source_text\":\"The measurement protocol follows an incremental loading sequence, sketched in Fig. 3.\",\"tag\":null,\"text\":\"The measurement protocol follows an incremental loading sequence, sketched in Fig. 3.\"},{\"bbox\":[317.61,390.1,243.88,141.89],\"bbox_pdf\":[317.61,260.01,561.5,401.9],\"float_ref\":\"S3.F3\",\"fragments\":[{\"bbox_pdf\":[317.61,260.01,561.5,401.9],\"words\":[]}],\"html\":null,\"kind\":\"figure\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":15,\"role\":\"figure\",\"source_ref\":\"S3.F3\",\"source_text\":\"\",\"tag\":null,\"text\":\"\"},{\"bbox\":[317.01,543.45,245.08,71.73],\"bbox_pdf\":[317.01,176.82,562.1,248.55],\"float_ref\":\"S3.F3\",\"fragments\":[{\"bbox_pdf\":[317.01,176.82,562.1,248.55],\"words\":[1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.983,\"verified\":true},\"order\":16,\"role\":\"caption\",\"source_ref\":\"caption32\",\"source_text\":\"Figure 3: Illustration of the experimental assembly. The male–female device is shown (colored gray and green, respectively), with the two magnets facing each other so that the magnetic force is repulsive and the upper magnet levitates. Three stages are shown that differ in the mechanical load applied to the top of the device, reducing the separation $h$ between the magnets.\",\"tag\":null,\"text\":\"FIG. 3. Illustration of the experimental assembly. The male– female device is shown (colored gray and green, respectively), with the two magnets facing each other so that the magnetic force is repulsive and the upper magnet levitates. Three stages are shown that differ in the mechanical load applied to the top of the device, reducing the separation h between the magnets.\"},{\"bbox\":[317.01,631.4,245.08,90.16],\"bbox_pdf\":[317.01,70.44,562.1,160.6],\"fragments\":[{\"bbox_pdf\":[317.01,70.44,562.1,160.6],\"words\":[1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":17,\"role\":\"paragraph\",\"source_ref\":\"paragraph33\",\"source_text\":\"The general procedure consists of the following steps: (i) the mass of the levitating system—initially consisting of the upper magnet and the male part—is determined to be $(28.740\\\\pm 0.005)~{}\\\\text{g}$ ; (ii) the setup is assembled as in Fig. 3, in a male–female arrangement with the magnets facing each other so that the resulting magnetic force is repulsive; (iii) the equilibrium separation $h$ is measured with a ruler (resolution of $0.5~{}\\\\text{mm}$ ); and (iv) known masses, consisting of a set of identical non-magnetic calibration weights with an individual mass of $m=(16.240\\\\pm 0.005)~{}\\\\text{g}$ , are sequentially added on the upper platform of the male part, and the new equilibrium separation $h$ is recorded for each added mass. In this way we obtain a table of equilibrium separations $h$ as a function of the additional mass $m_{\\\\text{add}}$ , from which the magnitude of the magnetic force $F_{\\\\text{mag}}$ can be computed.\",\"tag\":null,\"text\":\"The general procedure consists of the following steps: (i) the mass of the levitating system—initially consisting of the upper magnet and the male part—is determined to be (28.740 ± 0.005) g; (ii) the setup is assembled as in Fig. 3, in a male–female arrangement with the magnets facing each other so that the resulting magnetic force is repulsive; (iii) the equilibrium separation h is measured with a ruler (resolution of 0.5 mm); and\"},{\"bbox\":[557.11,26.17,4.98,9.96],\"bbox_pdf\":[557.11,755.87,562.09,765.83],\"fragments\":[{\"bbox_pdf\":[557.11,755.87,562.09,765.83],\"words\":[3706]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":18,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"2\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.28764\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.28764v2-p0001.pdf\",\"page_no\":2,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"74d191a23ad34924738893868c939081782fae733198661153ea602f96f2a0ae\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"caption29\",\"c_block\":0.955,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":0.993,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.955,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"a7061af0ac98f573b73a6dfdb23dbf150594133896fe2558a837d797d2dd547f\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/a7061af0ac98f573b73a6dfdb23dbf150594133896fe2558a837d797d2dd547f.pdf","asset_sha256":"a7061af0ac98f573b73a6dfdb23dbf150594133896fe2558a837d797d2dd547f","document_id":"arxiv:2608.28764","kind":"page","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.28764\",\"authors\":\"M. Flores, A. Kahrs, F. Rinderknecht, and Arturo C. Mart\\\\'i\",\"title\":\"Stable levitation of ring magnets: A low-cost, 3D-printed experiment on the magnetic force-distance law\",\"url\":\"https://arxiv.org/abs/2608.28764v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[116.07,74.14,183.01,11.46],\"bbox_pdf\":[116.07,706.41,299.08,717.86],\"fragments\":[{\"bbox_pdf\":[116.07,706.41,299.08,717.86],\"words\":[757,758,759,760,761,762,763]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"F_{\\\\text{mag}}=\\\\left(m_{\\\\text{magnet}}+m_{\\\\text{add}}\\\\right)g,\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"formula\",\"source_ref\":\"S2.E2\",\"source_text\":\"$F_{\\\\text{mag}}=\\\\left(m_{\\\\text{magnet}}+m_{\\\\text{add}}\\\\right)g,$ (2)\",\"tag\":\"(2)\",\"text\":\"Fmag = (mmagnet + madd) g, (2)\"},\"formula_index\":1,\"parent_page_sha256\":\"a7061af0ac98f573b73a6dfdb23dbf150594133896fe2558a837d797d2dd547f\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/a7061af0ac98f573b73a6dfdb23dbf150594133896fe2558a837d797d2dd547f.pdf","asset_sha256":"a7061af0ac98f573b73a6dfdb23dbf150594133896fe2558a837d797d2dd547f","document_id":"arxiv:2608.28764","kind":"formula","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.28764\",\"authors\":\"M. Flores, A. Kahrs, F. Rinderknecht, and Arturo C. Mart\\\\'i\",\"title\":\"Stable levitation of ring magnets: A low-cost, 3D-printed experiment on the magnetic force-distance law\",\"url\":\"https://arxiv.org/abs/2608.28764v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[93.77,294.64,205.31,31.02],\"bbox_pdf\":[93.77,466.33,299.08,497.36],\"fragments\":[{\"bbox_pdf\":[93.77,466.33,299.08,497.36],\"words\":[903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"B(z)=\\\\mu\\\\,g(z)=\\\\mu\\\\left[\\\\frac{\\\\mu_{0}}{2\\\\pi}\\\\frac{1}{(z^{2}+R^{2})^{3/2}}%\\n\\\\right].\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"formula\",\"source_ref\":\"S2.E3\",\"source_text\":\"$B(z)=\\\\mu\\\\,g(z)=\\\\mu\\\\left[\\\\frac{\\\\mu_{0}}{2\\\\pi}\\\\frac{1}{(z^{2}+R^{2})^{3/2}}% \\\\right].$ (3)\",\"tag\":\"(3)\",\"text\":\"[︃ ]︃ µ0 1 B(z) = µ g(z) = µ . (3) 2π (z2 + R2)3/2\"},\"formula_index\":5,\"parent_page_sha256\":\"a7061af0ac98f573b73a6dfdb23dbf150594133896fe2558a837d797d2dd547f\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/a7061af0ac98f573b73a6dfdb23dbf150594133896fe2558a837d797d2dd547f.pdf","asset_sha256":"a7061af0ac98f573b73a6dfdb23dbf150594133896fe2558a837d797d2dd547f","document_id":"arxiv:2608.28764","kind":"formula","page_number":2,"paper_attribution_json":"{\"arxiv_id\":\"2608.28764\",\"authors\":\"M. Flores, A. Kahrs, F. Rinderknecht, and Arturo C. Mart\\\\'i\",\"title\":\"Stable levitation of ring magnets: A low-cost, 3D-printed experiment on the magnetic force-distance law\",\"url\":\"https://arxiv.org/abs/2608.28764v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.28764\",\"authors\":\"M. Flores, A. Kahrs, F. Rinderknecht, and Arturo C. Mart\\\\'i\",\"title\":\"Stable levitation of ring magnets: A low-cost, 3D-printed experiment on the magnetic force-distance law\",\"url\":\"https://arxiv.org/abs/2608.28764v2\",\"version\":2},\"blocks\":[{\"bbox\":[72.32,53.07,226.76,29.89],\"bbox_pdf\":[72.32,709.04,299.08,738.93],\"continues_previous\":true,\"fragments\":[{\"bbox_pdf\":[72.32,709.04,299.08,738.93],\"words\":[3349,3350,3351,3352,3353,3354,3355,3356,3357,3358,3359,3360,3361,3362,3363,3364,3365,3366,3367]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":0,\"role\":\"reference\",\"source_ref\":\"bib.bib10\",\"source_text\":\"M. R. Osorio, D. E. Lahera, and H. Suderow, “Magnetic levitation on a type-I superconductor as a practical demonstration experiment for students,” Eur. J. Phys. 33(5), 1383–1395 (2012).\",\"tag\":null,\"text\":\"netic levitation on a type-I superconductor as a practical demonstration experiment for students,” Eur. J. Phys. 33(5), 1383–1395 (2012).\"},{\"bbox\":[54.0,84.45,245.09,40.35],\"bbox_pdf\":[54.0,667.2,299.09,707.55],\"fragments\":[{\"bbox_pdf\":[54.0,667.2,299.09,707.55],\"words\":[3368,3369,3370,3371,3372,3373,3374,3375,3376,3377,3378,3379,3380,3381,3382,3383,3384,3385,3386,3387,3388,3389,3390,3391,3392,3393,3394,3395]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":1,\"role\":\"reference\",\"source_ref\":\"bib.bib11\",\"source_text\":\"M. Giliberti, L. Perotti, and L. Rossi, “Motion of a superconducting loop in an inhomogeneous magnetic field: A didactic experiment,” Eur. J. Phys. 39(5), 055503 (2018).\",\"tag\":null,\"text\":\"[11] M. Giliberti, L. Perotti, and L. Rossi, “Motion of a superconducting loop in an inhomogeneous magnetic field: A didactic experiment,” Eur. J. Phys. 39(5), 055503 (2018).\"},{\"bbox\":[54.0,126.3,245.08,29.89],\"bbox_pdf\":[54.0,635.82,299.08,665.7],\"fragments\":[{\"bbox_pdf\":[54.0,635.82,299.08,665.7],\"words\":[3396,3397,3398,3399,3400,3401,3402,3403,3404,3405,3406,3407,3408,3409,3410,3411,3412]}],\"html\":null,\"kind\":\"list_item\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"reference\",\"source_ref\":\"bib.bib12\",\"source_text\":\"W. Meissner and R. Ochsenfeld, “Ein neuer Effekt bei Eintritt der Supraleitfähigkeit,” Naturwissenschaften 21(44), 787–788 (1933).\",\"tag\":null,\"text\":\"[12] W. Meissner and R. 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Magn. 47(8), 2045–2049 (2011).\"},{\"bbox\":[557.11,26.17,4.98,9.96],\"bbox_pdf\":[557.11,755.87,562.09,765.83],\"fragments\":[{\"bbox_pdf\":[557.11,755.87,562.09,765.83],\"words\":[3710]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":14,\"role\":\"page_number\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"6\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.28764\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.28764v2-p0005.pdf\",\"page_no\":6,\"page_size\":[612.0,792.0],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"74d191a23ad34924738893868c939081782fae733198661153ea602f96f2a0ae\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":null,\"c_block\":1.0,\"c_lines\":1.0,\"c_pdf\":1.0,\"c_region\":1.0,\"c_src\":1.0,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":1.0,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"e63e3354688864b4eeba2fa11fe17463df81ad40a7764adf76e8790382da7e06\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/e63e3354688864b4eeba2fa11fe17463df81ad40a7764adf76e8790382da7e06.pdf","asset_sha256":"e63e3354688864b4eeba2fa11fe17463df81ad40a7764adf76e8790382da7e06","document_id":"arxiv:2608.28764","kind":"page","page_number":6,"paper_attribution_json":"{\"arxiv_id\":\"2608.28764\",\"authors\":\"M. Flores, A. Kahrs, F. Rinderknecht, and Arturo C. Mart\\\\'i\",\"title\":\"Stable levitation of ring magnets: A low-cost, 3D-printed experiment on the magnetic force-distance law\",\"url\":\"https://arxiv.org/abs/2608.28764v2\",\"version\":2}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"annotates\":[\"heading\",\"list_item\",\"table\",\"paragraph\",\"furniture\"],\"attribution\":{\"arxiv_id\":\"2608.30178\",\"authors\":\"Lina Ji, Jie Xiong, Wen Xu, Xu Yang\",\"title\":\"Extinction and extinguishment properties for a nonlinear predator-prey branching model\",\"url\":\"https://arxiv.org/abs/2608.30178v1\",\"version\":1},\"blocks\":[{\"bbox\":[185.63,51.52,224.01,7.97],\"bbox_pdf\":[185.63,782.4,409.65,790.37],\"fragments\":[{\"bbox_pdf\":[185.63,782.4,409.65,790.37],\"words\":[5551,5552,5553,5554]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":0,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"NONLINEAR PREDATOR-PREY BRANCHING MODEL\"},{\"bbox\":[513.37,51.52,8.47,7.97],\"bbox_pdf\":[513.37,782.4,521.84,790.37],\"fragments\":[{\"bbox_pdf\":[513.37,782.4,521.84,790.37],\"words\":[5555]}],\"html\":null,\"kind\":\"furniture\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":null,\"verified\":true},\"order\":1,\"role\":\"page_header\",\"source_ref\":null,\"source_text\":\"\",\"tag\":null,\"text\":\"11\"},{\"bbox\":[145.02,81.48,305.24,35.65],\"bbox_pdf\":[145.02,724.76,450.26,760.41],\"fragments\":[{\"bbox_pdf\":[145.02,724.76,450.26,760.41],\"words\":[5561,5562,5563,5564,5565,5566,5567,5568,5569,5570,5571,5572,5573,5574,5575,5576,5577,5578,5579,5580,5581,5582,5583,5584,5585,5586]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"T_{*}<\\\\infty\\\\Longleftrightarrow\\\\int_{0}^{\\\\varepsilon}y^{-\\\\theta_{2}+(a_{1}/a_{%\\n2})\\\\kappa_{2}}\\\\,\\\\mathrm{d}y<\\\\infty\\\\Longleftrightarrow\\\\frac{a_{1}}{a_{2}}\\\\kappa%\\n_{2}>\\\\theta_{2}-1=\\\\kappa_{1}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S2.Ex23\",\"source_text\":\"$T_{*}<\\\\infty\\\\Longleftrightarrow\\\\int_{0}^{\\\\varepsilon}y^{-\\\\theta_{2}+(a_{1}/a_{% 2})\\\\kappa_{2}}\\\\,\\\\mathrm{d}y<\\\\infty\\\\Longleftrightarrow\\\\frac{a_{1}}{a_{2}}\\\\kappa% _{2}>\\\\theta_{2}-1=\\\\kappa_{1}.$\",\"tag\":null,\"text\":\"y ↓ 0. Moreover, ∫︂ ε −θ2+(a1/a2)κ2 dy < ∞ ⇐⇒ a1 T∗ < ∞ ⇐⇒ y κ2 > θ2 − 1 = κ1. 0 a2\"},{\"bbox\":[73.05,119.77,100.58,10.91],\"bbox_pdf\":[73.05,711.21,173.63,722.12],\"fragments\":[{\"bbox_pdf\":[73.05,711.21,173.63,722.12],\"words\":[5587,5588,5589,5590]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":3,\"role\":\"paragraph\",\"source_ref\":\"paragraph195\",\"source_text\":\"This proves all cases. ∎\",\"tag\":null,\"text\":\"This proves all cases.\"},{\"bbox\":[73.44,140.04,188.45,10.96],\"bbox_pdf\":[73.44,690.88,261.89,701.85],\"fragments\":[{\"bbox_pdf\":[73.44,690.88,261.89,701.85],\"words\":[5592,5593,5594,5595,5596]}],\"html\":null,\"kind\":\"heading\",\"latex\":null,\"level\":2,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":4,\"role\":\"heading\",\"source_ref\":\"title196\",\"source_text\":\"2.2. The lower comparison process\",\"tag\":null,\"text\":\"2.2. The lower comparison process.\"},{\"bbox\":[73.44,140.04,448.39,23.91],\"bbox_pdf\":[73.44,677.93,521.83,701.85],\"fragments\":[{\"bbox_pdf\":[73.44,677.93,521.83,701.85],\"words\":[5597,5598,5599,5600,5601,5602,5603,5604,5605,5606,5607,5608,5609,5610,5613,5614,5615]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":5,\"role\":\"paragraph\",\"source_ref\":\"paragraph197\",\"source_text\":\"Now we consider the following process driven by the same noises as $X$ :\",\"tag\":null,\"text\":\"Now we consider the following process driven by the same noises as X: ∫︂ ∫︂ ∫︂ ∫︂\"},{\"bbox\":[83.47,169.32,439.64,29.56],\"bbox_pdf\":[83.47,643.02,523.11,672.57],\"fragments\":[{\"bbox_pdf\":[83.47,643.02,523.11,672.57],\"words\":[5611,5612,5616,5617,5618,5619,5620,5621,5622,5623,5624,5625,5626,5627,5628,5629,5630,5631,5632,5633,5634,5635,5636,5637,5638,5639,5640,5641,5642,5643,5644,5645]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\displaystyle\\\\overline{X}_{t}=x-b_{10}\\\\int_{0}^{t}\\\\overline{X}_{s}^{r_{10}}\\\\,%\\n\\\\mathrm{d}s+\\\\int_{0}^{t}\\\\sqrt{2b_{11}\\\\overline{X}_{s}^{r_{11}}}\\\\,\\\\mathrm{d}B_{%\\n1}(s)+\\\\int_{0}^{t}\\\\int_{0}^{\\\\infty}\\\\int_{0}^{b_{12}\\\\overline{X}_{s-}^{r_{12}}}%\\nz\\\\,\\\\widetilde{N}_{1}(\\\\,\\\\mathrm{d}s,\\\\,\\\\mathrm{d}z,\\\\,\\\\mathrm{d}u).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"A1.EGx23\",\"source_text\":\"$\\\\displaystyle\\\\overline{X}_{t}=x-b_{10}\\\\int_{0}^{t}\\\\overline{X}_{s}^{r_{10}}\\\\,% \\\\mathrm{d}s+\\\\int_{0}^{t}\\\\sqrt{2b_{11}\\\\overline{X}_{s}^{r_{11}}}\\\\,\\\\mathrm{d}B_{% 1}(s)+\\\\int_{0}^{t}\\\\int_{0}^{\\\\infty}\\\\int_{0}^{b_{12}\\\\overline{X}_{s-}^{r_{12}}}% z\\\\,\\\\widetilde{N}_{1}(\\\\,\\\\mathrm{d}s,\\\\,\\\\mathrm{d}z,\\\\,\\\\mathrm{d}u).$ (2.5)\",\"tag\":\"(2.5)\",\"text\":\"∫︂ √︂ r12 t t t ∞ b12Xs− r10 r11 Xt = x − b10 Xs ds + 2b11Xs dB1(s) + z ˜︁N1( ds, dz, du). (2.5) 0 0 0 0 0\"},{\"bbox\":[73.44,203.36,448.8,49.98],\"bbox_pdf\":[73.44,588.55,522.24,638.53],\"fragments\":[{\"bbox_pdf\":[73.44,588.55,522.24,638.53],\"words\":[5646,5647,5648,5649,5650,5651,5652,5653,5654,5655,5656,5657,5658,5659,5660,5661,5662,5663,5664,5665,5666,5667,5668,5669,5670,5671,5672,5673,5674,5675,5676,5677,5678,5679,5680,5681,5682,5683,5684,5685,5686,5687,5688,5689,5690,5691,5692,5693,5694,5695,5696]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":7,\"role\":\"paragraph\",\"source_ref\":\"paragraph199\",\"source_text\":\"The process $\\\\overline{X}$ defined by (2.5) is a continuous-state nonlinear branching process studied systematically in [20]; see also Li [17]. These works provide a general analysis of its boundary and long-time behavior. The following proposition refines these results in the present setting by identifying the almost-sure logarithmic decay rate of $\\\\overline{X}$ .\",\"tag\":null,\"text\":\"The process X defined by (2.5) is a continuous-state nonlinear branching process studied systematically in [20]; see also Li [17]. These works provide a general analysis of its boundary and long-time behavior. The following proposition refines these results in the present setting by identifying the almost-sure logarithmic decay rate of X.\"},{\"bbox\":[73.44,262.12,448.3,25.91],\"bbox_pdf\":[73.44,553.87,521.74,579.77],\"fragments\":[{\"bbox_pdf\":[73.44,553.87,521.74,579.77],\"words\":[5697,5698,5699,5700,5701,5702,5703,5704,5705,5706,5707,5708,5709,5710,5711,5712,5713,5714,5715,5716,5717,5718,5719,5720,5721,5722,5723,5724,5725]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":8,\"role\":\"paragraph\",\"source_ref\":\"paragraph200\",\"source_text\":\"Proposition 2.2. Assume $b_{10}>0$ and $r_{1}\\\\geq 0$ . Then the solution of (2.5) is global and strictly positive. Moreover, $\\\\overline{X}_{t}\\\\rightarrow 0$ as $t\\\\rightarrow\\\\infty$ , and\",\"tag\":null,\"text\":\"Proposition 2.2. Assume b10 > 0 and r1 ≥ 0. Then the solution of (2.5) is global and strictly positive. Moreover, Xt → 0 as t → ∞, and\"},{\"bbox\":[188.19,290.61,201.38,21.62],\"bbox_pdf\":[188.19,529.67,389.57,551.28],\"fragments\":[{\"bbox_pdf\":[188.19,529.67,389.57,551.28],\"words\":[5726,5727,5728,5729,5730,5731,5732,5733,5734,5735]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\displaystyle\\\\lim_{t\\\\to\\\\infty}\\\\frac{1}{t}\\\\log\\\\overline{X}_{t}=-b_{1}\\\\mathbf{1}%\\n_{\\\\{r_{1}=0\\\\}}\\\\quad\\\\text{almost surely}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"A1.EGx24\",\"source_text\":\"$\\\\displaystyle\\\\lim_{t\\\\to\\\\infty}\\\\frac{1}{t}\\\\log\\\\overline{X}_{t}=-b_{1}\\\\mathbf{1}% _{\\\\{r_{1}=0\\\\}}\\\\quad\\\\text{almost surely}.$\",\"tag\":null,\"text\":\"1 lim log Xt = −b11{r1=0} almost surely. t→∞ t\"},{\"bbox\":[73.44,321.04,448.4,49.83],\"bbox_pdf\":[73.44,471.02,521.84,520.85],\"fragments\":[{\"bbox_pdf\":[73.44,471.02,521.84,520.85],\"words\":[5736,5737,5738,5739,5740,5741,5742,5743,5744,5745,5746,5747,5748,5749,5750,5751,5752,5753,5754,5755,5756,5757,5758,5759,5760,5761,5762,5763,5764,5765,5766,5767,5768,5769,5770,5771,5772,5773,5774,5775,5776,5777,5778,5779,5780,5781,5782,5783,5784,5785,5786]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":10,\"role\":\"paragraph\",\"source_ref\":\"paragraph202\",\"source_text\":\"Proof. If $b_{11}+b_{12}=0$ , the result follows directly from the corresponding ordinary differential equation. Hence we assume $b_{11}+b_{12}>0$ . The global existence, strict positivity, and convergence $\\\\overline{X}_{t}\\\\to 0$ follow from [20, Theorem 3.1 and Example 2.18]. It remains to prove the logarithmic limit.\",\"tag\":null,\"text\":\"Proof. If b11 + b12 = 0, the result follows directly from the corresponding ordinary differential equation. Hence we assume b11 +b12 > 0. The global existence, strict positivity, and convergence Xt → 0 follow from [20, Theorem 3.1 and Example 2.18]. It remains to prove the logarithmic limit. ∫︁ ∞\"},{\"bbox\":[73.05,372.69,449.06,24.08],\"bbox_pdf\":[73.05,445.12,522.11,469.2],\"fragments\":[{\"bbox_pdf\":[73.05,445.12,522.11,469.2],\"words\":[5787,5788,5789,5790,5791,5792,5793,5794,5795,5796,5797,5798,5799,5800,5801,5802,5803,5804,5805,5806,5807,5808,5809,5810,5811,5812,5813]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":0.941,\"verified\":true},\"order\":11,\"role\":\"paragraph\",\"source_ref\":\"paragraph203\",\"source_text\":\"For the stable Lévy measure $\\\\mu_{1}$ , we have $\\\\int_{0}^{\\\\infty}[\\\\log(1+v)]^{2}\\\\mu_{1}(\\\\,\\\\mathrm{d}v)<\\\\infty,$ and by Lemma A with $q=0$ , we have\",\"tag\":null,\"text\":\"2 For the stable L´evy measure µ1, we have [log(1 + v)] µ1( dv) < ∞, and by Lemma A.3 0 with q = 0, we have\"},{\"bbox\":[211.97,394.24,151.4,35.65],\"bbox_pdf\":[211.97,412.0,363.38,447.65],\"fragments\":[{\"bbox_pdf\":[211.97,412.0,363.38,447.65],\"words\":[5814,5815,5816,5817,5818,5819,5820,5821,5822,5823,5824,5825,5826,5827]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\displaystyle\\\\int_{0}^{\\\\infty}\\\\bigl{[}v-\\\\log(1+v)\\\\bigr{]}\\\\mu_{1}(\\\\,\\\\mathrm{d}v%\\n)=1.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"A1.EGx25\",\"source_text\":\"$\\\\displaystyle\\\\int_{0}^{\\\\infty}\\\\bigl{[}v-\\\\log(1+v)\\\\bigr{]}\\\\mu_{1}(\\\\,\\\\mathrm{d}v% )=1.$\",\"tag\":null,\"text\":\"∫︂ ∞ [︁ ]︁ v − log(1 + v) µ1( dv) = 1. 0\"},{\"bbox\":[73.04,434.59,448.8,23.86],\"bbox_pdf\":[73.04,383.44,521.84,407.3],\"fragments\":[{\"bbox_pdf\":[73.04,383.44,521.84,407.3],\"words\":[5828,5829,5830,5831,5832,5833,5834,5835,5836,5837,5838,5839,5840,5841,5842,5843,5844,5845,5846,5847,5848,5849,5850]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":13,\"role\":\"paragraph\",\"source_ref\":\"paragraph205\",\"source_text\":\"Applying Itô’s formula with jumps to $\\\\log\\\\overline{X}_{t}$ , first between interior localization times and then removing the localization using global strict positivity, yields\",\"tag\":null,\"text\":\"Applying Itˆo’s formula with jumps to log Xt, first between interior localization times and then removing the localization using global strict positivity, yields\"},{\"bbox\":[117.4,457.39,405.71,35.65],\"bbox_pdf\":[117.4,348.85,523.11,384.5],\"fragments\":[{\"bbox_pdf\":[117.4,348.85,523.11,384.5],\"words\":[5851,5852,5853,5854,5855,5856,5857,5858,5859,5860,5861,5862,5863,5864,5865,5866,5867,5868,5869,5870,5871,5872,5873,5874,5875,5877,5878]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\displaystyle\\\\log\\\\overline{X}_{t}={} \\\\displaystyle\\\\log x-\\\\int_{0}^{t}\\\\left[b_{10}\\\\overline{X}_{s}^{r_{10}-1}+b_{11}%\\n\\\\overline{X}_{s}^{r_{11}-2}+b_{12}\\\\overline{X}_{s}^{r_{12}-\\\\alpha_{1}}\\\\right]%\\n\\\\,\\\\mathrm{d}s+M_{t}^{B}+M_{t}^{J},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"A1.EGx26\",\"source_text\":\"$\\\\displaystyle\\\\log\\\\overline{X}_{t}={}$ $\\\\displaystyle\\\\log x-\\\\int_{0}^{t}\\\\left[b_{10}\\\\overline{X}_{s}^{r_{10}-1}+b_{11}% \\\\overline{X}_{s}^{r_{11}-2}+b_{12}\\\\overline{X}_{s}^{r_{12}-\\\\alpha_{1}}\\\\right]% \\\\,\\\\mathrm{d}s+M_{t}^{B}+M_{t}^{J},$ (2.6)\",\"tag\":\"(2.6)\",\"text\":\"∫︂ t [︂ ]︂ r10−1 r11−2 r12−α1 B J log Xt = log x − b10Xs + b11Xs + b12Xs ds + Mt + Mt (2.6) 0\"},{\"bbox\":[73.05,495.69,27.91,10.91],\"bbox_pdf\":[73.05,335.29,100.96,346.2],\"fragments\":[{\"bbox_pdf\":[73.05,335.29,100.96,346.2],\"words\":[5879]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":15,\"role\":\"paragraph\",\"source_ref\":\"paragraph207\",\"source_text\":\"where\",\"tag\":null,\"text\":\"where\"},{\"bbox\":[163.38,503.39,268.52,69.02],\"bbox_pdf\":[163.38,269.48,431.9,338.5],\"fragments\":[{\"bbox_pdf\":[163.38,269.48,431.9,338.5],\"words\":[5880,5881,5882,5883,5884,5885,5886,5887,5888,5889,5890,5891,5892,5893,5894,5895,5896,5897,5898,5899,5900,5901,5902,5903,5904,5905,5906,5907,5908,5909,5910,5911,5912,5913,5914]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle M_{t}^{B}:={} \\\\displaystyle\\\\int_{0}^{t}\\\\sqrt{2b_{11}\\\\overline{X}_{s}^{r_{11}-2}}\\\\,\\\\,\\\\mathrm{%\\nd}B_{1}(s), \\\\\\\\ \\\\displaystyle M_{t}^{J}:={} \\\\displaystyle\\\\int_{0}^{t}\\\\int_{0}^{\\\\infty}\\\\int_{0}^{b_{12}\\\\overline{X}_{s-}^{r%\\n_{12}}}\\\\log\\\\left(1+\\\\frac{z}{\\\\overline{X}_{s-}}\\\\right)\\\\widetilde{N}_{1}(\\\\,%\\n\\\\mathrm{d}s,\\\\,\\\\mathrm{d}z,\\\\,\\\\mathrm{d}u) \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":16,\"role\":\"formula\",\"source_ref\":\"A1.EGx27\",\"source_text\":\"$\\\\displaystyle M_{t}^{B}:={}$ $\\\\displaystyle\\\\int_{0}^{t}\\\\sqrt{2b_{11}\\\\overline{X}_{s}^{r_{11}-2}}\\\\,\\\\,\\\\mathrm{% d}B_{1}(s),$ $\\\\displaystyle M_{t}^{J}:={}$ $\\\\displaystyle\\\\int_{0}^{t}\\\\int_{0}^{\\\\infty}\\\\int_{0}^{b_{12}\\\\overline{X}_{s-}^{r% _{12}}}\\\\log\\\\left(1+\\\\frac{z}{\\\\overline{X}_{s-}}\\\\right)\\\\widetilde{N}_{1}(\\\\,% \\\\mathrm{d}s,\\\\,\\\\mathrm{d}z,\\\\,\\\\mathrm{d}u)$\",\"tag\":null,\"text\":\"∫︂ t √︂ B r11−2 Mt := 2b11Xs dB1(s), 0 ∫︂ ∫︂ ∫︂ r12 (︃ )︃ t ∞ b12Xs− J z ˜︁ Mt := log 1 + N1( ds, dz, du) 0 0 0 Xs−\"},{\"bbox\":[73.16,575.89,448.68,83.06],\"bbox_pdf\":[73.16,182.94,521.84,266.0],\"fragments\":[{\"bbox_pdf\":[73.16,182.94,521.84,266.0],\"words\":[5915,5916,5917,5918,5919,5920,5921,5922,5923,5924,5925,5926,5927,5928,5929,5930,5931,5932,5933,5934,5935,5936,5937,5938,5939,5940,5941,5942,5943,5944,5945,5946,5947,5948,5949,5950,5951,5952,5953,5954,5955,5956,5957,5958,5959,5960,5961,5962,5963,5964,5965,5966,5967,5968,5969,5970,5971,5972,5973,5974,5975,5976,5977,5978,5979,5980,5981,5982,5983,5984,5985,5986,5987,5988,5989,5990,5991,5992,5993,5994,5995,5996,5997,5998,5999,6000,6001,6002,6003,6004,6005,6006,6007,6008,6009,6010,6011,6012,6013,6014,6015,6016,6017,6018]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":17,\"role\":\"paragraph\",\"source_ref\":\"paragraph209\",\"source_text\":\"are local martingales. Since $r_{1}\\\\geq 0$ , every active exponent among $r_{10}-1$ , $r_{11}-2$ , and $r_{12}-\\\\alpha_{1}$ is nonnegative. The predictable quadratic variation of $M^{B}$ is $\\\\langle M^{B}\\\\rangle_{t}=2b_{11}\\\\int_{0}^{t}\\\\overline{X}_{s}^{r_{11}-2}\\\\,% \\\\mathrm{d}s.$ If $b_{11}=0$ , this bracket vanishes; otherwise $r_{11}-2\\\\geq 0$ , and $\\\\overline{X}_{t}\\\\to 0$ shows that its integrand is eventually bounded. Consequently, $\\\\int_{0}^{\\\\infty}\\\\frac{\\\\,\\\\mathrm{d}\\\\langle M^{B}\\\\rangle_{s}}{(1+s)^{2}}<\\\\infty$ almost surely. Therefore, $\\\\int_{0}^{t}\\\\frac{1}{1+s}\\\\,\\\\,\\\\mathrm{d}M_{s}^{B}$ converges almost surely as $t\\\\to\\\\infty$ . Similarly, for the jump term, the predictable quadratic variation is\",\"tag\":null,\"text\":\"are local martingales. Since r1 ≥ 0, every active exponent among r10 − 1, r11 − 2, and r12 − α1 ∫︁ t r11−2 B B is nonnegative. The predictable quadratic variation of M is ⟨M ⟩t = 2b11 0 Xs ds. If b11 = 0, this bracket vanishes; otherwise r11 − 2 ≥ 0, and Xt → 0 shows that its integrand ∫︁ ∞ d⟨M B ⟩s ∫︁ t 1 B is eventually bounded. Consequently, 0 2 < ∞ almost surely. Therefore, 0 1+s dMs (1+s) converges almost surely as t → ∞. Similarly, for the jump term, the predictable quadratic variation is\"},{\"bbox\":[168.07,656.44,259.14,67.38],\"bbox_pdf\":[168.07,118.08,427.21,185.45],\"fragments\":[{\"bbox_pdf\":[168.07,118.08,427.21,185.45],\"words\":[6019,6020,6021,6022,6023,6024,6025,6026,6027,6028,6029,6030,6031,6032,6033,6034,6035,6036,6037,6038,6039,6040,6041,6042,6043,6044,6045,6046,6047,6048,6049,6050,6051,6052,6053,6054,6055,6056,6057,6058,6059,6060]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\langle M^{J}\\\\rangle_{t}={} \\\\displaystyle b_{12}\\\\int_{0}^{t}\\\\overline{X}_{s}^{r_{12}}\\\\int_{0}^{\\\\infty}%\\n\\\\left[\\\\log\\\\left(1+\\\\frac{z}{\\\\overline{X}_{s}}\\\\right)\\\\right]^{2}\\\\mu_{1}(\\\\,%\\n\\\\mathrm{d}z)\\\\,\\\\mathrm{d}s \\\\\\\\ \\\\displaystyle={} \\\\displaystyle b_{12}\\\\left(\\\\int_{0}^{\\\\infty}[\\\\log(1+v)]^{2}\\\\mu_{1}(\\\\,\\\\mathrm{d}%\\nv)\\\\right)\\\\int_{0}^{t}\\\\overline{X}_{s}^{r_{12}-\\\\alpha_{1}}\\\\,\\\\mathrm{d}s. \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":18,\"role\":\"formula\",\"source_ref\":\"A1.EGx28\",\"source_text\":\"$\\\\displaystyle\\\\langle M^{J}\\\\rangle_{t}={}$ $\\\\displaystyle b_{12}\\\\int_{0}^{t}\\\\overline{X}_{s}^{r_{12}}\\\\int_{0}^{\\\\infty}% \\\\left[\\\\log\\\\left(1+\\\\frac{z}{\\\\overline{X}_{s}}\\\\right)\\\\right]^{2}\\\\mu_{1}(\\\\,% \\\\mathrm{d}z)\\\\,\\\\mathrm{d}s$ $\\\\displaystyle={}$ $\\\\displaystyle b_{12}\\\\left(\\\\int_{0}^{\\\\infty}[\\\\log(1+v)]^{2}\\\\mu_{1}(\\\\,\\\\mathrm{d}% v)\\\\right)\\\\int_{0}^{t}\\\\overline{X}_{s}^{r_{12}-\\\\alpha_{1}}\\\\,\\\\mathrm{d}s.$\",\"tag\":null,\"text\":\"∫︂ t ∫︂ ∞ [︃ (︃ )︃]︃2 J r12 z ⟨M ⟩t = b12 Xs log 1 + µ1( dz) ds 0 0 Xs (︃∫︂ ∞ )︃ ∫︂ t 2 r12−α1 = b12 [log(1 + v)] µ1( dv) Xs ds. 0 0\"},{\"bbox\":[73.05,726.93,450.9,41.52],\"bbox_pdf\":[73.05,73.44,523.95,114.96],\"fragments\":[{\"bbox_pdf\":[73.05,73.44,523.95,114.96],\"words\":[6061,6062,6063,6064,6065,6066,6067,6068,6069,6070,6071,6072,6073,6074,6075,6076,6077,6078,6079,6080,6081,6082,6083,6084,6085,6086,6087,6088,6089,6090,6091,6092,6093,6094,6095,6096,6097,6098,6099,6100,6101,6102,6103,6104,6105,6106,6107,6108,6109,6110,6111,6112,6113,6114,6115]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":null,\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":19,\"role\":\"paragraph\",\"source_ref\":\"paragraph211\",\"source_text\":\"If $b_{12}=0$ , this bracket vanishes; otherwise $r_{12}-\\\\alpha_{1}\\\\geq 0$ , and its integrand is eventually bounded. Thus, in either case, $\\\\int_{0}^{\\\\infty}\\\\frac{\\\\,\\\\mathrm{d}\\\\langle M^{J}\\\\rangle_{s}}{(1+s)^{2}}<\\\\infty$ almost surely. Hence $\\\\int_{0}^{t}\\\\frac{1}{1+s}\\\\,\\\\,\\\\mathrm{d}M_{s}^{J}$ also converges almost surely; see, e.g., Karatzas and Kardaras [13, Remark 7.2].\",\"tag\":null,\"text\":\"If b12 = 0, this bracket vanishes; otherwise r12 − α1 ≥ 0, and its integrand is eventually bounded. ∫︁ ∞ d⟨M J ⟩s ∫︁ t 1 J Thus, in either case, 0 2 < ∞ almost surely. Hence 0 1+s dMs also converges almost (1+s) surely; see, e.g., Karatzas and Kardaras [13, Remark 7.2].\"}],\"category\":\"scientific articles\",\"document\":\"arxiv:2608.30178\",\"licence\":{\"id\":\"CC-BY-SA-4.0\",\"share_alike\":true,\"url\":\"http://creativecommons.org/licenses/by-sa/4.0/\"},\"name\":\"arxiv-2608.30178v1-p0010.pdf\",\"page_no\":11,\"page_size\":[595.28,841.89],\"provenance\":{\"aligner\":\"a1\",\"aligner_git\":\"unknown\",\"engine\":\"pdfluent 1.2.0\",\"harvest_manifest\":\"ede24b96e97e828e1a120a32f871346f89e57284ca2b46317e784207902df465\",\"source_kind\":\"latexml-local\"},\"quality\":{\"block_why\":\"paragraph203\",\"c_block\":0.929,\"c_lines\":1.0,\"c_pdf\":0.997,\"c_region\":1.0,\"c_src\":0.995,\"figure_text_share\":0.0,\"furniture_share\":0.0,\"kept\":true,\"missing_lines\":0,\"missing_text\":[],\"order\":1.0,\"q\":0.929,\"reasons\":[],\"region_why\":[],\"unmatched_struct\":0},\"sha256\":\"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce\",\"source\":\"arxiv-aligned/dev\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce.pdf","asset_sha256":"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce","document_id":"arxiv:2608.30178","kind":"page","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.30178\",\"authors\":\"Lina Ji, Jie Xiong, Wen Xu, Xu Yang\",\"title\":\"Extinction and extinguishment properties for a nonlinear predator-prey branching model\",\"url\":\"https://arxiv.org/abs/2608.30178v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[145.02,81.48,305.24,35.65],\"bbox_pdf\":[145.02,724.76,450.26,760.41],\"fragments\":[{\"bbox_pdf\":[145.02,724.76,450.26,760.41],\"words\":[5561,5562,5563,5564,5565,5566,5567,5568,5569,5570,5571,5572,5573,5574,5575,5576,5577,5578,5579,5580,5581,5582,5583,5584,5585,5586]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"T_{*}<\\\\infty\\\\Longleftrightarrow\\\\int_{0}^{\\\\varepsilon}y^{-\\\\theta_{2}+(a_{1}/a_{%\\n2})\\\\kappa_{2}}\\\\,\\\\mathrm{d}y<\\\\infty\\\\Longleftrightarrow\\\\frac{a_{1}}{a_{2}}\\\\kappa%\\n_{2}>\\\\theta_{2}-1=\\\\kappa_{1}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":2,\"role\":\"formula\",\"source_ref\":\"S2.Ex23\",\"source_text\":\"$T_{*}<\\\\infty\\\\Longleftrightarrow\\\\int_{0}^{\\\\varepsilon}y^{-\\\\theta_{2}+(a_{1}/a_{% 2})\\\\kappa_{2}}\\\\,\\\\mathrm{d}y<\\\\infty\\\\Longleftrightarrow\\\\frac{a_{1}}{a_{2}}\\\\kappa% _{2}>\\\\theta_{2}-1=\\\\kappa_{1}.$\",\"tag\":null,\"text\":\"y ↓ 0. Moreover, ∫︂ ε −θ2+(a1/a2)κ2 dy < ∞ ⇐⇒ a1 T∗ < ∞ ⇐⇒ y κ2 > θ2 − 1 = κ1. 0 a2\"},\"formula_index\":2,\"parent_page_sha256\":\"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce.pdf","asset_sha256":"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce","document_id":"arxiv:2608.30178","kind":"formula","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.30178\",\"authors\":\"Lina Ji, Jie Xiong, Wen Xu, Xu Yang\",\"title\":\"Extinction and extinguishment properties for a nonlinear predator-prey branching model\",\"url\":\"https://arxiv.org/abs/2608.30178v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[83.47,169.32,439.64,29.56],\"bbox_pdf\":[83.47,643.02,523.11,672.57],\"fragments\":[{\"bbox_pdf\":[83.47,643.02,523.11,672.57],\"words\":[5611,5612,5616,5617,5618,5619,5620,5621,5622,5623,5624,5625,5626,5627,5628,5629,5630,5631,5632,5633,5634,5635,5636,5637,5638,5639,5640,5641,5642,5643,5644,5645]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\displaystyle\\\\overline{X}_{t}=x-b_{10}\\\\int_{0}^{t}\\\\overline{X}_{s}^{r_{10}}\\\\,%\\n\\\\mathrm{d}s+\\\\int_{0}^{t}\\\\sqrt{2b_{11}\\\\overline{X}_{s}^{r_{11}}}\\\\,\\\\mathrm{d}B_{%\\n1}(s)+\\\\int_{0}^{t}\\\\int_{0}^{\\\\infty}\\\\int_{0}^{b_{12}\\\\overline{X}_{s-}^{r_{12}}}%\\nz\\\\,\\\\widetilde{N}_{1}(\\\\,\\\\mathrm{d}s,\\\\,\\\\mathrm{d}z,\\\\,\\\\mathrm{d}u).\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":6,\"role\":\"formula\",\"source_ref\":\"A1.EGx23\",\"source_text\":\"$\\\\displaystyle\\\\overline{X}_{t}=x-b_{10}\\\\int_{0}^{t}\\\\overline{X}_{s}^{r_{10}}\\\\,% \\\\mathrm{d}s+\\\\int_{0}^{t}\\\\sqrt{2b_{11}\\\\overline{X}_{s}^{r_{11}}}\\\\,\\\\mathrm{d}B_{% 1}(s)+\\\\int_{0}^{t}\\\\int_{0}^{\\\\infty}\\\\int_{0}^{b_{12}\\\\overline{X}_{s-}^{r_{12}}}% z\\\\,\\\\widetilde{N}_{1}(\\\\,\\\\mathrm{d}s,\\\\,\\\\mathrm{d}z,\\\\,\\\\mathrm{d}u).$ (2.5)\",\"tag\":\"(2.5)\",\"text\":\"∫︂ √︂ r12 t t t ∞ b12Xs− r10 r11 Xt = x − b10 Xs ds + 2b11Xs dB1(s) + z ˜︁N1( ds, dz, du). (2.5) 0 0 0 0 0\"},\"formula_index\":6,\"parent_page_sha256\":\"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce.pdf","asset_sha256":"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce","document_id":"arxiv:2608.30178","kind":"formula","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.30178\",\"authors\":\"Lina Ji, Jie Xiong, Wen Xu, Xu Yang\",\"title\":\"Extinction and extinguishment properties for a nonlinear predator-prey branching model\",\"url\":\"https://arxiv.org/abs/2608.30178v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[188.19,290.61,201.38,21.62],\"bbox_pdf\":[188.19,529.67,389.57,551.28],\"fragments\":[{\"bbox_pdf\":[188.19,529.67,389.57,551.28],\"words\":[5726,5727,5728,5729,5730,5731,5732,5733,5734,5735]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\displaystyle\\\\lim_{t\\\\to\\\\infty}\\\\frac{1}{t}\\\\log\\\\overline{X}_{t}=-b_{1}\\\\mathbf{1}%\\n_{\\\\{r_{1}=0\\\\}}\\\\quad\\\\text{almost surely}.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":9,\"role\":\"formula\",\"source_ref\":\"A1.EGx24\",\"source_text\":\"$\\\\displaystyle\\\\lim_{t\\\\to\\\\infty}\\\\frac{1}{t}\\\\log\\\\overline{X}_{t}=-b_{1}\\\\mathbf{1}% _{\\\\{r_{1}=0\\\\}}\\\\quad\\\\text{almost surely}.$\",\"tag\":null,\"text\":\"1 lim log Xt = −b11{r1=0} almost surely. t→∞ t\"},\"formula_index\":9,\"parent_page_sha256\":\"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce.pdf","asset_sha256":"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce","document_id":"arxiv:2608.30178","kind":"formula","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.30178\",\"authors\":\"Lina Ji, Jie Xiong, Wen Xu, Xu Yang\",\"title\":\"Extinction and extinguishment properties for a nonlinear predator-prey branching model\",\"url\":\"https://arxiv.org/abs/2608.30178v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[211.97,394.24,151.4,35.65],\"bbox_pdf\":[211.97,412.0,363.38,447.65],\"fragments\":[{\"bbox_pdf\":[211.97,412.0,363.38,447.65],\"words\":[5814,5815,5816,5817,5818,5819,5820,5821,5822,5823,5824,5825,5826,5827]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\displaystyle\\\\int_{0}^{\\\\infty}\\\\bigl{[}v-\\\\log(1+v)\\\\bigr{]}\\\\mu_{1}(\\\\,\\\\mathrm{d}v%\\n)=1.\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":12,\"role\":\"formula\",\"source_ref\":\"A1.EGx25\",\"source_text\":\"$\\\\displaystyle\\\\int_{0}^{\\\\infty}\\\\bigl{[}v-\\\\log(1+v)\\\\bigr{]}\\\\mu_{1}(\\\\,\\\\mathrm{d}v% )=1.$\",\"tag\":null,\"text\":\"∫︂ ∞ [︁ ]︁ v − log(1 + v) µ1( dv) = 1. 0\"},\"formula_index\":12,\"parent_page_sha256\":\"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce.pdf","asset_sha256":"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce","document_id":"arxiv:2608.30178","kind":"formula","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.30178\",\"authors\":\"Lina Ji, Jie Xiong, Wen Xu, Xu Yang\",\"title\":\"Extinction and extinguishment properties for a nonlinear predator-prey branching model\",\"url\":\"https://arxiv.org/abs/2608.30178v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[117.4,457.39,405.71,35.65],\"bbox_pdf\":[117.4,348.85,523.11,384.5],\"fragments\":[{\"bbox_pdf\":[117.4,348.85,523.11,384.5],\"words\":[5851,5852,5853,5854,5855,5856,5857,5858,5859,5860,5861,5862,5863,5864,5865,5866,5867,5868,5869,5870,5871,5872,5873,5874,5875,5877,5878]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\displaystyle\\\\log\\\\overline{X}_{t}={} \\\\displaystyle\\\\log x-\\\\int_{0}^{t}\\\\left[b_{10}\\\\overline{X}_{s}^{r_{10}-1}+b_{11}%\\n\\\\overline{X}_{s}^{r_{11}-2}+b_{12}\\\\overline{X}_{s}^{r_{12}-\\\\alpha_{1}}\\\\right]%\\n\\\\,\\\\mathrm{d}s+M_{t}^{B}+M_{t}^{J},\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":14,\"role\":\"formula\",\"source_ref\":\"A1.EGx26\",\"source_text\":\"$\\\\displaystyle\\\\log\\\\overline{X}_{t}={}$ $\\\\displaystyle\\\\log x-\\\\int_{0}^{t}\\\\left[b_{10}\\\\overline{X}_{s}^{r_{10}-1}+b_{11}% \\\\overline{X}_{s}^{r_{11}-2}+b_{12}\\\\overline{X}_{s}^{r_{12}-\\\\alpha_{1}}\\\\right]% \\\\,\\\\mathrm{d}s+M_{t}^{B}+M_{t}^{J},$ (2.6)\",\"tag\":\"(2.6)\",\"text\":\"∫︂ t [︂ ]︂ r10−1 r11−2 r12−α1 B J log Xt = log x − b10Xs + b11Xs + b12Xs ds + Mt + Mt (2.6) 0\"},\"formula_index\":14,\"parent_page_sha256\":\"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce.pdf","asset_sha256":"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce","document_id":"arxiv:2608.30178","kind":"formula","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.30178\",\"authors\":\"Lina Ji, Jie Xiong, Wen Xu, Xu Yang\",\"title\":\"Extinction and extinguishment properties for a nonlinear predator-prey branching model\",\"url\":\"https://arxiv.org/abs/2608.30178v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[163.38,503.39,268.52,69.02],\"bbox_pdf\":[163.38,269.48,431.9,338.5],\"fragments\":[{\"bbox_pdf\":[163.38,269.48,431.9,338.5],\"words\":[5880,5881,5882,5883,5884,5885,5886,5887,5888,5889,5890,5891,5892,5893,5894,5895,5896,5897,5898,5899,5900,5901,5902,5903,5904,5905,5906,5907,5908,5909,5910,5911,5912,5913,5914]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle M_{t}^{B}:={} \\\\displaystyle\\\\int_{0}^{t}\\\\sqrt{2b_{11}\\\\overline{X}_{s}^{r_{11}-2}}\\\\,\\\\,\\\\mathrm{%\\nd}B_{1}(s), \\\\\\\\ \\\\displaystyle M_{t}^{J}:={} \\\\displaystyle\\\\int_{0}^{t}\\\\int_{0}^{\\\\infty}\\\\int_{0}^{b_{12}\\\\overline{X}_{s-}^{r%\\n_{12}}}\\\\log\\\\left(1+\\\\frac{z}{\\\\overline{X}_{s-}}\\\\right)\\\\widetilde{N}_{1}(\\\\,%\\n\\\\mathrm{d}s,\\\\,\\\\mathrm{d}z,\\\\,\\\\mathrm{d}u) \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":16,\"role\":\"formula\",\"source_ref\":\"A1.EGx27\",\"source_text\":\"$\\\\displaystyle M_{t}^{B}:={}$ $\\\\displaystyle\\\\int_{0}^{t}\\\\sqrt{2b_{11}\\\\overline{X}_{s}^{r_{11}-2}}\\\\,\\\\,\\\\mathrm{% d}B_{1}(s),$ $\\\\displaystyle M_{t}^{J}:={}$ $\\\\displaystyle\\\\int_{0}^{t}\\\\int_{0}^{\\\\infty}\\\\int_{0}^{b_{12}\\\\overline{X}_{s-}^{r% _{12}}}\\\\log\\\\left(1+\\\\frac{z}{\\\\overline{X}_{s-}}\\\\right)\\\\widetilde{N}_{1}(\\\\,% \\\\mathrm{d}s,\\\\,\\\\mathrm{d}z,\\\\,\\\\mathrm{d}u)$\",\"tag\":null,\"text\":\"∫︂ t √︂ B r11−2 Mt := 2b11Xs dB1(s), 0 ∫︂ ∫︂ ∫︂ r12 (︃ )︃ t ∞ b12Xs− J z ˜︁ Mt := log 1 + N1( ds, dz, du) 0 0 0 Xs−\"},\"formula_index\":16,\"parent_page_sha256\":\"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce.pdf","asset_sha256":"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce","document_id":"arxiv:2608.30178","kind":"formula","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.30178\",\"authors\":\"Lina Ji, Jie Xiong, Wen Xu, Xu Yang\",\"title\":\"Extinction and extinguishment properties for a nonlinear predator-prey branching model\",\"url\":\"https://arxiv.org/abs/2608.30178v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}
{"annotation_attribution":"Innovation Trigger B.V., PDFluent ground-truth annotations (2026), licensed CC BY-SA 4.0","annotation_json":"{\"block\":{\"bbox\":[168.07,656.44,259.14,67.38],\"bbox_pdf\":[168.07,118.08,427.21,185.45],\"fragments\":[{\"bbox_pdf\":[168.07,118.08,427.21,185.45],\"words\":[6019,6020,6021,6022,6023,6024,6025,6026,6027,6028,6029,6030,6031,6032,6033,6034,6035,6036,6037,6038,6039,6040,6041,6042,6043,6044,6045,6046,6047,6048,6049,6050,6051,6052,6053,6054,6055,6056,6057,6058,6059,6060]}],\"html\":null,\"kind\":\"paragraph\",\"latex\":\"\\\\begin{aligned} \\\\displaystyle\\\\langle M^{J}\\\\rangle_{t}={} \\\\displaystyle b_{12}\\\\int_{0}^{t}\\\\overline{X}_{s}^{r_{12}}\\\\int_{0}^{\\\\infty}%\\n\\\\left[\\\\log\\\\left(1+\\\\frac{z}{\\\\overline{X}_{s}}\\\\right)\\\\right]^{2}\\\\mu_{1}(\\\\,%\\n\\\\mathrm{d}z)\\\\,\\\\mathrm{d}s \\\\\\\\ \\\\displaystyle={} \\\\displaystyle b_{12}\\\\left(\\\\int_{0}^{\\\\infty}[\\\\log(1+v)]^{2}\\\\mu_{1}(\\\\,\\\\mathrm{d}%\\nv)\\\\right)\\\\int_{0}^{t}\\\\overline{X}_{s}^{r_{12}-\\\\alpha_{1}}\\\\,\\\\mathrm{d}s. \\\\end{aligned}\",\"level\":null,\"list_depth\":null,\"match\":{\"src_placed\":1.0,\"verified\":true},\"order\":18,\"role\":\"formula\",\"source_ref\":\"A1.EGx28\",\"source_text\":\"$\\\\displaystyle\\\\langle M^{J}\\\\rangle_{t}={}$ $\\\\displaystyle b_{12}\\\\int_{0}^{t}\\\\overline{X}_{s}^{r_{12}}\\\\int_{0}^{\\\\infty}% \\\\left[\\\\log\\\\left(1+\\\\frac{z}{\\\\overline{X}_{s}}\\\\right)\\\\right]^{2}\\\\mu_{1}(\\\\,% \\\\mathrm{d}z)\\\\,\\\\mathrm{d}s$ $\\\\displaystyle={}$ $\\\\displaystyle b_{12}\\\\left(\\\\int_{0}^{\\\\infty}[\\\\log(1+v)]^{2}\\\\mu_{1}(\\\\,\\\\mathrm{d}% v)\\\\right)\\\\int_{0}^{t}\\\\overline{X}_{s}^{r_{12}-\\\\alpha_{1}}\\\\,\\\\mathrm{d}s.$\",\"tag\":null,\"text\":\"∫︂ t ∫︂ ∞ [︃ (︃ )︃]︃2 J r12 z ⟨M ⟩t = b12 Xs log 1 + µ1( dz) ds 0 0 Xs (︃∫︂ ∞ )︃ ∫︂ t 2 r12−α1 = b12 [log(1 + v)] µ1( dv) Xs ds. 0 0\"},\"formula_index\":18,\"parent_page_sha256\":\"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce\"}","annotation_licence":"CC-BY-SA-4.0","asset":"assets/fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce.pdf","asset_sha256":"fc55914ddc5c622fd73c779dfde85aae25b5817c84fcf21f6846c9579b3bb2ce","document_id":"arxiv:2608.30178","kind":"formula","page_number":11,"paper_attribution_json":"{\"arxiv_id\":\"2608.30178\",\"authors\":\"Lina Ji, Jie Xiong, Wen Xu, Xu Yang\",\"title\":\"Extinction and extinguishment properties for a nonlinear predator-prey branching model\",\"url\":\"https://arxiv.org/abs/2608.30178v1\",\"version\":1}","paper_licence":"CC-BY-SA-4.0","source":"arxiv-aligned/dev"}