{ "schema_version": "1.0", "release": "2.0.0", "date": "2026-09-25", "title": "Eve Floquet Defect Rigidity: Resource-Constrained Extremality, Exact Inverse Theory, and Classical–Quantum Temporal Photonics", "status_definition": "All numbered statements have supplied internal proofs or explicit reductions under their literal hypotheses. This is not a confidence percentage, formal certification, independent review, or established novelty.", "numbered_statement_count": 44, "claims": [ { "id": "lem:linearize", "kind": "lemma", "number": "3.1", "page": 7, "title_latex": "An exact, unrelaxed linearization", "statement_latex": "Write $c=\\cos\\theta$, $s_\\theta=\\sin\\theta$, and define\n\\begin{equation}\\label{eq:z}\nz=\\frac{(u-1)v_B}{(B-1)v_u},\\qquad\n a=\\frac{(B-1)c^2}{v_B},\\quad b=\\frac{(B-1)s_\\theta^2}{v_B},\\quad\n c_0=\\frac{(B-1)s_\\theta c}{v_B}.\n\\end{equation}\nFor each angle, $u\\mapsto z$ is a continuous strictly increasing bijection\n$[1,B]\\to[0,1]$, and\n\\begin{equation}\\label{eq:linearized}\n\\frac1{v_u}=1-za,\\qquad \\frac{u}{v_u}=1+zb,\n\\qquad \\frac{w_u}{v_u}=zc_0.\n\\end{equation}\nThus a complete projective turn has time, resource, and gain\n\\begin{equation}\\label{eq:turnmoments}\n\\tau=\\pi-\\int_0^\\pi za,\\qquad U=\\pi+\\int_0^\\pi zb,\n\\qquad J=\\int_0^\\pi zc_0.\n\\end{equation}\nEvery measurable $z\\in[0,1]$ gives an actual control via its inverse map and\nthe clock $\\dd s=\\dd\\theta/v_u$; no randomized-control relaxation is used.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "lem:linearize", "pdf": "MANUSCRIPT.pdf", "page": 7 }, "dependencies": [ "foundation:projective", "foundation:fractional" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "lem:boundary", "kind": "lemma", "number": "3.2", "page": 7, "title_latex": "Boundary moments cannot carry positive gain", "statement_latex": "Every boundary point of $\\mathcal K$ has exactly one representing $z$\nup to null sets. That representative is invariant under\n$\\theta\\mapsto\\pi-\\theta$, and its gain $J$ is zero. In particular\n$\\mathcal K$ is strictly convex, and any moment represented with $J>0$\nlies in its interior.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "lem:boundary", "pdf": "MANUSCRIPT.pdf", "page": 7 }, "dependencies": [ "lem:linearize", "foundation:convex" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "lem:normal", "kind": "lemma", "number": "3.3", "page": 7, "title_latex": "Finite normal duals at every interior moment", "statement_latex": "Let $q\\in\\operatorname{int}\\mathcal K$. The largest gain above $q$ is\nattained and has finite real multipliers $\\eta,\\xi$. Its unique maximizing\nrepresentative is\n\\begin{equation}\\label{eq:resourceselector}\nz_*(\\theta)=\\boldsymbol1_{\\{K_{\\eta,\\xi}(\\theta)>0\\}},\\qquad\nK_{\\eta,\\xi}=\\sin\\theta\\cos\\theta+\\eta\\cos^2\\theta-\\xi\\sin^2\\theta.\n\\end{equation}\nThe quadratic is indefinite, $\\Delta:=1+4\\eta\\xi>0$, and has two simple\nzeros on the projective circle. The pair $(\\eta,\\xi)$ is unique.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "lem:normal", "pdf": "MANUSCRIPT.pdf", "page": 7 }, "dependencies": [ "lem:linearize", "lem:boundary", "foundation:convex" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "thm:resource", "kind": "theorem", "number": "4.1", "page": 9, "title_latex": "Global resource-constrained extremality and equality", "statement_latex": "Over every measurable $S$-periodic profile with $1\\le u\\le R^2$ and\n$S^{-1}\\int_0^S u=m$,\n\\begin{equation}\\label{eq:resourceF}\n\\boxed{\\sup_u L(u)=F(R,S,m):=\n\\max\\bigl(\\{0\\}\\cup\\{nG_m(S/n):n\\in\\mathcal N_m(R,S)\\}\\bigr).}\n\\end{equation}\nIf $\\mathcal N_m$ is nonempty, let $\\mathcal W_m$ be the set of all maximizing\nintegers. Every equality profile, and only those profiles, is up to a temporal\ntranslation and null-set changes $n$ repetitions of\n\\begin{equation}\\label{eq:resourcecell}\nu^*_{m,n}(s)=\n\\begin{cases}\nR^2,&0\\le s0$,\n\\begin{equation}\\label{eq:resourceDefect}\n\\boxed{nG_m(S/n)-L(u)=\n\\int_{\\theta_0}^{\\theta_0+n\\pi}\n\\frac{|K_{\\eta_n,\\xi_n}(\\theta)|\\,|\\widehat u(\\theta)-b(\\theta)|}\n{v_{\\widehat u}(\\theta)v_b(\\theta)}\\dd\\theta.}\n\\end{equation}\nThe same dual is the one associated with the expanding eigenline of the\nbinary cell of durations $pS/n$ and $(1-p)S/n$.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "thm:resourceDefect", "pdf": "MANUSCRIPT.pdf", "page": 10 }, "dependencies": [ "thm:resource", "lem:normal", "foundation:fractional" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "thm:concavity", "kind": "theorem", "number": "5.1", "page": 10, "title_latex": "Strict concavity and at most two adjacent winning windings", "statement_latex": "For $R>1$ and $10$. Let $Z$ be its two zeros on\n$\\R/\\pi\\mathbb Z$, and let $d_Z$ be distance to this set. Then\n\\begin{equation}\\label{eq:margin}\n|K(\\theta)|\\ge\\frac{2\\sqrt\\Delta}{\\pi^2}d_Z(\\theta).\n\\end{equation}\nFor the bounds-only calibration $\\xi=0$, this is exactly the uniform v1.0\nconstant $2/\\pi^2$, independent of $\\eta$.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "lem:margin", "pdf": "MANUSCRIPT.pdf", "page": 12 }, "dependencies": [ "foundation:fractional" ], "novelty_status": "V1_PRESERVED_OR_GENERALIZED; external priority unestablished", "corroborating_test_reports": [ "results/verification_v1_rerun.json", "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "thm:phasestability", "kind": "theorem", "number": "6.2", "page": 13, "title_latex": "Phase and clock stability from the exact defect", "statement_latex": "For a growing fixed-resource profile with winding $n$, put\n$d=B-1$, $E=nG_m(S/n)-L(u)$, and $\\Delta_n=1+4\\eta_n\\xi_n$.\nWith the canonical selector $b=b_{\\eta_n,\\xi_n}$,\n\\begin{align}\nQ:=\\int_{\\theta_0}^{\\theta_0+n\\pi}|\\widehat u-b|\\dd\\theta\n&\\le2\\pi B\\sqrt{ndE}\\,\\Delta_n^{-1/4},\\label{eq:phaseL1}\\\\\n\\inf_{t_0}\\int_0^S|u(s)-u^*_{m,n}(s-t_0)|\\dd s\n&\\le(1+2nd)\\,2\\pi B\\sqrt{ndE}\\,\\Delta_n^{-1/4}.\n\\label{eq:timeL1}\n\\end{align}\nHere $u^*_{m,n}$ denotes all $n$ repeated cells. The same assertions hold\nfor a bounds-only calibrated cell with $\\xi=0$, its branch defect, and\n$\\Delta_n=1$.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "thm:phasestability", "pdf": "MANUSCRIPT.pdf", "page": 13 }, "dependencies": [ "lem:margin", "thm:resourceDefect", "foundation:projective" ], "novelty_status": "V1_PRESERVED_OR_GENERALIZED; external priority unestablished", "corroborating_test_reports": [ "results/verification_v1_rerun.json", "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "thm:globalstability", "kind": "theorem", "number": "6.3", "page": 13, "title_latex": "Global resource-constrained square-root rigidity", "statement_latex": "Suppose $F=F(R,S,m)>0$. Let $F_n=nG_m(S/n)$ and let $\\mathcal O_*$ be\nthe union of all translates of every winning architecture. Define\n\\begin{align}\n\\delta_b&=\\min\\bigl(\\{F\\}\\cup\\{F-F_n:n\\in\\mathcal N_m\\setminus\\mathcal W_m\\}\\bigr)>0,\\nonumber\\\\\nC_0&=\\max_{n\\in\\mathcal W_m}\n\\frac{2\\pi B(1+2nd)\\sqrt{nd}}{S\\Delta_n^{1/4}},\\qquad\nC=\\max\\{C_0,d/\\sqrt{\\delta_b}\\}.\\label{eq:resourceC}\n\\end{align}\nEvery admissible measurable profile satisfies\n\\begin{equation}\\label{eq:globalstability}\n\\boxed{\\dist_1(u,\\mathcal O_*):=\n\\inf_{v\\in\\mathcal O_*}\\frac1S\\int_0^S|u-v|\\dd s\n\\le C\\sqrt{F-L(u)}.}\n\\end{equation}\nReplacing the distance by mean absolute permittivity error is valid after\nmultiplying $C$ by $\\varepsilon_+$.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "thm:globalstability", "pdf": "MANUSCRIPT.pdf", "page": 13 }, "dependencies": [ "thm:phasestability", "thm:resource" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "prop:resourceSharp", "kind": "proposition", "number": "6.4", "page": 14, "title_latex": "Sharpness while preserving the resource exactly", "statement_latex": "The exponent $1/2$ in \\eqref{eq:globalstability} cannot generally be improved,\neven when period, contrast, and mean are all fixed.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "prop:resourceSharp", "pdf": "MANUSCRIPT.pdf", "page": 14 }, "dependencies": [ "foundation:segments", "thm:resource" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "thm:main", "kind": "theorem", "number": "7.1", "page": 15, "title_latex": "Preserved exact bounds-only optimum", "statement_latex": "For all measurable $S$-periodic controls $1\\le u\\le R^2$,\n\\begin{equation}\\label{eq:F}\n\\sup_u L(u)=F(R,S):=\\max\\bigl(\\{0\\}\\cup\\{F_n:n\\in\\mathcal N(R,S)\\}\\bigr).\n\\end{equation}\nIf $\\mathcal N$ is empty, every profile has zero growth. Otherwise every\nmaximizer is exactly, up to translation and null sets, $n$ repetitions of\n\\begin{equation}\\label{eq:binary}\nu_n^*(s)=\\begin{cases}R^2,&0\\le s0$, the union $\\mathcal O_*$ of all winning translation orbits\nobeys\n\\[\n\\dist_1(u,\\mathcal O_*)\\le C\\sqrt{F(R,S)-L(u)},\n\\]\nwhere\n\\[\n\\begin{gathered}\n\\delta_b=\\min\\bigl(\\{F\\}\\cup\\{F-F_n:n\\in\\mathcal N\\setminus\\mathcal W\\}\\bigr),\\\\\nC=\\max\\left\\{\\max_{n\\in\\mathcal W}\n\\frac{2\\pi B(1+2n(B-1))\\sqrt{n(B-1)}}S,\n\\frac{B-1}{\\sqrt{\\delta_b}}\\right\\}.\n\\end{gathered}\n\\]", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "cor:boundstability", "pdf": "MANUSCRIPT.pdf", "page": 15 }, "dependencies": [ "thm:main", "thm:defect", "thm:phasestability" ], "novelty_status": "V1_PRESERVED_OR_GENERALIZED; external priority unestablished", "corroborating_test_reports": [ "results/verification_v1_rerun.json", "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "prop:sharp", "kind": "proposition", "number": "7.4", "page": 15, "title_latex": "The exponent $1/2$ cannot generally be improved", "statement_latex": "There exist fixed $R,S$ for which no estimate replacing\n$\\sqrt{F-L}$ in Corollary~\\ref{cor:boundstability} by $(F-L)^\\alpha$ can hold\nwith a finite constant when $\\alpha>1/2$.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "prop:sharp", "pdf": "MANUSCRIPT.pdf", "page": 15 }, "dependencies": [ "foundation:segments", "thm:main" ], "novelty_status": "V1_PRESERVED_OR_GENERALIZED; external priority unestablished", "corroborating_test_reports": [ "results/verification_v1_rerun.json", "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "thm:free", "kind": "theorem", "number": "8.1", "page": 16, "title_latex": "Sharp growth per unit time", "statement_latex": "There is a unique $\\gamma_R>0$ satisfying\n\\begin{equation}\\label{eq:gamma}\nG_R(\\gamma_R)=\\gamma_R\\tau_R(\\gamma_R).\n\\end{equation}\nEvery admissible periodic profile, of any period, obeys\n$L(u)/S\\leq\\gamma_R$. Equality holds precisely for translated repetitions\nof the canonical cell with $\\eta=\\gamma_R$. In particular equality at a\nspecified $S$ requires $S$ to be an integer multiple of\n$\\tau_R(\\gamma_R)$. The physical upper rate is $\\omega_-\\gamma_R$.\nMoreover, the same rate is the maximal asymptotic growth rate over\nnonperiodic measurable controls in $[1,R^2]$.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "thm:free", "pdf": "MANUSCRIPT.pdf", "page": 16 }, "dependencies": [ "thm:main", "thm:defect" ], "novelty_status": "V1_PRESERVED_OR_GENERALIZED; external priority unestablished", "corroborating_test_reports": [ "results/verification_v1_rerun.json", "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "thm:quantum", "kind": "theorem", "number": "9.1", "page": 18, "title_latex": "Exact finite-cycle and asymptotic quantum laws", "statement_latex": "Let $M\\in SL(2,\\R)$ be the one-period monodromy of any admissible program,\nwith the fixed reference above. Denote its $N$-period Bogoliubov coefficient\nby $\\beta_N$, its mean pair count by $\\nu_N=|\\beta_N|^2$, and its\nsqueezing parameter by $r_N=\\operatorname{arsinh}\\sqrt{\\nu_N}$.\nThen, with $U_j$ the Chebyshev polynomial of the second kind,\n\\begin{equation}\\label{eq:quantumCheb}\n\\beta_N=U_{N-1}(D)\\beta,\\qquad\nr_N=\\log\\|W_\\Omega M^N W_\\Omega^{-1}\\|_2.\n\\end{equation}\nIf $|D|>1$ and $L=\\arcosh|D|$, then\n\\begin{align}\n\\nu_N&=\\chi\\sinh^2(NL),\\qquad\n\\chi=\\frac{|\\beta|^2}{\\sinh^2L}\n=1+\\frac{(\\Im\\alpha)^2}{\\sinh^2L}\\ge1,\\label{eq:quantumhyper}\\\\\nr_N&=NL+\\tfrac12\\log\\chi+o(1).\\label{eq:quantumr}\n\\end{align}\nIf $|D|<1$, put $\\phi=\\arccos D$; then\n\\begin{equation}\\label{eq:quantumstable}\n\\nu_N=|\\beta|^2\\frac{\\sin^2(N\\phi)}{\\sin^2\\phi},\n\\end{equation}\nwhich is bounded for the fixed monodromy. If $|D|=1$, then\n\\begin{equation}\\label{eq:quantumparabolic}\n\\nu_N=N^2|\\beta|^2.\n\\end{equation}\nAt $M=\\pm I$ this is identically zero; otherwise this is polynomial photon\ngrowth, with $r_N=\\log N+O(1)$, not a positive Floquet exponent.\nIn every case,\n\\begin{equation}\\label{eq:quantumlimits}\n\\lim_{N\\to\\infty}\\frac{r_N}{N}=L,\\qquad\n\\lim_{N\\to\\infty}\\frac{\\log(1+\\nu_N)}N=2L.\n\\end{equation}", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "thm:quantum", "pdf": "MANUSCRIPT.pdf", "page": 18 }, "dependencies": [ "foundation:maxwell", "foundation:linear_algebra" ], "novelty_status": "ESTABLISHED_CLASSICAL_QUANTUM_IDENTITIES; independently derived here, not claimed new", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement.", "Fixed canonical reference vacuum; ideal prescribed material drive; no loss, dispersion, saturation, depletion or continuum ultraviolet claim." ], "independent_review": false, "formal_verification": false }, { "id": "cor:quantumargmax", "kind": "corollary", "number": "9.2", "page": 19, "title_latex": "Resource-constrained classical--quantum extremality", "statement_latex": "For the fixed $(R,S,m)$ admissible class,\n\\[\n\\operatorname*{arg\\,max}_u L(u)\n=\\operatorname*{arg\\,max}_u\\lim_{N\\to\\infty}r_N(u)/N\n=\\operatorname*{arg\\,max}_u\\lim_{N\\to\\infty}\\log(1+\\nu_N(u))/N.\n\\]\nWhen $F(R,S,m)>0$, this is exactly the union of the canonical binary\ntranslation orbits in Theorem~\\ref{thm:resource}. A deficit $\\delta$ in the\nasymptotic squeezing rate per cycle gives\n$\\dist_1(u,\\mathcal O_*)\\le C\\sqrt\\delta$; a deficit $\\delta$ in the\nlogarithmic pair-count rate gives $C\\sqrt{\\delta/2}$.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "cor:quantumargmax", "pdf": "MANUSCRIPT.pdf", "page": 19 }, "dependencies": [ "thm:quantum", "thm:resource", "thm:globalstability" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement.", "Fixed canonical reference vacuum; ideal prescribed material drive; no loss, dispersion, saturation, depletion or continuum ultraviolet claim." ], "independent_review": false, "formal_verification": false }, { "id": "prop:quantumfinite", "kind": "proposition", "number": "9.3", "page": 19, "title_latex": "Finite-cycle photon optimization is a different problem", "statement_latex": "Even within a single classical maximizing translation orbit, one-period\nsqueezing and photon production need not agree. Thus the finite-$N$ quantum\nargmax is not universally the full classical argmax.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "prop:quantumfinite", "pdf": "MANUSCRIPT.pdf", "page": 19 }, "dependencies": [ "foundation:segments", "thm:quantum", "thm:resource" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "thm:quantumstability", "kind": "theorem", "number": "9.4", "page": 19, "title_latex": "Finite-observation quantum rigidity", "statement_latex": "For a class with $\\|W_\\Omega M W_\\Omega^{-1}\\|_2\\le K_0$ uniformly,\n\\begin{equation}\\label{eq:finitequantumerror}\n0\\le r_N-NL\\le\\log\\{N(K_0+1)\\}.\n\\end{equation}\nFor the scalar bounds, one may take\n$K_0=\\max(\\Omega,\\Omega^{-1})e^{BS}$.\nLet $e_N=\\log\\{N(K_0+1)\\}/N$. For the fixed-resource class with $F>0$,\neither certificate\n\\[\nr_N(u)/N\\ge F-\\delta\n\\quad\\text{or}\\quad\n\\sup_v r_N(v)/N-r_N(u)/N\\le\\delta\n\\]\nimplies\n\\begin{equation}\\label{eq:finitequantumrigidity}\n\\boxed{\\dist_1(u,\\mathcal O_*)\\le C\\sqrt{\\delta+e_N}.}\n\\end{equation}", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "thm:quantumstability", "pdf": "MANUSCRIPT.pdf", "page": 19 }, "dependencies": [ "thm:quantum", "thm:globalstability", "foundation:ode" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement.", "Fixed canonical reference vacuum; ideal prescribed material drive; no loss, dispersion, saturation, depletion or continuum ultraviolet claim." ], "independent_review": false, "formal_verification": false }, { "id": "lem:boxselector", "kind": "lemma", "number": "10.1", "page": 20, "title_latex": "The exact corner selector", "statement_latex": "For every finite $\\eta$, the maximizer of $H_\\eta$ on the rectangle is a\nunique corner away from the following finite set of switching angles.\nFor $\\eta>0$, let\n$\\theta_1=\\arctan(b_+/\\eta)$ and\n$\\theta_2=\\pi-\\arctan(\\eta/a_+)$. In increasing angle order the arcs and\nstates are\n\\[\n\\begin{array}{c|c}\n(0,\\theta_1)&(a_-,b_+)\\\\\n(\\theta_1,\\pi/2)&(a_+,b_+)\\\\\n(\\pi/2,\\theta_2)&(a_+,b_-)\\\\\n(\\theta_2,\\pi)&(a_+,b_+).\n\\end{array}\n\\]\nFor $\\eta<0$, put $e=-\\eta$,\n$\\theta_1=\\arctan(e/a_-)$, and\n$\\theta_2=\\pi-\\arctan(b_-/e)$. The selector is\n\\[\n\\begin{array}{c|c}\n(0,\\theta_1)&(a_-,b_-)\\\\\n(\\theta_1,\\pi/2)&(a_-,b_+)\\\\\n(\\pi/2,\\theta_2)&(a_-,b_-)\\\\\n(\\theta_2,\\pi)&(a_+,b_-).\n\\end{array}\n\\]\nFor $\\eta=0$, only the two opposite corners are used:\n$(a_-,b_+)$ on $(0,\\pi/2)$ and $(a_+,b_-)$ on $(\\pi/2,\\pi)$.\nFor nonzero $\\eta$ there are exactly three distinct states and four switches\nper projective turn, counting the switch at the periodic cut.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "lem:boxselector", "pdf": "MANUSCRIPT.pdf", "page": 20 }, "dependencies": [ "foundation:maxwell", "foundation:fractional" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "lem:boxclock", "kind": "lemma", "number": "10.2", "page": 21, "title_latex": "A strictly monotone clock", "statement_latex": "The function $\\tau_\\Box$ is continuous and strictly decreasing, with\n\\begin{equation}\\label{eq:boxlimits}\n\\tau_\\Box(-\\infty)=\\frac\\pi{\\sqrt{a_-b_-}},\\qquad\n\\tau_\\Box(+\\infty)=\\frac\\pi{\\sqrt{a_+b_+}}.\n\\end{equation}\nFor every finite $\\eta$, $G_\\Box(\\eta)>0$. At zero,\n\\begin{equation}\\label{eq:boxzero}\nG_\\Box(0)=\\tfrac12\\log\\frac{a_+b_+}{a_-b_-},\\qquad\n\\tau_\\Box(0)=\\frac\\pi2\\left(\\frac1{\\sqrt{a_-b_+}}+\n\\frac1{\\sqrt{a_+b_-}}\\right).\n\\end{equation}", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "lem:boxclock", "pdf": "MANUSCRIPT.pdf", "page": 21 }, "dependencies": [ "lem:boxselector" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "thm:boxfixed", "kind": "theorem", "number": "10.3", "page": 22, "title_latex": "Complete fixed-period Maxwell corner-purity theorem", "statement_latex": "Let $S>0$ and both positive coefficient intervals be strict. Define\n\\[\n\\mathcal N_\\Box=\n\\left\\{n\\in\\N:\\frac{n\\pi}{\\sqrt{a_+b_+}}0$ with\n$G_\\Box(\\gamma_\\Box)=\\gamma_\\Box\\tau_\\Box(\\gamma_\\Box)$.\nIt is the largest amplitude growth per unit time over all periodic and\nnonperiodic measurable rectangle controls. Its periodic equality profiles\nare precisely repetitions of the canonical $\\eta=\\gamma_\\Box$ cell.\nFor strict intervals this free-rate optimizer uses three material corners\nand four switches per turn. Its asymptotic squeezing and logarithmic\npair-count optimizers coincide with the classical ones, with rates\n$\\gamma_\\Box$ and $2\\gamma_\\Box$ per unit time.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "cor:boxfree", "pdf": "MANUSCRIPT.pdf", "page": 22 }, "dependencies": [ "thm:boxfixed", "thm:quantum" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "prop:boxstability", "kind": "proposition", "number": "10.5", "page": 23, "title_latex": "Rectangle structural stability", "statement_latex": "For strict intervals and fixed $S$ with $F_\\Box(S)>0$, the distance in\nmean $|a-a_*|+|b-b_*|$ to the union of winning corner-program translates\nis bounded by $C_\\Box\\sqrt{F_\\Box-\\log\\rho(M)}$, with a finite constant\ncomputable from the arc formulas, the margins \\eqref{eq:boxmargin}, and\nthe finite winning-branch gaps. The finite-cycle quantum certificate of\nTheorem~\\ref{thm:quantumstability} holds with this distance and constant.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "prop:boxstability", "pdf": "MANUSCRIPT.pdf", "page": 23 }, "dependencies": [ "thm:boxfixed", "thm:phasestability", "thm:quantumstability" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "lem:polylinear", "kind": "lemma", "number": "11.1", "page": 23, "title_latex": "Fractional barycentric linearization", "statement_latex": "For any convex representation $A=\\sum_jp_jA_j$, put\n$q_j=p_jv_j/v_A$. Then $q_j\\ge0$, $\\sum q_j=1$, and\n\\begin{equation}\\label{eq:polylinear}\n\\frac1{v_A}=\\sum_j\\frac{q_j}{v_j},\\quad\n\\frac{\\mathbf r(A)}{v_A}=\\sum_j\\frac{q_j\\mathbf r_j}{v_j},\\quad\n\\frac{f_A}{v_A}=\\sum_j\\frac{q_j f_j}{v_j}.\n\\end{equation}\nConversely any angular simplex weights $q_j$ give actual convex weights\n$p_j=(q_j/v_j)/(\\sum_iq_i/v_i)$. Thus fixing angularly integrated time and\naffine resources is a finite-moment linear problem without a relaxation gap.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "lem:polylinear", "pdf": "MANUSCRIPT.pdf", "page": 23 }, "dependencies": [ "foundation:projective", "foundation:fractional" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "thm:polyresource", "kind": "theorem", "number": "11.2", "page": 24, "title_latex": "Affine-resource calibration, purity, and finite switching", "statement_latex": "Suppose the maximizing vertex $j_*(\\theta)$ is unique almost everywhere,\nno pair $H_i,H_j$ is identically equal, and its one-turn time, resource,\nand gain are $(\\tau,\\mathbf U,G)$ with $G>0$.\nFor every periodic measurable control having an expanding eigenline of\nwinding $n$, period $S=n\\tau$, and resource integral $n\\mathbf U$,\n\\begin{equation}\\label{eq:polyglobaldefect}\n\\boxed{nG-\\log\\rho(M)=\n\\int_{\\theta_0}^{\\theta_0+n\\pi}E_A(\\theta)\\dd\\theta\\ge0.}\n\\end{equation}\nEquality holds exactly for translated repetitions of the selected\none-turn corner program. It has at most $q(q-1)$ switches per turn.\n\nFor any feasible time/resource moment in the relative interior of its\nmoment body, a finite supporting pair $(\\eta,\\boldsymbol\\xi)$ exists for\nmaximal gain. If this pair additionally has the stated no-persistent-tie\nproperty and positive gain, it supplies the certificate above.\nUniqueness of general resource multipliers is not asserted.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "thm:polyresource", "pdf": "MANUSCRIPT.pdf", "page": 24 }, "dependencies": [ "lem:polylinear", "foundation:convex" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement.", "Interior/normality and persistent-tie exclusions are explicit; arbitrary polytope boundary strata are not classified." ], "independent_review": false, "formal_verification": false }, { "id": "thm:abstract", "kind": "theorem", "number": "11.3", "page": 25, "title_latex": "Preserved calibrated rotating-polytope extremality", "statement_latex": "Every periodic measurable $A(s)\\in\\mathcal A$ obeys\n$S^{-1}\\log\\rho(M)\\le\\lambda_*$. Equality holds exactly for translated\nrepetitions of the unique angular maximizing-vertex program; its one-turn\nperiod must divide $S$. There are at most $q(q-1)$ switches per turn.\nThe same sharp asymptotic rate holds for nonperiodic controls.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "thm:abstract", "pdf": "MANUSCRIPT.pdf", "page": 25 }, "dependencies": [ "thm:polyresource" ], "novelty_status": "V1_PRESERVED_OR_GENERALIZED; external priority unestablished", "corroborating_test_reports": [ "results/verification_v1_rerun.json", "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "thm:boxresource", "kind": "theorem", "number": "12.1", "page": 26, "title_latex": "Explicit two-resource Maxwell extremality", "statement_latex": "With the strict rectangle and interior means above, form the finite set\n\\begin{equation}\\label{eq:boxresourceN}\n\\mathcal N_{\\rm res}^{\\Box}=\n\\left\\{n\\in\\N:\\begin{array}{l}\nn\\pi/\\sqrt{a_+b_+}0$.\nThen over all measurable periodic rectangle controls satisfying\n\\eqref{eq:boxmeans},\n\\begin{equation}\\label{eq:boxresourceF}\n\\boxed{\\sup L=F_{\\rm res}^{\\Box}\n=\\max\\left(\\{0\\}\\cup\\{nG_*(S/n):n\\in\\mathcal N_{\\rm res}^{\\Box}\\}\\right).}\n\\end{equation}\nEvery positive-growth equality case is exactly a translation of repeated\ncopies of \\eqref{eq:boxresourceprogram}, with a winning winding number.\nThere are at most two winning numbers, and any tie is between adjacent\nintegers. There are no additional positive-growth optimizers.\nIf the set is empty, every admissible control has zero exponential growth.\n\nThus two opposite material corners are necessary and sufficient precisely\nwhen $P+Q=1$. Otherwise exactly three corners are used, with four switches\nper projective turn and two equal visits to the selected diagonal corner.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "thm:boxresource", "pdf": "MANUSCRIPT.pdf", "page": 26 }, "dependencies": [ "lem:boxnormal", "lem:boxpalindrome", "prop:boxresourceconcavity", "foundation:projective" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_maxwell_resource.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "lem:boxnormal", "kind": "lemma", "number": "12.2", "page": 26, "title_latex": "Boundary moments and normal duals for two Maxwell resources", "statement_latex": "A boundary moment of $\\mathcal K_{\\Box}$ whose two physical means are\ninterior can carry only zero gain. Every positive-gain moment with interior\nmeans therefore has a finite normal triple $(\\eta,\\xi,\\zeta)$ for\n\\begin{equation}\\label{eq:boxresourceH}\nH(\\theta,a,b)=\n\\frac{(b-a)\\sin\\theta\\cos\\theta-\\eta-\\xi a-\\zeta b}\n{b\\cos^2\\theta+a\\sin^2\\theta}.\n\\end{equation}\nIts maximizing corner is unique almost everywhere, and the normal triple\nis unique.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "lem:boxnormal", "pdf": "MANUSCRIPT.pdf", "page": 26 }, "dependencies": [ "lem:polylinear", "foundation:convex", "lem:boxselector" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_maxwell_resource.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "lem:boxpalindrome", "kind": "lemma", "number": "12.3", "page": 28, "title_latex": "Time reversal locks the two diagonal visits", "statement_latex": "The unique positive-gain one-turn maximizing program at an interior-mean\nmoment has equal durations for its two visits to the diagonal corner.\nConsequently it is exactly \\eqref{eq:boxresourceprogram}, up to translation.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "lem:boxpalindrome", "pdf": "MANUSCRIPT.pdf", "page": 28 }, "dependencies": [ "lem:boxnormal", "foundation:segments", "foundation:projective" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_maxwell_resource.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "prop:boxresourceconcavity", "kind": "proposition", "number": "12.4", "page": 29, "title_latex": "Resource-ray concavity and rate calibration", "statement_latex": "The feasible one-turn moments on the ray\n$\\tau(1,\\bar a,\\bar b)$ form a nondegenerate compact interval\n$[\\tau_-^{\\Box},\\tau_+^{\\Box}]$. Its endpoints carry zero gain. On its\ninterior the maximal gain $G_*(\\tau)$ is positive, analytic, and strictly\nconcave, and is the gain of \\eqref{eq:boxresourceprogram}. There is a unique\nmaximum of $G_*(\\tau)/\\tau$, at $\\tau_\\Box^\\circ$ in that interior.\nFixed-period winners lie among\n$\\lfloor S/\\tau_\\Box^\\circ\\rfloor,\\lceil S/\\tau_\\Box^\\circ\\rceil$,\nafter feasibility filtering.\n\nWriting $\\gamma_\\Box=G_*(\\tau_\\Box^\\circ)/\\tau_\\Box^\\circ$, its normal dual\nsatisfies $\\eta+\\xi\\bar a+\\zeta\\bar b=\\gamma_\\Box$ and gives a bounded\nperiodic projective gauge $V$ such that every trajectory obeys\n\\begin{equation}\\label{eq:boxresourcegauge}\n\\log\\frac{\\|x(s)\\|_2}{\\|x(0)\\|_2}\n\\le\\gamma_\\Box s+\\xi\\int_0^s(a-\\bar a)\\dd v\n+\\zeta\\int_0^s(b-\\bar b)\\dd v+\n V(\\theta(s))-V(\\theta(0)).\n\\end{equation}\nThus the same sharp rate holds even for nonperiodic controls whose two\nlong-run means converge to the declared resources.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "prop:boxresourceconcavity", "pdf": "MANUSCRIPT.pdf", "page": 29 }, "dependencies": [ "lem:boxnormal", "lem:boxpalindrome", "foundation:convex" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_maxwell_resource.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "cor:boxresourceStability", "kind": "corollary", "number": "12.5", "page": 30, "title_latex": "Two-resource Maxwell stability and quantum extremality", "statement_latex": "For every fixed strict rectangle, period, and pair of interior means with\n$F_{\\rm res}^{\\Box}>0$, there is an explicit finite $C_{\\Box,\\rm res}$ such\nthat\n\\begin{equation}\\label{eq:boxresourceStability}\n\\inf_{(a_*,b_*)\\in\\mathcal O_*}\\frac1S\\int_0^S\n\\bigl(|a-a_*|+|b-b_*|\\bigr)\\dd s\n\\le C_{\\Box,\\rm res}\\sqrt{F_{\\rm res}^{\\Box}-L}.\n\\end{equation}\nHere $\\mathcal O_*$ retains all translations of every winning explicit\nprogram. The optimal asymptotic squeezing rate is $F_{\\rm res}^{\\Box}$ per\nperiod; the optimal logarithmic pair-number rate is $2F_{\\rm res}^{\\Box}$.\nTheir maximizing controls are exactly the same programs. Near-maximal\nfinite-period squeezing yields \\eqref{eq:boxresourceStability} with the\nadditional $O(\\log N/N)$ spectral-deficit term of\nTheorem~\\ref{thm:quantumstability}.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "cor:boxresourceStability", "pdf": "MANUSCRIPT.pdf", "page": 30 }, "dependencies": [ "thm:boxresource", "prop:boxstability", "thm:quantum", "thm:quantumstability" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_maxwell_resource.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "prop:inversefailure", "kind": "proposition", "number": "13.1", "page": 31, "title_latex": "Exact two-layer isodiscriminant counterexample", "statement_latex": "Even when $T$ and common positive material bounds are known, the entire\nfunction $D(k)=\\frac12\\tr M(k)$ need not determine a two-layer medium up\nto translation or reversal.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "prop:inversefailure", "pdf": "MANUSCRIPT.pdf", "page": 31 }, "dependencies": [ "foundation:segments" ], "novelty_status": "V1_PRESERVED_OR_GENERALIZED; external priority unestablished", "corroborating_test_reports": [ "results/verification_v1_rerun.json", "results/verification_inverse_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "thm:inverse", "kind": "theorem", "number": "14.1", "page": 31, "title_latex": "Finite derivatives determine every reduced layer", "statement_latex": "Fix $T>0$, $\\mu>0$, and an integer layer bound $r\\geq1$. Let a marked medium\nhave at most $r$ positive-duration layers with positive constant speeds,\nand write $N=2^r$. The real dataset\n\\begin{equation}\\label{eq:jetdata}\n\\mathcal J_r=\n\\left\\{M_{11}^{(2j)}(0),M_{12}^{(2j)}(0):1\\leq j\\leq N-1\\right\\}\n\\end{equation}\nuniquely determines its reduced ordered layer sequence. The number of\nlisted real data is $2N-2=2^{r+1}-2$; $T$ and $M_{11}(0)=1$ are already\nknown. There is a constructive reconstruction by finite\nexponential-polynomial recovery followed by layer peeling.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "thm:inverse", "pdf": "MANUSCRIPT.pdf", "page": 31 }, "dependencies": [ "lem:prony", "lem:peel", "foundation:analytic_jets" ], "novelty_status": "V1_PRESERVED_OR_GENERALIZED; external priority unestablished", "corroborating_test_reports": [ "results/verification_v1_rerun.json", "results/verification_inverse_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "lem:prony", "kind": "lemma", "number": "14.2", "page": 32, "title_latex": "Finite exponential determination", "statement_latex": "An exponential polynomial with at most $N$ distinct real exponents is\nuniquely determined by its first $2N$ derivatives at zero, counting the\nzeroth derivative. Its exponents and coefficients can be constructed from\nthose derivatives by a finite Prony procedure.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "lem:prony", "pdf": "MANUSCRIPT.pdf", "page": 32 }, "dependencies": [ "foundation:linear_algebra" ], "novelty_status": "V1_PRESERVED_OR_GENERALIZED; external priority unestablished", "corroborating_test_reports": [ "results/verification_v1_rerun.json", "results/verification_inverse_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "lem:peel", "kind": "lemma", "number": "14.3", "page": 33, "title_latex": "Recovering the leading temporal layer", "statement_latex": "For a nonempty positive reduced medium, its first-row functions $f,g$\nuniquely determine its first speed and optical duration by\n\\begin{align}\nc_1&=\\lim_{z\\to+\\infty}\\frac{f(z)}{g(z)},\\label{eq:cpeel}\\\\\na_1&=-\\frac12\\lim_{z\\to+\\infty}\\frac1z\n\\log\\left|\\frac{f(z)-c_1g(z)}{f(z)+c_1g(z)}\\right|.\\label{eq:apeel}\n\\end{align}\nThe next first row is $(f,g)(E_1^{\\mathrm h})^{-1}$ and\n$\\tau_1=a_1/c_1$.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "lem:peel", "pdf": "MANUSCRIPT.pdf", "page": 33 }, "dependencies": [ "foundation:analytic_jets", "foundation:segments" ], "novelty_status": "V1_PRESERVED_OR_GENERALIZED; external priority unestablished", "corroborating_test_reports": [ "results/verification_v1_rerun.json", "results/verification_inverse_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "prop:datacount", "kind": "proposition", "number": "15.1", "page": 33, "title_latex": "Preserved regular-data lower bound", "statement_latex": "Any $C^1$ real measurement map that is injective on the open exactly-$r$-layer\nclass with known $T$ requires at least $2r-1$ scalar outputs. In particular\n\\begin{equation}\\label{eq:counts}\n2r-1\\le m_{\\min}^{\\rm jet}(r)\\le 2^{r+1}-2.\n\\end{equation}\nFor $r=1$, one jet value suffices: $c_1=\\sqrt{-M_{11}''(0)}/T$.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "prop:datacount", "pdf": "MANUSCRIPT.pdf", "page": 33 }, "dependencies": [ "thm:inverse", "foundation:inverse_function" ], "novelty_status": "V1_PRESERVED_OR_GENERALIZED; external priority unestablished", "corroborating_test_reports": [ "results/verification_v1_rerun.json", "results/verification_inverse_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "lem:projection", "kind": "lemma", "number": "15.2", "page": 34, "title_latex": "A direct generic projection bound", "statement_latex": "Let $F$ be a $C^1$ injective map on a finite or countable union of smooth\nparameter strata, each of dimension at most $d$, into $\\R^K$.\nFor any integer $m>2d$, almost every matrix $P\\in\\R^{m\\times K}$ makes\n$PF$ injective on the whole union.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "lem:projection", "pdf": "MANUSCRIPT.pdf", "page": 34 }, "dependencies": [ "foundation:inverse_function" ], "novelty_status": "ESTABLISHED_METHOD_OR_DIRECT_APPLICATION; not claimed as invention of the general method", "corroborating_test_reports": [ "results/verification_inverse_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement.", "Generic projection improves output count, not derivative order; no uniform noise bound.", "Global CAD projection search was not executed; its termination is a mathematical algorithm claim." ], "independent_review": false, "formal_verification": false }, { "id": "thm:jetCompression", "kind": "theorem", "number": "15.3", "page": 34, "title_latex": "Linear-count compression of exact transfer data", "statement_latex": "For the marked positive finite-layer problem with known $T$,\n\\begin{equation}\\label{eq:linearcount}\n\\boxed{2r-1\\le m_{\\min}^{\\rm jet}(r)\n\\le\\min\\{2^{r+1}-2,\\,4r-1\\}.}\n\\end{equation}\nWhen the time-average of $c(t)^2$ is also fixed and positive, define the\nanalogous minimum on that resource-constrained class. For $r\\ge2$,\n\\begin{equation}\\label{eq:resourcecount}\n\\boxed{2r-2\\le m_{\\min}^{\\rm jet,res}(r)\n\\le\\min\\{2^{r+1}-2,\\,4r-3\\}.}\n\\end{equation}\nThe lower bound requires a nonempty open reduced exactly-$r$-layer resource\nstratum. For one layer the known average fixes the material completely,\nso no additional spectral datum is needed. Since\n$c^2=(\\mu\\varepsilon)^{-1}$, this is precisely the inverse-permittivity\nresource when $\\mu$ is known and constant.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "thm:jetCompression", "pdf": "MANUSCRIPT.pdf", "page": 34 }, "dependencies": [ "thm:inverse", "lem:projection", "prop:datacount" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_inverse_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement.", "Generic projection improves output count, not derivative order; no uniform noise bound.", "Global CAD projection search was not executed; its termination is a mathematical algorithm claim." ], "independent_review": false, "formal_verification": false }, { "id": "prop:CAD", "kind": "proposition", "number": "15.4", "page": 35, "title_latex": "Certified projection search for algebraic inputs", "statement_latex": "When $T$ and any imposed resource $Q$ are positive real algebraic numbers,\na finite algorithm can find a rational matrix $P$ attaining either improved\nupper count in Theorem~\\ref{thm:jetCompression}. For exact real algebraic\ncompressed observations, the unique reduced medium can then be recovered\nby a finite real-algebraic decision and root-isolation procedure.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "prop:CAD", "pdf": "MANUSCRIPT.pdf", "page": 35 }, "dependencies": [ "thm:jetCompression", "foundation:analytic_jets", "external:real_QE" ], "novelty_status": "ESTABLISHED_METHOD_OR_DIRECT_APPLICATION; not claimed as invention of the general method", "corroborating_test_reports": [ "results/verification_inverse_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement.", "Generic projection improves output count, not derivative order; no uniform noise bound.", "Global CAD projection search was not executed; its termination is a mathematical algorithm claim." ], "independent_review": false, "formal_verification": false }, { "id": "prop:localInverse", "kind": "proposition", "number": "15.5", "page": 36, "title_latex": "Local inverse conditioning at regular points", "statement_latex": "For the full jet map, or any injective projected map above, the full-rank\npoints are open and dense within every smooth parameter stratum. At such\na point $x_0$, let $\\sigma>0$ be the smallest singular value of its Jacobian\nin the declared parameter/data norms. On a sufficiently small neighborhood,\n\\begin{equation}\\label{eq:localInverse}\n\\|x-y\\|\\le\\frac2\\sigma\\|F(x)-F(y)\\|.\n\\end{equation}\nA locally constrained fit whose data residual is at most the true noise\nbound $\\epsilon$ consequently has parameter error at most $4\\epsilon/\\sigma$,\nprovided the fit and true point remain in that neighborhood.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "prop:localInverse", "pdf": "MANUSCRIPT.pdf", "page": 36 }, "dependencies": [ "prop:datacount", "thm:inverse", "thm:jetCompression", "foundation:inverse_function" ], "novelty_status": "ESTABLISHED_METHOD_OR_DIRECT_APPLICATION; not claimed as invention of the general method", "corroborating_test_reports": [ "results/verification_inverse_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "thm:rigidityCompression", "kind": "theorem", "number": "15.6", "page": 36, "title_latex": "Saturation is an inverse certificate", "statement_latex": "Suppose $R,S,m$ are known, $F(R,S,m)>0$, and exact classical or asymptotic\nquantum extremality is certified. If the winning winding is unique, the\nentire temporal program modulo translation is determined with no further\nreal spectral measurements. If two windings tie, specifying one of those\ntwo integers determines the program. More generally, a known winding $n$\nand the branch saturation equality $L=nG_m(S/n)>0$ determine its canonical\nprogram even if that branch is not the global winner.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "thm:rigidityCompression", "pdf": "MANUSCRIPT.pdf", "page": 36 }, "dependencies": [ "thm:resource", "thm:concavity", "thm:quantum", "thm:boxresource" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false }, { "id": "prop:nearNoGo", "kind": "proposition", "number": "15.7", "page": 37, "title_latex": "No finite exact inverse for the entire near-extremal class", "statement_latex": "Fix a positive resource optimum $F$ and any $\\delta>0$. No fixed finite\nnumber of real observations that are $C^1$ on finite-dimensional control\nfamilies can uniquely reconstruct every measurable fixed-resource profile\nwith $L>F-\\delta$. This remains true if the growth itself is included among\nthe observations. Thus near-extremality gives approximate structural\ncompression, not exact finite-data reconstruction of arbitrary residual\nvariations.", "status": "PROVED_UNDER_STATED_ASSUMPTIONS", "proof_location": { "source": "MANUSCRIPT.tex", "label": "prop:nearNoGo", "pdf": "MANUSCRIPT.pdf", "page": 37 }, "dependencies": [ "thm:resource", "prop:datacount", "foundation:ode" ], "novelty_status": "NEW_TO_THIS_RELEASE; first-in-literature priority not established", "corroborating_test_reports": [ "results/verification_v2.json" ], "limits": [ "Hypotheses are the literal statement below and its referenced definitions. No independent or proof-assistant certification.", "Listed tests verify only finite consequences and are not proofs of the full statement." ], "independent_review": false, "formal_verification": false } ] }