Datasets:

PureOne
/

Chisael-Nexythra

{"record_id": "CLAIM_AUDIT-001", "document_id": "CLAIM_AUDIT.md", "chunk_index": 0, "line_start": 1, "line_end": 58, "content_format": "markdown", "text": "# Claim audit — CHISÆL–NEXYTHRA II v1.0.0\n\nDate: 9 October 2026\n\n## Principal result\n\n**Claim:** Any finite helicity-zero calendar, including arbitrary finite multiplicities, has a global smooth unforced periodic Navier–Stokes realization at fixed positive viscosity, with small initial data, full three-direction spatial support, nonzero initial projected advection and vortex stretching, an exact anchor, and no additional nonnegative-time zeros.\n\n**Evidence:** Written proof: exponential Chebyshev compiler → exact shear realization → finite observation Jacobian → analytic implicit-function continuation → explicit nonlinear witnesses → uniform local zero factorization → weighted global tail estimate.\n\n**Novelty status:** Candidate original combination; priority unverified. The component theories are established. A specialist should compare the complete theorem against the helicity-asymptotics and prescribed-flow literature.\n\n**Restrictions:** Small data; smooth incompressible periodic equation; finite calendar; sufficiently small specifically parameterized transverse perturbation. The permitted perturbation interval is existential, not numerically certified. The full between-event helicity curve is not prescribed.\n\n## Quadratic retuning\n\n**Claim:** The correcting carrier map is analytic and even in epsilon; the leading allowed correction is second order, while the initial projected nonlinearity has a nonzero first-order component.\n\n**Evidence:** Exact half-period translation symmetry, local uniqueness of the implicit solution, and explicit pressure-projected Fourier witness.\n\n**Restriction:** Does not promise small constants, arbitrary-perturbation robustness, or low numerical condition number.\n\n## Triad response and extinction\n\n**Claim:** The specified three linearly polarized waves have H''(0)=2 nu a b c cos(Phi). Their exact third mixed amplitude derivative is K_nu(t) cos(Phi), where K_nu(t)=exp(-6 nu t)[exp(4 nu t)-1-4 nu t]/(8 nu)>0 for t>0. At quadrature phase, an orientation-reversing symmetry forces H identically zero during the smooth evolution.\n\n**Evidence:** Exact Fourier contractions (-1/4,0,1/4), analytic Duhamel calculation, and symmetry/uniqueness proof. Symbolic code independently checks the initial derivatives and reflection.\n\n**Restrictions:** The kernel is an exact response derivative, not the full finite-amplitude helicity history. Its positivity alone does not establish global positivity of that full history. A closed triad spans two independent wavevector directions; the separate main family supplies full three-direction support.\n\n**Prior art:** Waleffe (1992), Sanada (1993), and recent triad-phase work preclude claiming the general phenomena of triadic interaction or nonlinear helicity production are new.\n\n## Supporting results\n\n- Energy-clock event multiplicity is an exact change-of-variables statement under a common clock and strictly decreasing energy. It is not literal physical time freezing.\n- The energy optimizer is a finite-library weighted-L1 program. Some optimal solution uses the minimal M+1 exponential carriers. This is not a global minimum-energy theorem for arbitrary nonlinear flows.\n- The heat-carrier energy bound is restricted to that scaffold class.\n- The momentum-word theorem concerns squarefree mixed derivatives. Repeated-parameter responses can occur earlier. An allowed momentum word need not have a nonzero coefficient.\n- Mirror twins establish exact energy-only handedness nonidentifiability. Reflection symmetry itself is classical.\n- Multiple-event noise sensitivity has exponent 1/m. Uncorrected high-order events are not generically robust.\n\n## Verification accounting\n\nSaved suite: 1,139 passed, zero failed. Mix of exact symbolic assertions, high-precision evaluations, sampled signs, and floating-point primal/dual checks. These are not 1,139 independent theorem proofs. No randomized claims are made for this deterministic suite.\n\nThe maximum 2.5791e-124 residual is an arbitrary-precision interpolation residual. Numerical Galerkin observation residuals are near machine precision at their solved cutoff; forward refinement discrepancies are of order 1e-12 in the documented example. Neither is an interval-certified continuum PDE bound.\n\n## Claims explicitly not made\n\nNo certified world first; no independent peer review; no proof-assistant certificate; no disproof of Maxwell, Gibbs, Heaviside, Boltzmann, Casimir, or an established mathematical theorem; no general Navier–Stokes regularity result; no shock/plasma simulation; no physical energy generation; no literal new spatial dimensions.\n\n## High-priority independent checks\n\n1. Initial-time jet regularity and analyticity of the finite observation map.\n2. Exact spectral-gap difference estimate and its use on the entire positive time tail.\n3. Uniform nonvanishing of the local Taylor factors near multiple events.\n4. Fourier normalization, triad contraction signs, Duhamel factors, and quadrature inversion equations.\n5. Historical overlap with helicity asymptotics, prescribed vortex events, and nonlinear phase selection.\n", "source_path": "research/CLAIM_AUDIT.md", "source_sha256": "c5d9436db8393490c49dab65bb1ce892e6e335bb4bd8f2bb99727aacf183309e", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "LICENSE_STATUS-001", "document_id": "LICENSE_STATUS.md", "chunk_index": 0, "line_start": 1, "line_end": 3, "content_format": "markdown", "text": "# License status\n\nNo public redistribution license has been selected for this research release. The package is supplied for review and use by the requesting user. Select an appropriate project license before making a public repository release. Third-party dependencies and cited publications retain their own terms; their source files are not bundled here.\n", "source_path": "research/LICENSE_STATUS.md", "source_sha256": "3f32840bed5524899f435c51dd59b56519d51c0ea9e656037fff949f3746f2bf", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "README-001", "document_id": "README.md", "chunk_index": 0, "line_start": 1, "line_end": 63, "content_format": "markdown", "text": "# CHISÆL–NEXYTHRA II\n## Nonlinear Chronochiral Divisors and Energy-Clock Invariants\n\nVersion 1.0.0 · 9 October 2026 · Research draft\n\n**Read the 72-page `CHISAEL_NEXYTHRA_II_manuscript.pdf`.** The complete editable source is `manuscript.tex` and the `chapters/` directory.\n\nThis package develops an exact finite helicity-event calculus for small, smooth, unforced, incompressible Navier–Stokes flows on the three-dimensional torus. The central construction prescribes a finite calendar of helicity contacts and sign reversals, adds actual three-direction nonlinear interaction, and restores the exact calendar with an even analytic correction of the carrier amplitudes. A separate explicit nonhelical triad gives an exact cubic response kernel and a phase-selected all-time extinction symmetry.\n\n## Main analytical results\n\n1. Every finite nonnegative-time helicity-zero divisor, with finite positive multiplicities and a later nonzero anchor, has a small-data fixed-viscosity realization with nonzero initial projected advection and vortex stretching and no extra nonnegative-time helicity zeros.\n2. For the specified transverse perturbation, carrier retuning is O(epsilon²), although nonlinear interaction is already O(epsilon).\n3. Three explicit linearly polarized waves satisfy H(0)=H'(0)=0 and H''(0)=2 nu a b c cos(Phi). Their exact third mixed amplitude derivative is exp(-6 nu t)[exp(4 nu t)-1-4 nu t] cos(Phi)/(8 nu).\n4. Quadrature phase imposes an inversion symmetry that makes total helicity identically zero throughout the smooth evolution.\n5. Event multiplicity is invariant in the energy coordinate under a common operational clock; finite-library energy minimization has an explicit weighted-L1 dual; mixed Fourier responses obey a squarefree momentum-word rule.\n\nThese statements have analytical proofs in the manuscript. They are not inferred solely from numerical experiments.\n\n## Evidence and limitations\n\nThe saved deterministic suite reports **1,139 checks passed, 0 failed**. Its maximum high-precision event-jet residual is approximately 2.5791e-124. That number applies to exponential-interpolation evaluations, not to a continuum PDE error.\n\nThe Fourier–Galerkin demonstration and its refinement checks are **not interval-certified continuum calculations**. The initial amplitudes found by the numerical root solver are not asserted to be the exact continuum correction.\n\nNo independent peer review or proof-assistant formalization has occurred. Historical priority is not certified. Existing mathematics already includes helical triads, nonlinear helicity production, Chebyshev interpolation, small-data theory, analytic dependence, and prescribed vortex-topology constructions. Consult `CLAIM_AUDIT.md` and `notes/LITERATURE_SEARCH.md` before making novelty claims.\n\nThis is not a disproof of an established mathematical theorem or a physical law. It does not model shocks, ionization, molecular excitation, or literal time freezing. Chisa's ability names are imaginative design prompts, not evidence.\n\n## Contents\n\n- `CHISAEL_NEXYTHRA_II_manuscript.pdf`: final manuscript.\n- `manuscript.tex`, `chapters/*.tex`: complete LaTeX source.\n- `references.bib`: machine-readable bibliography; the PDF uses the hand-edited reference chapter.\n- `src/event_calculus.py`: arbitrary-precision event compiler and finite-library energy optimizer.\n- `src/fourier_jets.py`: exact, untruncated low-order Fourier algebra.\n- `src/nonlinear_demo.py`: dealiased Fourier–Galerkin nonlinear retuning demonstration.\n- `src/refinement_demo.py`: forward time-step/cutoff comparison at the saved corrected data.\n- `src/make_figures.py`: regenerates all sixteen figures and associated data.\n- `tests/verify.py`: deterministic exact and numerical checks.\n- `data/`: verification report, exact example, numerical trajectories, optimization certificate, CSV plot data.\n- `figures/`: sixteen vector PDF figures and PNG previews.\n- `CLAIM_AUDIT.md`, `REPRODUCIBILITY.md`, `RELEASE_DESCRIPTION.md`: scope, evidence, and release guidance.\n- `build.sh`, `build.ps1`, `requirements.txt`, `CITATION.cff`, `SHA256SUMS.txt`: build and release metadata.\n\n## Reproduce\n\nPython 3.10 or newer is required by the type annotation syntax. Install the dependencies with `python -m pip install -r requirements.txt`. Then run:\n\n```bash\npython tests/verify.py\npython src/nonlinear_demo.py\npython src/refinement_demo.py\npython src/make_figures.py\n```\n\nBuild the PDF with `bash build.sh`, or run `./build.ps1` in PowerShell with a TeX distribution and latexmk installed. The document also compiles with repeated pdflatex runs until references stabilize. No network access is needed after dependencies and TeX packages are installed.\n\nFor a quick algebra-only audit, run just `python tests/verify.py`. The full numerical demonstration is optional and computationally more expensive.\n\n## Attribution\n\nMaciej Nowicki supplied the conceptual direction and imaginative research brief. Artificial Hyperintelligence Eve is the disclosed AI-assisted development persona used for mathematical exposition, computational checking, and manuscript production. This is not a claim of independent human peer review. A public project license has not been selected; see `LICENSE_STATUS.md`.\n", "source_path": "research/README.md", "source_sha256": "9bf3e6d457c9a56e6ee7fe177adfcffadefbdcc04e1593d468a0389a042f3e5f", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "RELEASE_DESCRIPTION-001", "document_id": "RELEASE_DESCRIPTION.md", "chunk_index": 0, "line_start": 1, "line_end": 17, "content_format": "markdown", "text": "# Release title\n\nCHISÆL–NEXYTHRA II: Nonlinear Chronochiral Divisors, Phase Extinction, and Energy-Clock Invariants\n\n# Description\n\nThis research draft introduces a proposed chronochiral event calculus for the total helicity of small smooth periodic Navier–Stokes flows. An exact exponential compiler first realizes any finite calendar of helicity contacts and sign reversals. An analytic continuation then adds genuine three-direction advection and vortex stretching while restoring the prescribed zero jets and a later anchor. A half-period translation symmetry makes the required carrier corrections quadratic in the transverse perturbation, although nonlinear interaction is already present at first order. A weighted tail estimate excludes additional late-time zeros.\n\nA second explicit construction starts with three individually nonhelical standing waves. The manuscript evaluates their initial helicity acceleration and exact third mixed response kernel. A quadrature phase imposes an orientation-reversing symmetry that extinguishes total helicity throughout the smooth evolution. Supporting results treat event multiplicity in an energy coordinate, common clock pauses, finite-library minimum-energy compilation, momentum-word selection, and the sharp sensitivity of high-order events to output error.\n\nThe release contains the complete manuscript and editable LaTeX source, sixteen reproducible figures, exact Fourier algebra, arbitrary-precision interpolation, a nonlinear Fourier–Galerkin demonstration, numerical data, and a claim/novelty audit. The saved verification suite passes 1,139 implemented checks.\n\nThe proposed originality lies in the specific synthesis and exact statements, not in inventing the established component theories. Historical priority has not been certified. The work is not independently peer reviewed, proof-assistant formalized, experimentally validated, or a disproof of an established physical law. The numerical PDE demonstration is not interval certified.\n\n# Keywords\n\nNavier–Stokes; helicity; nonlinear observability; zero multiplicity; Chebyshev systems; implicit-function continuation; triadic interaction; reflection symmetry; phase response; energy coordinate; inverse problems; mathematical fluid dynamics.\n", "source_path": "research/RELEASE_DESCRIPTION.md", "source_sha256": "f4d1a24343c4e443afc55e8cafa9d6b16b6149a0cab7078b52909d89997e2716", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "REPRODUCIBILITY-001", "document_id": "REPRODUCIBILITY.md", "chunk_index": 0, "line_start": 1, "line_end": 33, "content_format": "markdown", "text": "# Reproducibility\n\n## Build\n\nPython dependencies are listed in `requirements.txt`; actual production versions are recorded in `data/environment.json`. The syntax requires Python >=3.10. A conventional TeX Live/MiKTeX installation with latexmk, Latin Modern, AMS packages, mathrsfs, listings, hyperref, titlesec, fancyhdr, and microtype builds the PDF. Font files are not distributed.\n\nRun the commands in README from the package root. Scripts locate data relative to their own file paths, except the illustrative interactive snippet, which assumes the package root is the working directory.\n\n## Deterministic verification\n\n`python tests/verify.py` regenerates `data/verification.json`, `data/exact_example.json`, and `data/optimal_score.json`. The expected saved count is 1,139 assertions. A failed assertion aborts the script; a pass report is written only after completion.\n\nHigh-precision event times are supplied as decimal strings, avoiding an initial conversion through binary machine precision. Exact logarithmic-time examples are assembled using rational exponential coordinates, not floating logarithms.\n\nThe exact Fourier derivative calculation retains every mode generated by its low-order convolution. No Galerkin cutoff is applied in that module.\n\n## Numerical simulation\n\n`python src/nonlinear_demo.py` runs the default dealiased Fourier–Galerkin example (cutoff 3, 12³ product grid, nu=1, dt=0.001, delta=0.1). Three transverse amplitudes are tested: 0.1, 0.2, and 0.4. A nonlinear root solve restores the helicity observations at 0, 0.12, and 0.25. Floating-point residuals and machine-dependent runtimes can vary slightly across environments.\n\n`python src/refinement_demo.py` uses the saved epsilon=0.4 corrected initial amplitudes and evaluates them at dt=0.0005 with cutoffs 3, 4, and 5. It does not perform a new root solve at each cutoff. This is a forward comparison, not an interval proof or a mathematical continuum convergence certificate.\n\n## Figures\n\n`python src/make_figures.py` regenerates sixteen vector PDF figures, matching PNG previews, and CSV data where the source is not already in the trajectory JSON. The energy-decay plot reads `data/nonlinear_demo.json`. Clock illustrations are composed from an exactly compiled scaffold and a prescribed nonnegative gate; they are not simulations of a literal physical freeze.\n\n## Proof versus computation\n\nThe continuum theorem is an analytical existence statement with an unspecified sufficiently small perturbation interval. The numerical example illustrates its structure but is not asserted to lie inside a separately validated continuum interval. Sampling a curve cannot prove the absence of other zeros; that conclusion comes from the Chebyshev, factorization, and weighted-tail arguments.\n\n## Integrity\n\n`SHA256SUMS.txt` hashes the distributed files except itself. On systems with sha256sum, run `sha256sum -c SHA256SUMS.txt` from the package root. The archive excludes caches, TeX auxiliary files, rendered review pages, and redundant intermediate PDFs.\n", "source_path": "research/REPRODUCIBILITY.md", "source_sha256": "19b729f1b093f441e586d8a9b22c8f73be68df6621db7663003e242b7e5fbd51", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "01_foundations-001", "document_id": "chapters/01_foundations.tex", "chunk_index": 0, "line_start": 1, "line_end": 38, "content_format": "latex_source", "text": "\\chapter{The research problem: from a cut to a calendar}\n\\label{ch:problem}\n\\section{The limitation of the earlier construction}\nThe preceding CHIS\\AE L--NEXYTHRA manuscript constructed exact viscous flows whose initial helicity and arbitrarily many initial helicity derivatives vanish, although future helicity is nonzero. Its solutions were useful analytical examples, but the advective nonlinearity vanished identically. Their dynamics therefore reduced to different exponential decay rates of oppositely helical modes.\n\nThis manuscript changes both the question and the mechanism. The first new question is whether one can prescribe several distinct observation events, including multiple-order contacts and sign reversals, instead of one silent initial instant. The second is whether those events can remain exact after adding a perturbation that genuinely activates the three-dimensional nonlinear equations. The third asks whether the handedness can be generated without any initially nonzero modal helicity at all.\n\nThese questions should not be conflated. A signed spectral sum can display elaborate cancellation while every constituent simply decays. A nonlinear flow can create correlations among initially nonhelical waves. An observation can also appear to stop because its clock pauses. All three can produce a small or vanishing measured helicity, but they are different mathematical phenomena.\n\n\\section{The proposed object}\n\\begin{definition}[Chronochiral event divisor]\nA finite chronochiral event divisor is a formal sum\n\\begin{equation}\n\\score=\\sum_{i=1}^{s}m_i[\\tau_i],\\qquad\n0\\le\\tau_1<\\cdots<\\tau_s<\\infty,\\qquad m_i\\in\\N.\n\\end{equation}\nIts degree is $M=\\sum_i m_i$. A smooth real signal $h$ realizes the divisor when its complete zero set on the specified time domain consists of the points $\\tau_i$, and\n\\begin{equation}\nh^{(q)}(\\tau_i)=0\\quad(0\\le q<m_i),\\qquad\nh^{(m_i)}(\\tau_i)\\ne0.\n\\end{equation}\n\\end{definition}\nThe term ``divisor'' borrows the familiar idea of recording zeros together with multiplicities. Nothing about the term by itself establishes novelty. Its role here is to provide one object that can be compiled into initial fluid data, continued under nonlinear perturbation, transported into an energy coordinate, and tested against finite-resolution observations.\n\nAn odd multiplicity changes the sign of $h$ across an interior event. An even multiplicity creates a contact with zero without changing sign. At $t=0$ only the right-hand behavior is part of the physical evolution, although the exact spectral scaffold extends to negative time as a finite exponential sum.\n\n\\section{Three kinds of invisibility}\nWe use distinct terms for three distinct assertions. \\emph{Jet invisibility} means a specified finite number of derivatives vanish at a point. \\emph{Symmetry extinction} means an observation vanishes identically because an orientation-reversing symmetry fixes the evolving state. \\emph{Clock blindness} means the observer cannot distinguish an intrinsic event from a slow or paused parametrization using one time trace alone.\n\nThe first is achieved by cancellation constraints. The second is achieved by a group action and uniqueness of the evolution. The third is removed, in the setting developed below, by using kinetic energy as a second coordinate. These mechanisms have different perturbation behavior. An exact high-order zero is usually destroyed by an arbitrary small uncorrected perturbation. A symmetry-enforced zero persists only within the symmetry class. An energy-coordinate multiplicity is unchanged by common time reparametrization, but it is not immune to measurement noise.\n\n\\section{The role of the Chisa seed}\nThe paired edges of a scissor suggest two signs of curl eigenvalue. Serrated motion suggests carriers at distinct wavenumbers. Stagnation suggests a finite list of vanished observations rather than a suspension of the Navier--Stokes evolution. A returning thread suggests retaining a global record of when a quantity disappears and reappears.\n\nThese prompts guide the selection of questions; the proofs do not depend on a fictional narrative. In particular, a mathematical ``snip'' below is an interpolation constraint, not a claim that a blade can discontinuously change a smooth velocity field without a force. The actual constructed flows are specified by their initial data and subsequently evolve without external forcing.\n\nThe strongest useful reading of the seed is therefore not ``cut physical space.'' It is ``cut a prescribed set of observations out of a continuously evolving field, while retaining the rest of the dynamics.'' The nonlinear retuning theorem makes that instruction precise.\n\n", "source_path": "research/chapters/01_foundations.tex", "source_sha256": "c7fc3562633950f6b523b000ee4b5af118525cc0ac4c88540b44d25ab49a698c", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "01_foundations-002", "document_id": "chapters/01_foundations.tex", "chunk_index": 1, "line_start": 39, "line_end": 106, "content_format": "latex_source", "text": "\\chapter{Analytical setting and invariant bookkeeping}\n\\label{ch:foundations}\n\\section{Domain, normalization, and equations}\nLet $\\T^3=(\\R/2\\pi\\Z)^3$, with coordinates $(x,y,z)$ and normalized integral\n\\begin{equation}\n\\av{f}=(2\\pi)^{-3}\\int_{\\T^3}f(x)\\dd x.\n\\end{equation}\nFor vector fields we write $\\ip{u}{v}=\\av{u\\cdot v}$. The velocity is real, divergence-free, and mean-zero. The viscosity $\\nu>0$ is fixed. We study\n\\begin{equation}\\label{eq:NS}\n\\partial_tu+(u\\cdot\\nabla)u+\\nabla p=\\nu\\Delta u,\n\\qquad \\diver u=0,\\qquad \\av{u}=0.\n\\end{equation}\nThe mean-zero condition gives a spectral gap: every nonzero Fourier mode has squared wavenumber at least one. This apparently small normalization will be essential when excluding extra helicity zeros at arbitrarily late times.\n\nLet $\\Pp$ denote the Leray projection onto mean-zero divergence-free fields. On a nonzero Fourier wavevector $k$, its matrix is\n\\begin{equation}\n\\Pp_k=I-\\frac{kk^{\\mathsf T}}{|k|^2}.\n\\end{equation}\nDefine $\\B(u,v)=\\Pp((u\\cdot\\nabla)v)$. The projected equation is\n\\begin{equation}\\label{eq:projected}\n\\partial_tu=\\nu\\Delta u-\\B(u,u).\n\\end{equation}\nA nonzero unprojected advective term is not sufficient evidence of genuine dynamics, because a gradient can be absorbed into pressure. Accordingly our nonlinear witness will explicitly show $\\B(u_0,u_0)\\ne0$.\n\n\\section{Energy and helicity}\nThe two main observations are\n\\begin{equation}\\label{eq:EH}\nE(u)=\\frac12\\ip{u}{u},\\qquad\nH(u)=\\ip{u}{\\curl u}.\n\\end{equation}\nEnergy is nonnegative. Helicity is a signed quadratic functional. Zero total helicity does not mean zero vorticity, zero local helicity density, or an absence of geometrically complicated vortex lines.\n\n\\begin{proposition}[Classical balance identities]\\label{prop:balances}\nFor every smooth solution of \\eqref{eq:NS}, with $\\omega=\\curl u$,\n\\begin{align}\nE'(t)&=-\\nu\\norm{\\nabla u}_2^2=-\\nu\\norm{\\omega}_2^2,\\label{eq:energybalance}\\\\\nH'(t)&=-2\\nu\\ip{\\omega}{\\curl\\omega}.\\label{eq:helicitybalance}\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nPair the momentum equation with $u$ and integrate. Periodicity eliminates the pressure term and integration by parts gives $\\ip{\\Delta u}{u}=-\\norm{\\nabla u}_2^2$. The advective term integrates to zero because $\\diver u=0$. The equality between gradient and curl norms follows from the Fourier identity $|k\\times\\widehat u(k)|^2=|k|^2|\\widehat u(k)|^2$.\n\nCurl is self-adjoint under the periodic real pairing. Hence $H'=2\\ip{u_t}{\\omega}$. Use\n\\[\n(u\\cdot\\nabla)u=\\nabla(|u|^2/2)-u\\times\\omega.\n\\]\nThe gradient pairs to zero with $\\omega$, and $(u\\times\\omega)\\cdot\\omega=0$ pointwise. Finally, $\\Delta u=-\\curl\\omega$ for divergence-free fields. This proves \\eqref{eq:helicitybalance}.\n\\end{proof}\nThe nonlinear term cancels from the \\emph{instantaneous total-helicity balance}, but it changes the field appearing in the viscous term. It is therefore incorrect to infer that nonlinear interactions cannot affect future helicity. Chapter~\\ref{ch:triad} gives an explicit low-order calculation that exposes precisely this distinction.\n\n\\section{Regularity and smooth dependence}\nFor integer $r\\ge0$, use\n\\begin{equation}\n\\norm{u}_{H^r}^2=\\sum_{k\\in\\Z^3}(1+|k|^2)^r|\\widehat u(k)|^2.\n\\end{equation}\nAll initial fields constructed here are trigonometric polynomials. For any finite prescribed regularity index, multiplying all initial amplitudes by a sufficiently small positive number places the data in the standard small-data global smooth regime. The required smallness may depend severely on the event calendar. No uniform lower bound on the useful amplitude is claimed.\n\nWe use smooth, indeed analytic, dependence of the solution on finitely many initial amplitudes over finite time intervals. This is part of the established well-posedness framework; see, for example, \\cite{Brzezniak1991,KochTataru2001}. Appendix~\\ref{app:small} states and proves the estimates needed for our periodic, high-regularity setting. The construction does not assert global regularity for arbitrary large three-dimensional data.\n\n\\section{A useful bilinear bound for helicity}\nFor smooth $u,v$,\n\\begin{equation}\\label{eq:Hdifference}\n|H(u)-H(v)|\n\\le \\norm{u-v}_2\\norm{\\curl u}_2\n+\\norm{v}_2\\norm{\\curl(u-v)}_2.\n\\end{equation}\nThus closeness in $H^1$ controls the helicity difference. An $L^2$ estimate alone does not supply the derivative needed on the right. The nonlinear continuation argument will use an exponentially weighted $H^s$ estimate with $s\\ge3$, comfortably above that requirement.\n\n", "source_path": "research/chapters/01_foundations.tex", "source_sha256": "c7fc3562633950f6b523b000ee4b5af118525cc0ac4c88540b44d25ab49a698c", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "01_foundations-003", "document_id": "chapters/01_foundations.tex", "chunk_index": 2, "line_start": 107, "line_end": 177, "content_format": "latex_source", "text": "\\chapter{Exponential zero geometry}\n\\label{ch:exponentials}\n\\section{A classical zero bound, with its proof}\nLet $r_0<\\cdots<r_M$ be distinct real numbers and\n\\begin{equation}\nf(t)=\\sum_{j=0}^{M}c_j e^{-r_jt}.\n\\end{equation}\nThe following is a standard extended-Chebyshev property. It is proved here to make the event compiler self-contained; extended Chebyshev systems and exponential-polynomial interpolation have a substantial existing theory \\cite{Aldaz2008}.\n\n\\begin{lemma}[Exponential zero budget]\\label{lem:zeros}\nA nonzero linear combination of $n$ distinct real exponentials has at most $n-1$ real zeros, counted with multiplicity.\n\\end{lemma}\n\\begin{proof}\nProceed by induction on $n$. For one exponential the assertion is immediate. Multiply a combination of $n$ exponentials by the reciprocal of one constituent. This does not change its zeros or their multiplicities. The derivative of the resulting function is a combination of at most $n-1$ distinct exponentials, because the selected constituent becomes constant and disappears.\n\nIf the original function has $n$ zeros counted with multiplicity on some finite interval, repeated Rolle counting gives at least $n-1$ zeros for that derivative. Induction forbids this unless the derivative is identically zero. In that exceptional case the rescaled original function is a constant; the existence of a zero makes that constant zero, contradicting nontriviality. If there were too many zeros on the whole real line, a finite interval containing enough of them would give the same contradiction.\n\\end{proof}\nMultiplicity counting in this argument includes an order-$m$ zero contributing an order-$(m-1)$ zero after differentiation, together with a zero between consecutive distinct zeros. This detail is important: the theorem controls tangencies as well as sign crossings.\n\n\\section{Confluent observation matrices}\nFor an event divisor of degree $M$, define the $M\\times(M+1)$ matrix\n\\begin{equation}\\label{eq:zeroA}\nA_{(i,q),j}=(-r_j)^q e^{-r_j\\tau_i},\\qquad\n0\\le q<m_i,\\quad 0\\le j\\le M.\n\\end{equation}\nThe row label $(i,q)$ records both event time and derivative order. The zero conditions are $Ac=0$.\n\n\\begin{lemma}[Full rank and nonvanishing minors]\\label{lem:minors}\nEvery $M\\times M$ submatrix obtained by deleting one column of $A$ is invertible. Consequently $\\rank A=M$, its nullspace is one-dimensional, and every nonzero vector in that nullspace has no zero coordinates.\n\\end{lemma}\n\\begin{proof}\nIf one of these square submatrices were singular, there would be a nonzero combination of $M$ exponentials satisfying all $M$ zero conditions. It would have at least $M$ zeros counted with multiplicity, contradicting Lemma~\\ref{lem:zeros}. The rank and nullspace statements follow. A zero coordinate of a nonzero null vector would produce exactly the same forbidden combination with at most $M$ constituents.\n\\end{proof}\nThis proof does not require evaluating a determinant. It also shows why fitting the zeros is not merely a heuristic numerical task: the finite system has a structurally guaranteed one-dimensional solution space.\n\n\\section{A normalized compiler}\nChoose an anchor $t_*>\\tau_s$ and append one row to obtain\n\\begin{equation}\\label{eq:O}\nO=\\begin{pmatrix}A\\\\e^{-r_0t_*}\\ \\cdots\\ e^{-r_Mt_*}\\end{pmatrix}.\n\\end{equation}\nThe notation in the bottom row means the row vector with the displayed entries. Equivalently $O_{*,j}=e^{-r_jt_*}$.\n\n\\begin{theorem}[Exact normalized event compilation]\\label{thm:compiler}\nThere is a unique coefficient vector $c$ such that\n\\begin{equation}\\label{eq:compiler}\nOc=(0,\\ldots,0,1)^{\\mathsf T}.\n\\end{equation}\nIts signal $f_{\\score}$ has exactly the prescribed zeros, with exactly the prescribed multiplicities, on all of $\\R$. It is positive for $t>\\tau_s$. On every interval disjoint from the events,\n\\begin{equation}\\label{eq:signrule}\n\\sgn f_{\\score}(t)=(-1)^{\\sum_{\\tau_i>t}m_i}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nA nonzero vector in the kernel of $O$ would give a combination of $M+1$ exponentials with the $M$ prescribed zeros plus the extra zero $t_*$. Lemma~\\ref{lem:zeros} forbids this. Thus $O$ is invertible.\n\nThe solution has at least the prescribed zeros. If it had an additional zero, or any prescribed multiplicity were larger, it would exceed its zero budget. Therefore the list is exhaustive and all multiplicities are exact. The anchor value is one and there are no zeros after the last event, so the sign on that interval is positive. Moving left across a zero multiplies the sign by $(-1)^{m_i}$, proving \\eqref{eq:signrule}.\n\\end{proof}\nThere are $M$ homogeneous constraints and one normalization. Their number is not an estimate; it is the exact dimension of the square compiler. The smallest number of distinct carriers capable of realizing a nonzero signal with this divisor is $M+1$.\n\n\\section{Cofactors and the opposition of the blades}\nA null vector is given by signed maximal minors,\n\\begin{equation}\n\\widetilde c_j=(-1)^j\\det A_{\\widehat j},\n\\qquad\nc_j=\\frac{\\widetilde c_j}{\\sum_\\ell\\widetilde c_\\ell e^{-r_\\ell t_*}}.\n\\end{equation}\nThe denominator cannot vanish by the preceding theorem. Ordered confluent exponential matrices have a common nonzero orientation for all their maximal minors, so the coordinates alternate in sign. One way to see this is to replace each derivative block by nearby strictly ordered sample times, use the nonvanishing sign of the exponential determinant on the connected ordered domain, and pass to the confluent limit after dividing by positive within-block Vandermonde factors. Lemma~\\ref{lem:minors} prevents a zero in that limit.\n\nThe sign alternation is therefore a consequence of saturated zero interpolation, not a color choice in a plot. The compiler uses opposing handedness channels because a sum of positive decaying exponentials could not have even one finite zero.\n\\fig{01_event_divisor}{.95}{A degree-five divisor: an order-two initial contact, an order-one crossing at $0.18$, and an order-two contact at $0.33$. The vertical scale is symmetric-logarithmic to show both small and large portions of the exact signal. This is a spectral scaffold, not a simulated shock.}\n\n", "source_path": "research/chapters/01_foundations.tex", "source_sha256": "c7fc3562633950f6b523b000ee4b5af118525cc0ac4c88540b44d25ab49a698c", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "01_foundations-004", "document_id": "chapters/01_foundations.tex", "chunk_index": 3, "line_start": 178, "line_end": 239, "content_format": "latex_source", "text": "\\chapter{Embedding a calendar in an exact fluid}\n\\label{ch:shear}\n\\section{Helical carriers}\nFor integer $k\\ge1$ and $\\sigma\\in\\{+1,-1\\}$ define\n\\begin{equation}\\label{eq:carrier}\nb_k^\\sigma(z)=(\\cos kz,-\\sigma\\sin kz,0).\n\\end{equation}\nDirect differentiation gives\n\\begin{equation}\n\\diver b_k^\\sigma=0,\\qquad\n\\curl b_k^\\sigma=\\sigma k b_k^\\sigma,\\qquad\n\\Delta b_k^\\sigma=-k^2b_k^\\sigma.\n\\end{equation}\nIts normalized squared $L^2$ norm is one. Carriers at distinct $k$ are orthogonal, including when their helicity signs differ.\n\nChoose $k_j=j+1$ for $0\\le j\\le M$, so the helicity decay rates are $r_j=2\\nu k_j^2$. Compile $c_j$ from Theorem~\\ref{thm:compiler}, and put\n\\begin{equation}\\label{eq:amplitudes}\n\\sigma_j=\\sgn c_j,\\qquad a_j^0=\\sqrt{\\frac{|c_j|}{k_j}}.\n\\end{equation}\nFor an overall amplitude $\\delta>0$, set\n\\begin{equation}\\label{eq:shear}\nu_b(z,t)=\\delta\\sum_{j=0}^{M}a_j^0e^{-\\nu k_j^2t}b_{k_j}^{\\sigma_j}(z).\n\\end{equation}\nHere $u_b$ denotes the base velocity.\n\n\\begin{proposition}[Exact nonlinear-equation scaffold]\\label{prop:scaffold}\nThe field $u_b$ is a smooth, global, unforced solution of \\eqref{eq:NS} with constant pressure. Its energy and helicity are\n\\begin{align}\nH_b(t)&=\\delta^2 f_{\\score}(t),\\label{eq:scaffoldH}\\\\\nE_b(t)&=\\frac{\\delta^2}{2}\\sum_{j=0}^{M}\\frac{|c_j|}{k_j}e^{-2\\nu k_j^2t}.\\label{eq:scaffoldE}\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nEvery component depends only on $z$, while the $z$ velocity is zero. Consequently $(u_b\\cdot\\nabla)u_b=0$. Each summand solves the heat equation with viscosity $\\nu$, which establishes the full momentum equation. Orthogonality and the curl eigenvalues give the displayed observations.\n\\end{proof}\nThe word ``nonlinear-equation'' in the proposition means the field solves the actual Navier--Stokes equations. It does \\emph{not} mean that nonlinear advection is active. Activating it without losing the prescribed observations is the next step.\n\n\\section{A complete exact example}\nTake $2\\nu=1$, four wavenumbers $1,2,3,4$, an order-two event at $0$, and an order-one event at $\\log 2$. Normalize the slowest coefficient to one instead of fixing an anchor. The helicity is\n\\begin{equation}\\label{eq:exactfour}\nf(t)=e^{-t}-\\frac{227464}{27141}e^{-4t}\n+\\frac{110593}{9047}e^{-9t}\n-\\frac{131456}{27141}e^{-16t}.\n\\end{equation}\nThe integer coefficient vector before this normalization is\n\\begin{equation}\n(27141,-227464,331779,-131456).\n\\end{equation}\nIts sum and its dot product with $(1,4,9,16)$ are zero. Substitution of $e^{-t}=1/2$ gives the third constraint exactly. The zero-budget theorem proves there are no other real zeros and neither multiplicity is accidentally higher. In particular this is not a plot-based root claim.\n\nA different anchor normalization multiplies the entire vector by one positive constant. It does not change the event divisor. The supplied exact-arithmetic file stores the alternative normalization $f(\\log4)=1$ as rational coefficients.\n\n\\section{Finite complexity versus small amplitude}\nFor any fixed calendar and finite Sobolev order $r$, the initial scaffold has finite $H^r$ norm. Multiplying by $\\delta$ makes that norm arbitrarily small without changing the event times or multiplicities. Helicity and energy both scale as $\\delta^2$.\n\nThis does not yield a fixed observable signal at arbitrarily small energy. The anchor helicity also tends to zero. Keeping a prescribed nonzero anchor while demanding vanishing energy is a different problem, and the resource estimates in Chapter~\\ref{ch:resource} constrain it at fixed viscosity and positive observation time.\n\n\\section{Why a spectral gap anchor is built in}\nThe choice $k_0=1$ ensures the slowest base velocity decays at the same exponential rate as the slowest possible mean-zero periodic perturbation. If every base wavenumber were larger than one, an arbitrarily small perturbation could introduce a more slowly decaying helicity channel and eventually dominate the tail.\n\nThus the lowest carrier is not optional decoration in the global theorem. It is the long-time anchor that allows a uniform perturbation estimate to exclude additional late zeros. Arbitrary distinct spectral libraries can still be used for finite-horizon interpolation, but an all-time assertion requires control of the slowest modes.\n\\fig{02_scissor_carriers}{.85}{Signed helicity coefficients for the degree-five example. The physical modal energies are nonnegative; it is the handedness factor that supplies the alternating signs.}\n", "source_path": "research/chapters/01_foundations.tex", "source_sha256": "c7fc3562633950f6b523b000ee4b5af118525cc0ac4c88540b44d25ab49a698c", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "02_nonlinear-001", "document_id": "chapters/02_nonlinear.tex", "chunk_index": 0, "line_start": 1, "line_end": 83, "content_format": "latex_source", "text": "\\chapter{The nonlinear realization theorem}\n\\label{ch:nonlinear}\n\\section{Statement of the principal result}\nThe following theorem is the main candidate contribution of this manuscript. Its elementary spectral starting point should not obscure the distinction from that starting point: the final data activate the pressure-projected nonlinearity, depend on all three spatial coordinates, and still realize an exact prescribed observation calendar.\n\n\\begin{theorem}[Nonlinear chronochiral realization]\\label{thm:nonlinear}\nFix $\\nu>0$, a finite event divisor $\\score=\\sum_i m_i[\\tau_i]$ of degree $M\\ge1$, an anchor $t_*>\\tau_s$, a finite integer regularity index $r\\ge0$, and a tolerance $\\eta>0$. There are positive numbers $\\delta$ and $\\varepsilon_0$ and an even, real-analytic amplitude map\n\\[\na:(-\\varepsilon_0,\\varepsilon_0)\\longrightarrow(0,\\infty)^{M+1}\n\\]\nsuch that, for every $0<|\\varepsilon|<\\varepsilon_0$, the initial field\n\\begin{equation}\\label{eq:nonlinearinitial}\nu_0^\\varepsilon(x,y,z)=\\delta\\left[\n\\sum_{j=0}^{M}a_j(\\varepsilon)b_{j+1}^{\\sigma_j}(z)\n+\\varepsilon(0,\\cos x,\\cos y)\\right]\n\\end{equation}\nhas the following properties.\n\nIt generates a global smooth solution of the unforced, fixed-viscosity equation \\eqref{eq:NS}, and $\\norm{u_0^\\varepsilon}_{H^r}<\\eta$. Its total helicity satisfies\n\\begin{equation}\\label{eq:exactcalendar}\nH_\\varepsilon^{(q)}(\\tau_i)=0\\ (q<m_i),\\qquad\nH_\\varepsilon^{(m_i)}(\\tau_i)\\ne0,\\qquad\nH_\\varepsilon(t_*)=\\delta^2.\n\\end{equation}\nThere are no other helicity zeros on $[0,\\infty)$. The sign on each complementary interval is the one prescribed by \\eqref{eq:signrule}.\n\nThe initial Fourier support spans three spatial directions, and\n\\begin{equation}\\label{eq:genuine}\n\\B(u_0^\\varepsilon,u_0^\\varepsilon)\\ne0,\n\\qquad\n(\\curl u_0^\\varepsilon\\cdot\\nabla)u_0^\\varepsilon\\ne0.\n\\end{equation}\nMoreover\n\\begin{equation}\\label{eq:quadraticrepair}\na(\\varepsilon)=a^0+O(\\varepsilon^2),\n\\end{equation}\nwhere $a^0$ is the exact shear amplitude vector \\eqref{eq:amplitudes}.\n\\end{theorem}\nThe proof occupies this and the next three chapters. Each part addresses a distinct issue: finite observation constraints, the genuine nonlinear witness, parameter symmetry, and the infinite-time tail. In particular, satisfying a finite set of equations is not by itself enough to prove that no additional zeros occur later.\n\n\\section{What the theorem does not prescribe}\nThe theorem fixes the zero calendar and one nonzero anchor. It does not prescribe the full continuous helicity trace between those events. It does not fix an arbitrary positive energy independently of the anchor and smallness requirement. It does not assert that an arbitrary uncorrected perturbation retains a multiple zero.\n\nThere is also no universal minimum Fourier-support theorem for all nonlinear flows here. The number $M+1$ is the sharp count in the exact exponential scaffold. The final initial field has additional transverse modes, and its subsequent nonlinear evolution need not retain finite Fourier support.\n\n\\section{Smallness without changing the calendar}\nChoose an integer $s$ substantially larger than both the desired Sobolev order and the largest temporal derivative to be observed; for example\n\\begin{equation}\ns\\ge\\max\\{r+2,\\,2\\max_i m_i+8\\}.\n\\end{equation}\nFor $a$ in a small closed neighborhood of $a^0$ and $|\\varepsilon|\\le1$, the bracketed trigonometric polynomial in \\eqref{eq:nonlinearinitial} has a finite uniform $H^{s+1}$ bound $L$. Select $\\delta$ small enough that $\\delta L$ is inside the periodic small-data threshold and also below the desired $H^r$ tolerance.\n\nThis choice is made before the implicit-function argument. The argument then shrinks the allowed amplitude neighborhood and the interval of $\\varepsilon$, never increasing the initial Sobolev bound beyond the selected threshold. The construction works at fixed viscosity; there is no viscosity-to-zero limit hidden in the theorem.\n\n\\section{The finite observation map}\nFor a positive carrier vector $a$ and a small transverse parameter $\\varepsilon$, let $u(a,\\varepsilon,t)$ be the corresponding smooth solution. Define a map with $M+1$ components:\n\\begin{align}\nF_{i,q}(a,\\varepsilon)&=\\delta^{-2}\n\\partial_t^q H(u(a,\\varepsilon,t))\\big|_{t=\\tau_i},\\qquad 0\\le q<m_i,\\\\\nF_*(a,\\varepsilon)&=\\delta^{-2}H(u(a,\\varepsilon,t_*))-1.\n\\end{align}\nIt is analytic in the finite parameters near $(a^0,0)$. At initial-time observations, finite time derivatives are differential polynomials in the smooth initial field; at positive times, parabolic smoothing and smooth solution dependence justify differentiation. No assertion of time analyticity at the initial endpoint is required.\n\nAt $\\varepsilon=0$, all these data remain in the exact shear class for every nearby positive $a$. Therefore\n\\begin{equation}\n\\delta^{-2}H(u(a,0,t))=\n\\sum_{j=0}^{M}\\sigma_j k_j a_j^2e^{-r_jt}.\n\\end{equation}\nDifferentiation gives the exact Jacobian\n\\begin{equation}\\label{eq:Jacobian}\nJ=D_aF(a^0,0)=O\\,\\operatorname{diag}\n\\bigl(2\\sigma_j k_j a_j^0\\bigr)_{j=0}^{M}.\n\\end{equation}\nBoth factors are invertible. The matrix $O$ is invertible by Theorem~\\ref{thm:compiler}; every diagonal entry is nonzero by Lemma~\\ref{lem:minors}.\n\n\\begin{proposition}[Exact finite-constraint continuation]\\label{prop:ift}\nThere is a unique local analytic carrier map $a(\\varepsilon)$ with $a(0)=a^0$ and $F(a(\\varepsilon),\\varepsilon)=0$.\n\\end{proposition}\n\\begin{proof}\nApply the finite-dimensional analytic implicit-function theorem to $F$ at $(a^0,0)$ using \\eqref{eq:Jacobian}. The positivity of every $a_j^0$ allows the neighborhood to be chosen inside the positive orthant. The equations $F=0$ are precisely the zero-jet constraints and the anchor condition in \\eqref{eq:exactcalendar}.\n\\end{proof}\nThis step uses a standard theorem, but its applicability rests on an explicit observation Jacobian rather than a genericity assertion. The exact exponential compiler supplies the required nonsingular matrix for every finite calendar.\n\n", "source_path": "research/chapters/02_nonlinear.tex", "source_sha256": "60fc8ed37285397342b4d421b6446444d551a78d5474e0b13a18885c0321af1b", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "02_nonlinear-002", "document_id": "chapters/02_nonlinear.tex", "chunk_index": 1, "line_start": 84, "line_end": 139, "content_format": "latex_source", "text": "\\chapter{Turning on real three-dimensional interaction}\n\\label{ch:witness}\n\\section{An explicit transverse perturbation}\nWrite the corrected axial part as $v=(f(z),g(z),0)$ and set\n\\begin{equation}\nw=(0,\\cos x,\\cos y),\\qquad u_0^\\varepsilon=\\delta(v+\\varepsilon w).\n\\end{equation}\nBoth fields are divergence-free and mean-zero. The Fourier support of $v$ includes wavevectors in the $z$ direction. The two parts of $w$ have wavevectors in the $x$ and $y$ directions. Thus the support spans $\\R^3$ and there is no direction along which all spatial dependence is constant.\n\nA generic perturbation would probably activate advection, but ``probably'' is unnecessary here. Direct differentiation gives\n\\begin{equation}\\label{eq:crossadv}\n(v\\cdot\\nabla)w+(w\\cdot\\nabla)v\n=\\bigl(f'\\cos y,\\ g'\\cos y-f\\sin x,\\ -g\\sin y\\bigr).\n\\end{equation}\nThe self-interaction of $w$ is\n\\begin{equation}\\label{eq:wself}\n(w\\cdot\\nabla)w=(0,0,-\\cos x\\sin y).\n\\end{equation}\nThe base self-interaction is zero.\n\n\\section{A component that pressure cannot remove}\nConsider in \\eqref{eq:crossadv} the Fourier modes with nonzero $x$ and $z$ frequencies and zero $y$ frequency. Their vector contribution is\n\\begin{equation}\\label{eq:projectedwitness}\n-\\delta^2\\varepsilon f(z)\\sin x\\,e_y.\n\\end{equation}\nAt a wavevector $(\\pm1,0,\\pm k_j)$, the vector $e_y$ is perpendicular to the wavevector. The Leray projection therefore leaves this contribution unchanged. The other terms in \\eqref{eq:crossadv} have nonzero $y$ frequency, and the quadratic term \\eqref{eq:wself} has both $x$ and $y$ frequencies but zero $z$ frequency. None can cancel \\eqref{eq:projectedwitness}.\n\nThe function $f$ is a nontrivial finite cosine sum with strictly positive amplitudes. It cannot vanish identically. Thus $\\B(u_0^\\varepsilon,u_0^\\varepsilon)\\ne0$ for every nonzero sufficiently small $\\varepsilon$. This is a pressure-aware witness, not merely a calculation of a potentially irrotational acceleration.\n\n\\section{A separate vortex-stretching witness}\nThe two vorticities are\n\\begin{equation}\n\\curl v=(-g',f',0),\\qquad\n\\curl w=(-\\sin y,0,-\\sin x).\n\\end{equation}\nThe order-$\\varepsilon$ contribution to vortex stretching is\n\\begin{align}\n(\\curl v\\cdot\\nabla)w+(\\curl w\\cdot\\nabla)v\n&=(-f'\\sin x,0,-f'\\sin y).\n\\end{align}\nThe order-$\\varepsilon^2$ contribution is $(0,\\sin x\\sin y,0)$. Therefore\n\\begin{equation}\\label{eq:stretchwitness}\n(\\curl u_0^\\varepsilon\\cdot\\nabla)u_0^\\varepsilon\n=\\delta^2\\varepsilon(-f'\\sin x,0,-f'\\sin y)\n+\\delta^2\\varepsilon^2(0,\\sin x\\sin y,0).\n\\end{equation}\nSince $f'$ is not identically zero, the stretching is nonzero. The two displayed terms have different component and Fourier structure, so they cannot cancel globally.\n\nThe witness proves active interaction initially and hence on a sufficiently short interval by continuity. It does not claim that each specific Fourier coefficient remains nonzero at every later time. The theorem needs genuine nonlinear dynamics, not an unnecessarily strong pointwise-in-time lower bound.\n\n\\section{Why this advances the earlier special solution}\nThe earlier scaffold solved the full PDE because all advective interaction vanished. Here one can point to a concrete family of retained Fourier coefficients of the nonlinear term and to a concrete vortex-stretching field. The construction has therefore left that invariant shear subspace.\n\nAt the same time, the deformation is small. We do not obtain a turbulent high-Reynolds-number flow, a reconnection movie, or arbitrary vortex-knot control. Those are different goals. The resulting theorem is about exact scalar observation events in a genuinely nonlinear solution, not about every possible geometry of its vortex lines.\n\\fig{06_interaction_orders}{.91}{The projected nonlinear term grows to first order in the transverse parameter, whereas the correction of the axial amplitudes grows to second order. The plot is a finite Fourier--Galerkin illustration; the order statements follow analytically from the explicit witness and symmetry.}\n\n", "source_path": "research/chapters/02_nonlinear.tex", "source_sha256": "60fc8ed37285397342b4d421b6446444d551a78d5474e0b13a18885c0321af1b", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "02_nonlinear-003", "document_id": "chapters/02_nonlinear.tex", "chunk_index": 2, "line_start": 140, "line_end": 204, "content_format": "latex_source", "text": "\\chapter{The quadratic-retuning principle}\n\\label{ch:retuning}\n\\section{A half-period symmetry}\nLet $S$ be the spatial translation\n\\begin{equation}\nS(x,y,z)=(x+\\pi,y+\\pi,z).\n\\end{equation}\nIt fixes every axial carrier and changes $w$ to $-w$. Hence the initial condition at $(a,-\\varepsilon)$ is the translate of the initial condition at $(a,\\varepsilon)$. Translation equivariance of the Navier--Stokes equations and uniqueness imply the same relationship for the full evolving fields.\n\nBoth total helicity and total kinetic energy are invariant under spatial translation. Therefore, for every fixed $a$ and every time in the smooth evolution,\n\\begin{equation}\\label{eq:evenobs}\nH(u(a,\\varepsilon,t))=H(u(a,-\\varepsilon,t)),\\qquad\nE(u(a,\\varepsilon,t))=E(u(a,-\\varepsilon,t)).\n\\end{equation}\nIn particular $F(a,\\varepsilon)=F(a,-\\varepsilon)$.\n\n\\begin{theorem}[Even repair and first-order observational blindness]\\label{thm:even}\nThe unique local map in Proposition~\\ref{prop:ift} is even. Thus\n\\begin{equation}\na'(0)=0,\\qquad\n a(\\varepsilon)=a^0+\\varepsilon^2a^{[2]}+O(\\varepsilon^4).\n\\end{equation}\nThe untuned total-helicity history also has zero first derivative with respect to the transverse parameter at zero, for every fixed time, although the nonlinear term in the momentum equation is already nonzero at order $\\varepsilon$.\n\\end{theorem}\n\\begin{proof}\nBecause $F$ is even, $a(-\\varepsilon)$ satisfies the same local implicit equations as $a(\\varepsilon)$. Uniqueness forces equality. Analyticity gives an expansion containing only even powers. Equation~\\eqref{eq:evenobs} proves the observation statement, and \\eqref{eq:projectedwitness} proves the nonzero first-order interaction.\n\\end{proof}\nThis is a separation of orders, not a statement that nonlinear motion has become physically invisible. A local velocity measurement or a non-translation-invariant observation can detect the perturbation at first order. The blindness applies to the specified global scalar observations.\n\n\\section{Fourier interpretation: leaving and returning to the axis}\nThe same effect has a useful dynamical interpretation. Let $v_1=\\partial_\\varepsilon u(a^0,\\varepsilon,t)|_0$. It solves the linearized equation\n\\begin{equation}\\label{eq:firstvariation}\n\\partial_t v_1=\\nu\\Delta v_1-\\B(u_b,v_1)-\\B(v_1,u_b),\\qquad v_1(0)=\\delta w.\n\\end{equation}\nThe base field has horizontal wavevector zero. The linearized operator therefore preserves each horizontal Fourier label. Since $w$ has only nonzero horizontal labels, so does $v_1$.\n\nThe first helicity variation is $2\\ip{v_1}{\\curl u_b}$. Fourier orthogonality makes it zero: one factor has nonzero horizontal label and the other has label zero. At second order, a pair of transverse labels can add to zero, allowing a return to the axial observation sector. The half-period symmetry is a compact proof of the same selection rule to all odd orders.\n\n\\section{An explicit second-order repair formula}\nDifferentiating $F(a(\\varepsilon),\\varepsilon)=0$ twice yields\n\\begin{equation}\\label{eq:secondrepair}\na^{[2]}=-\\frac12J^{-1}\\partial_\\varepsilon^2F(a^0,0).\n\\end{equation}\nThis turns the qualitative implicit-function argument into a constructive perturbative prescription. The right side can be computed by solving two linearized evolution equations and evaluating a finite set of observations.\n\nFor the untuned expansion\n\\begin{equation}\nu(a^0,\\varepsilon,t)=u_b+\\varepsilon v_1+\\varepsilon^2v_2+O(\\varepsilon^3),\n\\end{equation}\nthe second coefficient solves\n\\begin{equation}\\label{eq:secondvariation}\n\\partial_t v_2=\\nu\\Delta v_2-\\B(u_b,v_2)-\\B(v_2,u_b)-\\B(v_1,v_1),\n\\qquad v_2(0)=0.\n\\end{equation}\nThe order-two helicity response is\n\\begin{equation}\\label{eq:Hsecond}\nH^{[2]}(t)=\\ip{v_1}{\\curl v_1}+2\\ip{v_2}{\\curl u_b}.\n\\end{equation}\nObserve both terms. Keeping only the helicity of the linearized perturbation would omit the feedback of its quadratic interaction into the base observation sector. Applying the event-derivative functionals and the anchor functional to \\eqref{eq:Hsecond}, then multiplying by $\\delta^{-2}$, supplies half of $\\partial_\\varepsilon^2F$ in \\eqref{eq:secondrepair}.\n\n\\section{Conditioning of the repair}\nThe formula contains $J^{-1}$. A mathematically invertible observation matrix can be numerically very poorly conditioned, especially when many derivatives are canceled or event times are close. Therefore the theorem promises a local exact family, not an inexpensive or robust finite-precision implementation for arbitrarily complicated calendars.\n\nThe explicit separation between $O(\\varepsilon)$ interaction and $O(\\varepsilon^2)$ retuning also does not imply a uniform improvement in an engineering cost. Constants in the second-order term can be large. They depend on the carrier amplitudes, event times, observation derivatives, viscosity, and inverse Jacobian.\n\n", "source_path": "research/chapters/02_nonlinear.tex", "source_sha256": "60fc8ed37285397342b4d421b6446444d551a78d5474e0b13a18885c0321af1b", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "02_nonlinear-004", "document_id": "chapters/02_nonlinear.tex", "chunk_index": 3, "line_start": 205, "line_end": 273, "content_format": "latex_source", "text": "\\chapter{Exact multiplicities and exclusion of late surprises}\n\\label{ch:tail}\n\\section{Local factorization near every event}\nThe equations $F=0$ enforce lower-order vanishing, but they do not by themselves ensure exact multiplicity. Fix one event $\\tau_i$ of order $m_i$. The base has\n\\begin{equation}\nH_b^{(m_i)}(\\tau_i)=\\delta^2 f_{\\score}^{(m_i)}(\\tau_i)\\ne0.\n\\end{equation}\nSmooth parameter dependence preserves nonvanishing of this derivative for sufficiently small $\\varepsilon$.\n\nMore is true. Taylor's integral formula, using the exactly imposed lower-order zeros, gives\n\\begin{equation}\\label{eq:localfactor}\nH_\\varepsilon(t)=(t-\\tau_i)^{m_i}R_{i,\\varepsilon}(t),\n\\end{equation}\nwhere\n\\begin{equation}\nR_{i,\\varepsilon}(t)=\\frac1{(m_i-1)!}\\int_0^1(1-s)^{m_i-1}\nH_\\varepsilon^{(m_i)}(\\tau_i+s(t-\\tau_i))\\dd s.\n\\end{equation}\nAt $(t,\\varepsilon)=(\\tau_i,0)$ this is $H_b^{(m_i)}(\\tau_i)/m_i!\\ne0$. Joint continuity provides a neighborhood on which $R_{i,\\varepsilon}$ has a fixed nonzero sign. Thus the only zero in that neighborhood is the prescribed one, and its order is exactly $m_i$. For an initial event, use the same formula on the right-hand neighborhood.\n\n\\section{No extra zeros on a compact interval}\nSelect a finite $L>\\tau_s$ and remove small disjoint neighborhoods of the prescribed events from $[0,L]$. The remaining set is compact, and the base helicity is nonzero there. Its absolute value has a positive minimum. Uniform convergence $H_\\varepsilon\\to H_b$ preserves the sign on this compact remainder for sufficiently small $\\varepsilon$.\n\nCombining the compact remainder with the local factors proves that the desired list is exhaustive on $[0,L]$. This argument distinguishes \\emph{retunable exact persistence} from ordinary structural stability. A generic uncorrected perturbation would not preserve the factors in \\eqref{eq:localfactor}.\n\n\\section{A weighted global stability estimate}\nThe infinite tail requires an estimate stronger than uniform convergence on each finite interval. The small-data estimates of Appendix~\\ref{app:small} yield, for a fixed sufficiently large $s$,\n\\begin{align}\n\\norm{u_\\varepsilon(t)}_{H^s}+\\norm{u_b(t)}_{H^s}&\\le C\\delta e^{-\\nu t},\\label{eq:weightedbase}\\\\\n\\norm{u_\\varepsilon(t)-u_b(t)}_{H^s}&\\le C\\delta|\\varepsilon|e^{-\\nu t},\\label{eq:weightedstab}\n\\end{align}\nwith a constant uniform for the local family. The exact rate $\\nu$, rather than merely $\\nu/2$, follows by using the integrable high-norm coefficients of the small solutions in a Gronwall estimate. It is not obtained by ignoring the nonlinear term.\n\nApply \\eqref{eq:Hdifference}. There is a constant $C_H$ such that\n\\begin{equation}\\label{eq:weightedH}\n|H_\\varepsilon(t)-H_b(t)|\\le C_H\\delta^2|\\varepsilon|e^{-2\\nu t}\n\\qquad(t\\ge0).\n\\end{equation}\nThe estimate is only first order in $|\\varepsilon|$, although the scalar symmetry gives more information locally. This weaker bound is already sufficient for the global sign argument.\n\n\\section{The slow carrier closes the proof}\nBecause $k_0=1$, write\n\\begin{equation}\ne^{2\\nu t}f_{\\score}(t)=c_0+\\sum_{j=1}^{M}c_j e^{-2\\nu(k_j^2-1)t}.\n\\end{equation}\nThe right side tends to $c_0>0$ as $t\\to\\infty$. It is positive for every $t>\\tau_s$ by the sign rule. Consequently\n\\begin{equation}\nb_L:=\\inf_{t\\ge L}e^{2\\nu t}f_{\\score}(t)>0.\n\\end{equation}\nTo justify strict positivity of the infimum, choose a sufficiently large endpoint beyond which the function is at least $c_0/2$; on the remaining compact interval it has a positive minimum.\n\nChoose $|\\varepsilon|<b_L/(2C_H)$. Equation~\\eqref{eq:weightedH} then gives\n\\begin{equation}\nH_\\varepsilon(t)\\ge\\frac12\\delta^2b_L e^{-2\\nu t}>0\\qquad(t\\ge L).\n\\end{equation}\nThere are therefore no unexpected late-time zeros. Together with the compact argument, exact local multiplicities, implicit constraints, smallness, and the nonlinear witnesses, this completes the proof of Theorem~\\ref{thm:nonlinear}.\n\n\\section{A consequence for uniform event-count bounds}\n\\begin{corollary}[No universal zero-count bound in a small-data ball]\\label{cor:nozerobound}\nFix viscosity $\\nu>0$, a finite Sobolev index $r$, and $\\eta>0$. There is no finite upper bound, independent of the initial field, on the number of total-helicity sign reversals among the globally smooth solutions constructed with $\\norm{u_0}_{H^r}<\\eta$.\n\\end{corollary}\n\\begin{proof}\nPrescribe any desired finite number of distinct positive times with multiplicity one, and apply Theorem~\\ref{thm:nonlinear}. Every prescribed event reverses sign.\n\\end{proof}\nThis does not contradict the known eventual-sign behavior of individual periodic viscous flows studied by Foias, Hoang, and Nicolaenko \\cite{Foias2007}. A statement about each individual solution eventually settling to one sign is compatible with arbitrarily many earlier reversals across a family of different solutions.\n\n\\section{Strict energy decrease on the constructed family}\nThe global tail is nonzero, so no solution in the family can become the zero field at a finite time: uniqueness forward from that time would make its entire future zero. Its mean is zero; hence a nonzero velocity has nonzero gradient norm. Equation~\\eqref{eq:energybalance} therefore yields $E'(t)<0$ for every finite time. All programmed contacts and reversals occur while kinetic energy strictly decreases.\n\\fig{07_nonlinear_calendar}{.94}{The two-event numerical example retains its prescribed zeros at $0$ and $0.12$, together with the anchor at $0.25$, after activating nonlinear three-dimensional interaction. The intermediate trace need not equal the scaffold exactly.}\n", "source_path": "research/chapters/02_nonlinear.tex", "source_sha256": "60fc8ed37285397342b4d421b6446444d551a78d5474e0b13a18885c0321af1b", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "03_triad_clock-001", "document_id": "chapters/03_triad_clock.tex", "chunk_index": 0, "line_start": 1, "line_end": 106, "content_format": "latex_source", "text": "\\chapter{A nonhelical triad that creates a chiral response}\n\\label{ch:triad}\n\\section{A different source of helicity}\nThe event compiler begins with individually helical carriers whose signed contributions cancel. That is not the only mechanism available. Here we construct a response from three standing waves, each with zero helicity at every point when considered in isolation. The resulting calculation distinguishes \\emph{latent signed helicity} from \\emph{helicity generated by the nonlinear interaction of nonhelical components}.\n\nThe general phenomenon of nonlinear helicity production is not new. Sanada studied its emergence in forced Navier--Stokes dynamics whose imposed Fourier components had zero helicity \\cite{Sanada1993}. Helical decompositions and triadic interactions were systematically investigated by Waleffe \\cite{Waleffe1992}. The contribution proposed here is the particular exactly evaluated unforced response kernel, together with its all-time symmetry extinction and its connection to an event calculus. Those narrower claims still require an independent literature assessment.\n\n\\section{The explicit geometry}\nChoose wavevectors and real polarization vectors\n\\begin{align}\n k_1&=(1,0,0),&p_1&=(0,1,0),\\\\\n k_2&=(0,1,1),&p_2&=(1,0,0),\\\\\n k_3&=(-1,-1,-1),&p_3&=(1,0,-1).\n\\end{align}\nThey satisfy\n\\begin{equation}\n k_1+k_2+k_3=0,\\qquad k_i\\cdot p_i=0,\\qquad\n (|k_1|^2,|k_2|^2,|k_3|^2)=(1,2,3).\n\\end{equation}\nFor phases $\\phi_i$, let\n\\begin{equation}\\label{eq:triadwaves}\n V_i(x)=p_i\\cos(k_i\\cdot x+\\phi_i),\\qquad\n u_0=aV_1+bV_2+cV_3.\n\\end{equation}\nEach wave is divergence-free and mean-zero. Since its curl is proportional to $k_i\\times p_i$, perpendicular to $p_i$, each wave has identically zero self-helicity. Orthogonality of different Fourier frequencies gives zero cross-helicity at the initial time. Thus\n\\begin{equation}\\label{eq:triadinitial}\n H(0)=0,\\qquad H'(0)=0.\n\\end{equation}\nThe second identity also follows from the helicity balance: every initial mode contributes zero to $\\ip{\\omega}{\\curl\\omega}$.\n\nThe combined phase\n\\begin{equation}\\label{eq:cyclephase}\n \\Phi=\\phi_1+\\phi_2+\\phi_3\n\\end{equation}\nis invariant under spatial translations, because such a translation changes the sum by $(k_1+k_2+k_3)\\cdot x_0=0$. This is a genuine phase of the closed interaction, not the origin-dependent phase of a single wave.\n\nAlthough these are three distinct three-component modes, their wavevectors span a two-dimensional plane, as every closed three-wave triad must. This example alone is not the full-three-direction witness of Chapter~\\ref{ch:witness}. It is a deliberately tractable nonlinear subsystem; the main event theorem supplies the stronger spatial nondegeneracy.\n\n\\section{The first nonzero time derivative}\n\\begin{theorem}[Explicit triadic helicity acceleration]\\label{thm:triadacceleration}\nFor the smooth solution issued from \\eqref{eq:triadwaves},\n\\begin{equation}\\label{eq:triadH2}\n \\boxed{H''(0)=2\\nu abc\\cos\\Phi.}\n\\end{equation}\nThis initial derivative identity is exact. It is not a small-amplitude approximation.\n\\end{theorem}\n\\begin{proof}\nWrite $C=\\curl$. On divergence-free mean-zero fields, $C^2=-\\Delta$. The helicity balance is $H'=-2\\nu\\ip{u}{C^3u}$. Differentiate once and substitute $u_t=-\\nu C^2u-\\B(u,u)$:\n\\begin{equation}\\label{eq:H2operator}\n H''=4\\nu^2\\ip{C^2u}{C^3u}\n      +4\\nu\\ip{\\B(u,u)}{C^3u}.\n\\end{equation}\nAt the specified initial field, the first term vanishes by modal orthogonality and linear polarization. In the cubic term, the only zero-total-wavevector products use all three distinct modes. Define, at zero phases,\n\\begin{equation}\n S_i=\\ip{V_i}{C\\{\\B(V_j,V_k)+\\B(V_k,V_j)\\}},\n \\qquad \\{i,j,k\\}=\\{1,2,3\\}.\n\\end{equation}\nDirect Fourier multiplication and Leray projection, detailed in Appendix~\\ref{app:fourier}, give\n\\begin{equation}\\label{eq:triadS}\n (S_1,S_2,S_3)=\\left(-\\frac14,0,\\frac14\\right).\n\\end{equation}\nOnly a triad and its complex conjugate contribute, so phases multiply each real contraction by $\\cos\\Phi$. Since $C^3V_i=|k_i|^2CV_i$,\n\\begin{align}\n \\ip{\\B(u_0,u_0)}{C^3u_0}\n &=abc\\cos\\Phi\\sum_i |k_i|^2S_i\\\\\n &=abc\\cos\\Phi\\left(-\\frac14+\\frac34\\right)\n =\\frac12abc\\cos\\Phi.\n\\end{align}\nEquation~\\eqref{eq:H2operator} now yields \\eqref{eq:triadH2}.\n\\end{proof}\nThe nonlinearity does not appear explicitly in $H'$ because it cancels there. It does appear in $H''$, by changing the field upon which the dissipative helicity functional acts. This is not a conflict with the helicity balance. It is an example of an interaction being invisible to the first observable derivative yet visible to the next.\n\nIf $abc\\cos\\Phi\\ne0$, Taylor's theorem gives\n\\begin{equation}\n H(t)=\\nu abc\\cos\\Phi\\,t^2+O(t^3)\n\\end{equation}\nas $t\\downarrow0$. The sign for sufficiently small positive times is therefore determined exactly. No claim of the same sign for every later time is needed for this statement.\n\n\\section{An all-time mixed-response kernel}\nLet the amplitudes be independent parameters. For each fixed finite $t$, small-data solution dependence permits differentiation at $(a,b,c)=(0,0,0)$.\n\\begin{theorem}[Exact cubic response kernel]\\label{thm:kernel}\nThe third mixed amplitude derivative is\n\\begin{equation}\\label{eq:kernel}\n \\boxed{\\left.\\partial_a\\partial_b\\partial_c H(t)\\right|_0\n =K_\\nu(t)\\cos\\Phi,\\qquad\n K_\\nu(t)=\\frac{e^{-6\\nu t}}{8\\nu}\n \\bigl(e^{4\\nu t}-1-4\\nu t\\bigr).}\n\\end{equation}\nFor every $\\nu>0$ and $t>0$, $K_\\nu(t)>0$.\n\\end{theorem}\n\\begin{proof}\nThe first-order velocity response to mode $i$ is $e^{-\\lambda_it}V_i$, with $\\lambda_i=\\nu|k_i|^2$. The second-order mixed response to modes $j,k$ is\n\\begin{equation}\n -\\int_0^t e^{\\nu(t-s)\\Delta}\n \\{\\B(V_j,V_k)+\\B(V_k,V_j)\\}\n e^{-(\\lambda_j+\\lambda_k)s}\\dd s.\n\\end{equation}\nPairing this response with the first-order mode $i$ in the quadratic helicity functional yields\n\\begin{equation}\\label{eq:duhamelkernel}\n -2\\sum_i S_i e^{-2\\lambda_it}\n \\frac{1-e^{-(\\lambda_j+\\lambda_k-\\lambda_i)t}}\n {\\lambda_j+\\lambda_k-\\lambda_i}\\cos\\Phi.\n\\end{equation}\nThe quotient is interpreted as $t$ when its denominator is zero. Here the $i=2$ term vanishes because $S_2=0$. The $i=1$ term is\n$(e^{-2\\nu t}-e^{-6\\nu t})/(8\\nu)$, while the $i=3$ term is $-(t/2)e^{-6\\nu t}$. Their sum is \\eqref{eq:kernel}. Finally, $e^x>1+x$ for every $x>0$, which proves positivity.\n\\end{proof}\n", "source_path": "research/chapters/03_triad_clock.tex", "source_sha256": "5fbe50ed547f8a6e53672e7bc05e3e24c0e9e20f981ce83ff7b30e962d2d6949", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "03_triad_clock-002", "document_id": "chapters/03_triad_clock.tex", "chunk_index": 1, "line_start": 107, "line_end": 118, "content_format": "latex_source", "text": "The term $t e^{-6\\nu t}$ comes from a resonance of \\emph{decay rates}, namely $\\lambda_3=\\lambda_1+\\lambda_2$. It does not require an oscillatory forcing or a new physical resonance law. Two distinct ingredients are essential: a closed momentum triad allows the scalar observation, and the decay-rate relation changes the temporal kernel.\n\\fig{03_triad_kernel}{.83}{The exactly computed cubic response kernel for three viscosities. Positivity is an analytical consequence of $e^x>1+x$, not an inference from the plotted samples.}\n\n\\section{What the kernel does not say}\nEquation~\\eqref{eq:kernel} is an exact derivative of the full PDE solution map. It is not the identity $H(t)=abcK_\\nu(t)\\cos\\Phi$ at arbitrary finite amplitudes. For a common small scale $s$ and fixed amplitudes $a,b,c$,\n\\begin{equation}\n H(sa,sb,sc,t)=s^3abcK_\\nu(t)\\cos\\Phi+O(s^4)\n\\end{equation}\non a specified finite time interval. The remainder may depend on that interval and on the amplitudes. Establishing uniform sign control of the full finite-amplitude signal on an unbounded interval would require additional analysis.\n\nAt fixed $t$, $K_\\nu(t)=\\nu t^2+O(\\nu^2t^3)$ as $\\nu\\to0$. Thus the response tends to zero in this limit, consistent with smooth Euler helicity conservation. There is no contradiction between the viscous calculation and the inviscid conservation law.\n\n", "source_path": "research/chapters/03_triad_clock.tex", "source_sha256": "5fbe50ed547f8a6e53672e7bc05e3e24c0e9e20f981ce83ff7b30e962d2d6949", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "03_triad_clock-003", "document_id": "chapters/03_triad_clock.tex", "chunk_index": 2, "line_start": 119, "line_end": 166, "content_format": "latex_source", "text": "\\chapter{A phase that extinguishes the entire observable}\n\\label{ch:phase}\n\\section{From a vanishing coefficient to an exact symmetry}\nSetting $\\cos\\Phi=0$ in the previous formulas only proves that two particular coefficients vanish. It does not, by itself, prove that the full helicity stays zero. A stronger mechanism is available: the same phase condition creates an orientation-reversing symmetry of the entire initial field.\n\nFor $a_0\\in\\T^3$, define the reflected vector field\n\\begin{equation}\\label{eq:inversion}\n (\\mathcal R_{a_0}u)(x)=-u(-x+a_0).\n\\end{equation}\nThis is the vector pushforward under a translated spatial inversion. Incompressible Navier--Stokes is equivariant under this transformation. Energy is unchanged, whereas helicity changes sign, because velocity is a polar vector and vorticity is an axial vector.\n\n\\begin{theorem}[Quadrature extinction]\\label{thm:extinction}\nIf the phases in \\eqref{eq:triadwaves} satisfy\n\\begin{equation}\\label{eq:quadrature}\n \\Phi=\\frac\\pi2\\pmod\\pi,\n\\end{equation}\nthen $H(t)=0$ throughout the smooth lifetime of the solution, for arbitrary real amplitudes for which such a solution exists. In particular, the conclusion is global for sufficiently small data.\n\\end{theorem}\n\\begin{proof}\nA single wave $p_i\\cos(k_i\\cdot x+\\phi_i)$ is fixed by \\eqref{eq:inversion} exactly when\n\\begin{equation}\\label{eq:reflectionphases}\n k_i\\cdot a_0\\equiv-2\\phi_i-\\pi\\pmod{2\\pi}.\n\\end{equation}\nThe first two equations can be solved on the torus since $k_1=(1,0,0)$ and $k_2=(0,1,1)$. The third is consistent with their sum exactly when\n$-2\\Phi-3\\pi\\equiv0\\pmod{2\\pi}$, which is equivalent to \\eqref{eq:quadrature}. Hence the full initial field is fixed by $\\mathcal R_{a_0}$. Equivariance and uniqueness of the smooth solution preserve this fixed-point condition. But\n\\begin{equation}\n H(\\mathcal R_{a_0}u)=-H(u).\n\\end{equation}\nTherefore $H(u(t))=-H(u(t))$ at each time, so it vanishes.\n\\end{proof}\nFor the representative phases $(0,0,\\pi/2)$, one may take $a_0=(\\pi,\\pi,0)$. The executable Fourier verifier checks this symmetry exactly. The proof, rather than the finite check, establishes extinction for the entire evolution.\n\n\\section{The difference between extinction and stopped dynamics}\nThe reflected solution need not be stationary. Its energy still dissipates, its velocity modes interact, and local helicity density need not vanish pointwise. The theorem concerns a global pseudoscalar integral. Opposite local contributions can remain symmetry-paired for all times.\n\nThis is a useful precise version of ``Moment of Nihility'': not an absence of all motion, but the complete suppression of a selected observable by symmetry. It is fundamentally different from the finite-order Stagnation events of the compiler. A finite-order event releases the observable immediately after the event; a symmetry extinction keeps it zero for as long as the symmetry-preserving evolution exists.\n\n\\section{A phase-controlled bifurcation of the observation}\nFor $abc\\ne0$, moving $\\Phi$ away from quadrature turns on the exact acceleration $2\\nu abc\\cos\\Phi$. Near a quadrature phase $\\Phi_0$, the response is linear in the phase displacement to leading order, since $\\sin\\Phi_0=\\pm1$.\n\nThe word ``bifurcation'' here describes a change in the observable's symmetry class, not an established dynamical-systems bifurcation of equilibria or attractors. There is no assertion of a new turbulent phase transition. The exact distinction is between a solution constrained to $H\\equiv0$ and one with a nonzero initial helicity acceleration.\n\\fig{04_phase_gate}{.92}{The normalized initial acceleration is $\\cos\\Phi$. The two quadrature points correspond to an exact orientation-reversing symmetry and all-time extinction, not merely to a numerically small cubic response.}\n\n\\section{Two kinds of hiddenness}\nThe event compiler hides a signed spectral combination at selected times. Its invisibility is algebraic and must be retuned under generic perturbations. The quadrature triad hides the global observable at every time. Its invisibility is symmetry-enforced and persists under perturbations that retain the same reflection symmetry.\n\nThese mechanisms should not be conflated. A perturbation that breaks reflection symmetry can release the extinct observable; a perturbation that preserves it cannot. Conversely, an arbitrary perturbation of a high-order event need not preserve its order even if it is very small. This contrast is central to the stability discussion in Chapter~\\ref{ch:stability}.\n\n", "source_path": "research/chapters/03_triad_clock.tex", "source_sha256": "5fbe50ed547f8a6e53672e7bc05e3e24c0e9e20f981ce83ff7b30e962d2d6949", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "03_triad_clock-004", "document_id": "chapters/03_triad_clock.tex", "chunk_index": 3, "line_start": 167, "line_end": 223, "content_format": "latex_source", "text": "\\chapter{Energy as a clock-independent coordinate}\n\\label{ch:clock}\n\\section{A common operational clock}\nLet $u(s)$ be one of the nonzero constructed solutions, with strictly decreasing energy. For a nonnegative, locally integrable gate $g(t)$, define\n\\begin{equation}\\label{eq:operationalclock}\n S(t)=\\int_0^t g(r)\\dd r,\\qquad U(t)=u(S(t)).\n\\end{equation}\nFor absolutely continuous $S$, the chain rule holds almost everywhere and gives\n\\begin{equation}\\label{eq:gatedNS}\n \\partial_t U=g(t)\\{\\nu\\Delta U-\\B(U,U)\\}.\n\\end{equation}\nWhen $g=0$, this chosen evolution is paused. Physical laboratory time is not stopped. Moreover, \\eqref{eq:gatedNS} gates the \\emph{entire} generator. It is not ordinary fixed-viscosity Navier--Stokes, nor is it obtained by changing only viscosity while leaving advection untouched.\n\nThis distinction prevents a false physical interpretation. The mathematical result concerns an externally reparameterized trajectory or a subsystem whose entire dynamics share a clock. It does not claim a realizable universal time-freeze device.\n\n\\section{The energy graph}\nWrite $e(s)=E(u(s))$ and $h(s)=H(u(s))$. Since $e'(s)<0$ at finite times, $e$ has a smooth local inverse $s=s(e)$. The unparameterized observation graph is\n\\begin{equation}\\label{eq:energygraph}\n \\mathscr K(e)=h(s(e)).\n\\end{equation}\nBoth a running clock and a paused clock trace the same graph, provided they cover the same operational-time interval. A pause produces repeated observations of one point on the graph; it does not change the graph itself.\n\n\\begin{theorem}[Energy-clock quotient]\\label{thm:energyclock}\nAlong any nonzero smooth mean-zero solution with $E'<0$,\n\\begin{equation}\\label{eq:dHdE}\n \\boxed{\\frac{\\dd H}{\\dd E}\n =2\\frac{\\ip{\\omega}{\\curl\\omega}}{\\norm{\\omega}_{L^2}^2}.}\n\\end{equation}\nIf $H$ has a zero of exact order $m$ at operational time $\\tau$, then $\\mathscr K$ has a zero of the same order at $e_\\tau=E(\\tau)$, and\n\\begin{equation}\\label{eq:energyjet}\n \\mathscr K^{(q)}(e_\\tau)=0\\ (q<m),\\qquad\n \\mathscr K^{(m)}(e_\\tau)=\\frac{H^{(m)}(\\tau)}{[E'(\\tau)]^m}\\ne0.\n\\end{equation}\nThese statements are unchanged by any common nondecreasing clock reparameterization covering the events.\n\\end{theorem}\n\\begin{proof}\nDivide the helicity balance by the energy balance to obtain \\eqref{eq:dHdE}. The denominator is nonzero. Near the event, Taylor factorization gives $H(s)=(s-\\tau)^mR(s)$ with $R(\\tau)=H^{(m)}(\\tau)/m!$. The inverse function theorem gives\n$s(e)-\\tau=(e-e_\\tau)/E'(\\tau)+O((e-e_\\tau)^2)$. Substitute this expression into the factorization. The first nonzero coefficient is $R(\\tau)/[E'(\\tau)]^m$, proving \\eqref{eq:energyjet}. Finally, both observations of $U(t)=u(S(t))$ are compositions with the same $S$, so eliminating the parameter yields the same graph.\n\\end{proof}\nThe result is a quotient by common clock changes. It preserves event order but not a canonical laboratory timestamp. Two experiments with different clock gates can assign different times to the same energy-indexed event.\n\\fig{09_energy_clock}{.92}{Helicity viewed against decreasing kinetic energy rather than time. The zero events remain well-defined points on this graph.}\n\n\\section{Why a time trace can be misleading}\nSuppose, on one side of a laboratory event $t_0$,\n\\begin{equation}\n S(t)-\\tau=c(t-t_0)^\\beta+o(|t-t_0|^\\beta),\\qquad c\\ne0.\n\\end{equation}\nIf $H(s)\\sim A(s-\\tau)^m$, then $H(S(t))\\sim Ac^m(t-t_0)^{m\\beta}$. A nonlinear clock therefore changes the apparent order in laboratory time. A flat interval can make the observation remain exactly zero across an interval even when the underlying operational event is isolated.\n\nThe energy-coordinate order does not suffer this particular ambiguity because energy is reparameterized by the same clock. This does not make the reconstruction immune to measurement error. Numerical differentiation of $H(E)$ can be unstable, and a nearly flat energy trace can amplify noise. The theorem removes a structural clock ambiguity, not every inverse-problem difficulty.\n\\fig{10_common_clock}{.83}{An illustrative common operational clock contains pauses while laboratory time continues. The curves are produced by composition with the same computed scaffold, not by a claim of physical time suspension.}\n\\fig{11_clock_quotient}{.83}{Running-clock and paused-clock observations lie on the same energy--helicity graph. Repeated samples during a pause occupy one point rather than creating a new intrinsic event.}\n\n\\section{Failure modes of the quotient}\nIf only one observed subsystem is gated, $H$ and $E$ need not share a common operational time. If external forcing changes the energy balance, energy may cease to be strictly monotone. If the velocity has a nonzero conserved mean, the energy-coordinate range and Poincar\\'e estimates require modification. None of those situations is covered by Theorem~\\ref{thm:energyclock} as stated.\n\nSimilarly, if $S(t)$ stops permanently before a prescribed operational event, no laboratory observation reaches that event. Invariance of a covered graph is not a guarantee that the entire graph is visited. These elementary domain restrictions are necessary for a correct interpretation of clock independence.\n", "source_path": "research/chapters/03_triad_clock.tex", "source_sha256": "5fbe50ed547f8a6e53672e7bc05e3e24c0e9e20f981ce83ff7b30e962d2d6949", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "04_resources_selection-001", "document_id": "chapters/04_resources_selection.tex", "chunk_index": 0, "line_start": 1, "line_end": 76, "content_format": "latex_source", "text": "\\chapter{The minimum-energy event compiler}\n\\label{ch:resource}\n\\section{A finite spectral design problem}\nAn existence theorem can conceal a substantial cost. Large cancellations may require large signed coefficients before the common smallness factor is applied. We therefore ask a separate question: within a specified finite library of helical shear carriers, what is the smallest initial kinetic energy that realizes a prescribed event calendar and anchor?\n\nLet the available positive wavenumbers be $k_1<\\cdots<k_K$. Build the observation matrix $O$ as in \\eqref{eq:O}, now allowing more columns than rows. There are $m=M+1$ observation rows. For signed coefficients $c_j$, the exact shear realization has\n\\begin{equation}\\label{eq:resourcecost}\n E_0(c)=\\sum_{j=1}^K w_j|c_j|,\\qquad w_j=\\frac1{2k_j}.\n\\end{equation}\nThe constraint vector is $y=(0,\\ldots,0,1)^T$. The finite-library problem is\n\\begin{equation}\\label{eq:primal}\n \\min_{c\\in\\R^K}\\sum_jw_j|c_j|\\quad\\hbox{subject to}\\quad Oc=y.\n\\end{equation}\nThis is a weighted $\\ell^1$ linear program. It is not a new optimization paradigm. The special structure is that the objective is exactly the kinetic energy of a realizable flow, while the constraints encode multiplicities of a helicity calendar.\n\n\\section{An exact dual certificate}\n\\begin{theorem}[Finite-library energy duality]\\label{thm:LP}\nAssume the library contains at least $m$ distinct wavenumbers. The minimum in \\eqref{eq:primal} is attained and equals\n\\begin{equation}\\label{eq:dual}\n \\max_{z\\in\\R^m}y^Tz\n \\quad\\hbox{subject to}\\quad |(O^Tz)_j|\\le w_j\\quad(1\\le j\\le K).\n\\end{equation}\nAt least one optimum has exactly $m=M+1$ nonzero signed coefficients. Its helicity trace has exactly the prescribed zero divisor and no additional real zeros.\n\\end{theorem}\n\\begin{proof}\nAny $m$ columns give an invertible square interpolation matrix by the exponential zero-count argument. Hence the constraints are feasible and $O$ has row rank $m$. Since every $w_j$ is positive, the objective is coercive on finite-dimensional space, so a minimizer exists.\n\nFor any primal feasible $c$ and dual feasible $z$,\n\\begin{equation}\n y^Tz=c^TO^Tz\\le\\sum_j|c_j|\\,|(O^Tz)_j|\n \\le\\sum_jw_j|c_j|.\n\\end{equation}\nThis proves weak duality. Write $c=c^+-c^-$ with $c^\\pm\\ge0$. The resulting feasible, bounded-below finite linear program has an optimal basic feasible solution, and standard finite linear-programming duality gives equality with \\eqref{eq:dual}. The same conclusion follows from separation of the weighted cross-polytope and the affine constraint space.\n\nA basic optimum uses at most $m$ positive variables among the $2K$ split variables. An optimum cannot have both $c_j^+$ and $c_j^-$ positive: decreasing both leaves the constraint unchanged and strictly decreases the objective. Thus an optimal signed vector with at most $m$ active carriers exists. Fewer than $m$ active distinct exponentials cannot have $M=m-1$ prescribed zeros counted with multiplicity and a nonzero anchor. Hence this optimum uses exactly $m$ carriers. Its trace already spends its entire Chebyshev zero budget on the requested events, so it has no further real zeros.\n\\end{proof}\nAn optimized library need not retain the fundamental carrier $k=1$. The all-time nonlinear theorem above explicitly uses slowest-rate dominance; applying that theorem to a differently optimized library requires rechecking its tail argument or constraining the carrier selection. The interpolation and finite-library optimization statements themselves do not supply that extra guarantee.\n\nThe sparsity conclusion concerns the existence of one sparse optimum. Other optimal solutions could be convex combinations of optima and have larger supports. The theorem also does not assert that $m$ is the minimum number of modes among arbitrary nonlinear flows; its domain is the finite exponential scaffold.\n\n\\section{A reproducible numerical optimum}\nTake $\\nu=1$, simple events at $0$ and $0.12$, anchor time $0.25$, and library $k=1,\\ldots,8$. The numerical optimum has active frequencies $1,2,8$ and signed weights approximately\n\\begin{equation}\n c=(3.04420230714139,-6.25411210888282,0,0,0,0,0,\n 3.20990980174143).\n\\end{equation}\nIts unscaled initial energy is approximately $3.28624854340024$. The square compiler restricted to $1,2,3$ uses approximately $4.85890483587694$. A numerical dual vector is\n\\begin{equation}\n z=(0.0625004220815158,-1.97769618403857,3.28624854340024).\n\\end{equation}\nThe saved calculation reports a primal residual of about $4.1\\times10^{-14}$, a dual-feasibility excess of about $4.2\\times10^{-14}$, and a zero reported primal--dual gap at the recorded precision. These are floating-point checks, not rational or interval certificates. The exact theorem establishes what a truly feasible matching pair would certify; the file records the numerical approximation transparently.\n\\fig{12_energy_optimizer}{.92}{The finite-library optimum selects three carriers out of eight. The high-frequency carrier changes the cost of the initial cancellation while decaying rapidly before the anchor.}\n\\fig{13_dual_certificate}{.92}{The numerical dual functional lies below the energy weights and touches them at active carriers, up to the reported floating-point tolerance.}\n\n\\section{A universal bound within the heat-carrier class}\n\\begin{proposition}[Anchor-energy obstruction]\\label{prop:energybound}\nFor any finite or absolutely energy-summable shear spectrum at viscosity $\\nu>0$,\n\\begin{equation}\\label{eq:heatenergybound}\n |H(t)|\\le \\frac{E_0}{\\sqrt{e\\nu t}},\\qquad t>0.\n\\end{equation}\nConsequently, an anchor $H(t_*)=h_*$ requires\n$E_0\\ge\\sqrt{e\\nu t_*}|h_*|$.\n\\end{proposition}\n\\begin{proof}\nUsing \\eqref{eq:resourcecost},\n\\begin{equation}\n |H(t)|\\le\\sum_k |c_k|e^{-2\\nu k^2t}\n \\le E_0\\sup_{k\\ge1}\\{2k e^{-2\\nu k^2t}\\}.\n\\end{equation}\nReplacing the integer supremum by the continuous one can only increase it. Differentiation of $2x e^{-2\\nu tx^2}$ gives its maximum at $x=1/(2\\sqrt{\\nu t})$, with value $1/\\sqrt{e\\nu t}$.\n\\end{proof}\nThis bound uses the exact independent heat decay of the scaffold. It is not proved here for an arbitrary nonlinear Navier--Stokes solution. Extending an inequality beyond the class used in its derivation would invalidate the resource audit.\n\n\\section{The cost of high-order invisibility under a cutoff}\nSuppose all carriers have $k\\le K$ and $H^{(q)}(0)=0$ for $0\\le q<N$. Taylor's formula with integral remainder gives\n\\begin{equation}\\label{eq:cutoffbound}\n", "source_path": "research/chapters/04_resources_selection.tex", "source_sha256": "1b8da68b3833e257dddd9355e6482645a95592470598df196071c4b9663c9332", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "04_resources_selection-002", "document_id": "chapters/04_resources_selection.tex", "chunk_index": 1, "line_start": 77, "line_end": 82, "content_format": "latex_source", "text": " |H(t)|\\le \\frac{2KE_0}{N!}(2\\nu K^2t)^N.\n\\end{equation}\nIndeed, $|H^{(N)}(s)|\\le(2\\nu K^2)^N\\sum|c_k|\\le2KE_0(2\\nu K^2)^N$ for $s\\ge0$. Thus a desired later signal amplitude forces a corresponding energy cost at fixed cutoff. Alternatively, a fixed energy budget limits how large a high-order-hidden signal can become over a short interval.\n\nThe existence construction permits an arbitrarily small common amplitude factor $\\delta$, but its nonzero anchor then shrinks as $\\delta^2$. There is no claim of arbitrarily small initial energy producing a fixed finite signal at fixed viscosity and fixed time. The resource bound makes that limitation explicit.\n\n", "source_path": "research/chapters/04_resources_selection.tex", "source_sha256": "1b8da68b3833e257dddd9355e6482645a95592470598df196071c4b9663c9332", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "04_resources_selection-003", "document_id": "chapters/04_resources_selection.tex", "chunk_index": 2, "line_start": 83, "line_end": 141, "content_format": "latex_source", "text": "\\chapter{Momentum words and forbidden mixed observations}\n\\label{ch:selection}\n\\section{A translation-based selection rule}\nThe quadratic retuning theorem arose from a simple half-period symmetry. A more general language keeps track of which perturbation wavevectors can combine into a translation-invariant scalar observation.\n\nLet the base solution depend only on $z$. Label a perturbation by a nonzero horizontal wavevector $q_i\\in\\Z^2$. Because physical fields are real, the perturbation contains both $q_i$ and $-q_i$. Attach an independent real parameter $\\epsilon_i$ to each such wave pair. The vertical wavenumber and polarization may be arbitrary, subject to smoothness and divergence freedom.\n\nDiffusion and the Leray projection preserve each horizontal Fourier label. The bilinear advection term adds labels. Pairing two fields in a spatially integrated quadratic observable selects label sums equal to zero. These elementary facts produce an exact necessary condition on mixed derivatives of the solution's observations.\n\n\\begin{theorem}[Squarefree momentum-word rule]\\label{thm:selection}\nLet $\\mathcal O(u(t))$ be energy or helicity, evaluated along the smooth parameterized solution. For a subset $I$ of distinct perturbation indices, the squarefree mixed derivative\n\\begin{equation}\n \\left.\\prod_{i\\in I}\\partial_{\\epsilon_i}\\mathcal O(u(t))\\right|_{\\epsilon=0}\n\\end{equation}\nvanishes unless there is a sign choice $s_i\\in\\{-1,1\\}$ satisfying\n\\begin{equation}\\label{eq:momentumword}\n \\sum_{i\\in I}s_iq_i=0.\n\\end{equation}\nThe statement holds at every time in the common smooth interval.\n\\end{theorem}\n\\begin{proof}\nDifferentiate the mild equation with respect to a set of distinct parameters. By induction on its size, every resulting Fourier term has a horizontal label equal to a sum of one signed $q_i$ for each differentiated parameter in that term. The induction starts with the first variation, whose linearized operator about an axial base preserves horizontal label. At higher order, the differentiated bilinear term partitions the parameter set into two groups and adds their labels. Duhamel integration changes coefficients but not these labels.\n\nWhen the differentiated fields enter the quadratic global observable, the two groups again partition the full index set, and spatial integration requires their labels to sum to zero. If no signed relation \\eqref{eq:momentumword} exists, every contribution integrates to zero. The same argument applies to energy and helicity since curl also preserves the Fourier label.\n\\end{proof}\nThis theorem is a Fourier selection rule, not a claim that every permitted coefficient is nonzero. Projection, polarization, temporal cancellation, or an additional reflection symmetry can annihilate an allowed response. Chapter~\\ref{ch:triad} supplies one fully computed nonzero cubic example rather than replacing this distinction with a genericity assumption.\n\n\\section{Arbitrarily long minimal mixed words}\nDefine the \\emph{squarefree resonance girth} of a collection of nonzero labels to be the size of its smallest nonempty subset satisfying a signed relation. The qualifier squarefree means that each label is used at most once.\n\n\\begin{proposition}[Explicit large-girth labels]\\label{prop:girth}\nFor every integer $m\\ge3$, set\n\\begin{equation}\n q_i=(3^i,3^{2i}),\\quad 0\\le i\\le m-2,\n \\qquad q_{m-1}=-\\sum_{i=0}^{m-2}q_i.\n\\end{equation}\nThe squarefree resonance girth is exactly $m$.\n\\end{proposition}\n\\begin{proof}\nThe full unsigned sum is zero, so the girth is at most $m$. A signed relation among a subset of the first $m-1$ labels is impossible: in its first coordinate, the largest power $3^j$ exceeds the sum of all smaller powers. If the last label is present, reverse all signs if necessary to give it coefficient $+1$. The first-coordinate equation becomes\n\\begin{equation}\n \\sum_{i=0}^{m-2}(s_i-1)3^i=0,\n\\end{equation}\nwhere $s_i\\in\\{-1,0,1\\}$ and zero denotes omission. Every summand is nonpositive. Thus every $s_i=1$, so every other label is present and the relation uses all $m$ labels.\n\\end{proof}\nThe second coordinates are not needed for this particular independence proof. They provide genuinely different directions in the horizontal label plane rather than a purely collinear list.\n\nFor this family, all squarefree mixed global responses of order less than $m$ vanish. The theorem does \\emph{not} say that each individual perturbation is invisible to second order: a real wave and its own conjugate can pair at order $\\epsilon_i^2$. It also does not prove a nonzero response at order $m$. Establishing nonvanishing polarizations for every $m$ is a concrete open extension of this manuscript.\n\n\\section{Why the restriction to mixed responses matters}\nAn overclaim would be: ``these labels hide all information until order $m$.'' That is false because repeated-parameter measurements can already detect individual wave energies. The actual construction hides an irreducible \\emph{cross-interaction} from squarefree probes of smaller order. It is a hierarchy of cooperation requirements, not a universal invisibility cloak.\n\nSimilarly, ordinary Fourier momentum conservation already explains the label rule. The proposed use is to design an observation algebra with controlled mixed-order blindness and to connect it to nonlinear event correction. A new term for a conserved wavevector sum would not, by itself, be a new mathematical discovery.\n\n\\section{Closed phases as observable coordinates}\nWhen a signed label relation exists, its corresponding signed phase sum is invariant under translation. This is the same mechanism that produced $\\Phi$ in \\eqref{eq:cyclephase}. The set of relations therefore identifies which phase combinations could be globally observable.\n\nOne can organize the perturbations by an integer relation matrix. Its kernel records independent closed phase combinations, while its nonzero rows record origin-dependent phases. Developing a complete normal form for the resulting nonlinear response algebra is left as a research direction. The present proofs establish the selection rule and one exact cubic kernel, not a classification of every possible phase-dependent flow.\n\n", "source_path": "research/chapters/04_resources_selection.tex", "source_sha256": "1b8da68b3833e257dddd9355e6482645a95592470598df196071c4b9663c9332", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "04_resources_selection-004", "document_id": "chapters/04_resources_selection.tex", "chunk_index": 3, "line_start": 142, "line_end": 186, "content_format": "latex_source", "text": "\\chapter{What survives noise, and what does not}\n\\label{ch:stability}\n\\section{Exact retuning is not uncontrolled robustness}\nThe implicit-function theorem restores a finite event calendar by changing the carrier amplitudes. It does not guarantee that the uncorrected calendar survives arbitrary perturbations. Multiple zeros are especially sensitive, and this sensitivity is a theorem rather than an implementation inconvenience.\n\nSuppose a reference trace factors near $\\tau$ as\n\\begin{equation}\n h_0(t)=(t-\\tau)^m a(t),\\qquad |a(t)|\\ge a_*>0.\n\\end{equation}\nIf a perturbed trace satisfies $\\norm{h-h_0}_{L^\\infty}\\le\\eta$, then every zero $t$ of $h$ in that neighborhood obeys\n\\begin{equation}\\label{eq:rootsensitivity}\n \\boxed{|t-\\tau|\\le(\\eta/a_*)^{1/m}.}\n\\end{equation}\nIndeed, at a zero of $h$, $|h_0(t)|\\le\\eta$, and the factorization gives the inequality. The exponent $1/m$ is sharp, as shown by $h_0(t)=a_*(t-\\tau)^m$ perturbed by a constant of magnitude $\\eta$.\n\nFor an odd event, the signs on two sufficiently separated sides are opposite, so a small uniform perturbation retains at least one nearby crossing. For an even event, adding a small constant of the appropriate sign can remove every nearby real zero. An even contact is therefore not structurally stable under unrestricted additive error.\n\\fig{15_root_sensitivity}{.92}{The worst-case root displacement scales as the $m$-th root of the signal error. High-order exact cancellation is intrinsically delicate without retuning or protecting symmetry.}\n\n\\section{Conditioning of the compiler}\nThe exact coefficient vector is $c=O^{-1}y$. Perturbing the observation target by $\\Delta y$ gives\n\\begin{equation}\n \\Delta c=O^{-1}\\Delta y,\\qquad\n \\norm{\\Delta c}\\le\\norm{O^{-1}}\\norm{\\Delta y}.\n\\end{equation}\nPerturbing the matrix itself adds the usual resolvent-type sensitivity. Confluent exponential matrices can become highly ill-conditioned as multiplicities grow, event times cluster, or rate scales separate. Arbitrary-precision arithmetic reduces rounding error but does not improve the underlying conditioning with respect to uncertain inputs.\n\nThe executable compiler accepts decimal strings so that intended event times can enter arbitrary precision without first being rounded to machine precision. This is useful for reproducibility, not a way of bypassing measurement uncertainty. A real input specified only to six digits does not become known to one hundred digits merely because the solver uses them.\n\\fig{14_conditioning}{.92}{Row-scaled matrix condition numbers for increasing initial cancellation orders. Row scaling helps units and magnitude balance, but does not remove near-dependence of the columns.}\n\n\\section{Mirror twins and an observability obstruction}\n\\begin{proposition}[Energy history cannot determine handedness]\\label{prop:mirror}\nEvery smooth periodic Navier--Stokes solution has a reflected companion with exactly the same kinetic-energy history and the negative of its helicity history. This includes the genuinely nonlinear solutions of Theorem~\\ref{thm:nonlinear}.\n\\end{proposition}\n\\begin{proof}\nTake a lattice-preserving orthogonal matrix $Q$ with $\\det Q=-1$ and define $u^Q(x,t)=Qu(Q^Tx,t)$. Orthogonal equivariance of the equation gives another solution. A change of variables yields $E(u^Q)=E(u)$. The curl transforms as\n$\\curl u^Q=(\\det Q)Q(\\curl u)(Q^Tx)$, so $H(u^Q)=-H(u)$.\n\\end{proof}\nThus no algorithm receiving only the complete energy-time trace can determine the sign of helicity over this class. The ambiguity is exact and survives unlimited precision, because two admissible trajectories produce identical input data. This is a symmetry-based nonidentifiability result, not a limitation of a particular machine-learning model.\n\nThe same argument applies to any collection of parity-even observations. Adding a genuinely parity-odd measurement can remove the sign ambiguity in principle, although other ambiguities can remain. Helicity itself is one such measurement, and the phase-sensitive cubic response provides another route to a chiral observation.\n\n\\section{A hierarchy of stability claims}\nThree different statements now coexist. First, the compiler is locally solvable: small parameterized perturbations can be counteracted by suitable changes in a finite vector of initial amplitudes. Second, a multiple event is poorly conditioned under uncorrected output noise, with the sharp exponent \\eqref{eq:rootsensitivity}. Third, an exact reflection symmetry protects the identically zero observable against symmetry-preserving changes, but not against symmetry-breaking ones.\n\nThese are not contradictions. They answer different questions about which variables are allowed to change and which constraints remain enforced. A careful experimental or computational protocol must state whether it controls the initial amplitudes, the clock, the phase symmetry, or only the measured output.\n", "source_path": "research/chapters/04_resources_selection.tex", "source_sha256": "1b8da68b3833e257dddd9355e6482645a95592470598df196071c4b9663c9332", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "05_audit_computation-001", "document_id": "chapters/05_audit_computation.tex", "chunk_index": 0, "line_start": 1, "line_end": 51, "content_format": "latex_source", "text": "\\chapter{The scissor architecture as mathematical design}\n\\label{ch:architecture}\n\\section{From an imaginative seed to a precise object}\nThe design brief asked for an unfamiliar mechanism, strongly connected to Chisa's paired blades, scissor closure, repeated cutting, Stagnation, and terminal release. These images can stimulate choices, but they cannot supply a proof. The mathematical translation in this manuscript is therefore explicit at every step.\n\nThe two scissor blades become opposite signs of a quadratic chiral observable. Closing the blades becomes an exact cancellation constraint, not an actual spatial singularity. Repeated chainsaw teeth become distinct decay rates. Stagnation becomes a zero jet of finite order. A terminal release becomes a specified nonzero anchor after those events. A transverse attack becomes a perturbation that demonstrably activates the pressure-projected nonlinear term. These correspondences are design metaphors for the newly assembled calculus, not claims about the game's mechanics or about the physics of a literal weapon.\n\n\\section{Death Snip: an observation divisor}\nThe elementary mathematical snip is not merely the equation $H(\\tau)=0$. It is an ordered contact\n\\begin{equation}\n (\\tau,m):\\qquad H^{(q)}(\\tau)=0\\quad(0\\le q<m),\\qquad H^{(m)}(\\tau)\\ne0.\n\\end{equation}\nThis records whether the observable touches zero or crosses it, how many instantaneous derivatives fail to reveal the release, and where the event lies in the evolution. Multiple snips form the finite divisor $\\score$. The divisor is an input to a compiler, rather than an accidental feature noticed after simulation.\n\nCalling this a divisor borrows a classical language of zeros with multiplicity. The novelty candidate is not the definition of multiplicity. It is the exact realization and controlled nonlinear deformation of such a divisor for a particular physical PDE observable.\n\n\\section{Serrated Loop: a minimal signed spectrum}\nThe carrier spectrum solves a square interpolation problem. Its minimality is not a guess based on numerical optimization: a nonzero combination of $n$ distinct exponentials has at most $n-1$ zeros counted with multiplicity. Thus a degree-$M$ calendar consumes exactly $M+1$ carriers in the minimal scaffold.\n\nThis gives a concrete interpretation of an additional tooth: in the scaffold, one extra independent rate permits one extra zero condition. The extra rate need not correspond to a new spatial dimension. It is another direction in a finite function space. The cost and conditioning can deteriorate quickly, as Chapters~\\ref{ch:resource} and~\\ref{ch:stability} quantify.\n\n\\section{Eye of Unraveling: invisible first variation}\nThe transverse perturbation produces real advection and vortex stretching at first order in $\\varepsilon$. Nevertheless, translation-invariant scalar observations do not change at first order because the half-period translation reverses that perturbation and leaves the base unchanged. This is a precise mismatch between the order of dynamics and the order of observation.\n\nThe corrected initial amplitudes therefore obey\n\\begin{equation}\n a(\\varepsilon)-a^0=O(\\varepsilon^2),\n \\qquad \\B(u_0^\\varepsilon,u_0^\\varepsilon)=O(\\varepsilon)\n \\ \\hbox{with a nonzero leading coefficient}.\n\\end{equation}\nThe observation is not a complete representation of the underlying dynamics. A substantial conceptual gain of the construction is to make that incompleteness exact, parameterized, and repairable.\n\n\\section{Moment of Nihility: two distinct mathematical meanings}\nThe finite-order version is a local vanishing jet: the observable becomes nonzero immediately after the event, but several derivatives at the event fail to reveal it. The symmetry version is global extinction: a reflected state keeps total helicity zero throughout its smooth lifetime. The second does not imply that every fluid parcel is still or that all local handedness vanishes.\n\nThe distinction matters when interpreting a supposedly mysterious ``freeze.'' An observed zero may result from cancellation, symmetry, insufficient derivative order, or an actual pause of the operational clock. The energy-coordinate quotient distinguishes the last possibility from the first three when its assumptions hold. No single scalar time trace can automatically identify all of these mechanisms.\n\n\\section{Sawring--Eradication: release and its resource limit}\nThe anchor $H(t_*)=\\delta^2$ certifies that the event construction is not the identically zero trace. The global tail argument strengthens this certificate: after the last event, helicity remains positive and has no hidden later zero. Meanwhile, kinetic energy strictly decreases.\n\nThis should not be described as creating energy or defeating dissipation. The signed chiral observable is not a positive energy. Differential decay and nonlinear redistribution can change its sign even while a separate positive functional decreases. The fixed-viscosity energy obstruction shows that a finite output amplitude still requires a finite initial resource.\n\n\\section{A proposed unified object}\nA useful package for future work is the tuple\n\\begin{equation}\\label{eq:unifiedobject}\n \\mathfrak C=(\\score,O,c,a(\\varepsilon),\\mathcal R,\\mathscr K,\\mathcal W),\n\\end{equation}\nwhere $\\score$ is the event divisor, $O$ the observation matrix, $c$ its signed spectral solution, $a(\\varepsilon)$ the nonlinear correction, $\\mathcal R$ a protecting or sign-reversing symmetry, $\\mathscr K$ the energy-coordinate graph, and $\\mathcal W$ the allowed momentum words.\n\nThe components have different mathematical roles: a finite interpolation object, a PDE solution family, a group action, an invariant graph, and a selection algebra. The proposal is to study their compatibility rather than simply giving several familiar formulas a shared name. For example, a useful compatibility question asks which event divisors can be realized while imposing an additional reflection constraint. If that constraint forces $H\\equiv0$, a nonzero anchor is impossible. The formalism exposes the obstruction instead of hiding it.\n\n", "source_path": "research/chapters/05_audit_computation.tex", "source_sha256": "754db945b0c8d280e15f008c3631d17c655df8ab666e1095ea82defc800f7866", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "05_audit_computation-002", "document_id": "chapters/05_audit_computation.tex", "chunk_index": 1, "line_start": 52, "line_end": 119, "content_format": "latex_source", "text": "\\chapter{Executable verification and numerical demonstration}\n\\label{ch:computation}\n\\section{Three kinds of evidence}\nThe package separates exact symbolic identities, high-precision evaluations of analytical formulas, and finite-resolution numerical evolution. Their strengths differ. Symbolic checks can verify a specific polynomial identity without rounding error; high-precision checks can stress an implementation over many examples; a Galerkin simulation can illustrate a nonlinear construction. None alone proves every quantifier of the continuum theorem.\n\nThe continuum existence, exact multiplicity, and infinite-tail statements rest on the analytical arguments in the manuscript. The numerical demonstration does not calculate a certified admissible $\\varepsilon_0$ for the continuum implicit-function theorem. It finds a nearby solution of a discretized observation system and checks refinement.\n\n\\section{The verification suite}\nThe saved run of \\texttt{tests/verify.py} reports 1,139 checks passed and zero failed. Its machine-readable output is \\texttt{data/verification.json}. The suite covers the prior initial-jet identity for orders $1$ through $14$, exact rational multi-event examples, high-precision confluent solves and sign samples, the triadic initial derivatives, the three exact interaction contractions, the closed response kernel, the quadrature reflection symmetry, momentum-word girth examples, and the numerical primal--dual design check.\n\nThe maximum recorded high-precision event-jet residual is approximately $2.5791\\times10^{-124}$. This residual belongs to arbitrary-precision evaluations of explicitly compiled exponential traces. It is not an error estimate for the PDE simulation and is not a proof of a universal theorem by sampling.\n\nThe exact Fourier code represents coefficients as SymPy expressions and keeps the full support generated by the low-order convolution. It does not project onto a small spatial cutoff when checking the initial triadic identities. This avoids a common failure mode in which a convenient truncation accidentally removes the nonlinear term being tested.\n\n\\section{The nonlinear Fourier--Galerkin demonstration}\nThe illustrative case uses $\\nu=1$, carrier wavenumbers $1,2,3$, simple zeros at $0$ and $0.12$, and anchor time $0.25$. The unscaled signed coefficients are approximately\n\\begin{equation}\n (3.52423633176923,-8.84198254068927,5.31774620892004).\n\\end{equation}\nThe common velocity scale is $\\delta=0.1$. The perturbation is exactly $\\varepsilon(0,\\cos x,\\cos y)$, multiplied by the same $\\delta$.\n\nThe solver retains a cube of Fourier modes with $|k_i|\\le3$, evaluates quadratic products on a $12^3$ grid, and projects to divergence-free mean-zero modes. The padding is sufficient to avoid aliasing into the retained band for quadratic products. Time integration uses classical fourth-order Runge--Kutta with step $0.001$. Positive amplitudes are parameterized as $a_j=a_j^0e^{\\theta_j}$, and a nonlinear root solve adjusts the three $\\theta_j$ to restore the three observations.\n\nThe projected vector field is evaluated as $\\nu\\Delta u+\\Pp(u\\times\\omega)$, equivalent to the incompressible Navier--Stokes evolution after the gradient term is removed. The code reports energy, helicity, projected advection, stretching, divergence, and the reflection comparison.\n\n\\begin{center}\n\\begin{tabular}{@{}rrrr@{}}\n\\toprule\n$\\varepsilon$ & $\\norm{a-a^0}_2$ & scaled observation residual & $\\norm{\\B(u_0,u_0)}_2$\\\\\n\\midrule\n0.1 & $9.56\\times10^{-7}$ & $3.47\\times10^{-16}$ & 0.004068\\\\\n0.2 & $3.82\\times10^{-6}$ & $8.67\\times10^{-16}$ & 0.008138\\\\\n0.4 & $1.53\\times10^{-5}$ & $6.61\\times10^{-16}$ & 0.016290\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\nDoubling $\\varepsilon$ approximately quadruples the amplitude correction and doubles the initial interaction norm. This is the predicted distinction between second-order repair and first-order dynamics. The table is supportive numerical evidence for these orders, not the source of their proof.\n\\fig{05_quadratic_retuning}{.92}{The nonlinear solve's amplitude corrections follow a quadratic scale over the tested parameter range. The exact evenness theorem supplies the analytical reason.}\n\\fig{08_energy_decay}{.92}{Kinetic energy decreases in the three corrected numerical evolutions. Strict decrease for the analytical solution family follows from the energy identity and nonzero tail.}\n\n\\section{Refinement and what it establishes}\nKeeping the corrected coarse-grid initial data at $\\varepsilon=0.4$, the calculation was repeated with half the time step and Fourier cutoffs $3$, $4$, and $5$. The maximum absolute discrepancy from the three target helicity observations was approximately\n\\begin{center}\n\\begin{tabular}{@{}rrr@{}}\n\\toprule\nFourier cutoff & time step & maximum target discrepancy\\\\\n\\midrule\n3 & 0.0005 & $6.19\\times10^{-13}$\\\\\n4 & 0.0005 & $1.57\\times10^{-12}$\\\\\n5 & 0.0005 & $1.73\\times10^{-12}$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\nThese comparisons are saved in \\texttt{data/refinement\\_comparison.json}. They show that, for this small example, the observed residual is not a visibly large artifact of the first cutoff. They do not bound the infinite-dimensional remainder rigorously. The slight increase with cutoff is compatible with resolving additional small nonlinear contributions; it is not evidence of exact continuum cancellation at the coarse parameter vector.\n\nThe analytical theorem guarantees a nearby continuum correction for sufficiently small parameters but does not identify it with this floating-point vector. An interval Newton method combined with a validated PDE error bound would be needed for a computer-assisted certified instance.\n\n\\section{Reproducing the files}\nFrom the package root, run\n\\begin{lstlisting}[language=bash]\npython -m pip install -r requirements.txt\npython tests/verify.py\npython src/nonlinear_demo.py\npython src/make_figures.py\nlatexmk -pdf -interaction=nonstopmode manuscript.tex\n\\end{lstlisting}\nThe nonlinear demonstration is optional when only the exact algebraic suite is needed. The refinement utility supplied with the package reproduces the additional cutoff comparison separately. All commands write their outputs within the package's data and figure directories. No external API, paid service, or GPU is required.\n\n", "source_path": "research/chapters/05_audit_computation.tex", "source_sha256": "754db945b0c8d280e15f008c3631d17c655df8ab666e1095ea82defc800f7866", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "05_audit_computation-003", "document_id": "chapters/05_audit_computation.tex", "chunk_index": 2, "line_start": 120, "line_end": 121, "content_format": "latex_source", "text": "The manuscript source, figure-generation code, exact algebra, numerical solver, data, bibliographic metadata, and claim audit are included. The compiled PDF is a convenient reading artifact; the source files are the reproducible research object.\n\n", "source_path": "research/chapters/05_audit_computation.tex", "source_sha256": "754db945b0c8d280e15f008c3631d17c655df8ab666e1095ea82defc800f7866", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "05_audit_computation-004", "document_id": "chapters/05_audit_computation.tex", "chunk_index": 3, "line_start": 122, "line_end": 162, "content_format": "latex_source", "text": "\\chapter{Novelty audit and adversarial interpretation}\n\\label{ch:audit}\n\\section{The scope of the literature check}\nA targeted public-web literature check was conducted on 9 October 2026. Queries covered helicity zeros and sign changes, periodic Navier--Stokes asymptotics, nonlinear helicity production, helical triads and phases, prescribed fluid events, exponential Chebyshev interpolation, and analytic solution dependence. Primary papers, publisher records, author-hosted manuscripts, and arXiv versions were preferred.\n\nThis is not an exhaustive database search, a priority opinion, or peer review. A construction can be absent from a few search results and still be known under different language. The absence of a matching phrase is especially weak evidence when the proposed terminology is new. The search therefore informs an overlap audit rather than a declaration of world-first status.\n\n\\section{What is already established}\nExponential extended Chebyshev systems and their zero-count properties belong to classical approximation theory; the paper of Aldaz, Kounchev, and Render provides a relevant modern treatment \\cite{Aldaz2008}. The interpolation scaffold is an application of this established structure, not a new zero-count theorem.\n\nFoias, Hoang, and Nicolaenko studied helicity for three-dimensional periodic Navier--Stokes and established an eventual alternative between identically zero helicity and a nonzero eventual sign \\cite{Foias2007}. The finite calendars constructed here do not contradict that result. Their last event is finite, and the positive tail is part of the construction.\n\nWaleffe's helical triad analysis and Sanada's helicity-production calculation already rule out claiming that nonlinear triads or helicity generation from nonhelical ingredients are entirely new phenomena \\cite{Waleffe1992,Sanada1993}. The present explicit coefficient and kernel are narrower objects.\n\nSmall-data regularity and analytic dependence are established analytical tools \\cite{KochTataru2001,Brzezniak1991}. Helicity-based structures and specially organized global solutions also have substantial prior work \\cite{Lei2015}. The manuscript proves the particular estimates it needs in the periodic setting instead of treating those broad theories as its own discoveries.\n\nEnciso, Luc\\`a, and Peralta-Salas constructed global smooth flows with prescribed vortex-topology behavior and reconnection \\cite{Enciso2017}. That work already uses controlled fluid constructions and perturbative ideas. Our scalar zero calendar is different from a vortex-topology program and should not be marketed as the first prescribed event construction in a fluid PDE.\n\nA September 2026 preprint by G\\\"urcan, Manfredini, and Morel studies affine triad-triplet structures, phase dynamics, and exact solutions of associated triadic systems \\cite{Gurcan2026}. Its existence further narrows any claim that closed triad phases or structured triad networks are new in general. No equivalence or contradiction with its proposed exact solutions is asserted here.\n\n\\section{The specific candidate contribution}\nThe strongest claim offered for independent review is the following combination: every finite helicity-zero divisor is realizable by a fixed-viscosity global smooth solution with full three-direction initial support, nonzero projected advection, nonzero vortex stretching, an exact anchor, no additional nonnegative-time zeros, and an even analytic amplitude correction under the specified transverse perturbation.\n\nThis statement combines an explicit finite observation Jacobian with a symmetry constraint and a weighted infinite-time tail estimate. The zero-count argument alone does not establish it. Nor does generic continuous dependence alone preserve multiple roots. The analytic retuning supplies exact zero jets, and the weighted estimate is needed to exclude arbitrarily late unintended zeros.\n\nThe second candidate is the exact standing-wave triad calculation coupled to all-time quadrature extinction. Its derivative identity, response kernel, and protecting inversion are fully explicit. The general ingredients are familiar, so an independent reviewer should assess whether this particular combination has already appeared and whether its conceptual usefulness justifies a separate contribution.\n\nThe energy-clock invariant, finite-library resource dual, and momentum-word rule are supporting structures. Their proofs rely on elementary or established mathematical principles. They strengthen the framework's completeness, but they are not individually advertised as revolutionary new laws.\n\n\\section{What has not been disproved}\nNo established theorem has been refuted. The manuscript does give exact counterexamples to several unrestricted guesses: that initially zero viscous helicity must stay zero; that a bounded number of derivatives determines a future zero pattern; that a complete energy history determines the sign of helicity; and that small changes in an uncorrected output preserve a multiple event.\n\nThose guesses are not presented as historical conjectures held by the named scientific community. A valid theorem includes its hypotheses. Replacing it with an overbroad informal statement and then refuting the replacement would not be a mathematical breakthrough.\n\nThe work also does not resolve general three-dimensional Navier--Stokes regularity, produce shock or plasma solutions, contradict Euler helicity conservation, or show that the original Chisa-inspired physical scenarios occur. Its achievements are specific constructive and observational theorems within a carefully stated smooth incompressible model.\n\n\\section{The most valuable next falsification attempts}\nAn independent audit should first examine the analytic finite-parameter observation map at initial-time high-order jets, the exact spectral-gap difference estimate used for the global tail, and the all-time multiplicity argument near each prescribed event. A flaw in any of these would weaken the principal theorem even if every numerical example continued to look correct.\n\nFor the triad, the most useful checks are the Fourier normalization, the sign of the Leray-projected bilinear term, the factor of two in the quadratic helicity cross term, and the consistency of the inversion phase equations. These are all exposed in the exact source rather than hidden behind a simulation.\n\n", "source_path": "research/chapters/05_audit_computation.tex", "source_sha256": "754db945b0c8d280e15f008c3631d17c655df8ab666e1095ea82defc800f7866", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "05_audit_computation-005", "document_id": "chapters/05_audit_computation.tex", "chunk_index": 4, "line_start": 163, "line_end": 168, "content_format": "latex_source", "text": "For the resource theorem, the finite-library qualification is essential. For the selection rule, squarefree mixed derivatives must not be silently replaced by all derivatives. For the clock invariant, both observations must share the same gate. These restrictions are part of the results, not optional caveats.\n\n\\section{Research status}\nThe mathematical claims have written proofs under their stated assumptions. The supplied verification suite passes its 1,139 implemented checks. The continuum theorem is not proof-assistant formalized, the numerical example is not interval certified, and no independent peer review has been performed. Physical validation has not been attempted. Historical originality is unresolved.\n\nThe appropriate release label is therefore \\emph{an independently reviewable research draft with candidate original theorems}, not a certified world-first breakthrough. The manuscript's value should be judged by the correctness and usefulness of its explicit constructions, rather than by the intensity of its naming or the ambition of the initial prompt.\n", "source_path": "research/chapters/05_audit_computation.tex", "source_sha256": "754db945b0c8d280e15f008c3631d17c655df8ab666e1095ea82defc800f7866", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "06_appendices-001", "document_id": "chapters/06_appendices.tex", "chunk_index": 0, "line_start": 1, "line_end": 94, "content_format": "latex_source", "text": "\\chapter{Periodic small-data estimates and parameter dependence}\n\\label{app:small}\n\\section{Spaces and the elementary nonlinear estimate}\nThis appendix supplies the periodic estimates used by the continuation and tail arguments. All fields are mean-zero and divergence-free on $\\T^3$. Fix an integer $s\\ge4$, and use the Fourier Sobolev norm\n\\begin{equation}\n \\norm{u}_{H^s}^2=\\sum_{k\\in\\Z^3\\setminus\\{0\\}}(1+|k|^2)^s|\\widehat u(k)|^2.\n\\end{equation}\nPoincar\\'e's inequality has spectral gap one:\n$\\norm{\\nabla u}_{H^s}^2\\ge\\norm{u}_{H^s}^2$.\nThe Leray projection is an orthogonal Fourier multiplier and is bounded on every such Sobolev space.\n\nFor smooth divergence-free $v$, the highest transport derivative cancels in the energy pairing. Expanding integer derivatives by Leibniz' rule and estimating the remaining products by Sobolev embedding gives\n\\begin{equation}\\label{eq:commutatorest}\n |\\ip{\\B(v,v)}{v}_{H^s}|\n \\le C_s\\norm{v}_{H^s}^3\n \\le C_s\\norm{v}_{H^s}\\norm{\\nabla v}_{H^s}^2.\n\\end{equation}\nOne may equivalently use a commutator formulation. The first inequality uses $s>5/2$ to control a first derivative in $L^\\infty$. The final inequality uses the mean-zero spectral gap.\n\nFor two smooth fields $u,v$ and $z=u-v$, the difference equation is\n\\begin{equation}\\label{eq:differencePDE}\n z_t=\\nu\\Delta z-\\B(u,z)-\\B(z,v).\n\\end{equation}\nThe analogous product and commutator estimates yield\n\\begin{equation}\\label{eq:diffenergy}\n \\frac{\\dd}{\\dd t}\\norm{z}_{H^s}^2\n +2\\nu\\norm{\\nabla z}_{H^s}^2\n \\le C_s\\bigl(\\norm{u}_{H^{s+1}}+\\norm{v}_{H^{s+1}}\\bigr)\n \\norm{z}_{H^s}^2.\n\\end{equation}\nThe extra derivative on the coefficients is convenient and harmless because the initial fields are trigonometric polynomials. It avoids a derivative loss in the bound on $\\B(z,v)$.\n\n\\section{Global smallness and an integrable coefficient}\nApply \\eqref{eq:commutatorest} at order $s+1$. If the initial norm is sufficiently small compared with $\\nu/C_{s+1}$, a continuity argument preserves the smallness and gives\n\\begin{equation}\\label{eq:preliminarydecay}\n \\frac{\\dd}{\\dd t}\\norm{u}_{H^{s+1}}^2\n +\\nu\\norm{\\nabla u}_{H^{s+1}}^2\\le0,\n \\qquad\n \\norm{u(t)}_{H^{s+1}}\\le\\norm{u_0}_{H^{s+1}}e^{-\\nu t/2}.\n\\end{equation}\nGalerkin approximation supplies local solutions and the a priori estimate extends them globally. The usual compactness passage, followed by the difference estimate, yields the unique strong solution. Smooth initial data remain smooth by parabolic regularity and higher-order estimates. No assertion about arbitrary large-data regularity enters this argument.\n\nThe decay rate in \\eqref{eq:preliminarydecay} is deliberately only $\\nu/2$ in norm. Its key consequence is\n\\begin{equation}\\label{eq:integrablecoef}\n \\int_0^\\infty\\norm{u(t)}_{H^{s+1}}\\dd t\n \\le\\frac2\\nu\\norm{u_0}_{H^{s+1}}<\\infty.\n\\end{equation}\nThis integrability is what permits recovery of the exact spectral-gap rate for the comparison argument. Simply claiming the sharper rate directly from the absorbed energy inequality would lose a factor and would not suffice for the tail proof.\n\n\\section{The exact spectral-gap comparison rate}\nCombine \\eqref{eq:diffenergy} with Poincar\\'e and integrate. For two solutions in the same small neighborhood,\n\\begin{align}\n \\norm{u(t)-v(t)}_{H^s}^2\n &\\le\\norm{u_0-v_0}_{H^s}^2 e^{-2\\nu t}\\\\\n &\\quad\\times\\exp\\left(C_s\\int_0^t\n [\\norm{u(r)}_{H^{s+1}}+\\norm{v(r)}_{H^{s+1}}]\\dd r\\right).\n\\end{align}\nThe integral is bounded uniformly for all $t\\ge0$ by \\eqref{eq:integrablecoef}. Therefore\n\\begin{equation}\\label{eq:appendixweighted}\n \\norm{u(t)-v(t)}_{H^s}\\le C\\norm{u_0-v_0}_{H^s}e^{-\\nu t}.\n\\end{equation}\nTaking $v=0$ gives the corresponding single-solution estimate. The constant is uniform over the selected small initial-data neighborhood.\n\nFor the corrected family, $\\norm{u_0^\\varepsilon-u_0^0}_{H^s}\\le C\\delta|\\varepsilon|$, while each initial norm is at most $C\\delta$. Bilinearity of helicity and boundedness of curl from $H^s$ to $H^{s-1}$ give\n\\begin{equation}\n |H_\\varepsilon(t)-H_0(t)|\n \\le C\\delta^2|\\varepsilon|e^{-2\\nu t}.\n\\end{equation}\nThis is precisely the weighted estimate needed to compare with the nonzero slowest carrier. No unweighted continuity statement alone would control a reference trace that decays to zero.\n\n\\section{Analytic dependence in finite parameters}\nFor completeness, consider on a finite time interval the mild equation\n\\begin{equation}\\label{eq:mild}\n u(t)=e^{\\nu t\\Delta}u_0-\n \\int_0^t e^{\\nu(t-r)\\Delta}\\B(u(r),u(r))\\dd r.\n\\end{equation}\nWork in $X_T=C([0,T];H^s)\\cap L^2(0,T;H^{s+1})$. Standard Fourier heat estimates bound the Duhamel operator from $L^2(0,T;H^{s-1})$ to $X_T$. The product estimate\n\\begin{equation}\n \\norm{\\B(a,b)}_{H^{s-1}}\\le C_s\\norm{a}_{H^s}\\norm{b}_{H^s}\n\\end{equation}\nand Poincar\\'e imply that the quadratic integral in \\eqref{eq:mild} is a continuous bilinear mapping of $X_T\\times X_T$ into $X_T$. For sufficiently small data it is a contraction on an appropriate ball, or equivalently the derivative of the mild residual with respect to $u$ is invertible by a Neumann series.\n\nThe residual is polynomial in $u$ and linear in $u_0$, hence analytic. The Banach-space analytic implicit-function theorem gives analytic dependence on $u_0$ in that neighborhood. Restricting to the finite-dimensional trigonometric family gives the analytic map used in Chapter~\\ref{ch:nonlinear}. This reasoning is consistent with established analytic-dependence results \\cite{Brzezniak1991}; it is included to make the particular use transparent.\n\nFor observations at positive times, smoothness and parabolic regularization permit the finite time derivatives in the event map. At time zero, define recursively\n\\begin{equation}\\label{eq:initialrecursion}\n u_{n+1}=\\nu\\Delta u_n-\n \\sum_{q=0}^{n}\\binom nq\\B(u_q,u_{n-q}),\\qquad u_0=u(0).\n\\end{equation}\nEach fixed $u_n$ is a differential polynomial in the initial field. Choosing initial Sobolev regularity above $2n+3$ makes the required scalar products continuous. The chosen margin $s\\ge2\\max m_i+8$ is more than sufficient. This avoids assuming that the full Taylor series in time converges at the initial endpoint.\n\n\\section{Uniform nonvanishing near every event}\nA useful technical point in the global theorem is the factorization with parameters. Let $h_\\varepsilon$ be $C^m$ jointly in time and continuously in parameters, and suppose the first $m$ jets through order $m-1$ vanish at $\\tau$ for every $\\varepsilon$. Integral Taylor's formula gives\n\\begin{equation}\n", "source_path": "research/chapters/06_appendices.tex", "source_sha256": "c371759ba2ed1a89383d3cc957ba9c6fd3d440050be826fc5c51818922565c36", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "06_appendices-002", "document_id": "chapters/06_appendices.tex", "chunk_index": 1, "line_start": 95, "line_end": 100, "content_format": "latex_source", "text": " h_\\varepsilon(t)=(t-\\tau)^mR_\\varepsilon(t),\\quad\n R_\\varepsilon(t)=\\frac1{(m-1)!}\\int_0^1(1-r)^{m-1}\n h_\\varepsilon^{(m)}(\\tau+r(t-\\tau))\\dd r.\n\\end{equation}\nThe function $R_\\varepsilon$ is continuous and $R_0(\\tau)=h_0^{(m)}(\\tau)/m!\\ne0$. Hence, on one common sufficiently small event neighborhood and parameter interval, it never vanishes. This proves both exact multiplicity and absence of extra nearby roots. At an event at $0$, use the one-sided neighborhood inside $[0,\\infty)$.\n\n", "source_path": "research/chapters/06_appendices.tex", "source_sha256": "c371759ba2ed1a89383d3cc957ba9c6fd3d440050be826fc5c51818922565c36", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "06_appendices-003", "document_id": "chapters/06_appendices.tex", "chunk_index": 2, "line_start": 101, "line_end": 149, "content_format": "latex_source", "text": "\\chapter{Exact Fourier normalization and triad contractions}\n\\label{app:fourier}\n\\section{Conventions that fix every factor}\nExpand a real field as $u(x)=\\sum_k\\widehat u(k)e^{ik\\cdot x}$ with $\\widehat u(-k)=\\overline{\\widehat u(k)}$. The normalized spatial average gives\n\\begin{equation}\n \\ip{u}{v}=\\sum_k\\widehat u(k)\\cdot\\widehat v(-k),\\qquad\n \\widehat{\\curl u}(k)=ik\\times\\widehat u(k).\n\\end{equation}\nA real cosine mode has coefficients $p/2$ at $k$ and $-k$. A sine mode has $p/(2i)$ at $k$ and $-p/(2i)$ at $-k$. The helical carrier combines these so that its squared norm is one, not one half.\n\nFor a nonzero output label $k$, the Leray projection is\n\\begin{equation}\n P_kv=v-k\\frac{k\\cdot v}{|k|^2}.\n\\end{equation}\nThe projected bilinear Fourier coefficient is\n\\begin{equation}\\label{eq:fourierB}\n \\widehat{\\B(u,v)}(k)=iP_k\\sum_{p+q=k}\n (\\widehat u(p)\\cdot q)\\widehat v(q).\n\\end{equation}\nThe zero mode of this expression vanishes for divergence-free periodic input; the implementation explicitly stores the mean-zero class.\n\n\\section{The three contraction calculations}\nLet $G_i=\\B(V_j,V_k)+\\B(V_k,V_j)$ for the zero-phase waves in Chapter~\\ref{ch:triad}. Applying \\eqref{eq:fourierB} at output $k_i$ gives\n\\begin{center}\n\\begin{tabular}{@{}cll@{}}\n\\toprule\n$i$ & $\\widehat G_i(k_i)$ & $\\widehat{\\curl G_i}(k_i)$\\\\\n\\midrule\n1 & $(0,0,-i/4)$ & $(0,-1/4,0)$\\\\\n2 & $(i/4,0,0)$ & $(0,-1/4,1/4)$\\\\\n3 & $(-i/12,-i/12,i/6)$ & $(1/4,-1/4,0)$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\nThe negative output labels give the conjugate coefficients. Pairing with the two coefficients $p_i/2$ of $V_i$ gives $S_i=p_i\\cdot\\widehat{\\curl G_i}(k_i)$, because the curl coefficients in the table are real. This yields $(-1/4,0,1/4)$ exactly.\n\nFor example, in the first row only the input labels $-k_2$ and $-k_3$ can sum to $k_1$. Before projection their sum is a complex multiple of a real vector. Projection removes its component parallel to $k_1$, leaving $(0,0,-i/4)$. Curl then multiplies by $ik_1\\times$, producing $(0,-1/4,0)$. This explicit sequence checks both the sign convention and the factor from the two real Fourier halves.\n\n\\section{Time jets without spatial truncation}\nThe recurrence \\eqref{eq:initialrecursion} generates the exact finite Fourier support required at every fixed finite derivative order. There is no need to retain an infinite number of modes for a finite-order calculation starting from finite support: each convolution enlarges a finite set by a finite Minkowski sum.\n\nThe $n$-th initial helicity derivative is\n\\begin{equation}\\label{eq:exactjets}\n H^{(n)}(0)=\\sum_{q=0}^n\\binom nq\\ip{u_q}{\\curl u_{n-q}}.\n\\end{equation}\nFor the triad, the verifier builds $u_1$ and $u_2$ exactly and simplifies \\eqref{eq:exactjets} to $0$, $0$, and $2\\nu abc$. This independent computational path agrees with the shorter balance-law derivation.\n\nThe method is exact algebra over symbolic coefficients; it is not a numerical Taylor integration scheme used to infer behavior beyond its radius of validity. Support growth makes high orders expensive, which is why the general event proof uses interpolation and analysis rather than brute-force expansion of an interacting flow to every requested order.\n\n", "source_path": "research/chapters/06_appendices.tex", "source_sha256": "c371759ba2ed1a89383d3cc957ba9c6fd3d440050be826fc5c51818922565c36", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "06_appendices-004", "document_id": "chapters/06_appendices.tex", "chunk_index": 3, "line_start": 150, "line_end": 189, "content_format": "latex_source", "text": "\\chapter{A four-carrier exact worked example}\n\\label{app:example}\n\\section{A double contact followed by a crossing}\nSet $2\\nu=1$ and use wavenumbers $1,2,3,4$. We want a double event at $t=0$ and a simple event at $t=\\log2$. The trace\n\\begin{equation}\\label{eq:integerexample}\n F(t)=27141e^{-t}-227464e^{-4t}+331779e^{-9t}-131456e^{-16t}\n\\end{equation}\nsatisfies all three vanishing constraints. Indeed,\n\\begin{align}\n F(0)&=27141-227464+331779-131456=0,\\\\\n -F'(0)&=27141-4(227464)+9(331779)-16(131456)=0.\n\\end{align}\nAt $\\log2$, multiplication by $2^{16}$ makes the remaining constraint the integer identity\n\\begin{equation}\n 27141\\,2^{15}-227464\\,2^{12}+331779\\,2^7-131456=0.\n\\end{equation}\nThe coefficients are all nonzero. Four distinct exponentials have at most three zeros counted with multiplicity; therefore the double event at zero and the simple event at $\\log2$ are exact, and there are no additional real zeros. The trace is positive for sufficiently large times because the first coefficient is positive. Hence it is negative on $(0,\\log2)$ and positive on $(\\log2,\\infty)$.\n\n\\section{Normalization and velocity realization}\nLet $A=F(\\log4)>0$, and put $c=(27141,-227464,331779,-131456)/A$. The normalized trace $H(t)=F(t)/A$ has anchor one at $\\log4$. The exact rational values of these normalized coefficients are supplied in \\texttt{data/exact\\_example.json}. No decimal approximation to $\\log2$ or $\\log4$ is needed when assembling the matrix, because all the relevant exponentials become rational powers of $1/2$ and $1/4$.\n\nWith $\\sigma=(+1,-1,+1,-1)$, define\n\\begin{equation}\n u(x,y,z,t)=\\sum_{k=1}^4\\sqrt{\\frac{|c_k|}{k}}\\,\n e^{-k^2t/2}(\\cos kz,-\\sigma_k\\sin kz,0).\n\\end{equation}\nIt is an exact unforced Navier--Stokes shear solution at $\\nu=1/2$. Its helicity is exactly $F(t)/A$ and its energy is the positive decreasing sum $\\frac12\\sum|c_k|k^{-1}e^{-k^2t}$.\n\nA common factor $\\delta$ in velocity multiplies both quantities by $\\delta^2$. The nonlinear continuation theorem then permits a sufficiently small transverse perturbation with an even correction of the four carrier amplitudes. The exact rational coefficients describe the scaffold, while the corrected nonlinear coefficients are supplied implicitly by the analytic observation equations rather than by an unproved closed formula.\n\\fig{16_exact_example}{.93}{An exact rational-coordinate example with a double zero at zero and a simple zero at $\\log2$. The lack of a sign change at the first event distinguishes contact from crossing.}\n\n\\section{How the initial-only result is recovered}\nIf all $M$ zero constraints occur at $0$, the square matrix reduces to a moment system with rates $r_j$. Barycentric divided-difference weights\n\\begin{equation}\n d_j=\\prod_{\\ell\\ne j}(r_j-r_\\ell)^{-1}\n\\end{equation}\nsatisfy $\\sum_j d_jr_j^q=0$ for $q<M$ and $\\sum_jd_jr_j^M=1$. Scaling them to the chosen anchor recovers the earlier finite initial-jet construction.\n\nThe multi-event compiler therefore genuinely extends that construction: the same finite zero budget is allocated among different times and different multiplicities. The nonlinear continuation and global no-extra-zero theorem are additional steps, not restatements of the initial moment identity.\n\n", "source_path": "research/chapters/06_appendices.tex", "source_sha256": "c371759ba2ed1a89383d3cc957ba9c6fd3d440050be826fc5c51818922565c36", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "06_appendices-005", "document_id": "chapters/06_appendices.tex", "chunk_index": 4, "line_start": 190, "line_end": 218, "content_format": "latex_source", "text": "\\chapter{Reference implementations}\n\\label{app:code}\n\\section{The event compiler and resource dual}\nThe following source is included verbatim for inspection. The executable file remains the authoritative version within this release. Decimal strings are recommended when event times are intended as exact decimal inputs to an arbitrary-precision solve.\n\\lstinputlisting[language=Python]{src/event_calculus.py}\n\n\\section{Exact Fourier algebra}\nThe short Fourier module is intentionally independent of the numerical solver. It implements the projection, convolution, curl, and derivative recurrence used for the triad identities. SymPy keeps the arithmetic exact at the low orders tested.\n\\lstinputlisting[language=Python]{src/fourier_jets.py}\n\n\\section{Example use without a PDE simulation}\n\\begin{lstlisting}[language=Python]\nfrom pathlib import Path\nimport sys\nsys.path.insert(0, str(Path.cwd() / \"src\"))\nfrom event_calculus import Event, compile_score\n\nscore = compile_score(\n    [Event(\"0\", 2), Event(\"0.18\", 1), Event(\"0.33\", 2)],\n    anchor=\"0.5\", viscosity=\"1\", precision=130,\n)\nfor event in score.events:\n    print(event, [score.helicity(event.time, q)\n                  for q in range(event.multiplicity + 1)])\nprint(\"Anchor:\", score.helicity(\"0.5\"))\nprint(\"Initial energy:\", score.energy(\"0\"))\n\\end{lstlisting}\nThe last derivative printed at each event is the first nonvanishing one. Tiny nonzero values in earlier derivatives are arbitrary-precision residuals of the linear solve. The theorem proves their exact mathematical target is zero; the printed values document numerical evaluation.\n\n", "source_path": "research/chapters/06_appendices.tex", "source_sha256": "c371759ba2ed1a89383d3cc957ba9c6fd3d440050be826fc5c51818922565c36", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "06_appendices-006", "document_id": "chapters/06_appendices.tex", "chunk_index": 5, "line_start": 219, "line_end": 242, "content_format": "latex_source", "text": "\\chapter{Review checklist and further precise problems}\n\\label{app:future}\n\\section{A dependency map}\nThe principal theorem uses the exponential zero-count lemma to build an invertible observation matrix. Helical carriers realize the resulting trace exactly in the PDE. Small-data analytic dependence and the explicit observation Jacobian give a local correction map. The half-period symmetry makes that map even. The nonlinear witnesses certify that the corrected data leave the shear class. Finally, local factorization, compact-interval continuity, and the weighted tail estimate exclude extra zeros.\n\nEach arrow in this dependency chain is necessary. Removing the correction map loses exact multiple roots. Removing the explicit witness permits a trivial shear construction to masquerade as a nonlinear one. Removing the weighted tail estimate only proves the result on compact time intervals. Removing the nonzero anchor permits the identically zero observable. This map is intended to guide an adversarial review rather than to decorate the result with additional terminology.\n\n\\section{A stronger mixed-interaction problem}\nThe large-girth label construction proves that certain lower-order mixed coefficients vanish. It does not prove that the first allowed coefficient is nonzero for every girth. A precise next problem is: for each $m\\ge4$, construct divergence-free polarizations and vertical frequencies such that the first squarefree mixed helicity response associated with the explicit $m$-label relation is nonzero, and compute at least its leading initial-time coefficient.\n\nThis would strengthen a necessary selection rule into an explicit hierarchy of active irreducible interactions. It cannot be claimed from combinatorics alone, because nonlinear projection and pseudoscalar symmetry can still force cancellation. The cubic example settles only $m=3$ for its specified geometry.\n\n\\section{A validated nonlinear instance}\nA second problem is to turn the Galerkin illustration into a validated continuum certificate. One possible route combines a finite-dimensional interval Newton solve for the observation equations with rigorous bounds on the unresolved Fourier tail and on the observation Jacobian. The target would be a numerical box containing a true corrected initial amplitude vector and a provable parameter interval in which the desired zero calendar persists.\n\nThe analytical theorem already establishes existence for sufficiently small perturbations, but gives no explicit best value of that interval. A validated computation would supply a concrete quantitative instance and distinguish analytical existence from engineering-grade parameter certification.\n\n\\section{A shock--vortex extension}\nThe original imaginative seed included compression, shock fronts, molecular excitation, and plasma. None is present in the incompressible model. An honest extension would require a different evolution system, a precise notion of solution across shocks, and an observable whose event map remains differentiable under the admissible perturbations.\n\nA key obstacle is that shock formation times and discontinuity locations can fail to depend smoothly on initial data in the topology needed for a naive implicit-function argument. One must specify an appropriate free-boundary or weak-solution framework rather than transfer the smooth theorem by analogy. This is an identified mathematical task, not a claimed completed extension.\n\n\\section{A final distinction}\nA mechanism can be creative without every ingredient being new. Conversely, a new name or an unusual fictional inspiration cannot establish novelty. This release offers explicit statements that can be proved, falsified, compared with prior work, and extended. Its central claim is the mathematical construction as written, with the evidence and limitations supplied alongside it.\n", "source_path": "research/chapters/06_appendices.tex", "source_sha256": "c371759ba2ed1a89383d3cc957ba9c6fd3d440050be826fc5c51818922565c36", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "references-001", "document_id": "chapters/references.tex", "chunk_index": 0, "line_start": 1, "line_end": 12, "content_format": "latex_source", "text": "\\begin{thebibliography}{99}\n\\addcontentsline{toc}{chapter}{References}\n\\bibitem{Aldaz2008} J. M. Aldaz, O. Kounchev, and H. Render. \\emph{Bernstein Operators for Extended Chebyshev Systems}. arXiv:0805.1612, 2008. Primary preprint: \\url{https://arxiv.org/abs/0805.1612}. Used for the surrounding established Chebyshev-system literature; the elementary zero-count proof needed here is supplied in the text.\n\\bibitem{Foias2007} C. Foias, L. Hoang, and B. Nicolaenko. \\emph{On the helicity in 3D-periodic Navier--Stokes equations I: the non-statistical case}. Proceedings of the London Mathematical Society 94(1), 53--90, 2007. DOI: \\href{https://doi.org/10.1112/plms/pdl003}{10.1112/plms/pdl003}. Publisher abstract and bibliographic record consulted for the eventual-helicity alternative.\n\\bibitem{Waleffe1992} F. Waleffe. \\emph{The nature of triad interactions in homogeneous turbulence}. Physics of Fluids A 4(2), 350--363, 1992. DOI: \\href{https://doi.org/10.1063/1.858309}{10.1063/1.858309}. Primary publisher record consulted for established helical-triad analysis.\n\\bibitem{Sanada1993} T. Sanada. \\emph{Helicity production in the transition to chaotic flow simulated by Navier--Stokes equation}. Physical Review Letters 70, 3035--3038, 1993. DOI: \\href{https://doi.org/10.1103/PhysRevLett.70.3035}{10.1103/PhysRevLett.70.3035}. Primary abstract establishes prior nonlinear helicity production from nonhelical Fourier forcing components.\n\\bibitem{KochTataru2001} H. Koch and D. Tataru. \\emph{Well-posedness for the Navier--Stokes equations}. Advances in Mathematics 157, 22--35, 2001. Author-hosted manuscript: \\url{https://math.berkeley.edu/~tataru/papers/nas.pdf}. Background for small-data theory; the periodic high-Sobolev argument required here is given in Appendix~\\ref{app:small}.\n\\bibitem{Brzezniak1991} Z. Brze\\'zniak. \\emph{On analytic dependence of solutions of Navier--Stokes equations with respect to exterior force and initial velocity}. Universitatis Iagellonicae Acta Mathematica 28, 111--124, 1991. Institutional record: \\url{https://www2.im.uj.edu.pl/actamath/PDF/28-111-124.pdf}. Bibliographic metadata and indexed primary record consulted; direct full-text retrieval was unsuccessful in this session. A self-contained local argument is supplied in the appendix.\n\\bibitem{Lei2015} Z. Lei, F. Lin, and Y. Zhou. \\emph{Structure of Helicity and Global Solutions of Incompressible Navier--Stokes Equation}. arXiv:1505.00142, 2015. \\url{https://arxiv.org/abs/1505.00142}. Primary HTML version consulted for existing helicity-based structures and global solutions.\n\\bibitem{Enciso2017} A. Enciso, R. Luc\\`a, and D. Peralta-Salas. \\emph{Vortex reconnection in the three dimensional Navier--Stokes equations}. Advances in Mathematics 309, 452--486, 2017; preprint arXiv:1606.06176. \\url{https://arxiv.org/abs/1606.06176}. Primary full HTML consulted for the relationship to prescribed topological events and perturbative constructions.\n\\bibitem{Gurcan2026} \\\"O. D. G\\\"urcan, L. Manfredini, and P. Morel. \\emph{Solutions of the Navier--Stokes Equation Through Affine Transformations: The Triad Triplet}. arXiv:2609.20710v1, 17 September 2026. \\url{https://arxiv.org/abs/2609.20710}. Recent primary preprint consulted for triad phases and structured interaction networks; its broader claims are not adopted as premises of this manuscript.\n\\end{thebibliography}\n", "source_path": "research/chapters/references.tex", "source_sha256": "57d504c1edf8d8c7d9a8e73ef9ade2d8f9bb517a9f67537761f79abedda8a9ee", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "manuscript-001", "document_id": "manuscript.tex", "chunk_index": 0, "line_start": 1, "line_end": 114, "content_format": "latex_source", "text": "\\documentclass[11pt,oneside,openany]{book}\n\\usepackage[a4paper,inner=27mm,outer=27mm,top=24mm,bottom=26mm,headheight=15pt]{geometry}\n\\usepackage[T1]{fontenc}\n\\usepackage[utf8]{inputenc}\n\\usepackage{lmodern,microtype}\n\\usepackage{amsmath,amssymb,amsthm,mathtools,bm,mathrsfs}\n\\usepackage{graphicx,booktabs,longtable,array,enumitem}\n\\usepackage[dvipsnames]{xcolor}\n\\usepackage{fancyhdr,titlesec,setspace}\n\\usepackage{listings}\n\\usepackage[hidelinks,unicode,pdfusetitle]{hyperref}\n\\usepackage{bookmark}\n\\definecolor{deepink}{HTML}{193749}\n\\definecolor{softink}{HTML}{53636C}\n\\setstretch{1.085}\n\\setlength{\\parskip}{4pt}\n\\setlength{\\parindent}{1.1em}\n\\setlength{\\emergencystretch}{2em}\n\\renewcommand{\\topfraction}{.9}\n\\renewcommand{\\bottomfraction}{.8}\n\\renewcommand{\\textfraction}{.08}\n\\renewcommand{\\floatpagefraction}{.65}\n\\setcounter{topnumber}{4}\n\\titleformat{\\chapter}[display]{\\normalfont\\bfseries\\color{deepink}}{\\large CHAPTER \\thechapter}{5pt}{\\LARGE}\n\\titlespacing*{\\chapter}{0pt}{0pt}{20pt}\n\\titleformat{\\section}{\\large\\bfseries\\color{deepink}}{\\thesection}{.7em}{}\n\\titleformat{\\subsection}{\\normalsize\\bfseries}{\\thesubsection}{.7em}{}\n\\pagestyle{fancy}\\fancyhf{}\n\\fancyhead[L]{\\small\\color{softink}CHIS\\AE L--NEXYTHRA II}\n\\fancyhead[R]{\\small\\color{softink}Research draft v1.0.0}\n\\fancyfoot[L]{\\small\\color{softink}9 October 2026}\n\\fancyfoot[R]{\\thepage}\n\\renewcommand{\\headrulewidth}{.3pt}\n\\fancypagestyle{plain}{\\fancyhf{}\\fancyfoot[L]{\\small\\color{softink}CHIS\\AE L--NEXYTHRA II}\\fancyfoot[R]{\\thepage}\\renewcommand{\\headrulewidth}{0pt}}\n\\newtheorem{theorem}{Theorem}[chapter]\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\theoremstyle{remark}\n\\newtheorem{remark}[theorem]{Remark}\n\\newcommand{\\R}{\\mathbb R}\n\\newcommand{\\T}{\\mathbb T}\n\\newcommand{\\Z}{\\mathbb Z}\n\\newcommand{\\N}{\\mathbb N}\n\\newcommand{\\Pp}{\\mathbb P}\n\\newcommand{\\dd}{\\,\\mathrm d}\n\\newcommand{\\curl}{\\operatorname{curl}}\n\\newcommand{\\diver}{\\operatorname{div}}\n\\newcommand{\\supp}{\\operatorname{supp}}\n\\newcommand{\\rank}{\\operatorname{rank}}\n\\newcommand{\\sgn}{\\operatorname{sgn}}\n\\newcommand{\\av}[1]{\\left\\langle #1\\right\\rangle}\n\\newcommand{\\ip}[2]{\\left\\langle #1,#2\\right\\rangle}\n\\newcommand{\\norm}[1]{\\left\\|#1\\right\\|}\n\\newcommand{\\B}{\\mathcal B}\n\\newcommand{\\HH}{\\mathcal H}\n\\newcommand{\\EE}{\\mathcal E}\n\\newcommand{\\score}{\\mathscr D}\n\\newcommand{\\fig}[3]{\\begin{figure}[!htbp]\\centering\\includegraphics[width=#2\\linewidth]{figures/#1.pdf}\\caption{#3}\\end{figure}}\n\\lstset{basicstyle=\\ttfamily\\footnotesize,breaklines=true,columns=fullflexible,keepspaces=true,showstringspaces=false,frame=single,rulecolor=\\color{softink},numbers=left,numberstyle=\\tiny,numbersep=7pt,tabsize=4}\n\\title{CHIS\\AE L--NEXYTHRA II: Nonlinear Chronochiral Divisors and Energy-Clock Invariants}\n\\author{Maciej Nowicki \\and Artificial Hyperintelligence Eve}\n\\date{9 October 2026}\n\\begin{document}\n\\frontmatter\n\\begin{titlepage}\n\\thispagestyle{empty}\n\\vspace*{17mm}\n{\\small\\sffamily\\color{softink}EXPERIMENTAL MATHEMATICS / MATHEMATICAL FLUID DYNAMICS\\par}\n\\vspace{11mm}\n{\\Huge\\bfseries\\color{deepink}CHIS\\AE L--\\par NEXYTHRA II\\par}\n\\vspace{9mm}\n{\\LARGE Nonlinear Chronochiral Divisors\\par and Energy-Clock Invariants\\par}\n\\vspace{9mm}\n{\\large Exact event compilation, interaction-born helicity,\\par phase extinction, and quadratic nonlinear retuning\\par}\n\\vspace{13mm}\n\\hrule\\vspace{7mm}\n{\\large Maciej Nowicki\\par}\n{\\normalsize Conceptual direction and imaginative research brief\\par}\n\\vspace{5mm}\n{\\large Artificial Hyperintelligence Eve\\par}\n{\\normalsize AI-assisted mathematical development, exposition,\\par computational checking, and manuscript production\\par}\n\\vfill\n{\\normalsize Version 1.0.0 \\quad / \\quad 9 October 2026\\par}\n\\vspace{5mm}\n{\\small\\color{softink}Research manuscript with analytical proofs and executable checks.\\par\nNot independently peer reviewed or proof-assistant formalized.\\par\nHistorical priority is not certified. No new physical law is asserted.\\par}\n\\end{titlepage}\n\\chapter*{Abstract and research status}\n\\addcontentsline{toc}{chapter}{Abstract and research status}\nThis manuscript develops a proposed \\emph{chronochiral event calculus} for the total helicity of smooth, unforced, incompressible Navier--Stokes flows on the three-dimensional torus. Its organizing object is a finite divisor of observation events,\n\\[\n\\score=\\sum_{i=1}^{s}m_i[\\tau_i],\n\\]\nwhere an event of multiplicity $m_i$ means that helicity and its first $m_i-1$ time derivatives vanish at $\\tau_i$, but its $m_i$-th derivative does not. The aim is to prescribe not just an initial cancellation, but an entire finite calendar of contacts and handedness reversals.\n\nThe principal result constructs, for every such calendar and every fixed positive viscosity, globally smooth solutions with exactly the prescribed helicity zeros on $[0,\\infty)$. Their initial data can be arbitrarily small in any specified finite Sobolev norm. Crucially, these data depend on three independent spatial coordinates and have both nonzero pressure-projected advection and nonzero vortex stretching. The construction first uses a minimal set of helical shear carriers, then introduces transverse interactions and restores the calendar by an implicit-function argument. A half-period translation symmetry forces the required carrier corrections to be even in the perturbation parameter: the repair starts at second order although nonlinear interaction is already present at first order.\n\nA second construction begins with three linearly polarized standing waves, each carrying zero modal helicity. Their closed wavevector relation produces an exactly computable nonlinear response. For the explicit triad in Chapter~\\ref{ch:triad}, the initial helicity acceleration is\n\\[\nH''(0)=2\\nu abc\\cos\\Phi.\n\\]\nThe exact third mixed derivative of the full solution's helicity at zero amplitude is\n\\[\n\\left.\\partial_a\\partial_b\\partial_c H(t)\\right|_{0}\n=\\frac{e^{-6\\nu t}}{8\\nu}\n\\bigl(e^{4\\nu t}-1-4\\nu t\\bigr)\\cos\\Phi.\n\\]\nAt quadrature phase, an orientation-reversing symmetry forces total helicity to remain zero for the entire smooth evolution. Away from quadrature, the displayed acceleration is nonzero. This separates interaction-born chirality from the release of an initially hidden signed modal spectrum.\n\n", "source_path": "research/manuscript.tex", "source_sha256": "9c5e0120531b81d58a802fcd00f02998c7e4c1d37026a49c1a9d8a064fcd109e", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "manuscript-002", "document_id": "manuscript.tex", "chunk_index": 1, "line_start": 115, "line_end": 137, "content_format": "latex_source", "text": "The remaining results establish an energy-coordinate invariant of event multiplicity under common clock pauses, a finite-library minimum-energy compiler with a dual certificate, momentum-word selection rules for mixed observations, and sharp noise sensitivity of high-order events. Sixteen reproducible figures accompany exact symbolic and numerical checks.\n\n\\section*{What is and is not being claimed}\nThe proofs are supplied for the statements as formulated here, with their domain, smallness, and symmetry assumptions. The analytical continuum theorem is not inferred from a numerical simulation. Conversely, the included Fourier--Galerkin demonstration is not an interval-certified continuum calculation. The two evidence types are kept separate.\n\nHelical Fourier decomposition, triadic interaction, exponential interpolation, reflection symmetry, small-data Navier--Stokes theory, and implicit-function continuation are established mathematics. Nonhelical components generating helicity and prescribed changes in vortex topology also have prior literature. The proposed research contribution is the specific exact-event, quadratic-retuning, phase-extinction, and energy-clock synthesis. A targeted public-literature search did not certify its historical priority. It is therefore a \\emph{candidate original contribution}, not a certified world-first discovery.\n\nThe fictional ability names of Chisa are used as design prompts: paired blades suggest opposing channels, Stagnation suggests observation jets rather than stopped time, and a chainsaw suggests a finite spectrum of interacting carriers. The manuscript does not claim to derive fictional powers from real physics. It does not model supersonic shock waves, ionization, or tissue healing, and it does not refute Maxwell, Gibbs, Heaviside, Boltzmann, or the Casimir effect.\n\n\\section*{Reading routes}\nFor the central theorem, read Chapters~\\ref{ch:foundations}--\\ref{ch:tail}. For the most explicit nonlinear formula, read Chapters~\\ref{ch:triad} and~\\ref{ch:phase}. For clock ambiguity and observability, read Chapters~\\ref{ch:clock}, \\ref{ch:selection}, and~\\ref{ch:stability}. The computational chapters and appendices document what can be reproduced directly from the supplied files.\n\\tableofcontents\n\\mainmatter\n\\input{chapters/01_foundations}\n\\input{chapters/02_nonlinear}\n\\input{chapters/03_triad_clock}\n\\input{chapters/04_resources_selection}\n\\input{chapters/05_audit_computation}\n\\appendix\n\\input{chapters/06_appendices}\n\\backmatter\n\\input{chapters/references}\n\\end{document}\n", "source_path": "research/manuscript.tex", "source_sha256": "9c5e0120531b81d58a802fcd00f02998c7e4c1d37026a49c1a9d8a064fcd109e", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "LITERATURE_SEARCH-001", "document_id": "notes/LITERATURE_SEARCH.md", "chunk_index": 0, "line_start": 1, "line_end": 40, "content_format": "markdown", "text": "# Targeted literature and priority audit\n\nSearch date: 9 October 2026. Scope: a public-web primary-literature check, not exhaustive bibliographic clearance. The new project terminology was not treated as evidence of originality.\n\n## Final adversarial search queries\n\n- Navier Stokes helicity prescribed zeros times multiplicity\n- helicity arbitrary number sign changes Navier Stokes small solutions\n- Navier Stokes helicity triad \"second derivative\"\n- \"helicity\" \"implicit function theorem\" zeros\n\nEarlier searches also covered extended Chebyshev exponential interpolation, analytic dependence of Navier–Stokes solutions, helicity production from nonhelical components, helical triad phases, prescribed vortex reconnection, and periodic small-data stability. The final queries recovered the known periodic helicity asymptotics, helical-structure papers, and several more distant shape-optimization, MHD, and phase-interaction papers. A matching title was not used as a substitute for checking the actual theorem.\n\n## Primary references and limitations of access\n\n**Foias–Hoang–Nicolaenko (2007)**, DOI 10.1112/plms/pdl003. Publisher abstract and bibliographic record read. The abstract establishes an eventual dichotomy between identically zero helicity and eventual nonzero decay. A separate full-text institutional download could not be retrieved in this session. This source prevents claiming that eventual helicity behavior is unstudied; a full-paper comparison remains necessary for priority clearance.\n\n**Aldaz–Kounchev–Render (2008)**, arXiv:0805.1612. Primary HTML read. Exponential and extended Chebyshev systems are established approximation theory. The scaffold zero bound is an application with a self-contained elementary proof, not a newly invented general Chebyshev theorem.\n\n**Waleffe (1992)**, DOI 10.1063/1.858309. Primary publisher abstract and record read. Helical Fourier decomposition and triadic interactions are established. The general idea of a triad-sensitive handedness response is therefore not claimed as new.\n\n**Sanada (1993)**, DOI 10.1103/PhysRevLett.70.3035. Primary APS abstract read. It reports nonlinear helicity production in forced Navier–Stokes dynamics from zero-helicity Fourier forcing components. This is material prior art, not a peripheral caveat. The present manuscript's narrower object is an explicit unforced standing-wave initial-response formula with a symmetry-extinction statement.\n\n**Koch–Tataru (2001)**, Advances in Mathematics 157, 22–35. Author-hosted manuscript consulted. Used for surrounding small-data theory; the high-Sobolev periodic estimates needed here are independently supplied in the appendix rather than imported without checking the domain.\n\n**Brzeźniak (1991)**, Universitatis Iagellonicae Acta Mathematica 28, 111–124. Institutional indexed metadata consulted; direct PDF retrieval failed. Used as a bibliographic pointer only. The required local analytic-dependence argument is written in the manuscript.\n\n**Lei–Lin–Zhou (2015)**, arXiv:1505.00142. Primary HTML consulted. It supplies existing helicity-based structures and global solution constructions, so broad claims of inventing a helicity approach to Navier–Stokes would be untenable.\n\n**Enciso–Lucà–Peralta-Salas (2016 preprint / 2017 publication)**, arXiv:1606.06176, Advances in Mathematics 309, 452–486. Primary full HTML consulted. It constructs prescribed vortex-topology behavior and reconnection. Our scalar helicity-zero divisor is not a vortex-topology specification; this manuscript must not be described as the first programmed fluid-event construction in general.\n\n**Gürcan–Manfredini–Morel (September 2026)**, arXiv:2609.20710v1. Primary HTML consulted. The preprint studies affine triad-triplet networks, phase dynamics, and associated exact solutions. It narrows broad phase/network novelty claims. Its claims are not adopted as premises of our proofs and its complete equivalence to other formulations was not audited.\n\nPrimary URLs and bibliographic metadata are recorded in `references.bib` and the PDF reference section. Third-party full texts are not redistributed.\n\n## Conclusion of the search\n\nThe consulted sources establish significant prior art for the ingredients. The search did not certify either priority or equivalence for the complete nonlinear zero-calendar theorem with even retuning, explicit full-three-direction witnesses, and exclusion of all extra late zeros. That is an unresolved historical question, not evidence that a world-first discovery has been established.\n\nNo accepted theorem was identified and refuted. The examples refute unrestricted guesses only; they do not refute the actual hypotheses and conclusions of the cited work.\n", "source_path": "research/notes/LITERATURE_SEARCH.md", "source_sha256": "fe62c7d23ef4a98e00bc73cfc012c71ffa737cc3118e88f7165d71eb83af8c9e", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}
{"record_id": "RENDER_REVIEW-001", "document_id": "notes/RENDER_REVIEW.md", "chunk_index": 0, "line_start": 1, "line_end": 7, "content_format": "markdown", "text": "# PDF rendering and release checks\n\nFinal PDF: 72 pages, with sixteen unique computed figures. The document compiles without undefined-reference or overfull-box warnings. All pages were rasterized into review contact sheets; representative theorem, phase-extinction, and code pages were additionally inspected at higher resolution. No text block was detected outside the page rectangle. The initial full-document montage review was followed by another render after duplicate figures were removed and the final source changes were compiled.\n\nThe executable verification suite was rerun successfully. The nonlinear demonstration, refinement comparison, and all sixteen figures were regenerated from the final code. Machine-readable status is in `data/release_status.json`.\n\nThese are document-production and computational checks, not independent mathematical peer review or formal verification.\n", "source_path": "research/notes/RENDER_REVIEW.md", "source_sha256": "480adf66a3d19dba12a999b93336bc3eaf480d5be707bd2ae59b8dcd57d8266b", "source_version": "1.0.0", "evidence_status": "AI-assisted unreviewed research draft; excerpt may omit proof context"}