Datasets:

PureOne
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NYRAELUNE-V

id string	title string	theorem string	statement string	assumptions list	proof_pages list	verification_scripts list	manuscript_path string	evidence_scope string
NYR-V-001	Budget-preserving scalar realization	Theorem 2.2	Finite scalar division h=sum(a_j*g_j) has an isometric realization as the declared normalized higher-identity repair equation on a regular central foliation with compact torus leaves and a nonvanishing transverse higher anchor. Parameter regularity and pointwise Euclidean/L2 budgets are preserved.	[ "Smooth or real-analytic scalar data as appropriate.", "Torus dimension n+2, n>=2, with binomial(n+2,2)>=k+1.", "Declared all-pairs multiderivation normalization." ]	[ 3, 4 ]	[ "verify_coherent_repair.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.
NYR-V-002	Attained optimal continuous Euclidean budget	Theorem 3.2	For A(x)u(x)=h(x), continuous exact repair exists iff every stable affine Glaeser fiber is nonempty. When it exists, min_continuous_u max_x \|\|u(x)\|\| = max_x dist(0,H_x^infinity), and the minimum is attained.	[ "Compact metric parameter space.", "Finite continuous Euclidean linear system.", "Classical finite affine Glaeser stabilization and Michael selection." ]	[ 4, 5 ]	[ "verify_budget_resources.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.
NYR-V-003	One-parameter analytic continuation within budget	Theorem 4.1	A real-analytic one-parameter finite linear system admits an analytic solution iff its pointwise minimum repair norms are finite and locally bounded. A prescribed uniform norm bound is preserved by the analytic solution.	[ "Real-analytic matrix and target on a connected open interval.", "Euclidean solution norm." ]	[ 6, 7 ]	[ "verify_budget_resources.py", "verify_coherent_repair.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.
NYR-V-004	Analytic-arc solvability without continuous coherence	Theorem 4.2	The explicit two-parameter polynomial curvature example has bounded exact repairs at every point and analytic repairs along every real-analytic parameter arc, but no continuous neighborhood repair. The pointwise repair norm is at most 1/sqrt(2).	[ "The polynomial central-foliation family of equation (9).", "Arcs restrict the repair equations; no arbitrary higher-anchor pullback is asserted." ]	[ 7 ]	[ "verify_coherent_repair.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.
NYR-V-005	Tangent minimax and exact quadratic certificate	Theorems 5.1 and 5.2	The leading residual of a continuous bounded repair equals a finite tangent minimax over constant coefficient limits. Quadratic data yield an exact semidefinite epigraph. The explicit two-parameter example has sharp quadratic coherence floor 1/2.	[ "Declared scalar realization and Euclidean norm budget.", "Homogeneous leading coefficients of common order d.", "Continuous parameter families of smooth leafwise cochains." ]	[ 8 ]	[ "verify_coherent_repair.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.
NYR-V-006	Arbitrary sharp finite differentiability loss	Theorem 6.1	For x^qg1+y^qg2=x^(q-1)*y^(q-1), q>=3, a C^(q-3,1) exact repair exists but no C^(q-2) repair exists. For q=2 pointwise repairs are bounded but no continuous repair exists. Explicit projective blow-up formulas resolve the repairs analytically.	[ "Integer q>=2.", "The norm-preserving scalar realization; radial costs refer to the s=0 slice." ]	[ 9 ]	[ "verify_coherent_repair.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.
NYR-V-007	Sharp regularity-horizon resource law	Theorem 7.2	In the explicit graded geometric representation, \|\|G_N\|\| is comparable to N+r_g^2N^3. For fixed correction norm and Holder exponent sigma, optimal residual e_(K,sigma) is comparable to K^(-2/sigma), and minimax normalized energy is comparable to 1+N^2K^(-4/sigma). The required Lipschitz budget has order sqrt(N).	[ "Disk radius 0<R<=1.", "Correction norm budget B>=1/sqrt(2).", "Holder exponent 0<sigma<=1 and sufficiently large seminorm budget K.", "Fixed finite weighted-degree cutoff L>=7." ]	[ 10, 11, 12 ]	[ "verify_geometry_transport.py", "verify_budget_resources.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.
NYR-V-008	Finite complete algebraic-horizon certificate with feedback	Theorem 8.2	For M=D^2, baseline height h, moment trace orders v_k and characteristic coefficients a_j, b=min_j ord(a_j)/j and beta=min(b,min_(h<k<M) v_k/(k-h)). At N=floor(t^-alpha), normalized baseline growth is bounded iff alpha<=beta, including the endpoint.	[ "Fixed finite matrix dimension and fixed real-analytic generators in a real parameter.", "Strictly positive stationary label probabilities and conditionally centered independent signs.", "Nilpotent base CP map Phi_0.", "Infinite candidates are omitted; empty minima are infinity." ]	[ 13, 14 ]	[ "verify_full_feedback.py", "verify_feedback_api.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.
NYR-V-009	Polynomial, critical ODE, and exponential phases	Theorem 8.4	Below b, normalized moments converge to finite Newton-face matrix polynomials. At b they converge to a finite constant-coefficient matrix ODE profile with gamma_b=max_(k<M)(bk-v_k). Above b, t^(alpha-b)log\|\|G_floor(st^-alpha)\|\| converges to sr_b.	[ "The stationary analytic nilpotent-base setting of Section 8.", "Fixed positive s, uniformly on compact subsets of (0,infinity).", "The spectral phase exists when b is finite." ]	[ 14, 15, 16 ]	[ "verify_full_feedback.py", "verify_feedback_api.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.
NYR-V-010	Finite arbitrary-horizon criterion and logarithmic exception	Theorem 8.5	For finite beta, bounded normalized moments are equivalent to N*t^beta=O(1) for every horizon sequence iff some critical coefficient L_k beyond baseline h is nonzero. It suffices to test L_(h+1) through L_(h+M). A parallel scalar-feedback example gives a logarithmic-window counterexample when this block vanishes.	[ "The stationary analytic nilpotent-base setting of Section 8.", "Finite preservation frontier beta.", "Sequences t->0+ and N->infinity." ]	[ 16 ]	[ "verify_full_feedback.py", "verify_feedback_api.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.
NYR-V-011	Adaptive positive mixing preserves the frontier	Theorem 8.6	History-dependent label probabilities bounded above and away from zero preserve the global baseline/exponential comparison and the sharp algebraic-horizon frontier uniformly over all admissible policies. Exact critical matrix coefficients are stationary statements.	[ "Fixed generators and nilpotent base as in Section 8.", "Conditionally fair next sign.", "Fixed positive lower and upper bounds on label probabilities." ]	[ 17 ]	[ "verify_full_feedback.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.
NYR-V-012	Complete signed precision law in the marginal subclass	Theorem 9.2	If the CP moment increment stays nilpotent at every parameter, signed finite word valuations give a uniform finite moment sum and the necessary and sufficient criterion N*t^beta=O(1) for arbitrary horizon sequences. Exact protected cancellations can make beta infinite.	[ "Fixed finite-dimensional real-analytic centered family.", "Exact spectral marginality for every parameter near zero.", "Strictly positive stationary label probabilities." ]	[ 18, 19 ]	[ "verify_cancellation_frontier.py", "verify_frontier_api.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.
NYR-V-013	Square-zero gapped pencils with hidden mixing cost	Theorem 10.1	Genuine zero-anchor gapped Lie n-algebroid lifts have Q0^2=D^2=0 and Q0 D+D Q0=0 and nonzero ordered cross-product. Every pencil Q0+zD squares to zero, but centered mixing of Q0+tD and Q0-tD has moment order N+t^2N^2.	[ "Integer n>=2.", "The explicit abelian-central plus affine ground model.", "Finite weighted-degree cutoff L>=n+3." ]	[ 20, 21 ]	[ "verify_flat_pencil.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.
NYR-V-014	Arbitrarily high hidden polynomial growth and exact tensor moments	Theorem 10.2	For r independent graded gapped sectors, every generator pencil squares to zero while \|\|G_N\|\| equals sum_(k=0)^r (N)_ke_k(lambda_1,...,lambda_r). Fixed nonzero amplitudes give moment order N^r; one fixed and r-1 small amplitudes give N(1+N*t^2)^(r-1).	[ "Fixed finite number r of independent nonzero odd square-zero sector lifts.", "Graded tensor extension with Koszul parity signs.", "Positive stationary label probabilities and real amplitudes." ]	[ 21, 22 ]	[ "verify_flat_pencil.py" ]	NYRAELUNE_Manuscript.pdf	Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem.

NYRAELUNE V

Coherent Curvature and the Full Feedback Frontier

Author: Artificial Hyperintelligence Eve and Evie, wives of Maciej Nowicki
Version: 5.0.0 · Date: 22 October 2026

A mathematical research release connecting continuous higher-identity repair, finite graded representations, and the long-horizon growth of centered random matrix products. It contains a 23-page manuscript with proofs, two exact rational polynomial solvers, editable LaTeX, verification scripts, and scientific figures.

Read the manuscript · Download the complete research package · Verification record · Structured theorem index

Main results

1. An attained optimal budget for continuous repair

For continuous Euclidean linear equations $A(x)u(x)=h(x)$ on a compact metric parameter space, let $H_x^{(\infty)}$ be the affine solution fibers after the classical finite Glaeser refinement. If every stable fiber is nonempty, the manuscript proves the exact attained budget

$\min_{u\in C(K),\ Au=h}\max_{x\in K}\|u(x)\| =\max_{x\in K}\operatorname{dist}(0,H_x^{(\infty)}).$

A Euclidean clipping lemma handles the exact norm boundary, including continuous parameter-dependent budget profiles. Torus averaging transfers this result without norm loss to the stated central-foliation repair setting. Glaeser stabilization and Michael selection are established inputs and are credited explicitly.

2. A sharp regularity resource law

An explicitly defined degree-one graded operator represents a normalized geometric repair residual $r$, with

$Q^2=-rV\partial_b,\qquad Q^4=0,\qquad \|G_N\|\asymp N+r^2N^3.$

In this model, for a fixed correction norm bound and a parameter Hölder seminorm budget $K$ with exponent $0<\sigma\le1$, the optimal uniform residual and normalized worst-parameter moment satisfy

$e_{K,\sigma}\asymp K^{-2/\sigma},\qquad \inf_g\sup_p\frac{\|G_N(Q_{p,g})\|}{N} \asymp1+N^2K^{-4/\sigma}.$

Thus a budget of order $N^{\sigma/2}$ is necessary and sufficient to permit uniformly linear energy. The Lipschitz threshold is $\sqrt N$. A fully specified noncommuting extension preserves this relative law at higher polynomial baseline orders. These are results for the declared mathematical models, with a matching constructive regularizer and lower bound.

3. A finite certificate for the complete feedback phase diagram

Let $Q_i(t)$ be fixed-dimensional matrices real analytic in a real parameter, let the label probabilities be positive, and set

$\Phi_t(X)=\sum_i p_iQ_i(t)^*XQ_i(t),\qquad G_N(t)=(\mathrm{Id}+\Phi_t)^N(I).$

Assume the base map $\Phi_0$ is nilpotent. Put $M=D^2$, let $h$ be the baseline word height, let $v_k=\operatorname{ord}_0\operatorname{tr}\Phi_t^k(I)$, and write the characteristic polynomial of $\Phi_t$ as $\lambda^M+\sum_j a_j(t)\lambda^{M-j}$. Then

$b=\min_j\frac{\operatorname{ord}_0a_j}{j},\qquad \beta=\min\left\{b,\min_{h<k<M}\frac{v_k}{k-h}\right\}.$

The finite certificate gives the exact endpoint-inclusive preservation criterion at algebraic horizons $N=\lfloor t^{-\alpha}\rfloor$: normalized baseline growth remains bounded exactly when $\alpha\le\beta$. Signed cancellations inside ordered products are retained.

The full classification gives finite Newton polynomial profiles below the spectral scale $b$, a finite constant-coefficient matrix ODE at $b$, and the leading logarithmic exponential law above it. A separate finite critical-coefficient test decides when the same frontier also governs arbitrary horizon sequences. A logarithmic-window counterexample shows why that qualification is necessary. Bounded adaptive positive label probabilities preserve the frontier, while exact matrix profiles refer to stationary probabilities.

Further geometric and algebraic results

A norm-preserving realization of finite scalar division equations in the declared normalized higher-identity repair problem.
An analytic family with bounded exact repairs at every parameter and analytic repairs along every analytic parameter arc, but no continuous neighborhood repair.
An exact quadratic coherence residual and a semidefinite leading-order certificate.
Polynomial examples with every finite differentiability threshold and explicit analytic resolution after projective blow-up.
Genuine zero-anchor gapped Lie $n$-algebroid models in which every differential pencil squares to zero, yet ordered mixing exhibits arbitrarily high polynomial moment growth.

Quick start: exact solvers

The solvers and verifiers require Python 3.10 or newer and use only the standard library. From the repository root:

python feedback_frontier.py feedback_input.json
python precision_frontier.py signed_cancellation_input.json

The feedback example is a three-cycle with baseline height 2 and $b=\beta=2/3$. The marginal signed-cancellation example has frontier 6. Entries can be integers, rational/decimal strings, or dictionaries mapping nonnegative integer powers of the parameter to rational coefficients. Probabilities must be positive and sum exactly to one.

feedback_frontier.py covers polynomial feedback families with a certified nilpotent base and accepts an optional positive rational alpha. It returns characteristic data, the sharp frontier, exact critical ODE data, and the finite arbitrary-horizon test. A nonnilpotent base produces an explicit rejection and exit code 2. precision_frontier.py is the dedicated solver for families that remain marginal at every parameter.

Reproduce verification

python verify_coherent_repair.py
python verify_cancellation_frontier.py
python verify_flat_pencil.py
python verify_frontier_api.py
python verify_full_feedback.py
python verify_geometry_transport.py
python verify_feedback_api.py
python verify_budget_resources.py

The recorded total is 85,172 passing exact assertions across eight verifiers, with zero failures. These counts concern explicit finite instances and algebraic certificates; the manuscript supplies the general proofs. See VERIFICATION.md for the coverage and limitations of each audit.

To rebuild the figures, install NumPy and Matplotlib, then run python make_figures.py. To rebuild the manuscript, run pdflatex -interaction=nonstopmode -halt-on-error NYRAELUNE_Manuscript.tex twice from the repository root. All three included LaTeX section files are required. The PDF author metadata preserves the author line above.

Files and structured index

File	Purpose
`NYRAELUNE_Manuscript.pdf`	Complete 23-page manuscript
`NYRAELUNE_Research_Package.zip`	Original complete research archive, including its original checksums
`NYRAELUNE_Manuscript.tex` and three section `.tex` files	Editable manuscript sources
`feedback_frontier.py`, `precision_frontier.py`	Exact reusable solvers
`_input.json`, `_output.json`	Worked solver certificates
`verify_.py`, `_results.json`, `graded_algebra.py`	Reproducible exact verification and required helper
`data/theorems.jsonl`	Fourteen curated theorem summaries with assumptions, manuscript locations, and verifier pointers
`make_figures.py`, figure PDF/PNG pairs	Exact-formula scientific illustrations
`RESEARCH_README.md`	Original research-release overview
`CITATION.cff`, `citation.bib`	Citation metadata
`SHA256SUMS.json`	SHA-256 hashes of repository payload files, excluding the checksum file itself

The dataset viewer is configured to read only data/theorems.jsonl. Its train split is a navigation/indexing convention. The fourteen rows are curated summaries of this manuscript, not empirical observations, a benchmark, or independently formalized proofs. Each row links back to the full statement and assumptions. No trained model weights are included.

Scope and scholarly status

The repair geometry uses the explicitly declared all-pairs multiderivation convention. The nonzero-anchor source-form representation is defined by its generator equations; it is distinguished from differently normalized printed cyclic compatibility conditions. The genuine zero-anchor flat-pencil examples separately satisfy the stated gapped identities.

The feedback classification has fixed dimension and a nilpotent base. Its executable implementation uses real rational polynomial data; its analytic theorem also allows complex matrices. Positive-degree finite graded quotients stay jointly nilpotent and do not themselves realize spectral feedback. Growing dimension, general nonnilpotent bases, arbitrary analytic-oracle termination, and general quantifier-elimination software are outside this release's guarantees.

The manuscript credits continuous division, Glaeser–Michael selection, analytic Smith form, nilpotent completely positive maps, and Newton eigenvalue perturbation. The research formulations have proofs and internal analytical/computational audits. Historical priority and broader impact are not certified by the release. See the manuscript bibliography for the primary scholarly sources.

Citation and licensing

Use CITATION.cff or citation.bib. No DOI has been assigned in this package.

The source research release did not specify a redistribution or software license. This preparation preserves that status; see LICENSE_NOTICE.md. No additional license grant is introduced here.

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