PureOne/nacre-9-odd-cycle-spectral-gate

NACRE–9 // The Odd-Cycle Spectral Gate

Minimal Pairwise De-Resonance of Mixed Sobolev Resolvents

Project byline: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Version: 1.0 — 29 September 2026
Release type: Research preprint, proofs, source code, and reproducible certificates. Not independently refereed.
Suggested Hugging Face repository: PureOne/nacre-9-odd-cycle-spectral-gate (dataset repository type; research artifact, not an ML training dataset).

Start here: Full manuscript · Theorem ledger · Prior-art audit · Adversarial audit.

What was solved

This release formulates and gives a complete theorem-level treatment of a specific sparse-regularization design problem. Starting with a full-support mixed Sobolev inverse on a torus, it classifies which pairwise penalties remove the logarithmic singular-value factor, proves the sharp minimum number of penalties and all equality cases, and determines spectral orders uniformly as the coefficients vanish.

It does not solve a recognized major open conjecture. It does not prove lossless compression of the original operator. Novelty of the exact combination remains unconfirmed. The proposed operator family is newly selected for this investigation; a substantial independent physical application has not been established.

The operator family

Let (Q_i=I-\partial_i^2) on (L^2(\mathbb T^d)). For a simple graph (G), define

[ A_{\boldsymbol\delta,G}=\prod_{i=1}^dQ_i+ \sum_{{i,j}\in E(G)}\delta_{ij}(Q_iQ_j)^{d/2}, \qquad T_{\boldsymbol\delta,G}=A_{\boldsymbol\delta,G}^{-1}. ]

All coefficients lie in ([0,1]). Every Fourier mode remains present. For even (d), the operator is a positive local differential operator of order (2d). In odd dimensions the same spectral formulas use Fourier functional calculus; locality is not claimed.

For a common coefficient (\delta), define the critical polytope

[ F_G={x\ge0:\ \textstyle\sum_i x_i=1,\quad x_i+x_j\le2/d\ \text{for every edge}},\qquad \nu=\dim F_G. ]

For small (\varepsilon), uniformly in (\delta\in[0,1]),

[ N_{T_{\delta,G}}(\varepsilon)\asymp \varepsilon^{-1/2}(1+\log(1/\varepsilon))^\nu \bigl(1+\min{\log(1/\varepsilon),\log(1/\delta)}\bigr)^{d-1-\nu}. ]

Here (N_T(\varepsilon)=#{n:s_n(T)>\varepsilon}), (\log(1/0)=\infty), and (\asymp) denotes two-sided bounds, not a leading-constant equivalence. Constants can depend on the fixed graph and dimension, but not the coefficient.

The optimal ten-dimensional design

Take (G=C_3\sqcup C_7), with one-based edges

(1,2) (2,3) (3,1)
(4,5) (5,6) (6,7) (7,8) (8,9) (9,10) (10,4)

For each fixed (\delta>0),

[ s_n(T_0)\asymp n^{-2}(\log n)^{18},\qquad s_n(T_{\delta,G})\asymp_\delta n^{-2}. ]

Ten pairwise summands are necessary and sufficient within this exact penalty family. The only ten-edge equality types are (C_3\sqcup C_7) and (C_5\sqcup C_5). “Ten” does not count expanded derivatives, FLOPs, or arbitrary alternative regularizers. Each pairwise summand in dimension ten expands into 36 derivative-energy terms.

Further results

The log-counting exponent equals (d-1) if the graph has no fractional perfect matching. Otherwise it equals the number of bipartite components in the graph of edges that can carry positive fractional-perfect-matching weight. Nonbipartiteness of the original graph alone is insufficient.

For an optimal graph and independent coefficients, put (\Delta=\sum_e\log(1/\delta_e)). The uniform law simplifies to

[ N(\varepsilon)\asymp\varepsilon^{-1/2} [1+\min{\log(1/\varepsilon),\Delta}]^{d-1}. ]

Deleting any single penalty in an optimal design restores the entire baseline logarithmic exponent. Exact deletion-infeasibility certificates are included for all ten edges of the selected design.

For even (d\ge4) and any graph with an edge,

[ |T_0-T_{\delta,G}|\asymp\delta^{2/(d-2)}. ]

Uniformly comparable positive variable-coefficient derivative forms inherit the spectral orders by the standard min–max principle. This is an explicitly bounded class of non-Fourier-diagonal operators, not a theorem about arbitrary variable-coefficient PDEs.

The essential limitation

In dimension ten, requiring (|T_0-T_{\delta,G}|\le K\varepsilon) forces (\delta=O(\varepsilon^4)). At accuracy (\varepsilon), the uniform crossover law then returns

[ N_{T_{\delta(\varepsilon),G}}(\varepsilon) \asymp\varepsilon^{-1/2}(\log(1/\varepsilon))^9. ]

Thus the improvement is for a different, regularized operator. It does not beat the optimal finite-rank approximation rate of the original target. It is not a demonstrated data-compression algorithm, GPU acceleration, or universal PDE solver.

Status and completeness

Item	Verified scope or status
Main theorem statements with written proofs	7/7; 100% of the stated theorem checklist
Automated tests in recorded run	27/27 passed
Independent graph-diagnostic comparisons	207 unlabeled graphs, orders 2 through 6
Exact optimal-design matrix certificate	Rank 10, absolute determinant 4, positive weights 1/2
Exact edge-deletion infeasibility certificates	10/10
Exact-rational symbol checks	400 finite cases
Formal proof-assistant verification	0/7 theorems
Independent referee reports	None
Recognized major open problem solved	None claimed
Exhaustive priority established	No

Percentages measure checklist coverage, not probabilities of correctness. Numerical LP checks are not exact proof certificates. Infinite asymptotic claims rest on the manuscript proofs, not the finite experiments.

A provisional editorial assessment is novelty 5–6/10, domain importance 6/10, and demonstrated world importance 3/10. These are judgments, not measurements; they do not justify “massive world-level breakthrough.”

Reproduce the checks

Python 3.10+ is required. From this directory:

python -m pip install -r requirements.txt
python -m pytest -q
python examples/reproduce.py

On Windows, double-click VERIFY_WINDOWS.bat to create a local virtual environment, install the requirements, and run the checks. The script pauses on completion or failure and never requests a token. Installation needs internet access; subsequent checks run locally.

verification/environment.json records the actual tested environment. The coefficient-counting CSV contains exact auxiliary dyadic sums, not exact PDE Fourier counts or engineering benchmarks. ratio_to_theta_proxy is a diagnostic relative to a comparison scale, not an estimate of a proved leading constant.

Rebuild the manuscript with a LaTeX distribution that includes latexmk:

cd paper
latexmk -pdf -interaction=nonstopmode -halt-on-error NACRE_9.tex

VERIFY_MANIFEST.py checks all packaged file hashes using only the Python standard library. Run it before reproduction; regenerating verification outputs can change their hashes.

Files

Path	Contents
paper/NACRE_9.pdf and .tex	Full manuscript and editable source
paper/references.bib	Bibliographic records; the manuscript also embeds its bibliography
THEOREM_LEDGER.md	Exact statements, assumptions, proof dependencies, novelty boundaries
PRIOR_ART_AUDIT.md	Closest literature and limits of the search
ADVERSARIAL_AUDIT.md	Failure modes explicitly tested or excluded
src/nacre9.py, tests/	Diagnostics, exact counters, and test suite
examples/reproduce.py	Regenerate certificates and finite examples
verification/	Recorded results and environment
PUBLIC_RELEASE_TEXT.md	Searchable title, abstract, and short announcement
metadata/ and CITATION.cff	Publication/citation templates, no invented DOI
SHA256SUMS.txt	File-integrity manifest

Provenance and publication

This work was prompted by ORYEON_QADESH_Xi10_v1.0_research_package(1).zip, SHA-256:

45f26a9852a3ffb2c36bdacc3773832553d755134aab6ee0761866e52828c50a

It is a new operator construction, not merely a renamed copy of the predecessor. The old archive is not redistributed. No external upload, submission, referee review, or DOI registration has been performed. See LICENSE_NOTE.md before selecting public reuse terms.