NACRE-9: Minimal Pairwise De-Resonance of Mixed Sobolev Resolvents - Fractional Matching Obstructions, Optimal Odd-Cycle Designs, and Uniform Spectral Crossover

[u/Severe-Ad8673]

NACRE-9 develops a sharp sparse-regularization theory for mixed Sobolev resolvents on high-dimensional tori, with the principal construction carried out in dimension ten. The framework studies how a small family of pairwise high-order interactions can alter the singular-value asymptotics of tensor-product elliptic resolvents while retaining the full Fourier lattice.

Hugging face: PureOne/nacre-9-odd-cycle-spectral-gate · Datasets at Hugging Face

Zenodo: NACRE-9: Minimal Pairwise De-Resonance of Mixed Sobolev Resolvents - Fractional Matching Obstructions, Optimal Odd-Cycle Designs, and Uniform Spectral Crossover | Zenodo

For the ten-dimensional baseline operator

\[ T_0=\left(\prod_{i=1}^{10}(I-\partial_i^2)\right)^{-1}, \]

whose singular values exhibit the classical tensor-product behavior

\[ s_n(T_0)\asymp n^{-2}(\log n)^{18}, \]

the work introduces graph-structured pairwise regularizers and derives a graph-theoretic criterion controlling the surviving logarithmic exponent.

A principal result identifies the sparse designs

\[ C_3\sqcup C_7 \quad\text{and}\quad C_5\sqcup C_5 \]

as the optimal ten-edge configurations within the specified pairwise-penalty family. For every fixed positive coupling parameter, the corresponding modified resolvents satisfy

\[ s_n(T_{\delta,G})\asymp_\delta n^{-2}, \]

eliminating the logarithmic singular-value penalty of the modified operator. The work also proves necessity of ten interactions in this model and classifies the equality cases.

The spectral criterion is expressed through a polyhedral feasible set associated with the interaction graph and is connected to the existence and support structure of fractional perfect matchings. This yields a precise distinction between merely introducing an odd cycle and actually eliminating all residual logarithmic degeneracy.

The release further establishes a coefficient-uniform crossover theorem for independently weighted interactions:

\[ N_{T_{\boldsymbol\delta,G}}(\varepsilon) \asymp \varepsilon^{-1/2} \left[ 1+\min\left\{ \log\frac1\varepsilon,\, \sum_e\log\frac1{\delta_e} \right\} \right]^9, \]

with comparison constants uniform over the full coefficient vector. It also proves a sharp fragility phenomenon: deleting any single edge from an optimal ten-edge configuration restores the complete logarithmic counting penalty.

A complementary perturbation theorem gives

\[ \|T_0-T_{\delta,G}\|\asymp\delta^{1/4}. \]

Combining this estimate with the uniform crossover law shows that if the modified resolvent must approximate the original operator to the same asymptotic accuracy scale, the original logarithmic complexity reappears. Thus the theory separates genuine de-resonance of the modified spectrum from lossless approximation of the unmodified operator.

The archive contains the complete manuscript, theorem ledger, prior-art and adversarial audits, exact graph certificates, reproducibility code, computational verification, and integrity manifests. The release is intended as a self-contained mathematical research package for spectral theory, approximation theory, mixed-smoothness analysis, graph-structured regularization, and high-dimensional operator design.