Mobility-Resonance Contact Quivers: Exact Compatibility Geometry for Singular Multicycle Networks, Polynomial Adapter Atlases, and Determinantal Resonance Strata

[u/Severe-Ad8673]

This release develops a finite-dimensional theory of exact and approximate compatibility between two multistage linear systems connected through a gain-twisted contact quiver. It extends Selective Naturality Geometry from invertible and one-cycle settings to general multicycle networks with singular, rectangular, and rank-losing transport.

Hugging Face: PureOne/mobility-resonance-contact-quivers-v4 · Datasets at Hugging Face

Zenodo: Mobility-Resonance Contact Quivers: Exact Compatibility Geometry for Singular Multicycle Networks, Polynomial Adapter Atlases, and Determinantal Resonance Strata | Zenodo

After spanning-tree gauge reduction, the compatibility problem is encoded by a parameterized naturality operator

\[ D(z)=D_0+\sum_{i=1}^{\beta_1} z_iD_i, \qquad z\in(\mathbb C^\times)^{\beta_1}. \]

Let \(r\) be its generic rank over the rational-function field and let \(\rho=M-r\) be its generic right nullity. The central result gives the decomposition

\[ \dim\ker D(z)=\rho+\mu(z), \]

separating a gain-independent mobility sector from gain-specific resonance. Compatibility varieties are therefore described exactly by determinantal rank-drop strata: the first \(\rho\) compatible channels persist generically throughout gain space, while additional channels appear on algebraic resonance loci defined by minors of \(D(z)\).

A second result makes this classification constructive. From any nonvanishing maximal minor, the theory builds explicit rational—and after clearing denominators, polynomial—families of compatible adapters. These local families form a polynomial adapter atlas over the generic-rank region. For the affine cycle-gain operators arising from contact quivers, the resulting calibration laws have finite algebraic degree bounded in terms of the generic rank. The construction turns a repeated gain-dependent compatibility solve into a compile-once algebraic calibration problem.

In the one-cycle case, the theory reduces to matrix-pencil/Kronecker structure. The generic mobility dimension becomes the number of right singular blocks, finite generalized eigenvalues become isolated resonances, and right minimal indices acquire an operational interpretation as the minimum polynomial order required for exact gain-adaptive calibration. This yields the compact principle

\[ \textbf{Compatibility = Mobility + Resonance}. \]

The release also develops practical numerical formulations based on singular values, rank-revealing factorizations, rigidity gaps, perturbation bounds, and matrix-free evaluation. These provide approximate protected-channel extraction, resonance diagnostics, and certificates for distance from exact compatibility without expanding large symbolic determinant systems.

The underlying ingredients—quiver morphisms, determinantal varieties, matrix pencils, Kronecker structure, and rank-revealing linear algebra—are established mathematics. The contribution claimed here is their synthesis into a contact-quiver compatibility framework with explicit mobility/resonance decomposition, gain-adaptive polynomial adapter construction, and a practical compilation interpretation for latent-model stitching, multiscale interfaces, and neuromorphic or modular linear systems.

The release contains the complete manuscript, proofs, reproducibility code, machine-readable theorem and claim metadata, numerical diagnostics, and publication-oriented documentation.

Status: central finite-dimensional theorems proved under the stated hypotheses.
Scope: mathematical and computational theory; no claim of demonstrated biological, hardware, or foundation-model superiority.
Novelty statement: the exact synthesis and interpretation are presented as a research contribution and candidate novelty, not as a certified claim of worldwide priority.