CHISA-RSI: Proof-Carrying Continuous Capability Closure for Recursive Self-Improving Networks - Global Stability-Plasticity, No-Valley Rewiring, and Certified Structural Self-Improvement

Abstract

CHISA-RSI develops a self-contained mathematical framework for capability-preserving structural self-modification in finite modular computational networks. It extends the Chernoff-Hybrid Information Severance Algebra (CHISA) into a continuous theory of recursive self-improvement in which architectural changes can be globally optimized, continuously executed, and accompanied by mathematical certificates of capability preservation or structural exhaustion. For every zero-sum computational workload \(q\), capability is defined from CHISA transport resistance by \[ \mathcal C_q=\frac{1}{q^\top L^\dagger q}, \] where \(L\) is the positive-conductance graph Laplacian. The theory proves that the worst capability multiplier over the entire modeled workload space is exactly characterized by the smallest generalized eigenvalue of the modified Laplacian relative to a baseline architecture. This turns capability from a binary preservation condition into a continuous spectral quantity with an associated capability spectrum and continuous-time growth rate. A central result is that CHISA’s Chernoff incompatibility cost becomes linear in conductance coordinates. Consequently, the optimal tradeoff between statistical coherence and universal structural capability can be formulated as a semidefinite program, eliminating nonglobal local optima from the continuous architecture-design problem. The release establishes several connected results: an exact universal continuous capability multiplier and capability spectrum; a continuous-time worst-case capability-growth rate; a no-valley theorem showing that every universally improving endpoint admits a continuous path with no transient capability loss; a globally optimal proof-carrying closure operator that either finds a strict capability-preserving coherence improvement or certifies that no such improvement exists in the available substrate; the Closure Matrix Eye, which compresses the infinite family of modeled workloads into a finite set of critical capability modes; an idempotent structural-RSI closure condition providing a mathematically certified stopping criterion rather than mere numerical convergence; a quantitative capability reserve measuring robustness to future degradation; an exact fixed-substrate capability ceiling identifying when further improvement requires new structural capacity; an exact spectral safety criterion for simultaneous severance and repair operations; integration with CHISA’s exact Cardano gate, Chernoff seam memory, Death Snip, reweaving, and transfer-resistance coupling. The resulting architecture separates global continuous optimization from discrete topology crystallization: a globally safe target architecture is first computed, a no-capability-valley trajectory realizes it continuously, and CHISA’s exact cubic gate machinery can subsequently crystallize continuous gates toward woven or severed states while retaining information-geometric seam memory. The release package contains the complete manuscript, source material, machine-readable theorem metadata, reproducibility and numerical-verification code, Hugging Face metadata, citation metadata, and explicit claims-and-scope documentation. Scope and claim boundary. The mathematical results are established for the explicitly defined finite, connected, positive-conductance CHISA structural model. Capability refers specifically to Laplacian transport capability \(1/(q^\top L^\dagger q)\). The work does not claim that this quantity is a complete measure of semantic or general intelligence, does not claim to solve unrestricted neural catastrophic forgetting or universal recursive self-improvement, and does not certify worldwide novelty or historical priority.