S-AI-Reasoning: Reasoning as Regulated Canonicalization

Abstract

Artificial reasoning systems are often evaluated by final-answer accuracy, although accuracy alone does not establish whether an internal trajectory has stabilized, uncertainty has been reduced, a candidate is formally admissible, or commitment is justified. This article generalizes S-AI-Reasoning as regulated canonicalization over a structured hypothesis space. The framework integrates logical-symbolic and perceptual-structural representations, active hypotheses, certified engram memory, a concrete six-hormone regulatory realization, sparse specialized-agent orchestration, explicit verification, the Extended Recursive Reasoning Cycle (RRC+), and an Accept-Clarify-Reject (ACR) commitment regime. The theory separates fixed-point existence, uniqueness, convergence, stability, finite termination, decidability, totality, verification, and correctness. These distinct guarantees are recomposed through a Conditional Joint-Consequence Principle: they may hold jointly when their respective assumptions and interfaces are satisfied, but no unconditional equivalence among them is asserted. Brouwer-type arguments are restricted to continuous self-maps on appropriate compact convex finite-dimensional domains; Banach-type convergence is invoked only in explicitly contractive regimes; and Lyapunov analysis supplies local coupled cognitive-hormonal stability under stated conditions. Consequently, convergence is not treated as correctness, and hormonal regulation is not assumed to make arbitrary operators contractive. The experimental evidence is layered accordingly. Deterministic controlled simulations exhibit net attraction on all tested trajectories, whereas bounded stochastic perturbation preserves net attraction while only approximately 62.06% of individual transitions decrease the Lyapunov-like diagnostic. On 80 exactly verifiable Maze instances, certified adaptive stopping preserves 100% resolution and certification while reducing mean depth from 20 to 12.1625 cycles, a 39.19% reduction. Sudoku exposes an operator-sufficiency boundary: elementary constraint propagation certifies 27 of 40 cold-start instances; on this valid recurrent subset, warm-start reduces mean depth from approximately 2.852 to 2.000 cycles. Across three supplied ProofWriter sources containing 59,220 instances, 9,048 instances fall within the declared positive unary Horn/CWA fragment, yielding 15.28% coverage and 100% Gold-label agreement within that fragment. On a supplied 500-instance PowerQA set with True, False, and Uncertain labels, an initial conservative symbolic compatibility run obtains 425/500 = 85.00% exact three-way agreement. All 174 Gold-Uncertain cases remain Uncertain, no True/False polarity reversal is observed, and the 251 committed True/False predictions all agree with Gold; however, 75 Gold-decidable cases remain unresolved. This PowerQA result is reported as a preliminary symbolic-kernel evaluation rather than as a completed six-hormone RRC+/ACR benchmark result. The framework is further extended to ARC-AGI-style perceptual abstraction through object, relation, geometry, symmetry, counting, invariant, and transformation hypotheses. The existing local ARC-style experiment is retained only as preliminary transformation evidence and is not presented as a public ARC-AGI benchmark result. Overall, the results support a bounded claim: when a problem admits an adequate structured representation, an admissible hypothesis language, regulated refinement, and explicit verification, reasoning can be organized as the controlled reduction of a hypothesis space toward a parsimonious canonical state whose commitment is separately certified. Keywords: Sparse Artificial Intelligence; S-AI-Reasoning; Regulated Canonical Reasoning; Canonicalization; Hypothesis Space; Hormonal Regulation; Sparse Multi-Agent Systems; Metacognition; Accept-Clarify-Reject; Certified Reasoning; Lyapunov Stability; Contraction; ProofWriter; PowerQA; ARC-AGI; Parsimony.