Towards The Mathematics of Autonomic Cognitive Organisation

Abstract

Abstract This paper proposes a candidate mathematical reduction of the Orchard Cognitive Framework’s autonomic stack: seven coupled mechanisms for lawful representation, adaptive dimensionality, pattern recognition, recursive pattern holding, alignment pricing, masked-geometry comparison, and curiosity-driven acquisition. Rather than reproduce the modules as an implementation specification, the paper asks whether a smaller set of substrate-agnostic primitives explains their recurring mechanics. Six candidates are extracted: (P1) typed distinction and non-collapse; (P2) conservative differentiation; (P3) conservative recursive composition; (P4) projection-preserving commutability; (P5) categorical admissibility with conditional alignment cost; and (P6) reciprocal unresolvedness–structure dynamics. Each primitive is defined statically and dynamically, assigned candidate mathematical forms, bounded by explicit failure conditions, and paired with an empirical isolation test and killer control. The paper then analyses all fifteen pairwise interactions, selected higher-order combinations, and criteria for independence, necessity, sufficiency, closure, minimality, and basis non-uniqueness. Two industry-relevant consequences receive particular attention. First, alignment is represented not as a single reward or compliance score but as geometry over an admissible domain: hard floors are exclusions, while lawful motion inside the domain is directionally priced and re-appraised when context changes. Second, anti-hallucination is treated as an anti-collapse architecture: unknown, unmeasured, prohibited, suspended, structural, and historically scarred states remain typed rather than being silently interpolated into asserted content. HOLD/abstention, claim-scope ceilings, provenance, replayable witnesses, and evidence-acquisition pathways are structural consequences of this representation. Existing toy-scale Orchard results support several local mechanisms, but the six-primitives reduction itself remains a falsifiable research programme. Keywords: AI alignment; hallucination; uncertainty; abstention; constrained optimisation; emergence; recursive composition; provenance; information geometry; cognitive architecture; dynamical systems; typed nulls; safety. Epistemic status and source discipline The source corpus contains document-level ratifications while preserving heterogeneous internal standings (proven within a fixture, conditional, open, deferred, or parked). This paper does not upgrade those internal standings. Canon-derived statements are cited to the relevant Orchard source. New reductions, equations, names, and generalisations introduced here are explicitly labelled candidate. Where a source commits only to a formal shape, this paper defines the boundary of a suitable mathematics and an empirical route for sharpening it rather than inventing an unearned closed form. The term “hallucination” is used in the AI-industry sense of plausible but false or unsupported model output, not as an analogy to human perceptual hallucination. In this paper, anti-hallucination is narrower than complete factuality: it denotes structural resistance to converting unresolved epistemic states into unsupported asserted content. This distinction is intentional.