Betti Numbers as Cognitive Integrity Metrics: Topological Health Diagnostics for Deterministic Memory Systems

Abstract

We introduce the Cognitive Integrity Score (CIS), an operational, real-time topological health diagnostic for deterministic simplicial associative memory networks. In parallel with significant advances in uncertainty quantification, semantic entropy, process-supervised verification, and metacognitive reinforcement learning for autoregressive foundation models, deterministic geometric memory complexes require exact, zero-overhead structural invariants. We demonstrate that the sequence of Betti numbers (β0, β1, β2, β3, β4), computed via harmonic eigendecomposition of combinatorial Hodge Laplacians Δk = ∂kT ∂k + ∂k+1 ∂k+1T, directly characterizes fundamental structural failure modes: β0 > 1 identifies concept space fragmentation; β1 >> 0 detects circular reasoning loops and associative trapping; and β2 >> 0 reveals relational voids (missing factual bridges). We formulate a multiplicatively gated Cognitive Integrity Score (CIS) combining low-order harmonic invariants with algebraic graph connectivity (λgap), designed to enforce strictly monotonic degradation under progressive topological decay. On grounded associative memory retrieval benchmarks across multi-seed trials (S = 10, N = 3,000 queries), CIS demonstrates strong predictive rank correlation with actual multi-hop query accuracy (ρ = 0.9710, p < 10-37). While global edge density provides an effective heuristic under uniform random attrition, CIS complements global density measures by remaining uniquely sensitive to targeted bottleneck failures that leave overall edge density largely unchanged. Furthermore, formal query chain admissibility evaluated via orthogonal projection onto harmonic retrieval subspaces, governed by boundary closure (∂2 ≡ 0), provides exact deterministic query refusal without statistical hallucination. Finally, we showcase sub-millisecond execution (< 0.05 ms on standard CPU hardware) via live discrete exterior calculus telemetry on an interactive 60 FPS engine. Related Work & Prior Art: The σ-Constant: A Universal Algebraic Invariant for Energy Propagation in d-Dimensional Simplicial Lattices (10.5281/zenodo.20350425) The xd = x + 1 Hierarchy: Cross-Dimensional Spectral Validation on Ad Root Lattices (10.5281/zenodo.20692936) The Lattice Octave: A Characteristic-Delay Model and Growth-Rate Partition on Simplicial Lattices (10.5281/zenodo.22181883)