The geometry
determines the answer.

We introduce the Cognitive Integrity Score (CIS), an operational, real-time topological health diagnostic for deterministic simplicial associative memory networks. We demonstrate that the sequence of Betti numbers (β₀, β₁, β₂, β₃, β₄), computed via harmonic eigendecomposition of combinatorial Hodge Laplacians Δ_k = ∂_k^T∂_k + ∂_k+1∂_k+1^T, 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). Monotonically gated with algebraic spectral connectivity (λ_gap), CIS achieves strong rank correlation (ρ = 0.9710, p < 10^-37) with multi-hop retrieval accuracy, while boundary nilpotency (∂² ≡ 0) enforces exact algebraic query refusal without statistical hallucination.

We prove that the algebraic polynomial family x^d = x + 1 (for integer d ≥ 2), whose roots are known as generalized golden ratios, characterizes dual-channel boundary-bulk energy transport on d-dimensional simplicial lattices under a dominant characteristic-delay model. We formulate and prove the Face-Poset Delay Decomposition Theorem, establishing that the combinatorial face-poset topology of a regular d-simplex admits exactly two maximal transit channel classes: a boundary facet transit channel of characteristic delay d−1 hops, and a bulk interior transit channel of characteristic delay d hops. Under the dominant-delay approximation, the unique positive real root c_d > 1 of x^d = x + 1 induces the growth-rate partition identity c_d^−(d−1) + c_d^−d = 1, balancing asymptotic energy transport between boundary and bulk.

We prove that the polynomial family x^d = x + 1 generates geometry-native organizing constants for A_d root lattices across dimensions — unifying the golden ratio (2D), plastic constant (3D), and σ-constant (4D) into a single hierarchy. Four independent lines of evidence: infinite-lattice Fourier analysis, algebraic recurrence, higher-order asymptotics with exact coefficients via Lagrange inversion, and a proof that the d-dimensional hyperbox subdivides into exactly C(2d−1, d−1) proportional types — extending Van der Laan's architectural system to four dimensions and beyond.

We derive σ ≈ 1.22074, the unique positive real root of x⁴ − x − 1 = 0, and demonstrate that it is the geometry-native organizing constant for energy propagation on the A₄ root lattice. The result is validated through two independent methods: Hopfield recall optimization and spectral Laplacian analysis. Zero free parameters. The lattice geometry determines the constant.