The Pell Equation from Circulant Criticality (z=4)

Preprint (February 2026). This short mathematical note isolates the number-theoretic step used in the AQEA chain. Starting from a standard circulant positivity bound with critical amplitude A_crit(z)=sec(π/z) and the conditional premise z=4 (simplicial face-adjacency), we obtain A_crit(4)=√2 and hence A^2=2. This introduces the quadratic field Q(√2) and the ring of integers Z[√2]. Units u=a+b√2 with norm N(u)=a^2−2b^2=±1 encode the Pell equation N^2−2d^2=1. Under a positive-norm (orientation-preserving) selection, the minimal unit is (1+√2)^2=3+2√2, yielding (N,d)=(3,2). Physics identifications are explicitly separated and labeled as structural (non-theorem) mappings. Reproducibility: all steps can be verified with standard mathematics and do not require proprietary software. Pell equation quadratic fields algebraic number theory units in \(\mathbb{Z}[\sqrt{2}]\) circulant matrices spectral positivity / eigenvalue bounds critical amplitude \(A_{\mathrm{crit}}\) discrete Hodge theory