On the Ramanujan Property of Paley Graphs and the Formalization of the Riemann Boundary

We present a connected body of machine-checked results, all generated by a single number-theoretic object: the quadratic-residue character chi_q. For every prime q = 1 (mod 4), the Paley graph P(q) is strongly regular, self-complementary, and Ramanujan. For q in {37, 41, 97, 101, 109} we give complete kernel-verified certificates in Lean 4 / Mathlib: the strongly-regular parameters (q, (q-1)/2, (q-5)/4, (q-1)/4), the exact non-trivial spectrum (-1 +- sqrt(q))/2, and the Ramanujan bound, each carrying an empty sorryAx audit. These certificates attest to the Ihara critical-circle condition - the graph Riemann Hypothesis - satisfied exactly by an infinite family of finite objects. The same character drives three deeper phenomena: the quadratic Gauss sum of magnitude sqrt(p) (Gauss, 1801); the square-root cancellation of character sums (Weil, Deligne); and, through quadratic reciprocity, the distribution of primes in the admissible 2,4,2 quadruplet, verified against the Hardy-Littlewood model with ratio tending to 1 through 10^8. Finally, we make the boundary of this ladder precise. Formalizing "P establishes Q" as the implication P -> Q, we prove not(P -> Q) iff (P and not Q); applied to the partition-function descriptions of zeta, this yields not(Facts -> RH) iff not RH, showing in machine-checked form that the partition-function route to the classical Riemann Hypothesis is confined to the zero-free half-plane Re(s) > 1, disjoint from the critical strip. The paper does NOT prove the Riemann Hypothesis, and no such claim is made. What is offered is an accurate, machine-checked map of the boundary. Contents: manuscript.pdf (6 pages), manuscript.tex (LaTeX source), README.md, and the nine Lean 4 / Mathlib formal developments referenced by the paper (all sorry-free; axiom audit: [propext, Classical.choice, Quot.sound]).