Pythagorean Angle Lattice: The Division-Ramification-Criticality Theorem (DRC theorem)

Abstract

Proves the Division-Ramification-Criticality (DRC) theorem, establishing that principal dyadic division, Gaussian ramification, and arithmetic criticality govern the exact arithmetic survival of Pythagorean angles under repeated halving : A primitive Pythagorean triple determines a rational point on the unit circle and hence a distinguished smaller acute angle. The Pythagorean Angle Lattice (PAL), a recently introduced angle-division framework, organizes cotangent coordinates obtained by integral subdivision of such angles. The problem studied here is arithmetic rather than merely analytic: when a Pythagorean angle is repeatedly divided by 2, for how long do the successive principal angles remain Pythagorean? For a primitive triple a^2 + b^2 = c^2, 0 < a < b < c, with smaller acute angle ∢, set δ = c - b and n = 2a/δ. We derive n = 2 cot(∢/2) and ∢ = 2 arctan(2/n). Writing the reduced rational parameter as n = p/q, gcd(p,q) = 1, the canonical Gaussian integer z = p + 2iq controls primitive reduction at the ramified prime 1 + i. We prove the exact gap law c - b = 2^(3 - ν_(1+i)(p + 2iq)) * q^2, which yields the three primitive gap strata and recovers c - b ∈ {1, 2, 8} precisely on the integral slice q = 1. A principal halving survives in the Pythagorean locus exactly when p^2 + 4q^2 is a square. In particular, the entire integral metallic slice has depth 0: no primitive triple with c - b ∈ {1, 2, 8} survives even one principal halving. Repetition produces a finite but unbounded division-depth filtration and the local transition law E → E or T, T → O, O → escape. Every strict acute Pythagorean angle has finite depth, every finite depth occurs infinitely often, and survivors to depth k admit a projective Gaussian 2^k-power representation. The resulting counting theory has a critical homogeneity. For k ≥ 2, if N_k(X) counts survivors to depth k of reduced denominator at most X, then N_k(X) ~ C_k * X^(σ_k), where σ_k = 2^(1-k) and 2^k * σ_k = 2. The first layer is marginal: N_1(X) = (3/π^2) X log X + B_PAL * X + O(√X log X). We also determine deepest-root ramification proportions, an analytic scaling function, the exact least denominator Q_k for every k ≥ 2, and the associated extremal depth frontier. These results form the Division-Ramification-Criticality (DRC) theorem: principal dyadic division, Gaussian ramification, and arithmetic criticality are three aspects of one self-contained survival theory for PAL.