Pythagorean Angle Lattice: Dyadic Collision Rings, Hasse Depth, and a Carry–Boundary Comparison for Circular Units

Abstract

In the Pythagorean Angle Lattice, the real carry word of a circular unit and its first dyadic boundary differ only by a reciprocal-unitary Gaussian-period automorphism and share the same Hasse depth on every compatible repeated-root primary block : We study a binary repeated-root structure occurring in a prime-conductor realization of the Pythagorean Angle Lattice (PAL). If a cyclic observation quotient has order M = 2^s m with m odd, then F₂[X]/(X^M − 1) = F₂[X]/(X^m − 1)^{2^s} contains non-semisimple primary blocks A_f = F₂[X]/(f^{2^s}). We describe their depth by Hasse derivatives, prove a finite-resolution theorem showing that the first 2^s Hasse jets depend only on coefficient collisions modulo 2^{sℓ}, and identify the resulting collision tower and its Frobenius pairing. For an odd primitive root g modulo an odd prime p, the normalized real circular unit ε_g = ζ_p^{(1−g)/2} (1 − ζ_p^g)/(1 − ζ_p) has a real-signature word given by the parity of the Euclidean carries k_j = ⌊g ⟨g^j⟩_p / p⌋. We compute, independently, the first dyadic boundary of the same circular unit. With g = 2r + 1 and Fibonacci polynomials F₀ = 1, F₁ = T, F_n = T F_{n−1} + F_{n−2} over F₂, its squared boundary is F_{r−1}(T)/(F_r(T) + F_{r−1}(T)) = (z + z² + ⋯ + z^{g−1})/(1 + z^g), T = z + z^{−1}. A finite residue-set expansion shows that this boundary is governed by exactly the same division-with-remainder parity cocycle as the real carry word, shifted by one exponent. After Frobenius and PAL norm folding, the remaining difference is convolution by a Gaussian-period polynomial E_Q. A characteristic-zero autocorrelation identity gives, after reduction modulo 2, that E_Q E_Q^∨ = 1 in the full, possibly nonreduced, collision group ring. Consequently, inside the observation quotient, the real carry realization and the dyadic circular-unit realization differ by a reciprocal-unitary automorphism and have the same Hasse depth on every compatible primary block. On a localized cyclic circular-unit block generated by the normed primitive-root source, the corresponding maps also have the same kernel and unit-equivalent image ideals. We give an exact depth-two computation at p = 325753. The scope is deliberately narrow, and Section 16 delimits it precisely: the comparison is a statement about the circular-unit block after projection to the observation quotient. It is not asserted for the full unit group, for arbitrary Selmer classes, or for depths computed in the unprojected Galois module.