Quasi-periodic prefactor at golden slope: period-2 structure, Zeckendorf-phase clustering, and separation from two controls

Abstract

This record reports the prefactor at the GOLDEN slope. It is a companion computation to the A076227 software deposit (Zenodo 22121363), on a different slope and a different sequence; it is not a new version of that record and retracts nothing in it. The same exact-integer dynamic programme is run at the golden slope 1/phi, where the prefactor exhibits an exact period-2 limit cycle along Fibonacci indices (last within-parity difference 0.001588, against 0.062063 for the gamma control through the identical code path), clustering of the detrended profile by Zeckendorf phase (eps_5/eps_1 = 0.198649 against a sealed threshold of 0.5), and separation from two controls. Relation to Zenodo 22121363. That record's subject is the sequence A076227 at the single slope gamma = 1/log2(3); its title, its README heading and its keywords name that slope and no other. The barrier used here is a different barrier and generates a different sequence: B(39) = 3611535862 there, 4574825460 here. The machinery is reused; the object is not the same object. This record therefore cites that one rather than versioning it. Cited, not derived: the barrier word's recursion s_(n+1) = s_n s_(n-1) is classical in combinatorics on words (Lothaire; de Luca). A parity-dependent "seam correction" that an intermediate draft derived is the translation cost of an index convention, not a phenomenon: on the classical characteristic word the identity holds exactly, with no correction, for all 14 of 14 tested scales. That was settled by construction, not by literature search. A further claim from that draft — that the prefactor's period-2 cycle and the seam defect were one fact — is WITHDRAWN. NOT CLAIMED: a solution to the irrational-slope problem; any result about A076227 itself. OPEN, named in the body: the observed decay rate 0.6676 against the predicted 1/phi = 0.6180, and a sup plateau of 0.093777 quoted in the working record which does not reproduce (the value derivable from the artifact is 0.094343; the discrepancy is recorded, not reconciled, and nothing rests on it). The prior-art scope is stated at what the gates actually covered, including that one sweep's arXiv arm screened zero documents while reporting itself operational.