Black-and-Gold Series Paper 1

Quasi-Equivalence Index as the Common Gate of Graded Integrity and Golden Scaling

C. J Tully Independent Researcher, Engineer Tamworth, NSW, Australia ORCID: 0009-00075661-7332

28th October 2026

Abstract

This paper is the first of the black-and-gold series. It fixes the Quasi-Equivalence Index as the single integrity gate that both the dark graded formalisms and the golden-spectrum arithmetic must respect. /5 The index is derived from the graded square of a five-dimensional Z Z-product whose structure constants encode golden-ratio weights. A hard threshold of 0.12 rejects incoherent states. Golden scaling by the fundamental unit of Q( 5) is proved to be an automorphisms that leaves the index exactly invariant on the coherent locus, so the gate remains inactive under iteration. The inverse recovers the original state. The construction is parameterfree once the threshold is fixed. No claim is made that preservation of the index constitutes cryptographic security beyond the gate itself, nor that algebraic reversibility implies metric boundedness.