One Number, Two Names: kappa = tau = 0.9539

A short genesis note on a single empirically-anchored number, 0.9539, which appears in the author's prior work under two distinct labels: tau, the cosine-similarity rejection threshold in the Anti-Drift Cognitive Control Loop (ADCCL); and kappa, the Sovereign Invariant of Geometrically Ordered Dynamics (GOD Theory), which couples redshift-dependent Information Tension and predicts a small shift in the first acoustic peak of the cosmic microwave background.

The note's purpose is to record the genesis order precisely. tau was discovered empirically during ADCCL development as the cosine-similarity stability boundary at which autonomous reasoning loops transitioned from coherent task pursuit to degraded output (hedging inflation, topic drift, circular reasoning). The number was fit by hand against a 19-case synthetic benchmark, post-hoc formalized as a trigonometric bound (arccos(0.9539) approximately 0.3040 rad, roughly 17.4 degrees), and mechanized in Lean 4 against Mathlib's Geometry.Euclidean.Angle library. kappa was not independently re-derived; it was adopted from tau as the empirical anchor available when constructing the cosmological coupling of GOD Theory. This is the opposite of a cross-paper coincidence: one number propagated from an engineering benchmark into a place where it can be independently tested.

The note states two falsifiers explicitly. F1: ADCCL re-fit against a fresh adversarial held-out corpus across multiple LLM providers may yield a materially different tau, indicating the 0.9539 value was a local artifact of the original development corpus. F2: the preregistered cosmological tests filed 2026-05-11 (JWST stacked flux at z=10, CMB first-acoustic-peak shift) may pin kappa to a different value than 0.9539, in which case the bridge between the two scales collapses while the engineering threshold survives independently.

This is a consolidation note. It introduces no new claims, no new mathematics, and no new physics. It documents provenance, surfaces a load-bearing dependency, and binds two independent falsification paths. Companion to the empirical Riemann gap-ratio note (DOI 10.5281/zenodo.20210870) released in the same batch.