Global Co-Identity Constraint (GCIC): Unity as a Gauge of Recognition Geometry
Summary
GCIC posits a single global phase \(\Theta\) that aligns all pinned recognition lengths on a \(\varphi\)‑geometric ladder \(\ell_k = L_0\,\varphi^{k+\Theta}\). It is a global gauge condition (co‑identity), not a signal; local dynamics remain finite‑rate.
Abstract
There exists a single global phase \(\Theta \in [0,1)\) such that every pinned or steady recognition state satisfies \(x(\ell) \equiv \Theta \pmod{1}\) with \(x(\ell)=\log_{\varphi}(\ell/L_0)\), implying allowed lengths \(\ell_k = L_0\,\varphi^{k+\Theta}\) for \(k \in \mathbb{Z}\). GCIC is a gauge condition defining the steady‑state manifold rather than a superluminal transport mechanism; local reproduction propagates at finite speed \(c\). A minimal log‑periodic cost \(C(\ell;\Theta)=C_{*}+\kappa\,[1-\cos(2\pi(x(\ell)-\Theta))]\) yields stationary minima at the ladder with positive curvature, enforcing the reciprocity constraint \(\ell(+n)\,\ell(-n)=L_0^2\) and leading to testable signatures including \(\varphi\)‑band clustering on a log‑axis, loop quantization (\(\sum \Delta x \in \mathbb{Z}\)), and unity under graph rewiring that preserves total loop phase.
Key Points
- \(\varphi\)‑geometric ladder: Allowed recognition lengths \(\ell_k = L_0\,\varphi^{k+\Theta}\) set by a single global phase.
- Gauge, not signal: GCIC defines the steady‑state manifold; local updates remain finite‑rate.
- Log‑periodic cost: Minimal \(C(\ell;\Theta)\) produces stationary minima with positive curvature and reciprocity \(\ell(+n)\,\ell(-n)=L_0^2\).
- Testable consequences: \(\varphi\)‑band clustering, loop quantization (\(\sum \Delta x \in \mathbb{Z}\)), and unity under rewiring with preserved loop phase.