gcic_canonical
plain-language theorem explainer
The canonical GCIC theorem asserts that any connected recognition lattice with vanishing reduced phase cost at lambda equal to ln phi forces all local phases to share the same wrapped value on the unit interval. Researchers citing the Anno Recognitionis §V argument invoke this result to establish global phase coherence from local rigidity. The proof is a direct specialization of the general uniqueness theorem at the pre-established nonzero canonical lambda.
Claim. Let $V$ be a vertex set with adjacency relation $adj$ such that the reflexive-transitive closure of $adj$ connects every pair of vertices. Let $Θ : V → ℝ$ assign a real phase to each vertex. If the reduced phase cost satisfies $J̃_{ln φ}(Θ(v) − Θ(w)) = 0$ for every adjacent pair $v, w$, then wrapPhase(Θ(v)) = wrapPhase(Θ(w)) for all $v, w ∈ V$.
background
The Global Co-Identity Constraint module derives global phase uniqueness from local graph rigidity on a connected lattice. The wrapping function wrapPhase projects any real onto [0,1) by subtracting its integer part, rendering phases modulo 1. The reduced phase cost J̃_lam(δ) vanishes exactly when the phase difference δ is an integer multiple of the period fixed by lam. The upstream theorem gcic_global_phase_unique states: suppose the lattice is connected, lam ≠ 0, and every adjacent pair satisfies J̃_lam(Θ(v) − Θ(w)) = 0; then all wrapped phases coincide.
proof idea
The proof is a one-line wrapper that applies gcic_global_phase_unique, supplying the connectedness hypothesis, the fact that lam_canonical is nonzero, and the given edge condition on the reduced phase cost at that lambda.
why it matters
This supplies the concrete instance of the Global Co-Identity Constraint required by the Anno Recognitionis §V argument, where the recognition lattice operates at the phi-ratio. It closes the derivation of global phase uniqueness from the local ingredients in GraphRigidity and ReducedPhasePotential. In the framework it realizes the phi-forced self-similarity and eight-tick octave by fixing the base to ln phi, ensuring the wrapped phase is independent of basepoint as stated in the master certificate.
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