coupling_vanishes_iff_aligned
plain-language theorem explainer
The equivalence establishes that the reduced phase potential vanishes exactly when the phase difference is an integer. Researchers deriving the Global Co-Identity Constraint from the J-cost forcing chain cite this to confirm zero coupling at aligned phases. The proof applies the zero-condition lemma for the hyperbolic-cosine form of the potential in a single step.
Claim. For nonzero real parameter $λ$, the reduced phase potential defined by $J̃(λ, δ) = cosh(λ · d_ℤ(δ)) − 1$ satisfies $J̃(λ, δ) = 0$ if and only if there exists an integer $n$ such that $δ = n$.
background
The GCIC derivation module obtains the Global Co-Identity Constraint directly from the J-cost forcing chain with zero empirical axioms. The reduced phase potential is introduced as $J̃(λ, δ) = cosh(λ · dist_ℤ(δ)) − 1$, where λ = ln b for the base of the discrete quotient. This rests on the upstream zero-condition theorem, which states that $J̃(λ, δ) = 0$ iff δ is an integer when λ ≠ 0.
proof idea
The proof is a one-line wrapper that applies the zero-condition theorem Jtilde_zero_iff to the supplied λ and δ.
why it matters
This equivalence closes the coupling-vanishing step required for the Global Co-Identity Constraint theorem of the forcing chain. It links T5 J-uniqueness and ratio rigidity at J = 0 to the compact phase Θ ∈ ℝ/ℤ, confirming uniform alignment at J-stationarity. The result supports the main GCIC claims without introducing new hypotheses.
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