gcic_derivation_cert
plain-language theorem explainer
The GCIC derivation certificate packages three properties derived from the J-cost forcing chain: phase alignment minimizes the reduced potential J̃ for every lambda and delta, the unit J-cost vanishes, and the aligned cost is strictly smaller than the sum of any positive collection of perturbed costs. Researchers replacing empirical axioms in the Global Co-Identity Constraint with theorems of Recognition Science would cite this packaging step. The proof is a one-line term that assembles the three component lemmas phase_alignment_minimizes_J̃, J
Claim. The certificate asserts the conjunction of three statements: for all real numbers λ and δ, J̃(λ, 0) ≤ J̃(λ, δ) where J̃(λ, δ) = cosh(λ · d_ℤ(δ)) − 1; for every list of positive real numbers of positive length, J(1) < sum of the list; and J(1) = 0, with J the J-cost function J(x) = (x − 1)²/(2x).
background
The GCIC Derivation module obtains the Global Co-Identity Constraint from the Recognition Science forcing chain with no empirical axioms. The reduced phase potential is defined by J̃(λ, δ) = cosh(λ · distZ δ) − 1, where λ = ln b for the discrete base b. The J-cost satisfies J(x) = (x − 1)²/(2x) and therefore vanishes at the unit element. Upstream lemma Jcost_unit0 records this vanishing by direct simplification of the definition. The module doc states the chain T5 (J-uniqueness) through ratio rigidity and compact phase Θ ∈ ℝ/ℤ to GCIC at J-stationarity.
proof idea
The proof is a term-mode construction that directly supplies the three conjuncts of the certificate: phase_alignment_minimizes_Jtilde for the first inequality, aligned_beats_perturbed for the strict collective subadditivity, and Jcost_unit0 for the unit vanishing. No further tactics or reductions are performed.
why it matters
This certificate completes the packaging of the GCIC derivation by bundling the minimization of J̃ under alignment, the strict subadditivity of collective J-cost, and the zero unit cost. It supports the main result gcic_from_forcing_chain in the same module, which asserts that the GCIC follows from the forcing chain. Within the framework it realizes the T5-to-GCIC step, replacing the empirical axiom of collective subadditivity with a theorem of J-cost.
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