shared_fixed_point
plain-language theorem explainer
The theorem establishes that the J-cost function vanishes at argument one, confirming a shared fixed point for language-model, photonic, and plasma substrates. Researchers unifying recognition-based models across these systems would cite it to anchor their consistency claims. The proof is a direct one-line reference to the unit lemma for J-cost.
Claim. $J(1) = 0$, where $J(x) = (x-1)^2/(2x)$ is the J-cost function.
background
The Three-Substrate Validation Certificate module records that language models, photonic qubits, and magnetized plasma all obey the same J-cost fixed point at unity together with descent from the J-cost gradient. The J-cost of a recognition event is defined as the derived cost of its comparator on positive ratios and equals the squared ratio $(x-1)^2/(2x)$. Upstream results include the equilibrium theorem stating that J equals zero at equilibrium and the Jcost_unit0 lemma obtained by simplification on that definition.
proof idea
The proof is a one-line wrapper that applies the Jcost_unit0 lemma. That lemma is discharged by simp on the definition of J-cost, which evaluates directly to zero at argument one.
why it matters
The result supplies the fixed_point component inside the threeSubstrateCert definition that bundles the shared properties of the three substrates. It realizes the J-uniqueness step (T5) of the forcing chain, where the fixed point at one forces the common descent direction and multi-channel extension observed in the empirical validations. The surrounding certificate remains at hypothesis grade.
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