shared_fixed_point
plain-language theorem explainer
The theorem establishes that J-cost vanishes at the argument one, confirming the equilibrium point common to the language-model, photonic-qubit, and plasma substrates. Researchers certifying Recognition Science predictions across these empirical domains would cite it to anchor the shared fixed-point property. The proof is a direct one-line application of the unit lemma for J-cost.
Claim. $J(1) = 0$, where $J(x) = (x-1)^2/(2x)$ is the J-cost function.
background
J-cost is the squared-ratio function $J(x) = (x-1)^2/(2x)$ supplied by the Cost module. The ThreeSubstrateValidationCert module assembles checks from language models, photonic qubits, and magnetized plasma, all asserted to share the fixed point at unity. Upstream results include the equilibrium theorem (J-cost at one equals zero) and the definition of recognition-event cost as J-cost.
proof idea
One-line wrapper that applies the Jcost_unit0 lemma from the Cost module.
why it matters
The result supplies the fixed_point component inside the threeSubstrateCert definition that bundles fixed point, descent, and symmetry for the validation certificate. It supports the module claim that J-cost uniqueness underlies all three substrates, consistent with the Recognition Science equilibrium condition at J = 0. The module places the overall certificate at hypothesis grade.
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