shared_fixed_point
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 in Recognition Science
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.
scope and limits
- Does not establish the 96.4 percent alignment rate reported for language-model layers.
- Does not derive the photonic code rate or plasma convergence value.
- Does not obtain the J-cost expression from the Recognition Composition Law.
- Does not address multi-channel J_n extensions beyond the unit case.
Lean usage
example : Jcost 1 = 0 := shared_fixed_pointformal statement (Lean)
34theorem shared_fixed_point : Jcost 1 = 0 := Jcost_unit0
proof body
Term-mode proof.
35
36/-- All three substrates exhibit J-cost descent: off-equilibrium costs positive. -/