shared_symmetry
The shared_symmetry theorem asserts that J-cost is invariant under inversion for any positive real argument. Researchers validating Recognition Science predictions across language-model layers, photonic codes, and plasma simulations cite this invariance to confirm consistency among the three substrates. The proof is a direct term application of the core Jcost_symm lemma.
claimFor every real number $r > 0$, the J-cost satisfies $J(r) = J(r^{-1})$.
background
J-cost is the Recognition Science cost function whose symmetry under inversion is the algebraic content of the present theorem. The ThreeSubstrateValidationCert module assembles empirical checks from language models (96.4 % layer alignment), photonic qubits (7/8 code rate), and magnetized plasma (convergence to 1.036), all sharing the fixed point at unity and the same descent direction. The upstream Jcost_symm lemma in the Cost module supplies the algebraic identity via field simplification and ring normalization.
proof idea
This is a one-line term proof that applies the Jcost_symm lemma from Cost, passing the positivity hypothesis hr directly.
why it matters in Recognition Science
The theorem supplies the symmetry field inside the ThreeSubstrateCert definition, which bundles fixed-point, descent, and alignment data for the three-substrate certificate. It anchors the claim that J-cost symmetry is substrate-independent, consistent with J-uniqueness in the forcing chain and the multi-channel extension. The surrounding certificate remains at hypothesis grade.
scope and limits
- Does not derive the explicit closed form of J-cost.
- Does not incorporate numerical data from any substrate.
- Does not extend the identity to non-positive or complex arguments.
- Does not address multi-channel J_n generalizations.
Lean usage
symmetry := shared_symmetryformal statement (Lean)
41theorem shared_symmetry {r : ℝ} (hr : 0 < r) :
42 Jcost r = Jcost r⁻¹ := Jcost_symm hr
proof body
Term-mode proof.
43
44/-- Language model validation: 7/8 layer alignment. -/