shared_descent
The shared_descent theorem asserts that J-cost is strictly positive for any positive real r different from 1. Researchers validating the three-substrate certificate in Recognition Science cite it to confirm common descent behavior across language-model layers, photonic codes, and plasma equilibria. The proof is a direct one-line application of the J-cost positivity lemma.
claimFor every real number $r > 0$ with $r ≠ 1$, the J-cost satisfies $0 < J(r)$, where $J$ is the cost function induced by a multiplicative recognizer comparator.
background
J-cost quantifies deviation from equilibrium in recognition events and satisfies J(1) = 0 with strict positivity off equilibrium. The upstream lemma Jcost_pos_of_ne_one states that J(x) > 0 for x ≠ 1 and x > 0; its proof rewrites Jcost via Jcost_eq_sq and applies positivity of squares over a positive denominator. The module ThreeSubstrateValidationCert assembles shared J-cost properties across three empirical substrates (language models, photonic qubits, magnetized plasma) that all exhibit descent from the fixed point at unity.
proof idea
The proof is a one-line wrapper that applies the lemma Jcost_pos_of_ne_one to the hypotheses hr and hne.
why it matters in Recognition Science
This theorem supplies the descent field to the threeSubstrateCert definition, which bundles fixed point, descent, symmetry, and alignment metrics. It supports the module claim that all three substrates share J-cost descent from x = 1, consistent with J-uniqueness in the forcing chain. The certificate remains at hypothesis grade pending further empirical closure.
scope and limits
- Does not establish numerical alignment fractions for language models.
- Does not derive the photonic code rate of 7/8.
- Does not address the plasma convergence value near 1.036.
- Does not prove uniqueness of the J function beyond positivity.
Lean usage
example (r : ℝ) (hr : 0 < r) (hne : r ≠ 1) : 0 < Jcost r := shared_descent r hr hneformal statement (Lean)
37theorem shared_descent {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
38 0 < Jcost r := Jcost_pos_of_ne_one r hr hne
proof body
Term-mode proof.
39
40/-- All three validate J-cost symmetry: J(r) = J(1/r). -/