threeSubstrateCert
The definition constructs a record bundling J-cost fixed-point, positivity, symmetry, and 7/8 alignment properties across three substrates. Recognition Science researchers cite it to reference the consolidated empirical support for the J-cost function in RS Exp B. The construction is a direct record assembly that references prior lemmas for each field.
claimLet $J$ be the J-cost function. The three-substrate certificate asserts that the set of validation substrates has cardinality 3, $J(1)=0$, $J(r)>0$ for all $r>0$ with $r≠1$, $J(r)=J(r^{-1})$ for all $r>0$, the language-model alignment fraction equals $7/8$, the photonic code rate equals $7/8$, and the alignment fraction equals $(2^3-1)/2^3$.
background
The J-cost function Jcost encodes recognition cost with a fixed point at unity, strict positivity away from equilibrium, and invariance under inversion. Module documentation states that RS Exp B validates these properties across language models (96.4% of MLP layers), photonic qubits (code rate 7/8, leakage 0.02%), and magnetized plasma (convergence to x=1.036), all sharing the same predictions at hypothesis grade.
proof idea
This definition is a direct record construction that populates each field of the ThreeSubstrateCert structure by referencing the corresponding shared lemmas: fixed_point from shared_fixed_point, descent from shared_descent, symmetry from shared_symmetry, lm_alignment from lm_fraction_eq, photonic_rate from photonic_code_rate_rfl, and f2_cube_connection from seven_eightths_from_F2_cube.
why it matters in Recognition Science
This certificate consolidates J-cost uniqueness (T5) across independent substrates and supplies the Lean interface for the hypothesis-grade validation in RS Exp B. It confirms the fixed point, descent, and symmetry that underpin the phi-ladder and eight-tick octave. The module documentation notes that the Lean content rests on J-cost uniqueness and touches the bridge from empirical observations to the T0-T8 chain.
scope and limits
- Does not contain a formal proof of the experimental outcomes.
- Does not derive the J-cost properties from first principles.
- Does not extend to substrates beyond the three enumerated.
- Does not claim numerical agreement with physical constants beyond the stated fractions.
formal statement (Lean)
68def threeSubstrateCert : ThreeSubstrateCert where
69 three_substrates := validationSubstrateCount
proof body
Definition body.
70 fixed_point := shared_fixed_point
71 descent := shared_descent
72 symmetry := shared_symmetry
73 lm_alignment := lm_fraction_eq
74 photonic_rate := photonic_code_rate_rfl
75 f2_cube_connection := seven_eighths_from_F2_cube
76
77end IndisputableMonolith.Foundation.ThreeSubstrateValidationCert