lm_fraction_eq
The equality fixes the language model alignment fraction at exactly 7/8. Workers building the three-substrate validation certificate cite this value to confirm the shared fixed point and descent properties across language models, photonic qubits, and magnetized plasma. The proof is a direct reflexivity on the constant definition of the fraction.
claimThe language model alignment fraction equals $7/8$.
background
The module documents a hypothesis-grade certificate for J-cost uniqueness across three substrates: language models (7/8 layer alignment), photonic qubits (code rate 7/8), and magnetized plasma (convergence to x=1.036). All substrates share the fixed point at x=1 where J(1)=0, descent along the J-cost gradient, and the multi-channel J_n extension. The language model alignment fraction is introduced as the rational constant 7/8 that records the proportion of MLP layers in which J-cost beats cross-entropy.
proof idea
The proof is a one-line wrapper that applies reflexivity to the definition of the language model alignment fraction.
why it matters in Recognition Science
This equality populates the lm_alignment field inside the threeSubstrateCert definition, which also records validationSubstrateCount, shared_fixed_point, shared_descent, and shared_symmetry. It supplies the concrete numerical anchor for the hypothesis-grade certificate that validates J-cost across substrates, consistent with the Recognition Science fixed point at x=1 and the eight-tick octave structure. The certificate remains open for further formalization of the reported 96.4 percent empirical alignment rate.
scope and limits
- Does not prove the reported 96.4 percent empirical alignment rate in language models.
- Does not derive the 7/8 value from the Recognition Composition Law or J-uniqueness axioms.
- Does not extend the equality to the photonic or plasma substrate measurements.
- Does not discharge the hypothesis-grade status of the overall certificate.
Lean usage
lm_alignment := lm_fraction_eqformal statement (Lean)
46theorem lm_fraction_eq : languageModelAlignmentFraction = 7/8 := rfl