planck_scale_matching_cert
The declaration assembles a verified certificate confirming that the recognition wavelength satisfies λ_rec ≈ 0.564 ℓ_P from equating bit cost J(φ) to curvature cost 2λ² at equilibrium. Researchers deriving constants from the Recognition Science ledger-curvature argument would cite it as the completed form of Conjecture C8. The construction is a direct record that supplies the four required fields from existing positivity, bound, extremum, and ratio lemmas.
claimThe Planck-scale matching certificate holds: $J(φ) > 0$, $0.11 < J(φ) < 0.12$, the extremum condition $J_{curv}(λ_{rec}) = J(φ)$ is satisfied, and the wavelength ratio obeys $λ_{rec}/ℓ_P = 1/√π$.
background
The module derives λ_rec from the unique cost functional J(x) = ½(x + x⁻¹) - 1 evaluated at the self-similar fixed point φ, yielding the bit cost J_bit = J(φ) = cosh(ln φ) - 1. Curvature cost is defined as J_curv(λ) = 2λ² by distributing a ±4 packet over the eight faces of the Q₃ hypercube. Equilibrium sets J_bit = J_curv(λ_rec); restoring SI dimensions via c³λ²/(ℏG) and face averaging introduces the factor 1/π, producing λ_rec = ℓ_P/√π.
proof idea
The definition constructs an instance of the PlanckScaleMatchingCert structure by direct field assignment. J_bit_ok is taken from the positivity theorem J_bit_pos, J_bit_numerical from the bound theorem J_bit_bounds, extremum_determines from extremum_condition, and planck_ratio from lambda_rec_over_ell_P. No further tactics or reductions are applied.
why it matters in Recognition Science
This definition completes the formal verification of the Planck-scale matching step required by Conjecture C8 in the Discrete Informational Framework paper. It closes the chain that begins with J-uniqueness (T5) and the forcing of φ (T6), supplying the numerical factor needed for downstream constant derivations. The certificate structure makes the entire ledger-curvature argument auditable as a single object.
scope and limits
- Does not derive the cost functional J from first principles.
- Does not address the physical mechanism of curvature packets.
- Does not extend the result beyond the specific Q₃ geometry and native RS units.
- Does not supply error bounds or higher-order corrections to the 0.564 factor.
formal statement (Lean)
298def planck_scale_matching_cert : PlanckScaleMatchingCert where
299 J_bit_ok := J_bit_pos
proof body
Definition body.
300 J_bit_numerical := J_bit_bounds
301 extremum_determines := extremum_condition
302 planck_ratio := lambda_rec_over_ell_P
303
304/-- Summary report for the Planck-Scale Matching derivation. -/