PlanckScaleMatchingCert
The PlanckScaleMatchingCert structure bundles four properties that close the loop from the unique cost functional J at the golden ratio to the Planck length ratio. Researchers closing Conjecture C8 in discrete informational frameworks would cite it to confirm the numerical match and extremum condition. The certificate is assembled directly as a structure from the definitions of bit cost, curvature packet, derived wavelength, and SI-scaled ratio.
claimThe certificate asserts that the bit cost satisfies $J>0$ and $0.11<J<0.12$, that the curvature cost equals the bit cost at the recognition wavelength, and that the SI recognition wavelength divided by the Planck length equals $1/√π$, where $J(x)=(x+x^{-1})/2-1$, the curvature cost is $2λ^2$, the recognition wavelength solves $J=2λ^2$, and the Planck length is $√(ℏG/c^3)$.
background
The module derives the recognition wavelength from the unique cost functional J(x) = ½(x + x^{-1}) - 1 evaluated at the self-similar fixed point φ. J_bit_val is defined as J(φ) and equals cosh(ln φ) - 1. J_curv(λ) is defined as 2λ² under the curvature packet hypothesis that distributes ±4 curvature quanta over the eight vertices of the 3-cube Q₃. lambda_rec_from_Jbit solves the extremum equation J_bit = J_curv(λ) to give √(J_bit/2). lambda_rec_SI restores dimensions via face averaging to produce √(ℏG/(π c³)). ell_P is the standard Planck length √(ℏG/c³).
proof idea
Structure definition that directly bundles the four properties. J_bit_ok and J_bit_numerical are taken from the positivity and bounds lemmas on J_bit_val. extremum_determines follows by substitution into the definition of J_curv and lambda_rec_from_Jbit. planck_ratio follows by algebraic simplification of lambda_rec_SI over ell_P using the face-averaging factor 1/π.
why it matters in Recognition Science
This certificate completes the C8 derivation chain in the module and supplies the four fields required by the downstream planck_scale_matching_cert definition. It links the T5 J-uniqueness result and the phi fixed point to the Planck scale via the eight-tick geometry of Q₃. The curvature packet axiom remains a physical hypothesis whose full derivation from Gauss-Bonnet on the 3-cube is left open.
scope and limits
- Does not derive the curvature packet axiom J_curv = 2λ² from first principles.
- Does not extend the matching beyond the 3-cube geometry.
- Does not compute the numerical value of λ_rec beyond the supplied bounds.
- Does not address renormalization or higher-order corrections to the ratio.
formal statement (Lean)
287structure PlanckScaleMatchingCert where
288 /-- J_bit is well-defined and positive -/
289 J_bit_ok : J_bit_val > 0
290 /-- J_bit ≈ 0.118 -/
291 J_bit_numerical : 0.11 < J_bit_val ∧ J_bit_val < 0.12
292 /-- The extremum determines λ_rec -/
293 extremum_determines : J_curv lambda_rec_from_Jbit = J_bit_val
294 /-- The Planck ratio is 1/√π -/
295 planck_ratio : lambda_rec_SI / ell_P = 1 / sqrt Real.pi
296
297/-- The Planck-Scale Matching Certificate is verified. -/