planck_scale_matching_report
plain-language theorem explainer
The declaration supplies a string report on the Planck-scale matching conjecture C8, equating the bit cost J(φ) ≈ 0.118 to the curvature cost 2λ² to obtain λ_rec ≈ 0.564 ℓ_P. Researchers in discrete spacetime models cite it to track the status of the recognition wavelength derivation. The implementation is a direct string concatenation of the five-step chain with status annotations.
Claim. The report asserts λ_rec = √(ℏG/(π c³)) = ℓ_P / √π, derived by solving J(φ) = 2 λ² where J(x) = ½(x + x^{-1}) - 1 and incorporating the 8-face averaging factor 1/π from Q₃ geometry.
background
In Recognition Science the cost functional is J(x) = ½(x + x^{-1}) - 1, with the self-similar fixed point at φ satisfying J(φ) = φ - 3/2. The module introduces J_curv(λ) = 2λ² as the cost of a ±4 curvature packet on the Q₃ hypercube. Upstream, the G constant is defined as λ_rec² c³ / (π ℏ) from this matching. The local setting is the ledger-curvature extremum argument for Conjecture C8.
proof idea
The definition constructs the report by concatenating fixed text blocks that list the derivation chain: J_bit equality, J_curv axiom, extremum solution, face-averaging, and the resulting ratio. No lemmas are applied; it is a static documentation string.
why it matters
This report documents the current status of the Planck matching step that supplies the expression for G in the constants module. It highlights the remaining gap in deriving J_curv from Q₃ geometry, which would close the hypothesis_interface for the curvature packet. It connects to the J-uniqueness theorem (T5) and the phi fixed point (T6) in the forcing chain, and to the eight-tick octave that fixes D = 3.
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