planck_gate_identity
plain-language theorem explainer
The Planck gate identity equates π ħ G to c³ λ_rec² in RS-native units where c = ℓ₀ = τ₀ = 1. Researchers deriving the Planck length from the recognition wavelength in Conjecture C8 would cite this when normalizing constants. The proof unfolds the definitions of G, ħ, λ_rec and applies field simplification after establishing the relevant non-zero conditions.
Claim. $π ħ G = c³ λ_rec²$
background
The Planck-Scale Matching module derives λ_rec ≈ 0.564 ℓ_P from the ledger-curvature extremum. Bit cost is J_bit = J(φ) = cosh(ln φ) - 1 from the unique cost functional. Curvature cost is J_curv(λ) = 2λ² over the 8 faces of the Q₃ hypercube, with equilibrium fixing λ_rec and face-averaging introducing the 1/π factor to give λ_rec = √(ħG/(πc³)). Upstream, lambda_rec is defined in Bridge.DataCore as √(ħ G / c³) while G is expressed as λ_rec² c³ / (π ħ) and ħ = φ^{-5} τ₀ with τ₀ the fundamental tick.
proof idea
The term proof unfolds G, lambda_rec, hbar, c, ell0, cLagLock, tau0 and tick, then applies simp with one_pow and mul_one. It introduces lemmas that π ≠ 0 and φ^{-5} ≠ 0 before invoking field_simp to reach the identity.
why it matters
This identity is invoked by the downstream theorem planck_gate_normalized to obtain c³ λ_rec² / (π ħ G) = 1. It completes the Planck ratio step of Conjecture C8 in the Discrete Informational Framework Paper, linking the recognition length to the Planck length via the phi-ladder and eight-tick octave. It touches the open numerical matching of alpha^{-1} inside (137.030, 137.039).
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