theorem
proved
planck_gate_normalized
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IndisputableMonolith.Constants.PlanckScaleMatching on GitHub at line 260.
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257 field_simp [hpi, hphi5]
258
259/-- Equivalent form: c³λ²/(πℏG) = 1. -/
260theorem planck_gate_normalized :
261 c^3 * lambda_rec^2 / (Real.pi * hbar * G) = 1 := by
262 have h := planck_gate_identity
263 have hne : Real.pi * hbar * G ≠ 0 := by
264 apply mul_ne_zero
265 apply mul_ne_zero
266 · exact Real.pi_pos.ne'
267 · exact hbar_pos.ne'
268 · exact G_pos.ne'
269 rw [div_eq_one_iff_eq hne]
270 exact h.symm
271
272/-! ## Summary: The Complete Derivation Chain -/
273
274/-- **PLANCK-SCALE MATCHING CERTIFICATE (C8)**
275
276The derivation chain is complete:
277
2781. ✓ J_bit = J(φ) = φ - 3/2 ≈ 0.118 (from unique cost functional)
2792. ✓ J_curv(λ) = 2λ² (from ±4 curvature on Q₃)
2803. ✓ Extremum: J_bit = J_curv(λ_rec) determines λ_rec
2814. ✓ Face-averaging gives 1/π factor
2825. ✓ λ_rec/ℓ_P = 1/√π ≈ 0.564
283
284**Gap Status**: The curvature packet axiom (J_curv = 2λ²) is hypothesized
285based on the Q₃ geometry. A fully rigorous derivation would require
286showing that ±4 curvature distributed over 8 vertices yields exactly 2λ². -/
287structure PlanckScaleMatchingCert where
288 /-- J_bit is well-defined and positive -/
289 J_bit_ok : J_bit_val > 0
290 /-- J_bit ≈ 0.118 -/