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theorem proved tactic proof

planck_gate_normalized

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formal statement (Lean)

 260theorem planck_gate_normalized :
 261    c^3 * lambda_rec^2 / (Real.pi * hbar * G) = 1 := by

proof body

Tactic-mode proof.

 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. -/

depends on (34)

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