`planck_gate_identity`

proved

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module: IndisputableMonolith.Constants.PlanckScaleMatching
domain: Constants
line: 251 · github
papers citing: none yet

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IndisputableMonolith.Constants.PlanckScaleMatching on GitHub at line 251.

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Derivations using this theorem

Planck Length in RS Units · derived / THEOREM
The Planck length is reconstructible as a derived rung of the phi-ladder
RS Lambda_rec Matches the Planck Length · derived / THEOREM
The recognition length lambda_rec coincides with the Planck length in SI

depends on

used by

planck_gate_normalized IndisputableMonolith.Constants.PlanckScaleMatching

formal source

 248    The physical content is the relationship λ_rec/ℓ_P = 1/√π.
 249
 250    The Planck gate identity: π · ℏ · G = c³ · λ_rec². -/
 251theorem planck_gate_identity :
 252    Real.pi * hbar * G = c^3 * lambda_rec^2 := by
 253  unfold G lambda_rec hbar c ell0 cLagLock tau0 tick
 254  simp only [one_pow, mul_one]
 255  have hpi : Real.pi ≠ 0 := Real.pi_pos.ne'
 256  have hphi5 : phi ^ (-(5 : ℝ)) ≠ 0 := (Real.rpow_pos_of_pos phi_pos _).ne'
 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

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