 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 -/
 291  J_bit_numerical : 0.11 < J_bit_val ∧ J_bit_val < 0.12
 292  /-- The extremum determines λ_rec -/
 293  extremum_determines : J_curv lambda_rec_from_Jbit = J_bit_val
 294  /-- The Planck ratio is 1/√π -/
 295  planck_ratio : lambda_rec_SI / ell_P = 1 / sqrt Real.pi
 296
 297/-- The Planck-Scale Matching Certificate is verified. -/
 298def planck_scale_matching_cert : PlanckScaleMatchingCert where
 299  J_bit_ok := J_bit_pos
 300  J_bit_numerical := J_bit_bounds
 301  extremum_determines := extremum_condition
 302  planck_ratio := lambda_rec_over_ell_P
 303
 304/-- Summary report for the Planck-Scale Matching derivation. -/
 305def planck_scale_matching_report : String :=
 306  "PLANCK-SCALE MATCHING (Conjecture C8)\n" ++
 307  "=====================================\n" ++
 308  "\n" ++
 309  "DERIVATION CHAIN:\n" ++
 310  "  1. J_bit = J(φ) = φ - 3/2 ≈ 0.118 [PROVED]\n" ++
 311  "  2. J_curv(λ) = 2λ² (curvature packet) [AXIOMATIZED]\n" ++
 312  "  3. Extremum: J_bit = J_curv → λ_rec [PROVED]\n" ++
 313  "  4. Face-averaging → 1/π factor [AXIOMATIZED]\n" ++
 314  "  5. λ_rec/ℓ_P = 1/√π ≈ 0.564 [PROVED]\n" ++
 315  "\n" ++
 316  "RESULT: λ_rec = √(ℏG/(πc³)) ≈ 0.564 ℓ_P\n" ++
 317  "\n" ++

papers checked against this theorem

No papers in the current corpus map onto this theorem yet.