SpectralViability

formal source

 393    1. Non-trivial linking (forces D = 3 by Alexander duality)
 394    2. At least 3 face pairs (for 3 generations)
 395    3. Gap-45 synchronization: lcm(2^D, 45) = 360 -/
 396structure SpectralViability (D : ℕ) where
 397  linking : D = 3
 398  sufficient_generations : 3 ≤ face_pairs D
 399  gap_sync : Nat.lcm (2 ^ D) 45 = 360
 400
 401/-- **THEOREM**: D = 3 satisfies all spectral viability requirements. -/
 402theorem D3_viable : SpectralViability 3 where
 403  linking := rfl
 404  sufficient_generations := by native_decide
 405  gap_sync := by native_decide
 406
 407/-- **THEOREM**: D = 1 fails (lcm(2,45) = 90 ≠ 360). -/
 408theorem D1_fails_sync : Nat.lcm (2 ^ 1) 45 ≠ 360 := by native_decide
 409
 410/-- **THEOREM**: D = 2 fails (lcm(4,45) = 180 ≠ 360). -/
 411theorem D2_fails_sync : Nat.lcm (2 ^ 2) 45 ≠ 360 := by native_decide
 412
 413/-- **THEOREM**: D = 4 fails (lcm(16,45) = 720 ≠ 360). -/
 414theorem D4_fails_sync : Nat.lcm (2 ^ 4) 45 ≠ 360 := by native_decide
 415
 416/-- **THEOREM**: D = 5 fails (lcm(32,45) = 1440 ≠ 360). -/
 417theorem D5_fails_sync : Nat.lcm (2 ^ 5) 45 ≠ 360 := by native_decide
 418
 419/-- **THEOREM**: Only D = 3 gives lcm(2^D, 45) = 360 for D ∈ {1,...,8}. -/
 420theorem gap_sync_unique :
 421    ∀ D : Fin 8, Nat.lcm (2 ^ (D.val + 1)) 45 = 360 → D.val + 1 = 3 := by
 422  decide
 423
 424/-- **THEOREM**: D = 3 is the unique spectral emergence dimension.
 425    For D ∈ {1,..,8}, only D = 3 satisfies gap-45 synchronization
 426    AND has face_pairs ≥ 3. -/