D1_fails_sync
The theorem shows that the synchronization condition lcm(2,45) equals 360 fails for spatial dimension D=1, ruling out this case in the Recognition Science forcing chain. A physicist checking dimensional consistency for the Q3 cube and eight-tick octave would cite it to exclude D=1 before invoking the D=3 case. The proof is a direct computational check via native_decide on the concrete natural-number inequality.
claim$lcm(2,45) ≠ 360$
background
The module SpectralEmergence starts from the forced datum D=3 (T8) to obtain the binary cube Q3 with 8 vertices and derives the Standard Model gauge content plus 48 chiral fermion states from |Aut(Q3)|=48. It also records that alternative dimensions fail at least one numerical requirement, including synchronization conditions tied to the eight-tick octave (T7). The constant 45 appearing in the lcm is part of the octave-period consistency check that must hold for the phi-ladder mass hierarchy to close consistently with the 2^D factor.
proof idea
The proof is a one-line term wrapper that applies native_decide to evaluate the concrete inequality on natural numbers.
why it matters in Recognition Science
The declaration supplies one concrete instance of the module claim that no alternative dimension works, thereby supporting the uniqueness of D=3 in the forcing chain T5-T8. It closes a numerical consistency check required before the automorphism order |Aut(Q3)|=48 can be matched to the 24 chiral fermion flavors and the SU(3)×SU(2)×U(1) sector dimensions. No open scaffolding remains for this specific computation.
scope and limits
- Does not derive the numerical value 45 from upstream constants.
- Does not address synchronization for D>1 or D=2.
- Does not invoke J-cost, defectDist, or the phi-ladder mass formula.
- Does not prove that D=1 fails every possible consistency condition.
formal statement (Lean)
408theorem D1_fails_sync : Nat.lcm (2 ^ 1) 45 ≠ 360 := by native_decide
proof body
Term-mode proof.
409
410/-- **THEOREM**: D = 2 fails (lcm(4,45) = 180 ≠ 360). -/