D2_fails_sync
plain-language theorem explainer
The declaration establishes that lcm(4,45) differs from 360, showing a synchronization failure for two spatial dimensions under the Recognition Science periodicity condition. Researchers verifying dimensional uniqueness in the derivation of the Standard Model from Q3 would cite this result when confirming that only D=3 satisfies the full set of algebraic and combinatorial constraints. The proof is a direct native computation of the LCM inequality with no intermediate steps.
Claim. For two spatial dimensions the required synchronization condition fails: lcm(2^D, 45) ≠ 360, since lcm(4, 45) ≠ 360.
background
The Spectral Emergence module derives SU(3) × SU(2) × U(1) gauge content, three generations, and 48 chiral fermion states from the binary cube Q3 = {0,1}^3 that arises once T8 forces D=3. The module documentation states that the cube symmetry group satisfies |Aut(Q3)| = 2^D × D! = 48 and equals the number of chiral fermionic states, with the eight-tick octave (T7) and phi-forced fixed point (T6) supplying the mass ladder. This theorem supplies one concrete numerical obstruction in the claim that no alternative dimension satisfies every requirement.
proof idea
The proof is a one-line wrapper that applies native_decide to evaluate Nat.lcm (2^2) 45 and confirm the inequality holds.
why it matters
The result closes the D=2 case in the module's exhaustive check that D ≠ 3 fails at least one requirement, supporting the self-consistency loop T8 (D=3) → Q3 → Aut(Q3) = B3 → gauge dimensions 3+2+1. It aligns with the framework landmarks that only three dimensions produce the observed 24 chiral flavors and phi-ladder mass hierarchy without free parameters.
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