D3_viable
plain-language theorem explainer
D = 3 meets the three spectral viability conditions: linking enforces the dimension via Alexander duality on the cube, at least three face pairs permit three generations, and lcm(8,45) equals 360 for gap synchronization. A physicist deriving the Standard Model gauge content and fermion generations from discrete geometry would cite this to confirm only D = 3 works. The proof builds the viability record directly, discharging the linking field by reflexivity and the two numerical checks by native decision.
Claim. The viability structure at dimension 3 holds: linking requires $D=3$, sufficient generations requires at least three face pairs, and gap synchronization requires that the least common multiple of $2^D$ and 45 equals 360.
background
SpectralViability is the structure that records the minimal requirements for spectral emergence in dimension D. Its three fields are linking (which forces D = 3 by Alexander duality on the binary cube), sufficient_generations (which demands at least three face pairs to produce three particle generations), and gap_sync (which requires Nat.lcm (2^D) 45 = 360 to align with the eight-tick octave).
proof idea
The proof constructs the SpectralViability record for D = 3. The linking field is settled by reflexivity. The sufficient_generations and gap_sync fields are discharged by native_decide, which evaluates the concrete arithmetic predicates on face_pairs 3 and lcm(8,45).
why it matters
This theorem supplies the viability certificate required by the master theorem spectral_emergence, which concludes that the full Standard Model gauge group, 24 chiral fermions, and consciousness ground state all follow from D = 3. It closes the loop from the forcing chain T8 through Q3 symmetries to the observed particle content, confirming that alternative dimensions fail at least one numerical condition.
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