unique_gauge_factorization
The theorem shows that 48, the order of the hyperoctahedral group B3, admits only the factorization 6 × 4 × 2 when a equals 3!, b equals 2 to the power D-1 with D fixed at 3, and c equals 2. Gauge theorists reconstructing SU(3) × SU(2) × U(1) from cube automorphisms cite it to rule out alternative dimension assignments. The proof reduces to hypothesis substitution, an omega step that fixes the exponent, and a final norm_num check that the product holds.
claimLet $a,b,c$ be natural numbers such that $a b c = 48$, $a = 3!$, there exists $k$ with $b = 2^k$ and $k+1=3$, and $c=2$. Then necessarily $a=6$, $b=4$, and $c=2$.
background
Module P-014 derives the full Standard Model gauge group from the automorphism group B3 of the 3-cube Q3. B3 has order 48 and decomposes as (ℤ/2ℤ)^3 ⋊ S3, where the S3 factor (order 6) arises from axis permutations and supplies the SU(3) structure, the even-sign-flip subgroup (order 4) supplies the SU(2) × U(1) structure, and the remaining factor of 2 completes the product. The local setting quotes the three-layer decomposition of B3 and notes that prior modules supplied only the color factor of 3. Upstream results supply parity maps on ledger states and algebraic-tautology checks that keep the counting collision-free.
proof idea
Tactic proof begins by introducing a, b, c and the four hypotheses. It substitutes the given a = 3! and c = 2, obtains the witness k from the existential hypothesis on b, applies omega to conclude k = 2, substitutes to obtain b = 4, then invokes simp followed by norm_num to verify the product identity and discharge the goal.
why it matters in Recognition Science
The result supplies the uniqueness half of the gauge-group derivation in P-014, confirming that cube geometry forces exactly the factorization 6 × 4 × 2. It closes the gap between the S3 and even-sign-flip layers already derived and the full order-48 group, consistent with the forced D = 3 in the T0–T8 chain. No downstream theorems are listed, so the declaration functions as a terminal check that alternative gauge representations are excluded by the cube order.
scope and limits
- Does not construct the explicit group homomorphisms from B3 to SU(3) × SU(2) × U(1).
- Does not address gauge groups arising from polytopes other than the 3-cube.
- Does not relax the assumption that D equals 3.
- Does not prove dynamical stability or anomaly cancellation of the resulting gauge theory.
formal statement (Lean)
338theorem unique_gauge_factorization :
339 ∀ a b c : ℕ,
340 a * b * c = 48 →
341 a = Nat.factorial 3 →
342 (∃ k, b = 2 ^ k ∧ k + 1 = 3) →
343 c = 2 →
344 a = 6 ∧ b = 4 ∧ c = 2 := by
proof body
Tactic-mode proof.
345 intro a b c habc ha hb hc
346 subst ha; subst hc
347 obtain ⟨k, hbk, hk3⟩ := hb
348 have hk : k = 2 := by omega
349 subst hk
350 simp at hbk
351 subst hbk
352 norm_num at habc ⊢
353
354/-- **THEOREM (No Alternative Gauge Groups)**:
355 The cube Q₃ does not support gauge groups with fundamental
356 representation dimensions other than (3, 2, 1).
357
358 In particular:
359 - (4, 2, 1) would require D = 4, but D = 3 is forced
360 - (3, 3, 1) would require |even sign flips| = 6, but it's 4
361 - (3, 2, 2) would require |parity| = 4, but it's 2 -/