Foundation Smgauge Algebra Smgauge Factor

The gauge-factor record

The Standard Model of particle physics describes three of the four known forces with a shared mathematical language: the strong force, the weak force, and electromagnetism (unified with the weak force at high energies). Each force has a gauge group, a set of continuous symmetries whose associated force carriers are counted by the group's generators. The strong force's group SU(3) has 8 generators, corresponding to 8 gluons; the weak force's SU(2) has 3, corresponding to the W and Z bosons; hypercharge U(1) has 1, corresponding to the photon. The total is 12 gauge bosons, a number confirmed by decades of particle physics experiments.

In Recognition Science, the declaration SMGaugeFactor (a discrete record of the three gauge groups) formalizes exactly this count. It defines three named factors: strong, weak, and hyperY, and attaches to each its generator count: 8, 3, and 1. The declaration proves, by direct computation, that these counts sum to 12 and that there are exactly three factors. The counts follow the standard formulas: SU(N) has N²−1 generators, and U(N) has N², so for N=3, 2, and 1 the counts are 8, 3, and 1. The result is a theorem in the framework's machine-checked library of formal theorems, with no unproved assumptions.

The declaration's significance lies in its connection to a larger claim. A separate result in the framework proves that the symmetry group of a cube decomposes as B₃ = (ℤ/2)³ ⋊ S₃, whose rank structure (3, 2, 1) matches the gauge group ranks. This declaration adds the Lie-algebra-level detail: the generator counts (8, 3, 1) match the dimensions of the three gauge groups. The structural prediction is that any gauge group derived from the same cube-automorphism rank decomposition has exactly 12 generators; a deviation would falsify the identification of gauge groups with cube symmetries.

What the declaration does not claim is equally important. It does not derive the gauge groups from first principles; the identification of cube automorphisms with gauge groups is a separate, honestly tagged STARTED result, not a completed proof. It does not predict the existence of new particles, the values of coupling constants, or any dynamical behavior of the forces. It only establishes a counting fact: the three named factors have 8, 3, and 1 generators, totaling 12, matching the empirical count of Standard Model gauge bosons. The empirical match is a check, not a derivation.

theorem strong_gen_count : factorGenCount .strong = 8 := by decide

theorem weak_gen_count : factorGenCount .weak = 3 := by decide

theorem hyper_gen_count : factorGenCount .hyperY = 1 := by decide

theorem sm_total_gen_count : smTotalGenCount = 12 := by decide

/-- Number of SM gauge factors = 3, matching the cube-automorphism
three-layer decomposition. -/
theorem factor_count : Fintype.card SMGaugeFactor = 3 := by decide

/-- Number of generators of `su(N)` is `N² - 1`. -/
def suGenCount (N : ℕ) : ℕ := N * N - 1

/-- Number of generators of `u(N)` is `N²`. -/
def uGenCount (N : ℕ) : ℕ := N * N

/-- SM gauge-algebra certificate. -/
def smGaugeAlgebraCert : SMGaugeAlgebraCert where
  strong := strong_gen_count
  weak := weak_gen_count
  hyperY := hyper_gen_count
  total := sm_total_gen_count
  factor_count := factor_count

What this page does not claim

It does not derive the gauge groups from first principles; the cube-to-gauge identification is a separate STARTED result. It does not predict any coupling constants, particle masses, or dynamical behavior of the forces. It does not claim the empirical match of 12 generators is a derivation; it is a check against measurement.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/SMGaugeAlgebra.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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