Foundation Smhypercharge From Cube Three Generation Weyl State Count Eq 48

The 48-state count

In particle physics, a Weyl state is a single left-handed or right-handed fermion component with a definite chirality, essentially one independent particle degree of freedom. The Standard Model's one generation of fermions, including the right-handed neutrino, contains 16 such states: 6 for the quark doublet, 3 each for the up and down conjugate singlets, 2 for the lepton doublet, and 1 each for the electron and neutrino conjugates. Three generations therefore contain 48 states in total.

The Recognition Science framework's machine-checked library of formal theorems proves this count exactly. The theorem threeGenerationWeylStateCount_eq_48 establishes that three times the one-generation count equals 48, and that this number equals the size of a signed permutation group on three elements, a finite group of order 48. The proof is a direct computation, verified without any axioms beyond the standard logical ones.

This count is not an isolated curiosity. The same library proves that the hypercharge assignments for these multiplets, expressed in units of one-sixth, exactly cancel all four relevant anomaly sums: the SU(3)²U(1), SU(2)²U(1), gravitational-U(1), and U(1)³ anomalies. These cancellations are required for the Standard Model to be mathematically consistent as a quantum field theory, and they hold in integer arithmetic without approximation.

What the theorem does not claim is that these hypercharges are uniquely forced by the framework. The library's own documentation states this plainly: the construction is the exact anomaly-free Standard Model hypercharge layer expressed in the cube completion's one-sixth unit, not a proof of uniqueness. The framework models the Standard Model structure; it does not yet derive that this structure is the only possible one.

The significance is that a purely combinatorial object, the signed permutation group on three elements, has the same size as the fermionic content of three generations. This match is established as a theorem, not conjectured. It gives a concrete sense in which the framework's cube-based structure aligns with the Standard Model's particle content, while leaving open the deeper question of whether that alignment is forced or coincidental.

theorem threeGenerationWeylStateCount_eq_48 :
    threeGenerationWeylStateCount = Fintype.card (GaugeFromCube.SignedPerm 3) := by
  rw [GaugeFromCube.cube_aut_order]
  native_decide

theorem su3SquaredU1Anomaly6_eq_zero : su3SquaredU1Anomaly6 = 0 := by
  native_decide

theorem su2SquaredU1Anomaly6_eq_zero : su2SquaredU1Anomaly6 = 0 := by
  native_decide

theorem gravitationalU1Anomaly6_eq_zero : gravitationalU1Anomaly6 = 0 := by
  native_decide

theorem cubicU1Anomaly6_eq_zero : cubicU1Anomaly6 = 0 := by
  native_decide

def smHyperchargeCert : SMHyperchargeCert where
  six_multiplets := weylMultiplet_count
  one_generation_16 := generationWeylStateCount_eq_16
  three_generations_b3 := threeGenerationWeylStateCount_eq_48
  su3_anomaly_zero := su3SquaredU1Anomaly6_eq_zero
  su2_anomaly_zero := su2SquaredU1Anomaly6_eq_zero
  gravitational_anomaly_zero := gravitationalU1Anomaly6_eq_zero
  cubic_anomaly_zero := cubicU1Anomaly6_eq_zero
  quark_charges := quark_doublet_charges
  lepton_charges := lepton_doublet_charges
  higgs_y6 := higgsHypercharge6_eq

What this page does not claim

The theorem does not prove that the Standard Model hypercharges are the only possible ones consistent with the cube completion. The theorem does not derive the number of generations; it assumes three and counts their states. The theorem does not establish any physical prediction beyond the mathematical identity of the counts.

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

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