Foundation Smhypercharge From Cube Generation Weyl State Count Eq 16

The sixteen-state generation

In particle physics, a generation of matter is the complete set of particles that participate in the weak force as a family: the up and down quarks, the electron and its neutrino, and their antiparticles. Each of these comes in variants, such as color charge for quarks or spin orientation for leptons. When physicists count every distinct left-handed state in one generation, the total is sixteen: six for the quark doublet, three for each of the two quark singlet types, two for the lepton doublet, and one each for the electron and neutrino singlets.

This count is not a new physical discovery but a well-established fact of the Standard Model. The framework's contribution is to derive the same number from a different starting point: a cube-completion scheme that assigns each particle type a multiplicity and a hypercharge value in units of one-sixth. The declaration generationWeylStateCount_eq_16 formally states that the sum of these multiplicities equals 16, and the machine-checked library of formal theorems verifies the arithmetic by computation. The same library also checks that the hypercharge assignments cancel all four anomaly sums exactly, which is the consistency condition that makes the Standard Model mathematically coherent.

What the declaration does not establish is that these hypercharges are uniquely forced by the cube structure. The docstring is explicit on this point: it presents the anomaly-free Standard Model hypercharge layer expressed in the cube's one-sixth unit, but it does not prove that no other assignment could work. The count of sixteen is a theorem; the uniqueness of the hypercharge values is not. The framework also does not claim that the cube scheme predicts the number of generations; the three-generation count of 48 is a separate statement that matches the cube's symmetry group order but does not derive the observed three generations from first principles.

For a reader, the practical upshot is a precise, verified accounting: one generation has sixteen Weyl states, three generations have forty-eight, and the hypercharge assignments cancel the anomalies exactly. This is a consistency check, not a derivation of the Standard Model's structure. The cube scheme reproduces a known fact in a new notation, and the proof ensures the arithmetic is correct.

theorem generationWeylStateCount_eq_16 : generationWeylStateCount = 16 := by
  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

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

What this page does not claim

The hypercharge values are uniquely forced by the cube structure. The framework derives the number of generations from first principles. The cube scheme predicts the Standard Model gauge group; it only reproduces the skeleton.

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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