Masses Mass Genesis Majorana Closure From Cube Charged Coupling Dimension Not Sp

The charged sector's fixed dimension

The Recognition Science framework models particles as patterns on a recognition 3-cube, a discrete record of events with eight vertices, one for each combination of three binary charges. A Dirac fermion, the kind that carries electric charge, spans all eight vertices. A Majorana fermion, its neutral self-conjugate cousin, spans only four, the vertices with even parity. The framework's library of machine-checked formal theorems proves that the charged lepton's internal dimension is fixed at 1, and the proof does not depend on which vertices the charged sector spans.

The theorem charged_coupling_dimension_not_span_determined states the result in precise terms. It assumes a dimension function on the cube's vertices, with the natural rule that reversing all three charges sends a vertex to its dual dimension, the dimension of the complementary face. It assumes the charged lepton's dimension is invariant under charge reversal and is at most 2. From these assumptions alone, the theorem forces the charged lepton's dimension to be exactly 1. The proof never uses the size of the charged sector's span, the set of vertices it occupies. The dimension is fixed by the symmetry of charge reversal and the bound, not by counting vertices.

This separation matters for the framework's account of particle masses. The charged sector's span, all eight vertices for a Dirac fermion, is a separate fact from its internal dimension. The theorem shows that the dimension is a structural necessity, forced by the cube's symmetries, while the span is a modeling choice. The framework's library also proves that the neutral Majorana sector, with its four-vertex span, has a closure factor of one half, the fraction of the cube it covers. The charged sector's dimension being 1 is the same kind of forced structural fact, but it does not by itself determine the charged sector's mass or its couplings.

What the theorem does not claim is just as important. It does not say that the charged lepton's mass is fixed by the dimension alone. It does not say that the charged sector's span is forced to be all eight vertices; that is a definitional choice, not a theorem. It does not say that the Majorana closure factor of one half applies to the charged sector; that factor belongs to the neutral sector. The theorem establishes one fact: under the symmetry and bound assumptions, the charged lepton's internal dimension is 1. The mass ladder that the framework derives for leptons rests on this dimension, but the dimension is not the whole story.

/-- **THEOREM (lepton coupling dimension forced).** Under the C=duality principle
    (`hequiv : cD (−s) = dualDim (cD s)` — charge conjugation acts as cube duality on the
    coupling dimension), a C-invariant lepton (`hCinv : cD sL = cD (−sL)`, electron and
    positron at the same dimension) with a valid coupling dimension (`≤ 2`) is FORCED to
    couple at the self-dual edge dimension `d = 1`. Derived from the symmetry principle —
    not asserted. -/
theorem lepton_dimension_forced
    (cD : ℤ → ℕ) (sL : ℤ)
    (hequiv : ∀ s, cD (-s) = dualDim (cD s))
    (hCinv : cD sL = cD (-sL))
    (hle : cD sL ≤ 2) :
    cD sL = 1 := by
  have e := hequiv sL
  unfold dualDim at e
  omega

/-- The three charged sectors are full-strand ⇒ their spans coincide. Any span-reading
    function takes one common value on all three. -/
theorem charged_spans_force_common_value
    (f : Finset Cube → ℕ)
    (upSpan leptonSpan downSpan : Finset Cube)
    (hup : upSpan = diracSpan) (hlep : leptonSpan = diracSpan) (hdn : downSpan = diracSpan) :
    f upSpan = f leptonSpan ∧ f leptonSpan = f downSpan := by
  subst hup hlep hdn; exact ⟨rfl, rfl⟩

theorem diracSpan_card : diracSpan.card = 8 := cube_card

theorem majoranaSpan_card : majoranaSpan.card = 4 := evenClass_card

What this page does not claim

The theorem does not determine the charged lepton's mass or its couplings. The theorem does not force the charged sector to span all eight vertices. The Majorana closure factor of one half does not apply to the charged sector.

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

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