Masses Mass Genesis Majorana Closure From Cube Charged Spans Force Common Value

The common value

A recognition cube, a discrete record of events arranged as the eight corners of a three-dimensional cube, carries a simple counting fact. The theorem charged_spans_force_common_value states that if three subsets of the cube's vertices are each equal to the full set of eight vertices, then any function that counts vertices assigns the same number to all three. That is, when the spans called up, lepton, and down are all the whole cube, their counts cannot differ. The proof is immediate: substituting equal sets into a function forces equal outputs.

The theorem is a statement about equality, not about physics. It does not say that up, lepton, and down spans are equal in nature; it says that if they are, any vertex-counting function treats them alike. The declaration does not compute a mass, a ratio, or a coupling. It does not mention the golden ratio, the neutrino sector, or any empirical value. It is a lemma about functions on finite sets, proved by reflexivity and substitution, and it carries no physical content beyond the equality it assumes.

In the Recognition Science framework, the declaration sits inside a larger derivation about Majorana closure. The framework models a neutral, self-conjugate fermion as spanning half the cube's vertices, four of eight, while a charged Dirac fermion spans all eight. The theorem here is a preliminary consistency check: it shows that if the charged spans are all the same set, their counts agree. It does not by itself establish the half factor, the mass ratio, or any physical prediction. Those results live in other declarations, such as evenClass_card and majoranaSpan_card, which prove that the even-parity class and the Majorana span each contain four vertices.

What the theorem changes is the reading of the derivation. A reader might worry that the framework assigns different counts to different charged spans; this declaration removes that worry by showing that equal spans force equal counts. The consequence is that the framework's later claims about charged sectors can rely on a single common counting value, provided the spans are indeed the full cube. The theorem does not prove that they are; it proves only the conditional equality.

/-- 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⟩

/-- 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⟩

/-- Majorana (self-conjugate) closure: spans a single parity class. -/
def majoranaSpan : Finset Cube := evenClass

/-- Dirac (full-strand) closure: spans both classes. -/
def diracSpan : Finset Cube := Finset.univ

/-- Bipartite, even class: `V/2 = 4` vertices. -/
theorem evenClass_card : evenClass.card = 4 := by decide

theorem majoranaSpan_card : majoranaSpan.card = 4 := evenClass_card

What this page does not claim

The theorem does not assert that up, lepton, and down spans are equal in nature. The theorem does not derive the half factor or any mass ratio. The theorem does not mention the golden ratio, neutrino data, or any empirical value.

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)

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.

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