Masses Mass Genesis Majorana Closure From Cube

The closure factor

In particle physics, a Dirac fermion, like an electron, has two independent chiral components, left and right. A Majorana fermion, a hypothetical neutral particle that is its own antiparticle, has only one. This is a statement about degrees of freedom, the number of independent ways a field can oscillate.

In Recognition Science, this distinction is derived from the geometry of a three-dimensional cube. The framework's model represents the possible states of a particle as the eight vertices of a cube, labeled by three binary coordinates. The framework proves that these eight vertices split into two classes of four, based on the parity of the sum of their coordinates. This is a basic property of a bipartite graph: every edge connects a vertex in one class to a vertex in the other.

Within this model, a Dirac fermion is identified with the full set of eight vertices, while a Majorana fermion is identified with just one of the four-vertex parity classes. The key result, proved in the machine-checked library, is that this identification makes the ratio of the Majorana to Dirac degrees of freedom exactly one half, a factor that is forced by the cube's structure rather than being a free parameter. The proof does not rely on any physical assumption about mass; it is a purely combinatorial theorem about the cube.

The module also shows this same factor of one half arises from a second, independent route. Charge conjugation is modeled as the map that flips all three coordinates of a vertex, an operation with no fixed points. This involution pairs the eight vertices into four orbits of two. A self-conjugate, Majorana state keeps one vertex from each orbit, again giving a factor of four over eight. This derivation requires only the general theorem that a fixed-point-free involution halves a finite set.

The importance of this result is that it resolves a previous obstacle. A predicted neutrino mass ratio was missing a factor of one half, and earlier attempts to supply it from external physics, such as winding numbers or the seesaw mechanism, failed. The module shows the missing factor was a combinatorial property of the framework's own three-dimensional structure all along.

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

/-- Bipartite, odd class: `V/2 = 4` vertices. -/
theorem oddClass_card : oddClass.card = 4 := by decide

theorem majoranaSpan_card : majoranaSpan.card = 4 := evenClass_card

/-- Conjugation is an involution. -/
theorem conj_involutive : ∀ v : Cube, conj (conj v) = v := by decide

/-- Conjugation has **no fixed point** (you can never have `v = v̄`). This is what
    forces orbits of size `2` and hence the exact halving. -/
theorem conj_no_fixedPoint : ∀ v : Cube, conj v ≠ v := by decide

What this page does not claim

This does not claim that Majorana fermions have been observed in nature. This does not claim that the full neutrino mass ratio is derived without any additional hypotheses. This does not claim that the identification of charge conjugation with the antipodal map is derived, rather than a modeling choice.

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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