Masses Mass Genesis Majorana Closure From Cube Colored Pair Unique Dual Dim Inva

The colored pair theorem

The 3-cube has eight vertices, each labeled by three bits. The Recognition Science framework's recognition structure, a discrete record of events, uses this cube to organize particle states. A duality operation maps each vertex to its opposite, flipping all three bits. This duality also acts on dimensions: it sends dimension 0 to 2, dimension 1 to 1, and dimension 2 to 0.

The theorem colored_pair_unique_dualDim_invariant asks a precise question: if you pick any two of the three dimensions (a "colored pair"), when does the duality map send that pair back to itself? The answer, proved by exhaustive checking in the machine-checked library of formal theorems, is that only one pair works: dimensions 0 and 2. Any other two-element subset, such as {0, 1} or {1, 2}, gets mapped to a different pair. This is a purely combinatorial fact about the finite set of three dimensions, with no physics assumed beyond the definition of the duality map itself.

This result is a keystone in the framework's account of Majorana particles, neutral fermions that are their own antiparticles. The framework models a Majorana particle as spanning only half the cube's vertices, the four vertices with even bit-parity. The duality map pairs each vertex with its opposite, and a self-conjugate closure keeps one vertex from each pair, giving the factor 1/2. The colored pair theorem identifies which pair of dimensions can remain invariant under this duality, a structural constraint that supports the derivation of the neutrino mass ratio.

What the theorem does not claim is broader significance. It does not by itself derive any particle mass, nor does it establish that the framework's identification of charge conjugation with the bit-flip map is physically correct. That identification is a modeling choice, not a theorem. The theorem only asserts a fact about finite sets and functions; its physical interpretation depends on the surrounding framework's assumptions, which remain open to scrutiny.

/-- **THEOREM (the colored pair `{0,2}` is the UNIQUE duality-invariant 2-subset).** Among all
    2-element subsets of the cell-dimension set `{0,1,2}`, exactly `{0,2}` (vertices/faces) is
    invariant under the cube self-duality `dualDimF` (`0↔2`, `1` fixed). The colored pair `{s,−s}`
    is conjugation-invariant as a set, so under the C=duality hypothesis its coupling dimensions
    form a duality-invariant 2-subset — which is therefore forced to be `{0,2}`, with no choice
    remaining once the polar structure is granted. This is the positive companion to
    `colored_pair_forced`: `{0,2}` is not an arbitrary target but the canonical duality-fixed pair.
    Kernel axioms only. -/
theorem colored_pair_unique_dualDim_invariant :
    ∀ S : Finset (Fin 3), S.card = 2 → (S.image dualDimF = S ↔ S = {0, 2}) := by
  decide

/-- **THEOREM (the colored pair `{0,2}` is the UNIQUE duality-invariant 2-subset).** Among all
    2-element subsets of the cell-dimension set `{0,1,2}`, exactly `{0,2}` (vertices/faces) is
    invariant under the cube self-duality `dualDimF` (`0↔2`, `1` fixed). The colored pair `{s,−s}`
    is conjugation-invariant as a set, so under the C=duality hypothesis its coupling dimensions
    form a duality-invariant 2-subset — which is therefore forced to be `{0,2}`, with no choice
    remaining once the polar structure is granted. This is the positive companion to
    `colored_pair_forced`: `{0,2}` is not an arbitrary target but the canonical duality-fixed pair.
    Kernel axioms only. -/
theorem colored_pair_unique_dualDim_invariant :
    ∀ S : Finset (Fin 3), S.card = 2 → (S.image dualDimF = S ↔ S = {0, 2}) := by
  decide

What this page does not claim

The theorem does not derive any particle mass or mass ratio by itself. The theorem does not prove that charge conjugation is the bit-flip map; that remains a modeling assumption. The theorem does not establish the physical correctness of the recognition cube as a model of particle states.

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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