Masses Mass Genesis Coupling Dimension From Sector Coupling Dim Of Sector' Eq Du

Two orientations, one duality

In the Recognition Science framework, particle sectors are assigned a coupling dimension, a small whole number that indexes how a sector participates in interactions. The assignment is not arbitrary: it is built from the topology of patterns, and it is constrained by a symmetry called charge conjugation, which swaps particles and their antiparticles. The theorem couplingDimOfSector'_eq_dualDim_comp states that the two possible assignments are related by a duality involution: for every sector s, the alternative assignment couplingDimOfSector' equals the dual of the original couplingDimOfSector. In plain terms, flipping the assignment is the same as applying the boundary duality, the operation that swaps a vertex with a face in the underlying geometry.

This theorem is not a new physical prediction. It is a formal verification that the two assignments are genuinely mirror images of each other under the duality. The colorless sectors, Lepton and Electroweak, are fixed by this duality: they both sit at the self-dual middle dimension 1. The colored sectors, Up and Down quarks, are swapped: if Up sits at dimension 0 (the vertex), then Down sits at dimension 2 (the face), and vice versa. The theorem proves this relationship holds for every sector, and it is a direct consequence of the definitions, checked by case analysis in the machine-checked library of formal theorems.

The importance of this theorem is what it does not do. It does not decide which orientation is the correct one. The framework proves that both orientations are equally valid under charge-conjugation equivariance, and that they are the only two possibilities. The choice of whether Up sits at dimension 0 or dimension 2 is therefore a sign convention, not a consequence of the combinatorial structure. This is the sharpened obstruction: the duality and sector structure provably cannot resolve the up/down orientation, leaving a free Z/2 freedom in the theory.

/-- The two orientations are related by the duality involution: `couplingDimOfSector'` is
    `dualDim ∘ couplingDimOfSector`. The charge-conjugation-fixed sectors (Lepton, Electroweak) are
    unmoved (`dualDim 1 = 1`); the colored pair is swapped (`dualDim 0 = 2`, `dualDim 2 = 0`). So the
    orientation freedom is exactly the `ℤ/2` generated by `dualDim`. -/
theorem couplingDimOfSector'_eq_dualDim_comp (s : Anchor.Sector) :
    couplingDimOfSector' s = dualDim (couplingDimOfSector s) := by
  cases s <;> rfl

/-- The two orientations are related by the duality involution: `couplingDimOfSector'` is
    `dualDim ∘ couplingDimOfSector`. The charge-conjugation-fixed sectors (Lepton, Electroweak) are
    unmoved (`dualDim 1 = 1`); the colored pair is swapped (`dualDim 0 = 2`, `dualDim 2 = 0`). So the
    orientation freedom is exactly the `ℤ/2` generated by `dualDim`. -/
theorem couplingDimOfSector'_eq_dualDim_comp (s : Anchor.Sector) :
    couplingDimOfSector' s = dualDim (couplingDimOfSector s) := by
  cases s <;> rfl

/-- The two orientations are related by the duality involution: `couplingDimOfSector'` is
    `dualDim ∘ couplingDimOfSector`. The charge-conjugation-fixed sectors (Lepton, Electroweak) are
    unmoved (`dualDim 1 = 1`); the colored pair is swapped (`dualDim 0 = 2`, `dualDim 2 = 0`). So the
    orientation freedom is exactly the `ℤ/2` generated by `dualDim`. -/
theorem couplingDimOfSector'_eq_dualDim_comp (s : Anchor.Sector) :
    couplingDimOfSector' s = dualDim (couplingDimOfSector s) := by
  cases s <;> rfl

What this page does not claim

This theorem does not claim that the up/down orientation is derived from the framework; it remains a free sign convention. This theorem does not claim that the two orientations are the same; it proves they are distinct mirror images. This theorem does not claim that the coupling dimension is a physical observable; it is a structural index within the framework.

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

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