Foundation Mass Weak Bases Edge Dressed Prefers Axis0

The flip-count asymmetry

The declaration edge_dressed_prefers_axis0 is a small theorem inside a larger framework that models three generations of quarks using an eight-dimensional space. The space is built from three binary axes, so each point is a vertex of a cube, labeled by three bits. The theorem states that one axis, axis 0, has more bit flips per cycle than the other two: axis 0 flips 4 times, while axes 1 and 2 each flip 2 times. This asymmetry, the flip count pattern [4, 2, 2], is a proved fact in the framework's machine-checked library of formal theorems.

The physical interpretation comes from the framework's account of mass. In this model, the three generations are distinguished by how they are "dressed": the first generation is the ground state, the second is dressed by edges, and the third by faces and edges. The theorem says that the edge-dressed generation, the second one, preferentially couples to the axis with the most flips, axis 0. The reason offered in the framework is that this minimizes the recognition cost, a quantity the framework derives from first principles. The declaration itself, however, only proves the numerical fact about bit flip counts; it does not prove the cost-minimization claim or the assignment of generations to axes.

The theorem is anchored to a prior result called GrayCodeChirality.bit0_most_flipped, which establishes the same inequality. The declaration is a formal statement in the framework's library, not an empirical measurement. It is a structural fact about the cube geometry the framework uses, and it serves as one ingredient in a larger story about why the mass and weak bases differ, which the framework claims is the origin of the CKM matrix. The theorem itself says nothing about the weak basis, the CKM matrix, or any measured mixing angle.

What the declaration does not claim is important. It does not claim that the second generation is the charm quark, or that the CKM matrix elements have been computed. It does not claim that the cost-minimization argument is proved; that remains a separate, unformalized step. It only establishes the arithmetic fact that one axis has more flips than the other two, within the framework's specific construction of an eight-dimensional space from three binary axes.

/-- The preferred axis for the edge-dressed generation (gen 2) is axis 0,
    because axis 0 has the most flips and hence the most "passive edge
    interaction" per cycle. -/
theorem edge_dressed_prefers_axis0 :
    bitFlipCount 0 > bitFlipCount 1 ∧ bitFlipCount 0 > bitFlipCount 2 :=
  GrayCodeChirality.bit0_most_flipped

/-- The preferred axis for the edge-dressed generation (gen 2) is axis 0,
    because axis 0 has the most flips and hence the most "passive edge
    interaction" per cycle. -/
theorem edge_dressed_prefers_axis0 :
    bitFlipCount 0 > bitFlipCount 1 ∧ bitFlipCount 0 > bitFlipCount 2 :=
  GrayCodeChirality.bit0_most_flipped

What this page does not claim

The cost-minimization argument that the edge-dressed generation prefers axis 0 is not proved by this declaration. The declaration does not identify the second generation as the charm quark or any specific physical particle. The declaration does not compute any CKM matrix element or mixing angle.

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

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