Foundation Gray Code Chirality Flip Asymmetry Nonzero

The asymmetry theorem

A Gray code is a way of ordering binary numbers so that consecutive values differ in exactly one bit. The canonical 3-bit Gray code visits the eight vertices of a cube in the order [0,1,3,2,6,7,5,4]. As the sequence advances, bit 0 flips four times, while bits 1 and 2 each flip only twice. The theorem flipAsymmetryNonzero states, in the framework's machine-checked library of formal theorems, that these counts are not all equal: bit 0 flips more often than the other two.

The framework calls this asymmetry chirality, meaning the directed walk distinguishes one axis from the other two. The proof is computational: the library checks the eight-step cycle directly and verifies the counts [4,2,2]. A companion theorem records the exact ratio: bit 0 flips twice as often as bit 1. The asymmetry vector, subtracting the average 8/3 from each count, has nonzero norm, which is the formal content of the declaration.

In Recognition Science, this unequal treatment matters because the framework identifies the three cube axes with particle generations. The theorem generation_coupling_asymmetry states that the first generation experiences twice as many flips as the second and third during one cycle. The framework presents this as the origin of flavor mixing and CP violation, though that physical identification is a modeling step, not a proved consequence of the counting theorem alone.

What the declaration does not claim is equally precise. It does not prove that the physical universe violates CP symmetry; it proves a combinatorial fact about a specific Gray code cycle. The framework's link from that fact to particle physics is a definitional choice, not a theorem. Nor does the declaration assert that all Gray codes are chiral, only this canonical one. The symmetry under particle-antiparticle exchange, expressed as J(x) = J(1/x), remains intact; the asymmetry appears only in the directed traversal.

/-- The flip counts are [4, 2, 2], not [8/3, 8/3, 8/3].
    This proves the S₃ axis-permutation symmetry is broken. -/
theorem flipAsymmetryNonzero :
    ¬(bitFlipCount 0 = bitFlipCount 1 ∧ bitFlipCount 1 = bitFlipCount 2) := by
  native_decide

/-- The ratio of flip counts between axis 0 and axis 1 is 2:1.
    This ratio, combined with the torsion gap Δτ₁₂ = 11, determines
    the Cabibbo angle. -/
theorem flip_ratio_21 : bitFlipCount 0 / bitFlipCount 1 = 2 := by
  native_decide

/-- **Generation coupling asymmetry**: generation 1 is driven twice as
    often as generations 2 and 3. This asymmetry is the kinematic
    source of flavor mixing — it forces the mass and weak eigenstates
    to be misaligned. -/
theorem generation_coupling_asymmetry :
    generationFlipCount 0 = 2 * generationFlipCount 1 ∧
    generationFlipCount 0 = 2 * generationFlipCount 2 := by
  constructor <;> native_decide

What this page does not claim

The theorem does not prove physical CP violation in the universe. The declaration does not assert that every Gray code is chiral. The framework's identification of axes with generations is a modeling choice, not a proved consequence.

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

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