Foundation Gray Code Chirality Bit0 Flips Four

The 4:2:2 flip split

A Gray code is a way of listing binary numbers so that each step changes exactly one bit. The canonical 3-bit Gray code cycle lists the eight vertices of a cube: 000, 001, 011, 010, 110, 111, 101, 100. The Recognition Science framework's library of machine-checked formal theorems proves that in this cycle, bit 0 flips four times while bits 1 and 2 each flip only twice. The theorem bit0_flips_four states this 4:2:2 split, and the total number of flips across all three bits is eight.

The 4:2:2 split is not an accident of notation. The framework's library proves the flip counts are unequal across the three axes, a property it calls chirality: the directed walk around the cube distinguishes clockwise from counterclockwise traversal. The three bits correspond to three particle generations in the framework's model, so different generations experience different numbers of flips during one recognition cycle. The theorem generation_coupling_asymmetry states that generation 0 flips exactly twice as often as generations 1 and 2.

In Recognition Science, this asymmetry is the geometric origin of CP violation, the observed difference between matter and antimatter behavior. The framework's cost function, which measures the cost of recognition, is symmetric under particle-antiparticle exchange: J(x) = J(1/x). The library proves this symmetry is preserved even while the cycle's chirality breaks CP. The combination, a preserved symmetry and a broken one, is the framework's explanation for why the universe treats matter and antimatter differently.

What the theorem does not claim: it does not prove that the Gray code cycle is the only possible cycle, nor that the 4:2:2 split is forced by the cost function alone. The chirality is proved by direct computation on this specific cycle, not derived from first principles. The identification of bits with particle generations is a definitional choice, not a theorem. The framework's library proves the arithmetic facts; the physical interpretation is a model built on top of them.

/-- Bit 0 flips 4 times. -/
theorem bit0_flips_four : bitFlipCount 0 = 4 := by native_decide

/-- Bit 1 flips 2 times. -/
theorem bit1_flips_two : bitFlipCount 1 = 2 := by native_decide

/-- Bit 2 flips 2 times. -/
theorem bit2_flips_two : bitFlipCount 2 = 2 := by native_decide

/-- **THEOREM**: The canonical Gray code cycle on Q₃ is chiral.

    PROOF: bitFlipCount 0 = 4 ≠ 2 = bitFlipCount 1, so the flip counts
    are not all equal.

    This is the foundational result for CP violation in RS: the 8-tick
    recognition cycle treats different axes (= generations) differently. -/
theorem cycle_is_chiral : IsChiral grayFlipCounts := by
  intro h
  have h4 : bitFlipCount 0 = 4 := by native_decide
  have h2 : bitFlipCount 1 = 2 := by native_decide
  have h01 := h 0 1
  simp only [grayFlipCounts] at h01
  rw [h4, h2] at h01
  norm_num at h01

/-- **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

/-- J-cost symmetry: J(x) = J(1/x) for all positive x.
    This is the algebraic statement of CPT invariance. -/
theorem jcost_symmetric (x : ℝ) (hx : 0 < x) :
    Cost.Jcost x = Cost.Jcost (1/x) := by
  simp [Cost.Jcost]
  ring

What this page does not claim

The theorem does not prove that the Gray code cycle is the only possible cycle on three bits. The identification of bits with particle generations is a definitional choice, not a derived result. The framework does not derive the numerical value of the CP violation phase from this chirality.

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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