Foundation Nine Parities Parities Flip Under Tick Reversal

The nine flips

In the Recognition Science framework, the ledger (a discrete record of recognition events) assigns nine independent binary labels, called parities, to every configuration. These labels track charge-parity, baryon minus lepton number, hypercharge, tick reversal, three color-charge signs, and two generation-mixing signs. The theorem parities_flip_under_tick_reversal proves that applying the operation of tick reversal combined with conjugation flips every one of these nine labels: a label that was 0 becomes 1, and one that was 1 becomes 0.

The proof is short and relies only on the arithmetic of binary values. In the integers mod 2, adding 1 always changes a value, so the operation tickReversalConjugate maps each parity to its opposite. The theorem also shows the operation is its own inverse: applying it twice returns the original configuration. The empty vacuum state, where every parity is 0, is the unique state not fixed by this reversal; flipping it produces a state where all nine labels are 1, which the library proves has Hamming weight 9.

The framework uses these nine parities to constrain which configurations are physically admissible. The count of nine is not numerology; the library proves it decomposes as 4 + 3 + 2, matching the three sources: spacetime, color, and generation. A separate theorem, total_parity_configs, shows there are 2^9 = 512 possible parity assignments in total.

What the theorem does not claim is that these parities correspond to measured physical quantities. The labels are defined within the framework's formal model; the library proves algebraic facts about them, not empirical facts about the universe. The connection to Tesla's "3, 6, and 9" is presented as a mapping in the documentation, not as a derived physical law. The theorem also does not claim that tick reversal is physically realizable; it is a formal operation on the ledger's labels.

/-- **THEOREM**: All nine parities flip under conjugation + tick reversal. -/
theorem parities_flip_under_tick_reversal (v : ParityVector) (p : ParityIndex) :
    tickReversalConjugate v p ≠ v p := by
  simp only [tickReversalConjugate]
  -- In ZMod 2, x + 1 ≠ x because 1 ≠ 0 in ZMod 2
  intro h
  have h2 : v p + 1 - v p = v p - v p := congr_arg (· - v p) h
  simp at h2

/-- Vacuum is NOT a fixed point of tick reversal (it maps 0 → 1). -/
theorem vacuum_not_fixed_by_tick_reversal :
    tickReversalConjugate vacuumParity ≠ vacuumParity := by
  intro h
  have := congr_fun h ParityIndex.P_cp
  simp [tickReversalConjugate, vacuumParity] at this

/-- Tick-reversed vacuum has Hamming weight 9 (all parities flipped). -/
theorem tick_reversed_vacuum_hamming_weight :
    hammingWeight (tickReversalConjugate vacuumParity) = 9 := by
  simp [hammingWeight, tickReversalConjugate, vacuumParity]
  decide

/-- The 4+3+2 decomposition sums to 9. -/
theorem source_decomposition : 4 + 3 + 2 = 9 := by norm_num

/-- Total number of parity configurations: 2⁹ = 512. -/
theorem total_parity_configs : Fintype.card ParityVector = 512 := by
  simp only [ParityVector]
  rw [Fintype.card_pi]
  simp only [Finset.prod_const, Finset.card_univ, ZMod.card]
  rw [parity_count_eq_nine]
  norm_num

What this page does not claim

The parities correspond to directly measured physical quantities. Tick reversal is a physically realizable operation. The mapping to Tesla's "3, 6, and 9" is a derived physical law. The theorem establishes that the nine parities are conserved under any physical process.

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

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