Foundation Nine Parities Tick Reversal Involutive

The tick reversal flip

In the Recognition Science framework, the ledger is a discrete record of recognition events, and each event carries nine independent binary labels called parities. These parities are simple yes or no flags, like a set of nine switches, that track properties such as charge parity, color charge sign, and generation mixing sign. The declaration tick_reversal_involutive establishes a precise fact about what happens when the direction of time, the tick, is reversed for all nine parities at once: doing it twice brings every switch back to its original position. In mathematical terms, the operation is an involution, its own inverse.

The proof is short and relies on the arithmetic of binary arithmetic, where adding one twice is the same as adding zero. The declaration shows that for any parity vector, applying the tick reversal conjugation twice yields the original vector exactly. This is a formal theorem in the machine-checked library of formal theorems, meaning the logic has been verified step by step. The result is not an assumption or a model choice; it is a derived consequence of how the tick reversal operation is defined on these nine binary labels.

The theorem connects to a broader structure. The nine parities are independent, and the vacuum state, where all parities are zero, is not fixed by tick reversal: reversing time on the empty ledger produces the opposite state where all nine flags are set. This is a separate theorem, but it highlights that the involution property is not trivial. The operation flips every switch, yet the double flip returns to the start, a symmetry that holds for every possible configuration, not just the vacuum.

What the declaration does not claim is equally important. It does not say that tick reversal is a physical symmetry of nature, nor does it assert that reversing time in the real universe is a valid operation. The theorem is about the formal structure of the ledger's parities, not about the actual direction of time. It also does not claim that the nine parities are conserved quantities; the involution property says nothing about whether the flags stay the same as events unfold. The result is a statement about the algebra of the labels, not about the dynamics of recognition.

/-- Double tick reversal is the identity. -/
theorem tick_reversal_involutive (v : ParityVector) :
    tickReversalConjugate (tickReversalConjugate v) = v := by
  ext p
  simp only [tickReversalConjugate]
  -- In ZMod 2: (x + 1) + 1 = x + 2 = x
  have : (2 : ZMod 2) = 0 := by decide
  calc v p + 1 + 1 = v p + 2 := by ring
    _ = v p + 0 := by rw [this]
    _ = v p := by ring

/-- **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

What this page does not claim

The theorem does not claim that time reversal is a physical symmetry of the universe. It does not claim that the nine parities are conserved quantities over time. It does not claim that the tick reversal operation is a symmetry of the ledger's dynamics, only of its parity labels.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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