Foundation Nine Parities Spacetime Parity Count

The four spacetime parities

In the Recognition Science framework, the double-entry ledger (a discrete record of recognition events) carries nine independent binary signs, called parities, that constrain which configurations are physically admissible. The declaration spacetime_parity_count establishes that exactly four of these nine parities belong to the spacetime group: charge-parity, baryon minus lepton number, hypercharge, and tick reversal. The theorem proves that this set has cardinality four and that the membership predicate matches the explicit list, with no other parity index counted among them.

The four spacetime parities are a subset of a larger algebraic structure. The full set of nine parities decomposes as 4 + 3 + 2: four spacetime parities, three color-charge sign flips from the SU(3) Cartan subalgebra, and two generation-mixing signs from the three-generation structure. The machine-checked library of formal theorems proves the decomposition source_decomposition : 4 + 3 + 2 = 9 and the total count parity_count_eq_nine. The spacetime count is one component of this decomposition, not an isolated fact.

The framework's library also proves that all nine parities flip under the combined operation of conjugation and tick reversal, and that the scalar vacuum page has all parities equal to zero. These are properties of the whole parity vector, not of the spacetime subset alone. The spacetime count theorem itself does not assert anything about how these parities behave under physical processes; it only fixes the cardinality and the membership of the four indices.

What the declaration does not claim is as important as what it proves. It does not claim that these four parities are conserved in any physical interaction, nor that they correspond to measurable quantum numbers in the standard model. It does not claim that the spacetime group is the only relevant symmetry, nor that the other five parities are less fundamental. The theorem is purely combinatorial: it counts a finite set and verifies a predicate. Any physical interpretation of the parities, such as their role in determining which configurations are admissible, is a separate claim that the declaration itself does not make.

/-- The modeled parity index has four spacetime constructors:
    C (charge) + P (parity in D=3) + T (tick reversal) + B-L = 4.
    The B-L parity exists because D = 3 supports non-trivial linking (Alexander duality). -/
theorem spacetime_parity_count :
    spacetimeParityIndices.card = 4 ∧
    ∀ p, isSpacetimeParity p ↔ p ∈ spacetimeParityIndices := by
  constructor
  · decide
  · intro p
    cases p <;> simp [isSpacetimeParity, spacetimeParityIndices]

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

/-- There are exactly 9 parity indices. -/
theorem parity_count_eq_nine : Fintype.card ParityIndex = 9 := by
  decide

/-- **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

/-- **THEOREM**: The scalar vacuum page has all parities vanishing.
    This is the unique ℤ₂-even configuration: the vacuum carries no
    charge, no color, no generation mixing, and is tick-symmetric. -/
theorem vacuum_parities_vanish (p : ParityIndex) :
    vacuumParity p = 0 := by
  simp [vacuumParity]

What this page does not claim

The four spacetime parities are conserved in any physical interaction. The parities correspond directly to measurable quantum numbers in the standard model. The spacetime group is the only relevant symmetry in the framework.

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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