Foundation Gap Derivation Parity Count Matches Enumeration

The parity count agreement

The declaration parityCount_matches_enumeration is a theorem in the machine-checked library of formal theorems used by Recognition Science. It states that when the spatial dimension D equals 3, the square of that dimension, 3² = 9, is exactly equal to the number of distinct parity states in a separate enumeration called NineParities.ParityIndex. The proof is a direct computation: it rewrites the dimension to 3, evaluates the square to 9, and then invokes a pre-existing lemma that the parity index has exactly nine elements. The entire chain is checked by the kernel, with no gaps and no extra axioms beyond the standard three that the ambient type theory supplies.

The theorem does not derive the number nine from first principles. It does not prove that three spatial dimensions must exist, nor that parity states must number nine. It only establishes a consistency between two definitions already present in the framework: the definition of parityCount as d² and the definition of the enumeration NineParities.ParityIndex as having nine elements. The agreement is a certificate, a cross-check that two independent-looking structures happen to align at D = 3. It is one small piece in a larger certificate called gap45_cert, which bundles several such checks together, including the coprimality of 2³ with 45 and the balance equation φ^(1−45) × φ^45 = φ.

What the theorem does not claim is broader than what it proves. It does not say that parity states are physically real, that the number nine has any intrinsic meaning, or that the enumeration is the only possible one. It does not claim that D = 3 is forced by the parity count; the dimension is set by a separate definition, and the theorem merely observes the numerical match. The theorem also does not establish any connection between parity states and the coherence energy E_coh = φ⁻⁵, even though both appear in the same file. Each fact stands on its own; the certificate assembles them, but the theorem itself is only the narrow equality between 9 and the cardinality of the parity index.

In plain terms, this is a bookkeeping check. The framework keeps a discrete record of events, and here it verifies that two of its own ledgers agree at one specific point. The value of the check is not that nine is surprising, but that the framework's internal definitions are mutually consistent at the dimension it uses. A stranger reading the framework can trust that this particular corner of the structure is sound, without needing to accept the larger claims about why D = 3 or what the gap of 45 means. The theorem is a small, solid tile in a much larger mosaic, and its honesty lies in how little it asserts.

/-- The parametric parity count matches the NineParities enumeration. -/
theorem parityCount_matches_enumeration :
    parityCount D = Fintype.card NineParities.ParityIndex := by
  rw [parityCount_at_D3, NineParities.parity_count_eq_nine]

/-- The parametric parity count matches the NineParities enumeration. -/
theorem parityCount_matches_enumeration :
    parityCount D = Fintype.card NineParities.ParityIndex := by
  rw [parityCount_at_D3, NineParities.parity_count_eq_nine]

noncomputable def gap45_cert : Gap45Cert where
  config_dim := configDim_at_D3
  parity_count := parityCount_at_D3
  parity_matches := parityCount_matches_enumeration
  gap := gap_at_D3
  coprime := coprime_at_D3
  ecoh := E_coh_gap_eq
  balance := gap_balance
  odd_coprime := coprimality_odd
  even_not_coprime := coprimality_even_fails

What this page does not claim

The theorem does not derive the number nine from first principles. The theorem does not prove that three spatial dimensions must exist. The theorem does not establish any connection between parity states and the coherence energy E_coh = φ⁻⁵.

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

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