Foundation Pair Kernel Canonical Generator Source S9 Recognition Production Grap

The posting multiplicity

A recognition production graph is a discrete object that records which pairs of sites are linked by a primitive posting. The theorem recognitionProductionGraph3_weight_eq_postingMultiplicity states that the weight of any edge in this graph equals the posting multiplicity of that pair: the weight is 1 if the pair is a realized primitive posting, and 0 otherwise. This is a proved theorem in the framework's machine-checked library of formal theorems, not a definitional choice.

The statement is a direct identity. For any two sites i and j, the graph's weight function returns the same number as the posting multiplicity function. The posting multiplicity function itself is defined to return 1 exactly when the pair belongs to the set of realized primitive posting pairs, and 0 otherwise. The theorem closes the loop: the graph's weight is not a separate fitted parameter but is forced to equal this multiplicity.

What the theorem does not claim is any numerical value for the weight beyond 0 or 1. It does not say which pairs are realized; that is determined by the set realizedPrimitivePostingPairs3, which comes from an earlier construction. It also does not assign a physical dimension to a posting. The theorem works with the abstract ledger action unit, and any later conversion to physical units remains an explicit parameter.

/-- The constructed generator coefficient is the real cast of posting-pair
multiplicity. -/
theorem recognitionProductionGraph3_weight_eq_postingMultiplicity
    {N : ℕ} [NeZero N]
    (i j : Fin (TorusCard3 N)) :
    (recognitionProductionGraph3 N).weight i j =
      (postingPairMultiplicity3 N i j : ℝ) := by
  classical
  by_cases h : (i, j) ∈ realizedPrimitivePostingPairs3 N
  · rw [recognitionProductionGraph3_weight_of_mem h]
    simp [postingPairMultiplicity3, h]
  · rw [recognitionProductionGraph3_weight_of_not_mem h]
    simp [postingPairMultiplicity3, h]

What this page does not claim

The theorem does not determine which pairs are realized primitive postings. The theorem does not assign a dimensional magnitude to a posting. The theorem does not involve any hydrogen or benchmark data.

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/PairKernelCanonicalGeneratorSourceS9.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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