Foundation Pair Kernel Weyl Event Center Attachment Finite Weyl Two Weight Actio

The two-weight action

The declaration finiteWeylTwoWeightAction27_one_one is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It concerns a specific finite structure: a 27-site carrier built from a three-dimensional torus over the field with three elements, which the framework uses to model a discrete event space. The theorem states that when two separate cost functions, one for shifts and one for clock occupations, are both given weight 1, their sum equals a single pre-existing action defined on the same carrier. In plain terms, it proves that the two-weight action with equal unit weights collapses exactly to the original one-weight action, with no residual terms.

The two cost functions being combined are themselves defined in the same module. The shift occupation cost measures how much an amplitude changes when moved one step along each of the three axes, while the clock occupation cost measures how much an amplitude concentrates at a designated center point. The theorem's proof is a direct computation: it unfolds the definitions, distributes the multiplication by 1, and reassembles the sums. It is a bookkeeping identity, not a physical law. It shows that the two-weight construction is a faithful generalization of the original action, in the sense that the original action is recovered exactly at the unit point.

The theorem does not claim that the two weights must be equal. The framework's library contains a separate, conditional theorem: if a finite Fourier exchange-invariance premise holds, then the weights are forced to be equal. But that premise is not currently supplied by any existing Recognition theorem. The physical equal-weight claim therefore remains OPEN, meaning it is a target for future derivation, not an established result. The declaration also introduces no dimensioned length and no empirical constant; it is a purely combinatorial and algebraic statement about the finite carrier.

What the theorem does establish is a clean compatibility property: the two-weight action, at unit weights, is exactly the original action. This matters because the framework uses such actions to attach a clock center to a realized posting, and the theorem guarantees that the two-weight formalism does not introduce a spurious new behavior at the unit point. It is a coherence check that the generalization is conservative. The reader can rely on the fact that the two-weight action at unit weights is not a new object, but a restatement of the old one.

theorem finiteWeylTwoWeightAction27_one_one
    (center : WeylEventSite27)
    (ψ : WeylEventAmplitude27) :
    finiteWeylTwoWeightAction27 center 1 1 ψ =
      finiteWeylEventAction27 center ψ := by
  simp only [finiteWeylTwoWeightAction27, one_mul]
  simp only [finiteWeylShiftOccupationCost27,
    finiteWeylClockOccupationCost27, finiteWeylEventAction27]
  rw [← Finset.sum_add_distrib]
  apply Finset.sum_congr rfl
  intro axis _
  rw [← Finset.sum_add_distrib]

What this page does not claim

The theorem does not force the two weights to be equal. The theorem does not introduce any dimensioned length or empirical constant. The theorem does not establish the physical equal-weight claim, which remains open.

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

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Derived articles

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