Foundation Pair Kernel Weyl Event Center Attachment Realized Posting Finite Weyl

The attachment theorem

The declaration realizedPostingFiniteWeylEventAction_center27 is a theorem in the Recognition Science framework's machine-checked library of formal theorems. In plain language it establishes this: when a realized posting event (a discrete record of a recognition event) is transported onto a 27-site carrier, the clock center of that event is exactly the image of the event's first production index under the enumeration equivalence. The theorem states that the action at the center, computed through the realized posting, equals the action computed directly from that same production index. This is a definitional identity, proved by reflexivity, so it is not an empirical claim and not a new physical law; it is a bookkeeping consistency result.

The module goes further. It proves that the other endpoint of the posting differs from the clock center by exactly one primitive axis shift along one of three axes. The production enumeration is transported back to its underlying (ZMod 3)^3 torus, and the other endpoint is shown to be either a forward shift or a backward shift by one step on that torus. This is a structural fact about how the discrete carrier is organized: the two endpoints of a realized posting are always adjacent along a single axis, never separated by a larger displacement.

The module also introduces two independent costs: a shift occupation cost and a clock occupation cost, each with its own weight. These are definitional choices, not derived facts. The theorem finiteFourierExchangeInvariant_forces_equal_weights shows that if a certain finite-Fourier exchange-invariance condition holds, then the two weights must be equal. But the docstring is explicit: existing Recognition theorems do not yet supply that exchange-invariance premise, so the physical claim that the weights are equal remains OPEN. The equal-weight result is conditional, not unconditional.

What the theorem does not claim matters as much as what it proves. It does not introduce any dimensioned length; the module explicitly states "No dimensioned length is introduced here." It does not assert that the two weights are equal in the physical theory; that remains a target. It does not claim that the 27-site carrier is the physical space of the framework; it is a finite combinatorial model used for this attachment construction. The theorem is a formal consistency check within a specific model, not a statement about the empirical world.

theorem realizedPostingFiniteWeylEventAction_center27
    (event : RealizedPostingEvent3 3)
    (ψ : WeylEventAmplitude27) :
    realizedPostingFiniteWeylEventAction27 event ψ =
      finiteWeylEventAction27
        (productionIndexEquivWeylEventSite27 event.1.1) ψ :=
  rfl

/-- A realized primitive posting remains exactly one positive or negative axis
step after transport to the Weyl torus. -/
theorem realizedPostingWeylCenter_respects_primitiveAxisShift27
    (event : RealizedPostingEvent3 3) :
    ∃ axis : Fin 3,
      realizedPostingWeylOtherEndpoint27 event =
          forwardShiftSite27 axis
            (realizedPostingWeylClockCenter27 event) ∨
        realizedPostingWeylOtherEndpoint27 event =
          shiftSite27 axis
            (realizedPostingWeylClockCenter27 event) := by
  obtain ⟨axis, haxis⟩ :=
    realizedProductionEvent_has_spatialWitness event
  refine ⟨axis, ?_⟩
  simpa [ProductionSpatialEventWitness3,
    realizedPostingWeylOtherEndpoint27,
    realizedPostingWeylClockCenter27,
    productionIndexEquivWeylEventSite27,
    PairKernelDeltaSpatialBridgeS5.framedForward,
    PairKernelDeltaSpatialBridgeS5.framedBackward,
    PairKernelPeriodic3.torusShift,
    forwardShiftSite27, shiftSite27, sub_eq_add_neg] using haxis

/-- A genuine finite-Fourier exchange symmetry would force equal shift and
clock weights.  The conclusion is conditional on `hinvariant`. -/
theorem finiteFourierExchangeInvariant_forces_equal_weights
    {shiftWeight clockWeight : ℝ}
    (hinvariant :
      FiniteFourierExchangeInvariantAxis3
        shiftWeight clockWeight) :
    shiftWeight = clockWeight := by
  have h := hinvariant deltaAxisAmplitude3
  rw [finiteDFTAxis3_delta] at h
  simp only [twoWeightOccupationCostAxis3,
    shiftOccupationCostAxis3_uniform,
    clockOccupationCostAxis3_uniform,
    shiftOccupationCostAxis3_delta,
    clockOccupationCostAxis3_delta,
    mul_zero, add_zero] at h
  linarith

What this page does not claim

The theorem does not claim that the two weights are equal in the physical theory. The theorem does not introduce any dimensioned length. The theorem does not claim the 27-site carrier is the physical space of 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/PairKernelWeylEventCenterAttachment.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.

Derived articles

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