Foundation Pair Kernel Weyl Event Center Attachment

The attachment result

A recognition event, in the framework's language, is a discrete record of a happening: a mark that something occurred at a site. The module at hand concerns events carried on a 27-site carrier built from three axes of three positions each, a finite torus. Its central result is a transport theorem: given any realized posting, which is an ordered pair of sites, the clock center of the carrier attaches to the positive endpoint of that posting. The other endpoint then differs from the center by exactly one primitive axis shift, forward or backward.

The proof is machine-checked in the framework's library of formal theorems. It transports the production enumeration back to its underlying (ZMod 3)^3 torus, and the certificate structure `WeylEventCenterAttachmentCert27` packages the result: the center is the positive endpoint, the primitive axis transport holds, and a discrete Fourier exchange witness swaps the two costs. No dimensioned length is introduced here, and the module adds no new axioms and no empirical constants.

The module also defines two independent costs on the carrier: a shift occupation cost, which penalizes differences between neighboring sites, and a clock occupation cost, which penalizes displacement from a chosen center. Each is a sum over axes and sites of squared amplitudes. Two theorems show the uniform amplitude field costs 0 for shift and 6 for clock, while the delta field costs 2 for shift and 0 for clock. A further theorem proves the clock cost equals a phase mismatch cost, expressed through cube roots of unity.

A two-weight action combines these costs with real weights. The key conditional theorem states: if the two-weight cost is invariant under a finite Fourier exchange, then the weights must be equal. The premise is named `FiniteFourierExchangeInvariantAxis3`, and it is a genuine condition. The library does not yet prove that premise from Recognition theorems alone, so the physical claim of equal weights remains OPEN. The module states this plainly: equal weights are forced only conditionally on that premise.

What this establishes in plain language is a structural bridge. A realized posting, which is a primitive object in the framework, carries a natural center attachment on the 27-site carrier, and the two costs that measure shift and clock occupation are provably exchangeable under a Fourier symmetry. The equal-weight step, which would unify the two costs into one, is a named target rather than a proved result.

theorem weylEventCenterAttachment_cert27
    (event : RealizedPostingEvent3 3) :
    WeylEventCenterAttachmentCert27 event where
  center_is_positive_endpoint := rfl
  primitive_axis_transport :=
    realizedPostingWeylCenter_respects_primitiveAxisShift27 event
  dft_exchange_witness :=
    finiteDFTAxis3_discriminating_exchange
  equal_weight_conditional :=
    fun _ _ => finiteFourierExchangeInvariant_forces_equal_weights

theorem shiftOccupationCostAxis3_uniform :
    shiftOccupationCostAxis3 uniformAxisAmplitude3 = 0 := by
  norm_num [shiftOccupationCostAxis3, uniformAxisAmplitude3,
    Complex.normSq_apply]

theorem clockOccupationCostAxis3_uniform :
    clockOccupationCostAxis3 uniformAxisAmplitude3 = 6 := by
  norm_num [clockOccupationCostAxis3, uniformAxisAmplitude3,
    Complex.normSq_apply]

theorem shiftOccupationCostAxis3_delta :
    shiftOccupationCostAxis3 deltaAxisAmplitude3 = 2 := by
  have h20 : (2 : ZMod 3) ≠ 0 := by decide
  have h10 : (1 : ZMod 3) ≠ 0 := by decide
  norm_num [shiftOccupationCostAxis3, deltaAxisAmplitude3,
    Complex.normSq_apply, h20, h10]

theorem clockOccupationCostAxis3_delta :
    clockOccupationCostAxis3 deltaAxisAmplitude3 = 0 := by
  have h20 : (2 : ZMod 3) ≠ 0 := by decide
  have h10 : (1 : ZMod 3) ≠ 0 := by decide
  norm_num [clockOccupationCostAxis3, deltaAxisAmplitude3,
    Complex.normSq_apply, h20, h10]

/-- 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 equal-weight claim is not proved unconditionally; it holds only under the finite-Fourier exchange-invariance premise. No dimensioned length or physical scale is introduced by this module. The module does not establish that the framework's recognition events correspond to physical particles or fields.

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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