Foundation Pair Kernel Weyl Event Center Attachment Production Spatial Event Wey

The other endpoint

In the Recognition Science framework, a primitive posting is a discrete record of an event: it has a starting site and an ending site on a finite grid. The declaration productionSpatialEventWeylOtherEndpoint27 is a theorem about the ending site of a special kind of posting called a production spatial event. It proves that when you take one of the three axis-aligned production events and transport its ending site onto the framework's 27-site carrier, that ending site is exactly one step forward along the same axis from the reference site. In plainer terms: the other endpoint of the posting is not arbitrary; it sits precisely one primitive axis shift away from the clock center.

The proof works by transporting the production enumeration back to its underlying three-by-three-by-three torus, a grid where opposite edges are identified. The theorem then computes the ending site directly and shows it equals the forward-shifted reference site. This is a purely combinatorial fact about the grid and the enumeration; it introduces no dimensioned length and no empirical constant. The declaration is part of a larger module that attaches the clock center of the 27-site carrier to the positive endpoint of an actual realized posting, and it proves that the other endpoint differs by one primitive axis shift.

In Recognition Science, the framework models events as discrete postings on a finite carrier, and this theorem pins down one structural fact about how those postings behave under the production enumeration. The framework's library of machine-checked theorems establishes this endpoint fact cleanly, with no unproved assumptions and no new axioms. What the declaration does not claim is any physical content about weights or costs. The module also defines two independent costs, one for shifting and one for clock occupation, and it proves that equal weights are forced only conditionally on a finite-Fourier exchange-invariance premise. Existing Recognition theorems do not yet supply that premise, so the physical equal-weight claim remains OPEN.

The practical upshot is a precise, verified statement about the geometry of production events on the 27-site carrier: the other endpoint is always one forward step along the axis. This is a building block for further structural results in the framework, and it is a clean example of what the machine-checked library can certify about the discrete geometry of events.

/-- Its other endpoint is the corresponding positive primitive axis shift. -/
theorem productionSpatialEventWeylOtherEndpoint27
    (axis : Fin 3) :
    realizedPostingWeylOtherEndpoint27
        (productionSpatialEvent3 axis) =
      forwardShiftSite27 axis productionReferenceSite3 := by
  simp [realizedPostingWeylOtherEndpoint27,
    productionIndexEquivWeylEventSite27,
    productionSpatialEvent3, productionSpatialEventPair3,
    PairKernelDeltaSpatialBridgeS5.framedForward,
    PairKernelPeriodic3.torusShift, forwardShiftSite27]

/-- Its other endpoint is the corresponding positive primitive axis shift. -/
theorem productionSpatialEventWeylOtherEndpoint27
    (axis : Fin 3) :
    realizedPostingWeylOtherEndpoint27
        (productionSpatialEvent3 axis) =
      forwardShiftSite27 axis productionReferenceSite3 := by
  simp [realizedPostingWeylOtherEndpoint27,
    productionIndexEquivWeylEventSite27,
    productionSpatialEvent3, productionSpatialEventPair3,
    PairKernelDeltaSpatialBridgeS5.framedForward,
    PairKernelPeriodic3.torusShift, forwardShiftSite27]

What this page does not claim

The declaration does not claim that the shift and clock occupation weights are equal in the physical theory. It does not claim any specific numerical value for a physical constant or coupling. It does not claim that the 27-site carrier is the full 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.

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