Foundation Pair Kernel Constructed Covector Event Occurrence Join Event Spatial

The spatial tie

In Recognition Science, a ledger is a discrete record of events, and a posting event is a single change to that record. The theorem eventSpatialOccurrenceTie_of_framed_choice establishes a precise link: if an event moves between two sites on a three-dimensional torus, and that movement is already known to be a forward or backward step along one of the three axes, then the event can be assigned a unique spatial occurrence, an axis and a direction (debit or credit). The theorem proves that this assignment is consistent: the direction is debit exactly when the step is forward, and credit exactly when the step is backward.

The theorem is a formal statement in the framework's machine-checked library of formal theorems. It does not prove that every posting event has such a framed step. That stronger claim, that every realized posting pair has a ledger state with nonnegative balances, remains an open target named MissingNonnegLedgerOnS8PostingWitness. The theorem only works when the framed step is given as a hypothesis. It also does not establish that the spatial occurrence is unique across all possible witnesses; the scale of the constructed source is witness-independent, but the poles may depend on the witness.

What the theorem does establish is a clean bridge between the ledger's internal bookkeeping and the spatial structure of the event. It shows that when a step is framed, the direction of the ledger entry (debit or credit) is not arbitrary but is forced by the geometry of the step. This is a step toward showing that the ledger's structure is not just a bookkeeping convenience but is tied to the spatial framework in which events occur.

theorem eventSpatialOccurrenceTie_of_framed_choice
    {N : ℕ} [NeZero N]
    (event : RealizedPostingEvent3 N)
    (k : Fin 3)
    (hframed :
      (torusSiteEquivFin N).symm event.1.2 =
          framedForward (Equiv.refl (Fin 3))
            ((torusSiteEquivFin N).symm event.1.1) k ∨
        (torusSiteEquivFin N).symm event.1.2 =
          framedBackward (Equiv.refl (Fin 3))
            ((torusSiteEquivFin N).symm event.1.1) k) :
    EventSpatialOccurrenceTie event k
      (canonicalSideOfFramedStep
        ((torusSiteEquivFin N).symm event.1.1)
        ((torusSiteEquivFin N).symm event.1.2) k) := by
  dsimp only [EventSpatialOccurrenceTie]
  rcases hframed with hfwd | hbwd
  · left
    refine ⟨hfwd, ?_⟩
    simp [canonicalSideOfFramedStep, hfwd]
  · -- Backward step: if it is also forward (possible only for tiny N), the
    -- debit/forward disjunct still holds; otherwise credit/backward holds.
    by_cases hfwd :
        (torusSiteEquivFin N).symm event.1.2 =
          framedForward (Equiv.refl (Fin 3))
            ((torusSiteEquivFin N).symm event.1.1) k
    · left
      refine ⟨hfwd, ?_⟩
      simp [canonicalSideOfFramedStep, hfwd]
    · right
      refine ⟨hbwd, ?_⟩
      simp [canonicalSideOfFramedStep, hfwd]

theorem eventSpatialOccurrenceTie_of_framed_choice
    {N : ℕ} [NeZero N]
    (event : RealizedPostingEvent3 N)
    (k : Fin 3)
    (hframed :
      (torusSiteEquivFin N).symm event.1.2 =
          framedForward (Equiv.refl (Fin 3))
            ((torusSiteEquivFin N).symm event.1.1) k ∨
        (torusSiteEquivFin N).symm event.1.2 =
          framedBackward (Equiv.refl (Fin 3))
            ((torusSiteEquivFin N).symm event.1.1) k) :
    EventSpatialOccurrenceTie event k
      (canonicalSideOfFramedStep
        ((torusSiteEquivFin N).symm event.1.1)
        ((torusSiteEquivFin N).symm event.1.2) k) := by
  dsimp only [EventSpatialOccurrenceTie]
  rcases hframed with hfwd | hbwd
  · left
    refine ⟨hfwd, ?_⟩
    simp [canonicalSideOfFramedStep, hfwd]
  · -- Backward step: if it is also forward (possible only for tiny N), the
    -- debit/forward disjunct still holds; otherwise credit/backward holds.
    by_cases hfwd :
        (torusSiteEquivFin N).symm event.1.2 =
          framedForward (Equiv.refl (Fin 3))
            ((torusSiteEquivFin N).symm event.1.1) k
    · left
      refine ⟨hfwd, ?_⟩
      simp [canonicalSideOfFramedStep, hfwd]
    · right
      refine ⟨hbwd, ?_⟩
      simp [canonicalSideOfFramedStep, hfwd]

What this page does not claim

The theorem does not prove that every posting event has a framed step; that stronger claim remains open. The theorem does not establish uniqueness of the spatial occurrence across all possible witnesses. The theorem does not by itself derive the three spatial dimensions of physical space.

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/PairKernelConstructedCovectorEventOccurrenceJoin.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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