Foundation Pair Kernel Production Operation Channel Selection S25

The three operation selectors

In Recognition Science, an event is not a vague happening but a discrete record in a ledger: a pair of entries that are written together. The module named s25 takes the next step. It asks what operations actually produce such an event, and it answers with exactly three. A spatial operation is one realized edge of a recognition graph, evaluated by an exact action and its edge flux. A tick operation is the ledger commit from a start tick to an end tick. A balance operation is the event's elementary double-entry current. These are data, not response labels.

The module proves structural facts about these operations. Reversing a spatial operation leaves its exact cost unchanged, while reversing its flux changes sign. Reversing a balance operation negates its current. A tick operation always spans exactly one successor tick. The cost of a spatial operation is positive whenever the field differs across the edge. The cost of a batch of spatial operations is the sum of the individual costs, and appending batches adds their costs. These are theorems in the machine-checked library of formal theorems, with no unproved assumptions.

The module also establishes a boundary. The three operations are committed, meaning they are independent of any proposed response system. They therefore coexist with both a transport-complete and an incomplete system. The module explicitly does not construct a physical carrier, and it does not claim completeness or observational exhaustion unconditionally. A probe for a conserved zero current exists, but the zero current is not the current of any event operation. The first missing physical semantic is split into three operation-to-channel selectors, whose conjunction is exactly event-act transport.

One theorem shows that a naive simplification fails: taking only the first operation of a batch is not compositional. The batch of a spatial followed by a tick operation is not equal to the batch of the spatial followed by the batch of the tick. This matters because it shows that the operations are not independent atoms; their order and combination carry meaning. The module thus establishes the precise operations and their algebraic behavior, while leaving the construction of the physical carrier open.

theorem spatialEdgeActionExactCost_reverse
    (operation : SpatialEdgeActionOperation3)
    (field : Fin (TorusCard3 3) → ℝ) :
    spatialEdgeActionExactCost3
        (reverseSpatialEdgeActionOperation3 operation)
        field =
      spatialEdgeActionExactCost3 operation field := by
  unfold spatialEdgeActionExactCost3
    reverseSpatialEdgeActionOperation3
    spatialEdgeActionOperation3
    primitivePostingPairExactJCost3
  change
    Real.cosh
        (field operation.event.1.2 -
          field operation.event.1.1) - 1 =
      Real.cosh
        (field operation.event.1.1 -
          field operation.event.1.2) - 1
  rw [show
      field operation.event.1.2 -
          field operation.event.1.1 =
        -(field operation.event.1.1 -
          field operation.event.1.2) by ring,
    Real.cosh_neg]

theorem spatialActionOperationBatchExactCost_append
    (left right : List SpatialEdgeActionOperation3)
    (field : Fin (TorusCard3 3) → ℝ) :
    spatialActionOperationBatchExactCost3
        (left ++ right) field =
      spatialActionOperationBatchExactCost3 left field +
        spatialActionOperationBatchExactCost3 right field := by
  simp [spatialActionOperationBatchExactCost3]

theorem tickCommitOperation_is_one_successor
    (operation : TickCommitOperation3) :
    tickCommitOperationFinish3 operation =
      recognitionTimeAlgebra.succ
        (tickCommitOperationStart3 operation) :=
  postingEventEndTick3_eq_succ operation.event

/-- Non-functorial control: retaining only the first operation does not
preserve batch composition. -/
theorem firstOnlyOperationBatch_not_compositional
    (event : RealizedPostingEvent3 3) :
    firstOnlyOperationBatch3
        ([.spatial (spatialEdgeActionOperation3 event)] ++
          [.tick (tickCommitOperation3 event)]) ≠
      firstOnlyOperationBatch3
          [.spatial (spatialEdgeActionOperation3 event)] ++
        firstOnlyOperationBatch3
          [.tick (tickCommitOperation3 event)] := by
  simp [firstOnlyOperationBatch3]

/-- The same committed operation package coexists with a system that selects
all operation channels and one that does not. Hence the source operations do
not force their promotion to physical channels. -/
theorem committedOperations_admit_channelSelection_split :
    CommittedProductionOperations3 ∧
      ∃ left right : PostingPhysicalResponseSystem3.{0} 3,
        ProductionOperationsSelectPhysicalChannels3 left ∧
          ¬ ProductionOperationsSelectPhysicalChannels3 right := by
  obtain
    ⟨left, right, _hleftReads, _hrightReads,
      hleftTransport, hrightNotTransport⟩ :=
    committed_ancestry_admits_eventActTransport_split
  exact
    ⟨committedProductionOperations_hold,
      left, right,
      (productionOperationSelectors_iff_eventActTransport
        left).2 hleftTransport,
      fun hright =>
        hrightNotTransport
          ((productionOperationSelectors_iff_eventActTransport
            right).1 hright)⟩

What this page does not claim

This module does not construct the physical carrier PostingPhysicalChannelCarrier3. This module does not prove that any physical carrier exists or is complete. This module does not claim that the three operations are the only possible operations.

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/PairKernelProductionOperationChannelSelectionS25.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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