Foundation Pair Kernel Production Operation Channel Selection S25 Production Ope

Channel selection without a carrier

The declaration in question, productionOperationSelectors_iff_observableExhaustion, is not a single statement but the name for a family of selectors defined in the framework's machine-checked library of formal theorems. The framework models a ledger, a discrete record of events, where each event is a realized posting between two points. Three kinds of operations are traced from those events: a spatial operation, which is an edge evaluated by an exact action and its flux; a tick operation, which is the ledger commit from a start to an end tick; and a balance operation, which is the event's elementary double-entry current. The selectors ask whether a proposed response system, one that maps events to physical channels, can realize each of these operations through its channels.

The key theorem, committedOperations_admit_channelSelection_split, proves that the committed operations admit a channel-selection split: there exists a response system for which all three selectors hold, and another for which they do not. This is a theorem, not a definitional choice. The selectors are data, not response labels. Their orientation, conservation, and finite-batch composition laws are theorem-backed. The theorem spatialEdgeActionOperation_cost_positive shows that a spatial operation has positive cost whenever the field differs across the edge; spatialEdgeActionFlux_reverse shows that reversing the operation negates the flux; balanceCurrentOperation_reverse shows the same for the balance current. These are the conservation and orientation facts that make the operations physical.

What the selectors do not do is construct the physical carrier itself. The docstring is explicit: the selectors still do not construct PostingPhysicalChannelCarrier3.Carrier. The first missing physical semantic is split into three operation-to-channel selectors, and their conjunction is exactly the event-act transport, not a weaker theorem. The committed operations are independent of a proposed response system; they coexist with both a transport-complete and an incomplete system. No physical carrier, completeness, or observational exhaustion is claimed unconditionally. The theorem firstOnlyOperationBatch_not_compositional shows that taking only the first operation in a batch is not compositional, a sharp negative result that rules out a naive simplification.

The practical consequence is a precise boundary. The framework proves that the three operation types can be assigned to channels in a way that respects their conservation and orientation, and that this assignment is not unique. But the physical carrier, the object that would make those channels concrete, remains an open target. The selectors are a necessary condition, not a sufficient one. A reader who wants to know whether the framework has derived a complete physical theory from the ledger must look elsewhere; this declaration establishes only the operation-to-channel split, and it does so with no sorry and no new axiom.

/-- 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)⟩

theorem spatialEdgeActionOperation_cost_positive
    (operation : SpatialEdgeActionOperation3)
    (field : Fin (TorusCard3 3) → ℝ)
    (hdrop :
      field operation.event.1.1 -
        field operation.event.1.2 ≠ 0) :
    0 < spatialEdgeActionExactCost3 operation field := by
  have hpositive :=
    realizedPrimitivePostingPair_exactJCost_positive
      operation.event.2 field hdrop
  simpa [spatialEdgeActionExactCost3,
    spatialEdgeActionOperation_weight_eq_one] using hpositive

theorem spatialEdgeActionFlux_reverse
    (operation : SpatialEdgeActionOperation3)
    (field : Fin (TorusCard3 3) → ℝ) :
    spatialEdgeActionFlux3
        (reverseSpatialEdgeActionOperation3 operation)
        field =
      -spatialEdgeActionFlux3 operation field := by
  change
    exactJEdgeFlux
        (recognitionProductionGraph3 3)
        field operation.event.1.2
          operation.event.1.1 =
      -exactJEdgeFlux
        (recognitionProductionGraph3 3)
        field operation.event.1.1
          operation.event.1.2
  exact
    exactJEdgeFlux_antisym
      (recognitionProductionGraph3 3)
      field
      operation.event.1.2
      operation.event.1.1

/-- 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]

What this page does not claim

No physical carrier is constructed by the selectors. No completeness or observational exhaustion is claimed unconditionally. The selectors do not prove that any particular response system is physically realized.

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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