Foundation Pair Kernel Response Ancestry S21 Elementary Posting Passes Balance C

The balance probe

The ledger, a discrete record of events, keeps accounts that change only when a posting moves value between them. The simplest possible move, an elementary posting, shifts one unit from a source account to a sink account. The balance current probe asks whether such a move satisfies two conservation conditions: the flow is antisymmetric, meaning the amount leaving the source equals the amount entering the sink, and the total divergence across all accounts is zero, meaning no value appears or disappears anywhere. The declaration elementaryPosting_passes_balanceCurrentProbe proves, in the framework's machine-checked library of formal theorems, that every elementary posting passes both conditions.

The proof is not a numerical check but a structural fact. For any choice of source and sink accounts, the elementary posting defines a flow that is antisymmetric by construction and whose total divergence vanishes. The theorem holds for any number of accounts, so the conservation law is not an accident of a particular size. A constant flow, one that moves the same amount out of every account, fails the probe because it is not antisymmetric; the library proves this separately.

In Recognition Science, the balance probe is one of five operational responses that form a coordinate basis for the response catalog. The framework derives these responses from a forced cost function, and the theorem shows the elementary posting is a legitimate witness for the balance response. The probe family separates responses: two responses that agree on all probes are identical. This makes the balance probe a reliable measurement instrument within the framework.

What the declaration does not claim is broader. It does not say that physical channels in an independently supplied carrier must realize this balance response, nor that the probe can identify points of such a carrier. The framework's library explicitly leaves open whether physical response systems realize the forced responses and whether physical channels are extensional under the probe family. The balance probe is a property of the posting structure itself, not a claim about how any external physical system must behave.

theorem elementaryPosting_passes_balanceCurrentProbe
    {n : ℕ}
    (source sink : Fin n) :
    BalanceCurrentProbe3
      (elementaryPosting source sink) :=
  ⟨elementaryPosting_antisym source sink,
    elementaryPosting_sum_div_zero source sink⟩

theorem constantFlow_fails_balanceCurrentProbe
    (n : ℕ) (hn : 0 < n) :
    ¬ BalanceCurrentProbe3 (constFlow n) := by
  intro hprobe
  exact constFlow_not_antisym n hn hprobe.1

/-- The committed response probes separate all five source responses. -/
theorem responseCoordinateProbes_separate
    (left right : RecognitionParentResponse3)
    (hindistinguishable :
      ObsEquiv responseCoordinateProbeFamily3
        left right) :
    left = right := by
  by_contra hne
  have hright : right ≠ left :=
    Ne.symm hne
  have hprobe :=
    hindistinguishable
      (responseCoordinateObservation3 left)
      ⟨left, rfl⟩
  simp [responseCoordinateObservation3,
    hright] at hprobe

What this page does not claim

The declaration does not claim that any physical carrier must realize the balance response. It does not claim the probe can identify points of an independently supplied physical carrier. It does not claim that the balance probe alone distinguishes all responses, only the full probe family.

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/PairKernelResponseAncestryS21.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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