Foundation Pair Kernel Scale Breaking Source Residual Native Posting Action Alon

What one premise cannot decide

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition is the act of registering one. The framework's library, a machine-checked collection of formal results, studies which properties of such a ledger force the rest of physics. One question is which source coordinate, the number attached to an event's origin, goes with the fundamental unit of action, the quantity that measures how much a process changes over time.

The declaration nativePostingAction_alone_admits_distinct_source_maps proves a precise negative result. It shows that assuming only one thing, that the minimum posting carries the native action quantum, does not determine the source coordinate. Two different source maps, two different assignments of numbers to events, both satisfy that premise and both appear among the realized primitive posting pairs. The result is a formal counterexample: the premise is true in a model, and yet the two source magnitudes differ. In plain terms, knowing which posting carries the fundamental action does not tell you which source number goes with it.

This is a statement about the limits of a single premise, not a failure of the framework. The library proves the premise alone is too weak. It also proves that adding a second statement, that the source coordinate is the multiplicative dual of the action measured in the Euler-J unit, selects exactly one of the two candidates. That second law, called NativeActionDualSourceLaw, is a hypothesis, a prediction with a named falsifier, not a result derived from the current premises. The library shows the current premises do not force it, and that one candidate satisfies every current premise while failing it.

The consequence is a clean separation of two physical statements. The action normalization and the source-action duality are independent scale-breaking facts. Each is needed, and neither follows from the other. The framework's contribution is to make that independence precise: it names the exact gap, proves the gap exists, and shows what closes it. A reader who wants to know what the framework claims about this corner of the ledger now has a sharp answer, one premise does not decide, two do.

/-- Even after the native posting-action assignment is fixed, a source map is
still free until the action-dual identification is supplied. -/
theorem nativePostingAction_alone_admits_distinct_source_maps :
    MinimumJPostingCarriesNativeAction3
        (@nativePostingActionCandidate3 3 inferInstance) ∧
      ∃ event : PostingPair3 3,
        event ∈ realizedPrimitivePostingPairs3 3 ∧
          unitSourceMagnitudeCountermodel3 event ≠
            nativeDualSourceMagnitudeCandidate3 event := by
  refine ⟨nativePostingActionCandidate_carries_nativeAction, ?_⟩
  obtain
    ⟨_hunitDual, _hnativeDual, event, hevent,
      _hA, _hB, hne⟩ :=
    sourceActionDuality_alone_admits_distinct_banked_magnitudes
  exact ⟨event, hevent, hne⟩

/-- The new law is not hidden in the old package: Candidate A satisfies every
current premise and fails the native-action dual law. -/
theorem currentPremises_do_not_force_nativeActionDualSourceLaw :
    ¬ (∀ sourceMagnitude : ℝ,
      CurrentRecognitionSourcePremises sourceMagnitude →
        NativeActionDualSourceLaw sourceMagnitude) := by
  intro hforce
  exact nativeActionDualSourceLaw_rejects_candidateA
    (hforce _
      candidateA_satisfies_currentRecognitionSourcePremises)

/-- HYPOTHESIS: one physical posting has native action and its positive source
coordinate is the multiplicative dual measured in the Euler-J action unit.
The statement is target-blind and contains no continuum readout. -/
def NativeActionDualSourceLaw (sourceMagnitude : ℝ) : Prop :=
  0 < sourceMagnitude ∧
    sourceMagnitude * Constants.hbar = eulerJActionUnit

What this page does not claim

The declaration does not claim that the native posting action premise is false. The declaration does not claim that the NativeActionDualSourceLaw is derived from the current premises. The declaration does not claim that the source coordinate is physically observable.

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

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