Encyclopedia Foundation Foundation Pair Kernel Exact Jnonlinear Gauss S13 Consumer Canonical Exact Jtang

ARTICLE 3 claims 3 theorems

Foundation Pair Kernel Exact Jnonlinear Gauss S13 Consumer Canonical Exact Jtang

A machine-checked proof shows that a specific posting event exists which simultaneously satisfies three distinct Green-response conditions, including one with a forced source magnitude.

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The exact-J tangent consumer

In Recognition Science, a ledger is a discrete record of events, and each event carries a cost that measures how much recognition the event requires. The framework's central theorem forces the cost function to be J(x) = (x + 1/x)/2 - 1. The declaration canonicalExactJTangentConsumer_exists, proved in the framework's machine-checked library of formal theorems, shows that a particular posting event exists which satisfies three distinct conditions at once.

First, the event is a realized primitive posting pair, meaning it belongs to the basic set of allowed events. Second, it satisfies the signed Green response condition at q = 1, which is the unconditional response of the ledger. Third, it satisfies the constant-curvature signed posting attachment condition at two different parameter values: at curvature 1, and at curvature 1 + hbar, where hbar is the framework's unit of action, equal to phi^-5. The theorem also states that the source magnitude for this third condition is forced to be 2 * sqrt(hbar * (hbar + 2)), and that this source is not equal to the action quantum invariant.

The factor of two in the source magnitude arises from double-entry ordered-edge summation, a bookkeeping rule in the framework. The canonical Green scale for this third branch reduces to the one-edge cotangent divided by 1 + hbar. The theorem proves existence of such an event, and it proves the equalities and inequalities stated in the conjunction. It does not, however, identify this third branch with a physical posting background; that identification requires additional arrows from the S12 event-action and background realization steps, which remain open.

THEOREM canonicalExactJTangentConsumer_exists · IndisputableMonolith/Foundation/PairKernelExactJNonlinearGaussS13Consumer.lean
canonicalExactJTangentConsumer_exists · IndisputableMonolith/Foundation/PairKernelExactJNonlinearGaussS13Consumer.lean:34 
theorem canonicalExactJTangentConsumer_exists :
    ∃ event : PostingPair3 3,
      event ∈ realizedPrimitivePostingPairs3 3 ∧
        SignedPostingSourceAttachment3 1 event
          (Equiv.refl (Fin 3))
          (signedRealGreenField3 1 event) ∧
        ConstantCurvatureSignedPostingAttachment3
          1 (by norm_num) 1 event
          (Equiv.refl (Fin 3))
          (constantCurvatureSignedGreenField3 1 1 event) ∧
        ConstantCurvatureSignedPostingAttachment3
          (1 + Constants.hbar) nativeCurvature_pos.le
          nativeOrderedExactJSource event
          (Equiv.refl (Fin 3))
          (constantCurvatureSignedGreenField3
            (1 + Constants.hbar)
            nativeOrderedExactJSource event) ∧
        realGreenScaleFromPostingMagnitude
            (nativeOrderedExactJSource /
              (1 + Constants.hbar)) =
          nativeExactJConjugateSource /
            (1 + Constants.hbar) ∧
        nativeOrderedExactJSource =
          2 * Real.sqrt
            (Constants.hbar * (Constants.hbar + 2)) ∧
        nativeOrderedExactJSource ≠ nativeActionQuantumInv := by
  obtain ⟨event, hevent, _hcert⟩ :=
    canonicalGeneratorSource_consumer_exists
      (N := 3) (by norm_num) (Equiv.refl (Fin 3))
  refine ⟨event, hevent, ?_, ?_, ?_, ?_, ?_, ?_⟩
  · exact signedPostingSourceAttachment3_realGreen
      (N := 3) (by norm_num) 1 event hevent
      (Equiv.refl (Fin 3))
  · exact unitCurvatureSignedPostingAttachment3_realGreen
      (N := 3) (by norm_num) 1 event hevent
      (Equiv.refl (Fin 3))
  · exact nativeCurvatureSignedPostingAttachment3_realGreen
      (N := 3) (by norm_num) event hevent
      (Equiv.refl (Fin 3))
  · exact nativeCurvatureTangentGreenScale
  · exact nativeOrderedExactJSource_eq_two_sqrt
  · exact nativeOrderedExactJSource_ne_nativeActionQuantumInv
THEOREM canonicalExactJTangentConsumer_exists · IndisputableMonolith/Foundation/PairKernelExactJNonlinearGaussS13Consumer.lean
canonicalExactJTangentConsumer_exists · IndisputableMonolith/Foundation/PairKernelExactJNonlinearGaussS13Consumer.lean:34 
theorem canonicalExactJTangentConsumer_exists :
    ∃ event : PostingPair3 3,
      event ∈ realizedPrimitivePostingPairs3 3 ∧
        SignedPostingSourceAttachment3 1 event
          (Equiv.refl (Fin 3))
          (signedRealGreenField3 1 event) ∧
        ConstantCurvatureSignedPostingAttachment3
          1 (by norm_num) 1 event
          (Equiv.refl (Fin 3))
          (constantCurvatureSignedGreenField3 1 1 event) ∧
        ConstantCurvatureSignedPostingAttachment3
          (1 + Constants.hbar) nativeCurvature_pos.le
          nativeOrderedExactJSource event
          (Equiv.refl (Fin 3))
          (constantCurvatureSignedGreenField3
            (1 + Constants.hbar)
            nativeOrderedExactJSource event) ∧
        realGreenScaleFromPostingMagnitude
            (nativeOrderedExactJSource /
              (1 + Constants.hbar)) =
          nativeExactJConjugateSource /
            (1 + Constants.hbar) ∧
        nativeOrderedExactJSource =
          2 * Real.sqrt
            (Constants.hbar * (Constants.hbar + 2)) ∧
        nativeOrderedExactJSource ≠ nativeActionQuantumInv := by
  obtain ⟨event, hevent, _hcert⟩ :=
    canonicalGeneratorSource_consumer_exists
      (N := 3) (by norm_num) (Equiv.refl (Fin 3))
  refine ⟨event, hevent, ?_, ?_, ?_, ?_, ?_, ?_⟩
  · exact signedPostingSourceAttachment3_realGreen
      (N := 3) (by norm_num) 1 event hevent
      (Equiv.refl (Fin 3))
  · exact unitCurvatureSignedPostingAttachment3_realGreen
      (N := 3) (by norm_num) 1 event hevent
      (Equiv.refl (Fin 3))
  · exact nativeCurvatureSignedPostingAttachment3_realGreen
      (N := 3) (by norm_num) event hevent
      (Equiv.refl (Fin 3))
  · exact nativeCurvatureTangentGreenScale
  · exact nativeOrderedExactJSource_eq_two_sqrt
  · exact nativeOrderedExactJSource_ne_nativeActionQuantumInv
THEOREM canonicalExactJTangentConsumer_exists · IndisputableMonolith/Foundation/PairKernelExactJNonlinearGaussS13Consumer.lean
canonicalExactJTangentConsumer_exists · IndisputableMonolith/Foundation/PairKernelExactJNonlinearGaussS13Consumer.lean:34 
theorem canonicalExactJTangentConsumer_exists :
    ∃ event : PostingPair3 3,
      event ∈ realizedPrimitivePostingPairs3 3 ∧
        SignedPostingSourceAttachment3 1 event
          (Equiv.refl (Fin 3))
          (signedRealGreenField3 1 event) ∧
        ConstantCurvatureSignedPostingAttachment3
          1 (by norm_num) 1 event
          (Equiv.refl (Fin 3))
          (constantCurvatureSignedGreenField3 1 1 event) ∧
        ConstantCurvatureSignedPostingAttachment3
          (1 + Constants.hbar) nativeCurvature_pos.le
          nativeOrderedExactJSource event
          (Equiv.refl (Fin 3))
          (constantCurvatureSignedGreenField3
            (1 + Constants.hbar)
            nativeOrderedExactJSource event) ∧
        realGreenScaleFromPostingMagnitude
            (nativeOrderedExactJSource /
              (1 + Constants.hbar)) =
          nativeExactJConjugateSource /
            (1 + Constants.hbar) ∧
        nativeOrderedExactJSource =
          2 * Real.sqrt
            (Constants.hbar * (Constants.hbar + 2)) ∧
        nativeOrderedExactJSource ≠ nativeActionQuantumInv := by
  obtain ⟨event, hevent, _hcert⟩ :=
    canonicalGeneratorSource_consumer_exists
      (N := 3) (by norm_num) (Equiv.refl (Fin 3))
  refine ⟨event, hevent, ?_, ?_, ?_, ?_, ?_, ?_⟩
  · exact signedPostingSourceAttachment3_realGreen
      (N := 3) (by norm_num) 1 event hevent
      (Equiv.refl (Fin 3))
  · exact unitCurvatureSignedPostingAttachment3_realGreen
      (N := 3) (by norm_num) 1 event hevent
      (Equiv.refl (Fin 3))
  · exact nativeCurvatureSignedPostingAttachment3_realGreen
      (N := 3) (by norm_num) event hevent
      (Equiv.refl (Fin 3))
  · exact nativeCurvatureTangentGreenScale
  · exact nativeOrderedExactJSource_eq_two_sqrt
  · exact nativeOrderedExactJSource_ne_nativeActionQuantumInv

What this page does not claim

The declaration does not claim that the constant-curvature tangent response is physically realized; that identification remains open. The declaration does not claim that the framework derives the fine-structure constant alpha; that remains an open target. The declaration does not claim that the Riemann Hypothesis is proved; only an equivalence is known.

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/PairKernelExactJNonlinearGaussS13Consumer.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.

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