Encyclopedia Foundation Foundation Pair Kernel Exact Jnonlinear Gauss S13 Consumer

ARTICLE 3 claims 3 theorems

Foundation Pair Kernel Exact Jnonlinear Gauss S13 Consumer

A machine-checked theorem shows that one accounting event can satisfy three distinct descriptions of the same underlying response.

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The three-branch consumer

The result is a formal object inside the Recognition Science library, a machine-checked collection of theorems. It concerns a ledger, a discrete record of recognition events, and the question of how a single event can be described in different ways. The central result, a proved theorem, establishes that there exists an event which simultaneously satisfies three separate conditions: it is a realized primitive posting pair, it attaches to a signed Green field, and it attaches to a constant-curvature signed Green field at two different scales.

The first condition is the unconditional signed ledger response at q = 1, the baseline accounting of the event. The second is the same response viewed as the zero-background exact-J tangent, meaning it is the linear approximation to the cost function J at a specific point. The third is the constant-native-curvature tangent response at full ordered-edge source, given by the formula 2 * sqrt(hbar * (hbar + 2)). The factor of two in this expression is forced by double-entry ordered-edge summation, a structural rule of the ledger. The canonical Green scale for this third branch reduces to the one-edge cotangent divided by 1 + hbar.

The theorem is precise about what it proves. It states that such an event exists, that it is a realized primitive posting pair, and that it satisfies the three attachment conditions listed above. It also states a specific equality: the real Green scale from posting magnitude at the native ordered exact-J source divided by 1 + hbar equals the native exact-J conjugate source divided by 1 + hbar. Finally, it records that the native ordered exact-J source is not equal to the native action quantum invariant.

What this establishes in plain language is that a single accounting event can be consistently described under three different interpretive frameworks: as a raw posting, as a linear approximation, and as a constant-curvature response. The third branch is explicitly a tangent model, a local linear approximation, not a full physical theory. Its identification with a physical posting background still requires the S12 event-action and background realization arrows, which are not part of this result. The theorem is a structural consistency result, not a physical derivation.

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

This result does not prove that the third branch corresponds to any physical posting background. The existence theorem does not establish uniqueness of the event satisfying the three conditions. The factor of two is a consequence of the ledger's summation rule, not an independent physical constant.

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)

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