Encyclopedia Foundation Foundation Pair Kernel Scale Covariant Observables S20 Consumer Prediction Ready

ARTICLE 5 claims 5 theorems

Foundation Pair Kernel Scale Covariant Observables S20 Consumer Prediction Ready

A machine-checked theorem pins down the ratio of source to curvature in the framework's recognition ledger, and shows which numbers survive a change of units.

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The exact-J Green ratios

A ratio is a comparison of two quantities of the same kind. In physics, ratios are often the only numbers that survive when you change your units: the ratio of a proton's mass to an electron's mass does not care whether you measure in kilograms or in pounds. The Recognition Science declaration predictionReady_exactJGreen_ratios is a machine-checked theorem that fixes one such ratio inside the framework's own ledger, the discrete record of recognition events that the framework uses to build physical quantities.

The theorem states, in the framework's native units, that the exact Green ratio at the canonical drop equals sqrt(hbar * (hbar + 2)) / (1 + hbar). Here hbar is the framework's reduced Planck constant, which the framework derives as phi^-5, where phi is the golden ratio. The same theorem also states that the Green ratio at drop 1 equals tanh(1), and that for every drop, the curvature times the Green ratio equals the one-edge source. These are exact equalities, not approximations. The declaration also includes a dimensional statement: the configuration dimension D equals 5, which is the framework's way of saying that the physical carrier dimension for the posting event is 5.

The theorem's practical content is scale covariance. The framework's consumer compiles a readout interface after dividing out the duration and energy boundary units. The theorem shows that the dimensionless action ratios survive this quotient, while an equality between a physical event's absolute action and the dimensionless numeral hbar does not survive. In plain terms: the framework's predictions that are ratios are stable under unit choices, but an equality that mixes a dimensioned quantity with a pure number is not. The numeral hbar and its D+2 exponent do survive, so the framework's constants remain meaningful.

What the declaration does not claim is just as important. It does not claim that the framework has measured the Green ratio against any experiment. It does not claim that the value sqrt(hbar * (hbar + 2)) / (1 + hbar) is a prediction of a new physical constant. It does not claim that the framework's hbar equals the measured Planck constant; the framework's hbar is a derived dimensionless number, not the SI constant. The theorem is a statement about the framework's internal consistency: given its definitions, these ratios hold exactly. Whether those ratios correspond to anything in the physical world is a separate empirical question, not settled by this declaration.

THEOREM predictionReady_exactJGreen_ratios · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean
predictionReady_exactJGreen_ratios · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean:101 
/-- Prediction-ready exact-J ratios. The native and ledger field-unit
branches remain separate evaluations of the same scale-covariant law. -/
theorem predictionReady_exactJGreen_ratios :
    exactJGreenRatioAtDrop nativeActionCanonicalDrop =
        Real.sqrt
            (Constants.hbar * (Constants.hbar + 2)) /
          (1 + Constants.hbar) ∧
      realGreenScaleFromPostingMagnitude
          (nativeOrderedExactJSource /
            (1 + Constants.hbar)) =
        exactJGreenRatioAtDrop nativeActionCanonicalDrop ∧
      exactJGreenRatioAtDrop 1 = Real.tanh 1 ∧
      (∀ drop : ℝ,
        exactJCurvatureAtDrop drop *
            exactJGreenRatioAtDrop drop =
          exactJOneEdgeSourceAtDrop drop) ∧
      GapDerivation.configDim GapDerivation.D = 5 := by
  refine ⟨?_, ?_, ?_, exactJGreenRatio_reciprocity, ?_⟩
  · rw [nativeExactJGreenRatio_eq,
      nativeExactJConjugateSource_eq_sqrt]
  · calc
      realGreenScaleFromPostingMagnitude
          (nativeOrderedExactJSource /
            (1 + Constants.hbar)) =
          nativeExactJConjugateSource /
            (1 + Constants.hbar) :=
        nativeCurvatureTangentGreenScale
      _ = exactJGreenRatioAtDrop
          nativeActionCanonicalDrop :=
        nativeExactJGreenRatio_eq.symm
  · exact exactJGreenRatioAtDrop_eq_tanh 1
  · exact GapDerivation.configDim_at_D3
THEOREM predictionReady_exactJGreen_ratios · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean
predictionReady_exactJGreen_ratios · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean:101 
/-- Prediction-ready exact-J ratios. The native and ledger field-unit
branches remain separate evaluations of the same scale-covariant law. -/
theorem predictionReady_exactJGreen_ratios :
    exactJGreenRatioAtDrop nativeActionCanonicalDrop =
        Real.sqrt
            (Constants.hbar * (Constants.hbar + 2)) /
          (1 + Constants.hbar) ∧
      realGreenScaleFromPostingMagnitude
          (nativeOrderedExactJSource /
            (1 + Constants.hbar)) =
        exactJGreenRatioAtDrop nativeActionCanonicalDrop ∧
      exactJGreenRatioAtDrop 1 = Real.tanh 1 ∧
      (∀ drop : ℝ,
        exactJCurvatureAtDrop drop *
            exactJGreenRatioAtDrop drop =
          exactJOneEdgeSourceAtDrop drop) ∧
      GapDerivation.configDim GapDerivation.D = 5 := by
  refine ⟨?_, ?_, ?_, exactJGreenRatio_reciprocity, ?_⟩
  · rw [nativeExactJGreenRatio_eq,
      nativeExactJConjugateSource_eq_sqrt]
  · calc
      realGreenScaleFromPostingMagnitude
          (nativeOrderedExactJSource /
            (1 + Constants.hbar)) =
          nativeExactJConjugateSource /
            (1 + Constants.hbar) :=
        nativeCurvatureTangentGreenScale
      _ = exactJGreenRatioAtDrop
          nativeActionCanonicalDrop :=
        nativeExactJGreenRatio_eq.symm
  · exact exactJGreenRatioAtDrop_eq_tanh 1
  · exact GapDerivation.configDim_at_D3
THEOREM predictionReady_exactJGreen_ratios · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean
predictionReady_exactJGreen_ratios · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean:101 
/-- Prediction-ready exact-J ratios. The native and ledger field-unit
branches remain separate evaluations of the same scale-covariant law. -/
theorem predictionReady_exactJGreen_ratios :
    exactJGreenRatioAtDrop nativeActionCanonicalDrop =
        Real.sqrt
            (Constants.hbar * (Constants.hbar + 2)) /
          (1 + Constants.hbar) ∧
      realGreenScaleFromPostingMagnitude
          (nativeOrderedExactJSource /
            (1 + Constants.hbar)) =
        exactJGreenRatioAtDrop nativeActionCanonicalDrop ∧
      exactJGreenRatioAtDrop 1 = Real.tanh 1 ∧
      (∀ drop : ℝ,
        exactJCurvatureAtDrop drop *
            exactJGreenRatioAtDrop drop =
          exactJOneEdgeSourceAtDrop drop) ∧
      GapDerivation.configDim GapDerivation.D = 5 := by
  refine ⟨?_, ?_, ?_, exactJGreenRatio_reciprocity, ?_⟩
  · rw [nativeExactJGreenRatio_eq,
      nativeExactJConjugateSource_eq_sqrt]
  · calc
      realGreenScaleFromPostingMagnitude
          (nativeOrderedExactJSource /
            (1 + Constants.hbar)) =
          nativeExactJConjugateSource /
            (1 + Constants.hbar) :=
        nativeCurvatureTangentGreenScale
      _ = exactJGreenRatioAtDrop
          nativeActionCanonicalDrop :=
        nativeExactJGreenRatio_eq.symm
  · exact exactJGreenRatioAtDrop_eq_tanh 1
  · exact GapDerivation.configDim_at_D3
THEOREM predictionReady_exactJGreen_ratios · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean
predictionReady_exactJGreen_ratios · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean:101 
/-- Prediction-ready exact-J ratios. The native and ledger field-unit
branches remain separate evaluations of the same scale-covariant law. -/
theorem predictionReady_exactJGreen_ratios :
    exactJGreenRatioAtDrop nativeActionCanonicalDrop =
        Real.sqrt
            (Constants.hbar * (Constants.hbar + 2)) /
          (1 + Constants.hbar) ∧
      realGreenScaleFromPostingMagnitude
          (nativeOrderedExactJSource /
            (1 + Constants.hbar)) =
        exactJGreenRatioAtDrop nativeActionCanonicalDrop ∧
      exactJGreenRatioAtDrop 1 = Real.tanh 1 ∧
      (∀ drop : ℝ,
        exactJCurvatureAtDrop drop *
            exactJGreenRatioAtDrop drop =
          exactJOneEdgeSourceAtDrop drop) ∧
      GapDerivation.configDim GapDerivation.D = 5 := by
  refine ⟨?_, ?_, ?_, exactJGreenRatio_reciprocity, ?_⟩
  · rw [nativeExactJGreenRatio_eq,
      nativeExactJConjugateSource_eq_sqrt]
  · calc
      realGreenScaleFromPostingMagnitude
          (nativeOrderedExactJSource /
            (1 + Constants.hbar)) =
          nativeExactJConjugateSource /
            (1 + Constants.hbar) :=
        nativeCurvatureTangentGreenScale
      _ = exactJGreenRatioAtDrop
          nativeActionCanonicalDrop :=
        nativeExactJGreenRatio_eq.symm
  · exact exactJGreenRatioAtDrop_eq_tanh 1
  · exact GapDerivation.configDim_at_D3
THEOREM actionQuotient_consumer · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean
actionQuotient_consumer · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean:134 
/-- The quotient result is intentionally asymmetric: dimensionless action
ratios survive, while equality between a physical event's absolute action and
the dimensionless RS `hbar = phi^-5` numeral does not survive the full
duration-energy unit quotient. The numeral and its D+2 exponent do survive. -/
theorem actionQuotient_consumer :
    (∃ left right : RecognitionPhysicalValuation3.{0} 3,
      SameRecognitionData3 left right ∧
        ∃ event : RealizedPostingEvent3 3,
          postingEventAction3 left.kinematics event.1 ≠
            postingEventAction3 right.kinematics event.1) ∧
      (∀ left right : RecognitionPhysicalValuation3.{0} 3,
        SameRecognitionData3 left right →
          ∀ first second : RealizedPostingEvent3 3,
            postingEventAction3 left.kinematics first.1 *
                postingEventAction3 right.kinematics second.1 =
              postingEventAction3 left.kinematics second.1 *
                postingEventAction3 right.kinematics first.1) :=
  ⟨absolute_eventAction_not_unit_invariant,
    fun _ _ hsame =>
      unitEquivalent_action_ratio_invariant hsame⟩

What this page does not claim

The declaration does not claim any experimental measurement or empirical confirmation. The declaration does not claim that the framework's hbar equals the SI Planck constant. The declaration does not claim that the Green ratio value corresponds to a known 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/PairKernelScaleCovariantObservablesS20Consumer.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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