Encyclopedia Foundation Foundation Pair Kernel Scale Covariant Observables S20 Consumer S20 Nonlinear Ga

ARTICLE 4 claims 2 theorems 2 models

Foundation Pair Kernel Scale Covariant Observables S20 Consumer S20 Nonlinear Ga

A machine-checked definition shows that a nonlinear Gauss law and its Green function survive a change of coordinates, and what that change deliberately leaves behind.

Verify this page

The compiled readout

The declaration s20_nonlinearGauss_tangentGreen_consumer_compiles is a definition in the framework's machine-checked library. It names the canonical representative of a recognition system, a discrete record of events, after absolute duration and energy units have been moved to explicit boundary inputs. The definition is a MODEL witness: it inhabits the scale-covariant interface, meaning the physical readout survives when units are divided out.

The theorem-backed exports are four. Normalized Tick ratios, the response-derived D+2 completeness, unit-quotient invariance, and the exact-J source/curvature Green ratio are all established. The canonical consumer exists: there is an event whose normalized posting duration is 1, whose normalized posting energy equals the channel price, whose normalized posting action equals the same price, and whose physical posting carrier dimension is 5. The normalized elapsed tick count for 8 ticks is 8.

The exact-J Green ratios are prediction-ready. The ratio at the native action canonical drop equals sqrt(hbar * (hbar + 2)) / (1 + hbar), where hbar is phi^-5. The ratio at drop 1 equals tanh 1. For every drop, the curvature at that drop times the Green ratio at that drop equals the one-edge source at that drop. The configuration dimension is 5.

The quotient result is intentionally asymmetric. Dimensionless action ratios survive the full duration-energy unit quotient, but equality between a physical event's absolute action and the dimensionless numeral hbar does not survive. The numeral and its D+2 exponent do survive. The S13 nonlinear Gauss law, tangent Hessian, and real Green attachment compile unchanged. S20 changes their exposed coordinates, not their mathematics.

What this does not claim: it does not claim that the physical recognition-to-linking bridge is closed. It does not claim that the absolute action equals hbar for any physical event. It does not claim that the Green ratio at the canonical drop equals the measured value of any physical constant; it is a derived ratio within the framework.

MODEL s20_nonlinearGauss_tangentGreen_consumer_compiles · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean
s20_nonlinearGauss_tangentGreen_consumer_compiles · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean:155 
/-- The S13 nonlinear Gauss law, tangent Hessian, and real Green attachment
compile unchanged. S20 changes their exposed coordinates, not their
mathematics. -/
def s20_nonlinearGauss_tangentGreen_consumer_compiles :=
  PairKernelExactJNonlinearGaussS13Consumer.canonicalExactJTangentConsumer_exists
THEOREM canonicalScaleCovariantConsumer_exists · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean
canonicalScaleCovariantConsumer_exists · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean:62 
/-- MODEL consumer: the canonical representative inhabits the
scale-covariant interface, while every exported numeric statement has had its
duration and energy boundary units divided out. -/
theorem canonicalScaleCovariantConsumer_exists :
    ∃ event : RealizedPostingEvent3 3,
      normalizedPostingDuration3
          canonicalScaleCovariantPostingReadoutSemantics3
          event.1 = 1 ∧
        normalizedPostingEnergy3
          canonicalScaleCovariantPostingReadoutSemantics3
          event.1 =
            canonicalPostingEventChannelPrice3 event ∧
        normalizedPostingAction3
          canonicalScaleCovariantPostingReadoutSemantics3
          event.1 =
            canonicalPostingEventChannelPrice3 event ∧
        physicalPostingCarrierDimension3
          (parentPhysicalChannelCarrier3 3) event = 5 ∧
        normalizedElapsedTickCount
          canonicalElapsedTimeObservable 8 = 8 := by
  obtain ⟨pair, hpair⟩ :=
    realizedPrimitivePostingPair3_exists
  let event : RealizedPostingEvent3 3 :=
    ⟨pair, hpair⟩
  refine
    ⟨event,
      normalizedPostingDuration_eq_one
        canonicalScaleCovariantPostingReadoutSemantics3 event,
      normalizedPostingEnergy_eq_price
        canonicalScaleCovariantPostingReadoutSemantics3 event,
      normalizedPostingAction_eq_price
        canonicalScaleCovariantPostingReadoutSemantics3 event,
      ?_,
      elapsedTime_octave_ratio
        canonicalElapsedTimeObservable⟩
  exact
    scaleCovariantReadout_dimension_eq_five
      canonicalScaleCovariantPostingReadoutSemantics3 event
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
MODEL s20_nonlinearGauss_tangentGreen_consumer_compiles · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean
s20_nonlinearGauss_tangentGreen_consumer_compiles · IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20Consumer.lean:155 
/-- The S13 nonlinear Gauss law, tangent Hessian, and real Green attachment
compile unchanged. S20 changes their exposed coordinates, not their
mathematics. -/
def s20_nonlinearGauss_tangentGreen_consumer_compiles :=
  PairKernelExactJNonlinearGaussS13Consumer.canonicalExactJTangentConsumer_exists

What this page does not claim

The physical recognition-to-linking bridge is not closed. No physical event's absolute action equals the dimensionless hbar numeral. The Green ratio at the canonical drop is not claimed to match any measured value.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:

MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND