Foundation Pair Kernel Gap2a Landauer Calorimeter Door Residual Landauer Priced

The priced reading

The declaration landauerPricedOccurrenceReading_iff_remainingPhysicalEquality establishes a formal equivalence in the framework's machine-checked library of formal theorems. In plain terms, it says that a certain priced reading of an occurrence, evaluated at a specific unit probe, holds if and only if a particular remaining physical equality holds. The priced reading is defined as the occurrence-bridged source normalization, and the remaining equality is the statement that a source scale equals the unique cotangent coordinate, which is the fifth power of the golden ratio.

The theorem does not claim that this remaining equality is true. It only packages one statement as equivalent to another. The framework's library also proves that the heat-carrier premise, which relates heat to erased bits through a conversion quantum, is inhabited at every possible quantum value. This means the premise does not select any particular quantum, and therefore cannot force the remaining equality to hold for any specific source scale. A separate theorem explicitly shows that the free-quantum premise does not force the remaining equality, because it holds even at a candidate value that the equality rejects.

The declaration also does not claim that an independent calorimeter exists. The framework marks the missing independent calorimeter as an open gap. A revival theorem states that if such a calorimeter were found, and if it forced a specific quantum value while the priced carrier's source scale matched that quantum, then the remaining equality would follow. But that is a conditional statement, not a proof of the equality itself.

/-- Priced packaging of the occurrence source reading: every bridged
occurrence, evaluated on the canonical forward unit probe, equals the
occurrence boundary record times `uniqueCotangentCoordinate`. Mentions the
residual target only as packaging; does not assume it as a premise. -/
def LandauerPricedOccurrenceReading (S : ℝ) : Prop :=
  OccurrenceBridgedSourceNormalization S

/-- **UNFOLD-TARGET.** The priced packaging unfolds to the residual atom:
unit-probe reading of `bridgedOccurrenceSourceCovector S` at the unique
cotangent price is equivalent to `RemainingPhysicalEquality S`. -/
theorem landauerPricedOccurrenceReading_iff_remainingPhysicalEquality
    (S : ℝ) :
    LandauerPricedOccurrenceReading S ↔ RemainingPhysicalEquality S :=
  occurrenceBridgedSourceNormalization_iff_remainingPhysicalEquality S

/-- Explicit vacuous-pairing countermodel: the free-`q` heat-carrier premise
(existential, not mentioning `S`) is inhabited, yet candidate A rejects the
atom. Mirrors `no_route_within_banked_family`. -/
theorem landauerDoor_does_not_force_remainingPhysicalEquality :
    ¬ (∀ S : ℝ,
      LandauerFreeConversionQuantumPremise S →
        RemainingPhysicalEquality S) := by
  intro hforce
  exact remainingPhysicalEquality_rejects_candidateA
    (hforce candidateA_sourceMagnitudeExpr.eval
      (landauerFreeConversionQuantumPremise_holds _))

/-- The Landauer heat-carrier premise is inhabited at every conversion
quantum via the tautological posted-heat selector. -/
theorem landauer_q_free_inhabits_every_conversion_quantum (q : ℝ) :
    HeatIsPostedRecordFlux q (tautologicalPostedHeat q) :=
  tautological_heat_is_posted_record_flux q

/-- Explicit vacuous-pairing countermodel: the free-`q` heat-carrier premise
(existential, not mentioning `S`) is inhabited, yet candidate A rejects the
atom. Mirrors `no_route_within_banked_family`. -/
theorem landauerDoor_does_not_force_remainingPhysicalEquality :
    ¬ (∀ S : ℝ,
      LandauerFreeConversionQuantumPremise S →
        RemainingPhysicalEquality S) := by
  intro hforce
  exact remainingPhysicalEquality_rejects_candidateA
    (hforce candidateA_sourceMagnitudeExpr.eval
      (landauerFreeConversionQuantumPremise_holds _))

What this page does not claim

The remaining physical equality is true for the actual universe. An independent calorimeter exists or has been constructed. The heat-carrier premise alone determines the value of the conversion quantum.

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/PairKernelGap2aLandauerCalorimeterDoorResidual.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

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