Foundation Pair Kernel Gap2a Landauer Calorimeter Door Residual Landauer Door Re

The conditional revival

In physics, Landauer's principle ties the heat a computer must release when it erases a bit of information to the energy cost of that erasure. The Recognition Science framework formalizes this as a ledger, a discrete record of events, in which heat is posted as a record flux. A stack of theorems in the framework's machine-checked library of formal theorems proves that this heat equals the number of erased bits times a conversion quantum, a single free parameter that the stack does not determine.

The declaration landauerDoor_revival_if_independent_calorimeter_selects_nativeActionQuantumInv is a conditional statement. It says: if a future, independent calorimeter (a device that measures heat) were to force the conversion quantum to equal a specific value, the framework's native action quantum, and if the priced carrier's source scale were identified with that same value, then a particular physical equality, RemainingPhysicalEquality, would follow. The theorem is an if-then statement; it does not assert that the calorimeter exists or that the equality holds.

The machinery also proves two negative results. First, the free conversion quantum premise, which holds at every quantum, does not mention the source scale S, so pairing it with any S cannot force the target equality. A countermodel shows this: the premise is inhabited yet candidate A, where S equals 1, rejects the equality. Second, the missing independent calorimeter type is cited as an open gap, not discharged. The revival theorem carries the missing premise as an explicit hypothesis only; it is not a default or an instance used elsewhere.

What this means in plain terms: the framework has formally recorded that a particular route to a physical conclusion is blocked, and has written down exactly what would unblock it. It does not claim the unblocking device exists, nor that the conclusion is true. The value of the declaration is precision about the missing premise, not progress on supplying it.

/-- **OPEN revival door** (not assumed elsewhere).

Cites `MissingIndependentCellCalorimeter`. The missing premise is carried
as an explicit hypothesis: a future independent calorimeter discharge that
forces selective conversion quantum `nativeActionQuantumInv` (not free
`q`), together with identification of the priced occurrence source scale
with that quantum. Under those hypotheses the priced carrier yields
`RemainingPhysicalEquality`.

The heat-carrier conjunct is retained as the named revival hypothesis even
though the present corpus already inhabits it tautologically at every `q`
(including `nativeActionQuantumInv`); independence of that inhabitation is
exactly the OPEN wall. The scale-identification conjunct is what closes the
atom once a selective independent forcing exists.

This theorem is the named revival condition only. It does not inhabit the
missing independent discharge, does not use a structure-field default, and
is not used as an instance elsewhere in this module. -/
theorem landauerDoor_revival_if_independent_calorimeter_selects_nativeActionQuantumInv
    (_cited_open_wall : MissingIndependentCellCalorimeter)
    (S : ℝ)
    (h_independent_forces_selective_quantum_nativeInv :
      ∃ heat : PhysicalStepHeat,
        HeatIsPostedRecordFlux nativeActionQuantumInv heat)
    (h_priced_carrier_scale_eq_that_quantum :
      S = nativeActionQuantumInv) :
    RemainingPhysicalEquality S := by
  -- Retain the OPEN heat hypothesis in the statement; closure uses the
  -- priced-scale identification with the already-proved cotangent equality.
  rcases h_independent_forces_selective_quantum_nativeInv with ⟨_heat, _hheat⟩
  change S = uniqueCotangentCoordinate
  exact h_priced_carrier_scale_eq_that_quantum.trans
    uniqueCotangentCoordinate_eq_nativeActionQuantumInv.symm

/-- 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 _))

/-- Named-gap citation without replaying the wall certificate's
`native_decide` / `ofReduceBool` payload. The door's independence residual
is the banked type `MissingIndependentCellCalorimeter` from
`LandauerCalorimeterForcing.lean` (certificate
`missingIndependentCellCalorimeterWallCert` remains the corpus authority). -/
theorem landauerDoor_cites_missingIndependentCellCalorimeter_type :
    MissingIndependentCellCalorimeter = MissingIndependentCellCalorimeter :=
  rfl

What this page does not claim

The theorem does not claim that an independent calorimeter exists or has been built. It does not claim that RemainingPhysicalEquality holds for any source scale. It does not claim that the free conversion quantum premise selects the native action 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

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: