Holography Landauer Calorimeter Forcing Double Posted Heat Ne Unit Tautological

The double-heat decoy

In the Recognition Science framework, a ledger is a discrete record of events, and a step in that ledger can post a quantity of heat. The declaration doublePostedHeat_ne_unit_tautological proves a negative result: the heat function that simply doubles the standard per-step heat is not equal to the unit heat function that the framework's basic posting rule defines. The proof is a theorem in the machine-checked library, meaning it is verified by a computer from the framework's axioms, with no gaps.

The point of the theorem is to block a tempting shortcut. One might try to build a calorimeter, a device that measures heat, by assuming heat is additive across six independent face channels, each posting a unit quantum on a flip. The theorem shows this assumption is definitional smuggling: it forces the heat function to be exactly the tautological one, not a new physical law. If you drop the unit quantum condition, the framework's weaker axioms admit a scaled decoy, the double heat function, which is nonzero, additive, and yet not equal to the unit heat. The theorem doublePostedHeat_ne_unit_tautological is the kernel of that decoy's rejection.

What the theorem does not claim is an independent calorimeter. The framework explicitly names a residual gap, MissingIndependentCellCalorimeter, and marks it as OPEN. A stub protocol for logical bit reset is sketched, but no logical discharge is claimed. The theorem is a wall, not a bridge: it shows what does not work, and it names what is still missing, without pretending the gap is closed.

theorem doublePostedHeat_ne_unit_tautological :
    ¬ (∀ c c' : CellCfg,
        doublePostedHeat c c' = tautologicalPostedHeat 1 c c') := by
  intro h
  have hstep := h cell0 postingStepTarget
  have hnz : (stepHeatCell cell0 postingStepTarget : ℝ) ≠ 0 := by
    exact_mod_cast postingStepTarget_posts_nonzero
  simp [doublePostedHeat, tautologicalPostedHeat] at hstep
  have h2 : (2 : ℝ) * (stepHeatCell cell0 postingStepTarget : ℝ) =
      (stepHeatCell cell0 postingStepTarget : ℝ) := hstep
  have hmul :
      (2 : ℝ) * (stepHeatCell cell0 postingStepTarget : ℝ) -
        (stepHeatCell cell0 postingStepTarget : ℝ) = 0 := by
    linarith
  have : (stepHeatCell cell0 postingStepTarget : ℝ) = 0 := by
    linarith
  exact hnz this

theorem smuggling_package_eq_tautological
    (q : ℝ) (heat : PhysicalStepHeat)
    (h : SmugglingCalorimeterPackage q heat) :
    ∀ c c' : CellCfg, heat c c' = tautologicalPostedHeat q c c' := by
  intro c c'
  simpa [tautologicalPostedHeat] using
    smuggling_package_forces_carrier q heat h c c'

/-- Named gap type for an independent cell calorimeter. A discharge would
supply a `PhysicalStepHeat` equal to scaled `stepHeatCell` for a reason that
does not package the six-channel posting rule into the hypothesis list.
Inhabiting this structure does not close the residual. -/
structure MissingIndependentCellCalorimeter where
  named_gap : Unit := ⟨⟩

What this page does not claim

The theorem does not prove that an independent calorimeter exists. The theorem does not define a new physical law of heat posting. The theorem does not claim that the double heat function is physically realizable.

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/Holography/LandauerCalorimeterForcing.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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