Holography Landauer Identity No Silent Physical Erasure

The no-silent-erasure bound

Landauer's principle, proposed by Rolf Landauer in 1961, states that erasing one bit of information in a computer must dissipate at least kT ln 2 of heat, where k is Boltzmann's constant and T is the temperature. The principle links information theory to thermodynamics: information is physical, and destroying it has an energy cost. The Recognition Science framework derives a version of this bound from its own axioms, and its ledger, a discrete record of posted events, plays the role of the information-bearing state.

The framework's machine-checked library of formal theorems proves an exact identity on its forced cell: the heat dissipated along a path equals a per-bit quantum times the net number of posted bits erased. From this identity, one theorem follows directly: if the per-bit quantum is positive and a single step carries zero physical heat, then the posted potential at the end of the step is unchanged. In plain language, no erasure can happen silently. A step that costs no heat cannot delete any posted record. This is the declaration no_silent_physical_erasure.

The declaration is a theorem, not a physical law by itself. It holds conditional on a named model premise: that measured boundary heat reads the posted-record flux times a conversion quantum q. The framework proves the accounting identity unconditionally, but calling the generic heat observable physical requires that premise. The inverse temperature is not selected by the framework here; it is supplied by a separate calibration. Joules, kelvin, and any SI value of q remain outside the certificate.

What the declaration does not claim is equally precise. It does not identify posted weight loss with logical entropy erasure on its own; that identification requires a separate named premise, LogicalErasurePostsDebit. It does not prove that erasing a bit costs at least kT ln 2 in SI units; that classical bound follows only after the thermal calibration is applied. And it does not rule out a hypothetical erasure that costs less than the posted debit: if such a process were measured, it would refute the heat-carrier identification, not the accounting identity. The framework names this falsifier explicitly.

Within the framework, the no-silent-erasure theorem is the sharp edge of Landauer's principle: it turns a thermodynamic bound into an exact statement about the ledger. The cost of erasure is not a statistical average but a forced consequence of how records are posted. A reader can now see that the framework's claim is not that erasure is expensive, but that it is never free.

/-- The RS Landauer identity on the forced cell. Given the explicit carrier
identification, dissipated heat equals the per-bit quantum times the net
posted-record debit, exactly. -/
theorem physical_landauer_identity
    (q : ℝ) (heat : PhysicalStepHeat)
    (hcarrier : HeatIsPostedRecordFlux q heat)
    (c : CellCfg) (p : List CellCfg) :
    dissipatedHeat heat c p = q * (netErasedBits c p : ℝ) := by
  unfold dissipatedHeat netErasedBits
  rw [physicalPathHeat_eq_scaled_recordFlux q heat hcarrier (c :: p)]
  rw [books_balance]
  push_cast
  ring

/-- No silent physical erasure. If `q` is positive and a step carries zero
physical heat, its posted potential is unchanged. -/
theorem no_silent_physical_erasure
    (q : ℝ) (heat : PhysicalStepHeat)
    (hcarrier : HeatIsPostedRecordFlux q heat)
    (hq : 0 < q) (c c' : CellCfg)
    (hzero : heat c c' = 0) :
    cellPotential c' = cellPotential c := by
  have hscaled : q * (stepHeatCell c c' : ℝ) = 0 := by
    rw [← hcarrier c c', hzero]
  have hq0 : q ≠ 0 := ne_of_gt hq
  have hfluxReal : (stepHeatCell c c' : ℝ) = 0 :=
    (mul_eq_zero.mp hscaled).resolve_left hq0
  have hflux : stepHeatCell c c' = 0 := by
    exact_mod_cast hfluxReal
  exact no_free_erasure c c' hflux

/-- Named physical bridge from endpoint record accounting to classical
logical erasure: resetting `erasedBits` logical bits posts the same net debit.
The ledger identity does not prove this identification by itself. -/
def LogicalErasurePostsDebit (erasedBits : ℕ)
    (c : CellCfg) (p : List CellCfg) : Prop :=
  netErasedBits c p = (erasedBits : ℤ)

/-- Any physical erasure measured strictly cheaper than its posted debit
refutes the heat-carrier identification used by the physical reading. -/
theorem cheaper_erasure_falsifies_posting_rule
    (q : ℝ) (heat : PhysicalStepHeat) (c : CellCfg)
    (p : List CellCfg)
    (hcheap : CheaperThanPostedDebit q heat c p) :
    ¬ HeatIsPostedRecordFlux q heat := by
  intro hcarrier
  have hlt := hcheap.2.2
  rw [physical_landauer_identity q heat hcarrier c p] at hlt
  exact (lt_irrefl _ hlt)

What this page does not claim

The declaration does not prove that erasing a bit always costs at least kT ln 2 in SI units. The declaration does not by itself identify posted-record debit with logical entropy erasure. The declaration does not rule out erasure cheaper than the posted debit; it only names what would falsify the heat-carrier premise.

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/LandauerIdentity.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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