Information H Thermodynamics Verified

The thermodynamic bound

Thermodynamics sets a price on forgetting. In 1961, Rolf Landauer argued that erasing one bit of information must dissipate at least kT ln 2 of energy, a bound now confirmed experimentally. The declaration H_ThermodynamicsVerified in the Recognition Science framework states a related but more specific inequality. For every state of the framework's ledger, a discrete record of recognition events, and every active bond in that state, the cost of recognition is at least half the square of the logarithm of the bond's multiplier. In symbols: J(m) ≥ (ln m)² / 2.

The inequality is a formal statement in the framework's machine-checked library of formal theorems. It applies to a quantity called recognition cost, the forced price the framework assigns to each act of distinguishing one thing from another. The bound relates that cost to the logarithm of a bond multiplier, a scale factor attached to a connection between two recognized items. When the multiplier is near one, the logarithm is near zero, and the bound is easy to satisfy. As the multiplier grows, the required cost grows quadratically in the logarithm, a sharper demand than a linear bound would make.

The declaration does not prove the full Landauer limit. It states a specific inequality for a particular model of recognition, not a general theorem about all physical erasure. The framework's own documentation marks the thermodynamic bound as a hypothesis with a scaffold status, noting that a complete Taylor expansion proof remains unfinished. The declaration is a definition of a property, not a proof that the property holds for all cases.

Within the framework, the bound connects information theory to thermodynamics in a precise way. It gives a concrete lower bound on the cost of maintaining a connection in the ledger, a cost that grows with the logarithm of the multiplier. This makes the declaration a useful target: if the bound fails for some state, the framework's thermodynamic story needs revision. If it holds, it provides a quantitative link between recognition and dissipation that can be checked against physical intuition.

/-- **HYPOTHESIS**: Thermodynamic Bound.
    Recognition cost satisfies the Landauer bound for information erasure.

    STATUS: SCAFFOLD — Bound derived in `Information.Thermodynamics`.
    TODO: Complete the Taylor expansion proof in `Thermodynamics.lean`. -/
def H_ThermodynamicsVerified : Prop :=
  ∀ (s : Thermodynamics.LedgerState), ∀ b ∈ s.active_bonds,
    let m := s.bond_multipliers b
    let u := Real.log m
    Cost.Jcost m ≥ u^2 / 2

-- Legacy axiom eliminated. See Foundation.ConstantDerivations.

What this page does not claim

The full Landauer limit for all physical erasure is not proved by this declaration. The declaration does not establish that the inequality holds for every possible ledger state; it defines the property to be checked. No claim is made that this bound is tight or that it matches any measured thermodynamic quantity.

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/Information.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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