Thermodynamic Limits of Proof

Every irreversible recorded distinction has a positive thermodynamic work floor. This positivity is the physical input that makes a finite-budget proof threshold exist; the arithmetic computes where the threshold falls. Landauer's principle supplies the universal ideal floor ε ≥ k_BTln 2 per irreversible bit, experimentally verified to ±10%. Proof available to an agent is checkable information for that agent, so some substrate must produce, retain, and expose evidence that excludes answer-changing alternatives. A finite detector array operating at temperature T for finite time has finite signal-acquisition capacity. Combining finite causal access, positive retained-record work cost, and exact lower bounds on required records gives the Physical Counting Impossibility Theorem: no fixed-budget physical substrate can provide universal exact proof once the retained-record lower bound exceeds the declared budget. The theorem requires exactly B < ∞ and ε > 0.

An answer reports a value; proof supplies checkable grounds for accepting it. The two carry different retained-record obligations: a reversible device may compute an answer and erase its scratch history, while proof requires retained, inspectable records that exclude answer-changing alternatives. A global answer register, oracle response, entanglement witness, finite survey catalog, or trusted device output supplies proof only under an interface that makes the relevant grounds available to the verifier. A proposed interface must therefore identify the retained-record lower-bound family R(n) it induces. Sound operational claims about efficient solvability inherit the same finite-budget obstruction when their acceptance would license universal exact proof. The result constrains proof-availability for agents. Substrate-free derivability has proof status only when a physical verification event makes it available to an agent. Parsing, checking, and retaining are physical operations with positive work cost per irreversible record.