Every irreversible recorded distinction has a positive thermodynamic work floor. Landauer's principle supplies the ideal bound $\varepsilon\ge k_B T\ln 2$ per irreversible bit, experimentally verified to $\pm 10\%$. Proof available to an agent is checkable information for that agent: 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 cost, and exact lower bounds on required records gives the Physical Counting Impossibility Theorem: no fixed-budget substrate can provide universal exact proof once the retained-record lower bound exceeds the declared budget. The theorem requires exactly $B<\infty$ and $\varepsilon>0$. An answer reports a value; proof supplies checkable grounds for accepting it. A reversible device may compute an answer and erase its scratch history, but proof requires retained, inspectable records. A global answer register, oracle response, entanglement witness, finite survey catalog, or trusted device output supplies proof only through an interface that exposes the relevant grounds to the verifier. A proposed interface must 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. Substrate-free derivability has proof status only when a physical verification event makes it available to an agent.