Thermodynamics of records gives a trade-off and no floor
Result, 2026-09-27 (a recorded failure of one route). Theorem I of the unit-and-indeterminacy note says that back-action indeterminacy is impossible for classical models with Liouville dynamics, product preparations obeying a prior density ceiling, and Bayesian conditioning on records. A natural attempt to escape it adds a heat bath and requires records to be physical: stored, and eventually erased, at temperature \(T>0\). This note shows that the attempt yields an exponential trade-off between dissipated energy and posterior area, and no positive floor:
\[\eta\ \ge\ A_0\,e^{-W/(k_BT)},\]
where \(A_0\) is the prior phase-space area of the body, \(\eta\) the area of its posterior support after the records, and \(W\) the total work spent on measurement and erasure (Theorem 1). For every \(\eta>0\) the construction of Theorem I reaches \(\eta\) with finitely many finite-precision readings, hence finite \(W\). Thermodynamics therefore prices resolution logarithmically, \(W\ge k_BT\ln(A_0/\eta)\), and cannot single out an action. The would-be unit \(k_BT\,\tau\) built from the bath and a record time rescales under the similarity of Theorem B of the dimensional note. So the thermodynamic route supplies neither ingredient (U) nor (I), and the question of STATE item 2 stays with the three denials listed in the unit-and-indeterminacy note.
The ingredients are standard (Landauer, Bennett, the Sagawa–Ueda bound for measurement plus erasure); the note records their consequence for this programme, with no novelty claimed.
1. The bound
Setting: the classical model of Theorem I, with body and pointers coupled to a heat bath at temperature \(T\). Records are pointer states that are read, used, and eventually reset to a standard state. The generalized second law for measurement and erasure (classical form of the bound of Sagawa and Ueda 2009, metadata; the erasure part is Landauer 1961 and Bennett 1982, metadata) states that the total work of a measurement that gains mutual information \(I\) (in nats) about the system, followed by erasure of the record, satisfies \(W_{\rm meas}+W_{\rm eras}\ge k_BT\,I\).
Theorem 1. Let the body’s prior be uniform on a phase-space region of area \(A_0\), and let a protocol of records leave, for almost every outcome, a body posterior supported in a region of area at most \(\eta\). Then the total work of measurement and erasure satisfies
\[W\ \ge\ k_BT\,\ln\frac{A_0}{\eta},\qquad\text{equivalently}\qquad \eta\ \ge\ A_0\,e^{-W/(k_BT)}.\]
Proof. The differential entropy of the uniform prior is \(\ln A_0\) (area measured in any fixed unit), and a density supported in area at most \(\eta\) has differential entropy at most \(\ln\eta\). The mutual information between body and records is the prior entropy minus the average posterior entropy, so \(I\ge\ln(A_0/\eta)\); the unit of area cancels. Apply the measurement-plus-erasure bound. \(\square\)
The construction of Theorem I is thermodynamically cheap. Its two pointers are read to precisions \(\epsilon'\) and \(\epsilon_2'\); each reading stores finitely many bits for a bounded pointer range, and erasing them costs a finite multiple of \(k_BT\ln2\). So every posterior area \(\eta>0\) is reached at finite work, and the bound of Theorem 1 is the only thermodynamic constraint.
2. What this says about the necessity question
- No floor. The minimal posterior area at work budget \(W\) is \(A_0e^{-W/(k_BT)}\), which tends to zero as the budget grows. A record of Newton’s sagitta at resolution \(s\) with impulse resolution \(\delta p\) costs at least \(k_BT\ln(A_0/(s\,\delta p))\): resolving a phase-space area a thousand times below \(\hbar\) costs only \(k_BT\ln1000\) more than resolving \(\hbar\) itself. Thermodynamics cannot distinguish \(\hbar\).
- No unit. The constants of the route are \(k_BT\) and whatever times the protocol uses. An action \(k_BT\,\tau\) rescales with \(\tau\), and the admitted class is closed under slowing the protocol, so Theorem B of the dimensional note gives floor zero. The universal unit of Theorem U enters through the equilibrium spectrum of radiation, which the record bound never uses.
- The exponential again. The trade-off is a Boltzmann factor, as in the exponential ladder of the dimension-ladder note, §4. Here the exponent contains no action constant, so the factor suppresses without setting a scale; in Wien’s tail \(e^{-h\nu/(k_BT)}\) the same factor sets one, because \(h\) appears in the exponent.
3. Consequence for STATE
The thermodynamic route to (I) is closed at theorem level: records with Landauer costs satisfy Theorem I with an exponential work–area trade-off. STATE item 2 keeps its three candidate denials (of Liouville dynamics, of product preparations, of Bayesian conditioning); the background and inflexion routes are under evaluation in the 2026-09-27 Astra run.