The thermodynamics of deadlines: one law for the work cost of acting in time

Douglas, Seth; Mghirbi, Nidhal

doi:10.5281/zenodo.23719180

Results · The thermodynamics of acting in time

The thermodynamics of deadlines: one law for the work cost of acting in time

Seth Douglas and Nidhal Mghirbi · version 1.0.0 · Zenodo, 28 October 2026 · 187 pages · doi:10.5281/zenodo.23719180 · PDF

In plain wordsA device that must release outputs before later inputs arrive pays a work cost however slowly it runs, and one law gives that cost for every schedule; Landauer’s erasure bound is one case of it.

The resultat three depths

In one sentenceA device that must let an output go for good before a later input arrives pays, in work, for every correlation its record keeps with what has left, however slowly it runs; one law gives that price for every schedule of arrivals, releases and returns.

As the paper states it, in briefFor every finite schedule in which each carrier returns with its Hamiltonian and the controller comes back, the optimal work per use is the least entropy one implementation must add to its own record, plus the energy it hands over, against the most demanding observer. The cost survives arbitrarily slow operation.

For a scientist
One paragraph

A quantum channel says what happens to each input, not when inputs arrive, when outputs leave for good, or what must be handed back. For every finite schedule in which each carrier returns with its Hamiltonian and the controller comes back, the paper shows that the optimal work per use at vanishing error is the least entropy one implementation must add to its own record, plus the energy it hands over, against the most demanding observer. Landauer’s bound, the thermodynamic capacity of Faist, Berta and Brandão in the time-covariant case, and the real-time costs of channels and processes are this law on different schedules, and the cost survives arbitrarily slow operation: order, not speed, sets it. Releasing each output of a repeated memoryless channel before the next input arrives costs nothing extra for classical channels, a positive amount for partial reset at every temperature, and $k_{B} T \ln 3$ per use for the Werner–Holevo qutrit at zero Hamiltonian, which is essentially free in batch. With a returned clock the batch cost of every channel is a single-letter quantity; freeness is decidable in batch for exact algebraic data and undecidable in real time. A two-qubit process that is free when held costs every early-release device at least $0.87 k_{B} T \ln 2$ per episode at one percent error; a three-arm feedback-trap protocol is proposed for a classical version, and no experiment has been run.

For an expert
The law, as stated in the paper

Definition 3.1 (Schedule)

A schedule $S$ is a finite, ordered list of arrival and release events. A visit $v$ brings a carrier $P_{v}$ with Hamiltonian $H_{v}$ into the device at its arrival and returns the same carrier, with the same Hamiltonian, at its release. Visits of the same carrier cannot overlap; visits of different carriers may overlap, and releases may come in any order. The device can act on a carrier only during its visit. A later loan of a carrier that the observer owns is a new visit, and its state may be correlated with everything the observer holds. The observer supplies the carriers and may choose them adaptively, using only outputs already released, its own memory and its quantum references; every receipt or message the observer sees is written on a carrier the observer has supplied, and its delivery is that carrier’s release.

Definition 3.2 (The schedule functional)

A private extension of a target strategy $T$ on $S$ consists of an initial device state $τ$ , an inert purification $F$ of it that is never acted on, and unitaries on the device and the carriers present, following $S$ , whose marginal strategy is within $ε$ of $T$ in the strategy distance, the largest trace distance over adaptive testers. These unitaries need not conserve energy; the extension is a mathematical object. For a purified tester $t$ , an observer allowed by the schedule that keeps its initial purifying systems and everything it discards coherently, let $Z$ be everything it owns at the end, including these systems, its references and every released carrier, with no duplicate copy of a carrier loaned again; the tester starts independent of the device and of $F$ . Let $E_{t} = \sum_{v} [Tr H_{v} ρ_{v, release} -$ $Tr H_{v} ρ_{v, arrival}]$ be the energy handed over. Set

k_{S, ε} (T)

= \inf_{private extensions} \sup_{purified t}

[\frac{\ln 2}{β} S (Z | F)_{t} + E_{t}]

and let $K_{β}^{S} (T) = β {\bar{k}}^{S} (T) / \ln 2$ , where ${\bar{k}}^{S} (T) = \inf_{ε_{n} \to 0} {lim sup}_{n} k_{S_{n}, ε_{n}} (T_{n}) / n$ is the regularized rate along the block schedules $S_{n}$ . As a check of the normalization, resetting a maximally mixed bit with zero Hamiltonian gives $k = k_{B} T \ln 2$ .

Theorem 3.3 (Work law for schedules)

Fix a temperature $0 < T < \infty$ , and let $S_{n}$ be the schedule of a block of $n$ uses, with the order of events across uses specified, in which each visit returns the same carrier with an unchanged Hamiltonian. The optimal rate of deterministic work for a device that returns its controller approximately is

W^{S} = \frac{\ln 2}{β} K_{β}^{S} .

The lower bound holds for every implementation; the rate is attained using the finite-source and phase modules of Appendix F. Appendix A states the full contract, including the battery’s absolute energy deviation. Resources that start correlated with the controller and must be returned in a prescribed joint state are not covered.

The contract of Theorem 3.3 is collected in the paper’s Table 2 (target, access, ports, free dynamics, controller, battery, limit); the results that depart from it are listed in its Table 3. The lower bound is a one-page accounting identity, given in full in Section 3.1.

The question

“Both results describe what a device does to its inputs. Neither says when the device may touch what, and that turns out to be half of the price.”

The paper, Section 1, on Landauer’s bound (1961) and the thermodynamic capacity of Faist, Berta and Brandão (2019, 2021)

Questionson the index

Answered Which unital channels are repeatable by one device with a finite memory and no resets? posed by others
Answered What does it cost, in work, to act on a schedule?
Open Does the predicted price of early release appear in a laboratory? left open

How this was checked

Blind audits: Nine whole-paper audits by a model from another developer, eight of them blind
Machine checks: Machine checks of the numbers and finite constructions.
Lean: Complete Lean 4 proofs of the lemmas named in the paper’s Table 7, standard axioms only; the main law is proved in the text.
Peer review: Not yet.

Each check, what it covers, and what it leaves to the written proof.

Each check, what it covers, and what it leaves to the written proof.
Check	Covers	Does not cover
Whole-paper audits	Several adversarial reviews by fresh instances of a model from the same developer as models that helped write it, and nine by a model from a different developer, eight of them blind audits that saw neither the earlier reviews nor the reports of their fixes; a separate review of the experimental design. The fixes are applied in version 1.0.0.	Errors shared by every reader. An audit is evidence, not a guarantee.
Machine checks	One command runs every script and prints a scorecard; deliberate corruptions must each be caught with a named diagnostic. The scripts check the numbers, worked examples and finite constructions listed in Table 7, in exact arithmetic where possible.	The general theorems. A script checks the instances it runs, not the general statement.
Lean 4	Complete proofs, with Mathlib and standard axioms only, of the lemmas named in Table 7: among them the coding form of the release rank, the energy-conserving reset, the matrix identities of the two-qubit certificate and the constant chains of the converses.	The main law, which is proved in the text.
Imported	Two results of the authors’ earlier preprints are used as stated: Lemma G.6, and the zero-Hamiltonian closure theorem used in the undecidability proof.	Their proofs, which are in those preprints.
Experiment	A three-arm feedback-trap protocol is proposed for the classical witness.	Any measurement: no experiment has been run.
Peer review	Not yet.	A preprint on Zenodo.

Every audit here is internal. Section 14 of the paper and verification/README.md list what is checked by machine, what is proved in Lean, and what is proved only in the text. How we check results.

Label checked 29 October 2026.

Novelty and credit

Novelty: Unifies named prior results as cases of one law: Landauer’s bound, the thermodynamic capacity of Faist, Berta and Brandão in the time-covariant case, and the real-time costs of channels and processes, credited.
Experiment: Mathematics proved under the stated contract. A three-arm feedback-trap protocol is proposed; no experiment has been run.

Landauer’s bound (1961) and the thermodynamic capacity of Faist, Berta and Brandão (2019, 2021) are cases of the law; their batch construction’s last step, erasing environments with the outputs as side information, is the step a real-time device cannot take.
The whole-block identity is a multipartite form of the finite-reservoir entropy-production identity of Esposito, Lindenberg and Van den Broeck and of Reeb and Wolf.
Rybár and Ziman introduced repeatable channels; the paper answers the question their paper closes with. Lie, Son, Boes, Ng and Wilming characterize thermal operations; the free-set theorem is a robust, work-cost form of their criterion.
The proposed protocol builds on the feedback-trap test of Landauer’s principle by Jun and coauthors; that access to a copy changes the cost of erasure is established by Bennett, by del Rio and coauthors and by Orlov and coauthors.

Recordthis paper and its companions

12 Oct 2026The four companion papers and the letter, versions 1.0.2 and 1.1.0; version 1.0.2 of each corrected one DOI in a citation (corrections).doi:10.5281/zenodo.23310622, 23310624, 23310628, 23310632; 23323808
28 Oct 2026Version 1.0.0 of this paper, which brings the four together.doi:10.5281/zenodo.23719180
30 Oct 2026Correction in a companion. The price of keeping time version 1.1.1 corrects a sentence of its batch-law appendix; no result depended on it.doi:10.5281/zenodo.23790746

Release dates are Zenodo publication dates. The paper cites version 1.0.2 of the first three companions and version 1.1.0 of The price of keeping time; Appendix K gives, for every result taken from them, its number there.

Versionsone, each with its own DOI

Every archived version, newest first. Dates are Zenodo publication dates. All versions: 10.5281/zenodo.23719179, which always resolves to the newest.
Version	Date	DOI	Notes
1.0.0	28 Oct 2026	10.5281/zenodo.23719180	PDF hosted here.

Paper, code and links

PDF

papers/deadlines-v1.0.0.pdf, version 1.0.0, 2,034,171 bytes, 187 pages.

SHA-256 18d14c616523c51a7426f9ae3fe95afa6e8d5b2e6dbe0b4404f3d80e19ff7a8f

Of paper/main.pdf, 2,034,171 bytes, in the Zenodo archive; identical in the repository at that commit. The same file is hosted here: papers/deadlines-v1.0.0.pdf.

DOI

10.5281/zenodo.23719180; all versions 10.5281/zenodo.23719179

Code

github.com/Apsiape/work-price-of-letting-go

Note

Brings together the four papers below

Commit

986689586d52b7d85c438cdf517734c56a49d80b

The commit archived by Zenodo (Apsiape/work-price-of-letting-go, tag v1.0.0).

Verify

From a clone of the repository, with Python 3.10 or later, as the README gives them:

python -m pip install -r verification/requirements.txt
cd verification
python verify.py              # every Python check; writes SCORECARD.md
python verify.py --negative   # deliberate corruptions, each must be caught
python verify.py --lean       # also compiles the Lean proofs

The Lean proofs need Lean 4.30.0 with Mathlib v4.30.0. verification/README.md explains what a passing check establishes and what it does not.

Lean, in detail

Complete Lean 4 proofs of the lemmas named in Table 7, each audited with #print axioms to use only the standard axioms; the main law is proved in the text.

Licence

Manuscript and repository content CC BY 4.0; software and machine-readable verification artifacts also under the MIT License, at your option.

Page checked against the paper, the release repository and the Zenodo record on 30 October 2026.

Corrections

None so far.

All corrections, and what counts as one.

Cite

BibTeX built from the repository’s CITATION.cff.

@misc{douglas2026deadlines,
  author  = {Douglas, Seth and Mghirbi, Nidhal},
  title   = {The thermodynamics of deadlines: one law for the work cost of acting in time},
  year    = {2026},
  version = {1.0.0},
  doi     = {10.5281/zenodo.23719180},
  url     = {https://github.com/Apsiape/work-price-of-letting-go}
}

deadlines.bib All papers as BibTeX