Does playing a nonlocal game more precisely require more heat? We study devices that start and finish in pure product blanks, receive thermal answer carriers, and release classical answers. The answer depends on when the devices may communicate again. If they reset locally first, their minimum cyclic work differs from the game's minimum mutual information by a bounded alphabet cost and is at least twice its one-way distillable entanglement demand. Published entanglement bounds therefore give fixed finite games with local work $2\alpha\log_2(1/\varepsilon)+O(\log\log(1/\varepsilon))$, for a dense set of coefficients $\alpha$, including transcendental values. Quantum reunion before reset keeps the work bounded. With coherent block recovery allowing $q$ qubits and $c$ dephased bits per play, the leading cost is $(\alpha L-q)_++(\alpha L-q-c)_+$, where $L=\log_2(1/\varepsilon)$. A relative-entropy potential proves the lower bound; compression proves the upper bound. An exact entropy-production identity records arbitrary schedules without solving their optimization. For exactly matched controls, a sector allocation theorem evaluates the entanglement and correlation costs. A W-state example shows why knowledge of accessible marginals cannot replace operational access. Perfect magic-square play has cyclic local work exactly four units. Work and bath heat agree at the zero-energy endpoints used here; one unit is $k_BT\ln2$.
Does playing a nonlocal game more precisely require more heat? We study devices that start and finish in pure product blanks, receive thermal answer carriers, and release classical answers. The answer depends on when the devices may communicate again. If they reset locally first, their minimum cyclic work differs from the game's minimum mutual information by a bounded alphabet cost and is at least twice its one-way distillable entanglement demand. Published entanglement bounds therefore give fixed finite games with local work 2αlog2(1/ε)+O(loglog(1/ε)), for a dense set of coefficients α, including transcendental values. Quantum reunion before reset keeps the work bounded. With coherent block recovery allowing q qubits and c dephased bits per play, the leading cost is (αL−q)++(αL−q−c)+, where L=log2(1/ε). A relative-entropy potential proves the lower bound; compression proves the upper bound. An exact entropy-production identity records arbitrary schedules without solving their optimization. For exactly matched controls, a sector allocation theorem evaluates the entanglement and correlation costs. A W-state example shows why knowledge of accessible marginals cannot replace operational access. Perfect magic-square play has cyclic local work exactly four units. Work and bath heat agree at the zero-energy endpoints used here; one unit is kBTln2.
In a nonlocal game of quantum value one that no finite-dimensional strategy wins, near-perfect play needs growing dimension, which separable states can have. We ask how much usable entanglement it needs: the demand at loss ε is the least one-way distillable entanglement of the original state of a strategy of loss at most ε. For synchronous games with tested diagonals and for constraint-variable consistency games, demand diverges exactly when no finite-dimensional strategy is perfect; then, for any number of Bell pairs and any fidelity below one, every sufficiently good mixed strategy yields that many at that fidelity by local isometries. For rational synchronous or binary constraint games, a loss guaranteeing this is computable under a promise, but boundedness of demand is undecidable among games of value one. Demand is at least logarithmic if near-winners have a sign relation and an even dilation mirrored by both players, as in a game from Abels' group. For agreement games the directions agree up to a factor three in loss, optimal among Bob-fixed deterministic local symmetrizations. The geometry of the group behind a game, not its presentation, sets the rates. Fixed games demand αlog(1/ε) for every α=21∑ilogmiqi with qi≥2 a power of two dividing mi, a dense set containing all positive rationals and some transcendentals, such as 21log62; a configuration-lamp game demands ε−1+o(1); Pauli lamps on the points of Houghton's three rays give demand between ε−0.0690 and ε−1/4. If group relations hold exactly and Alice matches Bob to error e, demand counts qubits addressable by short words: Θ(1/e), Θ(e−k) for integers k≥1, or 2Θ(1/e). Conditional on our attainment and tail bounds, the I3322 optimum needs bounded usable entanglement.
A physical device acts in time: inputs arrive, outputs are released and can no longer be touched, some systems are kept, and some must come back. This paper shows that the optimal work rate of such a device, over many uses at vanishing error, is fixed by one quantity, the least entropy that one implementation must add to its own record, plus the energy it hands over to the carriers, against the most demanding observer its schedule allows. One law for every schedule in which each borrowed system comes back with the Hamiltonian it arrived with: Landauer's bound, the thermodynamic capacity of time-covariant channels, the real-time cost of channels, interactive processes and the batch cost of a channel whose clock must be returned are special cases. Order, not speed. The cost survives arbitrarily slow operation; an operation can cost more merely because some of its outputs must leave sooner. Which carriers to hold. For linear tasks the price of a schedule is a rank, and deciding whether holding a given number of carriers lowers it at all is NP-complete. A witness. A two-qubit process that is free when held costs every early-release device at least 0.87 kT ln 2 per episode at 1% error, when the energy and entropy budgets of its controller and battery total at most 0.025 (energies in kT ln 2, entropies in bits); the error that enters is fixed by three Pauli measurements on one probe. A classical version needs no coherence. Beyond thermodynamics. Order has a price in entanglement and coherence theory too, with no single exchange rate between theories. Holding is decidable, releasing is not. In thermodynamics, whether a channel is free is decidable when outputs may be held and undecidable when they must be released; in the other theories examined, both are decidable. All rates here are optimal asymptotic work rates at vanishing error over the complete interaction, with hardware allowed to grow; where a controller comes back only approximately, its actual mean energy and entropy are also refunded up to sublinear changes. This record archives the manuscript, its TeX source and its machine checks (Python, and Lean with Mathlib). The manuscript is the authority for exact statements and hypotheses; the verification notes list what is checked by machine and what is proved only in the text. How the work was done, including the use of AI models, is described in the manuscript. This paper is one of four on the thermodynamics of time order: Only quantum channels save work by waiting (https://github.com/Apsiape/save-work-by-waiting); The work cost of answering in real time (https://github.com/Apsiape/answering-in-real-time); The price of keeping time (https://github.com/Apsiape/price-of-keeping-time). Scoped licenses: the manuscript, PDF and TeX source are licensed under Creative Commons Attribution 4.0 International (CC-BY-4.0). The software and machine-readable verification artifacts are additionally available under the MIT License. See RIGHTS.md in the archive. The aggregate Other (Open) category represents this mixed-license package.
A quantum channel says what happens to each input, not when inputs arrive, when outputs leave for good, or what must be handed back. We show that for every finite schedule in which each carrier returns with its Hamiltonian and the controller comes back, 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. 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 kBT ln 3 per use for the Werner–Holevo qutrit at zero Hamiltonian, which is essentially free in batch. For a classical task with zero Hamiltonians that releases a past and later answers a question about it, the real-time and held prices are logarithms of two fiber counts; for reversible questions they differ by the logarithm of the number of classes of pasts that later questions tell apart. With a returned clock, the batch cost of every channel is a single-letter quantity, equal to the capacity when the environment can carry the channel's energy change; otherwise the clock's spread must grow at least linearly with the number of uses. Freeness is decidable in batch for exact algebraic data and undecidable in real time, already in a fixed dimension. A two-qubit process that is free when held costs every early-release device at least 0.87 kBT ln 2 per episode at one percent error and budgets of 0.025 kBT ln 2; for a classical version we propose a three-arm feedback-trap protocol. This record archives the manuscript, its TeX source and its machine checks (Python, and Lean with Mathlib). The manuscript is the authority for exact statements and hypotheses; the verification notes list what is checked by machine and what is proved only in the text. How the work was done, including the use of AI models, is described in the manuscript. Most of the results first appeared in four papers, which this paper brings together: The price of time order (doi:10.5281/zenodo.23310622); Only quantum channels save work by waiting (doi:10.5281/zenodo.23310624; its title concerns repeated uses of one memoryless channel, and classical tasks with memory can pay for early release); The work cost of answering in real time (doi:10.5281/zenodo.23310628); The price of keeping time (doi:10.5281/zenodo.23310632). A short letter summarizes them (doi:10.5281/zenodo.23323808). Scoped licenses: the manuscript, PDF and TeX source are licensed under Creative Commons Attribution 4.0 International (CC-BY-4.0). The software and machine-readable verification artifacts are additionally available under the MIT License. See RIGHTS.md in the archive.
How expensive is an optimum that a finite quantum system can approach but never attain? For a fixed finite rational nonlocal game of quantum value one, write A_G(n)=log2 D_G(2^(−n)), where D_G is the least local dimension at loss 2^(−n). We prescribe log2 A_G(n) at every logarithmic-space, nondecreasing unit-increment scale F with F(n)/log n → ∞, up to an additive O(log n+log²(F(n)+2)) error, and match the one-way distillable entanglement of the original mixed strategy state. At every asymptotically flat such scale, the positive finite lower and upper normalized dimension endpoints are exactly the ordered pairs with lower endpoint in Σ⁰₂ and upper endpoint in Π⁰₂; existing positive limits are exactly Δ⁰₂. A single uniform spectral-tail construction supplies the density-free profile law behind these results. It also classifies stretched-scale pairs in the unit square and gives quasi-polynomial dimension windows of width three in the exponent. A computable unit-increment target outside every rational game's profile shows that computability alone is insufficient. We locate the boundaries through a quantitative locality cap, an honest-linear compression cap, and an explicitly conditional route to finite exponents above one. A separate exact-marginal correlation theorem identifies a Farey law and its Diophantine endpoint curve. The dimension arithmetic necessity applies to all rational games; the operational realization statements concern the constructed games.
A finite rational binary-scoring game can require infinite dimension for perfect play while admitting finite energy and finite entanglement entropy. Our game has ten Alice questions, eleven Bob questions, at most six answers, and 52 uniform question pairs. It is spatially perfect and never finitely perfect, with sharp dimension profile DΓ(ε)=10log2(1/ε)+O(loglog(1/ε)). Among pure perfect strategies, the displayed strategy attains the least entanglement entropy, 5.362740278498934 ebits, and least mean occupation, 275/18 per oscillator. We classify all perfect normal type-I marginal spectra and prove that this spectrum is the unique majorization maximum. These constants are also the small-loss limits of the entropy and one-sided oscillator energy prices. In the massive free scalar field we construct a pure perfect state in two strictly separated bounded regions, equal to the vacuum outside a bounded containing region and with every polynomial energy moment finite. In contrast, every fixed faithful split state has a positive loss floor; the complementary-wedge vacuum has supremum one without attainment. Energy-analytic states, bounded-region invertible vacuum preparations and product-Kraus preparations cannot be perfect. Every perfect pure spatial strategy has infinite Schmidt rank; finite entropy is attained but not forced. In every type-II1 factor the normalized trace has a loss floor, while bounded faithful prepared densities win perfectly with sharp trace-entropy deficit Δ∗=ln(7/6)−ln(2)/7. Type-II∞ permits every finite trace entropy. A graded modular ladder gives a normal semifinite hosting criterion. Its scalar fingerprint has sharp capacity and entropy-gap bounds; the capacity floor is approached but not attained in type I, and attained in type II.
A finite rational binary-scoring game can require infinite dimension for perfect play while admitting finite energy and finite entanglement entropy. Our game has ten Alice questions, eleven Bob questions, at most six answers, and 52 uniform question pairs. It is spatially perfect and never finitely perfect, with sharp dimension profile DΓ(ε)=10log2(1/ε)+O(loglog(1/ε)). Among pure perfect strategies, the displayed strategy attains the least entanglement entropy, 5.362740278498934 ebits, and least mean occupation, 275/18 per oscillator. We classify all perfect normal type-I marginal spectra and prove that this spectrum is the unique majorization maximum. These constants are also the small-loss limits of the entropy and one-sided oscillator energy prices. In the massive free scalar field we construct a pure perfect state in two strictly separated bounded regions, equal to the vacuum outside a bounded containing region and with every polynomial energy moment finite. In contrast, every fixed faithful split state has a positive loss floor; the complementary-wedge vacuum has supremum one without attainment. Energy-analytic states, bounded-region invertible vacuum preparations and product-Kraus preparations cannot be perfect. Every perfect pure spatial strategy has infinite Schmidt rank; finite entropy is attained but not forced. In every type-II1 factor the normalized trace has a loss floor, while bounded faithful prepared densities win perfectly with sharp trace-entropy deficit Δ∗=ln(7/6)−ln(2)/7. Type-II∞ permits every finite trace entropy. A graded modular ladder gives a normal semifinite hosting criterion. Its scalar fingerprint has sharp capacity and entropy-gap bounds; the capacity floor is approached but not attained in type I, and attained in type II.