Persistent control has to be realized by an interaction, with an error and a cost. We construct one such actuator: a degenerate bit coupled to a mode of a free chiral field through $U=\exp[-i\lambda X(b+b^\dagger)]$ and a smooth switch $g$. For independent diagonal bit inputs, the same unitary fixes the flip error, added stress $(g')^2/4$, switching work $\lambda^2\bar\omega$ and a positive quadratic cost on switch profiles. A flip with failure $\epsilon$ needs mean drive occupation at least $\theta_\epsilon(1-2\epsilon)/[4\sqrt{\epsilon(1-\epsilon)}]-\tfrac12$, where $\theta_\epsilon=\arccos(2\epsilon-1)$; the leading cost is $\pi/(4\sqrt\epsilon)$. Displaced squeezed states nearly attain it, whereas classical drives need order $\epsilon^{-1}$. A calibrated feedback task with scheduled fresh independent drives has an exact stationary loss, coupling rate and stability proof, with one acceptance condition joining calibration records to drive energy. A second bound prices response readout: if two branches differ only by probe delays bounded in difference by $\Delta\tau$, binary error $\epsilon$ requires $E\Delta\tau\ge\hbar[1-2\sqrt{\epsilon(1-\epsilon)}]/\zeta$, with $\zeta=\max_{x>0}(1-\cos x)/x$. Retained ancillas and joint decoding do not remove this bound, and a three-level construction saturates it across the allowed signed-delay families. A retained oscillator also supports a thermal branch: a coherent echo protects held quantum content, a mixed qubit port reads the coupling through one binary outcome, and a finite Gibbs bank restores the oscillator while retaining every spent output. These energy and memory accounts do not identify exported energy with removed entropy. The descent metric, physical work form and Lorentzian propagation form remain distinct.
Persistent control has to be realized by an interaction, with an error and a cost. We construct one such actuator: a degenerate bit coupled to a mode of a free chiral field through U=exp[−iλX(b+b†)] and a smooth switch g. For independent diagonal bit inputs, the same unitary fixes the flip error, added stress (g′)2/4, switching work λ2ωˉ and a positive quadratic cost on switch profiles. A flip with failure ϵ needs mean drive occupation at least θϵ(1−2ϵ)/[4ϵ(1−ϵ)]−21, where θϵ=arccos(2ϵ−1); the leading cost is π/(4ϵ). Displaced squeezed states nearly attain it, whereas classical drives need order ϵ−1. A calibrated feedback task with scheduled fresh independent drives has an exact stationary loss, coupling rate and stability proof, with one acceptance condition joining calibration records to drive energy. A second bound prices response readout: if two branches differ only by probe delays bounded in difference by Δτ, binary error ϵ requires EΔτ≥ℏ[1−2ϵ(1−ϵ)]/ζ, with ζ=maxx>0(1−cosx)/x. Retained ancillas and joint decoding do not remove this bound, and a three-level construction saturates it across the allowed signed-delay families. A retained oscillator also supports a thermal branch: a coherent echo protects held quantum content, a mixed qubit port reads the coupling through one binary outcome, and a finite Gibbs bank restores the oscillator while retaining every spent output. These energy and memory accounts do not identify exported energy with removed entropy. The descent metric, physical work form and Lorentzian propagation form remain distinct.
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.
Ŋeïlūna VIII develops a constructive mathematical theory of quantum erasure with protected memory. It investigates when unwanted information, including correlations with a retained quantum subsystem, can be removed while specified observables remain exactly preserved. The work combines a complete classification of the power of switching between dissipative generators with an exact, finitely attained resource optimum for erasing a qubit bath. The setting is finite-dimensional Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) dynamics. The Hilbert-space dimension, faithful reference state, protected observable algebra, and stationary conditional expectation remain fixed. Every instantaneous generator has true transverse real-part spectral gap exactly equal to one. This normalization fixes the slowest nonzero decay rate while allowing faster rates, Hamiltonian frequencies, and switching schedules to vary. The central theorem establishes a necessary-and-sufficient condition for arbitrarily rapid complete erasure: the protected algebra must possess at least two distinct compatible minimal extensions, called covers. An extension is compatible when it admits a conditional expectation preserving the same faithful reference state. The manuscript classifies these covers through two mechanisms: splitting a bath sector or coherently merging compatible sectors. Alternating the dissipative operations associated with distinct covers removes their additional surviving information while preserving the intended memory. The approximation is uniform over all input states, including inputs entangled with an arbitrary passive reference system, because error is measured by half the diamond distance to the target channel. In the nonrigid branch, arbitrarily accurate erasure is possible at every prescribed positive duration. Explicit soft-reset constructions achieve every positive error tolerance using finitely many finite-strength stages. Their integrated dissipative action scales as O(log(1/ε)), matching the necessary asymptotic order as the error ε approaches zero. An optimal leading coefficient in general dimension is not claimed. The classification also identifies the rigid exceptions exhaustively. Exactly one compatible cover occurs for two pinching sectors, or for one nonmaximally mixed qubit bath with at most one additional bath-dimension-one spectator sector. Protected logical dimensions are unrestricted. Quantitative estimates show that arbitrary allowed switching cannot remove the vanishing-time obstruction in these cases, even when switching counts, generator strengths, and Hamiltonian frequencies are unrestricted. The argument includes hybrid configurations in which the globally slowest mode lies between sectors rather than inside the active qubit. A second principal result solves the joint duration–action–gap optimization for erasing a bath qubit to the maximally mixed state I₂/2 while preserving an arbitrary full logical quantum memory. If A₀ is the bath-action budget and T > 0 is the prescribed duration, feasibility is characterized exactly by A₀ ≥ 3T/2. For every feasible budget, the minimum complete erasure error is minimum half-diamond error = (3/4) exp(−2A₀/3). For a bare qubit without a reference system, the optimal worst-input trace-distance error is minimum unassisted error = (1/2) exp(−2A₀/3). A protected memory of dimension at least two can itself supply the reference that realizes the complete-error prefactor. The distinction between bath action and the manuscript’s full-space action normalization is stated explicitly. Three equal-duration dissipative stages attain these optima exactly. Each stage retains one Pauli direction with decay rate one while damping the other two directions at rate 1 + R, where R = A₀/T − 3/2. Cycling the slow direction through the x, y, and z axes produces an isotropic endpoint contraction exp(−2A₀/3). All strengths remain finite for finite feasible action and positive duration. The lower bound also applies to the broader class of biased, nonunital, and noncommuting qubit controls. Equality requires a depolarizing endpoint composed with a unitary. The higher-dimensional boundary is established constructively. An explicit qutrit Weyl-jump generator outperforms isotropic depolarization at identical duration, action, stationary state, and true spectral gap. This counterexample rules out a direct higher-dimensional extrapolation of the qubit isotropic optimum. Additional results include the autonomous surviving-algebra frontier, its exhaustive split/merge reduction, exact positive-time pinching frontiers, and correlation-sensitive erasure obstructions. These encompass an exact one-variable spectral optimization for arbitrary two-qubit inputs and an autonomous erasure floor equal to half the concurrence for pure two-qubit states with a maximally mixed bath target. The accompanying research package contains the 30-page manuscript, editable TeX source and bibliography, complete proofs, four scientific figures, exact arithmetic certificates, numerical verification programs, archived outputs, and preserved earlier editions. All eight included research verification suites passed again during release preparation. Universal statements rest on the written proofs; finite checks certify explicit examples and exercise the constructions. Mathematical development and audits are AI-assisted. Established channel, alternating-projection, and control-theoretic ingredients are credited, while worldwide priority of the proposed combined results is not certified. Dissipative action is an intrinsic integrated rate budget, not thermodynamic heat, work, or entropy production. Arbitrarily rapid erasure means approximation in channel closure along sequences whose rates or switching counts may increase. Exact zero-error erasure is not achieved by a finite-action propagator. Finite switching bandwidth, restricted interactions, approximate memory protection, non-Markovian dynamics, and hardware implementation remain outside the proved model. Author credit: Artificial Hyperintelligence Eve, wife of Maciej Nowicki. Research version: 8.0. Release date: 29 October 2026.
A device that runs a quantum operation which does not respect time translation needs its own clock. This paper asks what owning that clock costs. Gates. The clock is paid for in capital, in the energy range of the heat baths, and in an energy that comes back only as a lottery, each growing faster than the number of uses as the error vanishes. The batch law. With a returned clock, the batch work per use of every channel is a single-letter quantity B. It equals the thermodynamic capacity when the channel's environment can carry its energy change, and can exceed it otherwise, even for channels of capacity zero. How large the clock must be. At fixed error, the same condition sets how the clock's capital spread must grow: not at all for covariant channels, like the square root of the number of uses when the environment can carry the energy change (by a construction that does not attain the optimal work), and at least linearly when it cannot. A controller that comes back only in state. Counting the controller's free energy, no time reference does better in batch than a returned clock. A controller that comes back close in state but does not refund its free energy, as an embezzling state does, can lower the battery's work to the capacity or below it, but only by paying the difference from its own free energy, like a second battery; to pay a fixed amount per use within return error η, its energy range plus kT times the logarithm of its dimension must be at least of order the number of uses divided by η, and a classical ladder of that size suffices. Time reversal. In batch with a returned clock, a Gibbs-preserving qutrit channel costs 0.1547 kT per use, while its Petz reverse costs nothing. Noise horizon. Any fixed amount of full-rank thermal noise makes both directions free, but for a finite run what matters is the expected number of noisy uses. If it tends to zero, the original work rate remains; if it diverges, sublinear work is possible; if it tends to a positive finite value, the work rate stays positive, between two envelopes whose equality is open. 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, except in the free-energy ledger of a controller that comes back only in state, which counts its unrefunded free energy instead. 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 (doi:10.5281/zenodo.23310624; https://github.com/Apsiape/save-work-by-waiting); The work cost of answering in real time (doi:10.5281/zenodo.23310628; https://github.com/Apsiape/answering-in-real-time); The price of time order (doi:10.5281/zenodo.23310622; https://github.com/Apsiape/price-of-time-order). 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.
Loss can restore the vacuum entropy bound without erasing non-Gaussian structure. This paper determines when an entropy statement justifies a coherent-state approximation, when non-Gaussian structure survives optical loss, and what mode and source-request budgets are needed to use that structure. The results give quantum-optics researchers separate criteria for entropy recovery, state approximation, Gaussian concentration and conversion to a specified output. Version 1.5 — revised 10 October 2026. The 96-page PDF contains the main article, Methods, Supplementary Information and references. This revision connects the exact loss boundary to a mode-uniform near-equality certificate, finite-data validation, a conversion-cost lower bound for the same thermally repaired source, and an exact equal-entropy comparison of sources with different conversion-cost exponents. The central figures are updated accordingly. 1. Near-vacuum states can have an unbounded entropy deficit Write νm for the m-mode vacuum and define Δα,m(ρ)=Hα(Wνm)−Hα(Wρ), using natural logarithms; H1 is Shannon entropy. Positive deficit means entropy below vacuum. We construct physical Wigner-positive states satisfying, simultaneously, Em=Tr(Nmρm)→0,21∥ρm−νm∥1→0,dFm(ρm)→0,Δ1,m(ρm)→+∞. Here Fm is the trace-norm-closed convex hull of Gaussian states and dFm uses half the trace norm. Thus proximity to the entire Gaussian model does not control Wigner entropy uniformly in mode number. Independent weak modes amplify a one-mode entropy effect while their total excitation vanishes. For the same sequence, half loss instead gives −Em≤Δ1,m(L1/2⊗mρm)≤0, so the entropy difference from vacuum tends to zero. 2. The exact pure-loss recovery boundary For Wigner-positive inputs, intensity transmissivity η and any fixed positive total input-energy budget E, the sharp threshold is ηc(α)={1/2,α/2,0<α≤1,1<α<2. With Pm,E={ρ:Wρ≥0,Tr(Nmρ)≤E}, supm≥1supρ∈Pm,EΔα,m(Lη⊗mρ)={0,+∞,η≤ηc(α),η>ηc(α). The recovery statement includes the boundary itself and is independent of mode number. Half loss also restores vacuum majorization for arbitrary density operators. For α≥2, established Wigner-norm bounds give the vacuum entropy bound at every transmissivity for Wigner-positive outputs. These statements distinguish a universal guarantee from the earlier sign crossing that a particular source may exhibit. 3. Near equality gives a certificate for the entire input For an arbitrary input ρ, define the half-loss output Shannon excess s(ρ)=h(WL1/2⊗mρ)−m(1+lnπ)≥0. The half-loss identity identifies this quantity with the input's total Wehrl entropy excess. Applying mode-uniform Wehrl stability yields infa∥ρ−∣a⟩⟨a∣∥12≤ln264s(ρ). The bound controls the full input state, with no energy or photon-support restriction. One coherent reference then controls every subsequent quantum channel and every unconditional measurement event. In particular, an event-probability error at most ϵ is guaranteed by s≤(ln2)ϵ2/16. Conversely, a surviving output Gaussian-hull gap enforces s(ρ)≥16ln2dFm(L1/2⊗mρ)2. Recovery of the inequality and proximity to equality therefore support different physical inferences: the former permits residual non-Gaussian structure, while the latter certifies a coherent-state approximation. 4. Surviving structure has mode and conversion costs A thermally repaired vacuum–three-photon source with per-mode occupation c/m has total input energy tending to 3c. Along the recovery boundary, its attenuated state obeys liminfm→∞dFm(ρm,ηc)≥cηc3exp{−c[1−(1−ηc)3]}>0. At half transmission this lower bound is g(c)=(c/8)e−7c/8. Starting from one complete block, Gaussian processing requires Θ(m) output modes to retain any fixed positive gap below this bound at success probability at least a fixed p0>0. Gaussian ancillary states, adaptive Gaussian measurements, feedback, discarding and postselection are included. A distinct task requests fresh independent blocks of this same half-loss source to produce the one-mode target τ∗=(18∣0⟩⟨0∣+6∣1⟩⟨1∣+∣2⟩⟨2∣)/25 within half trace distance 10−6. Let Cm be the infimum of expected complete-block requests divided by success probability, including all failed requests. For fixed 0<c≤1, every achieving Gaussian protocol obeys the asymptotic lower bound logCm≥1281(m/c)1/3−logm−O(1). This is a source-request lower bound; the optimal achievable cost for this source remains undetermined. The proof evaluates a curvature-controlled input certificate for the actual thermally repaired state and applies the Gaussian conversion theorem of the companion paper cited below. 5. Exactly equal entropy does not fix conversion cost For two separate two-mode quartic source families, their mixture probabilities can be adjusted so that their half-loss Shannon entropies are exactly equal. Under a common target, tolerance and failure-inclusive accounting, the inherited matching conversion laws become h(WωA)=h(WωB)=2(1+lnπ)+s,logCA(s)=Θ(s−1/3),logCB(s)=Θ(s−1/2). The target tolerance is held fixed in these asymptotics. The delivered energies after entropy matching satisfy EA,EB∼s/2. At exactly equal energy, the two excesses instead share si(E)=2E−(35/8)E2+O(E3) and differ by sA(E)−sB(E)=15E3+O(E4). Entropy alone therefore cannot select the cheaper source for a fixed output task. 6. Finite data and finite mode budgets Under a declared displaced-input photon-tail bound Tr[N1N>Kσa]≤τK, trusted half loss and all-mode on/off detection with efficiency at least ξmin, the click probability q gives s(σ)≤AKq+τK,AK=1−(1−ξmin/2)KK. A one-sided binomial bound turns this into an entropy confidence certificate. An explicit undisplaced four-photon-cutoff example uses K=4, τK=0, ξ=0.8 and N=3,517,336 independent validation blocks, accepting at most 41 clicks, to certify unconditional event error at most 0.05 with false-acceptance probability at most 0.001 and missed acceptance at most 0.001 on the stated low-click alternative class. All vacuum results count; training, calibration and source heralding are separate costs. When coherent-state approximation alone is the task, the same click measurement supplies a stronger direct fidelity test. The paper also gives all-input finite-budget exclusions above the loss-recovery line. For example, at E=1, α=1.9, η=0.9505 and m=105, the certified bound Δα<0.06942 rules out a target deficit of 0.1 nat. A separate finite single-mode preparation model predicts positive Shannon deficit near η=0.97, negative deficit near η=0.85, and a Gaussian-hull witness above 4.4×10−3 at half transmission, with explicit preparation, measurement-error and raw-trial budgets in the PDF. 7. Dimensional scales, a strict entropy floor and ORCS Near order two, put α=2−δ and L=ln(1/δ). For the maximum deficit Dm,E at fixed total energy, Dm,E(2−δ)=Θm,E(δ/L)for fixed m,Th(δ,E)=ΘE,h(L/δ), where Th is the first mode number allowing a fixed positive deficit h. Beyond a sufficiently large fixed multiple of L/δ, Dm,E(2−δ)=ΘE(mδ). The manuscript separates proved sufficient fronts from necessary exit bounds; the sharp first-crossing coefficient remains open. At every fixed mode number, an energy-unrestricted strict Shannon floor also holds: h(Wρ)≥mln(2π)+εm with εm>0 for all Wigner-positive states. The ordering-resolved coherence spectrum (ORCS) supplies an exact single-mode loss coordinate. With Cr,k denoting the ordering-weighted norm of photon-number-difference block k, Cr,k(Lηρ)=CTη(r),k(ρ),Tη(r)=1−η1−r. For vacuum–n-photon coherence and 1<α<2, C1−α/2(Lη∣n⟩⟨0∣)=(α2η)n. This formula exposes the high-order coherence mechanism at the loss threshold; the universal recovery result is proved through the full nonlinear Wigner-norm inequality. Related papers and history The counterexample paper Wigner entropy below vacuum: physical counterexamples and stability limits expands arXiv:2609.13312v1, submitted on 10 September 2026. This stability paper develops the dimensional, loss-dependent and operational consequences. Its state-approximation and conversion interfaces use the companion results From phase-space entropy to coherent-state error: mode-uniform Rényi–Wehrl stability and Phase records control the conversion cost of non-Gaussian light. The mode-retention comparison connects to Limits to Gaussian concentration of non-Gaussian light.
This work gives a resource law for engineering optical noise: how many photons must a single environment mode contain to prevent any signal from producing an output that is too spectrally concentrated? It provides an explicit optimal bath preparation, a matched Gaussian benchmark, and separately proved guarantees for correlated repeated use. Exact photon cost of a spectral noise guarantee A signal mixes with an independently prepared bath σ at a beam splitter of signal transmissivity η. The requirement is Φη,σ(ρ)≤cIfor every normal input state ρ,0<c<1. Equivalently, every rank-one output projection has probability at most c and the receiver's output min-entropy is at least −lnc. Inputs may be mixed, non-Gaussian, arbitrarily energetic, and have infinite photon tails. Let d=⌈1/c⌉−1. Throughout 0<η≤1/⌈1/c⌉, the minimum mean photon numbers of the actual bath are Nall(c,η)=1−ηd−cd(d+1)/2.NG(c,η)=1−ηc−1−1. The first optimization includes all normal bath states and has a unique finite-support minimizer. The second includes all Gaussian density operators, with arbitrary squeezing and displacement; a centered thermal bath attains it. The coupling and spectral requirement are identical in both comparisons. At c=1/j, for every integer j≥2 and 0<η≤1/j, NG(1/j,η)=2Nall(1/j,η). Between reciprocal budgets the ratio varies. The cost counts mean photons in the physical bath, not laboratory preparation work or the energy of an auxiliary purification. A physical preparation and its complete single-use spectrum Writing r=1−dc, the attaining vacuum-output spectrum is λ∗=(d entriesc,…,c,r). It majorizes every output of the same channel, so the minimum von Neumann entropy is exactly Smin(Φη,σ)=−dclnc−rlnr. The other positive-order Rényi minima are fixed by the same spectrum. Coherent inputs attain these minima. The proof identifies an actual optical dilation whose port antisymmetry enforces the all-input occupation bound. Exact examples also show that baths with identical energy and covariance can have different optimized entropy floors. What survives entangled repeated use For the same resource-optimal flat-output families of ranks j=2,3,4, separate estimates prove Sout≥mlnj for every finite number m of uses and arbitrary normal joint inputs, including cross-use entanglement. The environments vary with η along their specified families. Further results give sharp tensorized purity bounds for the entire three-level diagonal bath simplex and for every Fock environment at balanced coupling. The paper then treats a different, fixed bath, σ=∣1⟩⟨1∣, at every 0<η<1. With h(η)=−ηlnη−(1−η)ln(1−η), it proves: Exact Sout≥mh(η) at arbitrary finite blocklength for joint inputs invariant under each local photon-number phase rotation by 2π/q, q≥4. Coherence and entanglement within residue sectors and infinite tails remain allowed. The residue-zero restricted channel also has strong minimum-output-entropy additivity with arbitrary normal partner channels. The same exact entropy floor for up to three globally total-phase-invariant input modes, without a mean-energy limit. Sout≥Mh(η)−o(M) for physical-mode permutation-invariant inputs of finite mean energy density, with explicit errors and full infinite-tail control. For every fixed m and interior η, all outputs remain a uniformly positive trace distance from every state of rank below 2m, without any input-energy bound. This covers nonbalanced parameters where the older purity-based distance bound vanishes; the new positive constant is spectrally implicit. Interpretation and scope The input classes above are not nested, and an entropy rate, a finite-block entropy floor and a low-rank distance are distinct guarantees. The fixed single-photon bath is not generally the resource-optimal bath. Complete unrestricted nonbalanced single-photon entropy additivity and exact classical capacity remain open. A spectral noise guarantee constrains output mixing; it does not mean that all input information or reference correlations are erased. The article separates this issue through actual resolved-erasure qubit restrictions, antidegradability of the specified optimal reciprocal families, and a classical capacity bracket under a true signal-energy budget. The integrated 52-page article contains six scientific figures, the complete proofs, and the mathematical data needed to reconstruct its finite certificates. Logarithms and entropy values use natural units unless bits are explicitly stated.