A particle is confined by a high but finite potential wall. The wall's height is changed, and the particle's initial state is a complete thermal mixture. Almost all particles retain their ordered energy-level label as the wall becomes harder. That familiar measure of successful control nevertheless misses a small population at increasingly high energies. These rare outcomes dominate the leading dissipated work and make the work variance grow. The paper derives this rare-energy law from the actual continuum Hamiltonian, without discarding the initially occupied high-energy states or replacing the finite wall by a perfectly reflecting boundary. It then proves a limitation on pulse shaping: changing only the wall height with a bounded integrated height action cannot remove the leading energetic cost. This remains true when the waveform and duration change with the wall hardness. The mechanism is an endpoint displacement extending over an entire half-line in the boundary coordinate; the finite-norm correction produced by bounded-action control cannot cancel it. The leading thermodynamic information is also accessible in principle through final-energy measurements. For each fixed protocol, a finite set of appropriately chosen high-energy bands recovers arbitrarily much of the leading dissipative relative entropy. The energy-basis coherence contribution is smaller order. This does not make the measurement inexpensive: the energy range increases and the counts become rare. A separate finite-height continuum certificate supplies one concrete probability floor and work bound, uniformly over the stated action class.
A particle is confined by a high but finite potential wall. The wall's height is changed, and the particle's initial state is a complete thermal mixture. Almost all particles retain their ordered energy-level label as the wall becomes harder. That familiar measure of successful control nevertheless misses a small population at increasingly high energies. These rare outcomes dominate the leading dissipated work and make the work variance grow. The paper derives this rare-energy law from the actual continuum Hamiltonian, without discarding the initially occupied high-energy states or replacing the finite wall by a perfectly reflecting boundary. It then proves a limitation on pulse shaping: changing only the wall height with a bounded integrated height action cannot remove the leading energetic cost. This remains true when the waveform and duration change with the wall hardness. The mechanism is an endpoint displacement extending over an entire half-line in the boundary coordinate; the finite-norm correction produced by bounded-action control cannot cancel it. The leading thermodynamic information is also accessible in principle through final-energy measurements. For each fixed protocol, a finite set of appropriately chosen high-energy bands recovers arbitrarily much of the leading dissipative relative entropy. The energy-basis coherence contribution is smaller order. This does not make the measurement inexpensive: the energy range increases and the counts become rare. A separate finite-height continuum certificate supplies one concrete probability floor and work bound, uniformly over the stated action class.
Coherent boundary interference can purify the retained part of an increasingly mixed Gibbs state, while its work statistics probe inequivalent spectral sectors. We establish this connection for a confined particle subject to a coherently controlled potential wall. A common boundary source governs the conditional state, while its positive spectral measure determines the leading population--coherence allocation of dissipated work and the Ramsey response. Resonant interference produces an asymptotically pure retained state with an explicitly paid success probability. For a sharp wall, a discrete near-elastic sector contributes a logarithm to absolute work that is absent from the quadratic-weight continuum limit. Its contribution makes the ratio of absolute to network diverge on specified thermal paths. Smooth-wall thermodynamic limits, a finite smooth-device certificate with timing and readout errors, and a finite-count reconstruction scheme connect these results to operational resources.
Jarzynski's equality recovers an equilibrium free-energy difference from nonequilibrium work trajectories, but its exponential average can converge very slowly because rare trajectories carry large weight. We show that coarse thermodynamic information measured at the nonequilibrium endpoint can reduce this sampling burden while keeping the same microscopic trajectories and the same free-energy target. The endpoint mean energy and a coarse spatial record define a maximum-entropy state and select one member of an exact family of fluctuation relations. That physically selected member provides an exact statistical control for the ordinary Jarzynski estimator, even when the final energy and retained record do not commute. In a finite Bose--Hubbard system, this control substantially lowers the finite-confidence sampling cost, while a small controlled departure from exact endpoint matching produces a square-root information--sampling tradeoff.
Calibrating a confining wall at low energies does not specify the work of changing its microscopic profile. We quantify this distinction using positive finite potentials on a fixed Hilbert space and complete thermal initial states. For any fixed finite order of the wall's energy-dependent boundary response, a nonconstant finite-time drive can preserve that response throughout its motion while producing a strictly positive leading dissipated-work coefficient. The physical duration scales with the inverse wall height. A complete off-label transition law resolves rare outcomes of probability of order r3 at energies of order r−2, giving mean dissipation of order r and variance of order r−1. For sudden changes in arbitrary smooth bounded cavities, the leading work coefficients factor into thermal boundary pressure and microscopic profile energies. This factorization extends to canonical ideal Bose and Fermi gases at fixed particle number. A smooth, scalar-calibrated finite-wall example certifies positive dissipation and a bounded-energy detection event at nonzero duration, including full thermal tails and finite reporting errors. The preparation costs distinguish observing a rare event from estimating its moments. These results specify the dynamical information left outside stationary boundary calibration without implying a resource-free sensing advantage.
Strong coupling to a reservoir can do more than shift, broaden, or dress the work peaks of a driven quantum system. When the reservoir is infrared singular, a sudden change of a local control parameter can alter the boundary condition seen by infinitely many low-energy modes, converting a quasiparticle-like threshold line into a many-body edge. We demonstrate this mechanism for the inclusive work distribution of the biased spin-boson model under a sudden bias inversion. In the independent-boson limit, the problem is exactly solvable and gives a sharp infrared classification: a super-Ohmic bath can retain a finite elastic threshold weight, whereas Ohmic and sub-Ohmic baths extinguish the elastic line through boundary orthogonality. At the Ohmic fixed point, the same exponent controls both the vanishing elastic residue and the low-work continuum. We then ask how this edge is resolved away from the static-boundary limit. Using displaced-basis exact diagonalization of logarithmically discretized baths, we find that finite tunnelling leaves an edge-like continuum over the accessible energy window, while separating two operational diagnostics of the threshold: the cumulative-continuum exponent extracted from z-interleaved spectra lies above the elastic-overlap exponent extracted from z-averaged overlaps, θC>θZ. We interpret this separation as a finite-energy crossover away from the static-boundary fixed point, not as evidence for a new asymptotic fixed point. The separation survives fitting-window variation, oscillator-cutoff checks, spectrum-size checks, and leave-one-z-out tests, while time-domain characteristic functions provide a compatible but non-decisive diagnostic. Finally, the same threshold edge controls the sampling cost of Jarzynski-type exponential averages, making rare low-work events increasingly important at low temperature.
A frozen numerical record of a two-dimensional electron double slit in which the idealized impenetrable wall is replaced by a barrier of finite height and finite thickness, and the response is measured against a finite low-transmission control rather than against an infinite wall. Three artifacts. The primary record contains a preregistration of five cases, an independent split-operator solver, the closure evidence and a gate ledger; it was cryptographically committed before a blind exchange with an independent worker who had frozen his own calculation. A detector-position supplement varies only the plane at which the current is recorded, 1–14 nm, and identifies which planes are physically usable rather than forcing them. A wall-height record varies only the barrier height, 2.35–24 eV on both sides of the control, and both thicknesses. What the records establish: the finite-minus-control response separates into a through-thickness term, calibrated independently by the no-slit geometry, and a face and edge term that survives when the thickness kills tunnelling; a hard spatial partition into "material" and "aperture" contributions is not well defined, since shifting the mask by one or two grid cells reverses the sign of the attributed bulk contribution; the residual vanishes only when the two walls are identical and reverses sign when the test wall is the harder one, so it is the difference between two standard responses and not a property of one wall; and its form is one scalar function of the wall height times one geometry constant. What they do not establish, and prove they cannot establish in this geometry: which property of the wall carries that function, since a null regressor with no derivation behind it fits equally well and the measurement that would separate the two is smaller than the numerical systematics. No additional quantum path, no Born-rule violation and no physics beyond the time-dependent Schrödinger equation is claimed or supported. Withdrawn constructions, corrections and errata are retained in the records rather than removed. Each archive carries a manifest, a verifier that re-derives its summary from the raw output and rebuilds its report byte-identically, and a set of mutations that must each be caught — six, five and seven negative controls respectively. The two supplements reproduce values of the primary record's gate ledger bit for bit. Every claim carries its register: exact result, computed here, conditional, constitutive, open, or prediction.
Quantum work statistics differ from classical ones because initial energy coherence matters. The standard two-point measurement (TPM) gives a positive distribution but erases phase information. Coherence-retaining endpoint-work quasistatistics provide a compact probe of shortcut-to-adiabaticity performance. For work defined with respect to a reference Hamiltonian, an exact counterdiabatic shortcut pulls the final reference Hamiltonian back to an operator diagonal in the initial energy basis. Endpoint Kirkwood-Dirac or Margenau-Hill quasistatistics then lose sensitivity to initial coherence and reduce to the TPM result. Imperfect shortcuts restore this sensitivity: a non-commuting control error produces off-diagonal pulled-back Hamiltonian elements at first order in the error amplitude, whereas population-only transition probabilities change only at second order. Harmonic-oscillator and qubit benchmarks confirm this linear-versus-quadratic contrast. The result complements inclusive work-cost analyses: it does not measure the auxiliary field's energetic cost, but provides a phase-sensitive endpoint diagnostic of residual nonadiabaticity.