Convex-Entropy Reservoirs and Conditional Ground-State Preparation: An Operational Separation of Thermodynamic Curvature, Accessible Cooling, and Dynamical Reset, with a Spectral-Obstruction Case Study in the D1-D5 CFT

The paper has two components. Parts I–IV develop a conditional operational theory of reservoir-assisted cooling and exact reset at the level of channels; Part V is a self-contained Hamiltonian-level case study of exact reset embeddings in a D1–D5 CFT reservoir. The two notions of “exact reset” are different and are kept distinct throughout. We separate three logically distinct questions in reservoir-assisted cooling: what the reservoir entropy implies thermodynamically, which low-temperature ancillas are accessible, and what extra dynamical resource exact ground-state preparation needs. Convex microcanonical entropy is neither sufficient nor necessary for a microscopic cooling operation; it is a thermodynamic route to an ancilla supply, under a stated local-Gibbsian hypothesis. For a resonant qubit target we prove that every energy-conserving collision with a qubit ancilla acts as p' = p + s(q − p), 0 ≤ s ≤ 1, so cooling to error ε is possible exactly when ancillas with excited population at most ε can be supplied. We then remove the two-level restriction: for an arbitrary energy-incoherent sink, within the stated collision model, the least reachable excited population is a sector-wise majorization (sorting) problem with an explicit closed-form solution. Its consequences are: (a) a cooling criterion: a sink can cool a target of population p if and only if some pair of its levels one target gap apart has population ratio below p/(1−p); (b) a Gibbs sink at any finite temperature leaves the exact floor min{p, (1−p)e^{−β E_sys}} > 0, and this floor is stable under arbitrary sequences of fresh sinks at the same temperature (the sequence extension is proved by a concavity lemma); (c) the floor is a single-temperature statement: a hot and a cold thermal sink used together cool every p ∈ (0,1), and iterating fresh pairs drives p → 0 with no work store, measurement or record, while sequential single-sink collisions already reach n̄_c / (1 + n̄_c + n̄_h); (d) sharpness of the energy support, or one classical record of entropy h_2(p), is what buys exactness in finitely many steps. Exact reset has two independent resource obstructions in the stated model. The optimal residual excitation of any energy-conserving reset is the sink weight on directions that lack spectral room (the room floor), with a criterion for measurement-free exact reset; against limited control, any circuit of m gates with rank(G − I) ≤ ρ leaves the target excited with probability at least the sink-spectrum tail beyond ρm, so, within this gate model, the number of gates needed for exactness is at least the corresponding tail rank divided by ρ, with the operational integer requirement expressed by the ceiling where a finite target error is specified. Within the shell-ancilla model, an explicit dimension requirement dim ≥ 1/ε follows, with the corresponding bound attained by the stated construction. The room floor and the rank–error bound are energy-conserving, sector-wise and gate-rank refinements of the Ticozzi–Viola purification bound, and are presented as such. We also give a conditional zero-designated-work SWAP construction with N = O(log log(1/ε)) collisions for a quadratic-entropy family, a dimension-counting reset criterion with an exact ladder-sink cost–error curve, an error budget for measurement–feedback completion, and a finite birth–death noise model with a positive spectral gap. Black holes, Na⁺_147 nanoclusters, structured-vacuum engines and Casimir experiments are treated as delimited case studies, not demonstrations. The contribution is a conditional operational characterisation with falsifiable requirements; it is not a claim that the third law has been circumvented. Finite instances of the principal formulas were checked by exact arithmetic and numerical sampling (random energy-conserving unitaries against the closed forms, and exact evaluation of tabulated values; Section 27); these checks are consistency tests, not substitutes for the proofs. Part V. The case study proves two conditional obstructions to exact spectral reset embeddings in a D1–D5 CFT reservoir. The first is a sector-specific capacity bound: under explicit component-string, single-strand, graded-orbifold and constant-character hypotheses, the maximal-left-charge Ramond-ground-state sector has dimension 4N, so a larger source cannot embed into it. The second is a finite-dimensional perturbative criterion: for an exactly degenerate source block transported by a regular family, the second-order effective Hamiltonian must act as a scalar on the transported block, equivalently the block must lie in a single second-order eigenspace, which gives an eigenspace-multiplicity bound for a prescribed target sector. At level four, under an explicit Hermiticity and perturbative-matching hypothesis, fifteen long-multiplet primaries with distinct second-order lifts give a fixed-label non-scalar target block. All obstructions are conditional and local, no universal no-go theorem for the full D1–D5 Hilbert space is claimed, and no microscopic reset dynamics are constructed.