Sparse-oracle finite Gibbs purification procedure

Input: a nonzero finite-dimensional Hilbert space of dimension $d$, a Hermitian Hamiltonian matrix $K$, an inverse-energy parameter $\beta >0$ such that $H=\beta K$ is dimensionless, the ordered reference preparation $\mathrm{\mid }{\mathrm{\Phi }}_d⟩$ from Finite Gibbs density and its canonical purification, an output vector-error tolerance ${\epsilon }_{\mathrm{o}\mathrm{u}\mathrm{t}}>0$, and a rational amplification-failure target $0<\rho <1$. The input contract must also supply a rational shift-confidence target $0<{\delta }_{\mathrm{s}\mathrm{h}}<1$, a finite call cap $Q_{\mathrm{c}\mathrm{a}\mathrm{p}}$, and rational sampler precision $0<\nu \le {\delta }_{\mathrm{s}\mathrm{h}}^2\mathrm{/}({10}^{12}d(Q_{\mathrm{c}\mathrm{a}\mathrm{p}}+1{)}^2)$ as required by Coherent spectral minimum and a frozen Gibbs energy shift, as well as every sparse-oracle, block-encoding, controlled-evolution, inverse, norm-bound, energy-shift, coefficient-generation, implementation-error budget, synthesis-precision, reference-preparation, and cost object required by Sparse-oracle Gibbs-state purification. No matrix exponential, trace, exact spectral minimum, or gate is free input.

Procedure: invoke the normalized-output branch of Sparse-oracle Gibbs-state purification on $K$ with coefficient $\eta =\beta$, the stated vector-error and amplification-failure targets, the supplied shift-confidence target ${\delta }_{\mathrm{s}\mathrm{h}}$, and complete access/cost certificates. Retain its frozen energy shift and all declared finite subroutines. Measure only its specified good/bad flag. On a good branch return the normalized system-reference vector; on a cap failure return the explicit output FAIL. If the certified shift event fails, do not assert the Gibbs approximation guarantee; any overall failure probability includes the supplied ${\delta }_{\mathrm{s}\mathrm{h}}$ shift-confidence budget.

Output: either FAIL, or a normalized vector-register state in the successful branch, with the error and success guarantees supplied by Gibbs purification error and resource bound. An identity energy shift changes neither the normalized Gibbs density nor its canonical purification, since $e^{-(\beta K+cI)}=e^{-c}{e}^{-\beta K}$ and the trace is multiplied by the same scalar. This is a mathematical state-preparation procedure under the stated finite access model; it is not a physical bath or thermalization process.