M05-v1 — single-exponential temperature cost for one-dimensional Gibbs preparation
Frozen complete-original candidate, 22 October2026. Internally supported.
No external acceptance, historical priority, manuscript or publication is claimed.

Theorem (with input costs explicit).
Fix local dimension and finite interaction range. Given a chain Hamiltonian
H_N=sum_X h_X in the local-matrix input model of Rouze--Stilck Franca2610.03578v1
Theorem3.11/AppendixB.1, inverse temperature beta>=0 and epsilon>0, a uniform
randomized quantum algorithm prepares an output density matrix rho_hat with
 (1/2)||rho_hat-exp(-beta H_N)/Tr(exp(-beta H_N))||_1<=epsilon.
Put x=1+beta J_int and Q=1+log(N/epsilon), replacing epsilon by min(epsilon,1/8)
when necessary. For fixed constants C,c depending only on local parameters,
both elementary-gate count and classical preprocessing/expanded-circuit cost are
 bounded by exp(C x^3) N (1+b)^c Q^(C x+c).
Here b is the explicit local/scalar/raw-description bit bound described in the
proof. In the bounded source model with requested-precision descriptions, its
polynomial overhead is absorbed into exp(poly(x)) and a fixed additional power
of Q, giving the exact original's exp(Cx^c) N polylog(N/epsilon) shape. The
polylogarithmic exponent is O(x), not uniform in temperature. Arbitrarily longer
or larger-magnitude descriptions retain the displayed polynomial input factor;
no cost independent of reading such data is asserted. The proof additionally
permits arbitrary onsite energy magnitudes with their logarithmic bit costs.

The output density includes the algorithm's discarded classical random bits.
Every random seed has bounded computational work. No free Gibbs purification,
spectral oracle, postselected success or exponentially enumerated jump family
is used. The beta=0/zero-interaction cases and chains shorter than the reset
length are explicitly handled. The theorem concerns preparation, not the
stronger approximation of the whole global semigroup on every input.

Proof architecture, with complete stages enclosed under proof/.
1. The QBP coefficient is a positive real-time average. A reversible quantile
   register and interaction-picture simulation implement it from ordinary
   local matrices with polynomial bit cost; the huge time grid is not listed.
2. Sorted-time Dyson LCU builds boundary dressing and a separate inverse ODE.
   Fixed-point amplification charges every child preparation and inverse.
   Reset blocks have length exp(O(x)) and tree height O(x), yielding total
   exp(Cx^3) Q^(Cx+c) purification work. The larger localization region needs
   only dressing. A purification isometry encodes the reset jumps collectively.
3. The exact coherent correction uses a truncated odd csch kernel, with
   logarithmic LCU mass and coherent inverse CDF. Its compiled oracle is an
   exact Hermitian involution. The loss and correction define a genuine nearby
   Lindbladian, and CPTP diamond Duhamel controls errors linearly in duration.
   The imported stacked-jump simulator's actual classical/gate work is charged.
4. One global dimensionless input rounding gives a controlled Gibbs perturbation.
   Explicit conservative integer block/time parameters satisfy the source's
   influence inequality. Its discrete mixing theorem yields k=O(NQ) updates.
   Dyadic label sampling, local diamond errors and trace telescoping give the
   output guarantee. Offline sorting/joining templates and expanded gate lists
   charges all global classical assembly without a free quantum selector.

These statements compose exactly as proved in proof/05-global-composition.txt.
That frozen stage explicitly references the first four enclosed stages. Its
historical prospective-review wording records its status when authored; the
subsequent parent and owner review records supply the present supported verdict.
Earlier partial-stage statements of remaining gaps are likewise historical;
they are resolved by the later enclosed stages, not silently edited.

Attribution and scope.
Quantum belief propagation, DS mixing/locality, ordered Dyson LCU, fixed-point
amplification, interaction-picture simulation and stacked Lindbladian simulation
are imported and attributed. The complete charge for the proposed recursive
implementation is the original-scale contribution relative to Remark3.12.
Primary versions are pinned in dependency-map.json, including full local source
snapshots where used. Current official version records checked22October2026
remain v1 for2610.03578,2610.01190 and2609.18757. No worldwide novelty assertion.

Verification.
Owner independently read the original, all five proof stages and decisive source
interfaces; parent independently reviewed each stage and the global theorem.
Code-backed scalar, finite-matrix, resource-accounting and sampling controls
have two clean helper replays, with independent parent replay where stated.
Those controls are diagnostics supporting an analytic complexity proof; no
end-to-end compiled large-chain execution is claimed. All imported content is
hashed, and the bundle stores deduplicated immutable copies. Internal review
is fallible and is not external mathematical certification.
