M05-v1 — independent strict review
Date and literature cutoff: 22 October 2026
Reviewer: /root/top3_strict_independent (fresh-context child agent; no further agents)

VERDICT
Validity: supported internally.
Original coverage: complete in the exact original's local-matrix model and permitted temperature-dependent polylogarithmic exponent.
Literature: difference supported for the fully charged single-exponential implementation; the recursive architecture and analytic mixing results are inherited.
External status: no external acceptance or submission receipt established by this review.
Blocking gaps: none found. This is an analytic complexity review, not an execution of the entire large-chain circuit.

FROZEN IDENTITY AND REVIEW SCOPE
Original: campaign/originals/M05.tex
Original SHA256: 6252324602b02d4921873bf00ea200b88b4e6e194848d380c8276731de0f10f7
Candidate: campaign/continuation-R41-R60-2026-10-22/candidates/M05-v1
THEOREM.txt SHA256: 6d2361869b7c7772cd65ced42356e54069e065bd471ae0ccdd19e727dbbe2882
The entire five-stage proof, original mapping and relevant source interfaces were read. Exact hashes of all five proof stages and dependencies are in frozen-inputs.json. Historical partial-stage wording was interpreted in the composition explicitly stated by THEOREM.txt, not as independent final claims. Internal statuses were ignored as evidence. No frozen input was edited.

EXACT ORIGINAL MAPPING
Fix local dimension and finite range, let x=1+beta J_int and Q=1+log(N/epsilon). The candidate gives both elementary quantum gates and classical preprocessing/expanded-circuit work bounded by exp(C x^3) N (1+b)^c Q^(C x+c).
Here b includes the explicit per-local/scalar/raw-description input-bit costs. In the source model with bounded terms and requested-precision local descriptions, that factor is absorbed as stated. An arbitrarily long or high-magnitude descriptor does not become free. This is necessary to read the actual input, not a missing computational primitive.
The exponent of Q is O(x), explicitly temperature-dependent as the original permits. The result concerns state preparation, including the discarded classical random bits in the output density matrix, not uniform diamond approximation of the entire global semigroup.
Every random seed has bounded work. The algorithm includes beta=0, zero interaction and chains shorter than the selected reset block. There is no free block Gibbs oracle, free coherent global selector, uncharged jump enumeration or postselection.

DECISIVE ANALYTIC AUDIT
1. Coherent QBP coefficient, proof/01-coherent-filter.txt:45-156.
The real-time filter is a positive normalized law. Its Laplace-mixture/CDF representation supports reversible quantile evaluation. A midpoint grid has exponential cardinality but only logarithmic register length; no table over that cardinality is constructed. The monotone coupling bounds average time displacement, and derivative lower bounds on the truncated domain make bisection precision polynomial in requested bits.
The interaction-picture simulation is supplied with explicit local matrices: fixed-dimensional onsite diagonalizations, controlled phase arithmetic and boundary interactions are charged. Large onsite energies enter through their bit length rather than an unaccounted norm-dependent time grid.

2. Recursive normalization and inverses, proof/02-recursive-purification.txt:59-146.
The full-temperature purification transport uses the correct QBP coefficient and inverse-temperature parameter. A separate reversed/sign-changed ODE implements O^{-1}; the proof does not confuse it with O^*. Ordered Dyson terms include repeated grid nodes. The LCU normalization is exp(O(x)), and the finite-order and grid errors have explicit budgets.

3. Actual child cost and amplification, proof/02-recursive-purification.txt:147-269.
The lower success bound holds for every input, not just the exact ideal children. The pi/3 recursion cubes failure while preserving the successful direction. Reflections about the actual compiled child state use the child preparation and its exact circuit inverse, and all such calls are counted. Consequently child approximation enters through the successful target direction rather than being incorrectly charged once per ideal-oracle replacement.
The recurrence includes an exp(O(x)) amplification factor and a logarithmic precision factor at every tree level. Reset length ell=exp(O(x)) means height O(x), yielding exp(C x^3) Q^(C x+c). Applying that recurrence directly to a localization radius growing with log(N/epsilon) would be dangerous; the proof instead recursively prepares only the reset block. The larger region requires boundary dressing, not a larger Gibbs purification tree.
The collective reset isometry stores the old block and a purification environment. Projected-unitary construction of V^*V avoids enumerating an exponentially large list of jump operators.

4. Coherent correction, proof/03-coherent-correction.txt:5-194.
The odd csch representation has the correct imaginary sign and inverse-temperature scale. The lower cutoff uses the commutator bound, the upper tail decays exponentially, and the retained LCU mass grows logarithmically in the precision/energy scale. Explicit inverse-CDF arithmetic uses stable expm1/log1p forms and charged guard bits.
The assembled finite circuit is wrapped into an exactly Hermitian involution. Thus the compiled projected Hamiltonian and rectangular jump encoding define an actual nearby Lindbladian; positivity/CPTP structure is not merely approximate. The stated loss-operator errors propagate through the declared normalization.

5. Actual Lindbladian-simulator inputs, proof/04-local-simulation.txt:1-133.
The imported simulator is used with normalization alpha_G+alpha_V^2, the correct ancilla projectors, a Hermitian involution and a rectangular jump stack. Its coefficient support is O(q), while time-grid dependence is logarithmic. The candidate charges coefficient preparation, controlled calls/inverses, local register work, binary time arithmetic and circuit description.
The diamond perturbation bound is linear in e_G and (2 alpha_V+e_V)e_V. CPTP Duhamel gives a duration factor T; no artificial exponential in T||L|| is introduced. All oracles are built by the preceding stages, rather than assumed.

6. Global temperature and input precision, proof/05-global-composition.txt:38-129.
One rounded dimensionless K=beta H is used throughout, with a global Gibbs perturbation budget. Different updates therefore do not silently target incompatible Hamiltonians. Computable conservative X, u0, ell and T satisfy the source's explicit influence inequality; the three positive terms are separately bounded.
I checked the source's temperature-dependent clustering cutoff. The actual Appendix A expression includes the k-dependent denominator inside the square root; a secondary summary which dropped it was not a valid objection and was discarded. No source/candidate mismatch remains from that lead.

7. State preparation and classical assembly, proof/05-global-composition.txt:130-275.
The finite labelled covering retains repeated edge labels correctly. Its discrete mixture P is covered by the exact source contraction theorem; k=O(NQ) updates suffice for preparation. Approximating a whole Poissonized semigroup is unnecessary for the original success criterion.
Each local update has an explicit diamond error, dyadic label sampling has bounded TV error, and telescoping yields the claimed trace guarantee. The small-chain branch uses a height-O(x) recursive preparation. Preprocessing local templates, reading their neighborhoods, presampling labels and offline sorting/joining expanded gate lists cost N times the displayed local/polylog factors. This does not invoke a free coherent N-way lookup.

FRESH CONTROLS AND REPRODUCTION
Four supplied checkers (filter, recursive purification, correction and global composition) were inspected and each replayed twice in clean directories. All exits were zero with paired stdout agreement. The checks really calculate scalar, matrix or finite classical quantities. They do not implement and execute the full asymptotic quantum algorithm.
My independent 4-by-4 unitary-dilation control uses a nonscalar filter with singular values 0.21 and 0.63 and a nontrivial child input. At amplification levels 0,1,2,3 it makes 1,3,9,27 filter calls and checks the full matrix recurrence, failure (1-p)^(3^j), and successful-direction preservation up to phase. The final computed failure 0.0295889725734228 matches the recurrence to the stated 1e-13 tolerance. This floating diagnostic is not a certified arbitrary-depth theorem; the analytic recurrence carries that conclusion.

LITERATURE AND NOVELTY JUDGMENT
Source entries M05-S1 through M05-S6 identify exact primary theorem interfaces. The relevant contribution is completing the charged implementation, with normalization, recursive child calls, correction oracle, input arithmetic and global classical assembly all in one bound. Recursion, QBP, fixed-point amplification, interaction-picture evolution, locality, DS mixing and stacked Lindblad simulation must be attributed.
In particular, the closest source already proposes the recursive/collective strategy. Presenting that architecture itself as newly discovered would be misleading. Its detailed cost gap is the gap the candidate fills. Older recursive preparation and quasi-linear MPO construction do not already give the same N polylog size dependence with this temperature accounting.
No specific unverified oracle assumption, source-hypothesis mismatch or unresolved equivalent result remains. Ordinary search non-exhaustiveness is the remaining novelty limitation, not a blocking source gate.

NEXT JUSTIFIED ACTION
Retain the unchanged candidate for release assessment. Any manuscript must display the polynomial b overhead, the O(x) logarithmic exponent and the state-preparation scope, and clearly identify the implementation contribution relative to Remark 3.12. This review does not submit, publish or change catalog status.

