M08-v1 — independent strict review, 22 October 2026

Judgments
Validity: supported internally, with a computer-assisted global certificate.
Original coverage: complete positive resolution for the exact two-qutrit,
two-collision task, gaps in [1,3] with sum<=4, arbitrary U(9) controls,
unconditioned final tau2, and branchwise work budget. Classical randomization
and branch-dependent gaps are covered. No arbitrary-dimensional or arbitrary-
length cooling schedule theorem is claimed.
Source comparison: difference supported in the bounded current comparison;
no specific unresolved covering theorem found. External status: no external
expert acceptance established and no publication/submission performed here.

Target and difficult proof bridges
Original campaign/originals/M08.tex:7–23. The full proof.txt,
qutrit_exact.py, qutrit_local.py and check.py were read; frozen hashes are in
frozen-inputs.json. The exact a* is the unique root of q', not its decimal
display, and Q*=q(a*)=0.464789080273467... .
1. proof.txt:12–30: q''>0 on [1,2] proves the global swap benchmark. The
terminal affine witness Lambda=(lambda0,3,0) supports the exact target. A
branchwise inequality Q-Lambda.diag(rho_final)>=z* can be averaged, so
randomized branches need not individually reach tau2. Dropping the work
constraint is a safe lower-bound relaxation.
2. Lines 32–64: the second-stage optimized support is concave and invariant
under target input unitaries. Diagonalizing the first target marginal leaves
machine heat unchanged. The joint diagonal lies in the Birkhoff polytope;
concavity bounds it from below by a permutation vertex. Second-stage
rearrangement is exact. Row sorting is optimal for first heat and leaves
unordered row sums unchanged, giving exactly 280 label partitions. Thus the
finite problem covers quantum unitaries rather than assuming them classical.
3. Lines 66–83: the separate curvature ceilings 28 and 22 hold for every
fixed pair of permutations throughout the square. Subtracting the quadratic
makes each branch concave; the pointwise minimum remains concave. Bilinear
interpolation therefore supplies a lower bound even at ordering crossings.
4. Lines 85–112 and the exact checker: the dyadic cover reconstructs every
branch, rejects missing feasible cells and uses continuity on the budget
boundary. Every remote corner is recomputed, not read from search scores.
Thermal/multiplier floors at scale 2^60 are enclosed by Arb. The stated
127/S value-error budget is dominated by the subtracted 1000/S; integer
minimization/sorting preserves the uniform error bound.
5. Lines 114–132: inside B0 the designated partition is a full swap with
verified ordering. Its gradient vanishes exactly at (a*,2). Interval Hessians
on 400 rational cells are positive definite; strict convexity proves the
local lower bound. The other 279 branches are certified above z* on a
separate 31-leaf cover. They need only exceed z*, not the swap value at each
point. Combined with the 1006 remote leaves and 18 local cover leaves, this
proves the global support inequality.
6. Lines 134–143: the two swaps are feasible via principal logarithms under
the stated time/norm budget, reach tau2 exactly, obey the gap budget, and use
gross work 0.189062370657040...<2. They attain the relaxed lower bound and
therefore solve the original constrained infimum.

Primary-source comparison
• Taranto et al., 2404.06649v2, Appendix B.1: general optimal-unitary
swap arguments there are for qubits. The higher-dimensional thermodynamic
geodesic discussion optimizes a swap-based path and does not prove the exact
arbitrary-unitary two-qutrit optimum considered here.
https://arxiv.org/html/2404.06649v2
• The finite-molecule thermal-operation source 2505.10054, §II and Theorems
2/3, has energy-preserving collision restrictions and a different cooling
objective (see M06-review.txt); it does not cover this task.
https://arxiv.org/html/2505.10054
• Canzio et al., 2610.27948v1 (20 October), Eqs. (2)–(9), Proposition 1
and the qutrit example: slow-driving GKSL dissipation/thermodynamic length.
This fresh qutrit result has neither the two finite machines nor the exact
two-unitary heat minimization. https://arxiv.org/html/2610.27948v1

Reproduction, independent controls, manuscript
Both clean copies re-evaluate the complete global certificate successfully:
1006 remote leaves, local Hessian and competitor bounds, and a deliberately
raised support floor rejected. Newly written independent_controls.py imports
no candidate implementation: it enumerates all 362880 first-stage permutations
and independently groups their triples into 280 representatives. Four fixed
gap diagnostics give identical minima to float64 precision. One point is
outside the original gap domain as an algebra stress control; none of these
finite floating diagnostics substitutes for the interval global proof.
The exact root/thermal enclosures, rounding rules, local Hessian and cover
logic were inspected separately from the returned PASS values.
The manuscript M08-qutrit-heat/main.tex theorem (76), quantum reduction (132),
semiconcavity lemma (199), global certificate (227), local proof (260), and
attainment (330) match the checked claim. No mismatch in randomization or
work conventions found. Retain with its exact-task scope and computational
certificate attached; do not describe it as a general qutrit-schedule result.
