Independent parent review of M08-v1 — 2026-10-21

Decision: supported internally; complete with respect to the exact original
M08 task. No blocking mathematical or certificate gap was found in this review.
This is a second agent review within the same model family, not an external
referee decision or a claim of established global novelty.

Reviewed inputs
The frozen candidate is campaign/continuation-2026-10-21/candidates/M08-v1.
inputs.json records the seven checked input SHA-256 hashes. All calculations
were replayed in replay-input/, leaving the frozen candidate unchanged.
The original statement is campaign/originals/M08.tex, whose hash agrees with
the candidate's original.tex and claim.json. The assertion is Q*=Qswap for
two qutrit collisions with gaps in [1,3], g1+g2<=4, arbitrary joint unitaries,
classical randomization, exact unconditioned final target p(2), and gross
positive work at most 2. It is not a general-dimensional optimality theorem.

1. Quantum and randomized protocols
The optimized second-stage terminal support is an infimum of affine functions
of the target state, hence concave, and is invariant under target input
unitaries. After diagonalizing the first target marginal, Birkhoff reduction
therefore gives a valid lower bound at permutation vertices. The first
machine heat is linear in the joint diagonal. Relaxing the work constraint
enlarges the minimization set and is in the safe direction for this lower
bound. The terminal multiplier is affine in the final diagonal, so averaging
the branchwise bound covers randomized protocols whose unconditioned final
state is p(2), without requiring each branch to reach p(2).

2. The 280 partitions are complete
For each grouping of nine initial masses into three unordered triples,
passive sorting within each row minimizes first-machine heat without changing
the row sums. Row labels do not change the optimized second stage. Thus the
number of necessary representatives is 9!/(3!^3*3!)=280. This also remains
valid when input masses coincide. As an implementation control, an independent
full 9! enumeration at four gap pairs agreed with the reduced oracle to
within 2.1e-15. These binary64 controls are diagnostic only; the continuum
claim relies on the analytic reduction and certified bounds below.

3. Semiconcavity direction and numerical constants
For the qutrit Gibbs law, ||p'||_1<=1 and ||p''||_1<=2. For each fixed pair
of permutations, the coordinate second derivatives of the terminal support
are at most 28 and 22. In the terminal linear terms, centering a coefficient
vector around its midrange is justified by sum_j p''_j=0. This yields a bound
of 12 for the first-coordinate terminal term and lambda0<6 for the second,
which explains the constants without using an unjustified absolute-value
estimate. Subtracting the respective quadratic leaves a concave function;
the pointwise minimum of concave functions is concave. Consequently a box
of widths h1,h2 has lower bound equal to its minimum corner value minus
(28*h1^2+22*h2^2)/8. The inequality has the required lower-bound direction
and survives permutation-ordering changes.

4. Integer input error and interval enclosure
The common denominator is the lcm of rational gap denominators; the integer
objective has scale D*S^4, S=2^60. Flooring the primitive Gibbs inputs and
lambda0 gives a uniform 127/S error bound for every permutation branch:
54/S for first heat, 55/S for the terminal term, and 18/S for second-bath
heat. A 1000/S subtraction covers this error before minimization; ties and
changes of the minimizing branch do not invalidate a uniform bound.
The root ball encloses the exact rational bisection endpoints, with the
required derivative signs. All 40 coordinate intervals used for the local
Hessian grid enclose their exact rational endpoints. These containment
checks were performed independently in targeted_check.py.

5. Continuum cover and local minimum
The full replay passed. The split tree closes with 1006 remote leaves and
18 local leaves; no uncovered feasible interior remains. The g1+g2=4 edge
is obtained by continuity from the feasible interior. Remote lower margin:
2.7066193286321033e-6. The local box is [7/5,3/2] x [39/20,41/20]. Its
400 interval Hessian cells certify h11>=0.652338038363607 and determinant
>=0.20568643499388276. The active swap branch is stationary at (a*,2) and
strictly convex throughout that box. A separate 31-leaf cover bounds all
279 competing partitions strictly above it, with margin at least
5.714976218718871e-6. Thus this is not a finite-grid inference.

6. Attainment and original constraints
The exact root has a*=1.4459206660687878334... in (1,2), so the swap protocol
has admissible gaps (a*,2), reaches the exact target, and obeys the total-gap
constraint. Both stage works are positive; their gross sum is
0.1890623706570403958...<2. Its heat is
Qswap=0.4647890802734670691..., attaining the universal lower bound.

Reproduction evidence
replay.json: complete candidate replay, 200-bit Arb / exact integer arithmetic,
including a direct Arb control and a deliberately raised support-floor
negative control. targeted-results.json: exact endpoint containment and four
full-permutation controls. Both report pass. python-flint 0.9.0 / FLINT 3.6.0
was loaded from the existing strict-review dependency directory.

Source comparison and limits
The archived arXiv:2404.06649v2 source states a swap-optimality result for
qubit targets and machines in Appendix B.1; the present fixed qutrit task
requires the additional certificate reviewed here. Primary source:
https://arxiv.org/abs/2404.06649v2 . This comparison is not an exhaustive
novelty search. External review, broader novelty assessment, and any extension
beyond the stated two-collision task remain outside this decision. The frozen
candidate has not been edited or published.
