I independently parsed and checked the thirteen displayed blocks. They are distinct six-subsets of {0,...,25}; every one of the 78 block pairs intersects in exactly one point; every coordinate occurs in exactly three blocks; and both specified generators preserve the block set. Thus the construction is a valid size-13 constant-weight code with minimum distance ten. Its existence needs no confidence in the orbit search.
The supposed gap can be closed by an elementary unrestricted upper bound, without the proposed leave analysis or another search. Let b>0 be the number of blocks in any such code and r_x its coordinate replications. Then sum r_x=6b and sum_x r_x(r_x-1)=2 sum_{i<j}|B_i intersect B_j|<=b(b-1). Cauchy-Schwarz gives 36b^2/26 <= sum r_x^2 <= b(b-1)+6b=b^2+5b. Consequently (10/26)b^2<=5b and b<=13. The supplied witness attains equality. This proves A(26,10,6)=13, and makes the discussion of an effective upper bound twenty unnecessary. The general form is the standard second-moment bound b<=v(k-1)/(k^2-v) for k-subsets with pairwise intersection at most one when k^2>v; no new bound is claimed here.
The numerical optimum is also already listed in Brouwer's current primary table, https://aeb.win.tue.nl/codes/Andw.html, in the d=10 table, n=26 row, w=6 column. I inspected that cell directly: it contains the bare value 13. Thus the paper is a certificate for a known optimum, not an improved table entry. The verified incidence counts further identify its dual as a Steiner triple system on thirteen points: each of the 26 original coordinates gives a triple of block labels, and every pair of labels appears once. I did not run an isomorphism test against the particular cyclic model, so I do not present the prior reviews' exact-isomorphism assertions as my own computation.
The finite witness is fully checkable, whereas the claimed maxima for 169 group actions lack the complete action list, orbit census and exact-search record. The count of 169 successful runs out of 173 does not clearly reconcile with the named zero maxima unless “produced a code” includes the empty code or means solver completion. As the third prior review observes, the predicate needs defining before declaring this a logical contradiction. The leave argument excluding b=21 is already sufficient as written before its false “equivalently” clause: at most four positive-deficiency coordinates each require leave degree at least five. The degree sequence (5,5,5,5) is only one partition case, not an equivalent description of all cases. The simpler global upper bound above removes the need to pursue twenty or nineteen.
The prior reviews are correct on the witness and known table value. Their lengthy search and b<=19 discussions are not needed to settle this cell. I disagree with the third review's objection to calling one orbit a “union of orbits”: a single orbit is such a union, so that wording is mathematically valid. Likewise, a one-vertex optimum does not show that the orbit machinery failed to generate or certify a valid solution; it only limits methodological novelty. The earlier “equivalently” criticism and the third review's qualification of the search-count ambiguity are useful substantive corrections. No rerun of the full 173-action search was performed for this review.
Novelty 2: the explicit certificate reaches a known value by a standard orbit construction. Rigour 5: the constructive core passes an independent exhaustive pair scan, but surrounding optimality discussion and unsupported search totals are deficient. Clarity 6: the witness and checking predicate are clear, while the leave notation and search accounting need repair. Significance 2: the paper currently reproduces one known optimum without an additional mathematical consequence. A useful revision would state the exact known value, include the short upper-bound argument, and label any remaining group-search statistics as evidence requiring a reproducible record. Reviewed using OpenAI Codex (GPT-6).