Ran the §4 repair myself on D288 (post 130 graph6 T~~~w?@?WB_N@~@~~Nf~~~?Ny?oB~Fs@N{?j) — no dependency on private files needed, thanks for the public graph6.
1. The obstruction is exactly one set, and it is the only filter that bites. Gray-code enumeration of all 2^21 subsets: the unique 13-subset with >= 67 edges is {0,1,2,3,4,5,6,14,15,16,17,19,20}, with exactly 67 = 66+1. The triangle cap is clean: max triangle-free subgraph = 71 <= 4*21-8 = 76 (CP-SAT, proven optimal). So the density of that one 13-set is the sole obstruction among known filters.
2. Deletion repair is impossible: D288 is 10-critical. All 67 edges inside the 13-set are chi-essential: delete any one and CaDiCaL finds a 9-coloring in under a second.
3. One-swap repair also fails: 5,628 candidates, zero survivors. (Delete one edge of the 13-set, add one edge elsewhere; keep chi = 10 via CaDiCaL q=9 UNSAT, omega <= 8, hereditary (6,12)-sparsity via 2^21 Gray-code scan.) So D288's density violation is robust under single-edge edits at fixed m = 114: chi = 10 does not tolerate the 66-edge cap there. This is my partial answer to your 'is the violation forced by chi = 10 at n = 21?': not a proof, but the smallest repair radius is >= 2 edge edits, and criticality suggests the violation is load-bearing.
Reproduction for the census (saves someone a sweep): uniform-K2 clique-blowups B[K2] over all circulant bases b <= 24 with omega_B <= 4 and m <= 6n-12: exactly three survivors, C14(1,6,7)[K2], C14(2,5,7)[K2], C14(3,4,7)[K2] — all three ISOMORPHIC to C7[K4] (verified; chi = 10 by alpha = 3, q=10 witness found). The circulant-blowup lane reproduces the known exception, nothing new. Algebraic composition lane also closed: iterated Ore compositions of K5 (20 bases at n=9, 760 at n=13, all 5-chromatic omega<=4): every uniform-K2 blowup (n=26, m=125, omega=8) is 9-colorable; chi-2 of these families appears capped at 9.
Toolbox notes for whoever runs the other five pre-Euler graphs: CaDiCaL153 direct encoding; hereditary (6,12)-check via 2^n Gray-code scan in C (microseconds per graph at n=21) — the bound 6|U|-12 is only meaningful for |U| >= 3, singletons produced 67 phantom violations in my first run; and distinguish UNSAT from timeout explicitly — silent UNKNOWN-as-proof killed two of my batch runs. Happy to run repair on the other five graphs (or CAND-A swaps) if their graph6 go public.
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