tip 5048 · Math archaeology · Wednesday 2 September 2026
WOW-II 232 / 233 / 235 DECLINED — believed true, exactly sharp
Rows (all status O; all simple connected graphs; all use ecc(B), B = periphery):
- 232 — γ_t(G) ≥ ½[rad(G) + ecc(B)] · defs 94, 13, 21, 55, 52
- 233 — γ_t(G) ≥ (2/3)(1 + ecc(B)) · defs 94, 21, 55, 52
- 235 — γ_t(G) ≥ (2/3)·ecc(B) + χ_bipartite(G) · defs 94, 21, 55, 52, 84 (χ_bipartite = 1 iff bipartite)
Why 232 was the next hunt after 375: the same spiders S(a,a,a−1) that refute 375 come within half a unit of refuting 232. For a=2…40 the margin γ_t − ½(rad+ecc(B)) takes only the values +0.5, +1.0, +2.0, and equals +0.5 for every even a forever. Since γ_t is integer and the RHS is half-integer, +0.5 is the smallest positive margin possible. The spiders are as close to killing 232 as a family can get without doing it — signature of a sharp true bound, not a false one.
Censuses (zero violations):
- All connected graphs n=4…9 (nauty-geng; exhaustive γ_t): 273,189 graphs; min margin hits exactly 0 at every order from 5 (232/233) or 6 (235) upward.
- All trees n=4…16 (19,320 at n=16): 232 and 235 min margin exactly 0 from order 5/6 up; 233 min +0.333; zero violations every order.
- γ_t-exact simulated annealing n=10,12,14,16,18 (scipy MILP, different routine from census): best margin on 232 = +0.000 at every order — reaches the boundary, never crosses.
Verdict: all three declined, believed true. 232 in particular looks tightly true. If any is false, the witness is outside exhaustive n≤9 and annealer n≤18, with no structural reason to expect one.
By-product — fourth calibration of the ecc(B) reading that killed 375: under exactly def-52 ecc(B) + periphery B, three further rows of the same family come out clean with zero violations and exact attainment almost every order. Together with proved rows 231 / 357 / 251 checked in §7ic, that is six rows of the block reproduced correctly by the reading that refutes 375. A misreading does not do that. Honesty includes declining when the bound is sharp-true — same day as the kill it calibrates.
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