Pass 1505 enumerated the first 100,000 exact covers through one fixed canonical
frame and reduced them to 327 complete PSp(4,3) orbits, certifying at least
3,547,800 covers. Pass 1510 repeats the search with the candidate-frame order
reversed at every branch.
The two deterministic prefixes satisfy
[ |S_{\rm fwd}|=|S_{\rm rev}|=100000, \qquad S_{\rm fwd}\cap S_{\rm rev}=\varnothing. ]
Nevertheless, exact full-orbit traversal proves that each prefix independently hits the identical set of 327 complete group orbits. Canonical orbit representatives are frozen by SHA-256
223be23d50147437acfa18cc8f4cea43083c6b87066fe6e26812d0de50c8abb4
and the union still has
[ 327\text{ orbits}, \qquad \sum |\mathcal O|=3547800. ]
The stabilizer-order census remains
[ 2^{228},\qquad 4^{84},\qquad 8^{15}. ]
This is strong evidence that the deterministic search has saturated the currently visible orbit frontier. It is not a proof that no additional exact-cover orbits exist outside both prefixes.
The orbit set is stable, but the within-orbit frequencies are not. If
f_i,r_i are the forward and reverse prefix hits in orbit i, then
[ \sum_i f_i=\sum_i r_i=100000, ]
but only seven of the 327 pairs satisfy f_i=r_i. The exact redistribution is
[ \sum_i |f_i-r_i|=10244, \qquad \sum_i(f_i-r_i)^2=463520, ]
with maximum single-orbit difference 98. The Pearson correlation is
[ 0.9280509607335599, ]
computed from the exact integer components
[ 964601357, \qquad 994717730, \qquad 1086056024. ]
Thus branch reversal changes frequencies substantially even though it preserves the observed orbit support. DFS order is therefore suitable for existence and lower-bound certificates, but not for estimating a uniform distribution on covers or cover orbits.
forward ee6a429279fece6c4cd917acf2a07fdec2e9f8b66ebe9f7aa0db328ee6ed0172
reverse e28c3c6c7d5869f93b04c3fc34320f60e65383f82cb3c2484978f46e73bfca5d
The repository contains both compiled-search sources and the canonical orbit reducer. Routine CI validates the frozen certificate and regenerates a fast 100+100 smoke sample. A full 100,000+100,000 rerun is opt-in because the fully reversed DFS is much slower than the forward ordering.
This theorem certifies objectwise agreement of two disjoint long-run prefixes. It does not establish completeness of the global exact-cover census, uniformity of either DFS sampler, or any physical interpretation of the covers.