The empty cells of degree 24, with the field each one needs 1,926 cells still empty on the organizer's board, on 879 groups. Not one of them is refused by group theory. The leftover is an arithmetic problem throughout, and what sorts it is which tower a cell takes. Where the list comes from ------------------------- The leftover is the cells still empty on the organizer's discoveries board, not a subtraction from any one team's name list. The address of each empty cell comes from two exhaustive computations: the subfield lattice of all 25,000 transitive groups of degree 24, from a GAP block-system census the involution walk, packed as walktable.json.gz beside the reach screen: for each group, which base degree, base group and base signature delivers which real-root count Rebuild it against the organizer remaining list: python3 build_worklist.py --igp24 --site \ --remaining official_remaining.json The files --------- worklist.jsonl.gz One line per empty cell. Fields: t (the 24T number), r (the real-root count), job, and the bases. {"t":243,"r":0,"job":"quadratic","base_degree":12, "bases":[[41,12]],"w":[12],"other_base_degrees":[2,4,6]} bases are [base group, base real places] at base_degree. w is how many of the base's real places need a negative radicand. {"t":565,"r":24,"job":"other-tower","base_degrees":[2,4,6], "bases_by_degree":{"2":[[1,2]],"4":[[2,4]],"6":[[3,6]]}} worklist.json The same cells keyed by group then real-root count, with every base flattened to [base degree, base group, base real places]. One fetch, under a megabyte, for a browser or a one-line lookup. summary.json The counts, and the definition of each job. The three jobs -------------- quadratic 1,685 cells on 768 groups. The group has a subfield of degree 12, so the last step is a square root and its kernel is elementary abelian automatically. Get a base of the named group with the named number of real places, adjoin a square root, and aim the radicand negative at w of those places. r = 24 - 4s - 2w, where s counts the complex places of the base and w the real places where the radicand is negative. Equivalently r = 2(rho - w) for a base with rho real places, so w = rho - r/2. other-tower 235 cells on 108 groups. No degree-12 subfield, so no square root reaches these. Every one permits a base of degree 2, 3, 4 or 6, carrying a step of degree 24 over that base degree. Cells reachable from a degree-6 base number 220, from degree 2 167, from degree 3 118, from degree 4 12; a cell often appears in more than one. The step is not a quadratic, so its kernel is not abelian for free. That is the part nobody here has written, and it is where a new construction pays. no-opinion 6 cells on 3 groups. These sit on primitive groups, which have no proper subfield at all, so the screen has nothing to say. That is ignorance, not a refusal. What a listed base claims ------------------------- Permitted by group theory, and nothing more. Complex conjugation is an involution in the degree-24 group. Over a block system it fixes as many blocks as the subfield has real places, which fixes what it can fix upstairs. That law is exact and it is a refusal: a base absent from a cell's list cannot write that cell, whatever the arithmetic does. The allow direction is weaker. A base in the list means the group does not forbid the cell. It does not say a base field of that group and signature exists, that you can find one, or that the radicand can be aimed. That is arithmetic and this list does not settle it. The screen fails open. A group it has not censused, and the five primitive groups 24T7817, 24T10255, 24T24680, 24T24999, 24T25000, get no opinion rather than a refusal. A filter that refused what it had not measured would throw away exactly the work nobody has done yet. Empty cells as of the organizer's board, 20 August 2026. Stage 2 has not started. No group name in the published field table is a certified Galois-group computation. Same files: https://github.com/jonwashburn/galois Pages: https://recognitionphysics.org/igp24/worklist/