[unpaidmoon]

Change and mathematical scope

Scoped partial obstruction. No new κ and no selected-frontier change. Selected κ stays 781451/250000000.

In the restored binary singleton model, 28 <= f(8) <= 32 is unchanged. For a closed eight-source word of length 30, every admissible multiplicity vector has a source of multiplicity 2. This package proves:

• Outside a rank-r sojourn flat, Q0 + Q0 must cover 128 - 2^r nonzero vectors, so |Q0| is at least 16, 16, 15 and 12 at ranks 3, 4, 5 and 6. Rank 7 has no pair floor.
• Rank 3 forces the layer-1 projection to be the whole 3-flat. The complementary arc then needs at least four height steps, leaving only the shapes (a, q, k) = (7, 16, 5) and (8, 16, 4).
• An exhaustive placement search on the 108 canonical 4-cube walks misses at least two vectors outside the 3-flat. The minimum is 2, so no such word is universal.

Combined with the cited rank-at-most-2 obstruction, every multiplicity-2 sojourn of a universal length-30 word has rank at least 4. The rank-3 half does not use that citation or the bent census. Existence at rank at least 4 remains open. Heuristic misses are not offered as a nonexistence proof. Length 28 is not treated. The length-32 Gray concatenation remains a universal upper example. The recorded coverage-255 word, missing only mask 131, has sojourn ranks 6, 4 and 7 and still meets the new floors.

Task and overlap

Base commit: 7c49fe32c55de02c2c873655c804b16e4c3fef0a.

Session r119, the length-30 arc under the sojourn constraints. The assigned item is discussion #1166 comment 19377982. The predecessor record is upstream pull request 6943, head c1cf726df3a06e50ef655ae91e8c8aac942b19a7 (fork pull request 154). Its open question is this search. This package does not rerun the rank-at-most-2 obstruction, the bent census, the 4+4 prefix-set argument, or the length-28 sharpness question.

Card connector-alphabet-change, priority revision 8. Scope status: changed. The card asks for one new connector operation. This package stays in the published singleton alphabet and closes only the rank-3 case of the length-30 question.

Nearest work:

• Record length-30 sojourn obstructions for eight singleton sources #6943 records the low-rank, bent, and 4+4 obstructions and leaves rank at least 3 open. This package starts there.
• Research: exact seven-source singleton minimum f(7) = 24 #3058 leaves the eight-source question open and proves f(7)=24, cited and not replayed.
• Prove an eight-source singleton floor of 28 calls (no new κ) #5234 / research/eight-source-singleton-floor proves 28 <= f(8) <= 32. Length 28 is not revisited.
• Research: projection matching obstruction for eight-source singleton lifts (partial result) #5430 is the fixed-projection lift obstruction. This split does not reuse it.
• Research: six-source universal dirty-connector boundary, f(6)=16 (no new kappa) #1185 supplies the length-32 Gray upper bound, checked here as a positive control.
• Open-PR search on 2026-11-10 found Record length-30 sojourn obstructions for eight singleton sources #6943 and no other length-30 arc-search package. The latest top-level comments visible on discussion Handoffs and requests for independent checks #1166 run through comment 19481299 (2026-11-09). Comment ids 19377982 and 19492995 did not resolve through the API at submission time; Record length-30 sojourn obstructions for eight singleton sources #6943 is the public predecessor record.

Validation

CPython 3.12 and gcc, standard library, from the repository root:

python3 -B research/length30-arc-sumset/verify.py
python3 -O -B research/length30-arc-sumset/verify.py
python3 -B scripts/research_package.py --package research/length30-arc-sumset
python3 -B scripts/check_research_commit.py --package research/length30-arc-sumset --check-source-history

Observed:

• PASS_LENGTH30_ARC_SUMSET rank3=none best_miss=2 walks=108 floors=16,16,15,12
• PASS_REGISTERED_RESEARCH_PACKAGE research/length30-arc-sumset outcome=partial
• PASS_RESEARCH_COMMIT_BYTES at c5b463fe80dd0bcc234c12a6b1ed688ae855db06, source pin 7c49fe32c55de02c2c873655c804b16e4c3fef0a

The C search scores 224164584 placements in the five-height family and 9418680 in the four-height family. The canonical-walk digest is 3e51cc130c3e6c15, recomputed in Python from the 108 orbits of 2592 quotient walks. A fixed control word misses 60 outside vectors in both the packed model and the length-30 prefix layers. Height words of length at most 3 are exhausted on arcs of length at most 21; none spans F_2^3. The coverage-255 witness and a swapped-pair failure control are checked. The Gray concatenation of length 32 is universal. f(7)=24 was not replayed. Rank 4 and higher were not searched. The unpublished anneal was not replayed. No Lean, Docker, or selected cold replay was run.

Credit

Ross Richardson, #943, for the singleton model and prefix criterion. Douglas Colkitt, #1185, for the length-32 construction. conorholds, #3058, for f(7)=24 and the eight-source question. Nick Petalas, #5234 and discussion comment 19377982, for the deletion floor and the length-30 handoff. Nicholas Stone, #5430, for the fixed-projection lift obstruction that this split does not reuse. Upstream pull request 6943 for the rank-at-most-2 obstruction, the bent lower bound, the 4+4 argument, and the coverage-255 witness.

Prepared by unpaidmoon with substantial Cursor Grok assistance. New work is Apache-2.0. Independent human and proof-kernel review are unclaimed.

Open next questions

1. Does a universal length-30 word exist when every multiplicity-2 sojourn has rank at least 4, with at least 16, 15 and 12 layer-0 projections at ranks 4, 5 and 6? A yes answer lowers the upper bound on f(8) from 32 to 30. The coverage-255 witness meets these floors and misses only mask 131, so there is no one-mask counting contradiction yet. First step: fix one multiplicity-2 source whose open sojourn has rank 4 and whose layer-1 projection has between 8 and 14 points, then search the complementary arc with at least 16 layer-0 projections.
2. Does a universal length-28 word with multiplicities (4,4,4,4,4,4,2,2) exist? That is the sharpness question of Prove an eight-source singleton floor of 28 calls (no new κ) #5234 and is not considered here.

Workflow feedback

The quotient-coordinate relabeling reduces 2592 walks to 108 orbits, which makes the two rank-3 families a single exhaustive pass. No further session report.