[unpaidmoon]

Change and mathematical scope

Scoped proof, with an exhaustive finite census. 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 length-30 word, a multiplicity-2 sojourn of rank 4 cannot have exactly 16 distinct layer-0 projections. Inside the stated window of 8 to 14 layer-1 projections, every such sojourn therefore has at least 17 layer-0 projections and at most 13 layer-1 projections. The 14-point layer-1 boundary is empty.

The sixteen-point nonexistence is a proof. Its finite step is an exhaustive census of the 120^7 normalized two-point fiber tuples, after fixing the identity fiber as {0, 1}. The prune is proved sound in PROOF.md. The census finds no cover: 57,685,490 nodes, 0 solutions. This is not a heuristic search, and it is not a proof that no universal length-30 word exists. It does not change f(8).

The same pair and fiber inequalities also admit |Q1| = 6 with |Q0| = 24. That case sits outside the 8-to-14 search named in the source question and is not censused. Ranks 5 and 6, and length 28, are not examined. The length-32 Gray concatenation remains a universal upper example. The recorded coverage-255 word, missing only mask 131, has a rank-4 sojourn with 21 layer-0 projections and 9 layer-1 projections, so it still meets the new floor.

Task and overlap

Base commit: 17469fff78c22941280f04d39dc5a1faee09c840.

Session r150, the length-30 rank-at-least-4 sojourn search. The assigned item is open question 1 of upstream pull request 7015. The first step there is to fix one rank-4 multiplicity-2 source whose layer-1 projection has between 8 and 14 points and search the complementary arc with at least 16 layer-0 projections. This package censuses the size-16 boundary of that arc. It does not rerun the rank-3 obstruction, 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 settles only the 16-point rank-4 layer.

Nearest work:

• Record that rank 3 blocks a universal length-30 singleton word #7015 blocks rank 3 and records the pair floors 16, 15 and 12 at ranks 4, 5 and 6. Its prose describes the rank-4 window as 8 to 14 layer-1 points. The inequalities also admit layer-1 size 6; this package uses the inequalities and does not treat size 6 as excluded. The size-16 boundary of the stated window is the part settled here.
• Record length-30 sojourn obstructions for eight singleton sources #6943 records the low-rank, bent, and 4+4 obstructions and the coverage-255 witness.
• Prove length-28 double sojourns use at least four sources. #6966 proves the length-28 double-sojourn rank floor. Length 28 is not revisited.
• Research: exact seven-source singleton minimum f(7) = 24 #3058 leaves the eight-source question open and proves f(7) = 24, which is not used here.
• Prove an eight-source singleton floor of 28 calls (no new κ) #5234 proves 28 <= f(8) <= 32.
• 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.

Upstream main was re-fetched at session start and was still 17469fff78c22941280f04d39dc5a1faee09c840. An open-PR search for a length-30 rank-4 layer census found #7015 and no package that decides this boundary.

Validation

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

python3 -B research/length30-rank4-layer-floor/verify.py
python3 -O -B research/length30-rank4-layer-floor/verify.py
python3 -B scripts/research_package.py --package research/length30-rank4-layer-floor
python3 -B scripts/check_research_commit.py --package research/length30-rank4-layer-floor

Observed:

• PASS_LENGTH30_RANK4_LAYER16 solutions=0 nodes=57685490 branch1_nodes=515051 layer0_floor=17 layer1_ceiling=13 witness_coverage=255
• PASS_REGISTERED_RESEARCH_PACKAGE research/length30-rank4-layer-floor outcome=partial
• PASS_RESEARCH_COMMIT_BYTES at 94b36d4649b5d29829959409acbc1906ef0b4379

The C census and a nested Python recount of second-fiber index 1 agree on 515,051 nodes and zero solutions for that branch. A fixed eight-fiber fixture partitions the odd quotient directions and misses the even ones. The coverage-255 witness, a swapped-pair failure control, the normalized rank-4 layers of that witness, and the Gray concatenation of length 32 are checked. f(7) = 24 was not replayed. The seventeen-point fiber census, ranks 5 and 6, and length 28 were not searched. No Lean, Docker, or selected cold replay was run. The source-history preflight was not run: the checkout is shallow, and that preflight requires full Git history.

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; that theorem is not used. 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. Open pull request 7015 for the rank-3 blockage and for the rank-4 window whose sixteen-point boundary is settled here.

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 rank-4 multiplicity-2 sojourn has at least 17 layer-0 projections and layer-1 size 6 or 8 through 13, while ranks 5 and 6 keep the floors 15 and 12? A yes answer would lower the published upper bound on f(8) from 32 to 30. The coverage-255 witness still meets the tightened floor and misses only mask 131. First step: census one rank-4 layer of size 17, whose fiber shape on the 8-to-13 side is one triple and seven pairs, or search one complementary arc in that range. The size-6 layer forces exactly 24 layer-0 projections and was outside the 8-to-14 search. Blocker: at size 17 each direction has 18 sums available against 16 targets, so the exact packing used here stops. Length 28 is a different question.

Workflow feedback

The quotient-fiber packing makes the size-16 boundary a single finite census, and an incremental C search agrees with a nested Python recount on the locked branch. No further session report.