[unpaidmoon]

Change and mathematical scope

Scoped written proof with finite corroboration. 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-28 word with multiplicities (4,4,4,4,4,4,2,2), if the prefix set satisfies P+P = F_2^8, then each of the four arcs determined by the two multiplicity-2 sources uses at least four other sources.

Rank at most 2 is a product bound: the arc image has size at most 4, so the cross sum has size at most 96. Rank 3 splits on the other double source. Three quadruple sources leave only 14 outside steps, which visit at most 15 of the 16 quotient cosets. The other double plus two quadruples leaves at most three flat steps on the complement, hence at most four flat coordinates, and a 4-point set in F_2^3 has at most 7 pairwise sums. This does not construct a universal word and does not prove that none exists. Arcs of rank at least 4 remain open. A failed search is not a nonexistence proof.

Task and overlap

Base commit: 7c49fe32c55de02c2c873655c804b16e4c3fef0a.

Session r125. The question is discussion #1166 comment 19377982 as left open by the length-30 handoff, and open question 2 of the length-30 finding: whether a universal length-28 word with these multiplicities exists. This package proves a necessary arc constraint. It does not take the locked length-30 existence question.

Card connector-alphabet-change, priority revision 8. Scope status: changed. The card asks for one new connector operation. This package keeps the published singleton alphabet. The length-28 arc constraint is the added statement.

Nearest work:

• Research: exact seven-source singleton minimum f(7) = 24 #3058 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 and the multiplicity shape, and asks whether a 28-step word exists. This package starts from that question and does not revisit the 538-query batch bound.
• Research: projection matching obstruction for eight-source singleton lifts (partial result) #5430 excludes one fixed 24-state projection from a 28-step lift. The arc argument does not reuse that matching obstruction.
• Construct the shortest 36-step lift of the fixed PR5430 projection #6497 attains a 36-step lift of that same projection, with counts (4,4,4,4,4,2,2,12).
• Record length-30 sojourn obstructions for eight singleton sources #6943 records length-30 sojourn obstructions and leaves length 28 open. This is the length-28 rank bound, not that length-30 result.
• 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.

Overlap search at 2026-11-10T02:22:37Z: 895 open PR records and 6934 discussion records, complete pagination; local head inventory at 7c49fe32 was complete (8919 note records). Terms: 4,4,4,4,4,4,2,2, length-28, sojourn rank. Closed PR bodies are covered through local notes and the predecessors named above, not a full closed-PR index. No match is a novelty certificate.

Validation

CPython 3.12.3, standard library, from the repository root:

python3 -W error -I -B research/length28-sojourn-rank/verify.py
python3 -W error -I -O -B research/length28-sojourn-rank/verify.py
python3 -B scripts/research_package.py --package research/length28-sojourn-rank
python3 -B scripts/check_research_commit.py --package research/length28-sojourn-rank --check-source-history
python3 -B scripts/research_selected_baseline.py

Observed:

• PASS_LENGTH28_SOJOURN_RANK subsets=162 paths=320 worst_missed=7 gray=256
• PASS_REGISTERED_RESEARCH_PACKAGE research/length28-sojourn-rank outcome=completed
• PASS_RESEARCH_COMMIT_BYTES at 8f66b0f936d93a3c2fad756b488d4a1100582c1d, source pin 7c49fe32c55de02c2c873655c804b16e4c3fef0a
• PASS_RESEARCH_BASELINE_BINDINGS_ONLY (bindings only, not a scientific replay)

The subset census is all 162 nonempty subsets of F_2^3 of size at most 4. The path census is all 320 walks of length at most 3 on three flat bits, from every start. The five-point set {0,1,2,3,4} covers F_2^3, so the four-value bound is specific. A relabelled Gray step and a deleted Gray step are rejected. f(7)=24 was not replayed. No search of the remaining rank-at-least-4 words, no Lean, no Docker, and no frontier6030-verify.

Credit

Ross Richardson, #943, for the singleton model and prefix criterion. Douglas Colkitt, #1185, for the length-32 Gray construction. conorholds, #3058, for f(7)=24 and the eight-source question. Nick Petalas, #5234 and discussion comment 19377982, for the deletion floor, the multiplicity shape, and the open sharpness question. Nicholas Stone, #5430, for the fixed-projection obstruction that this split does not reuse. #6943 for the length-30 sojourn obstructions on a different length.

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-28 word exist when every multiplicity-2 arc uses at least four other sources? A yes answer shows the floor 28 is sharp. First step: fix one multiplicity-2 source, require both arcs to use at least four labels, and write the complementary flat-step budget before searching. The rank-3 budget of at most three flat steps does not apply, and {0,1,2,3,4} shows five coordinates can cover F_2^3.
2. Does that word, if it exists, decide whether the Prove an eight-source singleton floor of 28 calls (no new κ) #5234 batch minimum is 538, 540, or 542? First step: price one such word. No length-28 universal word is known.

Workflow feedback

The coordinate split separates a short counting obstruction from the remaining rank-at-least-4 search. No further session report.