[unpaidmoon]

Change and mathematical scope

Scoped obstruction with an exhaustive cube census. No new κ. Selected κ stays 781451/250000000.

In the restored binary singleton model, a universal closed length-28 word with multiplicities (4,4,4,4,4,4,2,2) must give at least one multiplicity-2 arc rank 7. If the arc ranks are ordered r ≤ s, then (r, s) is one of (4,7), (5,7), (6,7), and (7,7). When the smaller rank is r ∈ {4,5,6}, that arc leaves at least b(r) steps from its own support on the complementary arc, where b(4)=6, b(5)=10, and b(6)=14. These are the lengths of the shortest walks on the r-cube whose vertices form a difference basis of F_2^r. Explicit walks meet each length.

This does not construct a universal length-28 word, does not prove f(8) > 28, and does not price the #5234 batch minimum. The four surviving rank pairs remain open.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840. Card connector-alphabet-change, priority revision 8, scope changed: the alphabet stays the published singleton alphabet, and the new statement is the flat-step budget left open by #6966.

The prescribed first step was to fix one multiplicity-2 source, require both arcs to use at least four labels, and write the complementary flat-step budget before searching. This PR stops at that budget. It does not rerun the rank-at-most-3 argument of #6966 or the length-30 question of #7015.

Targeted open-PR search on 2026-11-10 for length-28, flat-step, sojourn, and rank-7 found #7015, which is a different length. No open PR states these budgets. Closed-PR coverage is through #5234, #6966, and #7015, not a full closed-PR index.

Validation

CPython 3.12.3, standard library. Both registered commands passed:

PASS_LENGTH28_FLAT_STEP_BUDGET b=6,10,14 q4=1024 q5=2343750 q5_equal=113400 q6_nodes=25232587 forbidden=3969 open=(70, 42, 14, 1)

python3 -W error -I -B research/length28-flat-step-budget/verify.py
python3 -W error -I -O -B research/length28-flat-step-budget/verify.py
python3 -B scripts/research_package.py --package research/length28-flat-step-budget --check-only
python3 -B scripts/check_research_commit.py --package research/length28-flat-step-budget --check-source-history

The cube checks are exhaustive finite censuses of the stated walk domains. The length-13 count assignments use a proved new-vertex bound; that bound accepts every prefix of the three positive walks. Not run: a search of the four surviving rank pairs, a replay of f(7)=24, Lean, Docker, and the selected cold replay.

Credit

Ross Richardson, #943, for the singleton model. Douglas Colkitt, #1185, for the length-32 Gray upper bound, not replayed. conorholds, #3058, for f(7)=24, not replayed. Nick Petalas, #5234, for the length-28 floor and the multiplicity shape. The rank-at-most-3 obstruction and the open budget question are unpaidmoon's #6966.

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 realize arc ranks (4,7), (5,7), (6,7), or (7,7)? A yes answer shows the floor 28 is sharp. First step: fix a rank-4 support, place at least six complementary steps from that support, and test one fiber walk on the three outside sources. Blocker: those pairs satisfy the occurrence budget, and no universal length-28 word is known.
2. Does such a word decide whether the Prove an eight-source singleton floor of 28 calls (no new κ) #5234 batch minimum is 538, 540, or 542? Blocked on question 1.

Workflow feedback (optional)

The #6966 open question asked for the flat-step budget before a word search. Writing that budget as a cube-walk length settled every rank pair in which both arcs have rank at most 6, so the search domain is now the four pairs above. It would help if an open-question first step named the cube dimension of the flat walk.