[petalas]

Result

Shortest unsigned word for the depth-five dirty chain connector, in a bounded range of characteristics. The calls are z_l += B_l z_{l-1} for l = 1..5. The source x = z_0 is immutable, z_1..z_4 and y = z_5 start dirty, and the maps are arbitrary. A correct word restores z_1..z_4 and adds c B5B4B3B2B1 x to y. This answers open question 2 of #4263 for depth five in this range.

m	shortest length	evidence
2	16 = 8m	exhaustive BFS over UT6(Z/2)
3	24 = 8m	exhaustive BFS over UT6(Z/3)
4	28 for c = 1, 3; 24 for c = 2	exhaustive BFS over 4^15 elements (opt-in run) plus explicit words
5 to 8	5m for (5,1), (6,3), (7,1..4), (8,1..4,6)	witnesses; the other 11 residues are a search failure, not a bound
9 to 20	5m for every c (162 pairs)	witnesses

• The depth-five premium is larger than at depth four: 8m at m = 2, 3, where depth four costs 6m (Prove the shortest unsigned depth-three chain connector in every characteristic (3m; 4m at c ≡ -1 for m ≤ 4; no new κ) #4249).
• At m = 4 the minimum depends on the residue c. Depth four has no such dependence there (Classify depth-four chain connector count vectors and unequal-price tariffs for m = 2..5 (no new κ) #4279's count vectors do not depend on c).
• The per-call floor 5m is attained for every residue from m = 9 to m = 20.

Claim types: exhaustive finite computation for the minima at m = 2, 3, 4, and a finite certificate for each witness. The m = 4 lower bound is an opt-in local run that hosted CI does not repeat. No all-m theorem, no new κ, no supplier admission and no selected-frontier change.

Only research/depth5-chain-connector/ is added, on base 12fa8c5c844be21bfefb38a19503fbd867a11244 (priority revision 7, card connector-alphabet-change, scope explicitly changed in TASK.json, as in #4249 and #4263).

Evidence and reproduction

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/depth5-chain-connector/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/depth5-chain-connector/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/depth5-chain-connector
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/depth5-chain-connector --check-source-history
# opt-in, about 9 minutes and 20 GB:
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/depth5-chain-connector/verify.py --exhaustive-m4

Locally (CPython 3.12.15, numpy 2.5.4, macOS arm64) both registered commands print PASS_DEPTH5_CHAIN_CONNECTOR bfs_cases=6 witnesses=173 m4_words=3 in about 7 s each, with a receipt bound to every package and input SHA-256. PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES pass. The --exhaustive-m4 run completed locally in 8 min 52 s and asserted the distances {1: 28, 2: 24, 3: 28}.

The verifier runs BFS over all of UT(K+1, Z/m): depth five at m = 2, 3, and depth four at m = 2..5 as a positive control (it reproduces #4249's 12, 18, 20, 25). For every witness it checks the length 5m, exactly m copies of each call, the interval subword counts, an independent UT6(Z/m) matrix product, and two literal replays with dirty registers and random 2x2 maps over Z/m. It checks the three m = 4 words the same way, requires full rows for 9 <= m <= 20, and runs five rejection controls.

search_witnesses.c (simulated annealing) found the 5m witnesses. The m = 4 words were read back from the BFS distance table. Neither search is part of the replay; the verifier checks the words themselves.

Limitations and unrun obligations

• The m = 4 lower bounds rest on the opt-in exhaustive run, which hosted CI does not repeat.
• The 11 residues at m = 5..8 without a 5m witness, and every m > 20, are not decided.
• Independent human and formal proof-kernel review are unclaimed.
• Every call carries its complete price. Admission, dispatch, routing of the four intermediates, precision, supplier caller and the outer multiplication theorem are not supplied.

Open next questions

1. What are the exact depth-five minima at m = 5..8 for the 11 residues with no 5m witness? A meet-in-the-middle search over balanced prefixes and suffixes could exclude 5m without the full m^15 BFS.
2. Is the depth-five minimum exactly 5m for every m >= 9?
3. Why does the m = 4 minimum depend on c? A 2-adic obstruction might be provable by hand and might recur at m = 8.

Related open work

• Attain the depth-four chain connector floor 4m for every 6 ≤ m ≤ 32 (no new κ) #4263: depth-four floor 4m for 6 <= m <= 32, and the source of this question (its open question 2). No shared number; different depth.
• Prove the shortest unsigned depth-three chain connector in every characteristic (3m; 4m at c ≡ -1 for m ≤ 4; no new κ) #4249: depth-three law and the depth-four minima at m = 2..5. This package's depth-four BFS control gives 12, 18, 20, 25, which agree.
• Classify depth-four chain connector count vectors and unequal-price tariffs for m = 2..5 (no new κ) #4279: depth-four count vectors and tariffs for m = 2..5. Its unit-price lengths 12, 18, 20, 25 agree with the control here. No depth-five number is shared.
• Attain the depth-four chain connector floor 4m for every 33 ≤ m ≤ 64, via an exact row model (no new κ) #4280: depth-four floor for 33 <= m <= 64. Different depth; no shared number.

CI notes

Hosted CI is green on head 87f4a21e02c83fab559dcf1470dd0a97dfb89b8f: run 39527014574. plan passed in 2m12s and the package replay passed in 1m45s. No check failed. CI runs only the registered commands; the opt-in --exhaustive-m4 BFS (the m = 4 lower bounds) is local-only evidence and was not run in CI.

Produced by Nick Petalas. Predecessor credit: #4249, #4263 and Ross Richardson's #943/#934 connector lineage. Apache-2.0.