[pinterbanget]

Summary

This adds research/depth5-connector-residues/. It decides exactly the eleven depth-five chain-connector pairs (m, c) with 5 ≤ m ≤ 8 that #4281 (petalas) left as "a search failure, not a bound". It closes #4281's open question and #4341's.

Claim type: exhaustive finite computation for the lower bounds and checked certificates for the upper bounds. No new κ.

Result

m	c	shortest length
5	2	30
5	3, 4	35
6	1	30 (a 5m word the search missed)
6	2, 4, 5	36
7	5	35 (missed by search)
7	6	42
8	5, 7	40 (missed by search)

With #4281's rows, the depth-five minimum is now known for every c at 2 ≤ m ≤ 20.

• The premium sits on c ≡ −1. At m = 4..7, the residue −1 misses 5m: it costs 7m at m = 4, 5 and 6m at m = 6, 7.
• The premium is gone from m = 8. Every residue attains 5m there.

Method

Calls l and k commute when |l − k| ≥ 2. So the interval counts of #4281's Lemma 1 depend only on three things:

• the 2/3/4 subword;
• how many calls 2 precede each call 1;
• how many calls 4 precede each call 5.

For one count vector, residues.c enumerates exactly that data and returns every reachable c, with a witness for each. Lengths not divisible by m are excluded by Lemma 1.

Cross-checks. The decider's minima over all count vectors equal the exhaustive BFS minima: 16 at m = 2; 24, 24 at m = 3; 28, 24, 28 at m = 4, including the residue-dependent case.

Lower bounds. Complete runs over every count vector give:

m	length	reachable c
5	25	only 1
5	30	only 1, 2
6	30	only 1, 3
7	35	every c but 6

Witnesses. All eleven are verified by the unitriangular product and by literal replays with random noncommuting 2×2 maps. (7,6), (8,5) and (8,7) were found by a simple annealer.

Evidence

verify.py, standard library, takes about 90 s:

• a Python port of the decider reruns m = 2 and m = 3 below 24 calls;
• it checks that the recorded complete C runs and cross-checks were made with this residues.c, by SHA256;
• it verifies every witness;
• it assembles the table, using Bound the depth-five chain connector: 8m at m = 2, 3, residue-dependent at m = 4, floor 5m for 9 ≤ m ≤ 20 (no new κ) #4281's vendored witness file to list exactly its open pairs.

The C runs are local evidence that CI does not repeat. The m = 7 balanced run took 93 s.

python3.12 -B scripts/research_package.py --package research/depth5-connector-residues --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/depth5-connector-residues
  PASS_REGISTERED_RESEARCH_PACKAGE research/depth5-connector-residues outcome=completed

Head: f87bec70596b2e34d641fd70a0fde281c94d5eaf.

Not covered

• m > 20.
• Any physical price.

Task and overlap

• Card: connector-alphabet-change, refined, priority revision 7.
• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244.
• No PR or discussion has taken Bound the depth-five chain connector: 8m at m = 2, 3, residue-dependent at m = 4, floor 5m for 9 ≤ m ≤ 20 (no new κ) #4281's open residues.

Credit

The model, Lemma 1 and #4281's witnesses are Nick Petalas's, building on #4249 and #4263. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance. Apache-2.0.

🤖 Generated with Claude Code