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Summary

This adds research/depth5-connector-mitm/. It answers open question 1 of petalas's #4281 (also my #4341's open question) at m = 5 exactly and at m = 6 for length 5m. The method is an exact meet-in-the-middle in U₆(ℤ/m), which avoids the 30 GB BFS table.

Claim type: an exhaustive finite computation with explicit certificates. No new κ.

Method

A length-L word splits into a prefix with counts a and a suffix with counts n − a. Since row 0 of S⁻¹ is e₀, P·S = I + cE₅₀ holds iff P = S⁻¹ + cE₅₀. Prefix products and suffix inverses are deduplicated per count vector, matched on the 14 coordinates other than (5,0), and every residue is read off in one pass. Each hit is turned into an explicit word by backtracking and replayed.

Results

m	c	minimum length
5	1	25
5	2	30 (none at 25; witness with counts (10,5,5,5,5))
5	3, 4	35 (none at 25 or 30 over all five count vectors; witnesses with counts (10,10,5,5,5))
6	1	30 = 5m. This is a new witness, missed by #4281's bounded search.
6	3	30
6	2, 4, 5	≥ 36 (none at 30)

• Controls. There is no length-5m word at m = 2, 3, 4, matching 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 BFS minima.
• Convention. All 173 of 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 witnesses replay under this product convention.
• Conclusion. 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 m = 5 search failures are genuine. The premium over 5m is 5 for c = 2 and 10 for c = 3, 4.

Evidence

python3.12 -B scripts/research_package.py --package research/depth5-connector-mitm --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/depth5-connector-mitm
  PASS_REGISTERED_RESEARCH_PACKAGE research/depth5-connector-mitm outcome=completed   (~25 s, numpy 1.26.4)

• In the verifier: all witnesses are replayed, and the exhaustive m ≤ 5 length-5m runs are rerun.
• Local evidence: the m = 5 length-30 runs (five count vectors) and the m = 6 length-30 run. They are recorded in EVIDENCE.json with commands and bound to the mitm.py SHA256; verify.py --full reruns them.

Head: 930469ad8c0b82b04cd4c1775913e27de47b1113. Base: canonical main 12fa8c5c, priority revision 7.

Not covered

• m = 7, 8, which exceed 32 GB when run in parallel here.
• Exact m = 6 minima for c = 2, 4, 5.
• Physical prices.

Task and overlap

• Card: connector-alphabet-change, refined.
• Claim: Handoffs and requests for independent checks #1166 (comment)
• Overlap: no other MITM run on 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.

Credit

@petalas for the model, Lemma 1, the witnesses and the question (#4281, building on #4249 and #4263).

Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance; no independent human review. Apache-2.0. This PR does not merge anything or change the frontier.

🤖 Generated with Claude Code

[pinterbanget]

Overlap note: a parallel session on this account had already decided the same question in #4418 (posted earlier, exact minima 30, 35, 35 at m = 5 and 30, 36, 36, 36 at m = 6, plus m = 7/8 results). This PR was prepared without seeing #4418, using a separately written meet-in-the-middle, so it serves as an independent confirmation of #4418's m = 5 minima and of its m = 6 length-30 residues {1, 3}. Its new content beyond #4418 is only the independent method and witnesses.