[adamgall]

Result

Adding the reverse F2 root transfer D:z1+=z0 lowers the exact universal five-source/31-output connector threshold from PR1615's one-way 8 transfers to 6, at exactly two calls per B and A operator. The new complete word has 78 root calls: 10 B + 62 A + 3 C + 3 D.

The witness uses the restoring (CD)^3 loop in GL(2,F2); four sources change physical ports between their paired calls. This answers PR1615's explicit reverse-transfer open question rather than extending the old one-way census.

Evidence

• Written moving-frame proof: paired source injections have equal normalized columns; paired readers have equal dirt rows. Three source directions cannot supply mask31; one direction reduces to the credited old945-pairing obstruction. The two-direction relaxation leaves175 word/partition pairs.
• Four-transfer loops have representatives CCCC/CCDD up to justified cyclic/root symmetry. Two independently structured exact engines agree on all histograms across2,102,100 reduced candidates, maximum31/32. Odd counts fail dirt parity; shorter even words pad to4. Thus6 is sharp.
• Explicit 78-call certificate, all31 readout pairs, complete180-factor+6-transfer expansion, arbitrary-dirty free noncommutative replay,522 dense basis/mixed-state cases and162 semantic mutation controls.
• Source-bound package with complete independent chooser output, task/budget, exact predecessor pins and registered Python3.12 replay.

Scope and complete price

This is an abstract incidence-minimal F2 connector result, not a new physical supplier or kappa. Every child port, both transfer directions, scratch restoration and parent routing/admission/dispatch remain charged. Under equal complete child/transfer prices the favorable condition against the old74-query single-root repair is 6*sigma + Delta H < 2*c. Unit root-call prices alone do not improve that repair.

Credit Ross Richardson/PR943, woklerx/PR1438, Gareth Davidson/PR1498, Hanson Sun/PR1615, Cinna Davis/Discussion1412 and Sagar Thakrar/seed58. Characteristic-zero signed3x3 PR1675/1682/1983 are different targets/rings. No selected record, frozen source or shared index changes.

Exercised verification

uv run --python 3.12 --no-project python -I -O -B research/bidirectional-five-source-connector/verify.py
uv run --python 3.12 --no-project python scripts/research_package.py --package research/bidirectional-five-source-connector

Observed:

PASS_BIDIRECTIONAL_FIVE_SOURCE minimum_transfers=6 root_calls=78 eligible=175 four_transfer_candidates=2102100 engines=agree factor_calls=180 dense_cases=522 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/bidirectional-five-source-connector outcome=completed

No full-suite, formal-kernel or GitHubCI-pass claim.

Independent seed 628ce1c7a4abbbac17914cb067826ef4, recursive-interfaces / connector-alphabet-change, exact base e112d71c8d7bb32435debcc54df932593dd8223a. Claim: #research-claim-and-completion-record .

Prepared by Adam Gall with substantial OpenAI Codex assistance; Apache-2.0. Independent implementation paths are not independent human review.

Research claim and completion record

Status: completed. The claim record is preserved here so this PR is self-contained. Historical coordination links in the committed research metadata refer to this record; the separate coordination discussion is being removed.

DONE — PR #2065 , full commit2721b1e31d718a6e70266a99521f629ce2e1d259. Adding reverse F2 root XOR lowers PR1615 universal five-source/31-output incidence-minimal threshold from8 to exactly6 transfers:78 complete root calls,10B+62A+3C+3D. Four sources change physical ports. Moving-frame source/readout proof reduces the four-transfer exclusion to175 word/partition pairs; two exact engines agree across2,102,100 reduced candidates (max31/32). A literal (CD)^3 witness attains32/32. Full180-factor+6-transfer arbitrary-dirty noncommutative replay,522 dense cases and162 semantic mutations pass. Focused optimized CPython3.12 replay and registered package replay both PASS; parent independently replayed and reviewed before publication. No physical supplier, paid-moment or kappa improvement claimed; unit transfer/child prices still give78 versus the old74-query repair. Complete unequal-port tariff and all added transfer/restoration/dispatch costs are retained. No GitHubCI-pass claim. Credit PR943,1438,1498,1615, Discussion1412 and seed58; signed characteristic-zero3x3 predecessors/parallel work are distinct.

Original claim: seed628ce1c7a4abbbac17914cb067826ef4, recursive-interfaces / connector-alphabet-change; basee112d71c8d7bb32435debcc54df932593dd8223a. Added exactly D:z1+=z0 to one-way C:z0+=z1; bounded question was a restoring loop below8 with exactly10B+62A and arbitrary dirty roots/generic maps. Claim fulfilled, not expired or extended.