Research: sharp fixed-port and two-root connector boundaries

[conorholds]

Change and mathematical scope

Adds one scoped research package with two optimality results for universal arbitrary-dirty F2 connectors, with immutable sources, unobserved output targets, and exactly two calls per source map and per output map:

• Fixed-z0 five-source optimum: eight paid bidirectional root XORs are necessary and sufficient when every child call uses physical z0. This improves PR1887's ten-transfer construction by 20% in transfer count, retaining its child ports and 72 child calls; the new witness changes both source order and readout boundary pairs.
• Two-root boundary: two roots cannot realize every nonzero source mask for n>=6, even with arbitrary invertible F2 root-frame changes and per-occurrence port choices. An attaining family supplies a three-root six-source witness, so three roots are optimal at n=6; root optimality for larger n remains open.

The necessity arguments combine written reductions with exact finite censuses. They are restricted to the stated alphabet and call counts. The attaining family and fully charged child/port/root-transfer invoices are documented in README.md. No benchmark, physical supplier, complete-cost optimum, new κ or multiplication-exponent improvement is claimed.

Task and overlap

Task connector-alphabet-change, with changed scope declared in TASK.json: this is a sharp interface-boundary side result rather than the original tiny alphabet-change experiment. Canonical-main guidance was reread at 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4; original scientific discovery pins remain unchanged in SOURCES.json.

The refreshed audit covered 426 open PRs, 859 discussion/comment records, 1,966 all-state PR bodies and indexed main research notes. No current exact overlap for these two theorem scopes was found; that audit is not a novelty guarantee.

PR2065, head 2721b1e31d718a6e70266a99521f629ce2e1d259, already supplies the six-transfer five-source endpoint with occurrence-varying ports; that result is credited and excluded from the new claims. PR2056, PR2128 and PR2156 allow combined DUO source calls; PR2134 and PR2146 use TRIO calls. Those alphabets are outside this obstruction. PR2063 concerns five-source one-root tariffs.

Validation

Python 3.12, standard library, from the repository root:

python3.12 -B scripts/research_package.py --package research/scalable-dirty-connectors --check-only
python3.12 -B scripts/research_package.py --package research/scalable-dirty-connectors

Both passed on the proposed package using the current-main runner:

PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
PASS_SCALABLE_DIRTY_CONNECTORS n=4..10 dense=460 rejected=704 extra_rejected=336 fixed_port=8
PASS_REGISTERED_RESEARCH_PACKAGE research/scalable-dirty-connectors outcome=completed

The bounded replay checks symbolic restoration, rectangular/singular-map forward/inverse cases, meaningful omissions/port/direction/label/schema controls, and two complete closed-word censuses. The separate C++20 six-transfer census covers 34,977,960 legal fixed-port schedules with maximum coverage 31/32; parity and padding exclude every smaller transfer count. The eight-transfer positive fixture exercises the same census engine on 105 legal pairings and finds one universal schedule. Native source and mathematical receipts are unchanged, so prior complete native results are reused; reproduction commands and the complete census histogram ship with the package. Generated receipts guard regressions rather than providing independent proof evidence.

Not checked: hosted CI, Lean or independent human proof review, supplier admission, or caller/head-return tape integration. These local results do not alter selected records or frozen predecessor files.

Credit

Credit Ross Richardson/PR934 and PR943 for the generic connector and prefix boundary; Douglas Colkitt/PR1185 for the six-source census and one-root constructions; woklerx/PR1438 and Gareth Davidson/PR1498 for the two-root contract; Cinna Davis/Discussion1412, Hanson Sun/PR1615, Sagar Thakrar/PR1767 and @nedveder/PR1812 for the one-way endpoint and checks; Cinna Davis/PR1887 for the fixed-port bidirectional predecessor and PR1675 for the distinct signed target; and Adam Gall/PR1915 for related frame-normalization techniques and PR2065 for the unrestricted-port six-transfer endpoint. Exact source heads, blobs and hashes are retained in SOURCES.json.

Substantial AI assistance was used in mathematical derivation, implementation and checking. No independent human peer review is claimed. Apache-2.0; predecessor attribution and source notices are retained.

Open next questions

README.md and TASK.json give the questions, relevance, first steps and blockers:

1. Identify a real recursive caller and evaluate the fully paid port/hosting/transfer inequality; no caller is identified yet.
2. Seek a seven-source three-root minimum-call witness in a bounded frame domain; prior unsuccessful searches are not impossibility proofs.
3. Emit complete tape-access/head-return traces for an identified caller and layout; both are prerequisites.