Research: prove four-root minimum for seven-source paired-call connectors

[mason-hale]

This proves that four dirty root rows are necessary and sufficient for all 127 seven-source masks when every source and output map is called exactly twice. It settles the seven-source root-minimum question explicitly left open in PR2182, pinned at dc252de1cc594829c3031abdb4de104713bc474a.

The new result is the three-root exclusion. The attaining four-root star is credited to PR2182, independently replayed here, and not claimed as a new construction.

Proof and scope

The model has immutable sources, arbitrary initial root/target dirt, independent fixed F2-linear child maps, unobserved output targets, arbitrary invertible scalar F2 root-row frames, and per-occurrence child ports. The proof allows arbitrarily many frame changes. It normalizes each twice-called source to one axis, uses the all-ones output to place all axes in a four-point affine plane, and derives mandatory prefix-difference masks.

Two separately written engines exhaust 135,135 paired source words × 651 class partitions = 87,972,885 cases. Every case misses mandatory masks; the smallest deficit is four. The C++ matching/prefix/incidence engine and Python restricted-growth/interval/subset-product engine agree on every histogram. The registered replay recomputes the complete Python census.

The inherited attaining program has 4 roots, 14 B calls, 254 A calls, and 56 paid root XORs. A literal witness and complete symbolic cost expression accompany the proof. No total-tariff optimum, physical caller admission, new supplier, kappa improvement, or proof-assistant verification is claimed. Extra child calls, clean roots, target feedback, and matrix-valued intra-row transformations are outside the exclusion.

Validation

On macOS arm64, CPython 3.12.14 and Apple Clang:

• Fresh Python/C++ census agreement; registered package replay and metadata/source-pin checks pass.
• Formal dirty-state forward/inverse replay and 4,362 concrete basis columns across 18 rectangular/singular-map cases pass.
• All 324 single-operation omissions and five additional mutations reject; six-source, neutral-transfer and odd-incidence controls pass.
• Read-only, no-Git exported package passes; a corrupted census receipt rejects.
• The emitted program exactly matches pinned PR2182's star(7). The original universal-connector control also passes.

Reproduce with Python 3.12 standard library:

python3.12 scripts/research_package.py --package research/seven-source-root-minimum

Exact base: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. Only the new research package is changed. Selected construction and upstream sources remain unchanged; selected cold replay was not run.

Coordination and credit

Discussion2275 records the scoped claim and completion. PR2229/2236 concern one-root sparse-output families; PR2063 concerns five-source one-root tariffs. They are distinct and credited, not resubmitted. Final overlap coverage and exact predecessor pins are in TASK.json/SOURCES.json.

Prepared by Mason Hale with substantial OpenAI Codex assistance. Credit @conorholds/PR2182 for the normalization method and star, Ross Richardson/PR934/943 for the generic connector and prefix criterion, and Douglas Colkitt/PR1185 for the six-source/Gray lineage. Apache-2.0; no independent human peer review is claimed.