[petalas]

Summary

P112, one seven-space and ONE common deletion set
  at most five cuts: at most two distinct directions
  six cuts: explicit independent orthogonal triple

Prove the sharp common-five deletion lemma over every characteristic-zero field, for directions whose residual binary label rank is at most seven. This resolves PR3230's named handoff; its separate-five counterexample has a different contract.

Adds only research/p112-common-five-deletion/, with the proof, literal witness, two finite implementations, complete receipts and replay instructions. Exact commit f27442f87cf1fd86a50b8ba15615d8d655d06a85, base 463113d675af7410ac91d53a4fed51ec6494eb9a. Canonical investigation. TASK.json records the scope change from generic helper synthesis to this explicit handoff.

Credit DreamingOfClouds/PR3154 for Lemma N and the decoder-floor reduction, Nick Petalas/PR3230 for the separate-five boundary and question, and the retained PR3161/3162/3169 E8/P112 geometry lineage, including joaqcoo. Development and verification used substantial automated assistance; no independent human review is claimed.

Evidence

• Before: separate five-line cuts admit three independent directions, but their common union has eight lines. The common-five question was open.
• After: all 43,435 seven-spaces, 121 live-trace classes, 1,200 candidate directions and 148 affordable completions prove the common-five bound. An independent lattice/RREF/dual-functional implementation reproduces the complete table. Six common cuts give a literal triple with Gram determinant 512.
• Registered Python 3.12.15/GCC14 replay and thirteen optimized behavioral groups pass. The exact committed read-only, Git-free export passes all three commands under optimized Python 3.13.5; Clang19 also reproduces the primary receipt. All fifteen publication files match committed bytes.

python3 -B scripts/research_package.py --package research/p112-common-five-deletion

The written classification uses the standard real Weyl-chamber theorem and a rational-nullspace field transfer. No new historical profile exclusion, full C5, decoder floor, supplier or kappa is claimed. Hosted planning and the complete package replay both passed at exact head f27442f87cf1fd86a50b8ba15615d8d655d06a85. Two unsuppressed Ruff S603 trust-boundary reports are documented in README.md and CHECKS.json.

Open next question: bind a still-open caller's literal single common support and retained code dimension before applying this lemma. Separately affordable cuts do not supply that hypothesis.

Merge Danger

Door: two-way

Blast Radius: research

One new opt-in package and one declared unchanged geometry input. Frozen predecessors and the selected result are unchanged. Retaining this package does not admit a supplier or certify a multiplication theorem.