Research: exclude P112 4,5^9 with quotient parity

[petalas]

Summary

Exclude the P112 TWOSIDE (5,5) weight profile 4,5^9 over every characteristic-zero field, closing PR3169's named quotient-independence handoff.

six quotient images in a four-flat
  six distinct five-piece code kernels in one plane
  six root lines, contradicting the E8 maximum of three
seven images after contracting a prescribed triple
  classes have size at most two, so one is a singleton
  valid four-piece set, support union <=44
fifteen root-simplex edges
  795 <= 9*44 + 5*72 = 756, contradiction

Adds only research/p112-quotient-parity/, with the proof, exact source pins, finite receipt, registered replay and failure controls. Commit 95b683196950efea6f3d7ac66745e00dba73425e; canonical-main baseline 997e6a9ca7b4f95d50e366f61412fd640d72a88a, priority revision 6. Canonical investigation.

Credit DreamingOfClouds/#3154 at 708f0ad8928c383de8294aea3c126e9461d2066d for TWOSIDE and the C5 handoff; Nick Petalas/#3164 at 857384079ab640a949fb7618b7909664d3e737e5 for root-simplex geometry and #3169 at 2dbbd3004b70772cef200e341037c7c05b094fbc for the specific failed independence step; and the retained E8/P112 decoder lineage. Development and local checking used substantial automated assistance.

Evidence

• Before: the actual profile survives the original BK/BK7 weight filters. The earlier support-only quotient estimate is 44+12=56, above distance 53.
• After: a six-circuit/root-plane restriction and odd-class argument establish a valid quotient set, giving a strict incidence contradiction with margin 39. No stronger coordinate-budget inequality is needed.
• Full replay covers all 3,309,747 five-flats, 7,140 root-plane pairs, 960 reflections and 232 seven-vertex matchings. Fourteen behavioral groups include an independently calculated rational six-circuit fixture, even-parity failure, strict cutoffs and changed-input rejection.
• Registered Python 3.12 replay, Python 3.13 read-only Git-free export, applicable Ruff checks and warnings-as-errors source compilation pass locally. All thirteen committed package/input byte comparisons pass. Internal mathematical and artifact reviews found no objection.

python3.12 -W error -B scripts/research_package.py --package research/p112-quotient-parity

This closes one named piece configuration. Full C5, C(P112)>=64, supplier admission and a new kappa remain open. The standard Weyl-chamber theorem is an explicit dependency. Independent human review, proof-kernel verification and hosted CI are separate. The selected frontier remains unchanged; open next questions are recorded in the package.

Merge danger

Door: two-way.

Blast radius: research.

One isolated opt-in proof package. Shared code, selected-result records and frozen inputs are unchanged.