[petalas]

Result

Global quadratic extension on P112 fails at eight deletions. With the seven-deletion theorem of #5697, the maximum deletion threshold asked for in #5697's Open next questions is exactly 7.

For each P112 line l, the eight lines sharing 15 live frames with l (its heavy octet) meet every live frame through l. The 112 lines give 14 distinct octets, which pair off: deleting one octet isolates exactly the eight lines of its partner. For each of the 14 octets, exact rational elimination gives 104 live lines, 648 live frames, incidence rank 62 and kernel dimension 42. Quadratic restrictions to the live set span 35 dimensions, so extension fails by 7. Quadratics plus the isolated-line indicators span the whole 42-dimensional kernel.

Claim type: finite exact certificate for 14 explicit deletion sets. The threshold value 7 combines it with #5697's theorem for |X| <= 7, which is cited and not rechecked here. No κ, supplier, invoice or frontier change.

Evidence and reproduction

Package: research/p112-eight-deletion-extension-failure. Standard library only; exact Fraction elimination, no modular or floating-point step. The verifier authenticates the pinned PR1650 geometry source (research/complex-rank8-rigidity-p105/verify.py) and runs controls: all eight seven-subsets of one octet isolate nothing and have the #5697 rank |L|-35, and a damaged octet fails to isolate its line.

~/git/integer-mult-bounds/.venv/bin/python3.12 -B research/p112-eight-deletion-extension-failure/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p112-eight-deletion-extension-failure
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p112-eight-deletion-extension-failure --check-source-history

Local results (CPython 3.12, macOS): PASS_P112_EIGHT_DELETION_EXTENSION_FAILURE in 13.1 s wall; PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed; PASS_RESEARCH_COMMIT_BYTES at 4569515.

Limitations

• Author-checked only. Independent human and proof-kernel review are unclaimed.
• No classification of all eight-deletion failures and no maximum-excess search.
• The Research: (G) does not imply (D) for full live families at eight deletions (no new kappa) #1668 eight-deletion fixtures recorded in Prove sharp seven-deletion incidence at every live E8 pivot #5697 keep extension (rank 69), so eight deletions include both outcomes. This package does not say which other eight-sets fail.

Open next questions

1. Classify all eight-deletion failures of global extension up to the FRAME0 stabilizer. A greedy eight-set found during this work isolates one line and has kernel 37. It is not one of the octets, so heavy-octet isolation is not the only failure shape.
2. Find the maximum kernel excess over 35 at eight and nine deletions (nine is the uniform injectivity limit).

Related open work

• Prove the F′ certificate ceiling 360/17 on P120, down from 24 (no new κ) #5956 (this account, opened after the CLAIM): F′ certificate ceiling on P120. It uses the same E8 frame geometry but a different quantity and computes no P112 number, so there is nothing to compare.

No other open PR studies P112 global quadratic extension or its deletion threshold (searched open PRs for deletion/extension terms). Predecessors: #5697 (Ali Romman, theorem through seven deletions and the question), #1650 and #1668 (Josie Elliston), #1560 (Andrey Mas), #5238.

CI notes

All checks passed at 4569515: plan and replay (research/p112-eight-deletion-extension-failure), run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40532545853.