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Change and mathematical scope

Finite exact classification of the 8-subsets of P112 that isolate at least one line in the live orthogonal-frame incidence. There are 686 such subsets: 14 heavy octets, each with rational kernel dimension 42 (excess 7 over the 35 trace-zero quadratics), and 672 double-isolation sets, each with kernel dimension 37 (excess 2). The kernel is spanned by the quadratic restrictions and the isolated-line indicators. No isolating 8-subset isolates exactly one line.

This is not a decoder, a supplier, a paid profile, or a κ improvement. The selected frontier is unchanged.

Remaining: non-isolating 8-subsets are not classified, so it is not proved that 7 is the maximum excess over every 8-subset. Nine-deletion excess is open. PR CrocSwap#5697 for |X| ≤ 7 is cited, not rechecked, except one 7-subset control.

Task and overlap

Base ef5f39403610d231a0c27e8a6b4f08c10209bcd6. Card complex-masked-decoder, scope key 2a6f2c726b78f9ae7d26, chooser seed b77666692a57b8d9b8ac33daaba6d4dd, priority revision 2. Scope is refined: the card asks for a masked-decoder budget; this package classifies the isolating frame-incidence kernels at the eight-deletion threshold left open by PR CrocSwap#5957, which is the constraint side of that masked even-entry exclusion. The eight-term all-pairs search is not repeated.

Nearest predecessor: Nick Petalas, PR CrocSwap#5957, which proves extension fails on the 14 octets (excess 7) and notes one greedy kernel-37 set. This census shows that set's family has 672 members and that each isolates two lines, and it identifies every isolating kernel. Overlap search on 2026-11-11 (545 open PRs, 7199 discussions, complete pagination) found no prior double-isolation classification.

Validation

python3 -B research/p112-isolating-eight-deletions/verify.py
python3 -O -B research/p112-isolating-eight-deletions/verify.py
python3 -B scripts/research_package.py --package research/p112-isolating-eight-deletions

CPython 3.12.3, standard library, PYTHONOPTIMIZE unset for the first run and set for the second. Both pass, as does the registered package replay. Final line:

PASS_P112_ISOLATING_EIGHT_DELETIONS octets=14 excess=7 doubles=672 excess=2 isolating=686 seven_control=1 damaged_control=1 generic_control=1

The rational ranks are the odd-prime sandwich at p = 1000003, together with the exact second-moment proof that the quadratics and isolated indicators lie in the integer kernel. The orientation control research/complex-decoder-field-extension-seed44/verify.py also passed. make frontier1094-verify was not run.

Credit

Prepared by Riquelme Pereira with substantial AI assistance. Geometry and the open question are Nick Petalas's PR CrocSwap#5957, following Ali Romman's PR CrocSwap#5697, Josie Elliston's frame censuses, Andrey Mas's PR CrocSwap#1560, and the retained E8 lineage. Apache-2.0.

Discussion writes to CrocSwap/integer-mult-bounds were rejected with Resource not accessible by integration and were not retried.

Open next questions

See research/p112-isolating-eight-deletions/README.md. The smallest useful follow-up is whether any non-isolating 8-subset has positive excess. It matters because that could raise the maximum above 7. Blocker: C(112, 8) = 475491062970, and no stabilizer-chain census was built here.

Workflow feedback (optional)

Task complex-masked-decoder at priority revision 2. The useful narrowing was PR CrocSwap#5957's explicit open classification, not another eight-term all-pairs search. Discussion posting was unavailable to this token, so the claim lives only in the package and this PR.