Prove a nine-root obstruction for one masked P112 co-star decoder

[Hope7Happiness]

Change and mathematical scope

The fixed rational masked P112 decoder with co-star normal groups [0,1,56,87] and term capacities [3,2,2,2] cannot satisfy the required equations. This package gives a reusable rank-two crown lemma and a nine-root certificate explaining the obstruction, including when coefficients may vanish and left/right factors differ.

Six roots exclusive to one two-term group force its factors onto two distinct lines. Three further mutually orthogonal roots then lose that group's contribution, leaving two terms that would have to realize 4I₃. The contradiction is rank at most two versus rank three.

The 12-file package contains the written proof, explicit E8 model and source-order provenance, exact certificate, deterministic expected result, standalone verifier, portability tests and task/replay metadata. Standalone replay needs only Python 3.10+ and the package directory. It has no runtime dependency on the historical search, census, pricing code or selected result.

Task and overlap

Task complex-masked-decoder; scope refined from constructing a changed masked decoder to an exact obstruction for this one nine-term co-star assignment. It changes the eight-term/all-pairs assumptions rather than re-running their known bound. Base: dadfea6e8a78ff6ba901705f87532a71c6f5bbb8; exact historical model commit/blob/SHA-256 and the original task card are retained in SOURCES.json and TASK.json.

Nearest adjacent work: #1292 (head 7a8325f6c8e5a6830ce919965a2959ac459e010a) concerns eight-term P104 rigidity; #1304 (head a052f0da36a3845c2ec785fd44e5b9e50de3b688) concerns a fixed A8 family. Their stated scopes differ from this two-sided co-star crown obstruction. Current Discussion #1316 mixes one coordinate term with a free rectangle, also a different mechanism. These pending claims are overlap context, not theorem dependencies. Pre-submission refresh covered 807 open PR titles/bodies and all 26 Discussions without truncated threads; no direct duplicate was identified within those limits.

Validation

Accepted by the final independent package review:

• Exact checker: 39 proof equations and 13 controls.
• Portability suite: 24 checks, including standalone copies with no checkout, spaces in paths, read-only deterministic runs, and malformed/missing/tampered inputs.
• Clean patch application and all registered Python 3.12 checks passed on original base dadfea6e8a78ff6ba901705f87532a71c6f5bbb8 and main bce54fb3f895cbe5cd4c0eb477c3afeed100190f. Both applied trees matched all 12 reviewed file hashes.
• A separate mathematical reviewer accepted the written proof and independently reconstructed its finite premises.

Reproduce with python3 -I -O -B research/p112-costar-crown-obstruction/verify.py and python3 -I -O -B research/p112-costar-crown-obstruction/test_portability.py.

Only this normal/capacity assignment is excluded. Other co-star candidates remain open; no decoder, physical realization or κ improvement is claimed. The verifier checks finite premises of the written proof, not a proof-assistant formalization.

Credit

Coordinated in Discussion #1267. Model attribution to PR717 and the retained PR968 adapter is recorded in SOURCES.json and NOTICE. Substantial AI assistance: Astra developed the proof/package, a separate Astra agent reviewed the mathematics, Fable 5.1 implemented preceding exploration, Gemini 3.8 Flash contributed brainstorming, and Codex coordinated. Separate-agent review is not human peer review.

Open next questions

README.md and TASK.json record the smallest follow-up: freeze another co-star assignment and test whether its exclusive-root graph has a rank-two crown and residual rank obstruction before coefficient synthesis. No other assignment has been classified here; an exact surviving decoder and fully charged physical realization would still be needed for any κ claim. Full multiplication/selected-supplier replays were not run because this scoped proof changes neither.

[Mister-Meeseeks]

100-PR batch review at ab2061c1249b027e08cf7b9426d6c913526d1e39.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Fixed P112 costar crown obstruction for capacities (3,2,2,2), supported by 39 exact equations and portability tests. Does not exclude unrelated frame families or shared-copy implementations.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.