[unpaidmoon]

Change and mathematical scope

Negative result. No new κ. The selected conditional exponent stays 62009/20000000.

Full monomial two-FULL routing and arbitrary restored identity scratch were previously proved only as separate extensions. For every finite scratch count n>=2, with shears only between the two main owners,

rank pi(T2 W T1) >= (n-1) dim B > (n-2) dim B = rank pi(diag(g0,g1,I,...,I)).

Monomial outer matrices L and R do not close the gap. The endpoint with two pure high-rank targets and identity scratch is impossible in this algebra, and therefore at every sufficiently large tensor width. This is not a scalar saving, a paid-profile improvement, or a conditional κ.

Remaining obligations: a FULL shear that reads a scratch owner, a non-identity transformed helper state, and any single small width below the predecessor's separating-polynomial bound.

Task and overlap

Base: de5ec2e7a6074068452f45b1a191b5c72832df4a (priority revision 8).
Dependencies: research/two-full-central-obstruction/PROOF.md blob 66df36c04d0f88018fd640aa095bb55eeacb17e1; grade/PROOF.md blob 4cdf1e1a29b2113e88513f7a7d983f1e52adb6db.

Task: complex-new-helper-inventory, scope refined. This is item 3 of the structured open-handoff list on discussion 1166 after excluding requests this account or a later merged PR already answered. It is handoff 8, discussion comment 19134766, sharpened by discussion 4902.

Nearest overlap is the predecessor package itself, which states that the two extensions are not one theorem. Merged #4920 is that predecessor. This note adds the simultaneous rank gap. It does not redo the raw shared-port cut invoice or the complementary shared-terminal bill.

Validation

Python 3.12, standard library only, from the repository root:

python3 -B scripts/research_package.py --package research/two-full-scratch-monomial-obstruction --check-only
python3 -B scripts/research_package.py --package research/two-full-scratch-monomial-obstruction

Both registered modes print PASS_TWO_FULL_SCRATCH_MONOMIAL_OBSTRUCTION witnesses=192 gap=6. The written proof is the universal statement. The 192 witnesses are a graded S3 corroboration, not a 17-axis Clifford enumeration. Controls: scratch-only input has no extra copy of B; zeroing the clean transverse factor drops the rank; a cutoff below 2t drops the rank.

Not run: native suite, Lean, selected cold replay, compiler/outer replay.

Credit

Chris R / Clear1nsight, research/two-full-central-obstruction, including Cinna Davis / 1337isnot1337, Cao Yifan, zxv, Jacob Alan Sussman, Ali Romman, Dugongue, Nick Petalas and Richard Hoekstra. The scratch question is JoyboyBrian's handoff on discussion 4902.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

See the package README. Smallest useful follow-up: one FULL shear that reads a scratch owner, with its projected image. The native two-FULL schedule does not contain that macro.