[petalas]

Summary

Scoped algebraic proof: allowing either of two elementary FULL shears to read or write any dirty scratch owner still leaves projected rank at least (n-1) dim B. Two pure high-grade main targets with restored identity scratch have rank (n-2) dim B, so this enlarged two-FULL class cannot attain them. The bound is sharp under the declared algebraic hypotheses. No new kappa is claimed.

This answers the scratch-reading follow-up in unpaidmoon's #5843 at 050b99f3f355a83f90189218c4c95c8131f8a991. The new proof chooses the actual first read-coordinate kernel:

x[q1] = 0
E1 x = x
u = W D1 x has low-grade coordinates
pi(D2 E2 u) = 0 forces u[q2] = 0, then u = 0

The theorem permits noncommuting query coefficients and free algebras with zero divisors. It is self-contained in research/scratch-reading-full-obstruction/PROOF.md. Credit unpaidmoon for the question and main-only injection argument, Chris R / Clear1nsight #4920 for the full-monomial Clifford grade and tensor-separation framework, thu-nmrc #5200 for the distinct joint-main-bank theorem, and their retained predecessor lineage. These are cited premises, not audit deliverables.

Evidence

The predecessor explicitly leaves scratch-reading FULL shears outside its result. The new theorem removes that location restriction while retaining the low-grade diagonal and monomial-algebra control assumptions.

A fresh verifier checks 736 explicit fixtures in the dimension-216 algebra F2[S3^3], including 512 with a scratch read, 512 with a scratch write and 176 with a scratch-to-scratch shear. These categories overlap. Direct elimination checks the full projected map, the actual first-read kernel, the target rank and outer free controls. All 736 whole maps have full projected rank. Separate equality examples attain the theorem's lower bound. It also checks 46,656 grade products, 689 matrices against an exhaustive span oracle, and 18 named sharpness/hypothesis/arithmetic controls. The finite fixtures do not enumerate arbitrary coefficients or free controls.

Local CPython 3.12.15 on macOS 27.0.1 arm64 passes normal and optimized modes with identical generated JSON, including proof/code hashes. The two registered commands took 6.11 and 6.15 seconds, each below its 120-second timeout. Metadata, the complete registered runner, committed-byte/source-history preflight, and git diff --check pass. The selected source-binding baseline also passes.

From the repository root, reproduce with:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -B -W error research/scratch-reading-full-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -B -W error research/scratch-reading-full-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/scratch-reading-full-obstruction
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/scratch-reading-full-obstruction --check-source-history

Expected scientific status is PASS_SCRATCH_READING_FULL_OBSTRUCTION, with cases=736, scratch_read=512, controls=18. The registered runner ends with PASS_REGISTERED_RESEARCH_PACKAGE. A reviewer on another machine can substitute their CPython 3.12 executable for the displayed interpreter path.

Limitations

The written theorem covers every field and finite owner count under its explicit filtered-algebra hypotheses. The finite checker uses an untwisted example; it is not a Clifford/native operator replay. The grade-17 application uses #4920's cited grade facts and all-large-width separation lemma and supplies no new sufficient width.

No native, compiler, precision/tape, outer, selected cold replay, formal proof-kernel verification or independent human acceptance is claimed. Multiple query-source blocks, productive transformed helper resources, clean COPY/CLEAR, noninvertible controls and changed transverse grades remain separate. No supplier admission or selected-frontier change is made.

Open next questions

Can a legal FULL macro with two independent query source coordinates evade the obstruction at a favorable complete tariff? First specify its complete target, transverse, restoration and paid-call contract, then compute its projected image. No physical macro or complete invoice for that changed premise is supplied here.

Related open work

Refreshed against this account's open PRs created since the advance CLAIM. No additional post-CLAIM PR studies this two-FULL projected-rank endpoint.

• Prove identity scratch does not unlock full monomial two-FULL routing. #5843 proves the main-only two-shear rank obstruction with identity scratch. This contribution answers its explicit scratch-reading follow-up and permits arbitrary distinct source/destination owners in either elementary shear. Its other native and physical obligations remain separate.

CI notes

At commit 133ab618a11c17b2b90ad5c1dba97bff8048a8c0, run 40495803169 failed in the sole failing step, plan / Onboarding helper tests. All ten reported failure/error entries are in unchanged tests/test_research_current_assignment.py; the step ends with FAILED (failures=8, errors=2, skipped=22). The errors are missing follow_up_steps and catalog_first_step keys; the failures concern frozen-card and current/historical first-step expectations. The command exits with code 1.

The same module and eight-failure/two-error summary are confirmed in PR #5887's plan job, another PR based on main de5ec2e7a6074068452f45b1a191b5c72832df4a. This shared onboarding failure is outside the package; no tests or other repository files were changed to address it.

The replay job was skipped. Hosted replay has not run or certified this package. Only the local scientific, metadata and source-history checks reported above have run for this contribution. CI is not green.

Merge Danger

Door: two-way.

Blast Radius: package.

The addition is confined to one research package. Existing packages, shared code and the selected record are unchanged.