[petalas]

Summary

Scoped written proof with exact finite corroboration: the autonomous row at the unchanged two-edge saved-bank interface forces F1=Q and F2=C1 Q, even when both feedback operators vary arbitrarily in the predecessor algebra.

R = B + Q^-1 F2 Q^-1 (K + F1 L)
L0 + L2 = 0, K0 + K2 = C1 D1
R0 + R2 - B2 = Q^-1 F2 Q^-1 C1 D1  => F2 = C1 Q
R1 = Q^-1 C1 (F1-Q) D1 = 0          => F1 = Q

The pair itself belongs to unpaidmoon #6961. That package proves the one-edge obstructions and explicitly leaves the all-pairs classification unproved. The new result is simultaneous uniqueness. The proof works in every associative unital ring with Q,D1 units, C1²=1 and QD1=D1Q. It survives quotients preserving the stated interface equations, without a faithful representation or word-length bound.

Produced by Nick Petalas. Credit unpaidmoon #6961 for the saved-bank identities and attaining pair; Cinna Davis #5220 for the active schedule and algebra; Dugongue / David Ganz #4948, Dillon #4933, Chris R / Clear1nsight and their retained native, compiler and precision lineage.

Card joint-query-controller, priority revision 8, changed scope to a fixed-interface classification. Canonical base is e54523517d0f03e58352e647c403a5af3e525a57; source documents are pinned as provenance. The selected #7000 frontier is unchanged. A larger two-edge operator search cannot supply a different solution to this row contract.

Evidence

• Before: Record that the C1 Q feedback edge leaves the four-input row coupled. #6961 supplies one pair and one-edge obstructions; its proof explicitly excludes classification of every pair.
• After: PROOF.md eliminates F1 from a two-column sum and forces both operators. The new standalone verifier checks 97,090 operator pairs across 75 fixtures, with exactly one solution each, 78,658 literal scalar inputs, three semantic controls and six invalid-fixture rejections. Fixtures include all 72 binary unit triples satisfying the reduced hypotheses, two ternary triples and a Z/4 triple with zero divisors.

Local CPython 3.12.15, macOS arm64, standard library only. Both warning-strict direct modes, both registered commands, metadata/source pins, AST compilation, whitespace checks, committed-byte/source-history preflight, official export/input-coverage check and both read-only Git-free exported modes passed. Normal and optimized generated hash-bound receipts agree exactly. Each registered mathematical command takes under one second locally, below its 120-second timeout.

Exact commands from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/two-edge-feedback-rigidity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/two-edge-feedback-rigidity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/two-edge-feedback-rigidity
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/two-edge-feedback-rigidity --check-source-history

Expected markers: PASS_TWO_EDGE_FEEDBACK_RIGIDITY, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES.

Limitations

The all-ring claim rests on the written proof. The finite fixtures corroborate the reduced interface; they are not native realizations or a replay of a predecessor scientific result. The inherited saved-bank row equations, chronology, phases, helper and owned-buffer contracts remain premises. The original witness is not a new construction, and its separate costs are not recomputed here.

No complete caller, independent target credit, native or physical tariff, improved κ or complete multiplication theorem is claimed. New saved sources, changed insertion points, different decoders or target rows are outside scope. All five selected assumptions remain. Independent human and formal proof-kernel review are unclaimed. No selected cold replay, predecessor scientific verifier, native, Lean or Docker suite ran.

Related open work

• Record the ordinary bill of the witness F1=Q, F2=C1 Q #7077 prices the same known pair and studies its unchanged scalar suffix; this PR proves uniqueness and does not recompute that bill.
• Record that saved bank bridge_query2:0 autonomizes the query-4 row #7089 changes the saved-bank source and adds an operator at the final query; that changed source is outside this fixed two-edge interface.
• Prove no D-exponent-0 owner index clears the x1 block #7033 excludes D-exponent-0 owner-indexed decoding for the published feed; it keeps the feedback operators fixed while this PR classifies them.
• Prove a D-exponent (0,-1) owner index leaves the x1 residual #7052 extends that decoding obstruction by a D3-inverse term; it changes decoder coefficients rather than these feedback operators.
• Prove D1^4 = I leaves the residual x1 monomial nonzero #7056 imposes D1^4=I on a residual decoder word; this uniqueness proof already tolerates extra relations preserving the interface and units, but does not evaluate its residual.

No same-number comparison with those cost or residual results was made.

Open next questions

• Change one chronological saved source, derive its four independent input columns and price the complete restored endpoint. Availability and physical costs remain blockers.
• Specify a consumer for the unique coupled row with its full ordinary and recursive invoice. Two jobs in one output bank are not independent target credit.

CI notes

CI did not certify this package. At head 1927d11aee9c32a76ccaf531d0b223c966d637e0, run 43012497455, plan job 130495338611, the Onboarding helper tests step ran 760 tests with exactly one failure and 24 skips. The exact failing test is test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record. Its AssertionError says that 'make frontier7000-verify' was not found in the current CI verification section.

The failure is in unchanged repository files: at canonical base e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json specifies make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. The record, document and test have identical bytes at the base and this PR head. This PR changes only its six package files. The replay job was skipped because plan failed, so no hosted mathematical replay or package certification occurred.

I read the logs and verified the same sole test failure, missing command, 760-test total and 24 skips in run 42905094314 and run 42903313790, and in this account's #7204 run 43012524632 and #7199 run 43001851217. Their replay jobs were also skipped.

Local registered, direct and exported checks remain the evidence described above. No fix, retrigger commit or job rerun was made for this repository-level failure; the head is unchanged. The own-open-PR refresh after the CLAIM found no additional PR studying this two-edge saved-bank row beyond the related work listed above.

Merge Danger

Door: two-way. Only a new isolated research package is added.

Blast Radius: package. No shared implementation, frozen source, selected certificate or predecessor credit is changed.