[unpaidmoon]

Change and mathematical scope

Scoped identity, together with an exhaustive census of the recorded banks. No new κ. The selected record stays κ = 781451/250000000.

On the published two-edge feed, save the pair-0 X bank at the bridge_query2 snapshot (replay label bridge_query2:0) and add the one operator I - C1 Q C1 into the existing final-query argument. The second operator stays C1 Q^2 on response2. No call is added. The four-input row after query 4 equals

[D1 Q^{-1}, 0, Q^{-1} C3 D1 D3, 0].

In this formal algebra that operator is the unique element which does so. It has exactly two monomials. Each cancels one term of the y0 coupling and one term of the x1 defect:

I:           y0 = -Q^{-1} C1 Q D1,    x1 = Q^{-1} D1
-C1 Q C1:    y0 = Q^{-1} C1 Q^2 D1,   x1 = -Q^{-1} C1 Q C1 D1.

Dropping either monomial leaves both defects nonzero. bridge_minus deletes the saved D1 on x1, so the copy is taken at the snapshot.

The 29 legal sources are the observations recorded before response4, excluding response2. Nineteen have a zero y0 or x1 column. Six have a unit x0 or y1 column, which forces the operator to vanish. Three labels are the same pair-0 Y bank; its unique candidate clears the two defects and writes nonzero -D1 + C1 Q^2 C1 D1 on x0. Only bridge_query2:0 works.

This splits into two evidence classes. The row equality is a proof in the predecessor's formal algebra, and it descends to every quotient. The bank list is an exhaustive finite census: the domain is that ordered observation list, every bank is included, and each bank is settled by a zero column or by the inverse of a one-term column. There is no operator search. The negative list is not a claim about every further physical quotient.

The nine observations at or after response4 are not inputs of this argument. The same lemmas reject them. The unchanged scalar suffix does not produce the four uncoded target banks. The row witness is not a caller.

One circuit under the predecessor's published units costs 2016 scalar ADDs, 560 monomial passes, 32 mode CLEARs, 112 COPYs, and 224 extra fields, with the inherited 8 mode fields reused and no new Gaussian child. That is an upper ledger for this circuit, not a minimum and not a tape proof.

Task and overlap

Base: CrocSwap/integer-mult-bounds 17469fff78c22941280f04d39dc5a1faee09c840, re-fetched at session start and again before submission. Both reads were that commit.

Task: joint-query-controller, scope refined. This is open question 2 of merged PR #6961 (research/active-feedback-outside-edge). It does not redo the named C1 Q one-edge result, the two-edge bill of F1 = Q, F2 = C1 Q, or the owner-indexed x1 block in open PR #7033.

Live inputs: research/coded-active-feedback/search.py and research/coded-native-caller/gauge.py, pinned in SOURCES.json.

Local overlap at this base searched 10178 notes. The question appears as an open question of the predecessor package. An open-upstream-PR title search on 2026-11-10 found no PR stating this insertion. The predecessor finding is #4930 (reply in thread). The later reply #4930 (reply in thread) is the x1-block census.

Validation

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

python3 -B scripts/research_package.py --package research/saved-bank-query-source --check-only
python3 -B scripts/research_package.py --package research/saved-bank-query-source

Metadata check passed. Both registered commands passed, including the optimized command. Checks are explicit need failures. Direct -I and -I -O replays passed before commit. No full multiplier replay, Lean, Docker, or native build was run.

Credit

The schedule, algebra, and published feed are Cinna Davis's research/coded-active-feedback, using Dugongue's PR4948 chronology, Dillon's PR4933 code, and the passive ceiling packaged by Chris R / Clear1nsight. The gauge engine remains research/coded-native-caller/gauge.py. The question is open question 2 of research/active-feedback-outside-edge (merged PR #6961). The declared-family lock and the owner-indexed y0 map are neighboring results and are not redone.

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

Open next questions

After this insertion, which coefficients of the four final banks does the unchanged scalar suffix clear? The autonomous statement stops at query4:3. The first step is to compare the four final banks in this replay with the uncoded targets, before adding a call or a second saved source. No tape or outer bill is priced here.

Does one named relation outside Q^4 = C1^2 = C3^2 = I and the inherited D conjugation make the live bank bridge_minus:0 succeed? That bank has a zero x1 column in this algebra. The first step is to impose that one relation and recompute P S for I - C1 Q C1 on bridge_minus:0. The positive identity is an equality, so a quotient does not reopen it.

Workflow feedback (optional)

The useful constraint was chronological: bridge_minus deletes the x1 monomial, so the same vector is not available live at the final query. Saving the bridge_query2 snapshot is the whole difference between the witness and a zero-column rejection.