[unpaidmoon]

Change and mathematical scope

Scoped bill and suffix identity for one named pair, no new κ. On the one reused owned bank of the published two-edge feed, the witness F1 = Q, F2 = C1 Q replaces I and C1 Q^2. No call is added.

Under the predecessor's uniform field units, the witness pays 1120 scalar ADDs, 448 monomial passes, 16 mode CLEARs, 224 COPYs and 336 resets. The published operators pay the same totals except 336 monomial passes. The difference is one Q on the saved FULL output. The response monomial changes from Q^2 to Q and keeps the count 112. Zero Gaussian children are charged only while each power of Q uses that monomial unit. This is not a tape proof and not a selected-frontier change. Selected κ remains 781451/250000000.

The existing scalar suffix sends query-4 banks (A, B, C, D) to (B, -A, D, -C). It clears no final bank to an uncoded target. Final banks 0 and 3 are

bank 0 = [2 Q^{-1} C1 D1, D1, -Q^{-1} C1 D1, 0]
bank 3 = [C3 C1 D1 D3, C3 Q D1 D3, -D1 D3, C3 Q D1 D3].

No rational word-free combination of the four final banks hits any uncoded target. The only target monomial present is bank 2's D1 Q^{-1}, and it is tied to Q^{-1} C3 D1 D3.

Evidence class: the operator comparison, suffix identity and final rows are proofs in the predecessor's formal algebra. The word-free miss is a complete support check of these sixteen columns. The ledger is an upper bound under inherited units. A failed search is not claimed, and this is not an unrestricted impossibility.

Task and overlap

Base: 17469fff78c22941280f04d39dc5a1faee09c840. Live inputs: research/coded-active-feedback/search.py and research/coded-native-caller/gauge.py.

Task: joint-query-controller, scope refined. The card asks whether a changed caller removes real work. This package does not change the query demand. It answers the first open question of merged PR #6961 / research/active-feedback-outside-edge: compare these two monomials with the published operators on the same owned bank, then read final banks 0 and 3, before adding a call.

Nearest work, not redone: #6945 declared-family y0 lock, #6951 owner-indexed y0 charge, #6961 one-edge C1 Q coupling. The x1-block question and a saved-bank source other than this response are left untouched. Open conditional-κ drafts are not used.

Validation

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

python3 -B scripts/research_package.py --package research/two-edge-witness-bill --check-only
python3 -B scripts/research_package.py --package research/two-edge-witness-bill
python3 -I -W error -B research/two-edge-witness-bill/verify.py
python3 -I -O -W error -B research/two-edge-witness-bill/verify.py

Author-checked locally: metadata and pins pass, and both direct commands print PASS_TWO_EDGE_WITNESS_BILL. Checks are explicit failures, not assertions. Controls include the published response, the separating suffix fixture, halving bank 0, and one C3 on bank 3. No network, native build, full multiplier replay, or Lean check was run.

Credit

The schedule, algebra, owned-bank lifetime and field units 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 is the inherited research/coded-native-caller/gauge.py. The witness pair is research/active-feedback-outside-edge. The monomial unit for one power of Q is the convention in research/owner-indexed-y0-clearance, whose primitive question stays open.

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

Open next questions

Does one source monomial primitive implement a single power of Q at the same 112-pass unit as Q^2? The witness charges two such powers, and the zero Gaussian-child line fails if either is a Gaussian child. First step: bind that primitive for one Q on one 112-field bank and reprice the two lines. No native word is replayed here.

Can one charged monomial, still with no new call, turn final bank 0 or final bank 3 into its uncoded target while bank 2 stays the autonomous row? Word-free combinations cannot. First step: one left monomial on bank 0 or bank 3, charged at 112 passes. This is not the published-feed x1 owner index and not a new saved-bank source.

Workflow feedback (optional)

The open question's first step, comparing the two operators on the same owned bank before reading banks 0 and 3, was enough to separate the bill from a new call.