Prove R437 node509 source-quotient obstruction to a third carrier deletion

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+# R437 node509: a source-quotient obstruction to a third carrier deletion
+
+**Scoped negative result; no new κ or physical supplier.** At current R437
+source `a24f9ff95c12fd0b6792e3f710569237efd72db2`, a promising next deletion is
+helper469, the SOURCE92 carrier. Even granting a free SOURCE77 read, its
+replacement by an early SOURCE92 injection into receiver444 cannot meet the
+fixed node509 output contract with an invertible linear register network.
+The source quotient has rank two at entry and rank one at the required exit.
+The obstruction persists with any number of borrowed registers returned modulo
+the allowed external source span. It excludes arbitrarily long networks in this
+specified alphabet, rather than only the six two-register matrices tried.
+
+This informs the next κ search: another GL rewrite at this unchanged boundary
+cannot supply the carrier saving. A candidate must change its accessible source
+space, retain an additional output dimension, move the receiver/source boundary,
+or supply and pay for a different operation. Such changes remain open.
+
+## Exact scope and evidence
+
+The verifier reads a byte-exact retained current R437 word. It independently
+checks its full 659-column arbitrary-DIRTY XOR identity and the rational closure
+of all 636 frames. Five helpers have exactly one single-SOURCE write in A/B,
+are not captured, and are otherwise only read: 462, 463, 466, 468 and469.
+At the first read of each, the destination's current frame is not contained in
+the SOURCE's current frame. Thus the immediate replacement by a read at that
+unchanged SOURCE frame fails. This census does not exclude earlier retiming or
+a coordinated rewrite.
+
+The focused cut concerns helper469, SOURCE92, node509 and writable receivers444
+and413. At entry before A1998, their SOURCE rows are respectively zero and
+`e87`; SOURCE92 is clean, at frame184. We optimistically replace the deleted
+carrier by an injection `444 += SOURCE92` at frame184. Frame184 is contained in
+509. The resulting source rows are `(e92,e87)`.
+
+Scanning all659 registers gives ten other roles whose frames can enter509 and
+then rejoin the next unchanged observation:
+
+`17, 67, 177, 185, 271, 293, 352, 371, 467, 468`.
+
+Targets177 and185 have zero SOURCE rows here. The inherited H1-H5 alphabet
+forbids target readout; allowing them as borrowed banks is an additional
+favorable relaxation and adds no quotient dimension. The other eight are
+SOURCE/helper roles.
+
+Their SOURCE rows span coordinates `17,22,33,38,50,60,67`. Add `e77` as an
+explicitly **free relaxed channel**, to avoid charging or assuming a solution to
+the incompatible early184/186 injection order. This larger span is called E.
+The two actual node outputs, modulo E, both equal `e87 + e92`. Hence the input
+quotient rank is two and the required output quotient rank is one.
+
+A second source census covers all 34 insertion cuts A2001 through A2034,
+before413 takes its next frame390 gate. The union of all newly eligible
+frame509 read channels adds no dimension to E. This is a finite eligibility
+fact. The theorem below still fixes the entry and exit SOURCE rows; it does not
+claim to rule out arbitrary delayed observations or a two-window program.
+
+The separately implemented checker propagates SOURCE87/92 one column at a time
+and checks all GL(2,2), GL(3,2), GL(4,2) matrices: 6, 168 and20,160 respectively.
+The added registers begin and end at zero **in the source quotient**, not with
+physically clean payloads. These finite checks support the implementation; the
+rank argument proves the arbitrary-register statement.
+
+## Theorem and proof
+
+Fix the retained entry/output rows and external space E described above.
+The allowed replacement uses any finite sequence of invertible linear updates
+among the writable registers, plus reads whose SOURCE row lies in E. Additional
+borrowed writable registers may be used if their entry and exit SOURCE rows lie
+in E. The replacement does not introduce an erasure, discard, new source
+channel, changed observation contract or separate physical primitive.
+
+Project every register's SOURCE row into `V/E`, where `V=F2^111`. Reads from E
+vanish. Every invertible linear register update left-multiplies the row matrix
+by an invertible matrix and therefore preserves its rank. Borrowed rows add
+zero rows at both boundaries. The initial matrix contains the independent rows
+`[e92]` and `[e87]`, so has rank two. The required final rows are both
+`[e92+e87]`, with all borrowed rows zero, so have rank one. Contradiction.
+
+Arbitrary dirty inputs cannot repair this source-column contradiction. Fresh
+D compensation only modifies target DIRTY responses; literal C restores the
+reversible word after its observations. Neither supplies a missing SOURCE
+identity at this fixed node boundary. This is a necessary-condition argument,
+not a proof that every source-correct candidate satisfies the full caller.
+
+The external-role test uses the exact rational closed-frame inclusions
+`current_frame <= 509 <= next_unchanged_observation_frame`. Captures are included
+before B when finding that next observation. Final frames cover roles with no
+later observation. This permits additional frame advances for free and is
+therefore favorable to a candidate. Exact closure reconstruction validates the
+set-inclusion representation of these retained frames.
+
+Three escape controls deliberately break the strict-loss certificate: grant
+an independent SOURCE87 channel; grant a SOURCE92 channel at the cut; or let a
+borrowed register retain the missing SOURCE87 quotient row. They are potential
+changes of contract, not free operations admitted by this package.
+
+## Budget gate, overlap and limits
+
+Pairing: reversible bit-circuit synthesis with quotient-space rank and frame
+partial orders. The initial objective was a fully paid κ improvement. Rohan
+Arun's pending PR3959 already realizes the unchanged width-eight pooled slot
+optimum supplied by mightyOil's PR1281 theorem, so another unchanged owner
+reshuffle was not pursued. The attempted changed local mechanism is an actual
+third-carrier deletion at this source. It fails before pricing. All hypothetical
+adapter and auxiliary costs were set to zero in the cut relaxation; no saving
+is booked when the necessary source contract fails.
+
+| Component | Status here |
+|---|---|
+| Current word's complete arbitrary-DIRTY linear identity | Checked, all659 columns |
+| Rational frames and this source boundary | Checked, all636 frames; exact ten-role frame cut (two target rows are zero) |
+| Immediate persistent-source replacements | Five candidates; all first-read gates fail |
+| Fixed node509 deletion with returned borrowed banks | Excluded by rank2-to-rank1 theorem |
+| Later eligible frame509 channel growth | 34 cuts checked; no new source dimension |
+| Native all-size realization, stock, preparation, tape/precision, full physical cost | Inherited and unresolved; not audited here |
+| New κ, a new local word, or a global carrier lower bound | Not established |
+
+Canonical main is `20ecf2891d80b5acf18d980b2164fb44ea7faa90`, priorities revision6.
+The selected κ remains0.001672660197076918. The team claim is
+[PR3967 comment](https://github.com/CrocSwap/integer-mult-bounds/pull/3967#issuecomment-6263066320).
+Live overlap included PR3647/3700/3950/3958/3959/3967/3982/3995 and the parent
+team's Discussion review; coverage is partial. PR1281 already supplies generic
+width-seven packing and conditional price tables; this package claims neither.
+
+## Reproduce
+
+From the repository root (Python3.12 standard library):
+
+```sh
+python3 -B scripts/research_package.py --package research/r437-carrier-quotient-cut
+```
+
+The two direct commands are `python3 -I -B research/r437-carrier-quotient-cut/verify.py`
+and `python3 -I -B research/r437-carrier-quotient-cut/test_package.py`.
+The latter performs an actual optimized Git-free subprocess replay and rejects
+changed source bytes, changed certificate bytes and a missing retained notice.
+`RESULT.json` is immutable during ordinary verification. `--write NEW_FILE`
+regenerates a certificate explicitly. The source export includes its original
+license and notice; `RETAINED.json` records exact origin Git blobs and SHA256s.
+Those foreign-head origins are documentary pins, not runtime Git dependencies.
+The verifier hashes the self-contained retained bytes before using them.
+
+## Open next questions
+
+1. Can a paid source87 or source92 channel reach this cut without disturbing
+   the inherited SOURCE writes and future frames? Start with the exact
+   `current <= cut <= next` failure and price a changed source/consumer pair.
+   Its physical preparation, restoration and recursive bill are missing.
+2. Can a two-window rewrite retain the missing quotient dimension in an
+   existing bank and consume it later? Start with a proposed changed future
+   observation map for413/444 and the borrowed bank. The fixed-output theorem
+   does not apply, but all subsequent reads, COPY windows and full dirty
+   cleanup must be checked and charged.
+3. Can a different third-carrier construction change the boundary rather than
+   merely resynthesize this GL network? Use the five-source census as an
+   entry map, then derive a whole paid profile before an owner search.
+
+Credit ai-nanachi/PR3647 for R437 and the helper464 GL deletion; Ali Romman/PR1043
+for the retained helper470/SOURCE77 deletion; Sagar Thakrar/PR3700 and PR2423 for
+independent checks and OQ4; mightyOil/PR1281 for the slot optimum; Rohan Arun/
+PR3959 for pooled R437; eumemic, DreamingOfClouds, DaysSky, Crow, chafreaky and
+the full notices for the original word, geometry, native, hosting and compiler.
+New proof, implementation and checks by Chris R / Clear1nsight with substantial
+OpenAI Codex assistance. Apache-2.0. No independent human or formal-kernel review
+is claimed.