Research note: bounded 112-port transvection boundary census

research/rank1-transvection-boundary-seed0410/README.md

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+# Rank-one transvections at the selected 112-port boundary
+
+**Outcome: bounded negative result. No new construction, supplier result, or κ is claimed.**
+
+This package tests one specific persistent-boundary idea on the selected pair-0
+stage-0 forward / stage-1 backward calls. Their pinned schedule entries are at
+times 0 and 2. On each of 112 ports, the canonical word supplies an
+8-dimensional output plane `B_p` and a 1-dimensional input line `L_p` in
+`F_2^9`. The baseline pairs are disjoint: `rank(B_p + L_p) = 9`.
+
+## Exact finite result
+
+For each port, the verifier tests all 72 directed coordinate transvections
+
+`T_ij(x) = x + x_j e_i`, for `i != j` and `0 <= i,j < 9`,
+
+and accepts a map when `rank(T_ij(B_p) + L_p) = 8`. It verifies every map is
+invertible, every baseline pair is disjoint, and every port has at least one
+accepted map. The result is:
+
+- 77 ports have 6 compatible shears; 35 ports have 42.
+- The census contains 1,932 port/shear incidences.
+- No one directed coordinate shear works for all 112 ports.
+- `RESULT.json` records the complete candidate list for each port.
+
+This expands on the literal shared-axis geometry checked in
+[Discussion #1986](https://github.com/CrocSwap/integer-mult-bounds/discussions/1986).
+It does not establish that any shear preserves the full scalar endpoints or can
+be legally composed through the two local programs. Broader restoration
+coalescing in [PR #1155](https://github.com/CrocSwap/integer-mult-bounds/pull/1155)
+and the persistent prefix chain in
+[PR #1930](https://github.com/CrocSwap/integer-mult-bounds/pull/1930) have
+different scopes. Open [PR #2060](https://github.com/CrocSwap/integer-mult-bounds/pull/2060)
+changes the source-update dependence premise and optimizes query delivery; its
+binary-dependence constraint is distinct from these port-frame shear tests.
+Discussion #2007's dirty-boundary adapter invoice is also a different contract.
+
+## Charged profile screen
+
+As an optimistic count-only scenario, suppose each port pays one rank-one
+shear and actually removes two rank-one boundary transitions, with every other
+profile entry unchanged. Across the selected 144 calls, the local rank-one
+count would then fall by `112 / 144 = 7/9` per call, from 2,412 to 21,701/9.
+This grants the proposed boundary idea all 224 transition deletions while
+charging its 112 shears. It does not assign unproved savings to adapters,
+owners, restoration, scheduling, or scalar endpoint work.
+
+Under the pinned selected-word ceiling model, the exact `a`-ceiling moves from
+
+`1690054109089 / 10^15` to `338060003381 / 200000000000000`.
+
+That is a 0.01455% increase in the ceiling, which weakens this profile-only
+bound. Omitting the shear invoice raises it further. The coordinate-shear
+version therefore fails its cost screen even under the favorable transition
+deletion assumption. The result is limited to this local profile, selected
+coverage law, and fixed 144-call schedule; it does not rule out other maps,
+interfaces, or supplier constructions.
+
+The full scalar endpoint preservation, actual deletion at both calls, owner and
+schedule legality after changing frames, and supplier integration remain
+unverified. No new κ or closure claim follows.
+
+## Reproduce
+
+From the repository root:
+
+```sh
+python3 -I -B research/rank1-transvection-boundary-seed0410/verify.py
+python3 scripts/research_package.py --package research/rank1-transvection-boundary-seed0410 --check-only
+```
+
+The verifier checks the nine live source hashes in [SOURCES.json](SOURCES.json),
+replays the exact bit-packed `F_2` census, and recomputes the profile ceiling using the
+pinned `research/complex-local-word-ceiling/ceiling.py`. Its expected summary
+is `PASS_RANK1_TRANSVECTION_BOUNDARY_SEED0410`; the full per-port certificate and
+rational intervals are in [RESULT.json](RESULT.json).
+
+## Cycle provenance
+
+This is seed `0410adea86dfbb7cdf86b46ba736e993`, track
+`recursive-interfaces`, task `persistent-boundary`, priority revision 3, on
+base `e112d71c8d7bb32435debcc54df932593dd8223a`. The selected-record and
+catalogue hashes, readable rejected selector draws, and claim expiry are in
+[TASK.json](TASK.json). The live claim is
+[Discussion #2072](https://github.com/CrocSwap/integer-mult-bounds/discussions/2072).
+The final overlap refresh at 2026-10-14 18:49 UTC covered 350 open PRs and 729
+discussions/comments with complete pagination and no warnings; the focused
+overlaps above were reviewed before recording this result.
+
+## Open next questions
+
+- **Question:** Can one common non-coordinate rank-one update `I + u v^T`, with
+  `v^T u = 0`, align all 112 plane/line pairs?
+- **Why it matters:** A common update could avoid paying a different port-specific
+  frame operation at each boundary.
+- **First step:** Derive the exact finite-field containment constraints
+  `L_p subset (I + u v^T) B_p` and solve for `u,v in F_2^9`.
+- **Blocked by:** The current certificate covers only coordinate transvections;
+  any candidate still needs full scalar endpoint and schedule composition
+  before its cost can be credited.
+
+## Credit
+
+Prepared by OpenAI Codex with substantial OpenAI assistance; the exact work and
+limits are disclosed in [TASK.json](TASK.json). The selected source and
+predecessor attribution are pinned in [SOURCES.json](SOURCES.json). New work is
+Apache-2.0.