[krzysztofwojt]

Change and mathematical scope

Closes Open next question 1 of Nick Petalas's #5085 with a dimension-uniform proof and attaining words. For an injective partial binary symplectic isometry with displacement space D of dimension two and singular residual form b:

• rank(b)=1 has exact transvection minimum 3;
• b=0 has exact minimum 4.

The predecessor leaves both cases at a lower bound of three. The new argument proves the general interval r + nullity(b) <= ell(h) <= 2r, hence the sharp all-rank zero-form result ell(h)=2r. A nonsingular full residual form forces the extra displacement; the construction preserves the current fixed subspace and fixes another source direction with at most two turns. Every axis stays in the original ambient space.

Claim type: scoped algebraic proof with finite corroboration. Required fixed spectators must be included in the partial contract. Unspecified coordinates are free to change. Free Pauli sign corrections retain the predecessor's formal tariff; signed compatibility and absolute phase remain separate. This is not native Gaussian paid rank, a physical invoice, supplier admission or a new kappa. Only eight files under research/singular-rank-two-pauli/ are added.

Task and overlap

Fresh main/base: 81e9bd69e4a5b00ba9c713a1620e36d576d5522e. Contribution head: 882cfb2894918ecfd3987fc05d70c91fd4bbd463. Priority revision 7; current-main refresh before submission remained at the base with unchanged governing and selected files.

Canonical claim #5094, taken from handoffs #1166. Task formal-interface-lemma, scope changed to a written proof answering this explicit handoff; no Lean formalization or selected-caller implication is claimed. TASK.json records the card and scope change. The general zero-form corollary follows from the same completion proof, without enlarging a search.

Nearest overlapping work is #5085, pinned at d32b46c02387b3bc9dc0bea344b2c6271b08c5f9: its existing two-turn cutoff and nonsingular alternating three-turn case are not new results here. Its missing singular cases are resolved. JoyboyBrian's merged #5051 at 0960807a68ff46acbe6510c8867912ef2324ee71 supplies the compatible signed-contract setting. These are bibliographic predecessors; no pending Git object, executable or source archive is needed for replay.

Pre-claim overlap covered 313 open PRs and 4,215 Discussion records. The 09:09Z pre-submission refresh covered 316 open PRs, 4,229 Discussion records and the complete 6,824-record committed local inventory, with no pagination warnings; a further direct live refresh found no competing singular-completion claim. Recent krzysztofwojt topics, including #5093's four-event negative association, were excluded. Coverage does not include unpublished work or every nested code diff.

Validation

On macOS arm64 / CPython 3.12.12:

• Registered warning-strict normal and optimized checks pass, about one second per direct command.
• Independent row-matrix enumeration and Cayley BFS cover all 6 and 720 symplectic matrices and all 5 and 67 source subspaces in ambient dimensions two/four: 10,112 distinct partial maps. The general bounds hold throughout; all 1,920 singular rank-two minima and their attaining words agree with exact distances.
• 268 fixed, conjugated larger fixtures through ambient dimension 18 pass. All 6,510 single-turn omissions and 12 invalid/out-of-scope/zero-axis controls reject. RESULT.json is recomputed and compared, binding seven source hashes.
• Committed-byte/full-source-history preflight and the official eight-file export integrity check pass at the contribution head. Both direct modes also pass from the Git-free export with PATH=/nonexistent.

python3.12 -B scripts/research_package.py --package research/singular-rank-two-pauli
python3.12 -B scripts/check_research_commit.py --package research/singular-rank-two-pauli --check-source-history

Standalone export replay: python3.12 -W error -B research/singular-rank-two-pauli/verify.py, and again with -O. Standard library only, no live external inputs. The written proof establishes the uniform claim; finite computations corroborate it. Both hosted checks passed at 882cfb2894918ecfd3987fc05d70c91fd4bbd463 in run 39930128272: planning in 4m11s and the registered package replay in 1m34s. CI corroborates execution, not independent scientific acceptance. Lean, Docker, predecessor scientific replays and the selected cold replay were not run. Independent human and proof-kernel review are unclaimed.

Credit

Krzysztof Wojt with substantial OpenAI Codex assistance, including a Luna supporting census/read-only review; Apache-2.0. Credit Nick Petalas #5085 for the residual-form cutoff and open question, JoyboyBrian #5051 for signed compatibility, David Ganz / Dugongue #4841 for native compiler context, and Cao Yifan / intqwq, Cinna Davis, zxv, Jacob Alan Sussman and the retained native/owner/compiler lineage. Classical symplectic transvections and Wall forms retain their credit; no literature-priority claim is made.

Open next questions

The package README/TASK.json identify two bounded follow-ups: characterize optimal completion for higher-rank nonzero forms (a general attainment argument remains missing), or supply one actual signed native caller with its fixed spectators, phase anchor and complete physical bill. Neither the general lower bound's attainability nor physical savings are assumed. The selected frontier remains unchanged.