[aliromman]

This construction and scoped lower bound gives the exact borrowed-space tariff for square-zero recoding. For a nonzero matrix N of rank r with N squared zero, implementing diag(I+N,I_q) with q arbitrary restored dirty coordinates requires exactly 2 ceil(r/q)+2 rectangular cross-bank shears. The lower bound permits every chronological word in that alphabet, and one grouped word attains both directional bounds. It also minimizes every positive weighted sum of gather/scatter counts. Clean borrowed input has the distinct sharp count 2 ceil(r/q)+1.

The proof and source application, attaining compiler, exact verifier, and frozen source context are pinned to this head. No complete kappa improvement is claimed. This result supplies an exact storage/ordinary-operation tradeoff; native realization, physical costs and a complete caller remain open.

The theorem extends to finite simultaneous-translation kernels over elementary binary groups in characteristic different from two. Replace r by the maximum character-block rank. A character-wise rank factorization and inverse character transform give an attaining finite kernel at every positive tensor width. The lower bound applies within the translation-kernel alphabet; unrestricted maps that mix characters have a different contract.

The grouped word uses no more coefficient ADD instructions than separate four-shear commutators on the same supplied factors. It also has proved bounds at every coefficient-ADD prefix and suffix. These instructions do not make rational multiplication, fixed-odd arithmetic, translations or routing free. In the order-four kernel fixture, one borrowed field uses six shears and 60 coefficient ADDs, while two fields use four shears and 72 ADDs. Fewer shears can therefore cost more scalar work.

Evidence includes 42 rational changed-basis cases and 2,947 coefficient ADDs with complete dirty-input replay, literal inverses and exact interval bounds. A separate BFS covers all 168 GL3(F2) states; all three rank-one square-zero targets have distance four. Complete binary balls of 481 states through depth two and 7,111 through depth three exclude every length-at-most-five word for a rank-two target with one dirty coordinate; the compiler attains six. Eight synthetic thirty-mode cases and eight translation-kernel tensor lifts also pass, with 90 controls. The synthetic thirty-mode tariff is 62,32,22,14,12,8,6,4 shears for q=1,2,3,5,6,10,15,30, each with 120 coefficient ADDs.

David Ganz (@Dugongue) supplies the direct reader application in PR #5634, head 0e263b0932a4a6936940d9bb8af48da269091d37. Its reader proof and provider interface supply the four-shear identity, reader substitution, section conjugacy and finite-character method. Both documents and their native/campaign notices are preserved byte for byte and authenticated. The new theorem does not compute that actual reader's character-rank maximum: rank(E-I)=30 alone is insufficient. Borrowing fewer fields during recoding also does not remove the thirty-field bank needed by its section. The sixteen-row joined services remain separate.

Python 3.12 and its standard library suffice:

python3.12 -B scripts/research_package.py --package research/borrowed-space-shear-tariff
python3.12 -B scripts/check_research_commit.py --package research/borrowed-space-shear-tariff

Expected markers:

PASS_BORROWED_SPACE_SHEAR_TARIFF
PASS_REGISTERED_RESEARCH_PACKAGE research/borrowed-space-shear-tariff outcome=completed
PASS_RESEARCH_COMMIT_BYTES

Normal and optimized registered checks, the ten-file committed-byte check, export validation and a Git-free optimized scientific replay passed. The first export correctly rejected Git-normalized CRLF source bytes; scoped attributes now preserve the original bytes, and the corrected commit was re-exported and checked. CI is pending at publication. The source-history preflight was not run in the shallow checkout. No actual Gaussian-kernel numerical replay, prior full source suite, physical arithmetic-interior/routing bill, selected cold replay, native/outer admission, proof assistant or independent human review was run.

The scientific base is 6cf7a683f447b9f1e2333791907fd92be5009a63, priority revision 7. The final canonical refresh reached 09062e91a942f205a529779b74c13c412afdb483 without changes to the read guidance, priorities or selected record. Fresh complete open-PR/Discussion overlap found the credited source and this distinct claim. No selected-frontier file is changed. New proof, implementation and checks: Ali Romman (@aliromman), with substantial OpenAI Codex assistance. Apache-2.0; the linked source attribution is retained.