Research: four coordinate-SWAP targets at r=4; calls 2, 4, 6 excluded, minimum narrowed to 8-12 (no new kappa)

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Change and mathematical scope

Four coordinate-SWAP targets at exactly four registers cannot use 2, 4 or 6 paid coordinate-CNOT calls in the formal F2[GL_2(F2)] rank-one-idempotent register model with free scalar GL_4(F2) mixing. The written sign-quotient invariants force any word of length below the 12-call upper bound to have even length, so the exact minimum is now narrowed to q in {8, 10, 12} for T = diag([S],[S],[S],[S]), S = aba = bab, with all four registers arbitrary dirty inputs restored exactly.

This is the first t = 4 case of the coordinate-CNOT line. #1414 proves t = 2 costs 6 calls at any finite workspace, #1491 proves t = 3 costs 9 at exactly three registers, and #1541 proves t = 3 costs 9 at any workspace and records "Four or more targets remain open". This package answers that boundary at the minimal register count, partially.

Three written invariants (README section 2). L1 (det parity): through F_2[G] -> F_2[eps]/(eps^2), det(I + eps N_i) = 1 + eps per call against 1 + t eps for the target, so q = t (mod 2); at t = 4, q is even. L2: free scalar mixing is forced to the identity for diagonal group-element targets, and sum_i N_i = D_t over F_2 (at t = r = 4, sum_i N_i = I_4, so q >= 4). L3: calls are involutions, so q-call balls are closed under inversion and meets need no matrix inverses.

Evidence. Complete finite meets over the exact natural-image balls exclude q = 2, 4, 6 at r = 4 (ball sizes 240 / 32,651 / 3,639,920 at depths 1-3; every meet empty at the natural level, which is already sufficient for exclusion). The 12-call upper bound is attained by four disjoint aba words and verified exactly in M_4(F_2[G]); a drop-one-call mutation fails. The same engine reproduces the known minima 3 (t=1,r=1), 6 (t=2,r=2), 6 (t=2,r=3 with one dirty register) and 9 (t=3,r=3), matching #1414 and #1491 at their boundary cases. Two independently written engines (bit-row packed vs tuple-row index loops) agree on every natural ball through r = 3 depth 4. A natural-alias control exhibits 252 pairs of distinct 3-call words with equal natural image and unequal formal value, so the natural image is used only for exclusion and every accepted word passes the full formal algebra (consistent with #1567 alias findings in the GL_3(F_2) model).

Recorded obstruction (not an impossibility statement). The residual q in {8, 10} needs a meet at the depth-4 ball or its successor. The complete depth-4 stream at r = 4 has 873,580,800 products; a 2^28-slot bitmap sketch over all of them estimates 2.74e8 distinct states (the same estimator reads 2.97e6 against the exact depth-3 count 3,639,920, an 18% low bias, so the true count is plausibly about 3e8 or more). One exact depth-4 ball is roughly 20 GB as a CPython set. The search stopped at the pre-declared compute cap with the residual domain and the exact measurement recorded.

Task and overlap

• Task ID / scope key: bit-new-register-model / 7f5f4426ef796683c485, refined to F2[GL2(F2)]/four-SWAP-targets/exactly-four-registers/rank-one-idempotent-calls/free-GL4-mixing. Chooser nonce b5ce6390976f7388a9834b2418b70942, track bit-mechanisms, priority revision 3. Scope refined inside the card's allowed variation ("a fixed larger register count or a specified nonabelian operation family"): t = 4 is the target count the line's own statements leave open. Not the excluded three-register abelian model.
• Base and heads: base 1f7a934975e5eacaf83511c9ac870263bf095e05 (origin/main at session start). No predecessor executable is imported; engines A and B are written in this session.
• Overlap. find_overlap.py with complete pagination at 2026-10-14T07:11Z (385 open PRs / 337 discussion records) for four coordinate swaps, four SWAP targets, four-swap, diag([S],[S],[S],[S]), r = 4 registers, 12 calls found no four-target coordinate-CNOT claim besides this one's own claim comment. No matches is not a novelty proof. Nearest work: Research theorem: exact paid cost for binary affine register words #1333 (model question), Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414, Finite certificate: three coordinate swaps need nine paid calls on three registers (no new kappa) #1491, Finite certificate: a third address coordinate does not cheapen two coordinate swaps (no new kappa) #1495, Finite certificate: two order-four GL3 targets need eight coordinate-CNOT calls (no new kappa) #1496 (commdata2338), Scoped negative theorem: three coordinate swaps need nine calls with arbitrary dirty workspace #1541 (Cinna Davis, open boundary statement).

Reproduce

Standard library only. Registered replay (Python 3.12):

python3 -B scripts/research_package.py --package research/four-swap-r4-cnot
python3 -I -B research/four-swap-r4-cnot/verify.py          # quick, ~60 s
python3 -I -B research/four-swap-r4-cnot/verify.py --full   # + obstruction sketch, ~9 min

Expected final lines:

PASS_FOUR_SWAP_R4_CNOT validation t2r2_min=6 t2r3_min=6 t3r3_min=9 t4r4 excluded={2,4,6} residual={8,10} upper=12 ball3=3639920 ball4_est=273823737 alias_pairs=252 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/four-swap-r4-cnot outcome=partial

--check-only metadata validation passes. python3 -B -m unittest discover -s tests -p 'test_research_*.py' runs 20 tests with 5 pre-existing sandbox-local errors in test_research_onboarding.PackageTests (Windows CRLF fixture scaffolding, documented by #1298; unaffected by this change).

Remaining obligations

• The exact minimum at t = 4 is not settled: q = 8 and q = 10 are neither excluded nor realized. Smallest next task: sharded or on-disk 4+4 meet at the measured depth-4 ball with formal resolution of any hit; then 5+5, or a written J^2-coefficient-layer bound (sum E_i + sum_{i<j} E_i E_j = T - I in F_2[G]/J^3) for q = 10.
• Not covered: r > 4 (restored dirty extras beyond the target registers), t > 4, other target tuples in F_2[GL_2(F_2)], non-rank-one couplings, and any alphabet change beyond the card's declared one.
• No physical compiler, hosting, precision/tape, supplier or analytic-transfer obligation is touched. No kappa, paid-moment or selected-record change; upstream/ and frozen inputs are unchanged. No full cold replay, Lean, independent human review or hosted-CI success is claimed.

Credit and attribution

Credit Gareth Davidson (@bitplane) for the model question (#1333); commdata2338 for #1414, #1491, #1495, #1496; Cinna Davis (1337isnot1337) for #1541 and its recorded open boundary; Ali Romman (#468), eumemic (#288) and David Govea (#720) for the register-model lineage. Hanson Sun (@bbMisk) #1567 natural-alias warning is independently consistent with this package's alias control.

Contributed by Aaron Wang with help from opencode. Apache-2.0.

[Mister-Meeseeks]

Batch review at ad6b7d641b3e8576a95add7981fa75b6a8278fbc.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Exact natural-image meets exclude two, four and six calls for four coordinate swaps at four roles, with formal witnesses and alias controls. Eight and ten calls remain unresolved in this package; the large depth-four sketch is author evidence, not a completed exclusion.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.