[jwr404]

Result

In #4939's formal model M_R(F2[GL_n(F2)]), t coordinate-SWAP targets cost exactly 3t paid calls, for every n >= 2 and R >= t. Free scalar mixing and restored dirty registers are allowed.

• For three swaps the minimum is 9. Spectator coordinates cannot reach the (3,3,2) eight-call row budget. This answers Prove six-call two-SWAP cost with arbitrary address spectators #4939 open question 1.
• For two swaps, the same cut reproves Prove six-call two-SWAP cost with arbitrary address spectators #4939 without its parabolic projection.
• At n = 2 it is PR1989's inversion cut.

Proof idea. Use the regular representation of GL_n(F2) and the cut A = {x : row0(x) < row1(x)} for a linear order on the nonzero vectors.

• The swap moves all of A out.
• CNOT(0,1) and CNOT(1,0) each cut exactly one increasing pair per line {a,b,a+b}, so each cuts |A|/3 for every order.
• CNOTs writing spectator rows cut nothing.
• A spectator read adds a vector uniform off the line. Its cut is a linear function of the line positions, and averaged over all orders it equals |A|/3 exactly, because the expected spread of a random 3-subset is (N+1)/2.
• Subadditivity of the crossing rank then gives calls >= 3t. The word a,b,a per register attains it.

No new κ. The selected exponent stays 60403/20000000. This is a negative formal-model screen, not a native construction.

Task and overlap

Card bit-new-register-model, scope changed (see TASK.json). Base: canonical main cef4b52f2a59009865bf8eabde1be3f159063703, priority revision 7.

find_overlap.py found PR1541 (three swaps at n = 2), the four-swap and label-saturation notes, and the GL2 corollary (3t at n = 2). None treats n >= 3 beyond #4939's t = 2. That is a search result, not a novelty proof. CLAIM.

The cut was found by column generation over subsets of the GL_3 regular representation (scipy/HiGHS, scratch only, not in the package). The written proof does not depend on that search.

Test plan

python3.12 -B scripts/research_package.py --package research/swap-row-order-cut --check-only
python3.12 -B scripts/research_package.py --package research/swap-row-order-cut
python3.12 -I -O -W error -B research/swap-row-order-cut/verify.py
python3.12 -B scripts/check_research_commit.py --package research/swap-row-order-cut

All four pass locally, on Linux with Python 3.12, in about one second.

The verifier checks:

• the exact cut counts in the regular representation for GL_2 and GL_3 over all orders, and GL_4 over eight orders;
• the line formula and the spread criterion;
• the all-order averages;
• the spread identities for N <= 63;
• the crossing-rank lemma on actual paid factors;
• random subadditivity checks;
• the attaining formal words.

Four mutations make it fail.

--check-source-history was not run locally because the checkout is shallow.

Open next questions

Do swaps on different coordinate pairs, or coordinate 3-cycles, admit the same per-target cut? See TASK.json.

Credit

• Mihir Srivastava (Prove six-call two-SWAP cost with arbitrary address spectators #4939), for the model and the open question.
• commdata2338 (PR1414/PR1495).
• Ross Richardson (PR1989) and the GL2 corollary.
• PR1541.
• Gareth Davidson (PR1333).

Prepared by James Reed with substantial Anthropic Claude (Claude Code) assistance. Apache-2.0. No independent human review.

🤖 Generated with Claude Code