Prove exact optimal QQ16 streaming sweep directions

[Arham-Tahir64]

Change and mathematical scope

PR #2722 prices its QQ16 bank-major streaming word with every scalar pass visiting sites in increasing order. This proves and supplies the exact optimum over choosing one increasing or decreasing full-site sweep per instruction: 11740N - 2714 closed two-tape word-address steps for every integer N >= 1, saving 126(N-1) against its 11866N - 2840 tariff.

The resulting scalar/native word is unchanged. Its full added local scalar allowance remains 17,472 primitives, including the predecessor's 24 extra unit ADDs; its native invoice is inherited. The construction retains the existing whole-window W/Z0 ownership conditions. This is a local access-protocol lemma and independent tariff check, with no new kappa or selected-frontier change.

Validation

One new opt-in package, research/qq16-bidirectional-sweeps/, contains the proof, exact schedules, standalone checker, receipts and source pins. The two windows improve from 3564N-896 to 3518N-850 and from 8302N-1944 to 8222N-1864.

The endpoint DP has at most four states. Every direction path has affine travel in N and the same cost at N=1, so minimizing at N=2 proves optimality for all positive integer N; the constants are not polynomial fits. The supplied path preserves each site's exact scalar order and dirty-Z0 restoration. Optimality applies only to these fixed separately closed instruction sweeps, not general scheduling, layouts, resynthesis or caching.

Validation on native macOS arm64 / CPython 3.12.14:

python3.12 -I -B scripts/research_package.py --package research/qq16-bidirectional-sweeps

Registered normal and optimized replays and five CLI groups pass. They cover independent literal address traces at N=1..32,64, exhaustive direction checks for 120 toy words / 6,216 path cases, all 223 bank deltas / 3,568 affine-edge cases, exact rational dirty-state comparisons, wrong-site and omitted-return controls, and malformed scope inputs. CLI groups cover Git-free read-only replay, source-byte corruption, typed result corruption, schedule corruption and equivalent gzip recompression. Metadata/pin checking, direct optimized CPython 3.14.7 replay and whitespace checks pass. The retained 202,523 source bytes match the independently fetched PR2709 Git blob exactly.

Task and overlap

The current-main task picker drew recursive interfaces, nonce 2b009b56cc2256f685cd5a8d0b321095, priority revision 5. joint-query-controller is refined to this specific access obligation. The task/source base is b017900dbb94d81afec343924c1a7fa6d540a7db; the isolated branch was fast-forwarded before submission to 84f8e607779b81c05edc962d4981d1f9b4b9bc9f, with guidance, selected record, runner and relevant inputs unchanged. Historical source pins remain intact.

Direct predecessors: Nick Petalas / PR2722 a9ec1a4f45f36d4ec131e43cf8528ea0e40ea4e8; ywz65 / PR2709 d7749c49e967b214537c0bd832e435e177e21d4f. PR2722 excludes other sweep schedules from its claim, so this extends rather than corrects that tariff. PR2651/2540 and the retained QQ16/native lineage keep their credit. TASK.json records scope key PR2722/fixed-bridge-word/bidirectional-full-instruction-sweeps. The final complete body snapshot covers 723 open PRs at 22:03 UTC and 1,434 Discussion records at 22:05 UTC, with no warnings. Exact predecessor heads are unchanged; no separate direction result was identified. This does not cover arbitrary diffs or unpublished work. No advance public reservation is claimed; the result will be reported in Discussion #2617.

Whole-window caller availability, retention, bit arithmetic, source-register implementation, instruction control, surrounding native movement, output consumption and the complete supplier moment remain open. No native-kernel or selected cold replay, formal-kernel proof or independent human review is claimed. Family decision: narrow and stop at this local lemma; unknown costs are not zero. The README gives the next whole-caller budget test.

Credit

Prepared by Arham Tahir with substantial OpenAI Codex assistance in task selection, derivation, implementation, checking and writing. New work is Apache-2.0. Credit Nick Petalas, ywz65, commdata2338, Ali Romman, starship-s, Sagar Thakrar, Ross Richardson and the retained geometry/supplier contributors. Bellman optimization and sweep reversal are standard; the contribution is this exact source-bound optimum and proof.

Open next questions

The README gives the next test: pin an actual producer/native/consumer schedule, verify whole-window W/Z0 availability and price the complete physical bill. The missing caller and unified movement/retention/control budget block a supplier conclusion; stop if that bill is unfavorable. Further local schedule searches are not the recommended next task.