[petalas]

Summary

A new complete-fiber CRT tail schedule uses one reusable suffix-slice buffer and at most 19V/2 payload actions. This saves at least 7V/2 against PR4187's cited 13V, and halves its staging capacity from 2S to S. The input is preserved, the buffer is erased, all payload heads return, and exactly one control is prepared per prefix.

control 0: source first slice -> output; source second slice -> output
control 1: source first slice -> buffer; source second slice -> output;
           buffer -> output and erase; return buffer head

Claim type: scoped written all-size construction with exact finite corroboration, produced by Nick Petalas. Exact reads/writes/moves are V+SB, V+2SB, 4V+4SB, where B counts switched fibers and S=2^H R. No traffic optimum, new kappa or selected-frontier change is claimed. Research stopped at this first verified improvement.

Credit ToikY / PR4187 at 45fc0750436ae5cc14b95fba81c030a3d7df3412 for the complete-fiber interface and two-buffer lowering; Cinna Davis / PR4100 for the elementary tail and amortization; Hiroki KUMAZAKI / PR4174 for placement context; Vlado Vojdanovski, Jose Manuel and the retained physical/compiler lineage. Buffering is classical. Predecessor claims are cited premises; no predecessor executable or scientific suite is imported, replayed or audited.

Evidence

Local CPython 3.12.15, macOS 27.0.1 arm64, standard library:

• Both warning-strict modes pass 23,786 cases, 787,968 forward payload-bit instances and 6,099,591 forward physical actions. Every case also executes the inverse. The separate sparse trace interpreter and address oracle agree. Fifteen rejection controls pass. Generated receipts bind all seven package-file hashes and agree between modes.
• All Boolean tables occur for the tested fiber counts at most eight; all binary payloads occur through volume eight and basis payloads through volume 32. Larger specified cases reach L=12, R=16 and H=10. These are not exhaustive larger-domain claims.
• Registered replay, metadata/pins, committed bytes, POSIX modes and declared source-history ancestry pass at 5b02379b9f60e7d3ff652a07ebec42fb0ec5648a. Individually timed final normal/optimized commands took 3.04/2.99 seconds, below separate 120-second limits. These are test durations, not runtime improvements.
• Official export integrity/input coverage passes. Both read-only, Git-free, empty-PATH replays match the worktree receipts; all eight exported files retain their bytes and modes. Actual-file mutations omitting erasure, omitting source return, and forcing a stale control fail respectively on scratch state, head state and the invoice.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/crt-tail-single-buffer/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/crt-tail-single-buffer/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/crt-tail-single-buffer
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/crt-tail-single-buffer --check-source-history --check-modes

Other hosts may substitute their CPython 3.12 path. Expected markers: PASS_CRT_TAIL_SINGLE_BUFFER, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES. The universal argument is in PROOF.md; finite checks do not replace its review.

Limitations and unrun obligations

The exact invoice covers payload reads, writes and unit head moves. Marker visits, encoded counters, evaluator preparation, both ordinary exchanges and intermediate erasure remain separately charged in the written conditional bound. The interpreter executes payload actions, not their complete finite-state controller. The actual numerical evaluator, full reservoir placement, ordinary callee and complete CRT/precision/outer budget are unrun. Sparse incomplete fibers are outside scope. All five selected framework assumptions remain. Independent human and formal proof-kernel review, a full rounded multiplier, selected cold replay, Lean and Docker are unclaimed.

Open next questions

• Can source retreat replace the remaining buffer with a favorable complete head-travel and encoded-control bill? No optimum is established here.
• Join the staging policy to an actual transported CRT layer, including its evaluator, ordinary exchanges and intermediate cleanup. Those implementations remain external premises.

Related open work

• Audit conditional CRT last-K two-slice streaming tariff #4187 studies the same complete-fiber middle stage. Its 13V/2S schedule is the credited comparison; this contribution changes the staging policy and proves 19V/2 and S.
• Audit uniform Beneš compiler with charged sequential arena scans #4192 supplies an encoded Beneš compiler. It addresses a missing evaluator component, not this payload staging bill.
• Reduce valid matched-packet parking by V+r steps (no κ) #6592 constructs matched-record stack parking. Its packet contract and quantities differ; this contribution supplies no competing parking value.
• Execute resident SCALE 2 on one ribbon with 14,212 fewer operations #6608 implements resident SCALE 2 on one ribbon. It uses a different physical entry/exit contract; its operation totals are not this complete-fiber tail tariff.

Merged #4100/#4174 supply caller context; merged #6185 supplies a different arbitrary-bit wrapper. Live handoffs, open/account PRs, main and merged object/formula searches, cached overlap and reservations were checked before claiming and committing. Canonical main stayed 7c49fe32c55de02c2c873655c804b16e4c3fef0a, revision 8/selected PR6030. Sparse notes, arbitrary diffs and unpublished work remain coverage limits.

The completion refresh examined all 16 account open PRs created since the advance CLAIM at 2026-11-03T12:08:14Z, including this PR. None of the other fifteen computes this complete-fiber tail-stage quantity. The adjacent #6608 physical circuit is distinguished above. No additional scientific computation compared unrelated workloads.

CI notes

Both hosted checks, plan and replay (research/crt-tail-single-buffer), passed at exact head 5b02379b9f60e7d3ff652a07ebec42fb0ec5648a in run 41634310964. The completed checks were read once with gh pr checks 6611 --repo CrocSwap/integer-mult-bounds; separate run metadata confirms the matching head, completed status and success.

All registered checks also passed locally. Hosted replay certifies execution of the declared normal/optimized finite checks and bindings. It does not provide independent human or formal proof-kernel acceptance of the written all-size construction, nor discharge the evaluator, ordinary exchange, encoded-controller or full CRT obligations. No CI failure remains.

Merge danger

Door: two-way. Blast radius: isolated additive research package. No shared source, frozen evidence, predecessor credit or selected result is edited.

Advance CLAIM: #1166 (comment)