Bind helper277 lifetime to exact residue arithmetic and charged decode

[Clear1nsight]

PR4017 supplied a reversible rounded residual, but local invertibility did not establish the real helper lifetime's linear cleanup or its recursive allowance. This contribution binds helper277 in both current pinned PR3945 providers to49 scalar touches, five rank-one physical climbs, and49 inverse-cleanup touches on private buffer981. It also explicitly corrects PR4017's rank-five-cost shorthand: rank5 is the completed residual/owner width, not one paid chronological call. The old macro theorems/counts remain intact.

The constructive addition is exact dyadic arithmetic over (Z/(2^b−1)Z)[i], a literal X/CNOT/Toffoli implementation with no additional workspace, and an explicit endpoint numerical decoder. Ring linearity preserves the complete source dirty-helper identities; no rounding record is needed. All unused bit codes have a total reversible extension, while ring/numerical semantics require canonical residues. For input component integer boundA, M>2048A recovers exact output on gridDelta/32. The optional decoder is fully charged. This finer-grid contract needs a real consumer before adoption.

At b32, the five actual physical climbs plus98 touching scalar instructions cost29,766,862,848 Toffoli before optional decode and all other provider work. This is a conservative constructive invoice, not a speedup, recursive fit, selected stock coefficient, new kappa or supplier admission. The source's five rank-one charges have no authenticated scalar-gate conversion here. Different numerical contracts/precision prevent treating8 modular additions versus18 earlier word primitives as an equal-work benchmark.

Validation: source-byte authentication; all2,560 phase-bearing physical-climb columns and512 terminal-residual columns; all1,360 small generalized-adder states and336 literal elementary-adder states;4,352 complete pair states including unused codes and398 elementary pair cases; actual source primal/dual/boundary/private-buffer identities; exact numerical endpoint recovery; phase, late-add, cleanup, missing modular-wrap, even-modulus and fixed-grid rejection controls. Normal and optimized registered replay, Git-free replay, exact committed-byte/source-history gate and official export receipts are reported in the publication handoff. Hosted CI is pending at submission.

Package: research/staged544-exact-residue-lifetime/. Scope, full ledger, reproducible commands, exact sources, explicit PR4017 correction and open next questions are included. The separate all-cut height bound assumes common-grid logical inputs and is not the b32 endpoint benchmark; physical nonempty entrance adapters need their own conversion.

Credit: Dugongue/PR3945; dataisfire; Sagar Thakrar's exact auditor and its Arham Tahir lineage; DreamingOfClouds; chafreaky; Cinna Davis; Jeremy Kothe; roger-sato; all retained native/owner/outer contributors; Barenco et al. Lemma7.2 via the credited PR4017 reconstruction; Chris R / Clear1nsight's PR3886/3987/3994/4017 source chain. Frozen notices remain exact. Chris R / Clear1nsight, substantial OpenAI Codex assistance. Same-team review is not independent human or proof-kernel validation.

Frozen publication: exact head 95c8edf315397a7f113d3a165f94716359c60c93, sole parent canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90, 29 added files. Root independently matched all 373594 published bytes. Registered normal/optimized, committed-byte/history, official export and read-only Git-free normal/optimized replays passed. Main and newest open-PR overlap delta refreshed at 11:25 UTC; no conflicting scope observed. Source claim: #4017 (comment). Public source-cost correction: #4017 (comment).