Research: bound rounded QQ16 producer errors with bounded dirty helpers

[commdata2338]

Change and mathematical scope

For the literal QQ16 F0→B_1_2 pair, a completed-operation relative tolerance u <= 2^(-beta-delta-102) is sufficient for PR #2965's two stage-output allowances when original data norms are at most H and dirty-helper norms are at most 2^delta H, for integers beta, delta >= 0. The error ledger permits nonzero ADD, SCALE, child, adapter and cover-map defects, preserves exact private CLEARs, and carries actual helper error from F0 into B1.

The new package is research/qq16-rounded-producer-envelope/: a conditional interface proof, exact finite certificate and standalone checks. It reproduces and credits the retained Horner magnitude bounds; the new delta is the relative-defect derivative ledger and noisy two-stage composition. Separate tolerances u_F <= 2^(-beta-delta-71) and u_B <= 2^(-beta-delta-102) also suffice. With PR #2965's separate decoder allowance epsD <= H 2^(-beta-1), its inherited completed-output theorem gives decoded error at most (7587/8192) H 2^-beta.

This is not a finite register-width theorem or an implementation meeting the local contracts. Exact helper-bit restoration, native internal arithmetic, physical hosting/movement and an affordable F17 consumer remain open. No child tariff, scalar word, supplier admission, selected frontier or kappa changes.

Task and overlap

Task card: joint-query-controller, track recursive-interfaces, priority revision 6; catalogue scope key 42581d429acc9509d503, refined scope qq16-f0-b12-rounded-completed-operations-dirty-bounded. The complete card is in the package's TASK.json. Its existing_checks was empty, meaning unknown coverage.

• Question/domain: quantify completed-operation error and intermediate norms on all 712 fields and all 26,099 F0 / 27,019 B1 events, for arbitrary positive payload tensor depth under the inherited native isometry and sweep contracts. Integers beta, delta are nonnegative; private initial dirt is arbitrary and cleared before use.
• Method: exact positive path polynomials at z=1 and their five relative-defect derivatives; a separate integer implementation checks all P/J cuts, and reverse adjoints check category totals. Native maps are bounded by their norms; no signed physical-state identity replacement is used.
• Scope status: refined to one conditional producer/dirty-state implication in PR #2965's handoff. The claim is in discussion #2727.
• Milestone and stopping rule: scoped interface lemma and finite certificate; stop after the numerical implication. The existing operation counts remain liabilities. Actual arithmetic, exact helper return, ownership, movement, storage, recursive tariffs, centre/coverage convention and F17 consumption remain unknown; no favorable complete cost is claimed.
• Success/falsification: complete source-bound cut coverage, separately implemented arithmetic agreement, explicit noisy-helper carry, and rejected omitted-defect/ownership/feedback mutants. A missing operation, unsupported native contract or uncharged extra handoff invalidates use of the resulting tolerance.

Exact scientific/submission base: 463113d675af7410ac91d53a4fed51ec6494eb9a. Canonical main was refreshed at 2026-10-17 17:46:17 UTC and again at 17:57:59 UTC; both checks found main unchanged and priority revision 6. Frozen inputs are preserved.

Exact dependency heads:

Source	Head
Literal retained joint-query package	a6e6ad3936f67b8f63b5b68c5e75d540610a65e7
PR #2965 completed producer-error transfer	526f2c242202b31b15e09a7502dfb8ff757b2bad
PR #2736 decoder completed-defect proof	eba0d6111a773cd13f5c0afb4354af2c85d04c3a
PR #2707 line-normalized decoder	efa51a6c7ffec6edd556a98deedfe9deeba65809

Seven live inputs, Git blob IDs and SHA256 digests are in SOURCES.json; these source bytes are retained on the stated base. No pending executable dependency is imported.

Nearest overlaps: #3181, head 80fc72d14546088e9fce3c06dc12bc67270da0af, computes signed scalar-projection cancellation/grid/adjoint quantities; this result charges nonzero scalar errors in a positive native-norm envelope. #3189, head 70537595e46444c8a63455523d9851a5efe55385, audits child-error lattice closure; no lattice or single-direction native lift is claimed here. #3017, head 0f493f4c264fbc70ee14364b68af260d858c143c, establishes a dirty-input obstruction, and #3067, head e1e28b4070ae52efa5a9d0aa826bc588be7206f2, supplies concrete F0 rounding evidence. This result adds a sufficient bounded-dirty contract for the pair. Two concurrent PRs on the same card have different premises and outputs: #3226, head 6b17b9b7c76d60726bd87123b9bf9905aa3151dd, proves an exact fixed grid for F0 alone (every primitive exact, no B1), and #3216, head af5b2ff1c0afa3100c961fa11b4d6e1fe18714cb, lifts one #3189 B1 error direction through the native suffix. The same submitter's merged #2651 concerns early-output retention, not this numerical implication.

Overlap checks of a cached, completely paginated snapshot searched the task, QQ16 with Horner, and QQ16 with producer error, plus current main and the canonical thread. The closing snapshot had 253 open PRs at 2026-10-17 17:49:11 UTC and 2,017 Discussion records at 17:52:04 UTC. A later snapshot (274 open PRs at 18:32:44 UTC) adds #3216 and #3226 above as the only newer QQ16 PRs. Cached coverage can miss newer changes, unindexed diffs or unpublished work; it is not proof of novelty.

Validation

Standard-library CPython 3.12.3, Linux x86_64:

python3 -Werror -I -B scripts/research_package.py --package research/qq16-rounded-producer-envelope --check-only
python3 -Werror -I -B scripts/research_package.py --package research/qq16-rounded-producer-envelope

Both pass; the full replay takes about 10 seconds. python3 -Werror -B scripts/check_research_commit.py --package research/qq16-rounded-producer-envelope also confirms all 15 package/input files match the committed bytes. The registered replay runs the complete primary checker normally and with -O, the separate integer/adjoint checker, and optimized behavioral tests. Principal markers:

PASS_QQ16_ROUNDED_PRODUCER_ENVELOPE stages=2 cuts=53098 offset=102 conditional_completed_operations no_new_kappa
PASS_QQ16_PRODUCER_ENVELOPE_INDEPENDENT integer_cuts=53098 reverse_rows=6 categories=5
PASS_REGISTERED_RESEARCH_PACKAGE research/qq16-rounded-producer-envelope outcome=completed

Exact evidence: 53,098 changed-field cuts; 53,034 permitted defect sites; the same complete P/J digest in rational and integer arithmetic; six reverse covectors matching five category totals; source-derived calibration of the inherited F0 endpoint bound 58. The all-cut/terminal data derivative bounds are J_F = 10632335516156845470267/4096 < 2^62 and J_B = 34778728470688788237067911/8 < 2^82.

Ten behavioral groups cover noncommuting isometries with coherent state-dependent errors, explicit positive polynomials, omitted feedback, dirty magnitude and noisy helper reuse, omitted native/map defects, erased private transients, ownership/uninitialized reads, invalid private sweep contents, scalar coefficient changes, exact tolerance inequalities, endpoint geometry and source corruption. A Git-free export passes direct optimized replay. Native physical semantics are inherited, not re-certified by these toy fixtures. The selected cold replay, full native operator replay and formal-kernel verification were not run. No independent human review is claimed.

Credit

Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance. Credit Nick Petalas for PR #2965's endpoint transfer and PR #2736's completed-defect analysis; Ali Romman and the retained QQ16/native team for the words and sweep contracts; ywz65 for PR #2707 and #2540; Arham Tahir for PR #2681, #2995 and #3036; Tekalign for PR #3017 and #3189; Ross Richardson for PR #3067 and earlier rounding/restoration contracts; Sricharan Vagvala for the native component-rounding reference (PR #3101); and dcharb78 for PR #3181's separate scalar-projection study. The predecessor packages' credit sections also name starship-s, Sagar Thakrar, DLamdata, Douglas Colkitt and the remaining source contributors; those notices are unchanged. Positive majorants are standard; no claim to invent that method is made. New material is Apache-2.0, and frozen predecessors are unchanged.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the derivation, the implementation, the checking and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

The README and TASK.json specify three follow-ups: implement one finite, priced completed native-child/scratch contract; decide and meet exact borrowed-helper restoration if required; and join a concrete F17 consumer and owner/sweep schedule to the full recursive invoice, including centre coverage. Start at an actual child/half-ADD interface and retain scalar rounding in the budget. The current result supplies conditional error allowances, not those implementation or cost premises. No further dependent search is proposed.

Workflow feedback (optional)

The current priority correctly points from decoder constants to producer obligations. The card's empty existing_checks requires reading a dense numerical chain to distinguish ideal grids, child-defect grids and primitive finite formats. A short coverage entry linking #2965, #3181 and #3189 would make the remaining implication easier to select without repeating existing work.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG