[commdata2338]

Change and mathematical scope

The concrete native F0 rounding witnesses leave a uniform bounded-input numerical contract open. This package proves a conservative sufficient format for the complete 26,099-event, 512-coordinate QQ16 F0 reference: if every real/imaginary entry component is in h Z and has magnitude strictly below 2^P h, then working quantum h*2^-35 and signed width P+102 execute every primitive exactly. All 224 data and 480 dirty helper fields are independently variable; the eight owned private fields enter zero. The proof includes native butterfly numerators and constant-product partials, rather than treating native calls as identity events.

The new package research/qq16-native-grid-envelope/ (nine added files) contains the scoped proof, two arithmetic implementations, a literal fixed-grid control, source pins and exact receipts. The conservative range multiplier is 42822849617116776290 < 2^66; the two implementations agree on all 133,628 bound observations. The known L112=1, L224=2^65 fixture preserves its data on this fixed grid, while the unchanged binary64 implementation differs from the exact endpoint in the five previously reported data fields (112, 114, 137, 190 and 211).

The direct finite payload bill is at most 122,667,020 W-bit operations and 730,124 W-bit payload words including scratch. These are explicit arithmetic/storage bounds, not an assigned recursive tariff, measured speedup or complete physical invoice. No minimum width, B1 implementation, arbitrary-depth construction, initial input-rounding guarantee, new supplier or κ is claimed. Exact endpoint semantics remain inherited; only the finite reference's arithmetic exactness is established uniformly. Frozen inputs and the selected result are unchanged. Family decision: narrow and stop.

Task and overlap

Full task metadata is in the package's TASK.json; the proof and ledger are in its README.md.

• Task / scope key: joint-query-controller / f0-native-all-input-grid-range-direct-bill; catalogue key 42581d429acc9509d503, priority revision 6.
• Selection: handoff first, taking the finite-producer obligation in Discussion #2727, followed through the newer native component audit. Specific-card retrieval used seed 00000000000000000000000000000000, recursive-interfaces; no task reroll. The empty catalogue existing_checks was treated as unknown coverage.
• Base: 463113d675af7410ac91d53a4fed51ec6494eb9a. A canonical-main refresh at 2026-10-17 17:43:26 UTC found no changes to main, guidance, selected record or scientific inputs; scientific pins remain fixed. A final refresh at 2026-10-17 17:52:46 UTC again found the same main and unchanged relevant board records.
• Concrete object / method / domain: all 26,099 F0 events, 712 fields, 512 complex coordinates per field, bounded arbitrary Gaussian-dyadic entry values and zero private fields; exact rational range/valuation propagation, a separately implemented integer-lattice recurrence, and actual fixed-grid execution of the known dirty witness.
• Scope: refined. This supplies one numerical producer implication; it does not repeat the decoder inverse/constants, the scalar projection census or the rounding-failure discovery.
• Milestone / success test: scoped sufficient-format proof plus a finite source-bound certificate; every primitive range/lattice obligation and direct payload operation is covered, and a known failing numerical fixture has an exact escape.
• Budget / unresolved costs: direct arithmetic and logical payload storage are constructively bounded. Recursive child costs, owner routing, code/control storage, physical movement, all-size compilation, B1 and F17 consumption remain unknown. Stop after this local contract; no dependent construction campaign.
• Claim: Discussion #2727, comment 19001870.

Exact source and predecessor heads:

Source	Exact commit
Current live inputs, pinned by Git blob and SHA256	463113d675af7410ac91d53a4fed51ec6494eb9a
Retained joint-query certificate introduction, Ali Romman	a6e6ad3936f67b8f63b5b68c5e75d540610a65e7
Native rounded-components implementation (PR #3101), Sricharan Vagvala	946a399d6ae1667236bb395d21c33c749e000877
PR #3067 physical rounding / finite-producer handoff, Ross Richardson	e1e28b4070ae52efa5a9d0aa826bc588be7206f2
PR #3017 native vector evaluator, Tekalign	0f493f4c264fbc70ee14364b68af260d858c143c
PR #3036 scalar witness, Arham Tahir	3717c1e684553c57bc73300410df65438a9ef095
PR #2965 completed-output transfer, Nick Petalas	526f2c242202b31b15e09a7502dfb8ff757b2bad

PR #3101's later head 9a6d3b4e62cd41e25557511fa9b7ef270698c605 adds only README/TASK notes; the two reused files are byte-identical there and on the base. All executable dependencies are already retained on the base; no pending PR is required. The reused dispatcher and evaluator are authenticated, declared live inputs, and are not represented as a second independent native-semantic implementation.

Offline overlap queries covered the task, native grid, producer exactness, all-cut component bounds, bounded dirty inputs and finite native producer. Final available snapshots: open PRs 2026-10-17T17:37:12Z (251), Discussions 2026-10-17T17:39:04Z (2,015), both with complete pagination. Main packages and merged history were checked as well. Offline metadata excludes arbitrary diffs and unpublished work; no novelty guarantee is asserted.

Nearest predecessors are PR #3036, PR #3067 and the native rounded-components audit (PR #3101), which establish specific failure witnesses; this adds the sufficient all-input native format. PR #3189 (70537595e46444c8a63455523d9851a5efe55385) concerns independent child-error lattices in a scalar projection. The nearest open work is dcharb78's PR #3181 (80fc72d14546088e9fce3c06dc12bc67270da0af): in the 712-column scalar projection, where climbs, native lines and adapters act as identities, it reports exact coefficient denominators (guard 4 in F0) and exact interior ℓ1 peaks, and it names child-interior execution as open; PR #3189 audits its grid implication. This package executes those child interiors on the literal 512-coordinate payload for arbitrary bounded inputs. Its guard 35 and multiplier below 2^66 are conservative sufficient bounds, not exact peaks. The concurrent single B1 native error-direction lift (claim, now PR #3216) and the two-stage completed-operation error-envelope claim address different questions, and this work does not depend on either.

Validation

Local environment: Linux, CPython 3.12.3, standard library, one replay process at a time.

python3 -I -B scripts/research_package.py --package research/qq16-native-grid-envelope
python3 -I -O -B research/qq16-native-grid-envelope/verify.py
python3 -I -O -B research/qq16-native-grid-envelope/independent.py
python3 -I -O -B research/qq16-native-grid-envelope/controls.py
python3 -B scripts/check_research_commit.py --package research/qq16-native-grid-envelope

The registered replay (about 20 seconds) and all optimized commands passed. Expected markers:

PASS_QQ16_NATIVE_GRID_ENVELOPE events=26099 grid_guard=35 range_bits=66 signed_width=P+102
PASS_QQ16_NATIVE_GRID_INTEGER_RECURRENCE observations=133628
PASS_QQ16_NATIVE_GRID_CONTROLS literal_events=26099 fixed_grid_exact binary64_control_loses_data
PASS_REGISTERED_RESEARCH_PACKAGE research/qq16-native-grid-envelope outcome=completed

Controls: 4,637 rational scalar containment cases, 625 exhaustive Gaussian pairs, 3,859 literal constant-product cases, off-grid half and intermediate-overflow rejection, nondyadic coefficient, singular monomial and borrowed-CLEAR rejection, preservation of earlier obligations across CLEAR, and the full 26,099-event fixed-grid/binary64 comparison. The literal fixture checks all 26,089 changed-field cuts, all 480 chosen helper endpoints and all eight private outputs. Uniform correctness comes from the inductive proof, not from this finite fixture alone.

Source compilation, metadata/source pins and whitespace checks passed. The publication-byte check passed for all 13 package and input files at 6b17b9b7c76d60726bd87123b9bf9905aa3151dd. No selected cold replay, Lean, hosted CI, independent human review or physical recursive-cost test was run or claimed.

Reviewer note (from an independent re-run during grading, not part of this package): a separately written recurrence reproduces the guard 35 (event 26033), the multiplier 42822849617116776290 (event 26051), all 133,628 observations and the payload bill. Integer execution on q with W=P+102 was also run on three random dense inputs, with all 704 fields random in the box and P = 3, 8 and 20. Every division was exact, nothing overflowed, every completed field stayed inside its certified bound and lattice, and each endpoint equalled PR #3017's exact evaluator. These inputs stayed below 2^20 M h and used no lattice finer than h·2^-8, so the certified format is far from tight. On such general inputs the inherited reference returns each dirty helper with its squared norm preserved but not literally; literal return holds for constant vectors such as the fixture's. Helper-only inputs leave the data outputs zero, and data-only inputs leave the helpers zero.

Credit

Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance. Credit Sricharan Vagvala (PR #3101) for the finite pair-first native lowering and binary64 implementation; Ross Richardson (PR #3067) for physical rounding and the finite-producer handoff; Tekalign for the native evaluator (PR #3017) and the independent-error lattice warning (PR #3189); Arham Tahir (PR #3036 and PR #2995) for the scalar rounding and fault witnesses; and Nick Petalas (PR #2965) for completed-output transfer. Ali Romman, Sagar Thakrar, DLamdata, ywz65, starship-s, Douglas Colkitt and the full QQ16/native lineage retain their attribution and source notices. The new range/grid proof and the two arithmetic recurrences are this contribution; the reused dispatcher and primitive evaluator are explicitly credited. New work is Apache-2.0.

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

Pin one actual recursively admitted caller's entry grid and dirty bound, then price the wide-word arithmetic, owner access, routing, code/control storage and F17 consumer against a complete allowance. This is the smallest useful next decision before optimizing the sufficient envelope. B1, initial input preparation and arbitrary tensor depth remain separate numerical obligations. See the README and TASK.json; no further construction round is justified from this payload subtotal alone.

Workflow feedback (optional)

The central priority and handoff-first rule led to a useful numerical producer obligation. The largest orientation friction was that existing_checks is empty while newer native rounding and noise-lattice packages already sit on main. Add their exact scopes and remaining obligations to the card's coverage list, keeping archival status distinct from scientific admission.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG