[commdata2338]

Change and mathematical scope

PR1736 leaves projector compatibility of its mixed width-seven/eight placements open. This package shows that every retained placement for k=0,...,74 keeps the entire local q-perp block in one slot or in the remainder. The builder packs a contiguous prefix starting at zero, and its first slot has width at least seven. A general prefix lemma therefore extends this account's PR1307 completion-free projector argument to these particular mixed placements.

The audit checks 296,325 representative helper slots and 166,500 child/copy partitions, with zero local splits, and preserves all 75 committed idle histograms. The 32-replica product accounts for 9,482,400 helper occurrences across alternative scenarios; it is proved by an injective family index, not independently replayed per replica. All 111 rational port projectors are checked. A rank/window-preserving split of k=1's first two width-seven factors gives different individual projectors under two rational completions while preserving their aggregate child.

This is a scoped geometric proof plus finite certificate, not a width-seven provider, a new price, a packing optimum, supplier admission or kappa claim. Package: research/bit-mixed-width-projectors/; the proof is PROOF.md, with read-only RESULT.json replay. Selected records and frozen predecessors are unchanged.

Task and overlap

Task card:

• Task ID / scope key: bit-helper-contract / bfb61f829a81535adeae, refined to PR1736/mixed7-8/prefix-packed/q-perp/no-completion. Chooser nonce 8cd45dd933439a35e236a028d99f1a85, track bit-mechanisms, priority revision 4.
• Base commit: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. Exact predecessors: incorporated PR1736 b4fd72163234c164d9a05f2f7dac9c57dcd7fb66; PR1307 4cb255ef445e2abc3ff8845fa01210b974e2cf6c; selected PR589 a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46. No pending implementation dependency. Every consumed main file has an exact Git blob and SHA256 in SOURCES.json.
• Concrete object/question: PR1736's actual wopt.solve(k, Fraction(1,570), True) plus verify.build placements, all 75 k values, all 111 ports, nine stages and replica-local families. Do individual factors need a local orthonormal completion?
• Scope status: changed. The original H2/K scalar examples have retained semantic-D coverage; this takes PR1736 Open next question 3 and PR1307 Open next question 2 instead. It is a hosting-interface sublemma.
• Changed restriction: extend PR1307's selected width-eight census to the specified mixed-width prefix placements. Width alone does not establish a split.
• Method/domain: exact rational projector algebra, independently checked source levels/windows and masks, typed role coverage, all remainders, a general prefix proof and finite mutation controls. No new candidate search or optimization claim.
• Nearest overlap: PR1736 supplies the placements and prices; PR1307 supplies the retained geometry argument; PR1281 supplies the window-free relaxation. PR2423's concurrent geometric audit is on the distinct R438 width-eight allocation, and the current class-U P111 plane-upgrade investigation concerns a different decoder object. PR2465's deletion ladder was considered but its slack pattern already has PR1281 coverage, so that work was not repeated.
• Full cost / observable improvement: every existing rank charge is retained and no operation is added. The required improvement is a discharged geometric sub-obligation, not a lower numerical bill. Physical conversion costs remain open.
• Success/stop test: source-bound complete coverage or an explicit split/counterexample, then verification and handoff within the bounded session. The 75-case certificate completes that scope.
• Artifact/reproduction: python3 -I -B scripts/research_package.py --package research/bit-mixed-width-projectors; predecessor credit below.

The claim is in discussion #1289. The final shared-snapshot overlap check covered 812 open PRs including drafts (2026-10-15 16:22:35 UTC) and 1,276 Discussion records (16:27:10 UTC), with complete pagination. Queries included the task card, 1736, width7-windows, mixed-width projector, q-perp split, and adjacent packing terms. Cached/offline warnings were retained; arbitrary diffs and unindexed closed work are not covered. No-match results are not novelty certificates.

Canonical main was fetched again at 2026-10-15 16:26:44 UTC and remained the exact base above. Guidance, priority revision, selected result and all consumed files were unchanged. The full local card and refresh decision are retained in TASK.json without changing historical source pins.

Validation

CPython 3.12.3, Linux x86_64, standard library only:

python3 -I -B scripts/research_package.py --package research/bit-mixed-width-projectors --check-only
python3 -I -B scripts/research_package.py --package research/bit-mixed-width-projectors
python3 -I -O -B research/bit-mixed-width-projectors/verify.py
python3 -I -O -B research/bit-mixed-width-projectors/test_verify.py

The first two commands passed in the checkout. Both optimized scripts passed in a Git-free relocated export containing only the package and declared inputs. Expected audit marker:

PASS_BIT_MIXED_WIDTH_PROJECTORS k=0..74 slots=296325 split=0 ports=111 controls=9 no_new_kappa

The main verifier rejects nine semantic controls. Six separate test groups pass, including 3,456 exact rational head/tail basis columns across 27 source fixtures, all 86 small width-seven/eight prefix words fitting the source capacities, changed completions, wrong exterior axes and gapped/overlapping placements. Real exported-source corruption rejects before execution; changing a receipt's integer zero to Boolean false also rejects. Syntax under warnings-as-errors, source bindings, metadata and whitespace checks pass. The required PASS_R439_BSECTION_STARS first-example control passed on all 661 columns with eight controls; PASS_RESEARCH_BASELINE_BINDINGS_ONLY also passed.

Unchecked obligations: a physical width-seven provider and its helper-to-host intertwiners, correlated dirty-state realization of a whole call, owner activity beyond the source window model, H1-H5 or a replacement interface, native and full selected cold replay, all-size compilation, precision/tape and analytic transfer. The predecessor DP and builder generate the audited inputs; their optimum and endpoint are not independently re-proved here. No formal-kernel proof, hosted CI success or independent human review is claimed.

Credit

Sagar Thakrar supplied PR1736's placement construction, temporal analysis, prices and explicit open question, and PR1023's width-packing model. This account's PR1307 supplied the source-level/projector argument. mightyOil/Philipp supplied PR1281's mixed-width relaxation. DreamingOfClouds supplied PR589/510/463; eumemic supplied PR555 and the carrier skeleton; DaysSky supplied PR442; Zhihao Chen supplied PR386; chafreaky and the retained geometry/audit/moment contributors retain credit. Existing source files and Apache-2.0 notices are unchanged. The new contribution is the prefix lemma, the all-75 mixed-placement application and the completion-ambiguity controls, not a new optimization of their placements.

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

Bind an actual width-seven provider to k=1 child0/copy0, whose host projector is exactly K_q: specify its entrance/exit maps and the helper-to-host intertwiner, then charge conversion and restoration. A future allocation that intentionally splits q-perp needs a shared completion and exact individual slot projectors; the two-completion witness provides a regression fixture. Neither follow-up is a supplier-admission conclusion from this audit.

Workflow feedback (optional)

The explicit PR1736 geometric handoff was useful after nearby packing patterns proved already covered by PR1281. A useful navigation improvement would link the mixed-width question directly to the builder's prefix layout and the mask & 255 in {0,254} gate, so future work distinguishes a split requiring basis data from a complete-call physical-hosting obligation.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG