[aliromman]

Change and result

A new six-field query caller raises conditional κ from 0.00315120 to 0.00328621476572836015146308, a 4.28455% increase over PR6375 with the same pinned native source. The resulting conditional time bound is O(n (log n)^(1−κ)). The selected frontier is unchanged.

The improvement comes from a different way to combine recursive suppliers. The caller uses four binary children of sizes 7,2,2,7, with full-stage projector ranks 2,3,3,2. It pays two additional whole idles and obtains six independent outputs with stage products ABCD, AB, CD, empty, B, C; every output also contains the central transform. The complete caller uses 18 native calls, 67,150 paid jobs and 13,652 productive fields, with rank-mass surplus 760.

Two steps make this possible:

1. Normalize the complete field. An ordinary translation-based map turns the native central query into an involution, then restores the required Gaussian outputs. Its scalar coefficients stay outside tensor powers. All normalization and basis-change work is charged.
2. Permit paid higher-rank stage projectors. The six-field root combines full and thin queries in a pattern unavailable to the preceding binary root. The literal 570-event caller charges every native call, helper debit, query, idle, adapter, owned lifetime and route.

The final composition includes PR6353, PR6372 and PR6375, with each source change and productive debit applied at all 18 calls. Those native mechanisms retain their original credit. The final refinement of completion/compiler parameters adds only 0.000449325% over the local 0.00328620 checkpoint; the 4.28455% comparison is the complete caller improvement over pinned PR6375.

Proof and how far the method goes

The package proves the exact raw funding limit for the fixed source and classified root family:

0.00328621476572836015146309 < κ* < 0.00328621476572836015146310

The complete claim is less than 2 × 10^-26 below this limit. The proof covers all positive binary child sizes and tree shapes. An exact recurrence, comparisons of all eleven seven-leaf tree histograms, and a concave infinite-tail bound establish the optimum. A direct rational comparison proves that the selected profile dominates throughout the crossing interval.

The family has six arbitrary fields, seven square-zero queries, four independent forward full stages and at most two extra whole idles. All outputs contain the central involution; scalar bases are independent of it and of the stage variables. The bound concerns the recursive invoice; the chosen word has its own complete ordinary-work budget. This is a family-specific limit. Equality at κ* is not certified by the strict completion argument.

Supplementary research includes the all-width normalization proof, the minimum-idle classification, a separate reverse caller, an active-space reduction for extra central-free outputs, and all-size price exclusions for seven specified fifth-stage extensions. It also preserves a helper-consuming module and a five-call reflection border. The border has two nonzero errors and no complete recovery/precision construction, so its favorable formal price is not a κ result.

Review map

Read	Purpose
Root proof and literal caller	New construction, whole-field normalization and exact endpoint
Final composition proof, receipt and source manifest	Native binding, all source changes and the complete invoice
Family proof and exact optimum	Quantified scope, all-size exclusion and strict approach to the limit
Research map, boundary modules and author review	Earlier advances, useful negative results, repaired proof gaps and remaining questions

Four scoped packages are added. The PR6353 intermediate composition retains the executable source closure. Earlier published results and nine exact null-witness rejections are documented with their scope and attribution; they are not counted as new gains here.

Verification

Local registered normal/optimized replays pass on CPython 3.12.13. Checks cover 799 native columns, 638,401 reverse coefficients, 7,488 physical caller contacts, two rational moment engines, completion/compiler bounds, ordinary costs, and 54 strict plus four functional outer predicates. The separate reverse caller checks 4,624 formal polynomials. Source and semantic rejection controls, committed-byte/source-pin checks, and isolated read-only exports pass. The new dominance guard rejects a deliberately invalid bound under optimized Python. Exact evidence and limits.

python3.12 -B scripts/research_package.py --package research/mixed-root-contact-composition
python3.12 -B scripts/research_package.py --package research/mixed-projector-root-headroom

Hosted CI passed at head dee2663ba0918be86993fd9d63c8d64b020f4c83: the planning job and all four package jobs are green. Each package passed normal and optimized replay and the unchanged-input checks on Ubuntu 24.04 / CPython 3.12.15.

The five inherited assumptions remain: native_all_columns, owned_erasure, compiler, precision_tape, analytic_recovery. The constants are theoretical; the uniform word and rounded multiplier are not materialized. Independent scientific review and formal-kernel acceptance remain open. The earlier PR6376 scalar audit does not discharge physical erasure. No practical speedup or unconditional multiplication theorem is claimed.

Task, overlap and credit

Task: joint-query-controller, refined to a new mixed-projector root. Base: canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8. Native source: PR6375 at 247b5c319b9f58fe9ee61dbf80697f8a9a429831; all other exact dependencies are in the manifests. Main, recent open PRs and active Discussions were refreshed before publication.

Nick Petalas's PR6380 studies the older nineteen-leaf binary topology; this PR changes the root and uses eighteen native calls. His PR6391 prices a different three-stage overlap proposal. The active PR6375 scalar audit remains separate. Canonical investigation.

New root, normalization, composition, proofs and author checks: Ali Romman with substantial OpenAI Codex assistance; no subagents. DreamingOfClouds supplied PR6375 with Claude Opus 5.5 via Claude Code. PR6348 supplies the binary module; PR6349 and its predecessors supply the native lineage. Credit Cinna Davis for the query/caller interfaces, Jacob Alan Sussman for productive recursion, and zxv and Cao Yifan for the interval/compiler lineage. All native, helper, geometry, BIT and analytic notices are retained. Apache-2.0.

Open next questions

Independent review should check the normalization, classification and complete native-to-caller bill. A larger raw gain must change a stated source or root hypothesis; a border extension first needs an affordable recovery and recursive precision proof. Concrete follow-ups.

This PR changes only the four research packages. It requests neither a merge nor a selected-frontier update.