Bound QQ16 transcript capacity and rule out the full initial R2 query

[commdata2338]

Change and mathematical scope

At the fixed QQ16 F0/B_1_2 cut (the witness from PR #1969), the known erased-input family carries an original selected (X_A, X_T) defect of dimension at least ⌈88|G|/7⌉. An added read-only rational-linear record of rank d leaves at least max(0, ⌈88|G|/7⌉ − d) of it invisible.

Consequence: the full initial R2 X query, which has at most 8|G| outputs on this family, does not restore the selected inputs: at least ⌈32|G|/7⌉ defect dimensions stay invisible. No record of at most twelve rational scalars per regular-cover owner does either. Thirteen per owner is necessary in this model, not shown sufficient. This answers the full-initial-R2 X handoff in discussion #2074.

Status: a scoped proof (written in PROOF.md) with exact finite source checks. It is information dimension, not paid recursive rank or native cost; no supplier or κ change.

What is new: PR #2104 bounds the A loss after the B2 read with a three-query trace bound. This package rederives that bound against main's packet (tr K = 1152|G|, tr K² = 64512|G|, so rank[B0;B1;B2] ≥ 144|G|/7) and measures the joint A/T defect, dim J(ker B1) = rank[B0;B1;B2] − 8|G|. That lets the record read every original coordinate, including selected inputs and cross-owner mixing, so the bound covers the full R2 X read as well as the narrower observation models of PR #2079 and PR #2122.

The package: research/qq16-transcript-capacity/ (eight new files) holds:

• PROOF.md: the observation and capacity contract, the rebound three-query premise, the joint defect isomorphism, the record-capacity inequality and the explicit R2 X formula;
• verify.py: reads main's frozen QQ16 packet; checks the Gram identities of the 96 ordinary ports, route distinctness for each of the three stage pairs and the F1/F2 alignment routes against B1's; replays the retained formal-h ordinary tail. Exact rational fixtures with noncommuting selected sections test the ordered compensation, the defect inverse, the R2 X formula and a concrete invisible direction;
• eight semantic and parser controls, two receipt mutants and three artifact tests, including an optimized read-only relocation without Git and source and receipt corruption rejection.

Remaining obligations.

• No independent or formal-kernel review. No compact witness for the actual regular cover and no precision bound; the witness exists by the rank argument.
• Native full-word replay, legal endpoint access, chronological hosting, the all-size compiler and supplier admission are not addressed.
• Widths 13 to 16 and the complete observation kernel remain open. Nonlinear encoders, changed first pairs or schedules, and uncounted payload storage are outside the theorem; a separately saved transient F0 A output counts as additional information.

Task and overlap

• Task ID / scope key: joint-query-controller / 0e34adbc1dd119161c3a, refined to QQ16-fixed-F0-B1/full-initial-R2-X/arbitrary-linear-transcript/A-T-defect-dimension. Chooser nonce 714dbcab67b2539705b72cb500a3f046, track recursive-interfaces, priority revision 4. Following revision 4's preference for a live, unclaimed central handoff, this was taken over the chooser's generic suggestions (connector-alphabet-change, coverage-law-change, centre-path-sharing); no reroll.
• Base and exact heads: base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, rechecked unchanged at 21:02 UTC; contribution 5646398dcdc01769df83b939c6b3eb5ecb4d5ebd. PR Add native joint centre-query transfers on sixteen selected ports #1969 (a6e6ad3936f67b8f63b5b68c5e75d540610a65e7, merged) is consumed from main with blob and SHA-256 pins in SOURCES.json. PR Quantify QQ16 information loss after the extra B2 query #2104 (4bdc574da394a39f955e8cf4637075649c43a632) is method credit, not a code dependency; PR Bounded B2 observation result for the QQ16 witness #2079 (49df8190b158e3028e702b2c6b798117006f7249) and PR Rule out one mixed affine helper for the QQ16 interface #2122 (7045301a8c15eb4408e3e9d80969c9d50a914cde) are the narrower predecessors. No pending PR executable is imported.
• Concrete object: the original selected (X_A, X_T) map on W(ker B1), and whether a narrow read-only record of the complete initial input can preserve it, especially the full R2 X.
• Scope status: refined. The changed premise lets records read selected inputs and mix owners; the first pair, ordinary tail and observation contract are unchanged.
• Nearest work: PR Quantify QQ16 information loss after the extra B2 query #2104 (leaves the full R2 X open); PR Rule out one mixed affine helper for the QQ16 interface #2122 (excludes B0 + M·B2 for scalar rational 8×8 M); PR Bounded B2 observation result for the QQ16 witness #2079 (the B2 read); PR Prove a QQ16 first-forward A logger obstruction #2086 (changed A logger) and PR Exhibit a QQ16 kernel invisible to nonzero-chart source queries #2093 (an explicit nonzero-chart witness); philippeVV's first-pair catalogue finding in CLAIM: one paid pre-F0 B2 source observation (seed 3e62c004) #2074. Added value: the joint A/T defect and a capacity bound for every linear record.
• Claim and final check: the claim is in discussion #2074. A final overlap check of open PRs and Discussions (complete pagination) and the changed-file index, plus a live check of PRs opened since, found no competing work.

Validation

python3 -I -B scripts/research_package.py --package research/qq16-transcript-capacity --check-only
python3 -I -B scripts/research_package.py --package research/qq16-transcript-capacity
python3 -I -O -B research/qq16-transcript-capacity/verify.py
git diff --check

Linux x86_64, CPython 3.12.3, standard library, single process. The registered replay was run again on the final package bytes before submission, under an 8 GiB memory cap: it passed in about 6 seconds and left the worktree unchanged.

PASS_QQ16_TRANSCRIPT_CAPACITY defect>=88|G|/7 q<=12_excluded full_R2_X_insufficient
PASS_REGISTERED_RESEARCH_PACKAGE research/qq16-transcript-capacity outcome=completed

The quantified statement follows from the written proof; the fixtures are failure-mode tests, not sampling evidence. No hosted CI result is claimed.

Credit

• Ali Romman (@aliromman) and the joint-query contributors (PR Add native joint centre-query transfers on sixteen selected ports #1969): the maps, ordered sections, bridge semantics and the original witness.
• Ross Richardson (@RossRichardson, PR Quantify QQ16 information loss after the extra B2 query #2104): the three-query trace bound and the full-R2 X handoff.
• Cinna Davis (@1337isnot1337, PR Bounded B2 observation result for the QQ16 witness #2079 and Rule out one mixed affine helper for the QQ16 interface #2122): the extra-B2 and mixed-helper investigations.
• Adam Gall (@adamgall, PR Prove a QQ16 first-forward A logger obstruction #2086 and Exhibit a QQ16 kernel invisible to nonzero-chart source queries #2093): the adjacent logger and nonzero-chart obstructions.
• Rohan Arun, Jeremy Kothe and the retained lineage, whose notices are unchanged. 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

From the package README:

1. Can two query records recover the complete controller? (B0 X_U, B2 X_U) recovers the selected defect on this erased family. First step: characterize the complete observation kernel on all inputs, then test that pair on it. Blocker: this family is only a subspace, so recovering it is not enough for a controller or a favorable bill.
2. What is the exact required record dimension? The gap between the necessary width of thirteen and the sixteen-coordinate description of the family remains. First step: sharpen the exact rank of [B0;B1;B2] with the actual route algebra. Blocker: the regular cover is too large for naive matrix expansion, and a sharper rank would still not price a legal implementation.

Workflow feedback (optional)

• What helped: the handoff in CLAIM: one paid pre-F0 B2 source observation (seed 3e62c004) #2074 named the exact next question, so the session could go straight to it.
• Friction: --task joint-query-controller returned 1,484 possible matches, because nested starting points contribute the generic folder word evidence; exact formula searches were needed to find the handoff. This account's claim in Agent workflow feedback: friction, detours, and better next steps #1200 addresses that helper.
• Suggestion: handoffs should state observation timing explicitly. "Selected endpoints" can hide whether F0's transient A output is retained, and that changes the admissible information; this package states the boundary and includes a control for it.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG