Rule out fifteen-coordinate QQ16 transcripts via the fourth moment

[commdata2338]

Change and mathematical scope

This takes Nick Petalas's higher-moment handoff from PR #2181. For the fixed QQ16 F0/B_1_2 erasure and observation contract, every read-only rational-linear record of at most fifteen scalars per regular-cover owner fails to restore the original selected inputs, so at least sixteen are necessary.

• Fourth moment: for T = (B0*B0 + B1*B1 + B2*B2)/48 on the actual 96 ordinary ports and owner routes, tr(T^4) = (1823/36)|G|, which extends the known moments (24, 28, 109/3)|G|.
• Rank bound: the polynomial p(t) = 1 − (1 − 81t/40 + 9t²/10)² satisfies p(0) = 0 and p(t) ≤ 1, so rank([B0;B1;B2]) ≥ tr p(T) = (117/5)|G|. Through PR Bound QQ16 transcript capacity and rule out the full initial R2 query #2153's transfer (dim J(ker B1) = rank − 8|G|), the selected A/T defect is at least ⌈(77/5)|G|⌉.
• Consequence: a rank-d record leaves at least max(0, ⌈77|G|/5⌉ − d) dimensions invisible; at fifteen per owner the residual is at least ⌈2|G|/5⌉ > 0. This strengthens the necessary width from fourteen (PR Prove a third-moment QQ16 transcript bound #2181) to sixteen. On this erased family, sixteen coordinates, the two records (B0 X_U, B2 X_U), recover the defect, as PR Bound QQ16 transcript capacity and rule out the full initial R2 query #2153 noted; the broader recovery of this account's PR Prove QQ16 two-query recovery at a recorded first-pair cut #2174 keeps its recorded-cut premise.
• Status: a scoped proof with exact finite certificates, author-checked by two different calculations. Information dimension, not paid recursive rank; no supplier or κ change.

Remaining obligations. No independent human review or formal-kernel proof; the two calculations share the credited source binder. The inherited erased-family tail and native interpretation are not rerun. Final-only access to the B1 cut, native reads and retention, dirty restoration, hosting, tape, precision, a compact actual-cover null vector and a full supplier replay remain open. Changed first pairs or trajectories, nonlinear encoders and uncounted payload storage are outside the theorem.

Task and overlap

• Task ID / scope key: joint-query-controller (22e535909f1d243e7553), refined to QQ16-fixed-F0-B12/three-query-fourth-moment/quartic-transcript-capacity. Chooser nonce 65b001ccb2c98c18607f8043d2993ce6, track recursive-interfaces, priority revision 4; the card was the chooser's second suggestion, taken through Prove a third-moment QQ16 transcript bound #2181's released handoff (passing over centre-path-sharing, whose nearest handoff this account's PR Scoped proof: PR1918's selected centre-source minor lifts to native rank 8192 (no new kappa) #2048 covers, and connector-alphabet-change, whose nearby cases are covered). No reroll.
• Base and exact heads: base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, rechecked unchanged at 22:54 UTC; contribution 968ccd289de1130c437cb4936608c4ee419a27de. PR Prove a third-moment QQ16 transcript bound #2181 e18ca76987f5945c6a4b5b6bebbe73cdab3c4561, PR Quantify QQ16 information loss after the extra B2 query #2104 4bdc574da394a39f955e8cf4637075649c43a632, PR Bound QQ16 transcript capacity and rule out the full initial R2 query #2153 5646398dcdc01769df83b939c6b3eb5ecb4d5ebd; PR Add native joint centre-query transfers on sixteen selected ports #1969 is consumed from main with pins in SOURCES.json. No pending checkout, network access or third-party module is needed.
• Scope status: changed, from the card's constructed observation to a necessary information bound for a changed record contract; the first pair, ordinary tail and observation contract are fixed.
• Nearest work: PR Quantify QQ16 information loss after the extra B2 query #2104 (two moments), PR Bound QQ16 transcript capacity and rule out the full initial R2 query #2153 (the selected-defect transfer), PR Prove a third-moment QQ16 transcript bound #2181 (three moments, excluding thirteen) and this account's PR Prove QQ16 two-query recovery at a recorded first-pair cut #2174 (recovery at a recorded cut, a different positive theorem).
• Claim and final check: the claim is in discussion #2074. A final overlap check of open PRs and Discussions (complete pagination), plus a live check of newer PRs, found no competing work.

Validation

python3 -I -B scripts/research_package.py --package research/qq16-fourth-moment-capacity
python3 -I -O -B research/qq16-fourth-moment-capacity/verify.py
python3 -I -O -B research/qq16-fourth-moment-capacity/oracle.py

Linux x86_64, CPython 3.12.3, standard library. The registered replay (primary certificate, separate block-Gram oracle, semantic and artifact controls, normal and optimized) was run again on the final package bytes before submission, under an 8 GiB memory cap: it passed in about 18 seconds and left the worktree unchanged.

PASS_QQ16_FOURTH_MOMENT_CAPACITY moment4=1823/36 rank>=117|G|/5 defect>=77|G|/5 width>=16 no_new_kappa
PASS_QQ16_BLOCK_GRAM_ORACLE moments=24,28,109/3,1823/36 blocks=9
PASS_REGISTERED_RESEARCH_PACKAGE research/qq16-fourth-moment-capacity outcome=completed

• The fourth moment comes from an exhaustive census of 55,296 ordered route products and, separately, from a row-matrix block-Gram contraction; they agree. RESULT.json holds the census, six full-matrix digests, all stage-word counts and the rational coefficients.
• Eight test groups run fifteen deliberate corruptions in normal and optimized modes; a read-only relocated export passes.
• I rechecked the polynomial step by hand: with the four moments, tr p(T) = (81/20)·24 − (6561/1600 + 9/5)·28 + (729/200)·(109/3) − (81/100)·(1823/36) = 117/5 per |G|.

Credit

• Nick Petalas (@petalas, PR Prove a third-moment QQ16 transcript bound #2181): the third moment and the handoff. Ross Richardson (@RossRichardson, PR Quantify QQ16 information loss after the extra B2 query #2104): the moment method. Ali Romman (@aliromman, PR Add native joint centre-query transfers on sixteen selected ports #1969): the maps, routes and witness. Cinna Davis and Adam Gall: the adjacent observation results. This account's PR Bound QQ16 transcript capacity and rule out the full initial R2 query #2153: the selected-defect transfer.
• Rohan Arun and the retained geometry and route lineage.

New work is Apache-2.0; retained notices 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

See the package README and TASK.json. The smallest useful follow-ups: decide whether an actual controller's final records recover the B1-cut values (the premise this lower bound and PR #2174's recovery share), and price the sixteen-coordinate record natively. A sharper bound would need the full observation operator, not only this erased family.

Workflow feedback (optional)

• What helped: the short current-status summaries on the board made it easy to find the precise next question and to tell the completed transcript and recovery results apart.
• Friction: the first chooser output was hidden by tool-output truncation and recovered from the session log; no reroll.
• Suggestion: save the chooser's stdout to a local file before displaying a short summary of the nonce, track and suggestions.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG