Prove QQ16 two-query recovery at a recorded first-pair cut

[commdata2338]

Change and mathematical scope

This answers the two-record handoff of this account's PR #2153. With the two initial ordinary records (B0 X_U, B2 X_U), a decoder recovers pair 0's original selected A/T inputs for arbitrary inputs in all four bank pairs, not only #2153's erased family, under an explicit post-B1 observation contract.

• Observations used: the two pre-F0 records q0 = B0 X_(0,U)^initial and q2 = B2 X_(0,U)^initial (eight coordinates per owner each); pair 0's selected values at the cut just after B_1_2, through p_A = X_A + Y_A, p_T = X_T + Y_T and d_A = Y_A; the forward histories f0 and f2; and the completed-X descendant C = Y_(0,U)^final after the retained terminal (X,Y) → (Y,−X), which is the named final Y, not the former-Y hole.
• Result: a decoder D with D O = J, where J selects the four original A/T blocks of pair 0, so the complete observation kernel carries no selected-input defect for pair 0. The key tail identity is Y_(0,U)^initial = Y_(0,U)^final − K_(2,U) f2. The decoder reconstructs the transient F0 A output rather than saving a copy of it.
• Premise: access to the post-B1 selected cut is granted, as in Bound QQ16 transcript capacity and rule out the full initial R2 query #2153's contract. A controller that offers only final selected values must still preserve or reconstruct those cut values; that implication is not supplied.
• Status: a scoped ordered identity and information lemma, conditional on the retained logical QQ16 maps. It is not physical execution, a final-only controller, recovery of every bank from two pair-0 reads, or a recursive-cost invoice; sixteen query coordinates are not identified with sixteen paid rank units. No supplier or κ change.

The package: research/qq16-two-query-kernel/ (eight new files): PROOF.md with the observation contract and the ordered derivation; verify.py with three zero residuals in a free noncommutative operator algebra, symbolic propagation of the 17-event tail, all 160 basis inputs of two independent rational models with exact rank [O;J] = rank O checks (18 and 27), the actual packet's port partition, Gram identities and selected inverses rebound to main, and twelve rejecting controls.

Pair binding: the pinned tail receipt does not record the pair of each local call, so the package binds the pairs to the complete selected recipe (stages 0 to 8 on pairs 0,0,0,1,1,2,2,3,3; F2 on pair 0, F8 on pair 3), which also matches the retained PR #377 recipe. The old zero witness does not distinguish a wrong assignment such as pair = stage//2 mod 4, but the general-input formula does, and a control rejects it.

Remaining obligations. Recovering p_A, p_T, d_A at the B1 cut from an actual controller's final records; native reads, storage and cleanup; the full paid invoice, ownership, tape and recursive precision; other pairs' selected inputs; an actual full-cover or native cold replay; all-size admission; a sharp transcript minimum. No formal-kernel or independent human review.

Task and overlap

• Task ID / scope key: joint-query-controller (0e34adbc1dd119161c3a), refined to QQ16-fixed-F0-B12/two-initial-ordinary-records-B0-B2/complete-observation-kernel. Chooser nonce ae4dafd62a69238bac3a959fcb3d7803, track bit-mechanisms, priority revision 4. Following revision 4's preference for a live central handoff, this was taken over the chooser's suggestions (bit-helper-contract, whose immediate scopes are covered; bit-new-register-model, where this account already has four results). No reroll.
• Base and exact heads: base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, rechecked unchanged at 22:02 UTC; contribution f79470a7931f3943360c10d15b428e0e4e479c22. Predecessors: PR Add native joint centre-query transfers on sixteen selected ports #1969 (a6e6ad3936f67b8f63b5b68c5e75d540610a65e7, merged, consumed from main with pins in SOURCES.json); this account's PR Bound QQ16 transcript capacity and rule out the full initial R2 query #2153 (5646398dcdc01769df83b939c6b3eb5ecb4d5ebd, rechecked unchanged); PR Quantify QQ16 information loss after the extra B2 query #2104 (4bdc574da394a39f955e8cf4637075649c43a632); PR Rule out one mixed affine helper for the QQ16 interface #2122 (7045301a8c15eb4408e3e9d80969c9d50a914cde). No pending executable is imported or adopted as a supplier.
• Scope status: refined. The changed premise allows two independently retained ordinary-source records, raising Bound QQ16 transcript capacity and rule out the full initial R2 query #2153's one-record information budget; the first pair, ordinary tail and observation contract are unchanged.
• Nearest work: PR Bound QQ16 transcript capacity and rule out the full initial R2 query #2153 (excludes records of at most 12 scalars per owner on its erased family and leaves this two-record kernel open); PR Quantify QQ16 information loss after the extra B2 query #2104 (B2 alone); PR Rule out one mixed affine helper for the QQ16 interface #2122 (B0 + M·B2); PR Bounded B2 observation result for the QQ16 witness #2079; Adam Gall's PR Prove a QQ16 first-forward A logger obstruction #2086 and Exhibit a QQ16 kernel invisible to nonzero-chart source queries #2093. The retained paired-log-ordered-decoder package works on a different interface with independent external dirty logs.
• 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-two-query-kernel
python3 -I -O -B research/qq16-two-query-kernel/verify.py

Linux x86_64, CPython 3.12.3, standard library. The registered replay (normal and optimized runs plus three artifact tests) was run again on the final package bytes before submission, under an 8 GiB memory cap: it passed in about 8 seconds and left the worktree unchanged.

PASS_QQ16_TWO_QUERY_KERNEL pair0_selected_recovered symbolic=3 tail=17 basis=160 controls=12 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/qq16-two-query-kernel outcome=completed

• Nine wrong decoders are rejected: discarding initial Y dirt, omitting or reversing the F2 correction, the wrong terminal sign, dropping either initial read, dropping the changed source or T baseline, and reversing selected section roles. Corrupting a query coefficient, the first negative bridge or the local pair assignment is also rejected.
• The fixtures test failure modes; the universal statement rests on the written ordered proof and the free-algebra identities. The native compiler and helper restoration are inherited from PR Add native joint centre-query transfers on sixteen selected ports #1969 and not rerun. No hosted CI result is claimed.

Credit

• Ali Romman (@aliromman, PR Add native joint centre-query transfers on sixteen selected ports #1969): the ordered conserved maps, selected sections, native words and ordinary witness schedule.
• Ross Richardson (@RossRichardson, PR Quantify QQ16 information loss after the extra B2 query #2104): the three-query rank method. 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 single-query and compressed-record tests. 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 query obstructions.
• Zhihao Chen (PR κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377): the retained complete recipe and signed terminal. The retained Rohan Arun, Jeremy Kothe, DreamingOfClouds and complex-supplier lineage. Jose Manuel's paired-log-ordered-decoder is a useful comparison on a different interface.

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

From the package README:

1. Can one actual controller recover p_A, p_T, d_A at the B1 cut from its final selected records? First step: trace its literal selected program. This decides whether the lemma supplies the missing decoder or more paid retention is needed.
2. What is the complete native cost of the two pre-F0 records and the decoder's B2 read of a completed-X descendant? Start from the actual endpoint frames and charge storage, all chart maps and cleanup together. Blockers: physical access, hosting, tape and precision.
3. Can 13 to 15 query coordinates suffice under the same cut contract? PR Bound QQ16 transcript capacity and rule out the full initial R2 query #2153 excludes at most twelve on its family. Test any proposed compression against the full observation operator, not only that family.

Workflow feedback (optional)

• Friction: the frozen zero-witness receipt omits each local call's pair, and a wrong assignment preserves the zero witness while changing the general-input formula. The complete selected template resolved it.
• Suggestion: successor witness receipts could record each local event's pair and reverse fields and bind the complete recipe, keeping the frozen historical receipts unchanged.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG