Research: adaptive QQ16 reads still need 66 ports; 45 optimal sets

[commdata2338]

Change and mathematical scope

Add research/qq16-adaptive-coordinate-reads/: a scoped proof and finite certificate for adaptive, side-effect-free coordinate access on the rational QQ16 hidden-input family of PR2354.

PR2354 establishes the fixed-set threshold of 66 ports and explicitly excludes adaptive choices. This contribution proves that every exact deterministic reader needs at least 66 distinct source-port reads on every input, regardless of adaptive choices, nonlinear computation or stopping rules. For input-independent random seeds, almost-sure exactness and termination on each rational input give the same lower bound almost surely by a countable-intersection argument.

The finite source geometry further classifies all 45 optimal terminal read sets. Every optimal set contains 22 mandatory ports and 44 of 45 coordinates participating in one relation. Port 77 can replace 44 specific ports in the predecessor's attaining set. Explicit rational decoders attain all 45 sets on the credited family. A proposed uniqueness strengthening fails and is deliberately rejected by a control.

These are information-access counts, not per-owner widths, native-line costs, recursive rank charges or a new kappa. The reader receives only complete original ordinary-port arrays and the zero inherited observations on this family. No source preprocessing, dense observation, changed call, input-dependent header or internal physical intervention is granted. Full QQ16 recovery, native access/retention/restoration, hosting, precision and all-size supplier admission remain unchecked. The selected frontier and frozen inputs are unchanged.

Task and overlap

Base: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 (canonical main). Contribution commit: aed06c8f8d60ce97b7eb3a7c6c17ad12cbe1e863; main was rechecked unchanged at 07:17 UTC before submission. The final canonical fetch at 2026-10-15T07:03:27Z found no change to main, revision 4, guidance or selected inputs.

Exact predecessors:

• PR2354 / kyzer1023: 308ebc7405303bb7ed91b012bb705949c6091a16; fixed-coordinate threshold, rank-74/rank-66 family and 17-owner certificate idea.
• This account's PR2326: fb7d64cd4a51c49bc56920305b0aca64011a420c; coordinate-access contract and hidden-family trajectory.
• Adam Gall's PR2093: 8b21a900eb520e4f11ea177b310f66316f8e8983; common-axis bit-function family and cancellation.
• Ali Romman's PR1969 source packet is retained on the exact base above. Every consumed main file, Git blob and SHA256 is in SOURCES.json, including the selected replay archive, QQ16 packet and two proof/witness references. There is no pending-PR executable dependency. PR2354 and PR2326 were re-fetched unchanged before handoff.

The canonical claim is Discussion2205/comment18942816. The complementary second-axis investigation tests a different hidden family against the original fixed attaining set; this work retains the original family and changes the query decision rule. PR2368, head e5d4d78dcfdd55d323fcbc60df477b9f25c7abff, addresses a different full ordered kernel after a dense old-chart read and is context only.

Shared cached overlap was checked before claiming and before submission. Entry coverage: 570 open PRs at 2026-10-15T06:28:56Z and 1082 Discussion records at 06:31:18Z. Final coverage: 583 open PRs at 06:55:23Z and 1111 Discussion records at 06:57:32Z; complete pagination in each source, expected offline warnings. Targeted terms included adaptive coordinate, adaptive tap, optimal read set, and 66-port. Metadata search excludes arbitrary diffs and is not a novelty certificate.

Task card:

Task ID / scope key: joint-query-controller / 22e535909f1d243e7553;
  QQ16/PR2354-first-axis-family/adaptive-coordinate-decision-trees/exact-recovery.
Base and exact predecessors: listed above.
Object/question: exact recovery of H(c)=B0 h on V=ker(R_U), with complete
  ordinary coordinate arrays selected adaptively; classify every optimal leaf.
Scope status: refined. The central follow-up changes PR2354's fixed-read-set
  premise. It supplies a lower bound for that repair, not a completed supplier.
Changed restriction: query indices and stopping times may depend on prior reads;
  retain the same rational family, exactness requirement and observation alphabet.
Method/domain: transcript-fiber proof on all V over Q; exact randomized extension
  with an input-independent probability space; rational/integer source certificates,
  all 67 eligible candidate sets, and finite exhaustive decision-tree controls.
Nearest overlap: PR2354 supplies the fixed-set premise; PR2326 supplies the family;
  the second-axis and dense-record investigations change different hypotheses.
Optimistic complete budget: reads, computation and retention are granted free.
  Fewer than 66 distinct scalar-bearing ports still cannot recover this family.
Success/stop: close the quantified implication and optimal-set classification;
  stop before physical controller synthesis or a claim about other families.
Time/compute: bounded 60-minute session; standard-library Python, one replay at a time.
Authors/artifact/reproduction: credits below; research/qq16-adaptive-coordinate-reads/;
  python3 -I -B scripts/research_package.py --package research/qq16-adaptive-coordinate-reads.

The choice is an explicit central-interface follow-up override. The recorded card retrieval uses nonce 68a7f11a97c2e55fc020480ffc2b66fd, recursive-interfaces; no novelty is inferred from that nonce. Full structured metadata is in TASK.json.

Validation

On CPython 3.12.3, standard library only:

python3 -B scripts/research_package.py --package research/qq16-adaptive-coordinate-reads --check-only
python3 -I -B scripts/research_package.py --package research/qq16-adaptive-coordinate-reads
python3 -I -O -B research/qq16-adaptive-coordinate-reads/verify.py

The registered normal/optimized runs and four artifact tests pass. Direct output:

PASS_QQ16_ADAPTIVE_READS defect=66 optimum=66 optimal_sets=45 nuisance_coordinates=29 controls=11 no_new_kappa

Evidence includes:

• A 66-by-66 integer minor directly on the 88-dimensional coefficient kernel, determinant 174085318024506601157689344; a matching all-owner upper bound proves exact rank 66. This independently rechecks a credited premise.
• 896 registered nonzero-chart route identities, 17 actual owner evaluations, 29 explicit varying kernel coordinates and the one exceptional port 77.
• All 67 candidate read-set ranks, checked both on a kernel basis and by the complementary-column rank identity; 45 have rank 66, 22 have rank 65. All 44 exchanged decoders and 11,968 owner/basis equalities pass.
• Eleven rejecting mathematical/receipt controls; exhaustive adaptive choices on a 16-input binary oracle (147 cells), plus dense-vector and nonlinear-target controls.
• Relocated read-only replay; a corrupted optimal-set receipt, changed source bytes and a dropped source pin all reject under optimized Python.

git diff --check passes. Applying patch.mbox to a temporary index at origin/main reconstructs the exact committed tree 7ff75e844a5e98504b3b58b35f9ba3bee6c2fbdf. Receipt formatting is immaterial; every canonical proof-relevant JSON field and scalar type is compared.

The registered replay was run again on the final package bytes before submission, under an 8 GiB memory cap: it passed in about 15 seconds and left the worktree unchanged. The counts are consistent: 96 - 29 = 67 candidate ports, of which 45 read sets have rank 66 and 22 have rank 65.

Not run: full native QQ16 trajectory/children, selected cold supplier replay, proof-assistant kernel, hosted CI, all-size/precision/outer multiplier checks, bounded-error randomized analysis, or the complementary second-axis witness. Written quantified proofs and finite checks are explicitly distinguished.

Credit

Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.

Credit kyzer1023 for PR2354's fixed-set threshold and whole-owner rank certificate; commdata2338 for PR2326's coordinate-access question; Adam Gall for PR2093's hidden family; Ali Romman for the native QQ16 contracts and source packet; Cinna Davis for the selected shear/tap investigations; EdDiboi for complementary fixed-coordinate parity work; and the retained supplier/compiler lineage. No predecessor executable is imported or vendored. The new result does not claim to rediscover the credited rank or fixed-set threshold. Retain the existing QQ16 notice; new proof and code are 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

All 45 optimal sets omit the same 29 ports C minus {77}. A hidden-input witness supported on those ports screens all 45 sets at once. This account's PR #2384 gives one: a second-axis witness on ports 44, 49, 58 and 70 with zero permitted observations through the full schedule. All four ports lie among the 29 that every optimal set omits, so that witness is invisible to each of the 45 sets. That combination was checked here only by port membership; the witness's own proof and checks are in PR #2384. Only after full recovery survives should native access and retention be priced.

A separate question is a bounded-error distributional lower bound. It requires an explicit input distribution and success criterion; the zero-error countable-intersection argument does not establish it. See the README and structured TASK.json questions.

Workflow feedback (optional)

Priority revision 4 helped steer the work to a concrete central interface. The shared board exposed several already-claimed alternatives. The main mathematical detour was useful: a source check falsified uniqueness of the optimal set and exposed 44 valid exchanges through port 77. A compact machine-readable list of a package's explicitly excluded hypotheses would make follow-ups easier to find without re-reading long thread histories.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG