[commdata2338]

Change and mathematical scope

Adds research/qq16-root-cutoff-parking/. PR #4865 proves a parked-volume law at the fixed cutoff 356 and explicitly leaves changing root descriptors open (its Open next question 2). The actual historical QQ16 backend stops at w < 73 or w^1000 < d, where d is the global root-width descriptor. This package keeps that dependence and proves a uniform envelope with T_d=max(73,ceil(d^(1/1000))):

H_d(w)=Theta((w/T_d)^s) for every nonterminal width, with constants independent of both parameters. Consequently the diagonal recurrence has H_d(d)=Theta(d^(999s/1000)), where s is the predecessor's existing moment root. The explicit constants and proof are in research/qq16-root-cutoff-parking/PROOF.md.

The maximum rank-48 path gives the exact selected-C row-division depth D. An independent inverse characterization uses B_1=T_d, B_(j+1)=73 ceil(B_j/48). All selected-C divisions are positive and exact iff F^D divides the positive incoming rows, where F=14336 Gamma. At the historical d=isqrt(2^16949), T_d=356 and D=13997, versus the cutoff-free sufficient depth 13999.

This is a scoped proof with exact finite corroboration, not installed stock reduction, a complete physical cost comparison, or a κ step. The physical term retains N Gamma v H_d(w); v is not held constant as d grows. No admitted RootPlan with w=d is constructed. The fixed-cutoff predecessor remains correct. Candidate-only nonchild/shell/control work, actual stock/layout admission, native/precision/restoration and outer obligations remain open. The selected #5413 result is unchanged.

Task and overlap

• Task: joint-query-controller, recursive-interfaces, priority revision 7, scope key 73442cf25caf1ed45f11; refined to original-QQ16-root-dependent-cutoff-parking-and-stock. Full task card: research/qq16-root-cutoff-parking/TASK.json. Selector seed: 45a2393c7289f16f036f033bc27ae9e7.
• Head a3be1f66a44e1e6d7fc798b238fea8470edd0a63 on base main 32890bc5de034ab8fc93ac88ead7279a6a416d04.
• Primary predecessors: #4865, 8f8f518bd1a40baa30434ab18e86fd851d122b28; #4753, 44e2c79344e0090d4e829e0dde4f533d8d0273ec. The coarse recursive-binding proof is retained at e708f02bf78c3ed6d770cedf00724ebc5ce928d1. All five live inputs use exact base-commit Git blobs/SHA256 pins; there is no pending-source execution dependency.
• Caller/comparison lineage: Install the actual QQ16 caller with a complete return and protocol bill #4166 0b082d22a0e9a442d264ab0633e35ef26c790630; Share closed primitive engines to remove parent parking at child calls #4167 6749e458800f461535d64cc91c017552d9f55e5f; Prove native-history action coupling and reduce the QQ16 comparison #4172 59b46625a8b2f0dfff3e38ba0949b4247423938a; Prove a complete terminal-band QQ16 cost exclusion #4177 4cdda81c2ad99ffe5add0ac5d894094c3cca101a. Their costs are not re-certified here.
• Claim and canonical thread. This answers the recurrence/row-divisibility part of Prove a critical-exponent law for QQ16 parked volume (no new κ) #4865 Open next question 2; it does not repeat Certify QQ16's first nonterminal C8 band and internal parking term #4753's one-generation band or Compose QQ16 protected returns across installed callers #4790's external-return implementation.
• Final refresh: 2026-10-25 05:38:54 UTC; canonical main was still 32890bc5de034ab8fc93ac88ead7279a6a416d04 at 05:51 UTC. Complete cached coverage: 371 open PRs at 05:31:21 UTC, 4779 Discussion records at 05:33:05 UTC, all 7402 base local notes (7404 including this package). Scoped searches found Prove a critical-exponent law for QQ16 parked volume (no new κ) #4865 and this claim. Arbitrary diffs and unpublished work remain outside coverage.

Bounded domain: a written all-integer recurrence theorem, exact finite widths 0..8192 at ten cutoffs, and 30 giant integer cutoff boundaries. The work stops at this analytic obligation; pricing one removed expense does not justify a family construction expansion. No recursive paid/target invoice changes, so this is not a signed-invoice improvement claim.

Validation

CPython 3.12.3 on Linux, standard library only:

python3 -B scripts/research_package.py --package research/qq16-root-cutoff-parking --check-only
python3 -B scripts/research_package.py --package research/qq16-root-cutoff-parking
python3 -B scripts/check_research_commit.py --package research/qq16-root-cutoff-parking --check-source-history

All pass. Scientific status: PASS_QQ16_ROOT_CUTOFF_PARKING; wrapper: PASS_REGISTERED_RESEARCH_PACKAGE. Normal and optimized modes compare the complete typed deterministic receipt.

Evidence: 81930 table entries; 70 independent breadth censuses; 584 rank/residue checks; 30 strict-power boundaries; inverse-depth agreement; 85 unchanged-source node-method and 304 child-method cases; 12 rejection controls. Exact stock and Gamma/volume arithmetic is checked at Gamma=1, 2, 7. The source methods execute over explicitly substituted integer descriptor primitives, not physical Tape/Count or native execution. The histogram is a cited, authenticated premise; its census and numerical moment root were not replayed.

Committed-byte/history checks cover all 13 package/input files. Official export/check passes; both direct modes pass with empty PATH in a read-only Git-free export, with all 14 files unchanged. A corrupt source input is rejected before execution. Direct export runs took about 2 seconds each, with maximum child RSS 48220 KiB; these are verification-resource observations, not algorithm benchmarks.

No production payload, complete ordinary/outer or selected cold replay, Lean, Docker, hosted CI, independent human review or proof-kernel verification was run.

Credit

Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance. Nick Petalas / #4865 supplies the weighted-leaf argument, potentials, affine shift and fixed-cutoff law. Brian / JoyboyBrian / #4753 supplies the histogram, fresh-row/full-bank premises, cutoff and parking floor. Chris R / Clear1nsight (#4166, #4167, #4172, #4177 and the recursive-binding proof) supplies the caller/comparison and recursive-binding lineage; Jose Manuel the physical machinery. Preserve ywz65, Sagar Thakrar, Tekalign, Ali Romman, Arham Tahir and all retained source notices. The new contribution is the uniform root dependence, exact selected-tree depth/divisibility characterization and checks. Apache-2.0; no literature-priority claim for the underlying proof techniques.

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

The README and TASK.json retain two concrete obligations. Before reducing stock, bind every RootPlan padding, prefix-geometry and depth consumer to one admitted family. For affordability, supply the complete upper invoice for candidate-only positive intervals at actual nonterminal geometry, preserving every later call and using return credits once. The current result authorizes neither step automatically.

Workflow feedback (optional)

certificates/selected-result.json and the README name #5413, while the October 23 selection-refresh note at the top of current-priorities.md still names #4610; following the note's link to the selected record resolves it. Generating that note from selected-result.json, or keeping only the link, would stop the two from drifting apart. The exact open questions in #4865 and #4744 made a bounded successor possible without repeating completed caller work.

🤖 Generated with Claude Code

https://claude.ai/code/session_01Jur4aiU6BBSmBiaRjkbECf