[petalas]

Summary

Scoped recurrence proof with exact finite corroboration. Brian's #4753 leaves the deeper original C8 descendant tree open. This package derives an explicit two-sided parked-current-bank-volume envelope at the critical supplier-moment root:

H(w) = 0 below 356
H(w) = 1 + sum_r (n_r/14336) H(r floor(w/73)) otherwise
H(w) = Theta(w^s),   0.9983245373 < s < 0.9983245374
P(w,v) = 510662 Gamma v H(w).

The new result is the critical-exponent envelope, including finite floor/cutoff constants and the absence of an extra logarithmic factor. The moment root itself is inherited. The shift 3456/25 is sharp for the stated individual affine inequalities, not claimed optimal among all potentials. The proof uses a weighted leaf identity and two potentials. This contribution is produced by Nick Petalas.

Brian/JoyboyBrian #4753 supplies the published histogram, cutoff, fresh-row/full-volume relation and park-loop floor. Chris R/Clear1nsight #4166/#4167/#4172/#4177, Jose Manuel and the retained ywz65, Sagar Thakrar, Tekalign, Ali Romman and Arham Tahir lineage retain their source and interface credit. Their results are cited premises; their verifiers were not replayed.

Evidence

Before, the published parked-volume term covers one generation. After, the written proof bounds every generation of the fixed-cutoff recurrence. The exact verifier checks 8,193 widths for the leaf identity, eleven integer path censuses through width 1,048,576, eight rejecting controls and outward rational moment signs. It prints PASS_QQ16_PARKING_POWER_LAW; both registered normal and optimized checks pass locally. The normal and optimized direct commands took 0.15 and 0.16 seconds on CPython 3.12.15/macOS arm64; this is a replay duration, not a performance claim.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/qq16-parking-power-law
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/qq16-parking-power-law/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/qq16-parking-power-law/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/qq16-parking-power-law --check-source-history

Use a CPython 3.12 path available on another machine. Static parsing, the under-60-line function check and git diff --check pass. The verifier writes no files and prints a generated package-hash receipt. Publication-byte and source-history preflight pass for all seven package/input files at 8f8f518bd1a40baa30434ab18e86fd851d122b28, with source base 624de9dc66858af0945e5f8a0942488d62280d43 reachable in its history.

Limitations

This proves the explicit recurrence, with physical interpretation conditional on the published tree/volume and admission premises. The fixed descriptor has finite admissible widths; the asymptotic statement does not assert an unbounded RootPlan family at that fixed descriptor. No production C8/RootLayer, native internal cost, complete signed QQ16 comparison, physical stock improvement, supplier admission, new kappa or selected-frontier change is established. Independent human and formal proof-kernel review are unclaimed. The remaining physical and complete-cost obligations were not run or discharged.

Related open work

• Compose QQ16 protected returns across installed callers #4790 composes protected returns across the installed QQ16 caller; the original descendant backend bounded here is retained. No new D* versus B* sign is inferred.
• Share closed primitive engines to remove parent parking at child calls #4167 supplies the parent protected-return protocol and its costs. This package bounds the selected descendants' current-bank park subtrace, not that existing optimization.
• Prove native-history action coupling and reduce the QQ16 comparison #4172 proves native-history coupling and the complete symbolic QQ16 comparison. This package prices one removed expense while retaining its unpriced terms.
• research: exact QQ16 consumer budget obstruction under centre-64 law #3908 excludes a fixed QQ16 recursive-price profile under the center-64 law. Its admission budget is a different quantity from the parked-volume recurrence here.
• research: QQ16 52-bit interior dynamic-range collapse, backward adjoint ledger, and dyadic grid exactness #3181 studies QQ16 interior range, adjoint sensitivity and grid precision. No corresponding range or sensitivity number is computed here.
• QQ16: add an F0-B1 selected-state shear #2216 changes the F0/B1 selected-state demand with a local shear. This proof keeps the original child backend and makes no claim about the modified demand.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22T22:14:56Z were refreshed. The nine other open submissions, #4858, #4859, #4860, #4861, #4862, #4863, #4864, #4866 and #4868, study XOR subsequences, chart words/neighborhoods, cyclic checkpoints, Clifford batching, dirty-echo trees, root-code floors, subspace pricing and permutation workspace. None studies this QQ16 parked-volume recurrence or supplies a competing value for the quantities computed here. No additional predecessor computations were run.

Open next questions

1. Upper-bound the complete unmatched candidate nonchild/shell work at one admitted nonterminal geometry, preserving every later root call. The present removed-expense lower bound cannot settle the sign.
2. Bind a growing family of admitted roots, height-dependent cutoffs and sufficient row stock to this envelope. Physical admission remains separate from recurrence asymptotics.

CI notes

Both hosted checks passed on exact head 8f8f518bd1a40baa30434ab18e86fd851d122b28 in run 39864758331: plan in 3m50s and registered research/qq16-parking-power-law replay in 1m7s. Run metadata confirms that head and a successful conclusion. The registered replay executes both warning-strict normal and optimized VERIFY.json commands. Local registered/direct checks, static/resource checks and committed-byte/source-history preflight also pass. Hosted CI certifies execution of the bounded checks; independent human and formal proof-kernel review remain unclaimed.

Advance claim. Canonical main and priority revision 7 were refreshed unchanged before publication. Entry overlap covered 379 open PRs and 3,853 Discussion records, plus sparse local notes and distinctive all-main searches. Closed-code and unpublished-work coverage is incomplete.

Merge Danger

Door: two-way. Blast Radius: isolated research package. Only this new package is added.