[petalas]

Summary

- Construct q^k in every least-borrowing trial.
+ Compare min(q^k, S_D(n)) with exact S_D(n).
+ Construct exact q^k once, after proving the minimum is small.

Closes PR3502's explicit saturation handoff for the unchanged covering predicate and fixed original width. Proves q^k < q*S_D(floor(e/w)) for the least fitted plan, including the legal-upper-bound case. The complete conditional setup bill improves from O(h^3+h^2(e+wh)^2) time / O(e+wh) cells to O(h^3+w^2 h^4) time / O(wh) cells, with h=1+ceil(log2(e+1)). For fixed width exponent below one, rejected trial powers no longer supply an e-sized term.

Only research/saturated-body-borrowing/ changes. The opt-in Count selector returns the same seven exact plan values. Exact canonical inputs and the complete frozen Count-source closure are declared; PR3502's unchanged proof and selector are separate byte-authenticated comparisons, not foreign ancestry dependencies. Credit Hiroki Kumazaki/PR3502, Nick Petalas/PR3486/3246, Jose Manuel/PR2899/2786/2808, Tekalign/PR3264, David Arias/PR2753, Ross Richardson/PR2420 and their retained lineage.

Evidence

• Before: at the reduced e=1000000 fixture, the old upper trial requests a 500000-bit power by arithmetic. That old request was not executed or timed.
  After: the literal full selector returns k=50 and exact rows 1125899906842624, with maximum observed runtime Count extent 63 cells. This is a Count-resource observation, not a runtime speed factor or Python heap measurement.
• 900 literal complete plans, 120 capped powers, 10 boundary/ownership controls and a separate 1920-case inverse-depth/slab census pass. The independent saturation-identity census checks 131072 positive triples. All 423 fitted integer cases require preserving nontrivial radix slack.
• Seven actual optimized public CLI corruptions reject wrong cap/power behavior, a saturated final output, omitted compatibility, source/comparison drift and a Boolean receipt. The initial incomplete allocation observer failed and is preserved separately; passing observations cover both Count allocation bindings.
• Python 3.12 normal/optimized registered replay, scoped Ruff/format, all 20 final-head committed-byte/source-history comparisons, export integrity and all four real committed Git-free commands pass at 99d969bdcaa03c655ff02b94ea40f372d6470d7d. All 21 exported files preserve bytes, modes and inode identities.

python3.12 -W error -B scripts/research_package.py --package research/saturated-body-borrowing
python3.12 -W error -B scripts/check_research_commit.py --package research/saturated-body-borrowing --check-source-history

Use an existing private external TMPDIR. Canonical base/submission refresh is 6f100d4bb5083f8c35d96fd95a539811c62f245c, priority revision 6, with unchanged guidance. Entry overlap covered 422 open PRs and 2346 Discussion records with complete pagination and no warnings; focused submission reads found the predecessor unchanged and no duplicate saturation work. Arbitrary diffs, closed PR bodies and unpublished work remain outside that coverage. Canonical scoped claim.

The setup arithmetic/refinement, completed callee and physical costs remain conditional. No payload callback, selected GL64/C0 execution, actual stock coefficient, supplier, cutoff, kappa or selected-frontier change is claimed. Full-cover, leaves, source access, fresh fields, pointwise restoration, precision/tape and outer transfer remain open. Independent human and formal proof-kernel review are unclaimed. Full selected cold replay, Lean/Docker and hosted CI were not run locally.

Merge Danger

Door: two-way

Seventeen isolated package files. Frozen sources and existing backends are unchanged.

Blast Radius: package