[petalas]

Summary

- Covering selector: min(q^k,S) reaches S.
+ Strict-volume selector: min(q^k,floor(S/2)+1) reaches floor(S/2)+1.
+ Return exact q^k only after proving 2q^k <= q*S_D(floor(e/w)).

Closes the complete Count-selector gap left by PR3503's strict row adapter and PR3518's covering-only saturation. The selector computes the least borrowing at actual body depth, with fixed original atom width, strict 2k<e, body at least m and exact S<2q^k. The proof extends the conditional setup bill O(h^3+w^2 h^4) operations and O(wh) cells to this wider predicate, with h=1+ceil(log2(e+1)) and a conservative 26-role allowance.

Only research/encoded-strict-borrowing/ changes. The full pending runtime/proof closure is retained byte-for-byte with origin commit/path/blob/SHA256 receipts. These are actual dependencies, not comparison-only sources or foreign ancestry pins. Credit Hiroki Kumazaki/PR3502, Nick Petalas/PR3503/3518/3486/3246, Jose Manuel/PR2899/2786/2808, Tekalign, David Arias/PR2753, Ross Richardson/PR2420 and the retained lineage.

Evidence

• Before: at reduced e=17,q=2,m=2,r=1,W=Q=1,w=1, covering needs k=8. At e=5 it has no fit.
  After: literal strict selection returns k=7 with rows 128 and stock 216, and k=2 with rows 4 and stock 6, respectively. The one-digit/fit comparison theorem belongs to PR3503; this contribution supplies the complete encoded selector and its setup proof.
• 990 complete literal plans preserve all seven fields, borrowed input/head and erased owned tapes. The independent exhaustive/slab census agrees on 2880 cases, including 680 one-digit savings and 53 newly fitted cases. Threshold checks cover 4096 decisions, 32 rejected equalities and 192 final-bound cases with 14 sharp equalities.
• The actual million-width selector returns k=49, depth 19, rows 562949953421312 and stock 609359740010496 with maximum observed Count extent 63 cells. The old full upper trial's 500000-bit power is derived arithmetic, not an executed or timed comparison.
• Strict Ruff/format, Python 3.12 normal/optimized registered checks, seven actual optimized CLI mutants, final-head committed-byte/source-history checks, export integrity and all four actual Git-free commands pass at c439222b9ab91aee16ed298663e0462cda080392. Export/source bytes, modes and inode identities remain unchanged.

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

Use an existing private external TMPDIR. Base/submission refresh is canonical 6f100d4bb5083f8c35d96fd95a539811c62f245c, priority revision 6, with unchanged guidance. Fresh entry coverage includes 434 open PRs and 2359 Discussion records with complete pagination and no warnings; focused submission checks find unchanged predecessor heads. Arbitrary diffs and unpublished work remain outside that coverage. Canonical scoped claim.

The old padding planner still rejects q^k<S; a later consumer must explicitly compose the strict-padding adapter. No payload/C0, selected GL64 execution, fixed-role microtransition table, actual stock coefficient, supplier, cutoff, kappa or selected-frontier change is claimed. Completed-callee, full-cover, source access, fresh fields, pointwise restoration, physical costs, precision/tape and outer transfer remain obligations. Full selected cold replay, Lean/Docker and hosted CI acceptance were not run locally. Independent human and formal proof-kernel review are unclaimed.

Merge Danger

Door: two-way

Seventeen isolated package files. Existing backends and frozen inputs are unchanged.

Blast Radius: package