[petalas]

Summary

fixed w = ceil(e^(17/10000))
n = floor((e-k)/w), 2k < e, n >= 64
S_d = 9475*(28416*|GL64(F_q)|*q^(4096(w-1)))^d

Every e >= 245767 admits S_D(n) <= q^k, for every prime q > 2^80.
At e = 245766, q = 213*2^80+1 admits neither covering nor S_D(n) < 2q^k.

Adds the sharp universal fit cutoff for PR3486/3523's existing actual-body selectors. Four exact depth intervals and a dyadic induction cover all later widths; a certified-prime/product enclosure proves sharpness for both predicates. This is a theorem about the prescribed reserve. Its application retains the completed-caller, fresh-row, pointwise restoration and physical-cost premises.

The original-width cutoff 311311 uses a different repeated-Q reserve at D(e) and remains unchanged. The difference 65544 is a fit-domain comparison, not a runtime speedup. No actual selected bootstrap coefficient, supplier, kappa or selected-frontier change is claimed. Only research/actual-body-fit-cutoff changes.

Evidence

• Before: the per-width actual-body selectors explicitly leave a universal cutoff unclaimed.
  After: every e>=245767 fits for every permitted prime, while 245766 fails at the exact certified prime 257501199577916014212415489.
• A separate forward-depth implementation checks all 1048575 widths through 2^20-1 at that prime, with covering/strict digits enclosed separately. Both have 802809 fitted widths and last failure 245766. Six actual optimized CLI mutants reject altered sources, duplicate/float receipts, a bad prime witness and a changed child rank.
• At final commit 25bed44, CPython 3.12.15 normal/optimized registered checks, separate per-command HEAD receipts, scoped Ruff/format and warnings-as-errors compilation pass. Final committed-byte/source-history checks cover 20 files. All five actual Git-free exported commands pass; source/export and shared-runtime identities remain unchanged.

python3.12 -W error -B scripts/research_package.py --package research/actual-body-fit-cutoff
python3.12 -W error -B scripts/check_research_commit.py --package research/actual-body-fit-cutoff --check-source-history

Canonical scoped claim. Base/submission canonical 20ecf28, priority revision 6, with unchanged guidance. Submission overlap covers 287 open PR bodies and 2407 Discussion records with complete pagination and no warnings; arbitrary diffs and unpublished work remain outside coverage. Nine exact canonical proof/source inputs close the standalone replay without foreign Git objects or network access.

Credit Nick Petalas/PR3486/3246, Hiroki Kumazaki's encoded-selector lineage, PR3503/3523, Tekalign/PR3264, Jose Manuel/PR2234/2809/2899, David Arias/PR2753 and Aakash's original-width theorem, dyadic method and prime certificate, with the retained source527/BIT/geometry/assembly lineage. Their reserve, selector, padding and consumer results are inherited.

The original independent attempt incorrectly assumed equal digit thresholds at every depth; the corrected independent implementation handles each predicate separately. Original diagnostics are preserved. Full selected cold replay, GL64 payload execution, universal physical/C0 refinement, complete invoice, precision/tape, analytic transfer, hosted CI acceptance and independent human/formal proof-kernel review remain unrun or unclaimed.

Merge Danger

Door: two-way

Blast Radius: package

Eleven isolated research files. Frozen inputs, default implementations and selected frontier remain unchanged.