Summary

Using the lemma set of arXiv:2601.22365 unchanged, we regenerated the per-case region partitions at a higher target bound and verified every region with the independently built, previously audited verification stack: all 1,104,177,103 regions across the four cases satisfy non-positivity of their designated splitting functions at the exact rational rho = 43/50 (= 0.860), with zero failures, zero invalid records, and zero inconclusive enclosures. Subject to the same lemma-layer obligations as the published proof (Section 5), this establishes the certificate-layer bound rho >= 0.860 — up from the published 0.8559, closing 40% of the remaining gap to the conjectured sqrt(3)/2 = 0.86602540…, with no new mathematics: the improvement comes entirely from regenerating partitions against the higher target and from verification semantics that check all box vertices and, where possible, whole boxes.

1. Claim and scope

Claim. For each of the four cases (d_regular / d_steiner, subcases f >= d and f <= d), the regenerated partition at target 0.860 covers its case domain (by the generator's construction, as in the published work) and every region's splitting function is certifiably non-positive at rho = 43/50 over the checked set (all finite box vertices for every region; the entire box for the 244,138,895 regions closed by full-box verification).

Scope. This is a certificate-layer result in the sense of the M1 audit: it inherits the published proof's lemma layer (trapped-point polygons, topology validity of the case decomposition, monotonicity of constituent lengths, and — only where a region is not full-box-closed — axis unimodality of the splitting functions). All inherited claims are rho-uniform: they concern lengths and geometry, never rho, so their transfer from 0.8559 to 0.860 is immediate. Nothing here should be cited as an unconditional theorem without those obligations stated.

Exactness. rho is held as the exact rational 43/50 end to end (python-flint fmpq); the generator's long-double 0.856/0.860 constants influence only which partition gets generated, never what is verified. Every verdict row records the rho it certified.

2. Method

1. Audited stack (M1). The verifier (Arb ball arithmetic, certified comparisons, all-2^n-vertices semantics, full-box-first per ADR-0004) independently re-verified the published 0.8559 certificate layer: 330,193,755 records, zero certificate-layer failures (verification-note.md).
2. rho as an exact parameter (ticket #15): the entire stack takes rho as a required exact rational; M1 behavior reproduces bit-identically at 8559/10000.
3. Exact fixed-tiling ceiling (ticket #16): F(rho) = rho·L_T* + (L_S+ − L_S−) is affine in rho with L_T* >= 0 and no lemma condition reads rho, so each record's critical rho is a certified division and the corpus minimum is one pass. The published tiling's exact ceiling is 0.8559000606 — eps-pinned to its target by the generator's termination slack, hence regeneration.
4. Regeneration at the target (ticket #17): the published generator, patched only in its rho constant (verified one-line diff against the pinned snapshot commit 709673a8), regenerated all four partitions at 0.860 in isolated work directories.
5. Claim-grade verification: every regenerated region verified at 43/50 under full-box-first semantics with the 128/256/512-bit precision ladder; zero not-ok outcomes.

3. Results

Claim-grade runs (16 workers; per-region verdict streams and run provenance under artifacts/verification/claim_0.860_*.run.json):

Certificates (regenerated locally; not redistributed — see Section 5):

Context (all sampled passes 50,000 records, zero bad outcomes): rho = 0.856 was additionally certified exhaustively for all four cases (559,929,762 regions) as the first record step. Sampled ladder evidence puts the existing lemma set's reach at >= 0.858 for d_steiner (both subcases; 0.862+ under probe) and >= 0.866 for d_regular — within 2.5e-5 of the conjectured optimum for that family.

4. Reproduction

From the released tooling (no redistributed upstream content):

# pin and fetch the upstream snapshot + dataset (hashes verified)
uv run python -m steiner_audit.acquire
# regenerate a partition at the target (their generator, patched rho)
uv run python -m steiner_audit.regen --family d_steiner --subcase f_ge_d --rho 0.860
# verify it at the exact rational
uv run python -m steiner_audit.verify --family d_steiner --subcase f_ge_d \
    --cert vendor/regen/d_steiner@0.860/certificate_rho=0.86_f_ge_d.bin \
    --rho 0.860 --workers 16
# exact ceiling / margin analysis of any partition
uv run python -m steiner_audit.ceiling --family d_steiner --subcase f_ge_d \
    --cert <path> --sample 50000

Generation cost at 0.860: 0.9–9.6 h per case (single core). Verification: ~4,600 records/s at 16 workers on Apple Silicon (measured); ~2.5 days for the largest case.

5. Limitations

• Lemma layer inherited, not verified. Identical to the published proof's posture; the M1 audit's findings (F1–F5) apply unchanged. The 244.1M full-box-closed regions do not depend on the unimodality claim; the remainder check vertices only, as their checker does — but at all 2^n vertices rather than their 2^(n-1).
• Generator trusted for coverage. That the partition tiles each case domain is the generator's construction, as in the published work; our verification establishes per-region non-positivity, not tiling completeness. The semantic-diff and regeneration reproducibility results of M1 support (but do not prove) generator determinism.
• Upstream content not redistributed. The PKU repository and dataset are unlicensed; reproduction fetches them from the public sources against pinned hashes (ADR-0002). Regenerated certificates are likewise derivatives and stay local; this note ships hashes only.
• Sampled ladder rungs are not claims. Only exhaustively verified rhos (0.856, 0.860) are claimed; ladder numbers above them are scouting evidence.

6. Artifact index

• artifacts/verification/claim_0.860_*.run.json — four claim-grade run records (counts, certificate sha256, code revision, exact rho).
• artifacts/verification/ceiling_full_*.json (+ .bottlenecks.jsonl) — exact fixed-tiling ceilings of the published partitions.
• artifacts/verification/probe_ceiling_*.json — sampled ladder passes.
• artifacts/acquisition/regen_manifest.json — content-addressed manifest of every regenerated certificate.
• artifacts/notes/observations.md — OBS-9 through OBS-19 (the campaign log, including generator scaling and non-monotonicity observations).