riemann

v1.59 local checkpoint — NS-60–66

RH, G2, the original Weil floor and the optimized NB correlation input remain open. This version is a local draft on top of the unpublished v1.58 checkpoint. Version numbering and integration must be rechecked before publication.

Results and scope

Exact identities / proved implications: explicit Euler-grid correction with every mixed term; fixed smoothed NB criterion; positive adjacent averages; raw trace at most 3*kappa/(8*N^2); full projected gain; endpoint correlation formula and complete remainder.
Scoped proved exclusions: uniform logarithmic norms for the specified Euler correction with sublinear-power grid product; full canonical sharp Mobius sequence after Mellin smoothing; the nonnegative smoothed-difference coefficient cone; using relative convergence in a changing smoothing norm as an RH test. Optimal coefficients and selected canonical subsequences are outside the sharp-sequence exclusion.
Certified finite computations: complete q=2 Grams through index 64, with a proved Bernoulli/Hurwitz remainder; 256/384-bit replays, doubled series cutoff, independent positive-cell integrals; a separate one-atom changing-norm control with complete quadrature tails.
Open input: a cofinal lower contraction estimate for the actual optimized residual in one fixed smoothing norm. Neither a small finite error nor the positive shapes of the atoms supplies this estimate.

proof.tex is integrated verbatim in the live manuscript. Its components are in ns60/ through ns66/. The diagnostic scans remain separately labelled in ../diag_ns60_nb_corrections/ and ../diag_ns61_smoothed_nb/. No diagnostic truncation enters a proof or an interval gate.

Replay

From the repository root, with the configured Python environment. The certified Python dependency is pinned in requirements-replay.txt; optional diagnostic packages are separately pinned in requirements-diagnostics.txt. The exact tested runtime versions are in runtime-provenance.json.

Replay commands:

PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns61/certify_smoothed.py --bits 256 --output /tmp/smoothed-256.json
PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns61/certify_smoothed.py --bits 384 --output /tmp/smoothed-384.json
PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns61/certify_smoothed.py --bits 256 --cutoff 256 --output /tmp/smoothed-cutoff256.json
PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns61/check_physical_gram.py --bits 256 --output /tmp/physical-256.json
PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns61/check_physical_gram.py --bits 384 --output /tmp/physical-384.json
PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns61/verify_replays.py
PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns63/certify_one_atom.py --bits 256 --output /tmp/one-atom-256.json
PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns63/certify_one_atom.py --bits 384 --output /tmp/one-atom-384.json
PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns63/check_q2_anchor.py --bits 256 --output /tmp/anchor-256.json
PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns63/check_q2_anchor.py --bits 384 --output /tmp/anchor-384.json
PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns63/verify_replays.py
PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns64/certify_separator.py --bits 256 --output /tmp/separator-256.json
PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns64/certify_separator.py --bits 384 --output /tmp/separator-384.json
PYTHONDONTWRITEBYTECODE=1 python evidence/v159/ns64/verify_replays.py
maxima --very-quiet -b evidence/v159/ns66/exact-checks.mac
maxima --very-quiet -b evidence/v159/ns65/exact-checks.mac
maxima --very-quiet -b evidence/v159/ns64/exact-checks.mac
maxima --very-quiet -b evidence/v159/ns60/exact-checks.mac
maxima --very-quiet -b evidence/v159/ns61/exact-checks.mac
maxima --very-quiet -b evidence/v159/ns62/exact-checks.mac
maxima --very-quiet -b evidence/v159/ns63/exact-checks.mac
./manuscript/build.sh
python tools/verify_manifest.py v159

The verifier checks the saved outputs and their source hashes; to audit a fresh run, compare its enclosures with the saved JSONs. All matrix solves use Arb’s certified preconditioned algorithm. No approx solve or midpoint inversion is used. The physical-integral and one-atom quadrature checks include the full omitted interval to infinity.

The HTML report and local self-review are the readable entry points. No independent review is claimed. The evidence is incremental, not a cumulative bundle of previous releases.

The q=2 anchor compares the one-atom Mellin quadrature with the separate Gram/load formula at both precisions. Its r=1 tail is deliberately coarse; it checks normalization and is not used for the high-accuracy finite claims.

NS-64 supplies the explicit dual separator for the nonnegative coefficient cone. Its complete six-function local Gram includes the entire x>1 tail exactly. The local cone distance is attained and is a lower bound for the full-space cone distance, with no equality claim for the latter. The current readable report and self-review are the v2 HTML files; v1 files preserve the earlier checkpoint before this quantitative witness.

NS-65 excludes fixed ordinary Cesaro iterates of the canonical sharp rule. NS-66 supplies its weaker fixed-order norm-growth criterion: the exponent is exactly beta_star-1/2. A direct subpolynomial bound or bounded cofinal subsequence is a new stated open target, not a bound established here. The analytic implications are separate from their finite exact checks.

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