jonwashburnRiemann

Artifact Verification Guide: Unconditional Far-Field Zero-Freeness ((\Re s\ge 0.6))

This repository contains the machine-checkable verification artifacts associated with Paper 1 of the three-paper split:

“A certified zero-free region for the Riemann zeta function in the half-plane (\Re s \ge 0.6)”
Jonathan Washburn (2026)

The goal of this guide is to make it easy for a skeptical referee to audit the unconditional far-field claim (the (\Re s\ge 0.6) zero-free region) by running a single, rigorous verifier based on ARB ball arithmetic (via python-flint).

Papers in this repository (three-paper split)

  • Paper 1 (Unconditional): paper1_farfield.tex / paper1_farfield.pdf
    A certified zero-free region for the Riemann zeta function in the half-plane (\Re s \ge 0.6)
  • Paper 2 (Effective / height-limited): paper2_energy_barrier.tex / paper2_energy_barrier.pdf
    Energy barriers and Carleson budgets for off-critical zeros of the Riemann zeta function
  • Paper 3 (Conditional / hypothesis-driven): paper3_cutoff_conditional.tex / paper3_cutoff_conditional.pdf
    A cutoff principle and conditional closure of the Riemann Hypothesis

To compile (run twice for references):

pdflatex -interaction=nonstopmode paper1_farfield.tex
pdflatex -interaction=nonstopmode paper2_energy_barrier.tex
pdflatex -interaction=nonstopmode paper3_cutoff_conditional.tex

Or, using the included Makefile:

make all

Overview: what is being discharged?

Paper 1’s unconditional far-field step is the hybrid certification summarized in:

  • Theorem “Certified far-field zero-freeness” (paper1_farfield.tex, Theorem thm:farfield)
  • Table “Certified far-field artifact data” (paper1_farfield.tex, Table tab:artifact-data)
  • Remark “Artifact reproducibility and verification” (paper1_farfield.tex, Remark rem:artifact-repro)
  • Appendix “Audit manifest (verifier commands and expected fields)” (paper1_farfield.tex, Appendix app:audit)

It establishes:

Unconditional far-field statement. The Riemann zeta function has no zeros in the region ({\Re s\ge 0.6}).

The verification uses rigorous interval arithmetic throughout. A printed PASS is intended to mean:
within the logic of ball arithmetic, the relevant inequality is certified and cannot be violated.

Prerequisites

  • Python 3.9+
  • python-flint (bindings to FLINT/ARB)

Install:

pip install python-flint

Files you should look at

  • verify_attachment_arb.py — the main verifier (ARB / ball arithmetic).
  • theta_certify_sigma06_07_t0_20_outer_zeta_proj.json — rectangle certification artifact for ([0.6,0.7]\times[0,20]) (this matches Table tab:artifact-data).
  • pick_sigma0599_raw_zeta_N16.json — Pick certificate artifact at (\sigma_0=0.599) with (N=16) (this matches Table tab:artifact-data; this is the primary half-plane certificate used to cover (\Re s\ge 0.6) without a separate tail lemma).
  • pick_sigma06_raw_zeta_N16.json — Pick certificate artifact at (\sigma_0=0.6) with (N=16) (cross-check; matches Table tab:artifact-data).
  • pick_sigma07_raw_zeta_N16.json — Pick certificate artifact at (\sigma_0=0.7) with (N=16) (cross-check; matches Table tab:artifact-data).

There are two sensible audit modes:

  1. Fast audit (recommended for referees): verify the shipped JSON artifacts match the manuscript tables.
  2. Full regeneration: rerun the certificate computations to reproduce the JSON artifacts from scratch.

Fast audit (verify shipped artifacts)

  • Rectangle artifact: open theta_certify_sigma06_07_t0_20_outer_zeta_proj.json and check:

    • results.ok is true
    • results.theta_hi < 1
    • results.processed_boxes = 380764

  • Pick artifact (primary): open pick_sigma0599_raw_zeta_N16.json and check:

    • pick.delta_cert matches Table tab:artifact-data
    • pick.P_spd_at_0 is true
    • pick.tail_weighted_l2_partial_hi matches Table tab:artifact-data

  • Pick artifacts (cross-checks): open pick_sigma06_raw_zeta_N16.json and pick_sigma07_raw_zeta_N16.json and check the same fields.

Full regeneration (optional)

Run the verifier in the repository root:

cd /path/to/this/repository
python verify_attachment_arb.py --help

The key audit modes corresponding to Table tab:artifact-data are:

  1. Rectangle audit (theta_certify) for ([0.6,0.7]\times[0,20]) in the far-field.
  2. Pick certificate audit (pick_certify) at (\sigma_0=0.599), (N=16), plus the tail bound check.

Verification Manifest (how Table tab:artifact-data is produced)

1) Rectangle Check (interval arithmetic subdivision)

Goal: certify (|\Theta(s)|<1) on the finite rectangle [ [\sigma_{\min},\sigma_{\max}]\times[t_{\min},t_{\max}] = [0.6,0.7]\times[0,20]. ]

Artifact file (included): theta_certify_sigma06_07_t0_20_outer_zeta_proj.json

Key fields in that JSON (expected):

  • results.ok = true
  • results.theta_hi = 0.9999928763...
  • results.processed_boxes = 380764
  • results.margin_1_minus_theta_hi ≈ 7.12e-6

Optional regeneration command (writes a new artifact):

python verify_attachment_arb.py \
  --theta-certify \
  --arith-gauge outer_zeta_proj \
  --arith-P-cut 2000 \
  --rect-sigma-min 0.6 --rect-sigma-max 0.7 \
  --rect-t-min 0.0 --rect-t-max 20.0 \
  --outer-mode midpoint \
  --outer-P-cut 2000 \
  --outer-T 50.0 --outer-n 2001 \
  --theta-init-n-sigma 10 --theta-init-n-t 50 \
  --theta-min-sigma-width 0.0001 --theta-min-t-width 0.001 \
  --theta-max-boxes 500000 \
  --prec 260 \
  --theta-out theta_certify_sigma06_07_t0_20_outer_zeta_proj.json \
  --progress

2) Pick Matrix Certificate (Schur on ({\Re s>0.599}))

Goal: certify a strict Pick gap at (\sigma_0=0.599) for the (16\times 16) arithmetic Pick minor (P_{16}(\sigma_0)): [ P_{16}(0.599) \succeq \delta I,\qquad \delta>0. ]

Artifact file (included): pick_sigma0599_raw_zeta_N16.json

Key fields in that JSON (expected):

  • pick.delta_cert = 0.5944375202...
  • pick.P_spd_at_0 = true
  • pick.tail_weighted_l2_partial_hi = 0.0137007383...

Optional regeneration command (writes a new artifact):

python verify_attachment_arb.py \
  --pick-certify \
  --pick-sigma0 0.599 \
  --pick-N 16 \
  --pick-coeff-count 32 \
  --pick-K 256 \
  --pick-rho 0.1 \
  --pick-rho-bound 0.2 \
  --arith-gauge raw_zeta \
  --arith-P-cut 2000 \
  --prec 260 \
  --pick-out pick_sigma0599_raw_zeta_N16.json

Optional cross-check regenerations can be done by changing --pick-sigma0 and --pick-out to reproduce pick_sigma06_raw_zeta_N16.json and pick_sigma07_raw_zeta_N16.json (see Table tab:artifact-data).

3) Tail Bound Audit (stability of the infinite Pick matrix)

Goal: ensure truncating to (N=16) does not invalidate the infinite-dimensional Schur conclusion.

This is the paper’s “tail perturbation” check: if (\varepsilon_N) is the coefficient tail bound and (C) is the perturbation constant, verify (C\varepsilon_N<\delta).

In Paper 1 (paper1_farfield.tex), it is summarized in Proposition prop:pick-gap and Table tab:artifact-data (see also Appendix app:audit).

Why this is rigorous (and not just numerics)

The verifier uses ball arithmetic:

  • Every number is stored as an interval (midpoint + radius).
  • Every arithmetic operation propagates errors automatically.
  • Certification is done by boolean checks like “upper bound < 1” or “Cholesky succeeds with positive pivots”.

So a PASS indicates there is no mathematical possibility (within the verified enclosures) that the inequality fails.

Troubleshooting

  • ImportError: No module named flint
    Install python-flint and ensure you’re using the right Python (python --version).

  • Runtime
    The rectangle subdivision is compute-heavy. If you only want to audit the shipped result, inspect theta_certify_sigma06_07_t0_20_outer_zeta_proj.json and check results.ok / results.theta_hi.

Citation

If using these artifacts, cite the manuscript:

Washburn, J. (2026). A certified zero-free region for the Riemann zeta function in the half-plane (\Re s \ge 0.6). Preprint.