Kernel-checked explicit Chebyshev bounds from a zero-free half-plane

We prove in Lean 4 with Mathlib that |ψ(x)-x|≤ 700 x^(61/64) for every real x≥ 2^256 and that |θ(x)-x|≤ x/(5 log² x) for every real x≥ 2^626, where ψ and θ are Mathlib's Chebyshev functions. The only analytic input about ζ beyond Mathlib is the formally proved zero-free half-plane Re s>7/8 of a public Lean release, which we use as a proved dependency, not as an axiom. Both theorems are checked by Lean's kernel and depend only on the axioms propext, Classical.choice and Quot.sound. The proof is a short, fully explicit route: an integrated Perron formula for ∫_0^x ψ, a Borel–Carathéodory bound |F'/F|≤72 192log (|t|+3) on 29/32≤Re s≤2 for the entire regularisation F of (s-1)ζ(s) (with F(1)=1), a contour shift to Re s=29/32, and a monotone unsmoothing step. It needs no zero counting, no truncated explicit formula and no verification of the Riemann hypothesis. The thresholds are far beyond those of published explicit estimates; the point is the formal status of a fully explicit statement, constants and starting point included. We also explain why the bounds obtained here do not reach the starting point of the explicit Chebyshev estimate used in our formal proof of Dusart's bound on Σ_(p≤x) 1/p, and identify the range that remains to be covered.

Contents of the release

• docs/paper/ — the manuscript (LaTeX source and PDF) and psi_halfplane_gen.py, which generates every derived decimal of the paper from exact rationals with outward rounding and checks each quoted constant against the Lean source; psi_halfplane_runs/ holds the output of the coverage script.
• research/qrh_unconditional/ — the Lean project: the library Psi (the proof), LittlewoodLean (the disk logarithm and Carathéodory's coefficient inequality it uses), lakefile.lean, lake-manifest.json and lean-toolchain (pins), fetch_oai.sh (fetches the imported 7/8 development from the public OpenAI math release at the commit in OAI_COMMIT and checks its tree against vendor_OAI.tree.sha256 and the two compatibility patches against PATCHES.sha256; no OpenAI code is redistributed), apply_patches.sh, run_gate.sh and gate/ (the release gate and the record of its passing run).
• research/dusart_ch/ch_coverage.py — the interval-arithmetic check of the coverage window of Chirre–Helfgott Corollary 1.2 shown in Figure 2 (not used in the proofs).
• REPLAY.md and replay/ — a clean replay from this bundle: 8 of 8 steps pass.
• requirements.txt — the Python packages; ORIGINAL_SHA256.txt (in the gate record) — original and shipped SHA-256 of files whose local paths were replaced by placeholders; SHA256SUMS — the SHA-256 of every file.

The imported 7/8 development is not redistributed: fetch_oai.sh obtains it from the public release at the pinned commit and verifies it file by file.

Verification

Lean via elan, Python 3.13 with pip install -r requirements.txt. In research/qrh_unconditional/: bash ./fetch_oai.sh, lake exe cache get, bash ./apply_patches.sh, lake build Psi.Final, then bash ./run_gate.sh (fresh-environment kernel replay with leanchecker, #print axioms checked against propext, Classical.choice and Quot.sound, statements under pp.all). In docs/paper/: python3 psi_halfplane_gen.py (rewrites psi_halfplane_numbers.tex; compare). In research/dusart_ch/: python3 ch_coverage.py. Integrity: shasum -a 256 -c SHA256SUMS.

Hardware

One MacBook Pro (Apple M3 Pro, 11 CPU cores, 18 GB memory); no GPU, no cluster.

License: Licensed by SmartLedger Solutions under CC BY-NC 4.0 (LICENSE.md). Commercial use requires a written license from SmartLedger Solutions (labs@smartledger.solutions). No patent license is granted.