Certified harmonic-measure barriers and explicit Littlewood bounds from zero-free regions (certificate release v1.0.0)

We give explicit bounds for holomorphic logarithms by propagating Euler-product information through zero-free regions, using a Cayley transform and harmonic measure. For the zero-free-disk logarithmic-control lemma used in recent claimed proofs of the quasi-Riemann hypothesis, at a=7/8, e=10^-3 and ε=1/10, this lowers a sufficient threshold on log C from 10^1778.3 to 10^28.6. In the analytic information model that imposes holomorphy on the zero-free disk together with the pointwise Euler majorant and the convexity profile, certified obstruction witnesses locate the optimal threshold in (10^28.5024,10^28.5339]. For the reduced one-chord model we prove an additive bracket and the exact limit of e·log x^*_red as the target approaches the boundary. The same propagation yields explicit Littlewood bounds under zero-freeness of a single primitive real character in Re s>a. For a=7/8 and q≥10^100 we obtain L(1,χ)≥0.0617/log log q; for every fixed a we obtain lower and upper bounds whose coefficients tend to ζ(2)e^-γ(1-a) and e^γ/(1-a) at an explicit rate, together with explicit bounds for L′/L(1,χ). Corollaries cover class numbers, regulators and Euler–Kronecker constants of quadratic fields. Ball-arithmetic programs certify the inequalities, and a separate fail-closed checker verifies the obstruction witnesses.

Contents of the release

• paper/ — the manuscript (LaTeX source and PDF), the generated tables and numbers, qrh_effectivity_gen.py, which writes every number in the text from the logs in paper/qrh_effectivity_runs/ and checks each against its defining inequality, and qrh_effectivity_figs.py, which draws the three figures (paper/figs/) from the same logs, byte for byte.
• paper/qrh_effectivity_runs/ — the output logs of every certified computation, the box certificates of the full-data and profile obstruction witnesses (full_data_boxes.jsonl, rational box corners) and the frozen parameter manifest (parameters.json, exact dyadic values of every parameter).
• programs/ — the distributed programs and checkers (Python 3 with python-flint): the standalone checker of every printed main-table threshold, the full-data/profile producer and its fail-closed checker, the Littlewood programs (fixed and moving parameters, L′/L), the direct bound at s = 1, the domain comparison and the exact-rational checker of the conditional parameter extension.
• lean/ — the Lean 4 project (the 29/30 maximum and the sharp-control lemmas) with Audit.lean, the axiom audit.
• SOURCES.md — transcriptions of the external statements used, with the versions and SHA-256 digests of the copies read.
• REPLAY.md and replay/ — a clean replay of every program and checker from this bundle: 11 of 11 steps pass, program outputs identical byte for byte to the shipped logs.
• SHA256SUMS — the SHA-256 of every file.

Verification

Install Python 3 and python-flint (0.9.0 was used). In programs/: python3 qrh_thresholds_check.py ../paper/qrh_effectivity_runs/three_circles.txt ../paper/qrh_effectivity_runs/cayley.txt, python3 qrh_full_data_check.py ../paper/qrh_effectivity_runs/full_data_boxes.jsonl and python3 qrh_ceiling_check.py each end with ACCEPTED and fail closed otherwise. The producers rerun with python3 <program>.py and reproduce the logs. In paper/, python3 qrh_effectivity_gen.py --check confirms every number in the manuscript. In lean/ (Lean 4 v4.34.1, Mathlib v4.34.1): lake build && lake env lean Audit.lean (only propext, Classical.choice, Quot.sound). Integrity: shasum -a 256 -c SHA256SUMS.

Hardware

Every computation ran on one MacBook Pro (Apple M3 Pro, 11 CPU cores, 18 GB memory), with no GPU and no cluster; the full replay takes a few minutes on one core.

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). The producer of the main-table thresholds is proprietary and not distributed; verification does not need it.