𝐀𝐁𝐎𝐔𝐓 𝐓𝐇𝐄 𝐑𝐈𝐄𝐌𝐀𝐍𝐍 𝐕𝐈𝐀 𝐁𝐋𝐔𝐑
(𝐇𝐞𝐥𝐬𝐨𝐧–𝐁𝐥𝐮𝐫 𝐏𝐫𝐨𝐠𝐫𝐚𝐦, 𝐇𝐁𝐏)
𝐖𝐡𝐚𝐭 𝐢𝐭 𝐢𝐬
The Helson–Blur Program develops a concrete, auditable route to RH built from two disciplined moves: blur by positive, normalized kernels; then deblur with tracked constants. This community hosts the core chain (Base → Main) together with essential background, visuals/diagnostics, and uniform extensions (e.g., Dirichlet families).
𝐌𝐞𝐭𝐡𝐨𝐝 (𝐨𝐧𝐞 𝐥𝐢𝐧𝐞)
Positive blur → monotone/contractive statement → auditable deblur with explicit error budgets.
𝐖𝐡𝐚𝐭 𝐭𝐡𝐢𝐬 𝐢𝐬 / 𝐢𝐬 𝐧𝐨𝐭
• Is: a checkable framework with explicit constants, stated assumptions, and reproducible diagnostics (including machine-verifiable steps where applicable).
• Is not: tacit heuristics. Items labeled Theorem/Lemma/Proposition include proofs; conditional statements list their hypotheses; program notes are marked as such.
𝐓𝐰𝐨 𝐥𝐞𝐧𝐬𝐞𝐬
• Zeta / Hilbert–Pólya lens. Helson ambient with logarithmic Rouché boundary guards, four summable flows, a blurred Cauchy/Herglotz bridge, and a canonical self-adjoint HP realization.
• Euler / Hopf lens. Hopf-algebraic structure and prime-band projectors make multiplicativity transparent on the Euler side; the Mellin/Fourier dictionary explains the role of scale.
𝐀𝐧𝐜𝐡𝐨𝐫𝐬 & 𝐧𝐚𝐦𝐢𝐧𝐠
Titles follow HBP | Track | number • Title (versioned on Zenodo). Example:
HBP | Main | 1.4 • Hilbert–Pólya Realizations via Blur (vN)
𝐓𝐫𝐚𝐜𝐤𝐬 (𝐰𝐡𝐚𝐭 𝐥𝐢𝐯𝐞𝐬 𝐡𝐞𝐫𝐞)
• Base (0.x): principles/axioms and the minimal toolkit for the main chain.
• Main (1.x): the Helson–Blur → Herglotz → Hilbert–Pólya chain (RH core).
• Observ (2.x): methods/visualization/diagnostics (e.g., phase heatmaps).
• Inter (5.x): interfaces (multiplicative choice, logical/categorical links).
• Aux / Question (4.x): applications, quantitative criteria, exploratory questions.
𝐑𝐞𝐩𝐫𝐞𝐬𝐞𝐧𝐭𝐚𝐭𝐢𝐯𝐞 𝐜𝐨𝐫𝐞 𝐞𝐧𝐭𝐫𝐢𝐞𝐬
• HBP | Main | 1.1 • Mittag–Leffler for Helson Zeta via Blur — a blur/Fourier Helson factorization yielding prescribed zero/pole patterns on domains touching ℜs>1/2.
• HBP | Main | 1.2 • Boundary Guards, Detectors, and Four-Flow Budgets — the guard mechanism, blurred Cauchy transform, and rigidity detector used across the chain.
• HBP | Main | 1.3 • Route: Hilbert–Pólya Realizations via Blur — compact route centered on BP2 (blur-positivity) on the critical line.
• HBP | Main | 1.4 • Hilbert–Pólya Realizations via Blur — from blurred phase → Herglotz limit → self-adjoint operator with spectrum at zero ordinates.
• HBP | Main | 1.5 • Local Blur Detection and Simplicity of Zeros — a one-pair removal + blur test that certifies simplicity locally.
• HBP | Main | 1.6 • Helson–Blur to Hilbert–Pólya — Uniform — uniform extension to Dirichlet L-functions (GRH, no Landau–Siegel zeros).
• HBP | Base | 0.1 / 0.2 / 0.3 — universal blur principle; multiplicative closure; foundations for the Helson–Blur RH chain.
• HBP | Observ | 2.1 — phase-heatmap methods for Fourier/Mellin and primes–zeros duality.
• HBP | Inter | 5.1 — RH vs. the multiplicative choice (axiomatic interface).
𝐒𝐮𝐠𝐠𝐞𝐬𝐭𝐞𝐝 𝐟𝐢𝐫𝐬𝐭 𝐫𝐞𝐚𝐝𝐬
Quick route: Base 0.1 → Base 0.3 → Main 1.3 → Main 1.4
Foundations of multiplicativity: Base 0.2; Inter 5.1; Inter 5.1.1
𝐕𝐞𝐫𝐬𝐢𝐨𝐧𝐢𝐧𝐠 & 𝐩𝐫𝐨𝐯𝐞𝐧𝐚𝐧𝐜𝐞
Each record is timestamped and versioned (v1, v2, …). Labels (Preprint / Report / Peer review) appear per record. Anchors remain stable; any reclassification is noted in the item history.
𝐂𝐥𝐚𝐢𝐦𝐬 & 𝐭𝐫𝐚𝐧𝐬𝐩𝐚𝐫𝐞𝐧𝐜𝐲
Items state their scope explicitly: proved results; conditional implications (with assumptions); computational/visual evidence; or program notes. Constants and dependencies on blur scales/windows are tracked. Numerical checks and finite certificates include a verification protocol when relevant.
𝐑𝐞𝐩𝐫𝐨𝐝𝐮𝐜𝐢𝐛𝐢𝐥𝐢𝐭𝐲
Visualizations (phase-heatmaps) and finite certificates include enough detail to replicate. Please report gaps; errata are versioned.
𝐇𝐨𝐰 𝐭𝐨 𝐜𝐢𝐭𝐞
Use the anchor in the title and include the DOI/license from the record. Example:
Perišić, A. (2025). HBP | Main 1.4 — Hilbert–Pólya Realizations via Blur. Zenodo. DOI: …
𝐂𝐨𝐧𝐭𝐚𝐜𝐭 & 𝐜𝐨𝐧𝐭𝐫𝐢𝐛𝐮𝐭𝐢𝐨𝐧𝐬
Author info and ORCID appear on each record. Feedback and independent checks are welcome; please reference the item’s anchor and, if helpful, section/line numbers. Contributions are encouraged—keep assumptions explicit, constants visible, and budgets auditable.
𝐍𝐨𝐭𝐞 𝐨𝐧 𝐫𝐞𝐜𝐞𝐧𝐭 𝐫𝐞𝐥𝐞𝐚𝐬𝐞𝐬
Zenodo’s “Recent releases” panel reflects the latest uploads/versions. Titles follow the anchor format; each record page includes abstract, history, DOI, and license.
𝐋𝐚𝐬𝐭 𝐮𝐩𝐝𝐚𝐭𝐞𝐝: 26 Sep 2025 • 𝐌𝐚𝐢𝐧𝐭𝐚𝐢𝐧𝐞𝐫: A. Perišić (ORCID)