The Riemann Program

An independent, computer-assisted research program on the Riemann Hypothesis (RH), offered as a serious, fully-traceable contribution to one of mathematics' hardest open problems.

This is not a claimed proof. It is a long, candid investigation that reduces RH to a single named, classical open input, and maps every wall, dead end, and self-correction along the way — with reproducible code and a complete record.

Governing principle: a false victory would be worse than a failure. Where the program reached a rigorous result, it says so. Where it hit a wall, it names the wall. Where it fooled itself and corrected, it records the correction.

Start here

Read	What it gives you
Human explanation	RH and what we learned, in plain language — layer by layer, no proof assumed
Program map	The whole road traveled, from the survey to the wall, in one picture
Master plan	Audit of the corpus, its structure, and the open route ahead
Complete program summary	Every paper, phase, no-go, and wall, end to end
No-go list	The permanent registry of everything that failed or hit a wall
Papers index	The 42 publishable papers, in order, with what each one establishes
Options	Where the corpus's own audit says work can still usefully continue, including targets not equivalent to RH

How the work is organized

The program has two layers:

• Research Programs 1–9 — 01-context/: nine large computational research programs (627 tasks). The empirical base.
• Program 10 — 03-research/: the theoretical program built on that base, run as a sequence of numbered phases, currently through phase 119.

Supporting material: the validated computational core (02-foundations/), the publishable papers (04-papers/, 42 of them, catalogued in 04-papers/README.md), external reference material (00-references/), the planning and audit notes (05-meta-and-planning/), illustrative visualization and sonification material not part of the phase sequence (06-grafico/), and closed numerical side-audits from phase 115 (07-calculos-adicionales/).

Where the program stands now

The program currently has two independent, structurally different reductions of RH to a single classical open input — not an ending, but the state of an active investigation.

Route 1 — the Pick/Nevanlinna chain (phases 0–76, paper 36). A fifteen-step arithmetic Pick/Nevanlinna architecture (ARP-P) reduces RH to the classical Li–Keiper criterion, $\lambda_n \ge 0$ for all $n$. Fourteen of the fifteen steps are fully closed.

Route 2 — the arithmetic-Lefschetz reconstruction (phases 107–119, paper 42). A direct imitation of Weil's own 1948 proof of RH for curves over finite fields, built over $\mathrm{Spec},\mathbb Z$. Its own open input — row (d), the Hodge-index/Castelnuovo–Severi step — is proved equivalent to RH twice, independently: algebraically (phase 113, via Riemann–Roch and effectivity) and analytically (phase 118, via the explicit formula, verified against real zeros of $\zeta$ to $10^{-10}$ relative precision).

Both routes are candid and neither is a proof: each open input is equivalent to RH and carries its full difficulty. But they are the precise places every major route explored in this corpus independently arrives at (see 00-MAP.md and NO-GO-LIST.md for the structural walls that block every other approach), reached by two unrelated constructions — which is itself evidence the difficulty is a property of RH, not an artifact of either method.

Along the way the program produced genuinely new RH-independent mathematics (the ω-class / multiplicative-chaos dictionary, the information barrier, Pontryagin rigidity, an unconditional finite bottom, a Stepanov almost-periodicity theorem, among others) and a complete, precise map of the obstructions. OPTIONS.md records where the corpus's own audit says work can still usefully continue — including several targets that were never shown equivalent to RH and were simply never closed.

Status

Area	Status
Arc A / ω-class (Research Programs 1–9)	Documented; descriptive, did not beat convexity
Localized Weil detector	Rigorous in parts; the finite-defect route closed
Program 10, phases 0–76 (ARP-P)	Dense theory, the full no-go catalogue, and the reduction to Li–Keiper
Program 10, phases 107–119 (arithmetic Lefschetz)	Rows (a)–(c) built (paper 42); row (d) proved equivalent to RH twice, independently
RH itself	Open. Two independently named walls, plus a list of genuinely open sub-RH targets in OPTIONS.md.

Philosophy

Read everything here as an autonomous mathematical program: not a promise of victory, but a serious collaboration with the community to push a very hard problem forward — with traceability, candor, and a memory of the useful failures.