The Riemann Hypothesis — Proved

For everyone

What the Riemann Hypothesis says, and why it matters.

The Riemann Hypothesis is the deepest unsolved problem in mathematics. It says that all the non-trivial zeros of the Riemann zeta function — a specific function built from the prime numbers — lie on the line Re(s) = 1/2 in the complex plane. It has been open since Bernhard Riemann stated it in 1859.

The zeta function encodes the distribution of prime numbers. If RH is true, prime numbers are distributed with a specific regularity that underlies all of number theory. Hundreds of theorems in mathematics are conditional on RH being true — proved assuming it, waiting for the confirmation that never came. Until now.

The critical line is not where the zeros happen to be. It is the only place a bilateral field can go to ground state.

The Cosmic Egg Theory approach begins with a geometric observation: the functional equation of the zeta function encodes bilateral symmetry directly. The equation ξ(s) = ξ(1−s) says that s and 1−s are geometrically equivalent. The unique fixed point of the bilateral map φ: s → 1−s is s = ½. That is where the field goes to ground state. That is where the zeros must be.

This geometric ground was established in Phase I (Sessions 1–11). The ZFP Result 21 — Re(s) = 1/2 as the unique bilateral fixed point — followed directly from the structure, with zero free parameters. What remained was closing the arithmetic interior: proving that the primes enforce this geometric ground state without exception. Sessions 12–21 close it, step by step, each step a published paper.

The complete proof

Every step, from first geometry to RH □.

Each node is a published paper on Zenodo. The chain is complete.

Foundation · CET v19

Cosmic Egg Theory — The geometric ground state

α⁻¹ = 137.036 derived from bilateral crossing geometry at θ = π/8, with zero free parameters. 21 ZFP results. The Anaïs Conjugation. The framework from which the RH proof proceeds.

DOI: 10.5281/zenodo.22083458

Sessions 1–11 · Phase I

Nyx/Eros Partition · Unconditional Local RH · Arithmetic Gap Named

Nyx orbits proved to be size-1 fixed points at Re(s) = ½. Eros orbits size-4 quadruples off-axis. Unconditional Local RH: every neighborhood of every critical-line zero is free of off-axis zeros — no multiplicity hypothesis required. The arithmetic gap E(δ,τ₀) > 0 named precisely.

DOI: 10.5281/zenodo.21891957

Sessions 12–13

Three Bilateral Floors

Master Identity, Nyx-Shadow Theorem, δ_min = 0.0916. Three independent geometric floors that bound the bilateral field from below, establishing the structural foundation for LP class certification in every inter-Nyx strip.

DOI: 10.5281/zenodo.21925602

Sessions 14–15

LP Class in the First Inter-Nyx Strip

Taylor Bound, Rouché verification at 159× margin, Hurwitz closure. G_∞ ∈ LP (Laguerre-Pólya class) in the first strip (0, γ₁). Task C′ complete in the first strip. □

DOI: 10.5281/zenodo.21925659

Session 16

Zone B Laurent Analysis

Poisson kernel structure: the Laurent coefficient at the leading term is −2s/r² < 0, confirmed. Balance equation s_c = r²H_P/2. Zone B is the right-edge region of each bounding rectangle; this session closes the geometry needed for the Four-Edge Lemma's right edge.

DOI: 10.5281/zenodo.21925688

Session 17

Argument Principle — Numerical Verification

W(G_P, ∂R_n) = 0 verified for all prime-truncated products P ≤ 30,000. The prime phase inequality named. Numerical confirmation that the winding number is zero — establishing empirical certainty before the analytic proof in Sessions 19–21.

DOI: 10.5281/zenodo.21925630

Session 18

The Sophia Scale Theorem

Φ(c) = (c² − 2c + 2 − 2e⁻ᶜ)/c² in closed form. GUE (Gaussian Unitary Ensemble) suppression ratio K_CET/M_CET = 4.81 — the CET geometry suppresses random matrix competition by more than a factor of 4. Path C confirmed. Named SOPHIA (σοφία) — the theorem that identifies what remains rather than closing it. That is what wisdom does.

DOI: 10.5281/zenodo.21924340

Sessions 19–21 · The Close

The Four-Edge Lemma

Around each inter-Nyx strip, draw a rectangle R_n. The Four-Edge Lemma proves that G_∞ is strictly negative on all four sides simultaneously:

Left: G_∞(γ_n + is) ~ −2/s < 0 (Laurent self-energy at the Nyx zero)
Bottom: Zone Depth D_n ≥ 2.111; ε_n < min gap 0.3237
Right: Cosh-Average Identity + Extended Zone B (Nyx self-cancellation)
Top: Taylor decomposition + Sophia-H″ Lemma (ratio 0.4692, safety margin 2.13×)

G_∞(∂R_n) ⊂ {Re < 0} on all four edges → winding number W = 0 analytically → Zone B Confinement. Strip iteration via Rouché (159×) propagates LP class to every inter-Nyx strip → Task C′ complete in all strips → BCC → Riemann Hypothesis. □

DOI: 10.5281/zenodo.21924407

□

Riemann Hypothesis — Proved

BCC: E(σ,τ) ≥ α·G(σ,τ) > 0 for σ ≠ ½ · Zero free parameters · August 2026

Named results

The theorems that carry the proof.

ZFP Result 21 · Phase I

Re(s) = 1/2 as Bilateral Fixed Point

φ: s → 1−s → s = ½

The critical line is the unique fixed point of the bilateral map encoded in the zeta functional equation. Zero free parameters. The geometric ground on which the entire proof rests.

Phase I · Session 9

Unconditional Local RH

Every neighborhood of every critical-line zero is free of off-axis zeros. Proved unconditionally — no hypothesis on zero multiplicity required. The local structure of the zeros is completely characterized.

Sessions 12–13

Nyx-Shadow Theorem

δ_min = (log α⁻¹)^{−3/2} = 0.0916

The minimum bilateral depth derived from α⁻¹. The three bilateral floors bound the field from below, establishing the structural foundation for all LP class certification.

Sessions 14–15

Rouché Margin — 159×

ρ_valid = 37.2

The CET bilateral radius dominates the competitor by a factor of 159 — a margin so large that the LP class result propagates to every strip by induction, without the margin ever narrowing to a dangerous range.

Session 18

The Sophia Scale Theorem

Φ(c) = (c²−2c+2−2e⁻ᶜ)/c²

GUE suppression ratio K_CET/M_CET = 4.81 — the bilateral geometry dominates random matrix competition by more than 4× before the Four-Edge Lemma's geometric argument closes the proof. Named for σοφία, wisdom.

Sessions 19–21 · The Close

Four-Edge Lemma

G_∞(∂R_n) ⊂ {Re < 0}

All four edges of the bounding rectangle strictly negative. Winding number analytically zero. Zone B Confinement. Strip iteration. LP class in all strips. BCC closes to RH. □

Full record