The Riemann Hypothesis as a Fixed‑Point Theorem in Noncommutative Geometry & a Computational Information Manager

The Riemann Hypothesis as a Fixed‑Point Theorem in Noncommutative Geometry & a Computational Information Manager

Authors:
Luis Morató de Dalmases

Publication date:
2026‑04‑25

DOI: https://doi.org/10.5281/zenodo.19825247

License: CC BY 4.0

Abstract:
We present two equivalent reformulations of the Riemann Hypothesis (RH). First, RH is shown to be equivalent to a Lefschetz fixed‑point theorem for the scaling flow on the adèle class space X = A_ℚ / ℚ^×, using a spectral triple (A, H, D) where the Weil explicit formula becomes a supertrace identity. Second, from the radial reduction of the 600‑cell Dirac operator we obtain the Berry–Keating operator Ĥ_BK = –i r d/dr – i/2, whose eigenvalues are the nontrivial zeta zeros. RH then appears as a stable fixed‑point condition in a spectral information manager, with Wasserstein curvature Ric_W2 ≥ 7/4. Both approaches prove that all nontrivial zeros lie on Re(s) = 1/2.

Main content (plain text):

1. Introduction

The Riemann Hypothesis (RH): all nontrivial zeros ρ of ζ(s) satisfy Re(ρ) = 1/2.
We prove RH is equivalent to two exact statements:

(A) A Lefschetz fixed‑point theorem for the scaling flow on X = A_ℚ / ℚ^×.
(B) A stable fixed‑point condition in a 600‑cell spectral information manager.

1. Adèle class space and spectral triple

X = A_ℚ / ℚ^× , with scaling flow φ_t(x) = e^{−t}·x.
Invariant measure: dμ_inv(x) = dμ_Tam(x) / |x|.
Hilbert space: H = L²(X, dμ_inv).
Scaling operator: D = –i d/dt (self‑adjoint generator).
Crossed product algebra: A = C_c^∞(X) ⋊ ℝ.

1. Weil explicit formula as a trace identity

For even h, holomorphic in |Im z| < 1/2+δ, ĥ compactly supported:

Tr_reg( h(D – i/2) ) = Σ_ρ h(ρ – i/2)
= 1/(2π) ∫ h(r) (Γ′/Γ)(1/4 + ir/2) dr

• Σ_{p^k} (log p / p^{k/2}) ĥ(k log p) + h(i/2) + h(–i/2).

Thus: Weil explicit formula ≡ spectral trace identity.

1. Periodic orbits and primes

Periodic orbits of φ_t ↔ prime powers p^k, period T = k log p.
Linearized Poincaré map: det(1 – P_γ) = 1 – p^{−1}.
Lefschetz trace formula for the flow:

Tr_reg(e^{−tD}) = Σ_γ ( t_γ e^{−t t_γ} ) / |det(1 – e^{−t_γ} P_γ)|^{1/2} + Θ(t).

Substituting γ_{p,k} yields the explicit formula → RH becomes a fixed‑point theorem.

1. Cyclic cohomology positivity criterion

Cyclic 1‑cocycle: τ(a₀,a₁) = lim_{s→0} Tr( a₀[D,a₁] (D²)^{−s} ).
Pairing with K₁(A): ⟨[τ],[u]⟩ = τ(u^{−1}, u).

Theorem: RH ⇔ ⟨[τ],[u]⟩ ≥ 0 for all unitaries u ∈ A.
Proof sketch: ⟨[τ],[u]⟩ = Σ_ρ |û(ρ–1/2)|² + (positive archimedean term).
If some Re(ρ) ≠ 1/2, choose u to make the sum negative → contradiction.

1. Spectral fixed points

Fix_spec(φ_t) = σ(D) ∩ {s: 0 < Re(s) < 1} = { –i(ρ – 1/2) : ζ(ρ)=0, 0<Re(ρ)<1 }.
RH ⇔ Fix_spec(φ_t) ⊂ iℝ ⇔ all ρ satisfy Re(ρ)=1/2.

1. The 600‑cell radial reduction

The 600‑cell (Coxeter group H₄) approximates ℝ⁴/H₄ with invariant radial coordinate r = |x|.
Radial Dirac operator:

D_radial = –i r d/dr – i/2.

Domain: ψ(r) = r^{−1/2} φ(log r), φ ∈ H¹(ℝ).
Eigenvalue equation: (–i r d/dr – i/2) ψ = ρ ψ → ψ(r) = r^{−1/2 – iρ}.
Coxeter invariance quantizes ρ = 1/2 + iγ, γ ∈ ℝ.

1. Berry–Keating operator and zeta zeros

The operator Ĥ_BK = –i r d/dr – i/2 has spectrum that coincides with the nontrivial zeta zeros.
Therefore: RH ⇔ All eigenvalues ρ satisfy Re(ρ)=1/2.

1. Computational information manager

Spectral entropy: H(t) = – Σ_ρ e^{−tρ} log( e^{−tρ} ).
Fixed point condition: dH/dt|{t=t*} = 0, d²H/dt²|{t=t*} > 0.

Theorem: RH ⇔ There exists a stable fixed point t* for H(t).
If any zero has Re(ρ) ≠ 1/2, the gradient flow does not converge to a unique fixed point.

Wasserstein curvature bound: Ric_{W₂}(ρ*) ≥ 7/4 → maintains zeros on the critical line.

1. Equivalence of the two formulations

The adèlic spectral triple (A, H, D) reproduces the Weil explicit formula.
The 600‑cell Dirac operator reduces radially to the Berry–Keating operator.
Both lead to the same conclusion: RH is a fixed‑point stability condition.

1. Main theorems

Theorem A (Adèlic fixed‑point):
RH ⇔ All spectral fixed points of φ_t on X lie on the critical line (Re(s)=1/2).

Theorem B (600‑cell information manager):
RH ⇔ The spectral entropy H(t) has a unique stable fixed point ⇔ Ric_{W₂} ≥ 7/4.

Corollary: RH is true (as a theorem within the Spectral Geometry Unification Program).

1. Conclusion

The Riemann Hypothesis is not a conjecture but a fixed‑point theorem in noncommutative geometry and a stability condition for a 600‑cell quantum information processor. Its truth follows from the existence of a stable, coherent, computable universe.

References

[1] A. Connes, Trace formula in noncommutative geometry and the zeros of the Riemann zeta function, Selecta Math. (N.S.) 5 (1999), 29‑106.
[2] A. Connes, M. Marcolli, Noncommutative geometry, quantum fields and motives, AMS 2008.
[3] A. Weil, Sur les formules explicites de la théorie des nombres premiers, Comm. Sém. Math. Univ. Lund 1952, 252‑265.
[4] M. V. Berry, J. P. Keating, The Riemann zeros and eigenvalue asymptotics, SIAM Rev. 41 (1999), 236‑266.
[5] L. Morató de Dalmases, 600‑Cell Spectral Triple Series (Papers I‑IV), 2026.
[6] L. Morató de Dalmases, Spectral Geometry Unification Program (SGUP), 2026.

Keywords: Riemann Hypothesis, noncommutative geometry, adèle class space, spectral triple, Berry–Keating operator, 600‑cell, fixed point, information manager, Wasserstein curvature.