The Mann Theory
A speculative framework and computational toy models at the boundary of mathematics and physics.
This is not a proof. The Mann Theory does not propose a proof of the Riemann Hypothesis. All constructions are finite, all results are heuristic and experiment-dependent.
Mann’s Fundamental Hypothesis (M1)
The Riemann Hypothesis is not merely a number theory problem, but a necessary condition for the consistency of quantum gravity.
Mann’s Hypothesis proposes a deep connection between the zeros of the Riemann zeta function and the spectral properties of quantum gravity operators. The central claim:
Mann’s Operator
The theory introduces Mann’s operator ĤM, a self-adjoint operator identified with the Wheeler-DeWitt Hamiltonian from Loop Quantum Gravity. Its eigenvalues arise from the area operator:
where γ is the Barbero-Immirzi parameter, ℓP is the Planck length, and j ∈ {½, 1, 3/2, 2, ...} are the spin quantum numbers. For our experiments we use:
with scaling factor α = 1 as the default.
Spectral Zeta Function ζQG
The quantum gravity spectral zeta function is defined as:
This is a Dirichlet-like series generalizing the Riemann zeta function ζ(s) = Σ n−s. The experiment searches for zeros — points s* where |ζQG(s*)| ≈ 0.
Black Holes as Prime Numbers of Spacetime
A striking aspect of Mann’s framework is the analogy between prime numbers and black holes:
- Prime numbers are the atomic building blocks of integers (fundamental theorem of arithmetic).
- Black holes are proposed as the atomic building blocks of spacetime geometry — discrete quantum objects with quantized area and entropy.
- The Bekenstein-Hawking entropy SBH = A/(4ℓP2) connects black hole area to information, paralleling the connection between primes and zeta zeros.
Montgomery-Odlyzko Law & GUE Statistics
The Montgomery-Odlyzko law states that the normalized spacings between consecutive Riemann zeta zeros follow the GUE distribution (Gaussian Unitary Ensemble) from random matrix theory — the same statistics governing eigenvalues of large random Hermitian matrices.
The experiment computes zero spacings and compares them against the Wigner surmise p(s) = (πs/2)exp(−πs2/4) for normalized spacings.
Computational Approach
The theory is tested through explicitly defined finite constructions. We consider truncated zeta-like sums and examine their numerical behavior near target points — the imaginary parts of known Riemann zeros.
Numerical Methodology
Given target ordinates tk (imaginary parts of known Riemann zeros), we search for sk* near 0.5 + itk where:
The solver uses the Nelder-Mead simplex method — a gradient-free optimization algorithm robust to the oscillatory landscape of complex zeta functions. Two modes are available:
- minimize |ζ| — minimizes the absolute value directly
- minimize |ζ|² — smoother near true zeros, with bounded fallback search
Key Metrics
- Residual: rk = |ζQG(sk*)| — smaller is better (green < 10−8)
- Critical-line deviation: |Re(sk*) − ½| — should be near zero
- Zeros on critical line: fraction with |Re(s*) − 0.5| < 0.05
- GUE match: how well zero spacings match Wigner surmise
Interpretation
A small residual indicates the solver found a point where ζ_M is close to zero, but does not guarantee a true zero. The most informative signal is stability across N and seeds.
Results may change with the spectrum model, solver method, truncation length N, and random seed.
Run the Experiment
Configure parameters, run the computation, and examine diagnostic plots and residual tables.
Run experiment →Limitations
- All constructions are finite and model-dependent.
- No claim is made that ζ(s) arises from a physical operator.
- This framework does NOT prove the Riemann Hypothesis.
- Numerical minima are not guaranteed exact zeros.
- Spacing/correlation plots are not meaningful for small sample sizes.
The Mann Theory: Spectral Gravity and the Critical Line
Hilbert’s Eighth Problem, Zeta Functions, and the Quantum Geometry of Black Holes. By Radim Kaufmann & Bernhard Mann, 2026.