Spectral Schism

In the paper's words

Standard analytic number theory treats the Riemann zeros as eigenvalues of a Hermitian operator acting on a rigid symplectic manifold. However, the asymptotic diffraction spectrum of the prime numbers exhibits fluid-like Lebesgue measure rather than the crystalline Bragg peaks required for such a spectrum. Here, a relativistic Rindler-Majorana Hamiltonian is subjected to the disorder potential of unbounded Ford-Maynard prime gaps.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Torquato, Zhang, and Martelli (2018) computed the structure factor $S(k)$ for large sets of primes and identified the presence of Bragg-like peaks alongside a diffuse background.

   from 2.2 Stealth Hyperuniformity

2. The critical threshold where the system transitions from a bound state to a scattering state was sought by sweeping through a parameter space of acceleration $a \in [0.01, 100]$ and disorder $\lambda \in [0.1, 1000]$.

   from 3.4 Simulation Methodology

3. (2014) proved that the gaps between consecutive primes grow without bound, exceeding any multiple of the average gap.

   from 4.3 The Asymptotic Drive

How it is done

1. The implications of this duality extend to the error term in the prime number theorem, which is controlled by the real part of the zeros.

   from 1.1 The Explicit Anchor

2. This statistical evidence is so strong that it has shifted the burden of proof onto finding the physical system that generates these statistics.

   from 1.2 The Statistical Tension

3. The zeros would then be the points where the determinant vanishes, corresponding to the bound states of the Hamiltonian.

   from 1.3 The Operator Hypothesis

4. The search for this elusive operator culminated in the proposal of the Berry-Keating Hamiltonian, $H=xp$, which describes a particle moving in a one-dimensional space with a hyperbolic potential.

   from 1.5 The Berry-Keating Model

What would falsify it

1. For the Riemann zeros, the form factor agrees with the GUE prediction for short times but deviates at the “Heisenberg time,” which corresponds to the period of the shortest periodic orbit.

   from 1.4 Semiclassical Chaos

2. A single counterexample to Robin’s inequality would disprove the conjecture.

   from 1.7 The Rarefaction Test

3. This would mean that the Riemann zeros are not discrete points on a line but a continuous band of resonances.

   from 2.1 The Point Process

Where the claim stops

1. However, the asymptotic diffraction spectrum of the prime numbers exhibits fluid-like Lebesgue measure rather than the crystalline Bragg peaks required for such a spectrum.

   from Crystalline

2. This finding suggests the Riemann Hypothesis is an effective field theory that fails at a specific Hagedorn temperature of arithmetic disorder.

   from Crystalline

3. However, the validity of this trace formula depends on the convergence of the sums involved, which is guaranteed only if the spectrum is sufficiently rigid.

   from 1.1 The Explicit Anchor

Where to read

1. Crystalline138 words · 1 min
2. 1.0 SPECTRAL RIGIDITY4,780 words · 21 min
3. 2.0 DIFFRACTIVE FLUIDITY4,580 words · 20 min
4. 3.0 RESONANT BREACH3,435 words · 15 min
5. 4.0 ENTROPIC PHASE4,469 words · 19 min
6. 5.0 ASYMPTOTIC HORIZON4,496 words · 20 min