Prime Numbers in Black Holes? Zeta Structures, Primitive Cycles, and a Possible FBA Perspective

Mar 12, 2026

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4 min read

Abstract

Recent popular science reports have suggested that “prime number patterns” might appear inside black holes. However, what lies behind this formulation is typically not a direct connection to the Riemann zeta function itself, but a deeper structural mechanism: primitive cycles in dynamical systems naturally generate zeta products whose zeros encode resonances or spectral fluctuations.

This architecture appears across many fields – from number theory to hyperbolic geometry and quantum chaos. In this article we explore the possibility that similar structures could also arise in horizon-near dynamics. Within the Frame Budget Approach (FBA), such structures can be interpreted as bookkeeping devices for irreducible budget loops in open, irreversible dynamical systems.

The resulting structural chain can be summarized compactly as:

primitive loops → zeta product → zeros → resonances → observables

Zeta chain — The “zeta chain”: primitive cycles generate weights, producing a zeta structure whose zeros encode resonances.

1. Starting Point: “Prime Numbers Inside Black Holes?”

Several recent papers and popular science articles have discussed the possibility that structures from number theory might appear in the context of black holes.

Examples include discussions of so-called “primon gases”, automorphic structures in BKL dynamics, or work relating zeta structures to spectral statistics and form factors in black-hole-related models.

Typical popular introductions include:

Der Standard: “Prime Number Patterns Discovered in Black Holes”
Scientific American: “Are Prime Numbers Hiding Inside Black Holes?”
LiveScience: “Exotic prime numbers could be hiding inside black holes”

However, the underlying research usually refers to mathematical objects such as:

automorphic L-functions
Selberg zeta functions
Ruelle zeta functions
resonance spectra of open dynamical systems

The key question therefore becomes:

What minimal structural conditions allow zeta functions to appear in the first place?

Riemann, Selberg, Ruelle, and possible horizon zeta functions belong to the same structural family: primitive irreducible objects generate product structures and zeros.

2. The Structural Class of Zeta Functions

Many seemingly different mathematical objects share the same structural form:

Riemann zeta: product over prime numbers
Selberg zeta: product over primitive geodesics
Ruelle zeta: product over primitive periodic orbits

In all these cases the same logic applies:

primitive irreducible objects generate a multiplicative product structure.

After analytic continuation this structure produces zeros or poles that control fluctuations, spectra, or resonances.

The prime number theorem, the Selberg trace formula, the Gutzwiller trace formula, and possible horizon explicit formulas share the same structural architecture.

3. Primitive Cycles and Euler Products

If a dynamical system possesses primitive cycles, repeated traversals can be written as powers of a primitive cycle.

The sum over repetitions can then be rewritten as a product over primitive cycles.

ℓγ – geometric cycle length
φγ – reversible phase
Λγ – irreversible damping

The analogy between prime factorization and primitive dynamical cycles.

Proposition – Universal Zeta Structure

Proposition. Any open dynamical system with primitive cycles and multiplicative repetition structure naturally admits a zeta product representation.

The zeros of this function correspond to resonances of the dynamics.

This observation underlies many results in number theory, hyperbolic geometry, and quantum chaos.

4. Determinants and Resonances

This relation connects primitive cycles directly to resonance spectra.

5. Interpretation within the Frame Budget Approach

Primitive budget loops
Part I – visible in the PDF in
Section I.2 “Primitive assumptions of the FBA” and
Section I.3 “Budget calculus”, in particular the subsections on global states, frames, minimal events, refinement invariance, and one-step budgets.
Time structure
Part II – Definition II.3.1 “Time as strictly increasing embedding”
Definition II.4.1 “Geometric proper time”
Definition II.4.2 “Aging (irreversible contribution)”
Open dynamics
Part IV – Definition IV.3.2.1 “Quantum Markov semigroup”
Formula Box IV.3.2.1 “Generator of a CPTP semigroup”
Lemma IV.3.3.1 “GKLS equation”
Gravitational proxy
Part VI – Formula Box VI.3.1.1 “Curvature proxy from budget lapse”

6. Horizon Zeta (Toy Model)

Primitive horizon-near budget loops generate a zeta structure with resonance poles.

7. Explicit Formula

8. Falsifiability

FBA references:

Part X – Formula Box X.7.2.1 “Generic bridge falsification criterion”
Part X – Formula Box X.10.2.1 “Bridge decision rule (binary) and goodness measure (continuous)”

References

Hartnoll & Yang – The Conformal Primon Gas at the End of Time
De Clerck, Hartnoll, Yang – Automorphic L-functions and complex primon gases
Basu, Das, Krishnan – An analytic zeta function ramp at the black hole Thouless time
Dyatlov, Guillarmou – Pollicott–Ruelle resonances
Cvitanović et al. – Periodic orbit theory
Frame Budget Approach – Parts I–X

Discussion

Zeta functions can be understood as compressed bookkeeping of irreducible cycles in open, irreversible dynamical systems.

In this perspective, black holes would not represent an exotic exception, but rather an extreme limiting case of a universal structural architecture.