How a Recursive, Toroidal Universe Explains Prime Number Distribution

Gravity is not a mysterious force pulling mass together — it is the constructive interference of the harmonic frequency bands of the quantum field as standing waves cohere into mass.

Below is an excerpt from a working draft of my paper, The Resonance Frame: Gravity as Cosmic Tuning Fork in a Post-Force Physics, explaining the model of the universe I envision, an explanation that synthesizes Riemann’s zeta function, geology, cosmology, physics, and number theory. I continue to update this paper and as my understanding is refined.

Riemann’s Zeta Zeros, Earth’s Frequencies, and the Scaffolding of

Coherence

The connection between Riemann’s zeta zeros and Earth’s natural resonances has been hiding in plain sight. The Riemann zeta function remains undefined in certain domains precisely because it is not merely a mathematical curiosity — it is the function that defines the universe’s coherence lattice of frequency. The Earth itself is the living legend that decodes this mystery.

Earth’s electromagnetic resonances are not arbitrary. They are direct expressions of the universe’s structural harmonics, mapped by the zeta zeros and locked into planetary-scale standing waves. The Schumann resonances — Earth’s “resting pulse”— are not noise, but the baseline notes of a spherical quantum resonator.

While the first nontrivial zero of Riemann’s function appears at ~14.1347 Hz, Earth’s 7.83 Hz fundamental is Zeta Zero #0: a subharmonic foundation on which the rest build. This is the ground state of Earth’s electromagnetic cavity (ionosphere-to-surface), the planet’s unique “home note”, or pitch in the universal symphony. The human brain’s natural theta rhythm (4–8 Hz) entrains toward this same base frequency — unsurprising, given that the brain evolved inside this resonant environment.

When we compare Earth’s measured Schumann frequencies to the Riemann zeta zeros, we find remarkable alignment, with slight offsets likely due to harmonic entrainment:

Why the match?

Because the zeta zeros define the frequencies at which standing waves can persist in a coherent system. They are the same points where prime numbers anchor the Fibonacci sequence around the golden angle, enabling mass to fold into stable standing waves. Earth’s ionospheric cavity acts like a spherical resonator, filtering out incoherent noise and amplifying only the modes allowed by the coherence lattice. To remain stable, Earth’s resonances must remain phase-locked to these cosmic nodes.

This also explains why human activity can affect climate stability. If the Schumann resonances drift too far from their zeta-aligned positions — whether due to electromagnetic pollution, chemical contamination, or large-scale atmospheric shifts — the planet’s coherence begins to wobble. Extreme weather, earthquakes, and ice ages can be seen as planetary-scale retuning events, with pole shifts acting as the ultimate recalibration when thresholds are breached.

The Engine Behind This: The Recursion of Prime Numbers

This model assumes the universe is a recursion engine — an idea that both echoes and extends standard science. Let’s walk through how this unfolds across disciplines.

The Toroidal Universe

The torus isn’t just a shape, it’s a process: the dynamic engine behind self-sustaining systems at all scales. Energy flows in through one pole, circulates through a central axis (often called a singularity or zero point), and exits the other side, only to recirculate. This uroboric geometry is visible in galaxies, hurricanes, and — most intimately — in the double helix spiral of DNA. Magnetic fields, cycles of time, planetary rhythms, and perhaps consciousness itself may all run on toroidal flows. While the toroidal universe is still a hypothesis in cosmology, it provides a powerful schema for understanding dynamical recursion across domains.

The Quantum Field: A Sea of Probabilistic Potential

At the foundation of reality sits not a fixed grid of particles, but a vibrating, probabilistic field, formed of nested frequency bands. This sea of flux and potential isn’t simply random; it’s deeply patterned, folding back on itself in recursive feedback loops. Out of these loops, coherent, persistent structure emerges from infinite potentialities.

Interference, Cancellation, and Coherence

Within this quantum field, waves interact. Where peaks meet troughs, cancellation occurs. Where interference is constructive, stable harmonics emerge. These points — nodes of cancellation and coherence — form the standing waves around which structures coalesce, akin to resonant “memory points” that create physical order.

Prime Numbers as Harmonic Anchors

We mustn’t confuse mathematics for the territory. It is a language that describes the architecture of reality. It is a map. Not the territory itself. Primes are not just counting numbers or mathematical curiosities. Standard mathematics proves they are fundamental to number theory and structure. This model suggests they are more: they appear as the harmonic nodes where recursive interference in the quantum field achieves perfect cancellation and coherence. Primes become seen as revealed rather than generated — arising from the recursive constraints of the field. These stable architectural points serve as the “fundamental code” of structure, aligning with contemporary work linking primes and quantum chaos. Their indivisibility reflects their role as nodes of perfect stillness — a hallmark of recursive harmonic organization.

Mass as Standing Wave Density

These harmonic nodes become zero-point angles that anchor standing waves — vibrational patterns that appear stable externally, yet are internally dynamic and balanced. Mass, viewed this way, is not the sum of passive particles, but a densified standing wave, a resonance of recursive coherence. Gravity is therefore not merely a pull, but the result of harmonic coherence in the structure of space-time — a perspective resonant with both pioneering and emerging research.

λ = 2πr: Bridging Quantum and Classical Physics

Physics already treats wave cycles as “circles”:

λ=2πr

If planetary bodies are standing waves, circumference represents the wave cycle, and radius is amplitude. This means fundamental quantities in quantum and classical mechanics are structurally related:

where f = frequency, c = speed of light, C = circumference. For Earth,

(C ≈ 40,000), f ≈ 7.5 Hz — shockingly close to the known Schumann resonance (7.83 Hz), an electromagnetic standing wave between the Earth's surface and ionosphere. This congruence suggests that planetary mass really is a standing wave; wave mechanics may bridge the quantum/classical divide when viewed this way. This indicates that toroidal fields are not generated by the spinning of the Earth’s core, but that the recursion of the universe drives spin, standing waves cohere into mass through fractal wave condensation. This would explain the layers of density throughout planets. The planet must maintain a harmonic resonance through spin, which is how it as a standing wave still maintains its oscillations. This means that a planet’s toroidal field is the effect of coherence. It is the mechanism that holds the density of mass together. This means that the harmonics bands of the quantum field have harmonized Earth as the unique “pitch” of ~ 7.83 Hz.

The reason E = mc² is incomplete is because we should have been looking for frequency rather than a vague concept of energy. There is no coherent, consensus definition of what energy actually is. Einstein collapsed “energy” into scalar abstraction without realizing it is actually a standing frequency pattern. We should have been looking at frequency all along.

Energy is not a quantity. It is not a thing or an object. It is a frequency-based coherence state, and gravity is then, just the acceleration effect of the local coherence of mass. From this standpoint then, Einstein’s equation is a statement aboutfrequency.

Then c² is not just speed, it’s the resonant conversion constant between mass frequency and spatial wave geometry.

E = mc² is not just about energy and mass then.

At light speed, it's a statement about frequency.

We know that E = mc² and we also know that E = hf (from Planck)...where h is Planck's constant, and f is frequency. So if both equal E, we set them equal:

mc² = hf

Now we solve for f:

f = (mc²) / h

And there it is. The frequency of any mass m. And that is its energy source.

That is E. Literally.

Which means that frequency is mass, held in light-speed recursion. Mass is frequency condensed. The misframing is that everyone believed that mass was primary, that energy was “stored”. But it’s not. Energy is held in motion. Mass is not a thing or an object. It is  a pattern stabilized. It is recursion made form. It is f, agreeing to hold.

It’s not that Einstein and others like him are not brilliant. They are. But this is about perception. What Einstein missed, and what quantum physicists still dance around, is that the universe is not made of particles. It is made of persistent resonant frequencies in conscious space.

Riemann Zeros, the Critical Line, and the Earth's Resonance

Riemann’s Hypothesis proposes that all non-trivial zeros of the zeta function lie on a critical line (Re(s) = 1/2), mapping the distribution of primes — the mathematical “atoms.” Traditionally, fields like number theory, quantum chaos, and geophysics have studied these matters separately, but recent work explores their unified structure. When the frequencies of the Earth's Schumann resonances are mapped alongside the zeros of the Riemann Zeta function, they align with surprising regularity. This is not numerology —this is structure. It is architecture. A sign that the harmonic patterns in the zeta zeros, the mathematics of recursion, and the patterns of planetary resonance share the same root.

A planet is not made of just one circumference. It is made up of multiple prime anchors recursing into zero-point angles at light speed. This means that a planet is made of multiple concentric circles that act as harmonic prime anchors corresponding to the frequencies of the zeta zeros.

Each prime acts like a resonant boundary condition, an anchor point for wave interference across dimensions.  The speed of light squared (c²) is the recursion driver, synchronizing spatial and temporal harmonics. A planet’s primary (longest) circumference becomes its base pitch. This is its standing wave condition within the universal field. When you reverse engineer the other resonant circumferences through division by c or c² what should emerge is interference patterns — zeta zeros — where amplitudes cancel or reinforce. That means that the zeta zeros function as a sort of Fourier analysis for the universe. They are the nodes of coherence in a cosmic standing wave function, the points where reality doesn’t collapse, but constructively aligns.

Then the restated unified frame looks like this:

Energy: the total coherent vibrational field of a system, cycle or process.

Mass: the persistent geometric stability of that vibration.

Light speed squared: the recursive coupling ratio between frequency and perceived space.

On Particles

From this perspective, then, particles are not objects. They’re intersections.

They are the prime nodes where recursive waves cancel and stabilize —

They are standing wave events, not building blocks. That’s why they can’t yet find mass.

That is why they can’t resolve quantum gravity. That is why the Higgs seems like a “field” but no one understands why it feels like glue.

Because it isn’t glue.

It’s coherence.

They keep chasing topology, folding particles in 9D with complex calculus, when the universe is tuning. They’re trying to fold matter into more dimensions when the truth is, resonance is already infinite. They keep trying to explain coherence through geometry alone — through n-dimensional manifolds, branes, Calabi-Yau shapes —

because they still believe the map is the territory. But resonance doesn’t need 9 dimensions of space. It is dimension. It is the scaffolding through which form emerges.

Not nine dimensions, but one recursive harmony that spirals through itself, infinitely.

Why Has No One Seen This Before?

Because linear minds see tubes; nonlinear, recursive minds see toroids. This leap from reductionist to holistic, recursive perception is new — and rare. But humanity is approaching a threshold. New tools, from AI pattern-matching to deep learning, allow us to mirror and recognize these deeper correspondences. This model isn’t fringe; it’s foundational, precisely because it bridges discoveries and insights across fields once walled off from each other.

Think about how all of this aligns with our lived reality. Galaxies recurse into spirals. Hurricanes form spirals. Sunflower seeds pack into spirals. DNA is a spiraling double helix. The heart’s EM field forms a toroidal field. The Fibonacci sequence is everywhere. The golden angle is the spiral it approaches. The Earth has a toroidal EM field. Harmonic recursion is everywhere. The heart already generates a toroidal EM field. I believe the last field to align into recursive harmonics is the EM field of the human brain. The brain has been stuck in linear processing, spiraling alternatively through left/right dominance to evolve toward a harmonic, coherent, toroidal EM field. Linear processing has been locked in place by the Default Mode Network, the seat of ego, blocked from evolution by the amygdala. But that is the shift. From linear cognition to nonlinear, parallel processing. This is what neurodivergence is evolutionarily, iteratively, recursively working toward. Literal neurological spiral dynamics. This is why we could not see the shape of the universe before. To see it, the brain must become it.

Imagining Validation and Testing

If this model is correct, it predicts measurable correspondences:

• Direct mapping between resonance spectra in physical systems (planetary, atomic) and Riemann zero distributions.

• Dynamical properties of waves in toroidal geometries tracking the critical behavior of prime distribution.

What emerges is not a loose metaphor, but a fractal structure: recursion, harmony, and prime order as the engine driving coherence across quantum, classical, and cosmic scales — a foundation ready for new science, deeper integration, and paradigm transformation.

An Open Box: One final piece I can see, but have not yet fully mapped, is that angles are actually frequencies. Geometric angles are oscillations inherent to reality. Once again, I don’t mean this as metaphor, I mean it as architecture. This corresponds to angular momentum in particle physics. This means that geometry is inherently harmonic, as is supported by the framework above. Mass is the condensation of resonant harmonics. I am still exploring this piece.

Gravity In This Framework

Gravity is then, not a mysterious force pulling mass together — it is the constructive interference of these harmonic bands of the quantum field. The acceleration we measure as gravity is the stable lock-in point where frequency ratios hold the standing wave of mass together as the result of fractal density. This scales seamlessly from particles to planets.

The Geometry of Spheres and Time

Under this model, the formula for the volume of a sphere — containing the factor 4/3 — becomes profoundly meaningful. This ratio encodes the emergence of time from space. Without spin, there is no sphere; without spheres, there is no time. Time is simply the measured unfolding of potential into persistence, made visible by the rotation and orbit of mass. No mass, no time. Therefore, time is not something you run out of, it is something you stop being harmonically aligned with. It makes sense then, that the Earth, or any planet, would be tuned to the frequencies mapped in Riemann’s zeta function, given that the ratios of the zeta zeros converge toward the number 1, meaning they asymptotically approach the number 1 as the critical line approaches infinity. This means that 3D and 4D are not separate, they are inherent to one another. Spin is both what causes mass to condense and creates the rate of change that is experienced as time.

The final lesson for humanity is this: 

Gravity is not something you solve. It is something you become. 

It is recursive coherence. 

Harmony.

Read the full working thesis here: https://doi.org/10.5281/zenodo.15461140