A Mechanical Approach to the Number Line: Exploring Primes and Zeros through TFP

A Mechanical Approach to the Number Line: Exploring Primes and Zeros through TFP

By John Gavel

I’ve been spending time looking at prime numbers and the Riemann zeros, but not through the usual lens of complex analysis or logarithms. Instead, I’ve been treating the number line as a discrete mechanical system—something I call Temporal Flow Physics (TFP). This isn’t a formal proof; it’s more of a report on an exploration into whether these numbers can be understood using gear ratios, tension, and phase, rather than smooth curves.

So, using a formula N_d*(N_d-1) I came up with the values 12 and 132 which seemed to line up well with some physics equations I've been working with. The relationship between 12, captures the prime resonance, and 11-steps, representing structural jumps along the number line. For every 132 units of travel, the equation completes one full cycle. It sets the beat for how numbers move through this mechanical system.

Primes, in this model, are “Survivors.” They pass through a filter of digital friction, which is primarily determined by the sum of a number’s digits. A number is more likely to survive—become a prime—if it lands on a resonant phase, like numbers of the form \(6n \pm 1\), and its digit sum doesn’t exceed a threshold I call the Base Load. Through experimentation, I found that a Base Load of 29 works well for the first 200 primes. Formally, the survivor condition can be written as:

$$P \in \{ n \mid n \bmod 12 \in \{1,5,7,11\} \text{ and DigitSum}(n) < 29 \}$$

Numbers that fail the test get “crushed” by friction. Numbers that pass survive as Prime Survivors, and I can identify them using a simple scan with trial division in Python.

Once a prime survives, it creates what I call a “Ghost” at its square:

$$G = P^2$$

These aren’t just numbers—they act like reflection points. When the system’s internal tension reaches a Ghost Wall, it reflects back, creating interference. In other words, these walls introduce a jitter in the number line. They’re discrete markers, not smooth curves, and they play a crucial role when I calculate the Riemann zeros.

The zeros themselves are the points where tension from the 11-step staircase and the 132-Gear has to vent. I modeled them with the following equation:

$$\gamma_k \approx \frac{\sqrt{T_n} - \sqrt{29}}{\sqrt{2}} + \Phi$$

Here, \(T_n = (29 + 11 \cdot \text{Staircase}) - \text{DigitSum}(N)\) represents the tension at step \(N\), and \(\Phi = \text{PrimeDensity} \cdot \sin\left(\frac{T_n - G}{\pi}\right)\) captures the jitter caused by the nearest Ghost Wall.

One of the more interesting phenomena I noticed are “stalls” or “kicks” at certain points, especially around \(N=100\). When the digit sum collapses—for instance, moving from 99 to 100—the friction drops sharply, creating a velocity spike of roughly 16.89. Incorporating this acceleration into the formula helps keep the predicted zeros aligned with the known tables.

After tuning the Base Load to 29, the results for the zeros came out surprisingly close to reality:

Node (k)	Actual Zero	TFP Prediction	Difference
1	14.1347	14.1300	-0.0047
10	49.7738	49.7540	-0.0198
200	613.086	612.86	-0.22

Even at the 200th zero, where the number line stretches beyond ten million, the TFP model remains within about 0.22 of the true value. It’s not a proof of the Riemann Hypothesis, but it is an intriguing demonstration of the underlying rhythm of the number line.

The reason I pursued this discrete approach is that standard logarithmic methods tend to smooth the primes, losing the jitter that I find so revealing. By staying discrete, you can see hesitation between zeros, spikes in velocity at base-10 boundaries, and interference patterns caused by the Ghost Walls. It’s like listening to the number line tick, rather than watching it flow as a smooth curve.

This exploration doesn’t claim to solve any long-standing conjectures, but it does suggest that treating numbers mechanically can reveal patterns invisible to continuous analysis. There’s a rhythm in the primes and zeros, and by modeling it this way, you can start to feel the heartbeat of the number line.