Temporal Flow, Pentagonal Constraints, and the Critical Line σ = 1/2
By John Gavel
This post explores the origin of key parameters in our temporal-relational framework, the role of primes as minimal survivors under geometric constraints, and the emergence of the critical line σ = 1/2.
Part 1: Where do K and H come from?
We start with the temporal throughput ratio for a 12-point system. In our framework:
- K comes from the 4-vertex unit:
K = N(N-1) = 4 × 3 = 12
- H comes from the 12-vertex system (icosahedral interactions):
H = N(N-1) = 12 × 11 = 132
Then the effective temporal ratio is:
\(\tau = \frac{H}{K^2 + H/2} = \frac{132}{12^2 + 132/2} = \frac{132}{144 + 66} = \frac{132}{210} = \frac{22}{35} \approx 0.629\)
This ratio appears repeatedly as the fundamental temporal throughput constraint for 12-point systems.
Part 2: Why Primes Are Survivors
Composite numbers must resolve interactions within their factors:
n = ab → a(a-1) interactions, b(b-1) interactions, plus inter-factor interactions
Under temporal constraints (one neighbor at a time), composites experience relational overflow — both neighbors may be occupied simultaneously, forcing temporary exclusion.
Primes, however, have no internal factorization. They require only:
p(p-1) interactions,
no internal sub-structure,
no temporal collision.
Thus, primes are the minimal temporal units that survive geometric closure.
Part 3: The Pentagonal Constraint
A number n survives the pentagonal constraint if:
\(\cot(n \omega) = C\)
where \(C\) is one of the golden-ratio-related values:
- Φ ≈ 1.618
- -1/Φ ≈ -0.618
- Φ - 1 ≈ 0.618
For composites, using the cotangent addition formula:
\(\cot(ab \omega) = \frac{C^2 - 1}{2C}\)
Setting this equal to C leads to:
\(C^2 - 1 = 2C^2 \Rightarrow C^2 = -1\)
No real solution exists — composites cannot phase-lock. Only primes (and 1) satisfy the constraint across all three pentagonal structures.
Part 4: The Survivor Fraction
Testing the fraction of primes satisfying the pentagonal constraint:
| ω | Total Primes ≤ 500 | Constrained Primes | Fraction | Theoretical Prediction |
|---|---|---|---|---|
| 10.0 ± 0.2 | 95 | 24 | 0.2526 | Φ⁻³ ≈ 0.236 |
The match is within ~7%, showing that the density of survivors is governed by the inverse cube of the golden ratio:
\(\text{Survivor fraction} = \Phi^{-3}\)
-Part 5: Ghosts and Zeros
Define the linear spectral function:
\(\Pi(\sigma, \omega) = \sum_{p \in \text{survivors}} p^{-\sigma} e^{i p \omega}\)
At σ = 1/2 (the critical line):
\(\Pi(1/2, \omega) = \sum_p p^{-1/2} e^{i p \omega}\)
Ghosts (zeros) occur when:
\(\sum_p p^{-1/2} \cos(p \omega_k) = 0\),
\(\sum_p p^{-1/2} \sin(p \omega_k) = 0\)
These are frequencies where survivor phases cancel exactly — geometric resonances, not random events.
Part 6: The Critical Line σ = 1/2
The gap factor scales as:
\(v \sim \frac{\sqrt{N}}{2}\)
The exponent 1/2 appears naturally as the geometric mean between linear relational load (N) and quadratic capacity (N²). In dual-pairing terms:
\(A(n) = n^{\sigma} \tilde{n}^{1-\sigma} = n^{\sigma}(N/n)^{1-\sigma} = N^{1-\sigma} n^{2\sigma - 1}\)
Scale invariance requires:
\(2\sigma - 1 = 0 \Rightarrow \sigma = 1/2\)
Part 7: Connection to the Riemann Zeta Function
The classical Riemann zeta function:
\(\zeta(s) = \sum_{n=1}^{\infty} n^{-s} = \prod_p \frac{1}{1 - p^{-s}}\)
The Riemann Hypothesis states that all non-trivial zeros have real part σ = 1/2. In temporal flow terms, these zeros are frequencies where phase accumulation among primes reaches exact destructive interference — the same phenomenon as ghost frequencies in Π(1/2, ω).
Part 8: Numerical Verification
Python code to verify the Φ⁻³ prediction:
from sympy import primerange
import numpy as np
PHI = (1 + np.sqrt(5)) / 2
CONSTRAINT_VALUES = [PHI, -1/PHI, PHI - 1]
def satisfies_constraint(p, omega, tolerance=0.2):
angle = p * omega
sin_val = np.sin(angle)
if abs(sin_val) < 1e-10:
return False
cot_val = np.cos(angle) / sin_val
return any(abs(cot_val - C) < tolerance for C in CONSTRAINT_VALUES)
primes = list(primerange(2, 501))
omega = 10.0
survivors = [p for p in primes if satisfies_constraint(p, omega)]
print(f"Total primes: {len(primes)}")
print(f"Survivors: {len(survivors)}")
print(f"Fraction: {len(survivors)/len(primes):.4f}")
print(f"Predicted (Φ⁻³): {np.sqrt(5)-2:.4f}")
Output:
- Total primes: 95
- Survivors: 24
- Fraction: 0.2526
- Predicted (Φ⁻³): 0.2361
Part 9: What This Means
- Primes are not random: They are minimal temporal units that survive relational constraints.
- Golden ratio matters: Φ appears naturally in gap scaling and volumetric resolution; the survivor fraction Φ⁻³ arises from 3D packing constraints.
- The critical line is structural: σ = 1/2 is the geometric mean of relational scaling, balancing temporal load and capacity.
- Zeros are resonances: Riemann zeros correspond to ghost frequencies — temporal resonances where accumulated phases cancel.
Conclusion
I didn't set out to work on the Riemann Hypothesis. I was studying temporal flow constraints and how geometry emerges from accumulated differences.
What I found is that the same equations governing N-point relational systems—N², N(N-1), and v = (√N+1)/2—appear to encode the structure of prime numbers.
Primes survive because they're minimal. The golden ratio appears because N=5 is pentagonal. The critical line σ=1/2 is the balance point. The zeros are resonances.
This is either a deep connection or an elaborate coincidence. The numerical tests suggest the former.
The work continues.
For more numerical examples and detailed tables (τ, N vs v, etc.), see the previous post.
