Temporal Flow, Pentagonal Constraints, and the Critical Line σ = 1/2

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.

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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:

\(\text{Survivor fraction} = \Phi^{-3}\)

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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.

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For more numerical examples and detailed tables (τ, N vs v, etc.), see the previous post.