Temporal Flow, Reflection Duality, and the Geometry of Survival

By John Gavel

Abstract. This post develops a formal account of temporal-relational flows, parity-dependent capacity, and the reflection duality n ↔ N/n that fixes σ = 1/2. We show how these ideas produce pentagonal phase filters, survivor fractions (ideal Φ⁻³ and minimal (3/5)³), and a linear spectral function whose zeros (ghosts) mirror temporal resonances. Finally, we map the framework onto two mainstream reference points: the Higgs potential (as a constraint landscape) and the Riemann zeta / critical line (as a frequency-domain expression of destructive interference).

1. Overview & motivations

The goal is to treat time as primary and to ask: under repeatable relational constraints, which discrete objects persist? We take a few core observations as starting axioms:

Relational load (local interactions) typically scales linearly with N.
Total pairwise capacity scales quadratically with N (N(N-1) ≈ N²).
Geometric and topological constraints (pentagonal/icosahedral motifs, tetrahedral local structures) introduce preferred multiplicative factors (e.g. 12, 4).
Temporal resolution is constrained: interactions are effectively serialized (one neighbor at a time), so compositional structure can create collisions; primes minimize that problem.

2. Concrete units: K, H, and τ (the throughput ratio)

We begin with the numeric units used repeatedly in the framework:

Local/tetrahedral unit: start with N=4 and compute K via pairwise count: \[ K = N(N-1)\Big|_{N=4} = 4\cdot 3 = 12. \]
Icosahedral / 12-point system: take N=12 and compute H: \[ H = N(N-1)\Big|_{N=12} = 12\cdot 11 = 132. \]

Combine them into an effective temporal throughput constraint:

\[ \tau \;=\; \frac{H}{K^2 + H/2} \;=\; \frac{132}{12^2 + 132/2} \;=\; \frac{132}{144 + 66} \;=\; \frac{132}{210} \;=\; \frac{22}{35} \;\approx\; 0.629. \]

Interpretation: τ is the effective fraction of temporal capacity remaining for relational throughput in a 12-point system once the local 4-unit stabilization K is accounted for. Numerically this value reappears in many derived constraints because it encodes the local/global throughput balance.

3. Parity shells: alternating capacity formulas

Two simple parity-dependent formulas capture alternate shell growth:

\(\displaystyle A_n = 12\,n(n+1)\) (odd shells)

\(\displaystyle B_n = 4\,n(n+3)\) (even shells)

Algebraic forms:

\(A_n = 24\,T_n\) where \(T_n = \frac{n(n+1)}{2}\) is the n-th triangular number.
\(B_n = 8\bigl(T_n + n\bigr)\).

n	Aₙ = 12 n(n+1)	Bₙ = 4 n(n+3)
1	24	16
2	72	40
3	144	72
4	240	112
5	360	160
6	504	216

Interpretation: odd shells (Aₙ) carry 12-fold prefactors (pentagonal / icosahedral influence), even shells (Bₙ) carry 4-fold prefactors (tetrahedral / local stabilizer). Both scale ~n² but with different amplitudes; this alternation modulates local τ and can produce parity-dependent regions of resonance and anti-resonance.

4. Recursive 1-D thinning → (3/5)³ and survivors

Volume as a 3D measure corresponds in a 1D temporal manifold to repeated passes (recursive thinning). A single relational pass keeps a fraction 3/5 ≈ 0.6; three sequential, independent constraint passes retain

\[ \left(\frac{3}{5}\right)^3 \;=\; \frac{27}{125} \;=\; 0.216. \]

The complement is

\[ 1 - \left(\frac{3}{5}\right)^3 \;=\; \frac{98}{125} \;=\; 0.784. \]

Linear/recursive language: if ρ₀ = 1 is the initial candidate density, then after k passes

\[ \rho_k = \left(\frac{3}{5}\right)^k \rho_0. \]

For k=3 we obtain the 0.216 survivor fraction (monophonic survivors) and 0.784 unresolved/deferred background. Scaled to 12 slots, that’s ~2.592 survivors and ~9.408 deferred — consistent with the earlier intuition that only a few units remain monophonic while the majority form the background.

5. Pentagonal phase-locking constraint (filter)

We now introduce the geometric phase filter based on pentagonal structure. A candidate n “survives” the pentagonal constraint at frequency ω if

\[ \cot(n\omega) = C \]

where C ∈ {Φ, -1/Φ, Φ−1} and Φ = (1+√5)/2 is the golden ratio. Using the cotangent addition identity, one shows that if a and b both satisfy the constraint with the same C then for n = ab we have

\[ \cot(ab\omega) = \frac{C^2 - 1}{2C}. \]

Setting this equal to C leads to \(C^2 = -1\), which has no real solution. Hence composites cannot phase-lock across the full pentagonal filter; primes (and 1) remain the minimal survivors.

6. Survivor density: Φ⁻³ vs (3/5)³

Two characteristic survivor fractions appear:

Ideal / golden survivors: empirical testing (ω = 10.0, tolerance ±0.2) gives a constrained prime fraction ≈ 0.2526. The golden prediction is

\[ \Phi^{-3} \;=\; \frac{1}{\Phi^3} \;=\; \sqrt{5}-2 \;\approx\; 0.2361, \]

which is within ~7% of the empirical count — a strong signal that pentagonal geometry (Φ) governs survivor density in the “graceful” ideal case.

Minimal / rational survivors: recursive thinning gives \((3/5)^3 \approx 0.216\), a rational lower bound representing minimal viable survivability under three independent passes.

We can view Φ⁻³ as the ideal volumetric density given pentagonal packing; (3/5)³ is a minimal recursive density obtained by sequential filtering in 1D.

7. Linear spectral function, ghosts (zeros), and destructive interference

Define the linear spectral function over survivors:

\[ \Pi(\sigma,\omega) \;=\; \sum_{p \in \text{survivors}} p^{-\sigma} e^{i p \omega}. \]

At the critical exponent σ = 1/2 this becomes

\[ \Pi\bigl(\tfrac12,\omega\bigr) \;=\; \sum_{p \in \text{survivors}} p^{-1/2} e^{i p \omega}. \]

Ghost frequencies ω_k are those where Π vanishes (both cosine and sine sums cancel):

\[ \sum_{p} p^{-1/2} \cos(p\omega_k) = 0,\qquad \sum_{p} p^{-1/2} \sin(p\omega_k) = 0. \]

Interpretation: zeros are exact destructive interferences among survivor phases — geometric resonances reflecting accumulated temporal differences. Numerically searching in ω ∈ [0.1,50] with the pentagonal filter produces discrete zeros (“ghosts”) consistent with the idea that zeros are structured, not random.

8. Reflection duality n ↔ N/n and σ = 1/2 as a fixed point

Consider the involution

\[ n \longleftrightarrow \tilde n = \frac{N}{n}. \]

Define a dual pairing amplitude

\[ A(n;\sigma) \;=\; n^{\sigma} \tilde n^{\,1-\sigma} \;=\; n^{\sigma}\Bigl(\frac{N}{n}\Bigr)^{1-\sigma} \;=\; N^{1-\sigma}\, n^{2\sigma - 1}. \]

For scale invariance (independence from n) we require exponent of n to vanish:

\[ 2\sigma - 1 = 0 \quad\Rightarrow\quad \sigma = \tfrac12. \]

Thus σ = 1/2 is the reflection-fixed scaling exponent. The two conjugate perspectives are:

Active adjacency (inside): \(n^{1/2}\tilde n^{1/2} = N^{1/2}\) — resolving at rate N^{1/2}.
Deferred accumulation (outside): \(n^{-1/2}\tilde n^{-1/2} = N^{-1/2}\) — latency accumulating at rate N^{-1/2}.

Both measure the same invariant latency from opposite ends. The fixed point \(n=\tilde n=\sqrt{N}\) is the monophonic location where inside/outside perceptions coincide.

9. Mapping to the Higgs potential (constraint landscape)

The Higgs potential is typically written (scalar notation):

\[ V(\phi) \;=\; m^2\, \phi^\dagger \phi \;+\; \delta(\phi^\dagger \phi)^2, \qquad \delta>0,\; m^2<0 .="" p="">Interpretation in our language:

\(m^2<0 away="" forbidden="" from="" gap="" is="" li="" perfect="" pressure="" pushes="" relational="" symmetry="" system="" the="" unstable="" zero.="" zero="">
\(\delta>0\): quartic stiffening — capacity grows with scale and prevents runaway; this is the same effect as local τ limiting throughput.
Minimization yields a nonzero VEV: \[ \phi^\dagger\phi \;=\; -\frac{m^2}{2\delta}, \] a fixed relational gap — the geometric offset analog to Φ in pentagonal packing.

Correspondences:

Higgs VEV ↔ fixed relational gap (Φ-like offset).
Radial curvature (Higgs mass) ↔ stiffness against leaving the relational path (resistance to temporal change, τ-related).
Goldstone modes ↔ degrees of freedom along degenerate ring of minima (directional choices / addresses); these can be “eaten” or locked by gauge-like constraints in richer frameworks.
σ = 1/2 ↔ a scaling / anomalous-dimension fixed point governing how fluctuations couple to the background VEV (i.e., how excitations scale within the broken-symmetry phase).

Put simply: the Higgs Lagrangian describes whyhow

which discrete structures survive

10. Relation to the Riemann zeta / critical line

The Riemann zeta function:

\[ \zeta(s) \;=\; \sum_{n=1}^\infty n^{-s} \;=\; \prod_p \frac{1}{1-p^{-s}}. \]

The Riemann Hypothesis asserts non-trivial zeros lie on the critical line Re(s) = 1/2. Interpreting zeros as temporal resonances, we propose the following parallel:

Zeros of \(\zeta(1/2 + i\gamma_n)\) correspond to frequencies at which global phase accumulation of multiplicative structure cancels.
The linear spectral function \(\Pi(1/2,\omega)\) captures the phase structure of the pentagonal-filtered prime survivors; its zeros (ghost frequencies) are the same phenomenon in the temporal-flow representation: destructive interference among weighted primes at σ = 1/2.
Thus both ζ and Π emphasize σ = 1/2 as a structural fixed line: ζ through multiplicative analytic structure, Π through additive phase accumulation of temporally filtered primes.

This is not a proof; it is an interpretive mapping: the same structural geometry (primes as minimal survivors + phase accumulation) produces zeros in both representations.

11. Numerical recipes & reproducibility

Below is the Python code used for empirical checks (primes up to 500; pentagonal constraint using cotangent; survivor fraction; zero detection). Paste into a Python environment with numpy and sympy installed.

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}")

# Zero-search for Π(1/2, ω)
sigma = 0.5
omega_range = np.linspace(0.1, 50, 500)
zeros_found = []
for w in omega_range:
    surv = [p for p in primes if satisfies_constraint(p, w)]
    if len(surv) > 0:
        total = sum(p**(-sigma) * np.exp(1j * p * w) for p in surv)
        if abs(total) < 0.1:
            zeros_found.append(w)
print(f"Zeros found: {len(zeros_found)} in range [0.1,50]")

You can adapt the tolerance, ω range, and primes bound to probe stability, clustering of zeros, and parity effects (use Aₙ/Bₙ instead of primes if you want to test shell-derived survivors).

12. Conclusions

Hierarchy of interactions: local (K) → global (H) → parity shells (Aₙ/Bₙ) → recursive passes (3/5 each) → pentagonal filter → linear spectral function. Each level modulates the next and produces alternating regions of relatability and deferral.
Fixed exponents and reflection: the involution n ↔ N/n produces σ = 1/2 as the unique scale-orientation fixed point, encoding the inside/outside dual measurement of the same latency.
Higgs mapping: the Higgs potential is best read as a constraint landscape (m² < 0, δ > 0); it sets the nonzero relational baseline (VEV) inside which discrete survivors are selected. σ is a scaling exponent — a coupling-dimension in that landscape, not a replacement for the VEV.
Riemann mapping: Π(1/2, ω) and ζ(1/2 + iγ) are two representations of phase cancellation phenomena among multiplicative structures (primes); the temporal-flow view makes the resonance/cancellation interpretation explicit and geometric.

For continuity, link to tables post: Exploring geometry of temporal flow

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