Introduction

Pattern Field Theory is built on primitive structural variables:

• motion,

• curvature,

• primes,

• adjacency,

• resonance,

• shell indexing,

• equilibrium and dissipation.

A system constructed from these primitives produces a consistent set of constants. These constants are not external inputs; they are determined by the lattice structure of the AOL and the resolutive geometry of QuantaHex.

Curvature and Ratio Constants

The constant \(\pi\)

Curvature closure on AOL shells forces a fixed angular constant. Whenever a resonance loop completes a rotation or a curvature shell closes, the associated angular measure converges to \(\pi\). Therefore, \[\pi \text{ is the closure constant of all AOL curvature loops.}\]

The constant \(\varphi\)

Ratio stabilization under discrete shell transitions yields the golden ratio \(\varphi\). Minimal distortion between inward and outward curvature on successive shells solves to the stable ratio: \[\varphi = \frac{1 + \sqrt{5}}{2}.\] Thus \(\varphi\) arises as the equilibrium ratio between shell-level curvature parameters.

Exponential Constants

The constant \(e\)

The exponential base \(e\) emerges from:

• prime-density asymptotics \(\pi(x) \sim x / \ln x\),

• exponential curvature accumulation,

• resonance decay on discrete shells.

The constant appears as the natural limit of scaled curvature increments.

The constant \(\zeta(2) = \pi^{2}/6\)

The value \(\zeta(2)\) is the universal packing density constant for hexagonal and spherical adjacency structures. On AOL shells, curvature overlaps generate density transitions proportional to \(\pi^{2} / 6\). This is the fundamental packing constant for second-order curvature coherence.

Prime-Driven Logarithmic Constants

Prime-spacing follows: \[p_{n+1} - p_{n} \sim \log p_{n}.\] Therefore logarithmic constants appear in:

• resonance timing,

• shell spacing,

• curvature jump lengths.

Constants such as \(\ln(2)\), \(\ln(10)\), and \(\ln(n)\) appear whenever prime-indexed structure is referenced.

Depth Exponents from AOL Structure

Volumetric depth exponent \(D_{\mathrm{vol}} \approx 2.7\)

Empirical galaxy surveys satisfy: \[N(<R) \propto R^{D_{\mathrm{vol}}}.\] With \(D_{\mathrm{vol}} \approx 2.7\). Prime-indexed curvature shells on the AOL generate multi-scale volumetric density transitions that converge to this exponent. Thus, \[D_{\mathrm{vol}} = 2.7 \text{ is a depth reading of prime-seeded volumetric complexity.}\]

Boundary depth exponent \(D_{\partial} \approx 1.4\)

Excursion boundaries of resonance fields on AOL projections satisfy a box-counting relation: \[N_{\partial}(\varepsilon) \propto \varepsilon^{-D_{\partial}},\] where empirical and theoretical values converge near \(1.4\). This exponent is the boundary complexity of curvature-defined resonance surfaces.

Unified Constants Table

Conclusion

The constants identified above arise from the deterministic structure of the Allen Orbital Lattice and the three-dimensional hexagonal discretization of QuantaHex. They are not imposed; they emerge from:

• prime-indexed curvature,

• discrete shell structure,

• resonance-loop closure,

• statistical depth readings.

This document establishes the invariant constants-base of Pattern Field Theory and provides the structural foundation for future developments.