1.1 — Effective Region Radius

Curvature-influence reach, propagated Region disturbance, and admissible equilibrium.

The Effective Region Radius \(R_{\mathrm{eff}}\)

The Effective Region Radius \(R_{\mathrm{eff}}\) defines the intrinsic curvature-influence envelope of a Region. It represents the distance at which the Region’s curvature contribution becomes indistinguishable from the dormant background.

\(R_{\mathrm{eff}}\) is not physical size, charge radius, or matter radius. It is a reach. Observable quantities such as nuclear charge radius, matter radius, or scattering form factors are derived from this geometric reach through additional projection rules not defined in this tier.

Every coherent entity in the EOTU lattice — CPP, nucleon, nucleus, or larger composite — generates curvature that propagates outward through the dormant corridor network.

Each structure therefore has a finite spatial limit beyond which it no longer raises dormant-cell amplitudes above the universal mean. This limiting distance defines the Effective Region Radius.

Across the post Freeze-Out framework, composite behavior follows the sequence:

\[ \mathrm{closure\ geometry} \rightarrow R_{\mathrm{eff}} \rightarrow \lambda_{\mathrm{Region}} \rightarrow \mathrm{curvature\ drop\text{-}off} \rightarrow \mathrm{admissible\ equilibrium} \]

Far-Field Curvature Drop-Off

A Region’s propagated curvature disturbance decreases with distance because its curvature inventory is distributed over expanding dormant-corridor shells. In the far-field inventory limit, the available shell area grows as \(4\pi r^2\), so the curvature contribution decreases as \(1/r^2\).

For a Region with effective reach \(R_{\mathrm{eff}}\), the geometric drop-off term is therefore:

\[ G_R(r) = \left( \frac{R_{\mathrm{eff}}}{r} \right)^2 \]

This expression describes the far-field curvature drop-off of the propagated Region disturbance. Local interaction is not determined by this geometric term alone. Near and intermediate Region interactions are filtered by closure geometry, corridor availability, and wavelength-phase compatibility.

Thus large-scale curvature behavior approaches inverse-square drop-off, while nuclei, atoms, ionization, bonding, and larger composite interactions are governed by the same drop-off filtered through admissible Region geometry and propagated wavelength structure.

Base Effective Reach Construction

The base \(R_{\mathrm{eff}}\) is a factor of \(k\), where \(k\) is the number of CPPs within a Region, squared, then multiplied by \(51L_0\).

\[ N_{\mathrm{proton}} = 6, \qquad N_{\mathrm{electron}} = 4, \qquad N_{\mathrm{neutron}} = 10 \]
\[ k_{\mathrm{CPP}}(N) = N_{\mathrm{cpp}}^2 \]
\[ R_{\mathrm{eff}}(k) = \left(51k_{\mathrm{CPP}}\right)L_0 \]

Summary

  • \(R_{\mathrm{eff}}\) defines the intrinsic curvature-influence reach of a Region.
  • \(R_{\mathrm{eff}}\) is not physical size, charge radius, or matter radius.
  • Every coherent entity generates curvature that propagates through the Dormant Corridor network.
  • Composite behavior proceeds through closure geometry, \(R_{\mathrm{eff}}\), propagated wavelength, curvature drop-off, and admissible equilibrium.
  • The far-field geometric drop-off term is \(G_R(r)=\left(R_{\mathrm{eff}}/r\right)^2\).
  • Near and intermediate interactions are filtered by geometry, corridor availability, and wavelength-phase compatibility.
  • The base effective reach construction is \(R_{\mathrm{eff}}(k)=\left(51k_{\mathrm{CPP}}\right)L_0\).