§2.1 — King Coherence Length

The King coherence length \(\lambda_k\) provides the SI bridge that converts native \(L_0\)-based lattice counts into measured distance.

Length Bridge Definition

The EOTU lattice is expressed internally using the primitive lattice quantum \(L_0\). Radial quantities within the theory are therefore represented as integer or fractional counts of \(L_0\). The King coherence length \(\lambda_k\) provides the physical scale that maps those lattice counts into SI distance for measured comparison.

\[ r_{\mathrm{SI}} = N_{L_0}\lambda_k \]

In this expression, \(N_{L_0}\) is the native lattice count and \(r_{\mathrm{SI}}\) is the corresponding measured-distance projection.

Proton Radius Mapping

The parameter \(a_z\) represents the proton triad radius expressed in units of \(L_0\). The measured proton radius is used only as the SI comparison bridge for assigning physical scale to the native lattice geometry.

\[ r_p \equiv \frac{a_z\lambda_k}{\sqrt{3}} \] \[ \lambda_k \equiv \frac{r_p\sqrt{3}}{a_z} \]

The bridge uses the proton radius comparison and the derived proton triad radius:

\[ r_p \approx 0.84247\,\mathrm{fm} \] \[ a_z = 467.6952\,L_0 \]

Derived Coherence Length

Substitution gives the King coherence length:

\[ \lambda_k = 2.809321648\times10^{-20}\,\mathrm{m} \]

This value is the physical spacing bridge for one King coherence interval. It is not a new independent physical assumption; it is the SI projection of the native lattice scale.

Native \(L_0\) Conversion

The primitive lattice quantum \(L_0\) is the native radial counting unit used throughout CPP, Region, shell, bonding, and curvature geometry. The corresponding SI conversion is:

\[ 1\,L_0 = 0.001797965855\,\mathrm{fm} \] \[ 1\,\mathrm{fm} = 556.1837316\,L_0 \] \[ 1\,\text{\AA} = 55{,}618{,}373.16\,L_0 \]

These conversions allow native EOTU distances to be compared with measured nuclear, atomic, and molecular scales without changing the underlying lattice construction.

Use Across the Framework

The King coherence length appears wherever native lattice distances are projected into measured units. It supports radius comparisons, shell radii, bonding lengths, corridor spacings, curvature coefficients, and time-step conversion.

\[ \lambda_k \rightarrow L_0\text{-to-SI conversion} \rightarrow \text{measured comparison} \]

The recurrence framework remains native to \(L_0\). The SI bridge enters only when the derived lattice value is expressed in meters, femtometers, angstroms, or other measured comparison units.

Summary

The King coherence length is \(\lambda_k=2.809321648\times10^{-20}\,\mathrm{m}\). It provides the physical bridge from native \(L_0\)-based lattice counts to measured SI distance.

The associated conversion is \(1\,L_0=0.001797965855\,\mathrm{fm}\), \(1\,\mathrm{fm}=556.1837316\,L_0\), and \(1\,\text{\AA}=55{,}618{,}373.16\,L_0\). These values allow the native EOTU geometry to be compared against nuclear, atomic, bonding, and curvature scales.