Appendix B — Recurrence-Scaled Horizon

Native Horizon and Comparison Horizon

The ledger horizon counts the actual Dormant-Corridor hops. A recurrence-scaled comparison horizon applies the Constellus recurrence scale between each emission epoch and the comparison endpoint.

Let the comparison endpoint be \(\tau_f\). The recurrence comparison factor is:

\[ S_{\tau \rightarrow \tau_f} = \frac{CS_k(\tau_f)} {CS_k(\tau)} \]

This is the same endpoint scaling used by the redshift relation.

\[ 1 + z_{\tau \rightarrow \tau_f} = \frac{CS_k(\tau_f)} {CS_k(\tau)} \]

Therefore:

\[ S_{\tau \rightarrow \tau_f} = 1 + z_{\tau \rightarrow \tau_f} \]

Scaled Horizon in Hop Units

For an integration over elapsed SI time, the recurrence-scaled horizon in hop units is:

\[ D_{\mathrm{scaled,hop}}(\tau_f) = \int_{t_i}^{t_f} \frac{S_{t \rightarrow t_f}} {\tau_0} \,dt \]

This expression counts the comparison-adjusted hop contribution across the elapsed interval.

Scaled Horizon in \(L_0\) Units

Since one King-coherence step corresponds to \(1/64\) of an \(L_0\) interval in the native horizon expression, the recurrence-scaled horizon in \(L_0\) units is:

\[ D_{\mathrm{scaled},L_0}(\tau_f) = \frac{1}{64} \int_{t_i}^{t_f} \frac{S_{t \rightarrow t_f}} {\tau_0} \,dt \]

Substituting the Constellus recurrence-scale ratio gives:

\[ D_{\mathrm{scaled},L_0}(\tau_f) = \frac{1}{64} \int_{t_i}^{t_f} \frac{1}{\tau_0} \frac{CS_k(t_f)} {CS_k(t)} \,dt \]

Formation-Coordinate Mapping

When the integration variable is the formation coordinate \(\tau\), the time-coordinate mapping is included explicitly.

\[ D_{\mathrm{scaled},L_0}(\tau_f) = \frac{1}{64} \int_{\tau_i}^{\tau_f} \frac{1}{\tau_0} \frac{CS_k(\tau_f)} {CS_k(\tau)} \frac{dt}{d\tau} \,d\tau \]

This form separates the recurrence-scale comparison from the mapping between formation coordinate and elapsed time.

Separated Horizon Forms

The native transport horizon and the recurrence-scaled comparison horizon are separate quantities.

\[ D_{L_0}(t) = \frac{t}{64\tau_0} \]

\[ D_{\mathrm{scaled},L_0}(\tau_f) = \frac{1}{64} \int_{\tau_i}^{\tau_f} \frac{CS_k(\tau_f)} {CS_k(\tau)} \frac{1}{\tau_0} \frac{dt}{d\tau} \,d\tau \]

The first expression counts native photon propagation through Dormant Corridors. The second expression applies the recurrence-scale comparison between each contributing epoch and the endpoint snapshot.

Appendix B Summary

The recurrence-scaled horizon is not the raw transport distance. It is the comparison horizon obtained by weighting each contributing interval by the Constellus recurrence ratio between that interval and the endpoint snapshot.

This keeps the native transport rule and the cross-epoch observational comparison distinct.