§3 — Epoch Energy Projection

Local Energy Projection

Local ledger-event energy is projected through the SI energy bridge. This projection applies within one formation-coordinate mapping.

\[ E_{\mathrm{SI}}(\tau) = \Gamma E_{\mathrm{intrinsic}}(\tau) \]

\[ \Gamma = 1\,\mathrm{eV} \]

The bridge \(\Gamma\) projects local intrinsic energy into electron-volts. It does not by itself compare energy across unlike formation-coordinate snapshots.

Cross-Epoch Energy Comparison

For cross-epoch comparison, the observed energy scale follows the inverse of the recurrence-period scaling. The redshift relation is:

\[ 1 + z = \frac{CS_k(\tau_o)} {CS_k(\tau_e)} \]

The corresponding energy comparison is:

\[ \frac{E_o}{E_e} = \frac{CS_k(\tau_e)} {CS_k(\tau_o)} \]

Therefore:

\[ E_o = \frac{E_e}{1 + z} \]

The observed energy is lower when the observation snapshot has a larger recurrence scale than the emission snapshot.

Emission Projection Substitution

Substituting the local projection at emission gives the observed comparison energy in terms of the emitted intrinsic ledger-event energy.

\[ E_o = \frac{\Gamma E_{\mathrm{intrinsic}}(\tau_e)} {1 + z} \]

Written directly in Constellus recurrence units, the same relation is:

\[ E_o = \Gamma E_{\mathrm{intrinsic}}(\tau_e) \frac{CS_k(\tau_e)} {CS_k(\tau_o)} \]

Thus, \(\Gamma\) projects local ledger-event energy, while the Constellus recurrence ratio compares that energy across unlike formation-coordinate mappings.

Wavelength Comparison

Photon speed is defined geometrically by the fixed step length and fixed update interval.

\[ c = \frac{\lambda_k}{t_k} \]

Observed wavelength satisfies:

\[ \lambda = cP \]

Since the observed period is scaled by the endpoint recurrence comparison, the observed wavelength comparison follows the same factor.

\[ \lambda_{\mathrm{obs}} = (1 + z)\lambda_{\mathrm{emit}} \]

This is an observational comparison between recurrence mappings, not a change in the packet’s emitted axial identity during transport.

Time Dilation

Any light-curve or temporal structure built from photon arrivals is dilated by the same recurrence-scale factor.

\[ \frac{P_{\mathrm{obs}}} {P_{\mathrm{emit}}} = 1 + z \]

Wavelength comparison and time dilation are paired observational consequences of endpoint recurrence scaling.

Packet Conservation Statement

Redshift is a ledger-based consequence of endpoint recurrence scaling. The photon packet preserves its emitted identity during propagation. The comparison arises when the same transported packet is interpreted through the recurrence scale of the absorption snapshot.

The conserved transport inventory may be written:

\[ \mathcal{C}(\tau_k) = \sum_i A_i + N_{\mathrm{ph}}\Delta\mu \]

with:

\[ \mathcal{C}(\tau_{k+1}) = \mathcal{C}(\tau_k) \]

For an emitted packet:

\[ E_e = \mathcal{E}_e(\Delta\mu) \]

and for the observed comparison:

\[ E_o = \mathcal{E}_o(\Delta\mu) \]

Because \(\Delta\mu\) is invariant, the energy ratio is an epoch-mapping effect.

Closure Statement

Cosmological redshift follows from invariant King-cycle timing, fixed Dormant-Corridor photon transport, and Constellus recurrence scaling.

The endpoint relation is:

\[ 1 + z = \frac{CS_k(\tau_o)} {CS_k(\tau_e)} \]

The corresponding energy relation is:

\[ E_o = \Gamma E_{\mathrm{intrinsic}}(\tau_e) \frac{CS_k(\tau_e)} {CS_k(\tau_o)} \]

This establishes the recurrence-scale redshift rule and its epoch-energy projection companion.