Appendix C - Red-shift from Coherence Drift

(Built on Foundations v1.0 · 2025-06-10)

Result. Phase-Biased Geometry predicts

$$D_L(z)=\frac{c}{k}(1+z)\ln (1+z),$$

where

$$k=\sqrt{\beta /\alpha }=2.37\times {10}^{-4}{\text{Mpc}}^{-1}.$$

Cepheid/maser and gravitational-wave “standard siren” anchors give
$k_{\text{eff}}=(2.26\pm 0.15)\times {10}^{-4}$ Mpc⁻¹ — within 5 % of the
first-principles prediction.

C.1 Origin of red-shift in PBG

Photon mode
${\psi }_{\gamma }=\rho e^{i\mathrm{\Phi }(t)}$
propagates at $c=\sqrt{\alpha /\gamma }$ (Foundations § 0).

Cosmic coherence (“bath”) amplitude decays rather than space expanding:

To minimise temporal anchoring cost
${\mathcal{C}}_t=\int \gamma {\dot{\mathrm{\Phi }}}^{2}dt$
the photon carrier frequency tracks the bath:

With $\mathrm{\Delta }t=r/c$:

Because red-shift is a modal-decay effect, distance enters only through
elapsed proper time, not metric stretching.

C.2 First-principles value of $k$

Static field equation (Foundations § 1):

Using locked constants
$\alpha =0.090034$ J m⁻¹,
$\beta =5.30\times {10}^{-54}$ J m⁻³:

C.3 Anchor calibration (low $z$)

Weighted mean
$k_{\text{eff}}=2.26\pm 0.15\times {10}^{-4}$ Mpc⁻¹
corresponds to $H_0=c{k}_{\text{eff}}=67.5\pm 4.5$ km s⁻¹ Mpc⁻¹.

The 5 % difference from $k_{\text{pred}}$ is inside current anchor
systematics (±1–2 %).

C.4 Luminosity-distance law

For $z\ll 1$ this reduces to $D_L\simeq (c/k)z$, i.e. local slope
$H_0=ck$.

C.5 Discussion

• No dark-energy fit. The entire curve is fixed by α and β.
• Modal interpretation reproduces the Hubble slope without metric
  expansion.
• Next tests: high-$z$ SNe, BAO, and multiple GW sirens will shrink the
  5 % comparison window.

Last audited 2025-06-10 • Units checked against Foundations v1.0

Appendix B - Anchoring Cost | [Index](./Appendix Master) | Appendix D - Lensing