Appendix Y — Derivation 25: Continuum Mechanics of Coherence Media

Overview

Unlike conventional frameworks, PBG does not assume the values of physical constants. Instead, it derives them from coherence structure, anchoring dynamics, and modal geometry.

This appendix summarises the key constants and parameters of the physical world as they arise naturally within the PBG framework.

1. The Speed of Light $c$

In PBG, the speed of light is not postulated. It arises from the maximum coherence propagation speed of a phase-preserving mode in a vacuum with minimal modal interference.

From the coherence cost minimisation of a free phase mode:

The least-cost propagation occurs when:

Thus the propagation speed is:

Both $\alpha$ and $\gamma$ arise from the underlying modal medium—specifically the resistance to phase curvature ($\alpha$) and the anchoring response inertia ($\gamma$).

(See Appendix K — Speed of Light from Coherence Anchoring.)

2. Coherence Kernel Decay Constant $k$

The coherence field $B(x)$ satisfies the Helmholtz equation derived from the anchoring cost functional:

Solving this yields the Yukawa-type kernel:

with decay constant:

This constant governs the coherence field falloff and replaces the gravitational inverse-square law with a bias-weighted exponential structure.

(See Appendix H — Coherence Kernel from Phase Structure.)

3. Photon Decoherence Sensitivity ${\gamma }_0$

Photons follow paths that minimise decoherence cost. The penalty field:

introduces a universal constant ${\gamma }_0$ that sets the coupling between photon coherence structure and external field gradients.

This constant calibrates solar lensing, grazing deflection, and deep-field arcs. It arises from the internal phase stability of the photon’s modal envelope.

(See Appendix D and Appendix J.)

4. Anchoring Coefficients $\alpha$, $\beta$

These constants appear in the fundamental anchoring cost:

• $\alpha$ quantifies resistance to coherence curvature
• $\beta$ sets the saturation tendency of the medium

Together they determine $k$, $c$, and modal binding energy thresholds. These are not inserted manually—they follow from the geometry and stability of the modal anchoring process.

(See Appendix H and Appendix AA.)

5. Effective Couplings: $G$, ${ϵ}_0$, ${\mu }_0$

PBG does not contain Newton’s constant $G$, the permittivity ${ϵ}_0$, or permeability ${\mu }_0$ as fundamental quantities.

Instead:

• Gravitational coupling arises from overlap of coherence fields:

  $$F\sim \mathrm{\nabla }({\rho }_c{B}^2)$$

  yielding Newton-like motion when $B\propto \frac{1}{r}{e}^{-kr}$

• Electromagnetic behaviour arises from modal interference:
  Phase gradients and anchoring asymmetries replace $E$ and $B$ fields.
  No intrinsic ${ϵ}_0$ or ${\mu }_0$ is required.

Their apparent values can be recovered statistically from modal ensemble behaviour in coherent matter clusters.

(See Appendix E, Appendix D, and Appendix AE.)

6. Agreement with Observation

Once coherence-derived constants are calibrated from a single system, PBG yields correct predictions across diverse phenomena—without parameter readjustment.

For example:

• The speed of light $c$ arises from phase anchoring geometry alone
• The photon decoherence penalty ${\gamma }_0$ is set by solar lensing
• The coherence kernel decay constant $k$ is fixed from Mercury’s orbit

Each of these is then applied unchanged to:

• Predict deep-field light deflection
• Model galactic rotation curves
• Quantify modal binding energies and thermodynamic turnover

This directional consistency across unrelated domains provides strong evidence that the constants are not tuned, but structurally anchored—emergent from the modal framework itself.

(See Sections 7.1–7.10 for predictions using these constants.)

Conclusion

All physical constants used in observable physics—$c$, $G$, $\hslash$-like scales, field sensitivities, and decay rates—are not assumed in PBG. They arise from anchoring geometry, coherence thresholds, and modal phase structure.

This closes the gap between observed parameters and first principles.

Appendix Y | [Index](./Appendix Master) | Appendix AA