Appendix AK — Derivation 37: Mass from Spin and Phase Anchoring

Appendix AK — Mass from Spin and Phase Anchoring

In Phase-Biased Geometry (PBG), mass is not an intrinsic scalar. It emerges as a resistance to modal displacement through coherence space, arising from internal spin structure, phase anchoring, and self-coherence drag.

This appendix formally derives the concept of inertial mass from the modal anchoring cost structure and coherence-preserving phase evolution.

Each mode $ψ$ is defined by:

A spatial coherence density $ρ (\vec{r})$
A structured internal phase $ϕ (\vec{r}, t)$
An anchoring field $B [ψ]$ which resists perturbation

To translate or accelerate a mode coherently, the phase structure must evolve:

ϕ (\vec{r}, t) = ϕ_{0} (\vec{r} - \vec{v} t)

This evolution imposes a coherence penalty, since the mode must continuously re-anchor to its moving coherence field.

2 · Anchoring cost of acceleration

Starting from the full time-dependent term

C_{kin} = \int γ | \partial_{t} ψ |^{2} d^{3} x,

a rigid translation with velocity v(t) gives

\partial_{t} ψ ≃ - (v \cdot \nabla) ψ ⟹ | \partial_{t} ψ |^{2} \approx ρ^{2} | v \cdot \nabla ϕ |^{2} .

For a slow acceleration a = d v/dt one finds

δ C_{accel} \approx γ \int ρ^{2} | a |^{2} d^{3} x,

so the coefficient of |a|² inside the integral plays the rôle of an inertial mass density.

3. Spin and Phase Topology

For spin-structured modes (e.g. electrons), the internal phase is not uniform but circulatory:

ϕ (θ) = n θ

Displacing such a mode requires re-anchoring a rotating phase structure within a moving coherence envelope.

The phase gradient's non-commutativity with spatial translation introduces modal drag, experienced as inertial mass.

4 · Definition of mass

Collecting the γ-weighted kinetic cost and equating the bulk term to ½ m v² yields

m = \frac{γ}{c^{2}} \int ρ^{2} | \nabla ϕ |^{2} d^{3} x

(the factor 1/c² appears because modal kinetic energy is stored in phase-rate units).
For a simple circular winding with radius R and winding number n, this reproduces

m \propto γ \frac{n^{2}}{R} .

5. Rest Mass and Composite Modes

For composite modes (baryons, atoms), the net phase structure leads to stabilised internal coherence patterns. These require anchoring synchrony, and the resistance to perturbation becomes collective:

m_{composite} \propto \sum_{i} m_{i} + Δ m_{binding}

The binding correction arises from shared anchoring regions and mutual coherence cost.

Summary

Mass in PBG is:

A structural resistance to modal translation
Arising from spin topology and coherence anchoring
Quantified via anchoring cost functionals
Emergent and mode-dependent, not scalar or fundamental

Appendix AJ - Lamb Shift | [Index](./Appendix Master) | Appendix AL - Solar Lensing & c