Appendix AC — Derivation 28: Coherence Class $\chi$ from Anchoring Structure

Appendix AC — Derivation 28: Coherence Class $χ$ from Anchoring Structure

Overview

In classical quantum theory, statistical behaviour is imposed by spin:

Fermions ( $χ = - 1$ ): antisymmetric states, exclusion principle
Bosons ( $χ = + 1$ ): symmetric states, shared occupation

In PBG, these distinctions arise from modal geometry.
The coherence class $χ$ of a mode is not a fixed label—it is a structural outcome of:

Anchoring cost growth under overlap
Phase interference stability
Saturation proximity

This appendix derives $χ$ directly from modal anchoring structure.

1. Overlap-Driven Anchoring Cost

Let two identical modes $ψ (x)$ attempt to anchor together:

ρ_{c} (x) = | ψ (x) |^{2} + | ψ (x) |^{2} = 2 | ψ (x) |^{2}

Anchoring cost becomes:

C = \int β (ρ_{c}) \cdot ρ_{c} d^{3} x

Using:

β (ρ_{c}) = \frac{1}{1 - ρ_{c} / ρ_{crit}}

If this diverges as $ρ_{c} \to ρ_{crit}$ , then doubling the mode is structurally forbidden.

2. Defining the Coherence Class $χ$

We define the coherence class $χ [ψ]$ as the limiting overlap penalty:

χ [ψ] = lim_{n \to 2^{-}} \frac{d^{2} C}{d n^{2}}

Where $n$ is the number of structurally identical copies of $ψ$ attempting to anchor in the same region.

If $χ [ψ] \to + \infty$ : strong self-exclusion → fermion-like
If $χ [ψ] \approx 0$ : no self-interaction → boson-like
Intermediate values: partial exclusion (e.g. parafermions)

This coherence class is thus not binary, but continuous and emergent.

3. Phase Interference Contribution

Modes with overlapping support and in-phase structure can reinforce each other. But if the phase winds destructively (e.g. out-of-phase lobes), anchoring instability increases.

Define:

I_{ϕ} = \int ψ^{*} (x) \tilde{ψ} (x) d^{3} x

as the interference overlap integral, where $\tilde{ψ}$ is a second copy with shifted phase.

Then:

χ [ψ] \propto \frac{1}{1 - | I_{ϕ} |}

For constructive interference: $| I_{ϕ} | \to 1$ → $χ$ small (boson-like)
For destructive interference: $| I_{ϕ} | \to 0$ → $χ$ large (fermion-like)

This shows $χ$ measures interference coherence compatibility, not symmetry under exchange.

4. Geometric and Topological Effects

The coherence class $χ$ is also influenced by:

Spatial phase winding (see Appendix M)
Modal topology (e.g. nodal surfaces, phase discontinuities)
Local saturation curvature (gradient of $ρ_{c}$ )

Thus, $χ$ is a functional of modal structure, not a quantum input.

5. Observational Signatures

Modes with large $χ$ resist aggregation: e.g. electrons, neutrinos
Modes with small $χ$ amplify each other: e.g. photons, coherence condensates
High $χ$ modes exhibit degeneracy pressure and quantised exclusion naturally

This explains quantum statistical behaviour as a consequence of geometry, not algebra.

Conclusion

The coherence class $χ$ is not a label—it is a structural property of anchoring geometry.

In PBG:

Fermions and bosons are modes with different overlap stability
Statistics emerge from saturation-driven coherence constraints
The class $χ$ measures how structure interferes with itself

Appendix AB | [Index](./Appendix Master) | Appendix AD

Appendix AC — Derivation 28: Coherence Class χ from Anchoring Structure