Appendix H — Derivation 8: Particle Statistics from Modal Anchoring

Overview

In quantum theory, particle statistics divide into two fundamental types:

• Bosons, which can occupy the same state
• Fermions, which obey the Pauli exclusion principle

These statistics are enforced axiomatically through field commutation relations.

In modal dynamics, there are no particles or fields. Yet bosonic and fermionic behaviour emerges naturally from the anchoring cost of overlapping coherence structures.

1. Structural Basis for Statistics

Let ${\psi }_i(x)$ and ${\psi }_j(x)$ be two modes attempting to anchor in the same coherence field $B(x)$. The anchoring cost is:

Where:

• $\mathcal{C}[\psi ]$ is the self-anchoring cost (see Appendix B)
• $\mathrm{\Gamma }({\psi }_i,{\psi }_j)$ is the interference cost due to structural overlap

If $\mathrm{\Gamma }$ is negligible or negative (stabilising), the modes can co-anchor—bosonic behaviour.

If $\mathrm{\Gamma }$ diverges (destabilising), the modes exclude each other—fermionic behaviour.

2. Anchoring Saturation

Each coherence field $B(x)$ supports a finite anchoring capacity, defined by the critical density ${\rho }_{\text{crit}}$.

If two modes with nearly identical phase structure attempt to anchor at the same location:

• Their overlap ${\rho }_c(x)=|{\psi }_i(x)+{\psi }_j(x){|}^2$ increases
• The cost $\mathcal{C}[{\rho }_c]$ exceeds the saturation threshold
• The configuration becomes unstable—structural exclusion occurs

This is the origin of Fermi exclusion: identical modes cannot stably co-anchor if their phase structures interfere destructively or saturate the coherence medium.

3. Phase Structure and Symmetry

Coherence functions $\psi (x)$ may differ not just in magnitude, but in internal phase topology.

If two modes are distinguishable by:

• Spinor rotation (e.g. opposite phase twist)
• Angular alignment (e.g. orthogonal windings)
• Structural offsets (e.g. coherence class)

Then anchoring may remain stable—multiple modes can occupy the same region without saturating the coherence envelope. This allows bosonic aggregation.

4. Modal Statistics Classification

• Bosons: Modes with compatible phase topology and stabilising interference (e.g. photons)
• Fermions: Modes with saturating or exclusive coherence profiles (e.g. electrons)

The key distinction is not particle identity, but anchoring compatibility:

Modes that amplify each other’s coherence can co-anchor.
Modes that disrupt coherence exclude each other.

5. No Operator Algebra

Traditional quantum statistics rely on operator identities:

• Bosons: $[a,a^{†}]=1$
• Fermions: $\{a,a^{†}\}=1$

Modal dynamics has no operators. The behaviour emerges directly from:

• The structure of $\mathrm{\Gamma }({\psi }_i,{\psi }_j)$
• The saturation behaviour of $B(x)$
• The total coherence overlap ${\rho }_c(x)$

There is no need to impose exclusion. It is a structural consequence of coherence field competition.

Conclusion

Particle statistics are not postulates.
They are manifestations of what coherence structures can and cannot anchor in the same space.

Appendix G | [Index](./Appendix Master) | Appendix I