Appendix AB — Derivation 27: Modal Statistics from Coherence Class

Overview

In quantum theory, particles are sorted into:

Fermions, obeying the Pauli exclusion principle
Bosons, capable of coherent occupation of the same state

These statistical behaviours are typically imposed through spin and symmetry assumptions.

In PBG, all structure arises from coherence anchoring.
This appendix shows that:

Modal exclusion and statistical behaviour emerge from structural overlap
The distinction between fermionic and bosonic statistics arises from saturation thresholds and anchoring compatibility
No quantum statistics are postulated—only anchoring geometry

1. Anchoring Structure and Overlap Constraint

Each mode $ψ_{i} (x)$ contributes to the coherence density:

ρ_{c} (x) = \sum_{i} | ψ_{i} (x) |^{2}

Anchoring is stable only if $ρ_{c} (x) < ρ_{crit}$ everywhere.
As overlap increases, the anchoring cost diverges:

β (ρ) = \frac{1}{1 - ρ / ρ_{crit}}

Thus, modes with overlapping support or incompatible phase structure begin to exclude each other structurally.

Let us define the coherence class $χ_{i}$ of a mode $ψ_{i}$ as:

The set of other modes with which $ψ_{i}$ can stably coexist under anchoring saturation.

This coherence class is not binary (fermion vs boson), but continuous and emergent.

Two classes naturally appear:

Exclusive (fermion-like): structure collapses if $ψ_{i}$ is doubled
Inclusive (boson-like): structure stabilises with increased overlap

3. Effective Occupation Limit

Let $n_{i}$ be the number of modes anchored with structure $ψ_{i}$ .
We define an effective modal saturation functional:

S [ψ_{i}] = \int \frac{n_{i} | ψ_{i} (x) |^{2}}{1 - ρ_{c} (x) / ρ_{crit}} d^{3} x

If $S [ψ_{i}]$ diverges as $n_{i} \to 2$ , then $ψ_{i}$ behaves like a fermion—it excludes itself structurally.

If $S [ψ_{i}]$ remains finite for arbitrary $n_{i}$ , then $ψ_{i}$ behaves like a boson—coherence is preserved in aggregation.

This directly yields modal occupation constraints from anchoring geometry, not quantum state labels.

4. Emergent Statistics

The structural partition function from Appendix AA is:

Z = \sum_{{n_{i}}} e^{- E [{n_{i}}] / T}

Modal energy increases nonlinearly with $n_{i}$ due to overlap penalties. The statistical weights become:

Fermi–Dirac analogue:

n_{i} = \frac{1}{e^{(ε_{i} - μ) / T} + 1}

when $S [ψ_{i}] \to \infty$ at $n_{i} = 2$

Bose–Einstein analogue:

n_{i} = \frac{1}{e^{(ε_{i} - μ) / T} - 1}

when $S [ψ_{i}]$ remains finite

Thus, modal statistics emerge entirely from anchoring saturation and coherence density geometry.

5. Thermal Ensembles from Structure Alone

No symmetry labels, spin assignments, or commutation relations are invoked.

Fermion-like exclusion is caused by saturation-driven decoherence
Boson-like aggregation is permitted by coherence amplification
The modal ensemble respects anchoring thresholds at all scales

This means PBG naturally recovers thermodynamic limits, blackbody distributions, degeneracy pressure, and condensation without postulating statistics.

Conclusion

Fermi–Dirac and Bose–Einstein behaviours are not fundamental—they are emergent properties of modal anchoring.

In PBG:

Structure determines statistics
Saturation defines exclusion
Stability enforces ensemble behaviour

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