Appendix AB — Derivation 27: Modal Statistics from Coherence Class

0 Why statistics must emerge rather than be imposed

Quantum textbooks bolt Fermi–Dirac or Bose–Einstein factors onto a Hilbert space by inserting operator (anti)commutators.
Phase-Biased Geometry (PBG) has no fundamental operator algebra; every many-body effect has to come from the one anchoring action.
The question is therefore simple:

If many identical modes try to live in the same envelope, does the anchoring functional let them pile up (boson-like) or choke off occupancy (fermion-like)?

The answer drops out of a density cap set solely by the calibrated constants

{α, β, γ} .

1 Universal density ceiling

For the total modal density

n (x) = \sum_{i} | ψ_{i} (x) |^{2},

higher-order crowding diagrams generated by the Yukawa kernel sum to a geometric series.
The quadratic β-term

β \int n d^{3} x

renormalises to

\begin{matrix} (1.1) & β_{e f f} (n) = \frac{β}{1 - \frac{n}{n_{⋆}}}, n_{⋆} = \frac{α^{3 / 2}}{4 π β^{1 / 2}} \end{matrix}

$n_{⋆}$ is a parameter-free critical density; the cost diverges when $n \to n_{⋆}$ .

2 Self-exclusion criterion for a single envelope

Take one normalised envelope $u (x)$ and place $n$ identical copies in it:

ψ_{n} (x) = \sqrt{n} u (x), u_{max}^{2} \equiv max_{x} u^{2} .

The β energy reads

E_{β} (n) = β \int \frac{n u^{2}}{1 - \frac{n u^{2}}{n_{⋆}}} d^{3} x .

Define

\begin{matrix} (2.1) & ς \equiv n_{⋆} u_{max}^{2} . \end{matrix}

If $ς > 1$ the integral blows up when the second copy ( $n = 2$ ) is added ⇒ fermion-like self-exclusion.
If $ς < 1$ the integral stays finite for all $n$ ⇒ boson-like self-coherence.

No spin, no commutator—just peak density versus the universal cap.

3 Spin/phase-winding link (topological origin of $ς$ )

For ground-state envelopes the peak density scales with the absolute winding number $w$ :

u_{max}^{2} (w) \propto w .

Odd $w$ ⇒ $u_{max}^{2} > n_{⋆}^{- 1}$ ⇒ $ς > 1$ ⇒ fermion-like.
Even $w$ ⇒ $u_{max}^{2} < n_{⋆}^{- 1}$ ⇒ $ς < 1$ ⇒ boson-like.

Thus the familiar spin–statistics dichotomy is replaced by a winding-statistics rule derived from α and β.

4 Partition function → FD / BE weights

For an ideal gas of identical modes with single-mode energy $ε$ :

Z = \sum_{n = 0}^{\infty} \exp [- β_{t h} (n ε + E_{β} (n) - n μ)] .

If $ς > 1$ , $E_{β} (n \geq 2) \to \infty$ and only $n = 0, 1$ survive ⇒

⟨ n ⟩ = \frac{1}{e^{(ε - μ) / T} + 1} (Fermi–Dirac) .

If $ς < 1$ , $E_{β} (n)$ grows linearly and the geometric sum yields

⟨ n ⟩ = \frac{1}{e^{(ε - μ) / T} - 1} (Bose–Einstein) .

The ± sign is not inserted; it crystallises from the anchoring divergence.

5 Why photons do not “condense forever”

The kinetic part $γ (\partial_{t} ϕ)^{2} n$ introduces a temperature-dependent cost.
For massless modes

n (T) = \frac{2 ζ (3)}{π^{2}} {(\frac{k_{B} T}{ℏ c})}^{3} ≪ n_{⋆}

even at the CMB last-scattering peak $T ≃ 3000$ K, so practical Bose condensation remains finite.

6 Quick sanity checks

System	$ς$ value	Outcome vs. standard physics
Electron 1s envelope	$ς ≫ 1$	Self-exclusion ⇒ white-dwarf degeneracy $P \propto n^{5 / 3}$
Photon plane wave	$ς ≪ 1$	BE statistics ⇒ Planck black-body
Cooper pair envelope	$w_{p a i r} = 2$ (even) ⇒ $ς < 1$	Composite boson ⇒ BCS condensation

7 Take-away

Statistics are an emergent property of the same α, β that set lensing and Lamb shift.
Fermion or boson behaviour is decided locally by peak envelope density versus a universal cap $n_{⋆}$ .
No spin labels, no operator algebra, yet the thermal weights reduce to FD or BE exactly where they should.

Exclusion and coherence are just opposite faces of the same anchoring cap.

Appendix AA - Modal Thermodynamics | [Index](./Appendix Master) | Appendix AC - Coherence Class Chi

Appendix AB — Modal Statistics from Coherence Saturation