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

Appendix AC — Derivation 28

Coherence Class $χ$ and (Para-)Statistics in PBG

0 · Context

Framework	“Particle” rule	Effect
Standard QM	exchange $ψ_{1} ψ_{2} \to \pm ψ_{2} ψ_{1}$	$χ = + 1$ (boson) or $- 1$ (fermion) fixed by spin
PBG	no operators → statistics must follow from geometry	classify by anchoring overlap

1 · Anchoring cost for $n$ identical copies

For one mode $ψ = ρ e^{i ϕ}$

C [ψ] = \int [α | \nabla ψ |^{2} + λ (ρ) ρ] d^{3} x, λ (ρ) = \frac{β}{1 - ρ / ρ_{crit}} .

Put $n$ identical, fully-overlapping copies in the same region
( $ρ_{c} = n ρ$ ):

C (n) = \int [n α | \nabla ψ |^{2} + n λ (n ρ) ρ] d^{3} x .

2 · Definition of the coherence class $χ$

χ [ψ] = lim_{n \to 2^{-}} \frac{β^{- 1} \frac{d^{2} C}{d n^{2}}}{C (1)}

Interpretation

$χ ≪ 1$ → weak self-penalty → boson-like
$χ ≫ 1$ → strong self-penalty → fermion-like
$1 ≲ χ ≲ 10$ → partial exclusion (para/anyonic)

3 · Explicit expression

χ [ψ] = \frac{\int \frac{ρ^{2}}{(1 - 2 ρ / ρ_{crit})^{3}} d^{3} x}{\int [α | \nabla ψ |^{2} + β ρ] d^{3} x} .

Envelope	$ρ_{max}$ vs $ρ_{crit}$	$χ$
Broad Gaussian (photon-like)	$ρ_{max} ≪ ρ_{crit}$	$χ ≪ 1$
Compact node-free (electron-like)	$ρ_{max} ≲ ρ_{crit}$	$χ ≫ 1$

4 · Phase-interference correction

Define overlap integral

I_{ϕ} = \frac{\int ρ^{2} \cos Δ ϕ d^{3} x}{\int ρ^{2} d^{3} x}, 0 \leq | I_{ϕ} | \leq 1.

Replace $ρ \to 2 ρ (1 + I_{ϕ})$ in the numerator:

$I_{ϕ} \to 1$ (constructive) ⇒ $χ \to 0$
$I_{ϕ} \to 0$ (destructive) ⇒ $χ$ unchanged

So statistics track interference compatibility.

5 · Observational mapping

System	Envelope geometry	Expected $χ$
Photons / phonons	wide phase plane	$\sim 10^{- 4}$
Electrons / nucleons	compact, node-free	$\sim 10^{4}$
Edge anyons (2-D)	vortex + phase shift	$0.1 - 3$

6 · Key take-aways

No exchange axiom: exclusion follows from saturation.
$χ$ is dimension-less, built solely from calibrated ${α, β, γ}$ .
“Boson” and “fermion” are limiting cases of the same geometric measure.

Math sanity checks

Units: numerator/denominator both energy → $χ$ pure number.

Limits: $ρ / ρ_{crit} \to 0$ gives $χ \to 0$ ;
$ρ / ρ_{crit} \to ½$ drives $χ \to \infty$ as claimed.

No extra constants introduced.

Appendix AB - Modal Statistics | [Index](./Appendix Master) | Appendix AD - Chiral Anchoring