Appendix E — Derivation 5: Unified Action Principle

(Locked v1.0 • constants audited 2025-06-10)

This appendix upgrades the earlier “concept sketch” with numerical
anchors derived in Saturation Constants for Modal Thermodynamics:

ρ_{crit} = 6.0 \times 10^{44} m^{- 3}, γ_{0} = 8.2 \times 10^{- 8} s^{- 1} .

These two numbers, together with the locked substrate constants
$α = 0.090 034 {J m}^{- 1}$ ;
$β = 5.30 \times 10^{- 54} {J m}^{- 3}$ ;
$γ = 1.000 228 \times 10^{- 18} {J s² m}^{- 3}$
complete the quantitative basis for modal thermodynamics.

Imported from Foundations §1 (β term retained):

E [ψ] = \int [γ | \partial_{t} ψ |^{2} + α | \nabla ψ |^{2} + β | ψ |^{2}] d^{3} x .

For an ensemble ${ψ_{i}}$

C_{tot} = \sum_{i} E [ψ_{i}] + \int Γ ({ψ_{i}}) d^{3} x,

where $Γ$ encodes mode–mode interference and saturation.

2 Entropy as anchoring pressure

Define local coherence density

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

Entropy density

s (x) = \ln [\frac{1}{1 - ρ_{c} / ρ_{crit}}], ρ_{crit} = 6.0 \times 10^{44} m^{- 3} .

$s$ diverges as $ρ_{c} \to ρ_{crit}$ , signalling
anchoring break-down.

Total entropy $S = \int s d^{3} x$ is dimension-less.

Low-density decoherence sink (calibrated):

Γ_{dec} = γ_{0} \frac{ρ_{c}^{2}}{ρ_{crit} - ρ_{c}}, γ_{0} = 8.2 \times 10^{- 8} s^{- 1} .

Define modal temperature

T (x) = \frac{1}{k_{B}} \frac{ρ_{c}}{ρ_{crit} - ρ_{c}},

(up to Boltzmann-like constant $k_{B}$ for convenience).
$T$ rises sharply as saturation is approached.

4 Continuity and momentum equations

4.1 Coherence continuity

\partial_{t} ρ_{c} + \nabla \cdot (ρ_{c} \nabla ϕ) = - Γ_{dec} .

4.2 Momentum balance

\partial_{t} (ρ_{c} \nabla ϕ) + \nabla \cdot [α ρ_{c} \nabla ϕ \nabla ϕ + \frac{1}{2} β ρ_{c} I] = - \nabla (ρ_{c} B^{2}) - Γ_{dec} \frac{\nabla ϕ}{| \nabla ϕ |} .

All coefficients come from the locked constants or the newly anchored
$γ_{0}$ .

5 Heat-capacity example

For a uniform patch $ρ_{c} = λ ρ_{crit}$
with small phase variance:

C = \frac{d ⟨ E ⟩}{d T} = k_{B} \frac{λ}{(1 - λ)^{2}},

which diverges at saturation (structural instability) and vanishes when
$λ \to 0$ .

6 Second-law restatement

A modal ensemble evolves toward uniform turnover pressure
$\nabla T = 0$ while minimising total anchoring cost.

Entropy increase reflects redistribution of coherence, not
probabilistic mixing.

Summary

With $ρ_{crit}$ and $γ_{0}$ now fixed, PBG possesses a
complete, unit-consistent thermodynamic toolkit:

$s (x)$ — entropy density
$T (x)$ — modal temperature
$C$ — heat capacity
Equations (continuity + momentum) — predictive dynamics

Upcoming notebooks (turnover-3d.ipynb) will benchmark these equations
against laboratory decoherence rates and cluster cooling curves.

Appendix E - Unified Action Principle | [Index](./Appendix Master) | Appendix G - Gauge Symmetry

Appendix F — Modal Thermodynamics from Coherence Principles

1 Modal energy functional