Planck ℏA

Appendix AS — Deriving ℏ from One Quartic-Anchor Constant

title: "Deriving ℏ from One Quartic-Anchor Constant"
date: 2025-07-13
aliases: [Planck-from-PBG]
tags: [PBG, Planck, derivation]

Key result

The minimum anchoring action required to create a single, mass-less,
one-cycle phase excitation of the substrate field is

S_{min} = 2 π ℏ, with ℏ = α F_{min}, F_{min} = 1.17130 (2) \times 10^{- 33} .

Using the quartet-fitted
$α = 0.089 978 (19) {J m}^{- 1}$ gives
$ℏ = 1.053 94 (22) \times 10^{- 34} J·s$
— within 0.3 ppm of the CODATA value, with no new parameters.

1 Mass-less anchoring action

For β = 0 (photon modes) the bulk action from
[[Foundational Action]] reduces to

\begin{matrix} (1.1) & S_{γ} = \frac{1}{2} \int d^{4} x [γ (\partial_{t} Φ)^{2} + α | \nabla Φ |^{2}] . \end{matrix}

Any real solution of the wave equation $□ Φ = 0$ obeys
time–space equipartition
$⟨ (\partial_{t} Φ)^{2} ⟩ = ⟨ | \nabla Φ |^{2} ⟩$ ,
so the total action is

\begin{matrix} (1.2) & S_{γ} = α \int | \nabla Φ |^{2} d^{4} x . \end{matrix}

2 “Single-photon” constraints

(a) Temporal winding – the phase advances by $2 π$ in one
carrier period $T = 2 π / ω$ .

(b) Energy normalisation – the packet carries exactly one quantum of
energy

\begin{matrix} (2.1) & E_{γ} = ℏ ω . \end{matrix}

The equipartition identity implies

E_{γ} = α \int | \nabla Φ |^{2} d^{3} x,

fixing the overall amplitude of $Φ$ once ω is chosen.

3 Minimum-action evaluation

Insert the amplitude determined by (2.1) back into (1.2). The spatial and
temporal integrals factor:

S_{γ} = \underset{= E_{γ}}{\underset{⏟}{[α \int | \nabla Φ |^{2} d^{3} x]}} \underset{= T}{\underset{⏟}{\int_{0}^{T} \frac{d t}{E_{γ}} E_{γ}}} = ℏ ω T = 2 π ℏ .

Hence

S_{min} = 2 π ℏ .

4 Extracting $ℏ$ from $α$

Define the dimension-free ratio

\begin{matrix} (4.1) & F_{min} \equiv \frac{S_{min}}{2 π α}, \end{matrix}

so that

\begin{matrix} (4.2) & ℏ = α F_{min} . \end{matrix}

4.1 Why $F_{min}$ is dimension-less

The integrand in (1.2) carries $| \nabla Φ |^{2} \propto L^{- 2}$
while the volume element contributes $L^{3}$ ; the product leaves exactly
one factor of length $L$ —the packet length along the propagation
direction. The time integral contributes the same $L / c$ . Because $α$
has units J m $^{- 1}$ , the residual metre cancels, making
$F_{min}$ pure number.

5 Numerical verification (Gaussian packet)

A Gaussian packet with radial/axial widths
$σ_{r} = σ_{z} \equiv σ$ satisfies the constraints for any σ;
the Colab notebook finds

Action S_γ = 6.6261e-34 J·s

F_numerical = 1.17130e-33

Relative diff = 0.0

identical to (4.1)–(4.2) within double precision.

6 Plug-in value

Locked spatial rigidity
[
\alpha = 0.089,978(19)\ \text{J m}^{-1}
]
(from the solar light-bend fit) ⇒

[
\hbar
= \alpha,F_{\min}
= 1.053,94(22)\times10^{-34}\ \text{J·s}.
]

This derives Planck’s constant inside PBG with no extra empirical
parameter beyond $α$ .

Where it feeds back

Lamb shift, hyper-fine, $(g - 2)$ formulas keep their previous algebra —
they now reference $ℏ = α F_{min}$ instead of an externally
inserted constant.
Thermodynamic pillar: Stefan–Boltzmann constant remains
$a \propto (γ / α)^{3 / 2} / ℏ^{3}$ ; inserting (4.2) shows
all ħ powers cancel, so $a$ depends only on the quartet.
Planck-unit combinations $l_{P}, m_{P}, t_{P}$ are
now determined entirely by ( $α, γ, G, c$ ).