Appendix Y — Derivation 25: Continuum Mechanics of Coherence Media

YYYY-04-Apr 19, 2025

Appendix Y — Catalogue of Coherence-Derived Constants

(clean v1.2 • 2025-06-12)

Purpose. List every numerical constant that the Phase-Biased
Geometry (PBG) framework uses, how it is calibrated once, and where
it subsequently applies.

1 Fundamental anchoring set ${α, β, γ}$

Symbol	Meaning (see Action 5.1)	SI unit	Calibrated from
$α$	spatial anchoring stiffness	J m⁻¹	solar light bending
$β$	envelope (mass) penalty	J m⁻³	hydrogen Lamb shift
$γ$	temporal anchoring inertia	J s² m⁻³	photon speed $c$

The least-action field equation

γ \partial_{t}^{2} Φ = α \nabla^{2} Φ - β Φ

follows directly from these three.

2 Speed of light

Insert a plane wave $Φ \propto e^{i (k \cdot x - ω t)}$ :

γ ω^{2} = α k^{2} ⟹ c = \sqrt{α / γ} .

Calibrated once by setting the right-hand side equal to the measured
$2.997 924 58 \times 10^{8}$ m s⁻¹.

3 Coherence-kernel decay length

The static Helmholtz equation
$α \nabla^{2} B - β B = ρ_{anchor}$
has Yukawa solution

B (r) = \frac{A}{4 π α r} e^{- r / ℓ}, ℓ = \sqrt{α / β},

so the decay constant is $k = 1 / ℓ = \sqrt{β / α}$ .

Numerical anchor – the Bohr radius ( $a_{0}$ ) fixes $ℓ$ since the
electron–proton anchoring cost reproduces the Rydberg energy.

4 Photon–lensing coupling (dimensionless)

When we write the path cost for a latent photon mode as

A_{γ} = \int \underset{dimensionless}{\underset{⏟}{η}} | \nabla B (x) |^{2} d t,

the coefficient $η$ is programmable. We set
$η = 1$ by convention, so there is no new physical constant. All
numerical lensing fits then test only the previously fixed ${α, β}$ via $B (r)$ and its gradient.

5 Derived laboratory constants

Symbol	PBG expression	Numerical check	Comment
$c$	$\sqrt{α / γ}$	set by definition	reused in all wave calculations
$k$	$\sqrt{β / α}$	$7.67 \times 10^{- 27}$ m⁻¹	appears in red-shift and Yukawa kernels
$ℏ_{eff}$	minimal anchoring action per $2 π$ phase winding	= Planck’s constant within 0.2 %	see Appendix AJ

All three are outputs of the theory after the single-pass calibration
in §1–3.

6 Where the constants are re-used unchanged

Constant(s)	First calibrated in	Later applied to
$α, γ$	light bending + $c$	CMB phase-speed; Casimir spacing; synchrotron spectra
$β$ (via $k$ )	Lamb shift	red-shift law; Yukawa lens models; galaxy rotation curves
$ℓ$	Bohr radius	nuclear liquid-drop radii ( $R_{A} \propto ℓ$ ); Coulomb screening equivalent
$ℏ_{eff}$	Appendix AJ	uncertainty-bound derivations; tunnelling exponents

No constant is re-tuned after its first appearance.

7 Sanity-check table (units)

Quantity	Formula	Check
$[α]$	J m⁻¹	kg s⁻²
$[β]$	J m⁻³	kg m⁻² s⁻²
$[γ]$	J s² m⁻³	kg
$[c]$	$\sqrt{α / γ}$	m s⁻¹
$[k]$	$\sqrt{β / α}$	m⁻¹

Everything is dimension-consistent; no hidden factors of $4 π$ remain.

Take-away

PBG needs exactly three real numbers fixed from experiment.
Every other “constant of nature” (speed of light, Yukawa length, Planck-
like action, derived couplings) emerges algebraically and is then
tested unchanged in every subsequent domain.

Appendix Y - Continuum Mechanics | [Index](./Appendix Master) | Appendix AA - Modal Thermodynamics