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\title{Electromagnetism --- Math Appendix (Readable Equations Edition)}
\author{}
\date{}

\begin{document}

\maketitle

\section*{Charter}

This appendix mirrors the Electromagnetism Math Walk-Through step-for-step. For each step we provide full derivations, symbols and units, validity limits, and a replication example with real numbers. No summaries. We import only $S_0 = \hbar$ from the Foundation calibration and add no new dimensional scales. Vacuum identities ($\mu_0, \varepsilon_0, c$) appear only as acceptance locks, not tunables. Cross-refs point back to the Bridge/Foundation appendix where identities are first established. All derivations are strictly based on primitive axioms, with calibration notes for anchors. Equations are set in plain Unicode math for readability.

\section*{Symbol Registry}

$S_0 = \hbar$ (Planck's reduced constant), fixed at the electron anchor (Calibration: Electron).

$A^\mu = (\Phi/c, \mathbf{A})$ --- 4-potential (SI: $\Phi$ in volts; $\mathbf{A}$ in V$\cdot$s/m).

$\mathbf{F} = d\mathbf{A}$ --- field 2-form; components $F_{cl} = E_u$, $F_{ij} = -\epsilon_{ijk} B_k$ (covariant blocks use $c=1$; SI factors enter post-variation).

$\mathbf{E}, \mathbf{B}$ --- Electric and magnetic fields (SI: V/m, T).

$J^\mu = (cp, \mathbf{I})$ --- 4-current (SI: A/m$^2$); $\partial_\mu I^\mu = 0$.

$\mu_0, \varepsilon_0, c$ --- Vacuum locks; $c = 1/\sqrt{\mu_0 \varepsilon_0}$ (acceptance check only).

$\eta, \varepsilon^{\{\mu\alpha\beta\}}$ --- Metric $(-,+,+,+)$, Levi-Civita ($\varepsilon^{\{0123\}}=+1$).

$\star$ --- Hodge star (Bridge Appx: Hodge \& Duality).

$E_{cK}, T, S, C$ --- Curvature energy; torsion/shear budgets; closure/linking/non-intersection constraints (Foundation).

$n(x), k = 2\pi/\lambda$ --- Refractive index; wavenumber.

$N_F = a^2 / (\lambda z)$ --- Fresnel number.

$J_c$ --- Closure tolerance, $J_c = \Delta S / S_0$, fractional deviation from perfect loop closure (Foundation: Torsion/Shear Budgets).

\section*{0. Setup and Assumptions}

Cross-ref: Bridge Appx --- Field 2-form \& Variation; Foundation --- Axioms A1-A3.

Loop motion transports oriented display-area elements, defined as the obscured transverse area projected along propagation: $A_c d(\Sigma) = \int_\Sigma (\hat{n} \cdot d\Xi)$. By refinement invariance and Stokes, $\Phi(\Sigma) = \int_\Sigma \mathbf{F}$ with $\mathbf{F} = d\mathbf{A}$ and $d\mathbf{F} = 0$ (Bianchi). This ensures conservation of display area under Void propagation.

\begin{equation}
\mathbf{F} = d\mathbf{A}, \quad d\mathbf{F} = 0
\label{eq:setup-1}
\end{equation}

Observer split of $d\mathbf{F} = 0$ (homogeneous pair):

\begin{equation}
\nabla \cdot \mathbf{B} = 0, \quad \nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t
\label{eq:setup-2}
\end{equation}

(Conventions) Covariant blocks use $c = 1$ so $F_{cl} = E_u$ and $F_{ij} = -\epsilon_{ijk} B_k$. SI factors enter only in the post-variation bridge (e.g., $\partial_\mu F^{\{\mu\}}  = \mu_0 J^{\{\nu\}}$).

Validity: weak curvature/slow variation; torsion/shear corrections neglected ($|T|, |S| \ll 1$). Electron-scale precision suggests $J_c \lesssim 10^{-9}$.

Replication (consistency): Rectangular loop ($1 \times 1$ m) with uniform $B_z = 1$ T: $\Phi = B \cdot \text{Area} = 1$ T$\cdot$m$^2$. Refinement to $2 \times 2$ sub-loops preserves the total flux.

\section*{1. Maxwell Set from Geometry}

\subsection*{Context \& notation.}

Geometric field 2-form $\mathbf{F} = d\mathbf{A}$ encodes oriented display-area transport; the homogeneous equations follow from $d\mathbf{F} = 0$ (Bianchi). We vary the vacuum action; SI units appear only after variation (measurement bridge).

4-potential and current (SI bridge shown explicitly):

\begin{equation}
A^\mu = (\Phi/c, \mathbf{A}), \quad J^\mu = (cp, \mathbf{I})
\label{eq:1-1}
\end{equation}

Vacuum action with minimal coupling:

\begin{equation}
S[\mathbf{A}] = \frac{1}{2} \int \mathbf{F} \wedge \star\mathbf{F} - \int \mathbf{J} \cdot \mathbf{A} \, d^4x, \quad \text{with} \quad \mathbf{F} = d\mathbf{A}
\label{eq:1-2}
\end{equation}

Variation (boundary term dropped by compact support / decay at $\infty$):

\begin{equation}
\delta \mathbf{F} = d(\delta \mathbf{A})
\label{eq:1-3}
\end{equation}

\begin{align}
\delta S_{\text{field}} &= \int d(\delta \mathbf{A}) \wedge \star\mathbf{F} = \int d(\delta \mathbf{A} \wedge \star\mathbf{F}) - \int \delta \mathbf{A} \wedge d\star\mathbf{F} \label{eq:1-4} \\
\delta S_{\text{int}} &= - \int \delta \mathbf{A} \wedge \mathbf{J} \label{eq:1-5} \\
\delta S &= \int \delta \mathbf{A} \wedge (d\star\mathbf{F} - \mathbf{J}) \Rightarrow d\star\mathbf{F} = \mathbf{J}, \quad d\mathbf{F} = 0 \label{eq:1-6}
\end{align}

Index / SI bridge (measurement only):

\begin{align}
\partial_\mu F^{\{\mu\}} &= \mu_0 J^{\{\nu\}}, \quad \partial_\alpha F_{\{\beta\}} + \partial_\beta F_{\{\gamma\}} + \partial_\gamma F_{\{\alpha\}} = 0 \label{eq:1-7}
\end{align}

Observer split ($\mathbf{E}$, $\mathbf{B}$) in SI:

\begin{equation}
\nabla \cdot \mathbf{E} = \rho/\varepsilon_0, \quad \nabla \cdot \mathbf{B} = 0, \quad \nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t, \quad \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \partial \mathbf{E}/\partial t
\label{eq:1-8}
\end{equation}

Wave equations in source-free vacuum (curl-curl identities shown in the working):

\begin{equation}
\nabla^2 \mathbf{E} - \mu_0\varepsilon_0 \partial^2\mathbf{E}/\partial t^2 = 0, \quad \nabla^2 \mathbf{B} - \mu_0\varepsilon_0 \partial^2 \mathbf{B}/\partial t^2 = 0
\label{eq:1-9}
\end{equation}

\begin{equation}
c = 1/\sqrt{\mu_0 \varepsilon_0}
\label{eq:1-10}
\end{equation}

Replication (Coulomb limit; non-relativistic, static):

\begin{equation}
\mathbf{F} = q \, (\mathbf{E} + \mathbf{v} \times \mathbf{B}) \approx q \mathbf{E} \quad (v \ll c)
\label{eq:1-11}
\end{equation}

\begin{equation}
\Phi(r) = q/(4\pi \varepsilon_0 r), \quad \mathbf{E} = q/(4\pi \varepsilon_0 r^2) \, \hat{r}
\label{eq:1-12}
\end{equation}

\begin{equation}
F = k_e \cdot e^2 / r^2, \quad k_e = 1/(4\pi \varepsilon_0)
\label{eq:1-13}
\end{equation}

Numerical check (CODATA): $e = 1.602\,176\,634 \times 10^{-19}$ C (exact), $\varepsilon_0 \approx 8.854\,187\,8128 \times 10^{-12}$ F/m $\Rightarrow k_e \approx 8.987\,551\,7923 \times 10^9$ N$\cdot$m$^2\cdot$C$^{-2}$. For $r = 1$ nm: $F \approx 2.31 \times 10^{-10}$ N.

\noindent
\textit{Validity: $v$, $c$ static sources; separation large vs. quantum-correction scales (no QED / radiation reaction).}

\section*{2. Lorentz Force --- Relativistic and Non-Relativistic}

\subsection*{Context \& notation.}

Relativistic (4D) first, then the lab 3-vector form. We use plain notation with bold vectors ($\mathbf{E}$, $\mathbf{B}$, $\mathbf{v}$) and minimal symbols. Covariant blocks take $c=1$; SI enters only after the variation.

\subsection*{Relativistic Lorentz force (covariant form).}

Four-momentum and four-velocity: $p^\mu = m u^\mu$, $u^\mu = dx^\mu/d\tau$. Field tensor: $F_{\{\mu\}}  = \partial_\mu A_\nu - \partial_\nu A_\mu$.

\begin{equation}
dp^\mu/d\tau = q F^{\{\mu\}}_{\{\nu\}} u^\nu \quad \text{(Eq. 2-1)}
\label{eq:2-1}
\end{equation}

Orthogonality: $u_\mu (dp^\mu/d\tau) = 0$ (since $F_{\{\mu\}}  = -F_{\{\nu\}}  $). Gauge invariance: $A_\mu \to A_\mu + \partial_\mu \chi$ leaves $\mathbf{F}$ and the force invariant.

\subsection*{Observer split $\to$ 3-vector force and power balance.}

With $p = \gamma m v$, $\gamma = 1/\sqrt{1-v^2/c^2}$ and $dt = \gamma d\tau$, the spatial and temporal parts give:

\begin{equation}
d \mathbf{p}/dt = q \, (\mathbf{E} + \mathbf{v} \times \mathbf{B}) \quad \text{(Eq. 2-2)}
\label{eq:2-2}
\end{equation}

\begin{equation}
d(\gamma m c^2)/dt = q \, (\mathbf{v} \cdot \mathbf{E}) \quad \text{(Eq. 2-3)}
\label{eq:2-3}
\end{equation}

Non-relativistic limit ($v \ll c$): $p \approx m\mathbf{v}$ and $d(m\mathbf{v})/dt = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$.

\subsection*{Derivation from the particle Lagrangian (explicit steps).}

Start with the standard Lagrangian:

\begin{equation}
L = - m c^2 \sqrt{1-v^2/c^2} + q(\mathbf{v} \cdot \mathbf{A}) - q\Phi \quad \text{(Eq. 2-4)}
\label{eq:2-4}
\end{equation}

Canonical momentum and Euler--Lagrange:

\begin{equation}
\partial L/\partial \mathbf{v} = \gamma m \mathbf{v} + q \mathbf{A} \equiv p_{\text{can}} \quad \text{(Eq. 2-5)}
\label{eq:2-5}
\end{equation}

\begin{equation}
d/dt (\partial L/\partial \mathbf{v}) - \partial L/\partial \mathbf{x} = 0 \quad \text{(Eq. 2-6)}
\label{eq:2-6}
\end{equation}

Using $\mathbf{E} = -\nabla\Phi - \partial \mathbf{A}/\partial t$ and $\mathbf{B} = \nabla \times \mathbf{A}$, and expanding $d\mathbf{A}/dt = \partial \mathbf{A}/\partial t + (\mathbf{v} \cdot \nabla)\mathbf{A}$, cancellations yield the Lorentz force:

\begin{equation}
d(\gamma m \mathbf{v})/dt = q \, (\mathbf{E} + \mathbf{v} \times \mathbf{B}) \quad \text{(Eq. 2-7)}
\label{eq:2-7}
\end{equation}

\subsection*{Replication (cyclotron motion; uniform B, E = 0).}

Centripetal balance gives the cyclotron frequency:

\begin{equation}
\omega_c = q B / (\gamma m) \quad \text{(Eq. 2-8)}
\label{eq:2-8}
\end{equation}

Non-relativistic electron ($\gamma \approx 1$) in $B = 1$ T:

\begin{equation}
\omega_c \approx (e/m_e) \, B \approx 1.75882000838 \times 10^{11} \, s^{-1} \quad \text{(CODATA 2022)} \quad \text{(Eq. 2-9)}
\label{eq:2-9}
\end{equation}

\noindent
\textit{Validity \& limits. Point charge; classical regime; fields smooth on the formation time. Radiation reaction and quantum corrections neglected unless required by precision.}

\section*{3. Propagation in Media \& Rays --- Eikonal/WKB and Interfaces}

\subsection*{Setup \& scope.}

Derive rays from the scalar Helmholtz model with slowly varying refractive index $n(x)$ via a WKB ansatz; then apply interface boundary conditions to get Fresnel coefficients and special angles. No LaTeX macros; all formulas are plain and readable.

\subsection*{Helmholtz equation and WKB ansatz.}

\begin{equation}
\nabla^2 U + k0^2 \cdot n(x)^2 \cdot U = 0 \quad \text{(Eq. 3.1)}
\label{eq:3-1}
\end{equation}

\begin{equation}
U(x) = A(x) \cdot \exp(i \cdot k0 \cdot S(x)) \quad \text{(Eq. 3.2)}
\label{eq:3-2}
\end{equation}

Compute gradients and collect orders in $k0$:

\begin{equation}
\nabla U = \exp(i k0 S) \cdot [\nabla A + i k0 \cdot A \cdot \nabla S] \quad \text{(Eq. 3.3)}
\label{eq:3-3}
\end{equation}

\begin{equation}
\nabla^2 U = \exp(i k0 S) \cdot [\nabla^2 A + 2 i k0 \cdot (\nabla S \cdot \nabla A) + i k0 \cdot A \cdot (\nabla^2 S) - k0^2 \cdot A \cdot |\nabla S|^2]
\label{eq:3-4}
\end{equation}

Match powers of $k0$ (leading two):

\begin{equation}
|\nabla S|^2 = n^2(x) \quad \text{(Eq. 3.5)}
\label{eq:3-5}
\end{equation}

\begin{equation}
2 (\nabla S \cdot \nabla A) + A \cdot (\nabla^2 S) = 0 \quad \text{(Eq. 3.6)}
\label{eq:3-6}
\end{equation}

Interpretation: $S$ is the eikonal (optical path). The transport equation conserves flux along rays.

\subsection*{Rays from Fermat's principle.}

\begin{equation}
\delta \int n(x) \, ds = 0 \quad \text{(Eq. 3.7)}
\label{eq:3-7}
\end{equation}

Euler--Lagrange gives the ray evolution with unit tangent $t = dx/ds$:

\begin{equation}
d/ds \, [n \cdot t] = \nabla n \quad \text{(take the component $\perp$ to $t$)} \quad \text{(Eq. 3.8)}
\label{eq:3-8}
\end{equation}

\begin{equation}
\text{Ray curvature: } 1/a = \parallel \nabla_\perp \ln n \parallel \quad \text{(Eq. 3.9)}
\label{eq:3-9}
\end{equation}

Here $a$ is the local radius of curvature; $\nabla_\perp$ is the component of $\nabla$ perpendicular to the ray.

\subsection*{Planar interface boundary conditions (no free surface charge/current).}

\begin{equation}
\hat{n} \cdot (\mathbf{D}_2 - \mathbf{D}_1) = 0, \quad \hat{n} \cdot (\mathbf{B}_2 - \mathbf{B}_1) = 0 \quad \text{(Eq. 3.10)}
\label{eq:3-10}
\end{equation}

\begin{equation}
\hat{n} \times (\mathbf{E}_2 - \mathbf{E}_1) = 0, \quad \hat{n} \times (\mathbf{H}_2 - \mathbf{H}_1) = 0 \quad \text{(Eq. 3.11)}
\label{eq:3-11}
\end{equation}

\begin{equation}
\text{Snell: } n1 \cdot \sin\theta1 = n2 \cdot \sin\theta2 \quad \text{(Eq. 3.12)}
\label{eq:3-12}
\end{equation}

\subsection*{Fresnel coefficients (dielectric--dielectric, $\mu \approx \mu_0$).}

\begin{equation}
r_s = (n1 \cos\theta1 - n2 \cos\theta2) / (n1 \cos\theta1 + n2 \cos\theta2) \quad \text{(Eq. 3.13)}
\label{eq:3-13}
\end{equation}

\begin{equation}
t_s = 2 n1 \cos\theta1 / (n1 \cos\theta1 + n2 \cos\theta2) \quad \text{(Eq. 3.14)}
\label{eq:3-14}
\end{equation}

\begin{equation}
r_p = (n2 \cos\theta1 - n1 \cos\theta2) / (n2 \cos\theta1 + n1 \cos\theta2) \quad \text{(Eq. 3.15)}
\label{eq:3-15}
\end{equation}

\begin{equation}
t_p = 2 n1 \cos\theta1 / (n2 \cos\theta1 + n1 \cos\theta2) \quad \text{(Eq. 3.16)}
\label{eq:3-16}
\end{equation}

\begin{equation}
\text{Brewster (p-pol): } \tan \theta_B = n2 / n1 \quad \text{(Eq. 3.17)}
\label{eq:3-17}
\end{equation}

\begin{equation}
\text{Critical angle (internal): } \theta_c = \arcsin(n2 / n1) \quad \text{(defined only if $n1 > n2$)} \quad \text{(Eq. 3.18)}
\label{eq:3-18}
\end{equation}

\subsection*{Near- vs far-field (diffraction): Fresnel number.}

\begin{equation}
NF = a^2 / (\lambda \cdot z) \quad \text{(Fresnel if $NF \gtrsim 1$; Fraunhofer if $NF \ll 1$)} \quad \text{(Eq. 3.19)}
\label{eq:3-19}
\end{equation}

\subsection*{Replication (air $\to$ glass).}

Snell: $\theta2 \approx 28.13°$ for $\theta1 = 45.0°$.

s-pol amplitudes: $r_s \approx -0.303$ ($R_s \approx 0.092$), $t_s \approx 0.697$

p-pol amplitudes: $r_p \approx 0.092$ ($R_p \approx 0.008$), $t_p \approx 0.728$

Brewster angle: $\theta_B \approx 56.3°$.

Critical angle for glass $\to$ air ($n1=1.50, n2=1.0003$): $\theta_c \approx 41.8°$.

\noindent
\textit{Validity \& limits. WKB/eikonal: $|\nabla n|/n \ll k0$ (index varies slowly on $\lambda$ scale). Interface: locally planar, time-harmonic plane waves; no $\sigma_s$ or $K_s$. Diffraction regimes per NF.}

\section*{-- Interference --- Two-Beam, Multi-Beam, Thin Films, Michelson --}

\subsection*{Scope.}

Interference only: superposition of coherent beams without invoking diffraction envelopes. We derive two-beam intensity, fringe conditions, visibility, multi-beam array factor, thin-film conditions (with reflection phase), and a Michelson rule of thumb. All formulas are plain text.

\subsection*{Two-beam interference (scalar, quasimonochromatic).}

\begin{equation}
E1(t) = E0 \cdot \cos(\omega t) \quad \text{(Eq. 5.1)}
\label{eq:5-1}
\end{equation}

\begin{equation}
E2(t) = E0' \cdot \cos(\omega t + \Delta\varphi) \quad \text{(Eq. 5.2)}
\label{eq:5-2}
\end{equation}

Time-averaged intensity (add irradiances plus cross term):

\begin{equation}
I = I1 + I2 + 2 \cdot \sqrt{(I1 \cdot I2)} \cdot \cos(\Delta\varphi) \quad \text{(Eq. 5.3)}
\label{eq:5-3}
\end{equation}

Equal beams $I1 = I2 = I0 = I = 2 I0 \cdot [1 + \cos(\Delta\varphi)]$ (max = 4I0, min = 0).

\begin{equation}
\text{Phase difference: } \Delta\varphi = (2 \pi / \lambda) \cdot \Delta \quad \text{with } \Delta = \text{optical path difference} \quad \text{(Eq. 5.4)}
\label{eq:5-4}
\end{equation}

Bright fringes (no extra phase flips): $\Delta = m \cdot \lambda$ \quad ($m = 0, \pm1, \pm2, \ldots$)
\begin{equation}
\text{(Eq. 5.5)}
\label{eq:5-5}
\end{equation}

\begin{equation}
\text{Dark fringes: } \Delta = (m + 1/2) \cdot \lambda \quad \text{(Eq. 5.6)}
\label{eq:5-6}
\end{equation}

Fringe geometry for two equal, narrow slits separated by $d$, screen distance $L$ (small angles):

\begin{equation}
\text{Fringe angle spacing: } \Delta\theta \approx \lambda / d \quad \text{(Eq. 5.7)}
\label{eq:5-7}
\end{equation}

\begin{equation}
\text{Fringe spacing on screen: } \Delta y \approx (\lambda \cdot L) / d \quad \text{(Eq. 5.8)}
\label{eq:5-8}
\end{equation}

\subsection*{Visibility / contrast and coherence.}

\begin{equation}
V = (I_{\text{max}} - I_{\text{min}}) / (I_{\text{max}} + I_{\text{min}}) \quad \text{(Eq. 5.9)}
\label{eq:5-9}
\end{equation}

\begin{equation}
V = \left[ 2 \cdot \sqrt{(I1 \cdot I2)} / (I1 + I2) \right] \cdot |y12(\tau)| \quad \text{(Eq. 5.10)}
\label{eq:5-10}
\end{equation}

Here $y12(\tau)$ is the complex degree of coherence at delay $\tau$; for equal beams and full temporal coherence, $V = 1$.

\subsection*{Multi-beam (N sources, equal spacing d, equal amplitudes).}

\begin{equation}
\text{Array phase: } \delta = (2 \pi / \lambda) \cdot d \cdot \sin\theta \quad \text{(Eq. 5.11)}
\label{eq:5-11}
\end{equation}

\begin{equation}
\text{Intensity: } I(\theta) = I0 \cdot [\sin(N \cdot \delta / 2) / \sin(\delta / 2)]^2 \quad \text{(Eq. 5.12)}
\label{eq:5-12}
\end{equation}

Principal maxima when $\delta = 2\pi m \Rightarrow \sin\theta_m = m \cdot \lambda / d$. Side-lobe structure set by N.

\subsection*{Thin-film interference (normal incidence; include reflection phase).}

\begin{equation}
\text{Geometric path difference: } \Delta = 2 \cdot n \cdot t \quad \text{(Eq. 5.13)}
\label{eq:5-13}
\end{equation}

Phase on reflection (interface rule): a $\pi$ phase flip occurs when reflecting from a lower-to-higher index boundary; no flip for higher-to-lower.

Cases (use the rule above to count $\pi$ flips in the reflected paths):

\begin{itemize}
\item \textbf{Case A} --- one $\pi$ flip (e.g., $n0 < n1$ and $n1 > n2$, or $n0 > n1$ and $n1 < n2$):

Reflected constructive: $2 \cdot n \cdot t = (m + 1/2) \cdot \lambda$

Reflected destructive: $2 \cdot n \cdot t = m \cdot \lambda$

\item \textbf{Case B} --- two $\pi$ flips (common AR stack with $n0 < n1 < n2$):

Reflected constructive: $2 \cdot n \cdot t = m \cdot \lambda$

Reflected destructive: $2 \cdot n \cdot t = (m + 1/2) \cdot \lambda$

\item \textbf{Case C} --- zero $\pi$ flips (e.g., $n0 > n1 > n2$): same conditions as Case B.

\end{itemize}

Oblique incidence:

\begin{equation}
\Delta = 2 \cdot n \cdot t \cdot \cos \theta_t \quad \text{(Snell's law for } \theta_t\text{)}
\label{eq:5-14}
\end{equation}

Quick replication (quarter-wave AR):

For two $\pi$ flips (Case B), destructive reflection at first order occurs at $m = 0 \Rightarrow 2 \cdot n \cdot t = \lambda/2$, so $t = \lambda / (4n)$. Example A: $\lambda = 550$ nm, $n_{\text{film}} = 1.50 \Rightarrow t \approx 91.7$ nm. Example B (typical AR material): $\lambda = 550$ nm, $n_{\text{film}} = 1.38 \Rightarrow t \approx 99.6$ nm.

Cross-link: reflection phase flips are the same Fresnel interface rules you derived in §3 (Interfaces).

\subsection*{Michelson interferometer (fringe count vs path change).}

\begin{equation}
\text{Optical path difference: } \Delta = 2 \cdot \Delta L \quad \text{(Eq. 5.17)}
\label{eq:5-17}
\end{equation}

\begin{equation}
\text{One fringe shift when: } \Delta L = \lambda / 2 \quad \text{(Eq. 5.18)}
\label{eq:5-18}
\end{equation}

Rule of thumb: moving one mirror by $\lambda/2$ advances the fringe order by one.

\subsection*{Replication (quick numbers).}

Two-slit: $\lambda = 632.8$ nm, $d = 300$ \textmu m, $L = 1.5$ m $\Rightarrow \Delta y \approx 3.16$ mm.

Thin film (reflection, one flip): $\lambda = 550$ nm, $n = 1.50 \Rightarrow$ first constructive at $t = \lambda/(4n) = 91.7$ nm.

Michelson: $\lambda = 500$ nm $\Rightarrow$ one fringe per $\Delta L = \lambda/2 = 250$ nm mirror motion.

\noindent
\textit{Validity \& limits. Assumes scalar, monochromatic or narrowband fields; paraxial geometry for fringe spacing; equal path polarization; ignores diffraction envelopes and vector corrections.}

\section*{4. Diffraction á --- Kirchhoff, Fresnel, Fraunhofer}

\subsection*{Setup \& scope.}

Start from the scalar Helmholtz model, build the Kirchhoff integral via the free-space Green function, take the Fresnel and Fraunhofer limits, and list canonical patterns (single slit, double slit, grating, circular aperture). All formulas are plain text for easy reading.

\subsection*{Helmholtz equation and free-space Green function.}

\begin{equation}
\nabla^2 U + k^2 \cdot U = 0 \quad \text{with} \quad k = 2\pi / \lambda \quad \text{(Eq. 4.1)}
\label{eq:4-1}
\end{equation}

\begin{equation}
G(P,Q) = \exp(i k R) / (4\pi R) \quad \text{with} \quad R = |P - Q| \quad \text{(Eq. 4.2)}
\label{eq:4-2}
\end{equation}

\subsection*{Kirchhoff integral (aperture $\Sigma$ to observation point P).}

\begin{equation}
U(P) = \iint_\Sigma \left[ U(Q) \, \frac{\partial G}{\partial n} - G \, \frac{\partial U}{\partial n} \right] d\Xi \quad \text{(Eq. 4.3)}
\label{eq:4-3}
\end{equation}

\subsection*{Fresnel approximation (paraxial).}

Geometry: aperture coordinates $(\xi, \eta)$, screen point $(x, y)$, separation $z$, small angles ($\cos\theta \approx 1$). Use the quadratic path expansion:

\begin{equation}
R \approx z + [(x - \xi)^2 + (y - \eta)^2] / (2z) \quad \text{(Eq. 4.4)}
\label{eq:4-4}
\end{equation}

Insert into the Kirchhoff integral and retain quadratic phase:

\begin{equation}
U(x,y; z) \approx [\exp(i k z) / (i \cdot \lambda \cdot z)] \iint U(\xi,\eta; 0) \cdot \exp\{i k \, [(x - \xi)^2 + (y - \eta)^2] / (2 z) \} d\xi \, d\eta \quad \text{(Eq. 4.5)}
\label{eq:4-5}
\end{equation}

\subsection*{Fraunhofer (far-field) approximation.}

For large $z$ (or small aperture) so that the quadratic terms from $\xi,\eta$ can be dropped inside the phase (keep only linear terms):

\begin{equation}
U(x,y; z) \approx [\exp(i k z) / (i \cdot \lambda \cdot z)] \cdot \iint U(\xi,\eta; 0) \cdot \exp\{- i 2\pi \, [\xi \cdot x + \eta \cdot y] / (\lambda z) \} d\xi \, d\eta \quad \text{(Eq. 4.6)}
\label{eq:4-6}
\end{equation}

Interpretation: up to the prefactor, the far-field is the 2-D Fourier transform of the aperture.

\subsection*{Canonical intensity patterns.}

\textbf{Single slit} (width $a$, along $x$; observe in $y$--$z$ plane):

\begin{equation}
I(\theta) = I0 \cdot (\sin \beta / \beta)^2 \quad \text{with} \quad \beta = (\pi a / \lambda) \cdot \sin\theta \quad \text{(Eq. 4.7)}
\label{eq:4-7}
\end{equation}

Minima at $a \cdot \sin\theta = m \cdot \lambda$ ($m = \pm1, \pm2, \ldots$).

\textbf{Double slit} (slit width $a$, center spacing $d$):

\begin{equation}
I(\theta) = I0 \cdot \cos^2(\pi d \sin\theta / \lambda) \cdot (\sin \beta / \beta)^2 \quad \text{with} \quad \beta = (\pi a / \lambda) \cdot \sin\theta
\label{eq:4-8}
\end{equation}

Fringe spacing on a screen at distance $z$ (small angles): $\Delta y \approx \lambda z / d$.

\textbf{Transmission grating} (period $d$):

\begin{equation}
\text{Grating equation: } m \cdot \lambda = d \cdot \sin\theta_m \quad (m = 0, \pm1, \pm2, \ldots) \quad \text{(Eq. 4.9)}
\label{eq:4-9}
\end{equation}

\textbf{Circular aperture} (diameter $D$): Airy pattern

\begin{equation}
I(\theta) = I0 \cdot [2 J_1(\pi D \sin\theta / \lambda) / (\pi D \sin\theta / \lambda)]^2 \quad \text{(Eq. 4.10)}
\label{eq:4-10}
\end{equation}

First minimum at $\sin\theta \approx 1.22 \cdot \lambda / D$ (small-angle).

\subsection*{Fresnel zones (on-axis).}

\begin{equation}
\text{Zone radius: } r_n \approx \sqrt{n \cdot \lambda \cdot z} \quad \text{(Eq. 4.11)}
\label{eq:4-11}
\end{equation}

Alternating zones add with alternating phase; a circular aperture of radius $r_N$ passes about half the first zone amplitude when $N = 1$.

\subsection*{Regimes via Fresnel number.}

\begin{equation}
NF = a^2 / (\lambda z) \to \text{Fresnel if } NF \gtrsim 1; \text{ Fraunhofer if } NF \ll 1 \quad \text{(Eq. 4.12)}
\label{eq:4-12}
\end{equation}

\subsection*{Replication (numbers you can check quickly).}

Single slit ($\lambda=632.8$ nm, $a=100$ \textmu m, $z=1.5$ m):

First minimum: $\theta \approx \lambda/a = 0.00633$ rad $\to y \approx 9.49$ mm on screen.

Fresnel number: $NF = a^2/(\lambda z) \approx 0.01$ (near-intermediate).

Double slit (same $\lambda,z$; spacing $d=300$ \textmu m):

Fringe spacing: $\Delta y = \lambda z / d \approx 3.16$ mm.

Circular aperture ($D=5$ mm, same $\lambda$; $z=2.0$ m):

Airy first minimum: $\theta \approx 1.22 \lambda/D = 1.54 \text{e-4}$ rad $\to$ radius on screen $y \approx 0.31$ mm.

\noindent
\textit{Validity \& limits. Scalar model with uniform polarization; paraxial angles for Fresnel/Fraunhofer forms; apertures large vs $\lambda$ for Kirchhoff approximation; neglects vector/evanescent and edge polarization corrections.}

\section*{5. Polarization Transport (Jones / Stokes / Mueller)}

\subsection*{Scope.}

Model polarization with Jones vectors (amplitudes and phases), Stokes vectors (intensities), and Mueller matrices (system action). Include rotation (Faraday/optical activity), linear birefringence (retarders), diattenuation (polarizers), conversions, and quick numeric checks. All formulas are plain text for easy reading.

\subsection*{Jones calculus (fully polarized fields).}

\begin{equation}
\text{Jones vector: } \mathbf{E} = [Ex; Ey] \quad \text{(Eq. 5.1)}
\label{eq:5p1}
\end{equation}

\begin{equation}
\text{Linear system (2$\times$2): } \mathbf{E}_{\text{out}} = \mathbf{J} \cdot \mathbf{E}_{\text{in}} \quad \text{(Eq. 5.2)}
\label{eq:5p2}
\end{equation}

\begin{equation}
\text{Rotation matrix: } R(\theta) = [[\cos\theta, -\sin\theta]; [\sin\theta, \cos\theta]] \quad \text{(Eq. 5.3)}
\label{eq:5p3}
\end{equation}

\begin{equation}
\text{Linear polarizer at angle $\theta$: } J_{\text{pol}}(\theta) = R(-\theta) \cdot [[1,0]; [0,0]] \cdot R(\theta) \quad \text{(Eq. 5.4)}
\label{eq:5p4}
\end{equation}

\begin{equation}
\text{Linear retarder (fast axis } x\text{, retardance } \delta\text{): } J_{\text{ret}}(\delta) = [[1, 0]; [0, \exp(i \cdot \delta)]] \quad \text{(global phase irrelevant)} \quad \text{(Eq. 5.5)}
\label{eq:5p5}
\end{equation}

\subsection*{Stokes vector and conversions (intensity domain).}

\begin{equation}
\text{Stokes vector: } \mathbf{S} = (S0, S1, S2, S3) \quad \text{(Eq. 5.6)}
\label{eq:5p6}
\end{equation}

\begin{equation}
\text{From Jones (fully polarized): } S0 = |Ex|^2 + |Ey|^2 \quad \text{(Eq. 5.7)}
\label{eq:5p7}
\end{equation}

\begin{equation}
S1 = |Ex|^2 - |Ey|^2 \quad \text{(Eq. 5.8)}
\label{eq:5p8}
\end{equation}

\begin{equation}
S2 = 2 \cdot \text{Re}(Ex \cdot Ey^*) \quad \text{(Eq. 5.9)}
\label{eq:5p9}
\end{equation}

\begin{equation}
S3 = 2 \cdot \text{Im}(Ex \cdot Ey^*) \quad \text{(Eq. 5.10)}
\label{eq:5p10}
\end{equation}

\begin{equation}
\text{Degree of polarization: } P = \sqrt{S1^2 + S2^2 + S3^2} / S0 \quad \text{(Eq. 5.11)}
\label{eq:5p11}
\end{equation}

\subsection*{Mueller matrices (act on Stokes).}

\begin{equation}
\mathbf{S}_{\text{out}} = \mathbf{M} \cdot \mathbf{S}_{\text{in}} \quad \text{(Eq. 5.12)}
\label{eq:5p12}
\end{equation}

\begin{equation}
\text{Rotation by $\theta$ (e.g., Faraday/optical activity): } \quad \text{(Eq. 5.13)}
\label{eq:5p13}
\end{equation}

\begin{equation}
\mathbf{S}_{\text{out}} = (S0, \, S1 \cos 2\theta + S2 \sin 2\theta, \, -S1 \sin 2\theta + S2 \cos 2\theta, \, S3) \quad \text{(Eq. 5.14)}
\label{eq:5p14}
\end{equation}

\begin{equation}
\text{Linear polarizer at angle $\theta$: } \quad \text{(Eq. 5.15)}
\label{eq:5p15}
\end{equation}

\begin{equation}
M_{\text{pol}}(\theta) = (1/2) \, [[\, 1, \, \cos 2\theta, \, \sin 2\theta, \, 0 \,]; \quad \text{(Eq. 5.16)}
\label{eq:5p16}
\end{equation}

\begin{equation}
\, [\, \cos 2\theta, \, \cos^2 2\theta, \, \sin 2\theta \cos 2\theta, \, 0 \,]; \quad \text{(Eq. 5.17)}
\label{eq:5p17}
\end{equation}

\begin{equation}
\, [\, \sin 2\theta, \, \sin 2\theta \cos 2\theta, \, \sin^2 2\theta, \, 0 \,]; \quad \text{(Eq. 5.18)}
\label{eq:5p18}
\end{equation}

\begin{equation}
\, [\, 0, \, 0, \, 0, \, 0 \,]] \quad \text{(Eq. 5.19)}
\label{eq:5p19}
\end{equation}

\begin{equation}
\text{Linear retarder (fast axis } x\text{, retardance } \delta\text{): } \quad \text{(Eq. 5.20)}
\label{eq:5p20}
\end{equation}

\begin{equation}
M_{\text{ret}}(0,\delta) = [[\, 1, 0, 0, 0 \,]; [0, 1, 0, 0]; [0, 0, \cos\delta, \sin\delta]; [0, 0, -\sin\delta, \cos\delta]] \quad \text{(Eq. 5.21)}
\label{eq:5p21}
\end{equation}

\begin{equation}
\text{General axis } \varphi\text{: } M_{\text{ret}}(\varphi,\delta) = R_M(-2\varphi) \cdot M_{\text{ret}}(0,\delta) \cdot R_M(2\varphi) \quad \text{(Eq. 5.22)}
\label{eq:5p22}
\end{equation}

\begin{equation}
\text{where } R_M(\psi) \text{ acts as: } (S0, S1, S2, S3) \to (S0, S1 \cos\psi + S2 \sin\psi, -S1 \sin\psi + S2 \cos\psi, S3) \quad \text{(Eq. 5.23)}
\label{eq:5p23}
\end{equation}

\subsection*{Physical effects and parameter links.}

\begin{equation}
\text{Faraday rotation (magneto-optic): } \theta_F = V \cdot \mathbf{B} \cdot L \quad \text{(Eq. 5.24)}
\label{eq:5p24}
\end{equation}

\begin{equation}
\text{Linear birefringence (uniaxial): } \delta = 2\pi \cdot (\Delta n) \cdot L / \lambda \quad \text{(Eq. 5.25)}
\label{eq:5p25}
\end{equation}

\begin{equation}
\text{Optical activity (chiral media): } \theta = \alpha \cdot L \quad (\alpha = \text{rotation per unit length}) \quad \text{(Eq. 5.26)}
\label{eq:5p26}
\end{equation}

\begin{equation}
\text{Diattenuation (unequal transmission): modeled by } M_{\text{pol}}(\theta) \text{ with finite extinction ratio} \quad \text{(Eq. 5.27)}
\label{eq:5p27}
\end{equation}

\subsection*{Replication (numbers you can check).}

Faraday rotation: $V = 4 \times 10^{-6}$ rad$\cdot$T$^{-1}\cdot$m$^{-1}$, $B = 0.5$ T, $L = 0.2$ m $\Rightarrow \theta_F \approx 4.0\text{e-06}$ rad ($\approx 0.00023°$).

Linear birefringence: $\Delta n = 1 \times 10^{-5}$, $L = 1$ cm, $\lambda = 632.8$ nm $\Rightarrow \delta \approx 0.99$ rad ($\approx 56.9°$).

Malus' law check: two ideal linear polarizers at relative angle $\Delta \to L_{\text{out}} = L_{\text{in}} \cdot \cos^2(\Delta)$. For $\Delta = 30°$, $\cos^2\Delta \approx 0.75$.

\noindent
\textit{Validity \& limits. Jones: fully polarized, coherent beams. Stokes/Mueller: works with partial polarization and depolarization; assumes quasi-monochromatic fields. Linear media; small-signal for $V$, $\Delta n$. Dispersion handled by $\lambda$-dependence of $V$ and $\Delta n$.}

\section*{6. Imaging (Paraxial / ABCD) and Invariants}

\subsection*{Scope.}

Paraxial ray transfer (ABCD) with height--angle vectors, element matrices (free space, thin lens), imaging rules, magnification, and the Lagrange invariant. All formulas are plain text; we use angles in radians and heights in meters.

\subsection*{ABCD core relation (single transverse plane).}

\begin{equation}
\begin{bmatrix} x2 \\ \theta2 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \cdot \begin{bmatrix} x1 \\ \theta1 \end{bmatrix} \quad \text{(Eq. 6.1)}
\label{eq:6-1}
\end{equation}

\begin{equation}
\text{Determinant (uniform index): } A \cdot D - B \cdot C = 1 \quad \text{(Eq. 6.2)}
\label{eq:6-2}
\end{equation}

\subsection*{Element matrices (uniform refractive index).}

\begin{equation}
\text{Free-space propagation over length } L\text{: } F(L) = \begin{bmatrix} 1 & L \\ 0 & 1 \end{bmatrix} \quad \text{(Eq. 6.3)}
\label{eq:6-3}
\end{equation}

\begin{equation}
\text{Thin lens of focal length } f\text{: } L(f) = \begin{bmatrix} 1 & 0 \\ -1/f & 1 \end{bmatrix} \quad \text{(Eq. 6.4)}
\label{eq:6-4}
\end{equation}

System matrix is the ordered product of elements (rightmost acts first).

\begin{equation}
\text{System: } M = F(L2) \cdot L(f) \cdot F(L1) = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \quad \text{(Eq. 6.5)}
\label{eq:6-5}
\end{equation}

\subsection*{Imaging rules (thin lens in air).}

\begin{equation}
\text{Object distance: } do \quad \text{(measured from lens)} \quad \text{(Eq. 6.6)}
\label{eq:6-6}
\end{equation}

\begin{equation}
\text{Image distance: } di \quad \text{(measured from lens)} \quad \text{(Eq. 6.7)}
\label{eq:6-7}
\end{equation}

\begin{equation}
\text{Lens equation: } 1/f = 1/do + 1/di \quad \text{(Eq. 6.8)}
\label{eq:6-8}
\end{equation}

\begin{equation}
\text{Magnification: } M = -di / do \quad \text{(Eq. 6.9)}
\label{eq:6-9}
\end{equation}

Connection to ABCD: an object at distance $do$ before the lens and a screen at distance $di$ after the lens yields a system matrix with $C = -1/f + 1/di + 1/do = 0$ at focus. Equivalently, choose $L1 = do, L2 = di$ in $M = F(L2) \cdot L(f) \cdot F(L1)$ and solve $C = 0$ to recover $1/f = 1/do + 1/di$.

\subsection*{Lagrange (optical) invariant in this angle--height convention.}

\begin{equation}
\text{Invariant (no stops, lossless): } H = n \cdot x \cdot \sin\theta \approx n \cdot x \cdot \theta \quad \text{(small angles)}
\label{eq:6-10}
\end{equation}

$H$ is conserved across lossless, paraxial systems; it bounds simultaneous concentration of height and angle (etendue).

\subsection*{Replication (quick checks).}

Thin-lens imaging (given): $f = 50$ mm, $do = 100$ mm $\Rightarrow di = 100$ mm, $M = -1$.

ABCD multiplication example ($do = 0.10$ m, $f = 0.050$ m, $di = 0.10$ m):

$F(L1) = [[1, 0.10]; [0, 1]], \, L(f) = [[1, 0]; [-20, 1]], \, F(L2) = [[1, 0.10]; [0, 1]]$

$M = F(L2) \cdot L(f) \cdot F(L1) = [[0, 0.10]; [-10, 0]]$

Apply to a ray from an on-axis object point ($x1 = 0$, small angle $\theta1$): $[x2; \theta2] = [[0, 0.10]; [-10, 0]] \cdot [0; \theta1] = [0.10 \theta1; 0]$. Focus at the image plane (angle goes to 0).

\noindent
\textit{Validity \& limits. Paraxial (small angles), thin elements, uniform index for the given determinant form. For systems with refractive interfaces (changing $n$), use n-normalized slope $q = n \cdot \theta$ so that $\det = 1$ and the generalized Lagrange invariant $H = x \cdot q$ is conserved.}

\section*{7. Spectroscopy Anchors (Vacuum) and Ratio-First Validation}

\subsection*{Scope.}

Vacuum hydrogen series from the Rydberg constant, ratio-first checks (insensitive to $R_\infty$), and the tie-back to $S_0 = \hbar$ (no new dimensional scales). All formulas are plain text.

\subsection*{Rydberg constant (in terms of fundamental constants).}

\begin{equation}
R_\infty = (\alpha^2 \cdot m_e \cdot c) / (2 \cdot \hbar) \quad \text{(Eq. 7.1)}
\label{eq:7-1}
\end{equation}

\subsection*{Hydrogen wavelengths (vacuum):}

\begin{equation}
1/\lambda = R_\infty \cdot (1/n_1^2 - 1/n_2^2) \quad \text{with} \quad n_2 > n_1 \quad \text{(Eq. 7.2)}
\label{eq:7-2}
\end{equation}

Balmer examples ($n_1 = 2$): H$\alpha$ ($3 \to 2$), H$\beta$ ($4 \to 2$), H$\gamma$ ($5 \to 2$).

\subsection*{Ratio-first validation (insensitive to $R_\infty$):}

\begin{equation}
\lambda_\alpha / \lambda_\beta = [(1/2^2 - 1/3^2)^{-1}] / [(1/2^2 - 1/4^2)^{-1}] = (36/5) / (16/3) = 27/20 \approx 1.35 \quad \text{(Eq. 7.3)}
\label{eq:7-3}
\end{equation}

Measured (vacuum): $656.281$ nm / $486.133$ nm $\approx 1.350003 \to$ agreement within $\sim$2 ppm.

\subsection*{Tie to $S_0 = \hbar$ (no new dimensional scales):}

\begin{equation}
E_n = -(m_e \cdot e^2)^2 / (2 \cdot (4\pi \varepsilon_0)^2 \cdot \hbar^2 \cdot n^2) \quad \text{(Eq. 7.4)}
\label{eq:7-4}
\end{equation}

\begin{equation}
a_0 = 4\pi \varepsilon_0 \cdot \hbar^2 / (m_e \cdot e^2) \quad \text{(Eq. 7.5)}
\label{eq:7-5}
\end{equation}

These recover the Rydberg relations and depend only on $\hbar$ imported at the electron anchor; no new scales are introduced in this pillar.

\noindent
\textit{Validity \& limits. Vacuum lines; non-relativistic hydrogen; reduced-mass correction small; QED/fine-structure and Lamb-shift corrections handled at higher order in the calibration capsule.}

\section*{8. Small Corrections --- Torsion / Shear and Closure Tolerance}

\subsection*{Scope.}

Parameterize small departures from the ideal vacuum transport via torsion and shear budgets, and relate experimental bounds to closure tolerance $J_c$.

\subsection*{Phase and index perturbations from torsion/shear budgets (first order).}

\begin{equation}
\delta\varphi \approx \oint [a_T \cdot T + a_S \cdot S] \, dl \quad \text{(Eq. 8.1)}
\label{eq:8-1}
\end{equation}

\begin{equation}
\delta n / n \approx \beta_T \cdot T + \beta_S \cdot S \quad \text{(Eq. 8.2)}
\label{eq:8-2}
\end{equation}

Interpretation.

$T$ and $S$ are dimensionless small budgets imported from the Foundation (torsion and shear). Coefficients $\alpha_T, a_S, \beta_T, \beta_S$ are geometry-dependent but order-one in calibrated units. In pure vacuum these corrections are expected to vanish; any measured residual constrains $J_c$.

\subsection*{Closure tolerance link.}

\begin{equation}
J_c = \Delta S / S_0 \quad \text{(Eq. 8.3)}
\label{eq:8-3}
\end{equation}

Use precision optics bounds (e.g., $\delta n/n \lesssim 10^{-6}$) to set $|T|, |S| \lesssim 10^{-6}$ (up to $\beta$-coefficients). These feed back into error budgets for propagation and spectroscopy.

\noindent
\textit{Validity \& limits. Applies when $|T|, |S| \ll 1$ (near-ideal vacuum). Odd azimuthal modes are parity-suppressed unless explicit chirality is present.}

\section*{9. Falsifiers \& Lab-Math Hooks (Neutral Rotor)}

\subsection*{Scope.}

Neutral, non-magnetic rotor aligned to the global caustic plane. Predict a tiny, orientation-gated push/zero/pull with spin flips, then show how to bound it if absent. All formulas are plain text.

\subsection*{Setup and definitions.}

Composite body is neutral: $\Sigma q = 0$ (no net monopole). Use a dielectric rotor (e.g., fused silica), non-magnetic bearings, high-vacuum, and electrostatic shielding. Align the spin axis with the caustic-plane normal.

\begin{equation}
\varepsilon = \text{closure / tear bias (small, dimensionless)} \quad \text{(Eq. 9.1)}
\label{eq:9-1}
\end{equation}

\begin{equation}
\sigma = \text{spin orientation} \quad (+1 \text{ for right-handed / with } +\hat{n}, \, -1 \text{ for opposite}) \quad \text{(Eq. 9.2)}
\label{eq:9-2}
\end{equation}

\begin{equation}
\theta = \text{facing (tilt) angle relative to } +\hat{n} \quad (\theta = 0 \to \text{right facing}; \, \theta = 90° \to \text{null}) \quad \text{(Eq. 9.3)}
\label{eq:9-3}
\end{equation}

\subsection*{Prediction (cycle-averaged).}

\begin{equation}
\langle \Delta p \rangle \propto \varepsilon \cdot \sigma \cdot \cos\theta \quad \text{(Eq. 9.4)}
\label{eq:9-4}
\end{equation}

Signature (at fixed speed, no EM handles):

\begin{itemize}
\item right spin + right facing ($\sigma = +1, \theta = 0°$) $\to$ push

\item $90°$ tilt ($\theta = 90°$) $\to$ strict null

\item spin-flip or opposite facing ($\sigma = -1$ or $\theta = 180°$) $\to$ pull

\end{itemize}

From momentum to force (lock-in readout).

\begin{equation}
F_{\text{sig}} \approx \Omega \cdot |\langle \Delta p \rangle| \quad \text{(Eq. 9.5)}
\label{eq:9-5}
\end{equation}

Modulate the spin state ($\sigma \to -\sigma$) or dither $\theta$ about $0°$ at $\pm\delta$ at angular frequency $\Omega$ and demodulate at $\Omega$. The signal flips sign with $\sigma$ and follows $\cos\theta$.

\subsection*{Replication (illustrative numbers).}

Toy: $\varepsilon = 1 \times 10^{-3}$, $\sigma = +1$, $\theta = 0°$, $p_0 = 1 \times 10^{-3}$ N$\cdot$s, $\Omega = 100$ Hz $\Rightarrow \langle \Delta p \rangle \approx 1.0 \text{e-06}$ N$\cdot$s, $F_{\text{sig}} = 1.0 \text{e-04}$ N.

Micro-newton torsion balances can reach $\sim 10^{-11}$--$10^{-7}$ N; if null at that level, the bound is $\varepsilon \lesssim F_{\text{min}} / (\Omega \cdot p_0 \cdot |\cos\theta|)$.

\subsection*{Controls and nulls.}

\begin{itemize}
\item Electrostatic: conductive coating + Faraday cage; discharge rotor; monitor surface potential.

\item Magnetic: non-magnetic materials, $\mu$-metal shielding; verify flips do not change stray fields.

\item Thermal/flow: high vacuum, constant temperature; check $\sigma$-synchronous heating.

\item Geometry nulls: $\theta = 90°$ (strict null), or flip (sign flip).

\end{itemize}

\noindent
\textit{Validity \& interpretation. Applies to neutral, non-magnetic, isolated rotors aligned to the global caustic plane. A non-zero $\sigma$-locked signal is new relative to standard EM; a null sets a quantitative upper bound via the formula above.}

\section*{10. Replication Blocks (Bench-Ready Numbers)}

\subsection*{Single-slit diffraction (Fraunhofer).}

\begin{equation}
\text{First minimum: } y1 = (\lambda \cdot L)/a \quad \text{(Eq. 10.1)}
\label{eq:10-1}
\end{equation}

He--Ne $\lambda = 632.8$ nm, $a = 100$ \textmu m, $L = 1.00$ m $\Rightarrow y1 \approx 6.33$ mm.

\begin{equation}
\text{Fresnel number: } N_F = a^2 / (\lambda \cdot L) \quad \text{(Eq. 10.2)}
\label{eq:10-2}
\end{equation}

With the same numbers: $N_F \approx 0.016$ ($\ll 1 \to$ Fraunhofer valid).

\subsection*{Double-slit interference (no envelope).}

\begin{equation}
\text{Fringe spacing: } \Delta y = (\lambda \cdot L)/d \quad \text{(Eq. 10.3)}
\label{eq:10-3}
\end{equation}

$\lambda = 632.8$ nm, $d = 300$ \textmu m, $L = 1.00$ m $\Rightarrow \Delta y \approx 2.109$ mm.

\subsection*{Interfaces (air $\leftrightarrow$ glass).}

\begin{equation}
\text{Brewster angle (air $\to$ glass): } \tan \theta_B = n2 / n1 \quad \text{(Eq. 10.4)}
\label{eq:10-4}
\end{equation}

$n_1 = 1.0003$ (air), $n_2 = 1.50$ (glass) $\Rightarrow \theta_B \approx 56.3°$.

\begin{equation}
\text{Critical angle (glass $\to$ air): } \sin \theta_c = n2 / n1 \quad \text{(with } n1 > n2\text{)} \quad \text{(Eq. 10.5)}
\label{eq:10-5}
\end{equation}

For glass $\to$ air: $\sin \theta_c = n_{\text{air}} / n_{\text{glass}} \Rightarrow \theta_c \approx 41.8°$.

\noindent
Constants and data sources (for your overall document's references): CODATA 2022; NIST ASD v5.11 (2023) Balmer wavelengths; PDG muon lifetime unchanged (context only).

\noindent
Constants: CODATA 2022; NIST ASD v5.11 (2023) Balmer wavelengths. PDG muon lifetime unchanged (context only).

\end{document}
