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\title{Electromagnetism Math Walk-Through}
\author{}
\date{}

\begin{document}

\maketitle

We pick up directly from the canonical Foundation Math Walk-Through. We import a single dimensional scale $S_0 = \hbar$ from the electron anchor---no new dimensional scales are introduced here. All field structure descends from the base variational framework; we do not retune base locks.

-Foundation Bridge (2-form F): In the main Math Appendix, display-area flux is encoded as a 2-form F with $\mathrm{d}F = 0$ (Bianchi). With action $S[\mathbf{A}] = \frac{1}{2} \int \mathbf{F} \wedge \mathbf{*F}$, Euler---Lagrange yields $\mathrm{d}\mathbf{*F} = \mathbf{J}$.

-The Helmholtz potentials $\Phi, \mathbf{A}$ here are the local representatives, giving $\mathbf{E} = -\nabla\Phi - \partial \mathbf{A}/\partial t$ and $\mathbf{B} = \nabla \times \mathbf{A}$ in the source-free Maxwell limit (see Math Appendix §1---§1.5).

\section*{0. Setup and Assumptions}

\subsection*{Notation \& Objects.}

Spacetime coordinates $x = (t, \mathbf{x}')$; potentials $\mathbf{A} = (\Phi, \mathbf{A}^*)$; field 2-form $\mathbf{F} = \mathrm{d}\mathbf{A}$ with components $F_{0i}=E_i$, $F_{ij}=-\epsilon_{ijk} B_k$.

\subsection*{Units \& Locks.}

Import a single dimensional scale $S_0 = \hbar$ (electron anchor); all other locks are acceptance targets. No retune of base constants.

\subsection*{Continuity from symmetry.}

Gauge invariance $\mathbf{A} \to \mathbf{A} + \nabla\chi$, $\Phi \to \Phi - \partial \chi/\partial t$ implies $\partial \rho / \partial t + \nabla \cdot \mathbf{J} = 0$. We use this repeatedly in conservation steps.

Display-area flux is encoded as a 2-form $\mathbf{F}$ with $\mathrm{d}\mathbf{F} = 0$ (Bianchi). With action $S[\mathbf{A}] = \frac{1}{2} \int \mathbf{F} \wedge \mathbf{*F}$ and minimal coupling $\int \mathbf{J} \cdot \mathbf{A} \, \mathrm{d}^4x$, Euler---Lagrange yields $\mathrm{d}\mathbf{*F} = \mathbf{J}$. Writing $\mathbf{F} = \mathrm{d}\mathbf{A}$ and splitting $\mathbf{A} = (\Phi, \mathbf{A})$, we obtain $\mathbf{E} = -\nabla\Phi - \partial \mathbf{A}/\partial t$ and $\mathbf{B} = \nabla \times \mathbf{A}$, with continuity $\partial \rho/\partial t + \nabla \cdot \mathbf{J} = 0$ (from gauge invariance). (See Math Appendix §1---§1.5.)

\subsection*{Geometry $\to$ Potentials $\to$ Fields (derivation sketch).}

We pick up directly from the canonical Foundation Math Walk-Through. We import a single dimensional scale $S_0 = \hbar$ from the electron anchor---no new dimensional scales are introduced. Base locks are not retuned; all EM structure is inherited via the geometric variational framework.

\subsection*{Imports \& Units (explicit).}

We begin with the anchor constant of the entire framework:
\[
S_0 = \hbar
\]

Geometric operators are carried from the base pillar: curvature descriptor $\mathcal{K}$ and oriented loop density $\Omega$.

Define pillar-local potentials: $\Phi[\mathcal{K}, \Omega]$ and $\mathbf{A}[\mathcal{K}, \Omega]$.

Fields: $\mathbf{E} = -\nabla\Phi - \partial \mathbf{A}/\partial t$, $\mathbf{B} = \nabla \times \mathbf{A}$.

Sources are geometric descriptors: $(\rho, \mathbf{J})$ from orientation and topology of loops.

Continuity follows directly: $\partial \rho/\partial t + \nabla \cdot \mathbf{J} = 0$.

Limit regime: weak curvature / slow variation $\to$ Maxwell equations.

\section*{1. Maxwell Set from Geometry}

\subsection*{Action and variation (explicit).}

Field action: $S = \frac{1}{2} \int \mathbf{F} \wedge \mathbf{*F}$; interaction: $S_{\text{int}} = \int (\delta \mathbf{A}) \cdot (\mathrm{d}^* \mathbf{F} - \mathbf{J}) \, \mathrm{d}^4x$ $\Rightarrow$ $\mathrm{d}^* \mathbf{F} = \mathbf{J}$; geometric identity $\mathrm{d}\mathbf{F} = 0$.

\subsection*{Split into E, B equations.}

$\nabla \cdot \mathbf{B} = 0$; $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$; $\nabla \cdot \mathbf{E} = \rho/\varepsilon_0$; $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \partial \mathbf{E}/\partial t$.

\subsection*{Wave equation and dispersion.}

Take curl of Faraday, substitute Ampère---Maxwell: $\nabla^2 \mathbf{E} - \mu_0 \varepsilon_0 \partial^2 \mathbf{E}/\partial t^2 = 0$; plane waves $\mathbf{E} = \mathbf{E}_0 \mathbf{e}^{\mathrm{i}(k\mathbf{x} - \omega t)}$ enforce $\omega = c |\mathbf{k}|$ with $c = (\mu_0 \varepsilon_0)^{-1/2}$.

\subsection*{Energy---momentum (for later optics power).}

Clarification (measurement bridge): We are not importing ``energy'' or ``momentum'' as primitives. In this pillar they arise from the same geometric action via Noether; the familiar energy density $u = \frac{1}{2}(\varepsilon_0 E^2 + B^2/\mu_0)$ and Poynting vector $\mathbf{S} = (1/\mu_0) \mathbf{E}\times\mathbf{B}$ are used only as measurement bridges to lab observables (power, intensity, flux).

The only dimensional scale admitted is $S_0 = \hbar$ (set once at the electron anchor); vacuum identities $(k_e = 1/(4\pi \varepsilon_0)$, $c = (\mu_0 \varepsilon_0)^{-1/2})$ are limit locks, not inputs to the dynamics.

$u = \frac{1}{2}(\varepsilon_0 E^2 + B^2/\mu_0)$, $\mathbf{S}^* = (1/\mu_0) \mathbf{E}^* \times \mathbf{B}^*$. These transport relations are used in diffraction and imaging budgets.

The stress---energy from S$_{\text{field}}$ yields the Poynting vector $\mathbf{S} = (1/\mu_0) \mathbf{E}\times\mathbf{B}$ and energy density $u = \frac{1}{2}(\varepsilon_0 E^2 + B^2/\mu_0)$, used later for optical power budgets (Appendix §1.5).

\subsection*{Energy---momentum \& Poynting (one line).}

Start: $\delta(S_{\text{field}} + S_{\text{int}}) = 0$ with $S_{\text{field}} = \frac{1}{2} \int \mathbf{F} \wedge \mathbf{*F}$, $S_{\text{int}} = \int \mathbf{J} \cdot \mathbf{A} \, \mathrm{d}^4x$. Variation $\delta \mathbf{A}$ gives $\mathrm{d}\mathbf{*F} = \mathbf{J}$; closure $\mathrm{d}\mathbf{F} = 0$ is geometric. In components: $\nabla\cdot\mathbf{B}=0$, $\nabla\times\mathbf{E}=-\partial \mathbf{B}/\partial t$, $\nabla \cdot \mathbf{E} = \rho/\varepsilon_0$, $\nabla\times\mathbf{B}=\mu_0 \mathbf{J}+\mu_0\varepsilon_0 \partial \mathbf{E}/\partial t$. Taking $\nabla \times$ of Faraday and using Ampère---Maxwell yields the wave equation $\nabla^2 \mathbf{E} - \mu_0\varepsilon_0 \partial^2 \mathbf{E}/\partial t^2 = 0$ with $c = (\mu_0\varepsilon_0)^{-1/2}$. Plane-wave ansatz $\mathbf{E}=\mathbf{E}_0 \mathbf{e}^{\mathrm{i}(\mathbf{k}\mathbf{x} - \omega t)}$ enforces $\omega=ck$.

\subsection*{From action to Maxwell (steps).}

We recover the Maxwell set under weak curvature / slow variation.

Gauss-B: $\nabla \cdot \mathbf{B} = 0$ (vector identity).

Faraday's Law: $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$.

Gauss-E: $\nabla \cdot \mathbf{E} = \rho/\varepsilon_0$; in statics gives Poisson's equation.

Ampère---Maxwell: $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \partial \mathbf{E}/\partial t$.

Wave equation: $\nabla^2 \mathbf{E} - \mu_0\varepsilon_0 \partial^2 \mathbf{E}/\partial t^2 = 0$, speed $c = (\mu_0\varepsilon_0)^{-1/2}$.

\begin{mdframed}
\textbf{Box 1 (Maxwell's Equations, see Math Appendix §1):}
\begin{align*}
\nabla \cdot \mathbf{E} &= \rho/\varepsilon_0\\
\nabla \cdot \mathbf{B} &= 0\\
\nabla \times \mathbf{E} &= -\partial \mathbf{B}/\partial t\\
\nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \partial \mathbf{E}/\partial t
\end{align*}
\end{mdframed}

\begin{mdframed}
\textbf{Box 2 (EM Wave Equation, see Math Appendix §1.5):}
\[
\nabla^2 \mathbf{E} - (1/c^2) \partial^2 \mathbf{E}/\partial t^2 = 0
\]
\end{mdframed}

\section*{2. Lorentz Force and Coulomb Limit}

\subsection*{Minimal coupling derivation.}

$L = \frac{1}{2} m v^2 + q \mathbf{v} \cdot \mathbf{A} - q \Phi \Rightarrow \mathbf{m a} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B})$. Gauge choices (Coulomb/Lorentz) do not change observables.

\subsection*{Poisson $\to$ Coulomb.}

Static: $\nabla \cdot \mathbf{E} = \rho/\varepsilon_0$, $\mathbf{E} = -\nabla\Phi = \nabla^2\Phi = -\rho/\varepsilon_0$. For $\rho = q \, \delta^3(\mathbf{x})$: $\Phi = k_e q/r$, $\mathbf{E} = k_e q \mathbf{r}/r^2$, $k_e = 1/(4\pi \varepsilon_0)$.

Static: $\nabla \cdot \mathbf{E} = \rho/\varepsilon_0$ and $\mathbf{E} = -\nabla\Phi = \nabla^2\Phi = -\rho/\varepsilon_0$. Point charge $\rho = q \, \delta(\mathbf{x}) \Rightarrow \Phi = k_e q/r$ and $\mathbf{E} = k_e q \mathbf{r} / r^2$ with $k_e = 1/(4\pi \varepsilon_0)$ (Appendix §2.2).

\subsection*{Coulomb from Poisson.}

With $L = \frac{1}{2} m v^2 + q \mathbf{v} \cdot \mathbf{A} - q\Phi$, Euler---Lagrange gives $\mathbf{m a} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})$. Gauge invariance $\mathbf{A} \to \mathbf{A}+\nabla\chi$, $\Phi \to \Phi - \partial \chi/\partial t$ leaves equations invariant (Appendix §2).

\subsection*{Minimal coupling $\to$ Lorentz force.}

Start from minimal-coupling Lagrangian: $L = \frac{1}{2}m v^2 + q \mathbf{v} \cdot \mathbf{A} - q\Phi$.

Gauge invariance is preserved under transformations $\Phi \to \Phi - \partial \chi/\partial t$, $\mathbf{A} \to \mathbf{A}+\nabla\chi$.

Euler---Lagrange equations yield: $\mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})$.

\begin{mdframed}
\textbf{Box 3 (Lorentz Force, see Math Appendix §2):}
\[
\mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})
\]
\end{mdframed}

Static point source solution of Poisson's equation gives Coulomb's law.

\begin{mdframed}
\textbf{Box 4 (Coulomb's Law, see Math Appendix §2.2):}
\[
\mathbf{F} = k_e q_1 q_2 / r^2
\]
\end{mdframed}

\section*{3. Propagation in Media \& Rays}

\subsection*{Constitutive relations.}

Linear, isotropic media: $\mathbf{D} = \varepsilon \mathbf{E}$, $\mathbf{B} = \mu \mathbf{H}$; refractive index $n^2 = \varepsilon_r \mu_r$. In vacuum, $\varepsilon_r=\mu_r=1$.

\subsection*{Eikonal and ray equation.}

Ansatz $\mathbf{E}(\mathbf{x}) = \mathbf{A}(\mathbf{x}) \mathbf{e}^{\mathrm{i} \mathbf{S}(\mathbf{x})/\lambda}$. Leading order: $|\nabla \mathbf{S}|^2 = n^2$ (eikonal). Rays follow $\mathrm{d}/\mathrm{d}s$ ($\mathbf{n} \hat{\mathbf{n}}$); curvature balance $1/a = |\nabla \perp \ln n|$.

\subsection*{Interfaces (Snell, Fresnel, TIR, Brewster).}

Snell: $n_1 \sin\theta_1 = n_2 \sin\theta_2$. Fresnel: $r_s = (n_1 \cos\theta_1 - n_2 \cos\theta_2)/(n_1 \cos\theta_1 + n_2 \cos\theta_2)$, $r_p = (n_2 \cos\theta_1 - n_1 \cos\theta_2)/(n_2 \cos\theta_1 + n_1 \cos\theta_2)$. Brewster: $\tan\theta_B = n_2/n_1$ (for p-pol). TIR: $\sin\theta_c = n_2/n_1$ ($n_1>n_2$).

\subsection*{Wavelength convention.}

Spectroscopy uses vacuum wavelengths unless stated; air values differ by $n(\lambda)$. Balmer anchors follow NIST ASD vacuum lines; reconciled in Calibration.

Spectroscopy uses vacuum wavelengths unless stated; air values differ by $n(\lambda)$. Numeric anchors (e.g., Balmer) follow NIST ASD vacuum lines; reconciliation appears in the Calibration doc.

\subsection*{Vacuum vs air wavelengths (convention).}

Tangential E,H and normal D,B continuity give Fresnel coefficients; Brewster angle $\tan\theta_B = n_2/n_1$; TIR at $\sin\theta_c = n_2/n_1$. These feed the polarization section (Appendix §3).

\subsection*{Fresnel/Brewster boundary set.}

In linear media: $\mathbf{D} = \varepsilon\mathbf{E}$, $\mathbf{B} = \mu\mathbf{H}$ with $n^2 = \varepsilon_r \mu_r$. Eikonal ansatz $\mathbf{E}(\mathbf{x}) = \mathbf{A}(\mathbf{x}) \mathbf{e}^{\mathrm{i} \mathbf{S}(\mathbf{x})/\lambda}$ yields $|\nabla \mathbf{S}|^2 = n^2$ and ray equation $\mathrm{d}/\mathrm{d}s$ ($\mathbf{n} \hat{\mathbf{n}}$) $=\nabla \perp \ln ||$. Snell's law $n_1 \sin\theta_1 = n_2 \sin\theta_2$ follows from stationary phase at interfaces (Appendix §3).

\subsection*{Constitutive \& eikonal chain.}

Constitutive relations: $\mathbf{D} = \varepsilon\mathbf{E}$, $\mathbf{B} = \mu\mathbf{H}$; refractive index $n^2 = \varepsilon_r \mu_r$.

Eikonal derivation via WKB ansatz gives $|\nabla \mathbf{S}|^2 = n^2$.

Snell's Law: $n_1 \sin\theta_1 = n_2 \sin\theta_2$.

Curvature balance: $1/a = |\nabla \ln n||$.

Boundary conditions (tangential E,H; normal D,B) $\to$ Fresnel equations.

Brewster's angle: $\tan\theta_B = n_2/n_1$.

Critical angle for total internal reflection: $\sin\theta_c = n_2/n_1$.

\begin{mdframed}
\textbf{Box 5 (Snell's Law, see Math Appendix §3):}
\[
n_1 \sin\theta_1 = n_2 \sin\theta_2
\]
\end{mdframed}

\section*{5. Polarization Transport}

\subsection*{Representations.}

Jones vectors for fully coherent beams; Stokes/Mueller for partial polarization. Stokes $\mathbf{S} = (S_0,S_1,S_2,S_3)$; Mueller matrix M gives S$_{\text{out}} = \mathbf{M} \mathbf{S}_{\text{in}}$.

\subsection*{Medium effects.}

Faraday rotation (small-signal): $\theta = \mathbf{V} \mathbf{B} L$; birefringence via $\Delta n$ induces phase retardance $\delta = 2\pi \Delta n L / \lambda$. Link to e$_{\text{eff}}$, $\mu_{\text{eff}}$ as geometry-to-medium mapping.

Jones: $\mathbf{E}^* = (\mathbf{E}_x, \mathbf{E}_y)$. Stokes $\mathbf{S} = (S_0,S_1,S_2,S_3)$ with Mueller matrices M so that S$_{\text{out}} = \mathbf{M} \mathbf{S}_{\text{in}}$. Faraday rotation in magnetoactive media: $\theta = \mathbf{V} \mathbf{B}$ (small-signal). These models parameterize polarization changes induced by geometry via e$_{\text{eff}}$, $\mu_{\text{eff}}$ (Appendix §5).

\subsection*{Jones/Stokes/Mueller pipeline.}

Jones calculus: field as $(\mathbf{E}_x, \mathbf{E}_y)$.

Stokes parameters $S_0,S_1,S_2,S_3$ define polarization state.

Mueller matrices connect lab measurements to polarization state.

Faraday rotation: $\theta = \mathbf{V}BL$.

\begin{mdframed}
\textbf{Box 8 (Faraday Rotation, see Math Appendix §5):}
\[
\theta = \mathbf{V} \mathbf{B} L
\]
\end{mdframed}

\section*{4. Wave Optics}

\subsection*{Kirchhoff/Huygens integral.}

Scalar diffraction integral reduces to Fresnel (near-field) and Fraunhofer (far-field) regimes by stationary-phase. Replace $\lambda \to \lambda/n$ in media.

\subsection*{Canonical patterns.}

Single slit: $I(\theta) = I_0 \left(\sin\beta/\beta\right)^2$, $\beta = (\pi a/\lambda) \sin\theta$. Double slit: fringe spacing $\Delta y = \lambda L / d$. Grating: $d \sin\theta_m = m \lambda$.

Kirchhoff integral yields Fresnel (near-field) and Fraunhofer (far-field) limits. Single-slit: $I(\theta) = I_0 \left(\sin\beta/\beta\right)^2$ with $\beta = (\pi a/\lambda) \sin\theta$. Double-slit: $\Delta y = \lambda L/d$; grating: $d \sin\theta_m = m \lambda$. These carry over to media by replacing $\lambda \to \lambda/n$ (Appendix §4---§4.3).

\subsection*{Huygens---Fresnel and regimes.}

Huygens---Fresnel integral $\to$ diffraction patterns.

Single-slit diffraction minima: a $\sin\theta = m\lambda$.

Double-slit interference spacing: $\Delta y = \lambda L/d$.

Diffraction grating: $d \sin\theta_m = m\lambda$.

\begin{mdframed}
\textbf{Box 6 (Double-Slit Interference, see Math Appendix §4):}
\[
\Delta y = \lambda L / d
\]
\end{mdframed}

\begin{mdframed}
\textbf{Box 7 (Diffraction Grating, see Math Appendix §4.3):}
\[
d \sin\theta_m = m\lambda
\]
\end{mdframed}

\section*{6. Imaging (Paraxial Optics)}

\subsection*{ABCD law.}

Ray vector $(x, \theta)$ propagates by $[x_2; \theta_2] = [[A,B],[C,D]] [x_1; \theta_1]$ with determinant AD $-$ BC = 1. Thin lens: $1/f = 1/d_o + 1/d_i$; magnification $M = -d_i/d_o$. GRIN handled by slice composition.

Paraxial propagation through elements via ABCD matrices; thin-lens equation $1/f = 1/d_o + 1/d_i$; magnification $M = -d_i/d_o$. Determinant constraint AD $-$ BC = 1 preserves optical invariant; extends to GRIN via slice composition (Appendix §6).

\subsection*{ABCD matrices \& invariants.}

Ray transfer (ABCD) matrices in paraxial optics.

Thin lens equation: $1/f = 1/d_o + 1/d_i$.

Magnification: $M = -d_i/d_o$.

\begin{mdframed}
\textbf{Box 9 (Thin Lens Equation, see Math Appendix §6):}
\[
1/f = 1/d_o + 1/d_i
\]
\end{mdframed}

\section*{7. Spectroscopy Anchors}

\subsection*{Rydberg formula.}

$1/\lambda = R_\infty (1/n_1^2 - 1/n_2^2)$. Balmer anchors (vacuum; NIST ASD): H$\alpha$ 656.281 nm, H$\beta$ 486.133 nm, H$\gamma$ 434.047 nm.

\subsection*{Ratio-first use.}

Use ratios (e.g., H$\beta$/H$\alpha$ = 20/27 ideal) when they beat absolute calibration error; carry uncertainties through to acceptance bands.

Hydrogen: $1/\lambda = R_\infty (1/n_1^2 - 1/n_2^2)$. Balmer anchors (vacuum; NIST ASD): H$\alpha$ 656.281 nm, H$\beta$ 486.133 nm, H$\gamma$ 434.047 nm. Prefer ratio-first predictions (e.g., H$\beta$ from H$\alpha$ via 20/27) when they beat absolute calibration error (Appendix §7).

\subsection*{Rydberg/anchors (vacuum).}

Hydrogen spectrum via Rydberg formula: $1/\lambda = R_\infty (1/n_1^2 - 1/n_2^2)$.

Balmer anchors: H$\alpha$, H$\beta$, H$\gamma$. Ratio-first prediction e.g. H$\beta$ from H$\alpha$ via 20/27.

\begin{mdframed}
\textbf{Box 10 (Rydberg Formula, see Math Appendix §7):}
\[
1/\lambda = R_\infty (1/n_1^2 - 1/n_2^2)
\]
\end{mdframed}

\section*{8. Corrections (Torsion/Shear)}

\subsection*{Parameterization \& bounds.}

Small, dimensionless budgets enter phase and transport: $\delta\phi = \phi \left(\alpha_\mathrm{T} T + \alpha_\mathrm{S} S\right)\mathrm{d}l$; $\delta n/n \approx \beta_\mathrm{T} T + \beta_\mathrm{S} S$. Bounds traced by closure tolerance $J_c$ vanish in pure vacuum.

Introduce small, dimensionless budgets for torsion/shear---entering as phase and transport corrections: $\delta\phi = \phi \left(\alpha_\mathrm{T} T + \alpha_\mathrm{S} S\right)\mathrm{d}l$, $\delta n/n \approx \beta_\mathrm{T} T + \beta_\mathrm{S} S$. These are bounded by closure tolerance $J_c$ and are testable; they vanish in pure vacuum (Appendix §8).

\subsection*{Bounded perturbations (how).}

Torsion/shear operators define bounded corrections.

Perturbations appear in refractive index, boundary phases, polarization transport.

Bounds provide falsifiers if exceeded.

\section*{9. Falsifiers \& Lab-Math Hooks}

\subsection*{Neutral-rotor fingerprint.}

Composite neutral (monopole sum zero) aligned to global caustic plane; spin picks orientation. With $\varepsilon>0$, half-cycles aligned with $+\hat{\mathbf{n}}$ are weighted slightly more than opposite half-cycles, giving $(\Delta p) \times \varepsilon \cos\theta$ (push/zero/pull). Spin-flip reverses sign; 90 tilt is a true null. Absence sets a sharp upper bound; presence is new physics vs. point-charge intuition.

A composite neutral (net monopole zero) still aligns to the global caustic plane. Spinning a non-magnetic rotor in that plane gates orientation; the model predicts a tiny cycle-odd response: $(\Delta p) \times \varepsilon \cos\theta$ (push/zero/pull). Spin-flip reverses sign; wrong facing yields null. Absence sets a sharp upper bound; presence is a clean falsifier of ``point-charge only'' intuition (Appendix §9).

\subsection*{Neutral-rotor (explicit).}

Neutral rotor predicts orientation-gated forces.

Push / zero / pull triad arises from asymmetry $\times \varepsilon \cos\theta$.

Spin-flip reverses sign. Null cones appear as discrete angular zeros.

Spectroscopy ratio bands: calibrated vs. ratio-first predictions.

\begin{mdframed}
\textbf{Box 11 (Neutral Rotor Prediction, see Math Appendix §9):}
\[
(\Delta p) \propto \sigma \cos\theta
\]
\end{mdframed}

\section*{10. Appendix Cross-Refs}

\subsection*{Exact mapping to Math Appendix.}

Maxwell \& wave eqn (§1---§1.5); Lorentz \& Coulomb (§2---§2.2); Snell/Fresnel/GRIN (§3); Fresnel---Kirchhoff \& gratings (§4---§4.3); Faraday (§5); Paraxial (§6); Rydberg/spectroscopy (§7); Torsion/Shear bounds (§8); Neutral-rotor falsifier (§9). Keep numbers synchronized with the main Appendix index.

\subsection*{Lock statement.}

Classical vacuum relations appear here only as limit checks/locks; the pillar's dynamics are derived from geometry with no new dimensional scales beyond $S_0 = \hbar$ (set once at the electron anchor; no retune).

Classical vacuum relations appear here only as limit checks/locks; the pillar's dynamics are derived from geometry with no new dimensional scales beyond $S_0 = \hbar$ (set once at the electron anchor; no retune).

\subsection*{Exact mapping.}

Full Fresnel derivation, Kirchhoff integral, Jones$\to$Stokes conversions, air-index models.

Worked examples: Coulomb potential, Fresnel/Brewster, diffraction, Faraday rotation, GRIN curvature.

Optional: thin-film transfer matrices.

Wavelength convention: spectroscopy uses vacuum wavelengths unless stated; air values differ by $n$. Numeric anchors (e.g., Balmer) follow the NIST ASD vacuum list and are reconciled in the pillar Calibration doc.

Balmer anchors (vacuum; NIST ASD): H$\alpha$ 656.281 nm, H$\beta$ 486.133 nm, H$\gamma$ 434.047 nm. Ratio-first predictions (e.g., H$\beta$ from H$\alpha$ via 20/27) are preferred when they beat absolute calibration error.

Cross-References to Foundation \& Appendix (exact): Maxwell set \& wave equation (Math Appendix §1---§1.5); Lorentz force \& Coulomb (§2---§2.2); Snell/Fresnel/GRIN (§3); Fresnel---Kirchhoff \& gratings (§4---§4.3); Faraday (polarization) (§5); Paraxial optics (§6); Rydberg/spectroscopy (§7); Torsion/Sheaf corrections \& bounds (§8); Neutral-rotor falsifier (§9). Keep these section numbers synchronized with the main Appendix index.

Classical vacuum relations appear here only as limit checks/locks; the pillar's dynamics are derived from geometry with no new dimensional scales beyond $S_0 = \hbar$ (set once at the electron anchor; no retune).

\end{document}
