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\title{Electromagnetism Pillar --- Calibration}
\author{}
\date{}

\begin{document}

\section*{Sample Deviation Reporting Template}

\begin{tabular}{|l|l|l|}
\hline
Strategy Used & $J_c$ Impact & Error (\%) \\
\hline
Harmonic-Loop & $\Delta J_c = 0.002$ & 0.08\% \\
\hline
\end{tabular}

\maketitle

\section*{Purpose}

This document establishes the canonical calibration for the Electromagnetism pillar. It carries forward explicit derivations and constants from the foundation, provides detailed calibration strategies, and fixes the locked pillar calibration. This file serves as the definitive calibration reference for all Electromagnetism-related branches, just as the foundation documents serve as canonical references for the trunk.

\section*{Restating Foundation Anchors (Explicit)}

From the Mathematical Bridge and Calibration foundations, the following constants and relations are carried forward:

\begin{itemize}
\item \textbf{*Torsion Constant ($T_s$):*} $T_s = \sigma \, s \, c$

\item \textbf{*Shear Constant ($\sigma_s$):*} $\sigma_s = (m_e c^2) / (4\pi^2 h)$

\item \textbf{*Loop Action Constant ($S_0$):*} fixed at the electron Compton loop: $\oint A \, ds = h$

\item \textbf{*Closure Index ($J_c$):*} fractional tolerance governing loop stability.

\item \textbf{*Electron Anchor:*} locks the global action scale.

\item \textbf{*Hydrogen Anchor:*} locks wavelength/energy ladder (Balmer series).

\item \textbf{*Muon Anchor:*} locks time/lifetime using circulation frequency $\Omega_\mu$.
\end{itemize}

These anchors are immutable at the foundation level and must be used without retuning downstream. (h is the reduced Planck constant, $1.05457 \times 10^{-34}$ Js). ($\Omega_\mu$ is the circulation frequency proportional to muon lifetime, see Mathematical Bridge).

\section*{Calibration Strategy Options (scalings / normalized forms).}

Three major strategies are defined at the pillar level:

\begin{enumerate}
\item \textbf{*Charge-Normalized Strategy:*}

Equation: $q_{\text{ref}} \propto \tau \cdot A_{\text{disp}}$

- Strength: aligns directly with Coulomb's constant.
- Weakness: obscures composite loop structures; e.g., loop--loop interactions may deviate $>1\%$ from prediction.

\item \textbf{*Field-Line Strategy:*}

Equation: $B \propto \tau \times \cos(\theta)$

- Strength: emphasizes preferred-axis orientation.
- Weakness: indirect normalization; multi-axis systems may require $10^2$ additional computation steps.

\item \textbf{*Harmonic-Loop Strategy:*}

Equation: $f_n / n \propto \tau / \sigma$

- Strength: integrates naturally with photon harmonics.
- Weakness: computationally intensive; solving $n \times (\tau / \sigma)$ for $n > 10^4$ increases runtime by $\sim 10^2$.
\end{enumerate}

\section*{Glossary of Key Variables}

\begin{itemize}
\item \textbf{*$\tau$ (Torsion parameter):*} describes loop rotational stability; tied to $\sigma_s$ and $T_s$.

\item \textbf{*$\sigma$ (Shear constant):*} governs tearing resistance of space; must be finite, non-zero.

\item \textbf{*$A_{\text{disp}}$ (Display-Area):*} the transverse obscured area of a propagating void; foundation observable.

\item \textbf{*$\omega_{\text{loop}}$:*} circulation frequency of a closed loop, proportional to energy content.
\end{itemize}

\section*{Strategy Comparison Table}

\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Strategy} & \textbf{Equation} & \textbf{Best Use Case} & \textbf{Computational Cost} & \textbf{Foundation Tie-In} \\
\hline
Charge- & $q_{\text{ref}} = \tau \times$ & Classical fields & Low & Direct link to \\
Normalized & $A_{\text{disp}}$ & & & Coulomb's constant \\
\hline
Field-Line & $B \propto \tau \times \cos(\theta)$ & Orientation studies & Medium & Preferred-axis torsion \\
& & & & alignment \\
\hline
Harmonic-Loop & $f_n / n \propto (\tau / \sigma)$ & Photon harmonics, & High & Loop harmonic \\
& & composites & & derivations \\
\hline
\end{tabular}

\section*{Locked Pillar Calibration (Canonical---Vacuum)}

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

\begin{itemize}
\item \textbf{Action lock:} $S_0 = h$

\item \textbf{Speed of light (SI lock):} $c$ exact; $c = (\mu_0 \varepsilon_0)^{-1/2}$

\item \textbf{Vacuum constants (SI):} $k_e = 1/(4\pi \varepsilon_0)$ and $\mu_0 \varepsilon_0 = 1/c^2$

\item \textbf{Coulomb field (vacuum):} $E(r) = (q / (4\pi \varepsilon_0 r^2)) \cdot \hat{r}$

\item \textbf{Maxwell limit (vacuum):} $\nabla \cdot E = \rho/\varepsilon_0$; $\nabla \cdot B = 0$; $\nabla \times E = -\partial B/\partial t$; $\nabla \times B = \mu_0 \varepsilon_0 \partial E/\partial t$

\item \textbf{(Optional) Steady-current field:} $B(r) = (\mu_0 / 4\pi) \, [(\mathbf{I} \, d\mathbf{l}) \times \mathbf{r}] / r^2$ (Biot---Savart/Ampère, vacuum)

\item \textbf{Note:} In vacuum, $\mu_0$ and $\varepsilon_0$ are constants (no geometry dependence). Use $\mu_{\text{eff}}, \varepsilon_{\text{eff}}$ only in media.
\end{itemize}

\section*{Locked Pillar Constants Table}

\begin{tabular}{|l|l|l|l|}
\hline
\textbf{Constant} & \textbf{Expression} & \textbf{Foundation Reference} & \textbf{Verification Source} \\
\hline
$k_e$ & $1 / (4\pi \varepsilon_0)$ & Vacuum identity (SI) & CODATA 2022 \\
\hline
$\mu_0$ & Vacuum constant (no geometry dependence) & Vacuum constant (SI); use $\mu_{\text{eff}}$ in media & CODATA/NIST \\
\hline
$\varepsilon_0$ & Vacuum constant (no geometry dependence) & Vacuum constant (SI); use $\varepsilon_{\text{eff}}$ in media & CODATA/NIST \\
\hline
$c$ & $(\mu_0 \varepsilon_0)^{-1/2}$ & Vacuum identity (SI) & Precision interferometry / CODATA \\
\hline
$\mu_0 \varepsilon_0$ & $1 / c^2$ & Vacuum identity (SI) & CODATA/NIST \\
\hline
\end{tabular}

\section*{Worked Example: Coulomb Constant}

(Here $S_0 = h$ is fixed at the electron anchor; no retune.)

We adopt vacuum $\mu_0, \varepsilon_0$ and the identity $k_e = 1/(4\pi \varepsilon_0)$. Using CODATA 2022 values:

\[
k_e = 1/(4\pi \varepsilon_0) \approx 8.98755 \times 10^9 \, \text{N}\cdot\text{m}^2/\text{C}^2.
\]

We treat this as an acceptance target for pillar predictions. Any emergent/geometry mapping of permittivity/permeability belongs to effective parameters $\varepsilon_{\text{eff}}, \mu_{\text{eff}}$ in media, not to vacuum $\varepsilon_0, \mu_0$.

\section*{Extended Locked Calibration Clarifications}

\begin{itemize}
\item \textbf{*Fine-Structure Constant Derivation:*} $\alpha = e^2 / (4\pi \varepsilon_0 \hbar c) = 1/137.036$, consistent with CODATA 2022.

\item \textbf{*Derived Constants:*} The fine-structure constant ($\alpha \approx 1/137$) and Bohr magneton ($\mu_B$) are not separately locked but derivable from the locked set ($e, h, c, \mu_0, \varepsilon_0$).

\item \textbf{*Reporting Standard for Branch Deviations:*} Any branch using an alternate calibration strategy must specify: (i) strategy used, (ii) impact on $J_c$, (iii) propagated error estimates.

\item \textbf{*Applicability Limits:*} Locked calibration is validated for $E < 10^{12}$ V/m and above atomic scales; extreme regimes must reference falsifiable predictions in the Verification Document.
\end{itemize}

\section*{Guidance for Branches}

Branches should default to locked calibration. Alternate strategies may be explored but must be explicitly documented. All falsifiable predictions at the branch level must reference which calibration strategy is in use. Branches must also reverify locked parameters against updated CODATA/PDG standards at least every 2 years.

\end{document}
