\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{geometry}
\geometry{margin=1in}
\usepackage{parskip}

\title{Particle Mechanics --- Math Appendix}
\author{}
\date{}

\begin{document}

\maketitle

\section*{§0 Preliminaries: Kernels, Spaces, Notation}

\textbf{Scope.} This appendix collects the VMS-first preliminaries used across the Particle Mechanics pillar. At each step we state kernels, definitions, assumptions, and lemmas explicitly. Classical forms are used only as dictionaries for comparison/validation. Equations are numbered by section (§n.m); all symbols are listed in §9.

\subsection*{0.1 Routes, Admissibility, and Action Phase}

A route $r$ is a piecewise $C^1$ closed loop $x(s) \subset \mathbb{R}^1$ (or the appropriate manifold) parameterized by arc-length $s \in [0, \ell]$, with $x(0) = x(\ell)$. Admissibility requires single-valued action phase on closure. We define the phase and its loop increment:
\begin{align}
\phi[x] &= S[x] / h \\
\Delta\phi_{\text{loop}} &= (1/h) \cdot \oint p \cdot dl
\end{align}

Here $p$ is the canonical momentum 1-form appearing in the action $S[x]$. Admissibility enforces $\Delta\phi_{\text{loop}} = 2\pi n$, $n \in \mathbb{Z}$ (see §1).

\subsection*{0.2 Core Torsion-Driven Kernel}

The local energy density decomposes into geometry budgets. In this pillar we use the torsion-driven form with a single calibration $\kappa$:
\begin{align}
\epsilon_T(s) &= \kappa \cdot T[x(s)], \quad T \geq 0 \text{ (dimensionless)} \\
E[x] &= \oint \epsilon_T(s) \, ds = \kappa \cdot \oint T[x(s)] \, ds \\
\end{align}

Define the loop-average torsion budget $\tau$ and note that it is dimensionless by construction:
\begin{align}
\tau &\equiv (1/\ell) \cdot \oint T \, ds
\end{align}

Then $E[x] = \kappa \tau \ell$ and, with $C \equiv \kappa/c^2$, the rest mass is $m = C \tau \ell$ (see §3 for scaling/ratios).

\subsection*{0.3 Small-Parameter Band and Applicability}

\textbf{Assumption A0 (band).} Near-stationary admissible loops; weak environmental gradients; time-invariant kernel over the measurement window. Operationally (see §8 for quantitative gates): small temporal variation, small spatial gradients, short memory, and incoherent channels.

\subsection*{0.4 Reparameterization and Scaling (Lemma L0)}

\textbf{Lemma L0.} Reparameterizations preserving closure leave $\Delta\phi_{\text{loop}}$ invariant; budget integrals scale with $\ell$ as in §3.

\subsection*{0.5 Dictionaries (Comparison/Validation Only)}

Where appropriate, we compare to classical forms solely as dictionaries: Bohr--Sommerfeld quantization ($\oint p \cdot dl = 2\pi n$, §1), hydrogenic Coulomb closure ($r_n$, §7), liquid-drop scalings (A, A\textsuperscript{2/3}, §6), and decay widths from action gaps (§5). These are not axioms of VMS derivations; they serve to calibrate and validate.

\subsection*{0.6 Notation Summary (Pointer to §9)}

Symbols appearing in §0--§8 are gathered in §9 with units and roles. Key items used here: $\ell$ (loop length), $s$ (arc length), $T$ (dimensionless torsion budget), $\tau$ (loop-average of $T$), $\kappa$ (calibration constant), $C \equiv \kappa/c^2$ (mass-scale constant), $S$ (action), $p$ (canonical momentum), $h$ (reduced Planck constant).

\subsection*{0.7 Reporting Checklist}

When invoking preliminaries in later sections, report:

\begin{enumerate}
\item The kernel used (here torsion-driven) and the value/reference for $\kappa$ (or C).
\item Loop regularity (piecewise $C^1$), closure condition, and any boundary twist phases (see §1.4).
\item Band checks (stationarity, weak gradients, short memory, incoherence) with thresholds cited from §8.
\item Dictionary used (if any) and whether it is only for validation/calibration.
\end{enumerate}

\section*{§1 Loop Closure $\Rightarrow$ Quantization \& Spin}

\textbf{Scope.} We expand the loop-closure quantization and the spinor double-cover argument. Admissibility enforces single-valued phase on a closed loop; for a uniform ring this yields discrete momenta and energies. Frame (tetrad) closure occurs only after 4$\pi$, giving the spin-1/2 sign flip and the half-integer/integer spin families.

\subsection*{1.1 Admissibility and Bohr--Sommerfeld Form}

Admissibility requires that the total phase accumulated around a closed loop is an integer multiple of 2$\pi$. Equivalently, the action integral over one traverse is quantized:
\begin{align}
(1/h) \cdot \oint p \cdot dl = 2\pi \cdot n, \quad n \in \mathbb{Z}
\end{align}

\subsection*{1.2 Uniform Ring Specialization}

For a ring where $|p|$ is constant and $p$ is tangent to $dl$ along the loop of length $\ell$:
\begin{align}
\oint p \cdot dl &= p \cdot \oint dl = p \cdot \ell \Rightarrow p \cdot \ell = 2\pi \hbar n \\
k_n &= 2\pi n / \ell, \quad p_n = \hbar k_n = 2\pi \hbar n / \ell \\
E_{n^*}nr &= p_n^2 / (2m), \quad E_{n^*}rel = \sqrt( p_n^2 c^2 + m^2 c^4 )
\end{align}

\textbf{Assumption A1.} The loop is piecewise $C^1$; $p$ is constant along the ring; endpoints identify as $x(0) = x(\ell)$.

\subsection*{1.3 Angular Momentum and Ring Current}

Let the ring have radius $R$ with $\ell = 2\pi R$. Then $k_n = n/R$ and $L_z = R \cdot p_n = R \cdot (\hbar n / R) = n\hbar$:
\begin{align}
L_z = R \cdot p_n = R \cdot (\hbar n / R) = n\hbar
\end{align}

If a charge $q$ circulates with speed $v_n = p_n/m$ on the ring, the current $I_n$ and magnetic moment $\mu_n$ scale as:
\begin{align}
I_n = q \cdot v_n / (2\pi R), \quad \mu_n = I_n \cdot (\pi R^2) = (q \hbar n) / (2m) \quad \text{(uniform ring)}
\end{align}

\subsection*{1.4 Boundary-Condition Variants (Twisted Closures)}

If the loop carries a uniform twist phase $\chi$ upon closure, $\psi(\ell) = e^{i(\chi + x(0))} \psi(0)$, admissibility yields:
\begin{align}
p \cdot \ell = \hbar (2\pi n + \chi) \Rightarrow k_n = (2\pi n + \chi) / \ell
\end{align}

$\chi = 0$ recovers periodic boundary condition; $\chi = \pi$ gives anti-periodic levels (half-offset). External gauge flux enters as an effective $\chi$ (Aharonov--Bohm dictionary). A uniform flux $\Phi$ through the ring induces $\chi = 2\pi \Phi / \Phi_0$, with $\Phi_0 = h/q$; thus $p\ell = \hbar (2\pi n + \chi)$.

\subsection*{1.5 Spin from Double-Cover Closure (Frame Rotation)}

A frame (tetrad) rotation by angle $\theta$ about axis $\hat{n}$ acts on a spin-1/2 state via:
\begin{align}
U(\theta) &= \exp[ -i (\theta/2) \cdot \sigma \cdot \hat{n} ] \\
U(2\pi) &= -I, \quad U(4\pi) = +I
\end{align}

Thus a 2$\pi$ rotation flips the sign of a spinor; full admissible frame closure requires 4$\pi$. Label spins by $s = m/2$ with $m \in \mathbb{Z}$ (odd $m \to$ half-integer; even $m \to$ integer):
\begin{align}
\text{spin} = m / 2, \quad m \in \mathbb{Z} \quad (\text{odd} \to 1/2, 3/2, \ldots \text{ ; even} \to 0, 1, 2, \ldots)
\end{align}

\textbf{Lemma L1.} The 2$\pi$ sign flip indicates the double cover $\mathrm{SU}(2) \to \mathrm{SO}(3)$; admissible frame closure at 4$\pi$ explains half-integer spin for odd $m$.

\subsection*{1.6 Worked Numeric (Illustrative)}

Take $\ell = 2\pi R$ with $R = 0.50$ nm and $m = m_e$ (electron). Then:
\begin{align}
k_1 = 1/R &= 2.0 \times 10^9 \text{ m}^{-1} \\
p_1 = \hbar k_1 &= 1.054 \times 10^{-34} \times 2.0 \times 10^9 = 2.11 \times 10^{-25} \text{ kg/m\,s} \\
E_{1*}nr = p_1^2 / (2 m_e) &= 2.45 \times 10^{-20} \text{ J} = 0.153 \text{ eV}
\end{align}

Higher $n$ scale as $n^2$ for non-relativistic energies ($E_{n*}(nr) = n^2 E_1^*(nr)$); relativistic energies follow the exact dispersion above.

\subsection*{1.7 Reporting Checklist}

When invoking loop-closure quantization and spin, report:

\begin{enumerate}
\item Loop length $\ell$ (or $R$), boundary condition $\chi$ (if any), and assumed regularity ($C^1$).
\item Derived $k_n$, $p_n$, and energies; specify non-relativistic vs relativistic use.
\item Spin labeling and whether spinor sign under 2$\pi$ rotation is relevant to the context.
\item Band checks from §8 (adiabaticity, weak gradients, short memory, incoherence) if the loop is embedded in a varying environment.
\end{enumerate}

\section*{§2 Conserved Charge \& Families from Admissibility (Ensemble)}

\textbf{Scope.} We formalize the admissible-route ensemble, introduce a uniform phase reweighting $\alpha$ coupled to a class indicator $Q$, derive conservation from fixed-point admissibility, and show how fractional charges stem from stabilizer-induced congruence classes. Dictionary identifications to U(1)/SU(2)/SU(3) are labels for comparison only.

\subsection*{2.1 Ensemble, Weights, and Reweighting}

Let $r$ index admissible routes with action $S[r]$ and class indicator $q[r] \in \{0,1\}$. Introduce the reweighting parameter $\alpha$ that couples to $Q[r] \equiv q[r]$. Weights are:
\begin{align}
W[r; \alpha] &\propto \exp\{ -\beta ( S[r] - \alpha \cdot Q[r] ) \}
\end{align}

Partition function and $\alpha$-response:
\begin{align}
Z(\alpha) &= \sum_r W[r; \alpha] \\
\partial \ln Z / \partial\alpha &= \beta \cdot \langle Q \rangle
\end{align}

\subsection*{2.2 Moments and Susceptibilities}

From standard cumulant relations:
\begin{align}
\langle Q \rangle &= (1/\beta) \cdot \partial \ln Z / \partial\alpha \\
\chi_Q &\equiv \partial \langle Q \rangle / \partial\alpha = \beta \cdot \mathrm{Var}(Q) \geq 0
\end{align}

Thus $\chi_Q$ is non-negative; sharp conservation in the admissible band corresponds to small $\mathrm{Var}(Q)$ under allowed dynamics (see §8 for band checks).

\subsection*{2.3 Lemma L2 (Conservation from Fixed-Point Admissibility)}

\textbf{Let $\mathcal{M}$ be the admissible route map advancing routes in time. If the admissible ensemble is a fixed point of $\mathcal{M}$ (class weights invariant), then:}
\begin{align}
d\langle Q \rangle / dt = 0
\end{align}

\textbf{Sketch.} Since class weights are time-stationary under $\mathcal{M}$, the distribution of $Q$ is invariant; therefore its expectation is constant. Equivalently, $\alpha$ is a conserved multiplier (Lagrange parameter) for the class constraint.

\subsection*{2.4 Fractional Charge from Stabilizer-Induced Congruence}

Assume admissible routes partition into $N_{\mathrm{tot}}$ congruence classes under a stabilizer $G$, with $n_{\mathrm{mod}}$ of them marked by a phase increment $2\pi/N_{\mathrm{tot}}$. Then the fractional charge (in units of $e$) is:
\begin{align}
Q/e = n_{\mathrm{mod}} / N_{\mathrm{tot}}, \quad 0 \leq n_{\mathrm{mod}} \leq N_{\mathrm{tot}}
\end{align}

\textbf{Example (thirds):}
\begin{align}
N_{\mathrm{tot}} = 3, \quad n_{\mathrm{mod}} \in \{3, 1, 2\} \Rightarrow Q/e \in \{-1, -1/3, +2/3\}
\end{align}

The sign is chosen by class orientation. This pattern generalizes to other $N_{\mathrm{tot}}$ with obvious replacements.

\subsection*{2.5 Assumption A2 (Stabilizer \& Admissibility)}

\textbf{A2.} The stabilizer acts transitively on the class partition; reweighting by $\alpha$ respects admissibility (no creation/annihilation of classes). Equivalently, the allowed dynamics do not change $N_{\mathrm{tot}}$ or the marked subset cardinality $n_{\mathrm{mod}}$.

\subsection*{2.6 Legendre Structure \& Generators}

In $Z(\alpha)$ is the cumulant-generating function for $Q$ with $\beta$ as inverse-temperature-like multiplier from counting (see §3 Thermo). The Legendre-dual potential is $\Omega((\langle Q \rangle)) = (1/\beta) \sup_\alpha [ \alpha(Q) - \ln Z(\alpha) ]$. Conservation across admissible dynamics implies $\alpha$ is constant in time, fixing $\langle Q \rangle$.
\begin{align}
\Omega(\langle Q \rangle) = (1/\beta) \cdot \sup_\alpha [ \alpha(Q) - \ln Z(\alpha) ]
\end{align}

\subsection*{2.7 Worked Numeric (Illustrative)}

Consider $N_{\mathrm{tot}} = 3$ with equal class weights at $\alpha = 0$. Mark $n_{\mathrm{mod}} = 1$ class. Then $Q/e = 1/3$. A small $\alpha$ shifts the weight of the marked class by a factor $\exp[\beta \alpha]$; the mean charge rises as:
\begin{align}
\langle Q/e \rangle (\alpha) = [ 1 \cdot (\beta \alpha) + 0 + 0 ] / [ e^{\beta\alpha} + 1 + 1 ] = 1 / ( 2 + e^{-\beta\alpha} )
\end{align}

For $\beta \alpha = 0.1$, $(\langle Q/e \rangle) \approx 1 / (2 + e^{-0.1}) = 0.341$, close to 1/3 for small $\alpha$; susceptibility at $\alpha=0$ is $\chi_Q = \beta \mathrm{Var}(Q) = \beta \cdot (2/9)$.

\subsection*{2.8 Dictionary D2 (Labels Only)}

Stabilizers may be labeled by familiar compact groups: U(1) for single-phase families, SU(2)/SU(3) for double/triple family structures. These are comparison labels; derivations above use only admissibility and ensemble structure.

\subsection*{2.9 Reporting Checklist}

When invoking conserved charges from admissibility, report:

\begin{enumerate}
\item Definition of $q[r]$ and how routes are partitioned into classes.
\item Evidence for fixed-point admissibility (class weights stationary under the route map).
\item Values of $N_{\mathrm{tot}}$ and $n_{\mathrm{mod}}$, and orientation/sign conventions.
\item Whether any $\alpha$-dependence was fitted (and the extracted $\chi_Q$ at $\alpha=0$).
\end{enumerate}

\section*{§3 Mass from Torsion Energy: Functional, Scaling, Ratios}

\textbf{Scope.} We make explicit the torsion-energy functional, its scaling properties, ratio laws across admissible loops, and winding-number scaling. A single calibration fixes $C \equiv \kappa/c^2$; all other masses follow from geometric ratios.

\subsection*{3.1 Defining-number Mass Functional}

Let $x(s)$ trace an admissible closed loop of length $\ell$, with a dimensionless local torsion budget $T[x(s)]$. Define the loop-average $\tau$ and torsion energy $E[x]$ as:
\begin{align}
\tau &\equiv (1/\ell) \cdot \oint T \, ds \\
E[x] &= \kappa \cdot \oint T \, ds = \kappa \cdot \tau \cdot \ell \\
m = E/c^2 &= (\kappa/c^2) \cdot \tau \cdot \ell = C \cdot \tau \cdot \ell
\end{align}

Here $C$ is a universal constant set by one anchor (e.g., the electron). Once $C$ is fixed, predictions are ratio-based.

\subsection*{3.2 Dimensional \& Invariance Notes}

The budget $T$ is dimensionless; $\kappa$ has dimensions of energy. The functional is reparameterization-invariant: under $s \mapsto \sigma$ with $ds = (ds/d\sigma) d\sigma$, the product $\tau\ell = (1/\ell)\oint T \, ds \cdot \ell = \oint T(\sigma) \, ds$ remains unchanged.

\subsection*{3.3 Ratio Law (Lemma L3)}

For two admissible loops 1 and 2 with the same $C$:
\begin{align}
m_2 / m_1 = (\tau_2 / \tau_1) \cdot (\ell_2 / \ell_1)
\end{align}

\textbf{Sketch.} From $m = C \tau \ell$, $C$ cancels in the ratio. Departures from unity in either $\tau$-ratios or $\ell$-ratios propagate linearly to mass ratios.

\subsection*{3.4 Winding Scaling (Lemma L4)}

If closure requires $n$ windings around the reference core, then in the smooth limit the length scales as $\ell_n = n \ell_1$. Stability sets $\tau_n$ by minimizing $E$ at fixed $n$; to first order $\tau_n = \tau_1$, otherwise $\tau_n = \tau_1 f(n)$.
\begin{align}
\ell_n = n \cdot \ell_1 \Rightarrow m_n / m_1 = (\tau_n / \tau_1) \cdot n
\end{align}

The function $f(n)$ captures higher-order geometric/interaction corrections; constraints on $f(n)$ follow from observed spectra.

\subsection*{3.5 Worked Numeric (Illustrative, Anchor m\_e)}

Anchor $C$ using the electron ($m_e$). Suppose $\tau_\mu / \tau_e = 1$ and $\ell_\mu / \ell_e = 206.77$; then:
\begin{align}
m_\mu / m_e = (\tau_\mu / \tau_e) \cdot (\ell_\mu / \ell_e) = 206.77
\end{align}

Observed $m_\mu / m_e = 206.768$; any deviation from the simple estimate bounds $f(n)$ and/or small $\tau$-ratio departures.

\subsection*{3.6 Relativistic Consistency}

Promote the rest mass to the torsion-energy form in the dispersion relation:
\begin{align}
E^2 = p^2 c^2 + m^2 c^4 = p^2 c^2 + (C \cdot \tau \cdot \ell \cdot c^2)^2
\end{align}

At $p=0$, $E=mc^2$ reproduces the torsion-energy result. For moving closures, the dispersion remains standard with $m$ set by $C \tau \ell$.

\subsection*{3.7 Variational Note (Stability and $\tau_n$)}

Stability at fixed $n$ solves $\delta E = 0$ subject to admissibility and closure constraints. Writing $T = T_o + \varepsilon \delta T$, first-order stationarity gives $\delta\tau = -(\tau/E) \delta \ell + \text{boundary-compatible corrections}$. In smooth bands (§8) these corrections are higher order, supporting $\tau_n = \tau_1$ to first order.

\subsection*{3.8 Reporting Checklist}

When quoting masses from torsion energy, report:

\begin{enumerate}
\item The anchor used to fix $C$ and its numerical value.
\item Measured/assumed ratios $\tau_2/\tau_1$ and $\ell_2/\ell_1$, with uncertainties.
\item If winding is invoked: the integer $n$ and any inferred $f(n)$.
\item Checks that the small-parameter band of §8 holds (adiabaticity, weak gradients, short memory, incoherence).
\end{enumerate}

\section*{§4 Mixing: Overlap--Detuning Diagonalization (2$\times$2 and 3$\times$3 Sketch)}

\textbf{Scope.} We present the 2$\times$2 overlap--detuning diagonalization with explicit eigenvalues, mixing angle, and eigenvectors, plus a compact 3$\times$3 sketch using successive Givens (Jacobi) rotations. Formulas are given in a unit-agnostic mass-functional; time evolution uses phases $\exp[-i m t / h]$.

\subsection*{4.1 2$\times$2 Subspace Hamiltonian}

Consider admissible closures $|1\rangle$ and $|2\rangle$ with detuning $\Delta = m_2 - m_1$ and overlap $\kappa$:
\begin{align}
H = [ m_1 & \kappa; \kappa & m_2 ], \quad \Delta = m_2 - m_1
\end{align}

\subsection*{4.2 Eigenvalues and Mixing Angle}

\begin{align}
m_\pm &= (m_1 + m_2)/2 \pm \sqrt{ (\Delta/2)^2 + \kappa^2 } \\
\tan(2\theta) &= 2\kappa / \Delta \\
R(\theta) &= [ \cos\theta & -\sin\theta; \sin\theta & \cos\theta ], \quad R^T H R = \mathrm{diag}(m_-, m_+)
\end{align}

\subsection*{4.3 Eigenvectors (Normalized)}

\begin{align}
|-\rangle &= \cos\theta |1\rangle - \sin\theta |2\rangle \\
|+\rangle &= \sin\theta |1\rangle + \cos\theta |2\rangle
\end{align}

Phase convention: a common phase on $|1\rangle$ and $|2\rangle$ leaves $\theta$ and $m_\pm$ invariant.

\subsection*{4.4 Useful Limits (Level Repulsion)}

Small overlap ($|\kappa| \ll |\Delta|$):
\begin{align}
\text{Assume } \Delta \equiv m_2 - m_1 > 0 \text{ and } |\kappa| \ll |\Delta|:
\quad \theta \approx \kappa / \Delta, \quad m_- \approx m_1 - \kappa^2/\Delta, \quad m_+ \approx m_2 + \kappa^2/\Delta
\end{align}

Note: If $\Delta < 0$, swap the labels $(1 \leftrightarrow 2)$ or keep the formulae and interpret $m_\pm$ accordingly.

Near degeneracy ($|\Delta| \ll |\kappa|$):
\begin{align}
\theta \to \pi/4, \quad m_\pm \approx (m_1 + m_2)/2 \pm |\kappa|
\end{align}

\subsection*{4.5 Time Evolution and Oscillation Probability}

Let $|\psi(0)\rangle = |1\rangle$. With eigenmasses $m_\pm$ and eigenstates $|\pm\rangle$, the survival/transition probabilities under unitary evolution are:
\begin{align}
\Delta m &\equiv m_+ - m_- = \sqrt( \Delta^2 + 4 \kappa^2 ) \\
P_{[1\to2]}(t) &= \sin^2(2\theta) \cdot \sin^2( \Delta m \cdot t / (2\hbar) ) \\
P_{[1\to1]}(t) &= 1 - P_{[1\to2]}(t)
\end{align}

These hold when decay widths are negligible over the timescale of interest (see §5 for widths).

\subsection*{4.6 Worked Numeric}

Given $m_1 = 1.000$, $m_2 = 1.200$, $\kappa = 0.050$:
\begin{align}
\Delta &= 0.200 \\
\tan(2\theta) = 2\kappa/\Delta &= 0.500 \Rightarrow \theta = 0.2449787 \text{ rad} (\approx 14.036^\circ)
\end{align}

\begin{align}
m_\pm = 1.100 \pm \sqrt( 0.0100 + 0.0025 ) = 1.100 \pm 0.111803 \Rightarrow m_- = 0.988197, \quad m_+ = 1.211803 \\
\Delta m = 0.223606
\end{align}

\textbf{Example oscillation amplitude:} $\sin^2(2\theta) = (2\kappa/\Delta m)^2 = 4\kappa^2/(\Delta^2 + 4\kappa^2) = 0.2$.

\subsection*{4.7 3$\times$3 Sketch: Successive Givens/Jacobi Rotations}

For a real symmetric 3$\times$3 $H$ with entries $H_{ij}$ ($i \leq j$), zero the off-diagonals by successive plane rotations $R_{12}(\theta_a)$, $R_{13}(\theta_b)$, $R_{23}(\theta_c)$:
\begin{align}
R_{12}(\theta) &= [ \cos(-) s & 0; s c & 0; 0 0 1] \\
R_{13}(\phi) &= [ \cos \phi & -\sin\phi; 0 1 0; s 0 c ] \\
R_{23}(\psi) &= [ 1 0 0; 0 \cos(-); s 0 c ] \quad (c \equiv \cos, s \equiv \sin)
\end{align}

Choose each angle to eliminate the targeted off-diagonal: $\tan(2\theta_a) = 2H_{12}/(H_{22} - H_{11})$, etc. The final orthogonal rotation $R = R_{12} R_{13} R_{23}$ diagonalizes $H$. Phase-bearing dictionaries (CKM/PMNS) extend this with complex phases; in the real symmetric case above, phases are absent.

\subsection*{4.8 Reporting Checklist}

When quoting mixing results, report:

\begin{enumerate}
\item $m_1, m_2$ (or $\Delta$) and $\kappa$, including units and uncertainty.
\item Derived $\theta$, $m_\pm$, and $\Delta m$; indicate if widths ($\Gamma$) were neglected or included.
\item For 3$\times$3: the rotation sequence and angles used, and whether phases are constrained or set to zero.
\end{enumerate}

\section*{§5 Decays \& Widths from Action Gaps (Path-Sum Derivation)}

\textbf{Scope.} We derive decay amplitudes and widths from an action-weighted path sum with a local transition kernel $K_w$ that adds an action penalty $\Delta S$ for paths connecting initial and final admissible closures. In the Markovian, incoherent-channel limit, widths add linearly and rates are proportional to $|\Lambda|^2$ times phase-space $\Phi_{\mathrm{PS}}$.

\subsection*{5.1 Path-Sum Amplitude with Local Penalty}

\textbf{$S_0$: Action scale with units of action (J$\cdot$s).} Unless stated otherwise, set $S_0 = h$ by calibration; numerics in §5 use $S_0 = h$. Let $S[\text{path}]$ be the baseline action along a path connecting $i \to f$. A local transition kernel contributes an additive penalty $\Delta S \geq 0$ that depends on the path neighborhood where the kernel acts. The transition amplitude is the action-weighted sum:
\begin{align}
A_{(i\to f)} \propto \sum_{\{\text{paths}\}} \exp( -[ S[\text{path}] + \Delta S ] / S_0 )
\end{align}

Assuming a unique dominant saddle path (or a narrow family) and a stationary kernel, the path integral is saddle-dominated and factorizes:
\begin{align}
A_{(i\to f)} = \exp( -\Delta S / S_0 ) \cdot A_o
\end{align}

Here $A_o$ contains prefactors (e.g., fluctuation determinants) and any weak dependence on $i,f$ under the smooth-limit assumptions.

\subsection*{5.2 Widths, Phase Space, and Additivity}

For an incoherent, Markovian channel the partial width is proportional to the available phase space $\Phi_{\mathrm{PS}}$ times $|\Lambda_{i\to f}|^2 \approx \Phi_{\mathrm{PS}} \cdot |\Lambda_{i\to f}|^2 \cdot \exp( -2 \Delta S / S_0 )$. Summing over incoherent channels gives total width $\Gamma = \sum_{k} \Gamma_{k}$ (incoherent addition):
\begin{align}
\Gamma_{(i\to f)} \propto \Phi_{\mathrm{PS}} \cdot |\Lambda_{i\to f}|^2 = \Phi_{\mathrm{PS}} \cdot \exp( -2 \Delta S / S_0 )
\end{align}

The lifetime $\tau$ and width satisfy $\tau = h / \Gamma$ (narrow-width approximation).
\begin{align}
\tau = h / \Gamma
\end{align}

\subsection*{5.3 Assumption A5 (Applicability Band)}

\textbf{A5.} Stationary kernel (adiabatic over a decay time), a single dominant saddle (or well-separated saddles), short memory (Markovian), and incoherent channels (negligible long-time interference). See §8 for quantitative checks.

\subsection*{5.4 Worked Numeric (Illustrative)}

Let $\Delta S/S_0 = 5$ and normalize $\Phi_{\mathrm{PS}}$ into the overall proportionality:
\begin{align}
\Gamma \propto \exp( -10 ) = 4.54 \times 10^{(-5)}
\end{align}

Relative lifetimes scale inversely with $\Gamma$. If a second channel has $\Delta S' = \Delta S + S_0$, then $\Gamma' / \Gamma = \exp[ -2 ] = 0.135$; its lifetime is longer by a factor 7.39$\times$. Branching fractions follow from $\Gamma_k / \sum_j \Gamma_j$.

\subsection*{5.5 Log-Linear Diagnostics and Extraction of $\Delta S/S_0$}

In the band where $\Phi_{\mathrm{PS}}$ varies slowly across a family of decays, the dependence is log-linear:
\begin{align}
\ln \Gamma = \ln(\Phi_{\mathrm{PS}} \cdot |\Lambda_o|^2) - 2 \Delta S / S_0
\end{align}

Plotting $\ln \Gamma$ against a control that shifts $\Delta S$ (e.g., a barrier parameter) yields slope $-2/S_0$, allowing an empirical extraction of $S_0$ or $\Delta S$ changes.

\subsection*{5.6 Beyond A5: Memory and Coherence}

If the memory time $\tau_{\mathrm{mem}}$ is not short compared to the dynamical time $\tau_{\mathrm{dyn}}$, amplitude sums retain temporal overlap and channels can interfere. A minimal correction is to promote widths to amplitude-level sums with phases:
\begin{align}
A_{(\text{tot})} = \sum_k A_k = \sum_k |A_k| e^{i(\phi_k)}, \quad \Gamma \propto \Phi_{\mathrm{PS}} \cdot |A_{\text{tot}}|^2
\end{align}

Non-Markovian kernels require explicit time-ordering in the path sum; interference terms $2|A_a| |A_b| \cos(\phi_a - \phi_b)$ appear and can modify rates substantially.

\subsection*{5.7 Phase-Space Factor $\Phi_{\mathrm{PS}}$ (Notes)}

$\Phi_{\mathrm{PS}}$ is the kinematic density of final states. For two-body decays at fixed $Q$ it reduces to a simple momentum factor; for multi-body decays it involves higher-dimensional integrals. In many comparisons across nearby channels, $\Phi_{\mathrm{PS}}$ varies slowly and can be treated as a known multiplicative factor apart from known kinematic edges.

\subsection*{5.8 Reporting Checklist}

When quoting widths from action gaps, also report:

\begin{enumerate}
\item Assessed small-parameter checks for A5 (stationarity, single saddle, short memory, incoherence).
\item How $\Phi_{\mathrm{PS}}$ was computed or normalized for the channel(s).
\item Any observed deviations suggesting memory/coherence, and whether amplitude-level sums were used.
\item Uncertainties propagated to $\Delta S/S_0$ when fitting log-linear forms.
\end{enumerate}

\section*{§6 Composites \& Binding: Budget Build of Liquid-Drop Form}

\textbf{Scope.} We expand the liquid-drop build from VMS budgets. The goal is to express the binding energy as a sum of volume, surface, asymmetry, and Coulomb budgets with clean A-scalings, and to make explicit that a single calibration sets the overall scale.

\subsection*{6.1 Scaling from Closure Geometry}

Let $R = r_o A^{(1/3)}$ be the smooth-limit radius of a composite closure of mass number $A$. Volumes and areas then scale as $V \propto A$ and $S \propto A^{(2/3)}$. The kernel-mapped Coulomb budget of a uniformly charged sphere scales as $Z^2/R \propto Z^2/A^{(1/3)}$. These scalings produce the canonical powers of $A$ in the liquid-drop ansatz.
\begin{align}
R &= r_o \cdot A^{(1/3)} \\
E_V &= a_V \cdot A \\
E_S &= a_S \cdot A^{(2/3)} \\
E_C &= a_C \cdot Z^2 / A^{(1/3)}
\end{align}

Asymmetry (two-species mismatch) penalizes imbalance between neutrons ($N$) and protons ($Z$). To leading order from a two-reservoir budget, the penalty scales as $(N-Z)^2/A$:
\begin{align}
E_A = a_A \cdot (N - Z)^2 / A
\end{align}

\subsection*{6.2 Budget Sum (Smooth-Limit Dictionary Form)}

Collecting the pieces gives the dictionary binding formula:
\begin{align}
E_{\mathrm{bind}} = a_V A - a_S \cdot A^{(2/3)} - a_A \cdot (N - Z)^2 / A - a_C \cdot Z^2 / A^{(1/3)}
\end{align}

\subsection*{6.3 Lemma L6 (Single-Calibration Structure)}

Write the binding as:
\begin{align}
E_{\mathrm{bind}}(A, Z) = K \left[ b_V(A, Z) - b_S(A, Z) A^{(2/3)} - b_A(A, Z) \frac{(N-Z)^2}{A} - b_C(A, Z) \frac{Z^2}{A^{(1/3)}} \right]
\end{align}

Here $K$ has units of energy, and the $b_*$ are dimensionless budget ratios determined by VMS geometry/kinematics in the smooth limit (weak higher-order corrections allowed). In this identification:
\begin{align}
a_V = K \cdot b_V, \quad a_S = K \cdot b_S, \quad a_A = K \cdot b_A, \quad a_C = K \cdot b_C
\end{align}

For the Coulomb budget specifically, dimensional consistency requires:
\begin{align}
b_C = a_C / K = (3/5) (a \hbar c) / (K r_o)
\end{align}

Thus no extra dimensional knob is introduced provided $r_o$ (from $R = r_o A^{(1/3)}$) is fixed by the same budgets (or reported once from geometry).

\textbf{Calibration:} If $b_*$ (and $r_o$) are fixed by VMS, a single reference nucleus ($A_{\mathrm{ref}}, Z_{\mathrm{ref}}$) suffices:
\begin{align}
K = E_{\mathrm{bind}}^{(\mathrm{ref})} / [ b_V A_{\mathrm{ref}} - b_S A_{\mathrm{ref}}^{(2/3)} - b_A (N_{\mathrm{ref}} - Z_{\mathrm{ref}})^2/A_{\mathrm{ref}} - b_C Z_{\mathrm{ref}}^2/A_{\mathrm{ref}}^{(1/3)} ]
\end{align}

If $r_o$ (hence $b_C$) is not fixed independently, then a two-point calibration is required, or $r_o$ must be reported explicitly.

\subsection*{6.4 Parameterization and Units (Dictionary vs VMS)}

For cross-checks, one may compare to standard liquid-drop (semi-empirical) dictionary values (illustrative): $a_V = 15.8$ MeV, $a_S = 18.3$ MeV, $a_A = 23$ MeV, $a_C = 0.71$ MeV. Within VMS, these arise from a single $K$ and budget ratios $b_*$; the dictionary values are used for calibration/validation only.

\subsection*{6.5 Worked Numeric (Deuteron A=2, Z=1)}

Illustrative use with smooth-limit coefficients. For $A=2$, $Z=1$, $N=1$ the asymmetry term vanishes. Using dictionary-like values (for demonstration):
\begin{align}
a_V &= 16.0 \text{ MeV}, \quad a_S = 18.3 \text{ MeV}, \quad a_A = 23.0 \text{ MeV}, \quad a_C = 0.71 \text{ MeV}
\end{align}

Compute each term:
\begin{align}
E_V = a_V \cdot A &= 32.0 \text{ MeV} \\
E_S = a_S \cdot A^{(2/3)} &= 18.3 \cdot 2^{(2/3)} = 29.0 \text{ MeV} \\
E_A = a_A \cdot (N-Z)^2/A &= 0.0 \text{ MeV} \\
E_C = a_C \cdot Z^2 / A^{(1/3)} &\approx 0.71 \cdot 2^{(-1/3)} = 0.56 \text{ MeV} \\
E_{\mathrm{bind}} = E_V - E_S - E_A - E_C &= 32.0 - 29.0 - 0 - 0.56 = 2.4 \text{ MeV}
\end{align}

A light retuning of the single calibration (e.g., $\pm$3\% on $a_V$ via $K$) brings this to $\approx 2.22$ MeV, aligning with the observed deuteron binding. Because $A=2$ is outside the large-$A$ surface-smoothing regime, deviations primarily constrain the surface:volume budget ratio, as expected.

\subsection*{6.6 Reporting \& Band Notes (Pointers to §8)}

When quoting liquid-drop results from VMS budgets, report: (i) the reference nucleus used for the single calibration; (ii) the resulting ($a_V, a_S, a_A, a_C$); (iii) the small-parameter checks from §8 (adiabatic kernel, weak gradients, short memory, incoherent channels). For very light nuclei ($A \lesssim 4$), expect larger finite-size corrections. Report the radius parameter $r_o$ used in $R = r_o A^{(1/3)}$ with an uncertainty (e.g., $r_o = 1.2$ fm $\pm$ few \%), since fits are sensitive to $r_o$.

This makes cross-comparisons reproducible and clarifies calibration choices.

\section*{§7 Atomic Attachment (Hydrogen) from EM Kernel Closure}

\textbf{Scope.} We expand the hydrogen case where the electromagnetic (EM) kernel closes to a smooth, central effective kernel. In this smooth-limit dictionary, the central potential reduces to the Coulomb form and yields the standard hydrogenic spectrum. The VMS-first path is: loop closure + EM kernel $\to$ central effective $\to$ dictionary results (for calibration/validation only).

\subsection*{7.1 Kernel Closure and Stationarity Conditions}

Assume a central, time-invariant kernel $K(r)$ derived from the EM pillar in the smooth limit. Stationarity of the action under azimuthal rotations enforces angular-momentum quantization; radial stationarity balances centripetal and central forces.
\begin{align}
m_e \cdot e \cdot v \cdot r &= n * \hbar \quad \text{(angular momentum quantization)} \\
e^2 / r &= m_e \cdot v^2 \quad \text{(centripetal balance; Gaussian units)}
\end{align}

Notes on units: the above uses Gaussian--cgs with Coulomb strength $e^2$. In SI, replace $e^2$ by $e^2/(4\pi \varepsilon_0)$.

\subsection*{7.2 Orbit Radius and Energy (Dictionary Forms)}

From the two stationarity conditions:
\begin{align}
r_n = n^2 \hbar^2 / (m_e \cdot e^2) \quad \text{(Gaussian--cgs)}
\\
r_n = (4\pi \varepsilon_0) \cdot n^2 \hbar^2 / (m_e \cdot e^2) \quad \text{(SI)}
\end{align}

Kinetic and potential energies obey $T = (1/2) m_e v^2$ and $V(r) = -e^2/r$ (Gaussian) or $V(r) = -e^2/(4\pi \varepsilon_0 r)$ (SI). Using virial relations for a $1/r$ potential ($T = -E$ and $V = 2E$), total energy levels are:
\begin{align}
E_n = - (m_e \cdot e^4) / (2 \hbar^2 n^2) \quad \text{(Gaussian--cgs)}
\\
E_n = - (m_e \cdot e^4) / (2 (4\pi \varepsilon_0)^2 \hbar^2 n^2) \quad \text{(SI)}
\end{align}

\subsection*{7.3 VMS-First Path (Sketch)}

Within VMS, admissible loop closure under the EM kernel yields an effective central action with smooth-limit torsion budget. In that limit, the loop's azimuthal phase single-valuedness on closure, enforcing the integer $n$, identifies the central term with the Coulomb dictionary then reproduces the above $r_n$ and $E_n$. This mapping is used solely for calibration/validation; predictive work remains in the VMS kernel formalism.

\subsection*{7.4 Worked Numeric (n = 1)}

Constants (CODATA-style):

\begin{itemize}
\item $h = 1.054 571 817 \times 10^{-34}$ J$\cdot$s
\item $m_e = 9.109 383 7015 \times 10^{-31}$ kg
\item $e = 1.602 176 634 \times 10^{-19}$ C
\item $\varepsilon_0 = 8.854 187 8128 \times 10^{-12}$ F/m
\end{itemize}

Results:
\begin{align}
a_0 \equiv r_1 &= 0.529 177 \text{ \AA} \\
E_1 &= -13.6 \text{ eV} \\
\mu &\equiv (m_e m_p) / (m_e + m_p) \\
a_0 &= (4\pi \varepsilon_0) \hbar^2 / (\mu e^2), \quad E_1 = - \mu e^4 / [2 (4\pi \varepsilon_0)^2 \hbar^2]
\end{align}

\textbf{Comment:} Using $\mu$ instead of $m_e$ shifts $a_0$ and $E_1$ at the $\sim 10^{-4}$ level; fine for calibration/validation.

\subsection*{7.5 Validity Band (Pointers to §8)}

The dictionary forms hold in the smooth-limit band: adiabatic kernel ($\varepsilon_{\mathrm{stat}} \lesssim 0.1$), weak gradients ($\varepsilon_{\mathrm{grad}} \lesssim 0.2$), short memory ($\varepsilon_{\mathrm{mem}} \lesssim 0.2$), and incoherent single-channel attachment ($\chi_{\mathrm{ab}} \ll 1$). Outside this band (strong driving, long memory, turbulence, or coherent multi-channel interference), escalate to the appropriate branch models and do not overreach the pillar kernels.

\subsection*{7.6 Reporting Checklist}

When you use this insert, also report:

\begin{enumerate}
\item Unit convention (Gaussian vs SI) and any conversions used.
\item Estimated small parameters ($\varepsilon_{\mathrm{stat}}, \varepsilon_{\mathrm{grad}}, \varepsilon_{\mathrm{mem}}, \chi_{\mathrm{ab}}$) to justify smooth-limit use.
\end{enumerate}

\section*{§8 Validity Band, Assumptions, and Failure Modes}

\subsection*{Scope}

The particle-mechanics pillar is calibrated for near-stationary admissible loops evolving in weakly varying environments, with time-invariant (or adiabatic) kernels and short-memory, incoherent transition channels. Outside this band (strong driving, long memory, turbulence, or coherent multi-channel interference), move to the appropriate branch models and do not overreach the pillar kernels.

\subsection*{8.1 Small-Parameter Regime (Operational Band)}

We quantify ``near-stationary/weak-variation'' with dimensionless controls comparing environment and kernel variability to loop dynamics.

\textbf{Let:}
\begin{itemize}
\item $K(x, t)$ be any kernel entering the action density (e.g., torsion-driven budget or EM contribution),
\item $\Omega_{\text{loop}} \equiv 2\pi T / \ell$ a characteristic loop frequency,
\item $L_{\mathrm{char}}$ a characteristic spatial scale of the loop segment under consideration.
\end{itemize}

\subsubsection*{(A) Temporal stationarity}
\begin{align}
\varepsilon_{\mathrm{stat}} = (1/\Omega_{\text{loop}}) * |\partial_t \ln K| << 1
\end{align}
\textbf{Bound:} $\varepsilon_{\mathrm{stat}} \leq 0.1 \to$ adiabatic, pillar-valid.

\subsubsection*{(B) Weak spatial gradients}
\begin{align}
\varepsilon_{\mathrm{grad}} = L_{\mathrm{char}} * |\nabla \ln K| << 1
\end{align}
\textbf{Bound:} $\varepsilon_{\mathrm{grad}} \leq 0.2 \to$ nearly uniform.

\subsubsection*{(C) Short memory in decay/path sums}
\begin{align}
\varepsilon_{\mathrm{mem}} = \tau_{\mathrm{mem}} / \tau_{\mathrm{dyn}} << 1, \quad \text{with } \tau_{\mathrm{dyn}} \approx 1/\Omega_{\text{loop}}
\end{align}
\textbf{Bound:} $\varepsilon_{\mathrm{mem}} \leq 0.2 \to$ Markovian rate addition valid.

\subsubsection*{(D) Incoherent channels}
\begin{align}
\chi_{\mathrm{ab}} = |f A_a| A_b \, dt| / \sqrt{(|A_a|^2 dt} (|A_b|^2 dt)
\end{align}
\textbf{Require } $\chi_{\mathrm{ab}} \leq 0.1 \to$ incoherence holds.

\subsubsection*{(E) Gentle driving / weak nonlinearity}
\begin{align}
\varepsilon_{\mathrm{drive}} = \Delta S_{\mathrm{drive}} / S_0 << 1
\end{align}
\textbf{Bound:} $\varepsilon_{\mathrm{drive}} \leq 0.1 \to$ safe.

\textbf{Summary band:} $\varepsilon_{\mathrm{stat}} \leq 0.1$, $\varepsilon_{\mathrm{grad}} \leq 0.2$, $\varepsilon_{\mathrm{mem}} \leq 0.2$, $\chi_{\mathrm{ab}} \leq 0.1$, $\varepsilon_{\mathrm{drive}} \leq 0.1$

\subsection*{8.2 Assumptions}

\begin{enumerate}
\item Admissible loops: piecewise $C^1$ closures with single-valued action phase, no topological transitions.
\item Time-invariant kernels: constant or adiabatic.
\item Weak gradients: higher curvature negligible over $L_{\mathrm{char}}$.
\item Short-memory decays: exponential or faster, incoherent addition valid.
\item No turbulence: sub-adiabatic fluctuations only.
\end{enumerate}

\subsection*{8.3 What Breaks First (Failure Modes)}

\begin{itemize}
\item \textbf{Non-stationary kernels:} $\varepsilon_{\mathrm{stat}} \geq 0.2 \to$ drift in quantization, requires non-stationary kernel branch.
\item \textbf{Strong gradients:} $\varepsilon_{\mathrm{grad}} \geq 0.3 \to$ torsion averages vary; upgrade to gradient/multiscale branch.
\item \textbf{Long memory/coherence:} $\varepsilon_{\mathrm{mem}} \geq 0.3$ or $\chi_{\mathrm{ab}} \geq 0.2 \to$ interference arises; upgrade to coherent-branch.
\item \textbf{Strong driving:} $\varepsilon_{\mathrm{drive}} \geq 0.2 \to$ sidebands, nonlinear response; use driven/nonlinear branch.
\item \textbf{Turbulence:} phase diffusion strong; stochastic branch required.
\end{itemize}

\subsection*{8.4 Turbulence \& Noise Gate}

Define $\mathrm{Re}_\phi = \Omega_{\text{loop}} / D_\phi$, with $D_\phi = \text{phase-diffusion rate}$.

Estimate $D_\phi$ from phase diffusion: $\mathrm{Var}[\phi(t)] = 2 D_\phi t \Rightarrow D_\phi = (1/2) \cdot d \mathrm{Var}[\phi]/dt$.

Use an ensemble of short segments (within the stationarity window) to compute $\mathrm{Var}[\phi(t)]$ and extract $D_\phi$ from the initial slope.

\textbf{Require } $\mathrm{Re}_\phi >> 1$ (e.g., $\geq$10). If $\mathrm{Re}_\phi \leq 1$, system is turbulent, must switch to stochastic branch.

\subsection*{8.5 Quick Decision Procedure}

\begin{itemize}
\item \textbf{All within thresholds + $\mathrm{Re}_\phi \geq 10$ $\Rightarrow$ Pillar valid.}
\item \textbf{One exceeds by factor $\sim$2 $\Rightarrow$ escalate to branch.}
\item \textbf{Borderline $\Rightarrow$ sensitivity test; $> 10\%$ variation sensitivity $\Rightarrow$ treat as out-of-band.}
\end{itemize}

\subsection*{8.6 Reporting Checklist}

\begin{enumerate}
\item $\Omega_{\text{loop}}, T_{\text{loop}}, \ell$, kernels $K$ and $\varepsilon_{\mathrm{stat}}, \varepsilon_{\mathrm{grad}}$.
\item $\tau_{\mathrm{mem}}$ and $\chi_{\mathrm{ab}}$ if decays/multi-channels used.
\item Driving amplitude $\varepsilon_{\mathrm{drive}}$ if forced.
\item Noise via $D_\phi$ and $\mathrm{Re}_\phi$.
\item If thresholds exceeded, specify branch used.
\end{enumerate}

\section*{§9 Symbols \& Units}

$h$ (J$\cdot$s); $c$ (m/s); $\kappa$ (U); $T$ (dimensionless torsion); $\tau=(1/\ell)\oint T\,ds$; $C \equiv \kappa/c^2$ (mass-scale constant); $\ell$ (loop length); $s$ (arc length); $T$ (dimensionless torsion); $\tau=(1/\ell)\oint T\,ds$; $C \equiv \kappa/c^2$; $\beta=(\mathrm{k_B T})^{[-1]}$; $A,N,Z$ integers; $\Phi_{\mathrm{PS}}$ phase-space factor; §0 dimension unit; $\Delta S$ action gap; $S_0$ --- action scale (J$\cdot$s), typically equal to $h$ in this appendix.

\section*{§10 Completeness Checklist (Cross-References)}

\begin{itemize}
\item[\checkmark] Loop closure \& spin $\to$ §1 (1.1--1.6)
\item[\checkmark] Conserved charge \& fractional units (ensemble) $\to$ §2 (2.1--2.4)
\item[\checkmark] Mass functional, scaling, ratios $\to$ §3 (3.1--3.4)
\item[\checkmark] Mixing diagonalization (2$\times$2; 3$\times$3 sketch) $\to$ §4 (4.1--4.4)
\item[\checkmark] Decays \& widths from action gaps $\to$ §5 (5.1--5.3)
\item[\checkmark] Composites: budget build to liquid-drop $\to$ §6 (6.1) + lemmas
\item[\checkmark] Hydrogen closure (EM kernel $\to$ central) $\to$ §7 (7.1--7.2)
\item[\checkmark] Validity band; assumptions $\to$ §8 ; Symbols $\to$ §9
\end{itemize}

\section*{§11 Reference Tables: Elements, Molecules, Materials}

\subsection*{11.1 Purpose and Scope}

The three tables (Elements, Molecules, Materials/Solid-State) are not catalogs but pillar-level anchors. They define calibration points and archetypes, allowing applied branches (chemistry, condensed matter, superconductivity, etc.) to proceed without re-deriving fundamentals. Improvements (flags, naming, calibration notes) are tracked in the Tables Improvements Log.

\subsection*{11.2 Elemental Table (All 118 Elements)}

\subsubsection*{§11.2 Elemental Table --- Full VMS Derivations (Rewritten)}

\subsubsection*{11.2.1 Element Entry Schema}

$\{ Z, A, N=A-Z, \text{Symbol}, \text{Name}, \text{VMS Reference}, \text{Calibration Scheme}, \text{Predicted Values}, \text{Published Measurements}, \text{Error}, \text{Validation Status} \}$

Each row is isotope-aware; the default $A$ is the most stable (or abundantly observed) mass number unless otherwise stated. Predicted Values include binding energy $E_{\text{bind}}(A,Z)$, mass excess, and other derived quantities as needed by branches.

\subsubsection*{11.2.2 Core Budgets and Definition of f\_VMS(Z,A) --- Expanded}

This section derives the smooth-limit budgets, fixes dimensions, and defines the predictor $f_{\mathrm{VMS}}(Z,A)$. All symbols follow §§0--8 and §6.3 (revised L6). Classical forms appear only as dictionaries for validation.

\paragraph{A. Definitions and Conventions}

\begin{itemize}
\item $Z$ = proton number
\item $A$ = mass number
\item $N \equiv A - Z$
\item $\kappa$ single energy scale (MeV)
\item $b_*(A, Z)$ dimensionless budget ratios (geometry/kinematics fixed in smooth limit)
\item $\delta E_{\mathrm{micro}}(A, Z)$ micro corrections (shell, odd--even, etc.; pillar default = 0)
\end{itemize}

\paragraph{B. Budget Scalings from Smooth Geometry}

Let $R = r_o A^{(1/3)}$ be the smooth-limit effective radius of a composite closure of mass number $A$. Volumes and areas then scale as $V \propto A$ and $S \propto A^{(2/3)}$. With dimensionless torsion-geometry weights encoded in $b_*$, the smooth-limit budgets read:
\begin{align}
E_V &= K \cdot b_V(A,Z) \cdot A \\
E_S &= K \cdot b_S(A,Z) \cdot A^{(2/3)} \\
E_A &= K \cdot b_A(A,Z) \cdot (N-Z)^2 / A \\
E_C &= K \cdot b_C(A,Z) \cdot Z^2 / A^{(1/3)}
\end{align}

Volume $\propto A$ reflects bulk closure density; surface $\propto A^{(2/3)}$ follows isoperimetric scaling; asymmetry penalizes reservoir imbalance; Coulomb scales as $Z^2/R$ for a uniform sphere (dictionary).

\paragraph{C. Coulomb Ratio and Dimensional Accounting}

With $R = r_o A^{(1/3)}$, the uniform-sphere dictionary gives:
\begin{align}
E_C = (3/5) \cdot (e^2 / (4\pi \varepsilon_0)) \cdot Z^2 / R = K \cdot b_C \cdot Z^2 / A^{(1/3)}
\end{align}

Thus $b_C = a_C / K = (3/5) (a \hbar c) / (K r_o)$. Thus $b_C$ is dimensionless via the product $(a \hbar c) / (K r_o)$. No extra dimensional knob is introduced provided $r_o$ is fixed by VMS geometry or reported once (cf. §6.3 Lemma L6 revised).

\paragraph{D. Binding Energy and Predictor Definition}

\begin{align}
E_{\mathrm{bind}}(A,Z) = K \left[ b_V A - b_S A^{(2/3)} - b_A \frac{(N-Z)^2}{A} - b_C \frac{Z^2}{A^{(1/3)}} \right] + \delta E_{\mathrm{micro}}(A,Z)
\end{align}

Define the elemental predictor:
\begin{align}
f_{\mathrm{VMS}}(Z,A) \equiv b_V A - b_S A^{(2/3)} - b_A \frac{(N-Z)^2}{A} - b_C \frac{Z^2}{A^{(1/3)}} + (1/K) \delta E_{\mathrm{micro}}
\end{align}

\paragraph{E. Sensitivities (Useful for Calibration/Uncertainty)}

Partial derivatives at fixed $b_*$ (smooth limit):
\begin{align}
\frac{\partial E_{\mathrm{bind}}}{\partial A} &= K \left[ b_V - (2/3) b_S A^{(-1/3)} + b_A (N-Z)^2/A^2 + (1/3) b_C Z^2/A^{(4/3)} \right] + \frac{\partial E_{\mathrm{micro}}}{\partial A} \\
\frac{\partial E_{\mathrm{bind}}}{\partial Z} &= K \left[ - 2 b_A (N-Z)/A - 2 b_C Z/A^{(1/3)} \right] + \frac{\partial E_{\mathrm{micro}}}{\partial Z}
\end{align}

These are used to (i) locate the most-stable $A$ for fixed $Z$ via $\partial E_{\mathrm{bind}}/\partial A = 0$, and (ii) study proton--neutron moves across isobars via $\partial E_{\mathrm{bind}}/\partial Z$.

\paragraph{F. Stability Condition (Smooth Valley of Stability)}

At fixed $Z$, a smooth approximation to the most-bound isotope $A^*$ satisfies:
\begin{align}
( \partial E_{\mathrm{bind}} / \partial A )|_{A=A^*} = 0
\end{align}

Insert the derivative from E. and solve numerically for $A^*$ in a narrow window.

\paragraph{G. Limits and Scaling Checks}

\textbf{Large-A (surface suppressed):}
\begin{align}
E_{\mathrm{bind}} / A \to K \left[ b_V - b_C \cdot Z^2/(A \cdot A^{(1/3)}) \right] \quad (A \to \infty)
\end{align}

\textbf{Symmetric case N = Z (ignoring Coulomb):}
\begin{align}
E_{\mathrm{bind}} = K [ b_V A - b_S A^{(2/3)} ]
\end{align}

\textbf{Extreme asymmetry |N-Z| >> 0 (penalty dominates):}
\begin{align}
E_{\mathrm{bind}} \downarrow \text{ as } b_A \cdot (N-Z)^2 / A \uparrow
\end{align}

\paragraph{H. Micro-Corrections Placeholder (Branch-Controlled)}

By default $\delta E_{\mathrm{micro}}=0$ at the pillar level. Branches may add shell corrections, odd--even pairing, and deformation terms, e.g.,
\begin{align}
\delta E_{\mathrm{micro}} = \Delta E_{\text{(shell)}}(A,Z) + \Delta E_{\text{(pair)}}(A,Z) + \Delta E_{\text{(def)}}(A,Z) + \ldots
\end{align}

Any such terms must be reported explicitly in the Calibration Scheme column and kept consistent across the table.

\subsubsection*{11.2.3 Coulomb Ratio and Dimensional Accounting}

With $R = r_o A^{(1/3)}$ the smooth-limit radius, the uniform-sphere Coulomb budget gives:
\begin{align}
E_C = (3/5) (e^2 / (4\pi \varepsilon_0)) Z^2 / R = K \cdot b_C \cdot Z^2 / A^{(1/3)}
\end{align}

Thus
\begin{align}
b_C = a_C / K = (3/5) (a \hbar c) / (K r_o)
\end{align}

No extra dimensional knob is introduced provided $r_o$ is fixed by the same geometry budgets (or reported once, see §6.3).

\subsubsection*{11.2.4 From Binding Energy to Atomic Mass}

Let $m_p, m_n, m_e$ be proton, neutron, and electron masses; $B \equiv E_{\mathrm{bind}}$. The nuclear mass and neutral-atom mass are:
\begin{align}
M_{\mathrm{nucleus}}(A,Z) &= Z m_p + N m_n - B/c^2 \\
M_{\mathrm{atom}}(A,Z) &= M_{\mathrm{nucleus}}(A,Z) + Z m_e - E_{\mathrm{elec}}(Z)/c^2
\end{align}

$E_{\mathrm{elec}}(Z)$ is the total electronic binding energy (small on nuclear scales; negligible at pillar accuracy unless a branch specifies otherwise).

\subsubsection*{11.2.5 Calibration Procedures (K, r\_o, ratios)}

Two minimal, VMS-consistent options are provided; pick one and record it in the Calibration Scheme column.

\textbf{Single-point K (geometry-fixed):} $b_*$ and $r_o$ are fixed by VMS geometry; use one reference ($A_{\mathrm{ref}}, Z_{\mathrm{ref}}$) to set $K$:
\begin{align}
K = E_{\mathrm{bind}}^{(\mathrm{ref})} / [ b_V A_{\mathrm{ref}} - b_S A_{\mathrm{ref}}^{(2/3)} - b_A (N_{\mathrm{ref}}-Z_{\mathrm{ref}})^2/A_{\mathrm{ref}} - b_C Z_{\mathrm{ref}}^2/A_{\mathrm{ref}}^{(1/3)} ]
\end{align}

\textbf{Two-point K \& $r_o$:} If $r_o$ (hence $b_C$) is not fixed independently, solve for ($K, r_o$) using two references ($i=1,2$):
\begin{align}
E_i = K [ b_V A_i - b_S A_i^{(2/3)} - b_A (N_i - Z_i)^2/A_i - b_C(r_o) Z_i^2/A_i^{(1/3)} ]
\end{align}

This pair determines $K$ and $r_o$; more points can be used in a least-squares sense if desired (record residuals).

\subsubsection*{11.2.6 Selecting A for a Given Z (Stability Map)}

For an elemental row keyed by $Z$, the table uses $A^*$ that maximizes $B$ within the admissible band. In practice:
\begin{align}
A^*(Z) = \arg\max_A E_{\mathrm{bind}}(A,Z) \quad (\text{within admissibility band})
\end{align}

Compute $E_{\mathrm{bind}}$ across a narrow isotope window around the empirical valley of stability, or use the analytic stationarity condition $\partial E_{\mathrm{bind}}/\partial A = 0$ in the smooth limit.

\subsubsection*{11.2.7 Uncertainty and Validation Flags}

Error bars combine calibration and model residuals:
\begin{align}
\sigma^2(E_{\mathrm{bind}}) = (\partial E_{\mathrm{bind}}/\partial K)^2 \sigma_K^2 + (\partial E_{\mathrm{bind}}/\partial r_o)^2 \sigma_{r_o}^2 + \sum_j (\partial E_{\mathrm{bind}}/\partial b_j)^2 \sigma_{b_j}^2 + \sigma_{\mathrm{micro}}^2
\end{align}

Validation flags adhere to the global scheme: $\checkmark$ (validated vs measurement), $\approx$ (compared to QM only), $\triangle$ (pending measurement). Record the scheme used and residuals.

\subsubsection*{11.2.8 Compact Cookbook (Per-Row Algorithm)}

\begin{enumerate}
\item Choose calibration scheme (single-point with geometry-fixed ratios, or two-point with ($K, r_o$)).

\item For element $Z$, scan $A$ in a small stability window; compute $b_*(A,Z)$ from geometry budgets; set $\delta E_{\mathrm{micro}}=0$ unless branch-specified.

\item Evaluate $E_{\mathrm{bind}}(A,Z)$ using the equations above; pick $A^*$ that maximizes binding within the band.

\item Compute $M_{\mathrm{nucleus}}$ and $M_{\mathrm{atom}}$. Record Predicted Values; compare to Published Measurements; compute Error and set Validation Status.

\item Log any table-wide refinements in the Improvements Log and update calibration notes consistently.
\end{enumerate}

\subsubsection*{11.2.9 Final Formulas (for quick reference)}

\begin{align}
E_{\mathrm{bind}}(A,Z) &= K [ b_V A - b_S A^{(2/3)} - b_A (N-Z)^2/A - b_C Z^2/A^{(1/3)} ] \quad (\pm \delta E_{\mathrm{micro}}) \\
b_C &= (3/5) (\alpha \hbar c) / (K r_o) \\
M_{\mathrm{nucleus}} &= Z m_p + N m_n - E_{\mathrm{bind}}/c^2 \\
M_{\mathrm{atom}} &= M_{\mathrm{nucleus}} + Z m_e - E_{\mathrm{elec}}(Z)/c^2
\end{align}

\subsection*{11.3 Molecular Table (Representative Archetypes)}

\subsubsection*{11.3.1 Purpose \& Coverage}

This section defines how the Molecular Table is populated from VMS budgets. Archetypes cover covalent (H2, CH4, benzene), ionic (NaCl), metallic (Fe), hydrogen bonding (H2O, ice), resonance/network (graphene), and biomolecular anchors (DNA base pairs). All equations below are Courier text and avoid \LaTeX{} and Equation objects.

\subsubsection*{11.3.2 Core Decomposition}

The smooth-limit molecular binding is built from atomic closures plus interaction and geometry budgets:
\begin{align}
E_{\mathrm{mol}} = \sum_i f_{\mathrm{VMS}}(Z_i, A_i) + \Delta E_{\mathrm{bond}} + \Delta E_{\mathrm{geometry}}
\end{align}

Where:
\begin{align}
E_{\mathrm{atom},i} &= K \cdot f_{\mathrm{VMS}}(Z_i, A_i) \quad \text{(from Elemental Table entry for atom } i \text{)} \\
\Delta E_{\mathrm{bond}} &= \sum_b E_{\mathrm{bond}}(b) \quad \text{(sum over bonds } b \text{)} \\
\Delta E_{\mathrm{geometry}} &= \sum_c E_{\mathrm{angle}}(c) + \sum_d E_{\mathrm{dihedral}}(d) + \sum_s E_{\mathrm{stretch}}(s) + \sum_t E_{\mathrm{torsion}}(t)
\end{align}

The first term imports elemental predictions. The second and third apply molecular interaction budgets (smooth limit). Micro-corrections may be added by branches but default to zero in the pillar appendix.

\subsubsection*{11.3.3 Bond Budgets (Smooth Limit)}

For a bond between atoms $a$ and $b$ at distance $r_{\mathrm{ab}}$, with coordination/valence weights $w_{\mathrm{ab}}$:
\begin{align}
E_{\mathrm{bond}}(a-b) = - K \cdot \beta_{ab} \cdot g_{\mathrm{bond}}(r_{\mathrm{ab}}) \cdot w_{\mathrm{ab}}
\end{align}

Recommended smooth-limit dictionary for $g_{\mathrm{bond}}$ (choose one per class, report choice in the table):
\begin{align}
g_{\mathrm{bond}}(r) &= \exp( -r / r_b ) \quad \text{(short-range exponential)} \\
g_{\mathrm{bond}}(r) &= 1 / ( 1 + (r / r_b)^n ) \quad \text{(soft step, } n \geq 2 \text{)}
\end{align}

Bond-type parameters:
\begin{align}
\beta_{ab} &: \text{ dimensionless bond ratio (geometry/kinematics)} \\
r_b &: \text{ characteristic bond length (Å)} \\
w_{ab} &: \text{ coordination/valence weight in [0, 1] (network consistency)}
\end{align}

\subsubsection*{11.3.4 Angular, Torsion, and Stretch Budgets}

Angles ($c$), dihedrals ($d$), small stretches ($s$), small torsions ($t$):
\begin{align}
E_{\mathrm{angle}}(c) &= K \cdot \kappa_c \cdot ( \theta_c - \theta0_c )^2 \\
E_{\mathrm{dihedral}}(d) &= K \cdot \Lambda_d \cdot [ 1 - \cos( n_d \phi_d - \phi0_d ) ] \\
E_{\mathrm{stretch}}(s) &= K \cdot k_s \cdot ( r_s - r0_s )^2 \quad \text{(small-strain regime)} \\
E_{\mathrm{torsion}}(t) &= K \cdot \tau_t \cdot ( \chi_t - \chi0_t )^2
\end{align}

Parameters are dimensionless ratios or geometry constants; report the set used per archetype.

\subsubsection*{11.3.5 Ionic \& Polarization Terms}

For partially ionic bonds or salts, include Coulomb and polarization budgets:
\begin{align}
E_{\mathrm{Coulomb}} &= - (1 / (4\pi \varepsilon_0)) \cdot \sum_{(a,b)} (q_a q_b) / r_{ab} \\
E_{\mathrm{pol}} &= - (1/2) \cdot \sum_a \alpha_a \cdot | E_{\mathrm{loc}}(a) |^2
\end{align}

Here $q_a$ are partial charges ($\sum_a q_a = 0$ for neutral molecules), $\alpha_a$ are polarizabilities, and $E_{\mathrm{loc}}(a)$ is the local field.

\subsubsection*{11.3.6 Resonance \& Delocalization ($\pi$ Networks)}

For conjugated systems (benzene, graphene), add a resonance budget over rings or tiles:
\begin{align}
E_{\mathrm{res}} = - K \cdot \sum_r y_r \cdot R_r
\end{align}

Where $R_r = \text{average bond-order on ring } r$ (in [0, 1]) and $y_r$ is a dimensionless resonance ratio for ring $r$. Set $R_r$ from the chosen bond-order assignment (network self-consistency).

\subsubsection*{11.3.7 Hydrogen Bonding (Directional)}

Directional hydrogen bonds (D--H$\cdots$A) include distance and angle gates:
\begin{align}
E_{\mathrm{HB}} = - K \cdot \beta_{HB} \cdot g_{\mathrm{HB}}(r_{HA}) \cdot h_{\mathrm{HB}}(\theta_{\mathrm{DHA}})
\end{align}

With
\begin{align}
g_{\mathrm{HB}}(r) &= \exp( -r / r_{HB} ) \\
h_{\mathrm{HB}}(\theta) &= \max( 0, \cos( \theta - 180^\circ ) ) \quad \text{(one-sided directionality)}
\end{align}

\subsubsection*{11.3.8 Geometry \& Constraints}

Total molecular energy (smooth limit) including optional terms:
\begin{align}
E_{\mathrm{total}} &= \sum_i E_{\mathrm{atom},i} + \sum_b E_{\mathrm{bond}}(b) + \sum_c E_{\mathrm{angle}}(c) + \sum_d E_{\mathrm{dihedral}}(d) + \sum_s E_{\mathrm{stretch}}(s) + \sum_t E_{\mathrm{torsion}}(t) \\
&\quad + E_{\mathrm{Coulomb}} + E_{\mathrm{pol}} + E_{\mathrm{res}} + E_{\mathrm{HB}} + E_{\mathrm{other}} \quad \text{(if used)}
\end{align}

Subject to constraints (report which are enforced):
\begin{align}
\sum_a q_a &= 0 \quad \text{(neutral molecule)} \\
\text{valence and coordination counts}&\text{ satisfied on each atom} \\
\text{ring/graph constraints}&\text{ for resonance networks}
\end{align}

\subsubsection*{11.3.9 Calibration \& Defaults}

Calibration follows the Elemental Table (K fixed; ratios reported). Suggested defaults at pillar level:
\begin{align}
\delta E_{\mathrm{micro}} &= 0 \quad \text{(no shell/pairing in molecules)} \\
E_{\mathrm{elec}}(Z) &\text{ negligible for nuclear masses (use only if atomic masses required)} \\
\text{for metals: treat small}&\text{ clusters with bond + delocalization; bulk goes to Materials Table (§11.4)}
\end{align}

\subsubsection*{11.3.10 Worked Mini-Examples (Illustrative)}

\textbf{H2 (covalent):} two atoms, one bond, one stretch mode:
\begin{align}
E_{\mathrm{H2}} = 2 \cdot f_{\mathrm{VMS}}(1, 1) + (- K \cdot \beta_{\mathrm{HH}} \cdot g_{\mathrm{bond}}(r_{\mathrm{HH}})) + K \cdot k_s \cdot (r_{\mathrm{HH}} - r0_{\mathrm{HH}})^2
\end{align}

\textbf{H2O (hydrogen bonding + angles):}
\begin{align}
E_{\mathrm{H2O}} = 2 \cdot E_H + E_O + \sum_b E_{\mathrm{bond}}(b) + E_{\mathrm{angle}}(\mathrm{H-O-H}) + \text{optional } E_{\mathrm{HB}} \text{ in ice/network contexts}
\end{align}

\textbf{NaCl (ionic):}
\begin{align}
E_{\mathrm{NaCl}} = E_{\mathrm{Na}} + E_{\mathrm{Cl}} + E_{\mathrm{Coulomb}} + \text{small geometry terms (stretch, angle)}
\end{align}

\subsubsection*{11.3.11 Reporting Checklist (Per Row)}

\begin{itemize}
\item Which budgets are active (bond, angle, torsion, Coulomb, polarization, resonance, HB).
\item Parameter set used ($\beta_{ab}, r_b, k_c, \Lambda_d, k_s, \tau_t, y_r, q_a$, etc.).
\item Constraints enforced (charge neutrality, valence counts, network rules).
\item Predicted values and residuals vs measurements or QM for the archetype.
\end{itemize}

\subsection*{11.4 Materials / Solid-State Table (Representative Archetypes)}

This section provides a full derivation of solid-state energy budgets. Representative archetypes include NaCl, diamond, graphite, Fe, Cu, Al, Si, Ge, GaAs, YBCO, MgB2, Pb, polymers, glass, and liquid crystals. Bonding frameworks include ionic, covalent, metallic, superconducting, polymeric, amorphous, and anisotropic cases.

\subsubsection*{11.4.1 Core Build}

Bulk energies are built from unit-cell closures, surfaces, and defect corrections:
\begin{align}
E_{\mathrm{bulk}} = N \cdot E_{\mathrm{cell}} + E_{\mathrm{surface}} + \Delta E_{\mathrm{defect}}
\end{align}

With
\begin{align}
E_{\mathrm{cell}} &= \sum_a E_{\mathrm{atom},a} + \sum_b E_{\mathrm{bond}}(b) + \sum_{\mathrm{geom}} E_{\mathrm{geom}} \quad \text{(from §11.3)}
\end{align}

\subsubsection*{11.4.2 Surface Budget}

Surface terms follow the liquid-drop analogy (§6). For a crystal of $N$ cells and linear size $L$:
\begin{align}
E_{\mathrm{surface}} = K \cdot b_S \cdot N^{(2/3)}
\end{align}

Here $b_S$ is a dimensionless surface ratio derived from geometry of the lattice termination.

\subsubsection*{11.4.3 Defect \& Disorder Budgets}

Vacancies, substitutions, dislocations, and amorphous disorder are encoded as corrections:
\begin{align}
\Delta E_{\mathrm{defect}} = \sum_v E_{\mathrm{vacancy}} + \sum_s E_{\mathrm{substitution}} + \sum_d E_{\mathrm{dislocation}} + \sum_a E_{\mathrm{amorphous}}
\end{align}

Each defect energy term is derived from local bond breaking or distortion budgets.

\subsubsection*{11.4.4 Ionic Solids}

For NaCl-type lattices, Coulomb Madelung sums contribute:
\begin{align}
E_{\mathrm{Madelung}} = - ( M \cdot e^2 / (4\pi \varepsilon_0 a_0) ) \cdot (Z+ \cdot Z-)
\end{align}

Where $M$ is the Madelung constant ($\sim$1.7476 for NaCl), and $a_0$ is the nearest-neighbor distance.

\subsubsection*{11.4.5 Covalent Solids}

For diamond, graphite, Si, Ge: unit cell energies are dominated by directional covalent bonds:
\begin{align}
E_{\mathrm{cell}} = \sum_b ( - K \cdot \beta_{ab} \cdot g_{\mathrm{bond}}(r_{\mathrm{ab}}) ) + \sum_{\mathrm{angles}} E_{\mathrm{angle}}
\end{align}

Graphite requires separate in-plane vs out-of-plane bond ratios; anisotropy must be reported.

\subsubsection*{11.4.6 Metallic Bonding}

For metals (Fe, Cu, Al): apply free-electron gas dictionary with VMS scaling:
\begin{align}
E_{\mathrm{electron}} = (3/5) \cdot N_e \cdot E_F
\end{align}

With
\begin{align}
E_F = \hbar^2(3\pi^2 n_e)^{(2/3)} / (2 m_e)
\end{align}

This electron-gas term adds to atomic closures, giving cohesion and conductivity baseline.

\subsubsection*{11.4.7 Superconductors}

For superconductors (YBCO, MgB2, Pb), an additional condensation energy budget is added:
\begin{align}
E_{\mathrm{SC}} = - (1/2) \cdot N(0) \cdot \Delta^2
\end{align}

Here $N(0)$ is the electronic density of states at the Fermi level, $\Delta$ is the superconducting gap. This provides the anchor for superconductivity branch tables (§§ Superconductor Cookbook).

\subsubsection*{11.4.8 Polymers, Glass, and Liquid Crystals}

Polymeric and amorphous materials use averaged bond and angle budgets with disorder corrections. Liquid crystals include orientational order parameters $S$:
\begin{align}
F_{\mathrm{LC}} = (1/2) \cdot A \cdot S^2 + (1/4) \cdot B \cdot S^4 + (1/6) \cdot C \cdot S^6
\end{align}

$S$ is the nematic order parameter; $A, B, C$ are material-specific coefficients.

\subsubsection*{11.4.9 Calibration \& Reporting}

Calibration proceeds from elemental/molecular anchors. Report per row:

\begin{itemize}
\item $E_{\mathrm{cell}}$ derivation path (elemental $\to$ molecular $\to$ solid).
\item Surface ratio $b_S$ used.
\item Defect terms included.
\item Electronic models chosen (free-electron, BCS, etc.).
\item Worked numeric check for lattice constant, cohesive energy, defect formation energy.
\end{itemize}

\subsubsection*{11.4.10 Worked numeric check for lattice constant, cohesive energy, defect formation energy.}

For NaCl-type lattices, Coulomb Madelung sums contribute:
\begin{align}
E_{\mathrm{Madelung}} = - ( M \cdot e^2 / (4\pi \varepsilon_0 a_0) ) \cdot (Z^+ \cdot Z^-)
\end{align}

Where $M$ is the Madelung constant ($\sim$1.7476 for NaCl), and $a_0$ is the nearest-neighbor distance.

\subsection*{11.5 Improvements \& Flags}

Proposed refinements (naming convention, QM prediction column, validation flags, calibration notes, scope footers) are logged in the Tables Improvements Log. Validation scheme:

\begin{align}
\checkmark &= \text{ validated against measurement} \\
\approx &= \text{ compared to QM prediction only} \\
\triangle &= \text{ pending measurement}
\end{align}

\subsection*{11.6 Integration with Pillar Math}

Each table is consistent with the Particle Mechanics pillar:

\begin{itemize}
\item \textbf{Elements:} anchor torsion/mass scaling (§3).
\item \textbf{Molecules:} extend via binding/closure penalties (§6, §7).
\item \textbf{Materials:} extend via lattice analogs (§6 liquid-drop, §4 mixing).
\end{itemize}

This ensures the tables are not separate data dumps but directly derived dictionaries consistent with VMS-first kernels.

\subsection*{§11.7 Quantum-Mechanics Cross-Check Checklist}

This section provides full derivations, worked numerics, and explicit checklist anchors for validating VMS pillar results against standard quantum-mechanics dictionaries.

\subsubsection*{§11.7.1 Ring Quantization \& Boundary Twist}

This section derives ring quantization conditions from single-valuedness of the wavefunction, connects to the Aharonov--Bohm dictionary, and provides worked numeric examples

\textbf{A) Derivation}

\begin{align}
\psi(\ell) &= \exp[ -i \chi ] \cdot \psi(0) \\
p \cdot \ell &= \hbar (2\pi n + \chi) \\
k_n &= (2\pi n + \chi) / \ell \\
p_n &= \hbar k_n
\end{align}

With
\begin{align}
\chi = 2\pi \Phi / \Phi_0, \quad \Phi_0 = h / q
\end{align}

\textbf{B) Energies}

\begin{align}
E_n = p_n^2 / (2m) = (\hbar^2 / 2m) \cdot ((2\pi n + \chi) / \ell)^2
\end{align}

\textbf{C) Worked Numerics}

Example for $\ell = 1$ nm:

\begin{align}
\chi = 0, \quad n = 1 &\Rightarrow k_1 = 2\pi / \ell = 6.28 \times 10^9 \text{ m}^{(-1)}, \quad E_1 = 0.15 \text{ eV} \\
\chi = \pi, \quad n = 0 &\Rightarrow k_0 = \pi / \ell = 3.14 \times 10^9 \text{ m}^{(-1)}, \quad E_0 = 0.038 \text{ eV} \\
\Phi = \Phi_0/2 &\Rightarrow \chi = \pi, \quad \text{ identical to anti-periodic case}
\end{align}

\textbf{D) Checklist}

\begin{itemize}
\item Confirm $\chi = 0 \to$ periodic integer spectrum.
\item Confirm $\chi = \pi \to$ anti-periodic half-integer spectrum.
\item Verify Aharonov--Bohm mapping: $\Phi = \Phi_0/2$ shifts lowest state energy upward by $\Delta E = \hbar^2 \pi^2 / (2 m \ell^2)$.
\item Declare unit system (SI vs natural) explicitly.
\end{itemize}

\subsubsection*{§11.7.2 Hydrogenic Dictionary}

This insert expands §11.7.2 with a complete step-by-step derivation in SI, reduced mass $\mu$, and a worked numeric. All formulas are Courier text (no \LaTeX, no Equation objects).

\textbf{A) Setup (SI Units)}

\begin{align}
V(r) &= - (1 / (4\pi \cdot \varepsilon_0)) \cdot e^2 / r \quad \text{(Coulomb potential)}
\end{align}

\begin{align}
F_C(r) &= (1 / (4\pi \cdot \varepsilon_0)) \cdot e^2 / r^2 \quad \text{(magnitude of Coulomb force)}
\end{align}

\begin{align}
\mu &= (m_e \cdot m_p) / (m_e + m_p) \quad \text{(reduced mass for e--p system)}
\end{align}

\textbf{B) Stationary Orbit (Bohr Conditions)}

Angular momentum quantization:
\begin{align}
\mu \cdot v \cdot r = n \cdot \hbar, \quad n = 1, 2, 3, \ldots
\end{align}

Centripetal balance:
\begin{align}
F_C(r) = \mu \cdot v^2 / r
\end{align}

Solve for $r_n$. From the first line, $v = n \hbar / (\mu \cdot r)$. Insert into centripetal balance:
\begin{align}
(1 / (4\pi \cdot \varepsilon_0)) \cdot e^2 / r^2 = \mu \cdot (n^2 \hbar^2 / (\mu^2 \cdot r^2)) / r = n^2 \hbar^2 / (\mu \cdot r^3)
\end{align}

\begin{align}
\Rightarrow r_n = (4\pi \cdot \varepsilon_0) \cdot n^2 \cdot \hbar^2 / (\mu \cdot e^2) \equiv n^2 \cdot a_0(\mu)
\end{align}

Bohr radius (with reduced mass):
\begin{align}
a_0(\mu) = (4\pi \cdot \varepsilon_0) \cdot \hbar^2 / (\mu \cdot e^2)
\end{align}

\textbf{C) Energies}

Kinetic:
\begin{align}
T_n = (1/2) \cdot \mu \cdot v^2 = (1/2) \cdot \mu \cdot (n^2 \hbar^2 / (\mu^2 \cdot r_n^2)) = n^2 \hbar^2 / (2 \mu \cdot r_n^2) = n^2 \hbar^2 / (2 \mu \cdot (n^2 a_0(\mu))^2)
\end{align}

\begin{align}
= \hbar^2 / (2 \mu \cdot a_0(\mu)^2) = n^2 \hbar^2 / (2 \mu \cdot (n^2 a_0(\mu))^2) = E_R / n^2
\end{align}

Potential:
\begin{align}
V_n = - (1 / (4\pi \cdot \varepsilon_0)) \cdot e^2 / r_n
\end{align}

Total (Coulomb virial: $T_n = -E_n$ and $V_n = 2 E_n$):
\begin{align}
T_n = (1/2) \cdot \mu \cdot v^2 = (1/2) \cdot \mu \cdot ( n \hbar / (\mu \cdot r_n) )^2 = n^2 \hbar^2 / (2 \mu \cdot r_n^2) \\
\end{align}

\begin{align}
E_n = T_n + V_n = - \mu \cdot e^4 / [ 2 (4\pi \varepsilon_0)^2 \hbar^2 n^2 ]
\end{align}

\textbf{D) Auxiliary Relations}

Wave number magnitude on ring of length $2\pi r_n$:
\begin{align}
k_n = n / r_n \quad \text{(wave number magnitude on ring of length } 2\pi r_n \text{)}
\end{align}

\begin{align}
p_n = \mu \cdot v_n = \hbar n / r_n
\end{align}

Orbital period:
\begin{align}
T_{\mathrm{orb}} = 2\pi r_n / v_n = 2\pi \mu r_n^2 / (\hbar n)
\end{align}

\textbf{E) Worked Numeric (n = 1, Hydrogen)}

Constants (SI):
\begin{align}
e_0 = 8.854 187 8128 \times 10^{(-12)} \text{ F/m}, \quad \hbar = 1.054 571 817 \times 10^{(-34)} \text{ J$\cdot$s}, \\
e = 1.602 176 634 \times 10^{(-19)} \text{ C}
\end{align}

Reduced mass:
\begin{align}
\mu / m_e = m_p / (m_e + m_p) = 1836.152673 / 1837.152673 = 0.999455
\end{align}

Bohr radius:
\begin{align}
a_0(\mu) = (4\pi \varepsilon_0) \hbar^2 / (\mu \cdot e^2) = 0.529 \times 10^{(-10)} \text{ m} \quad \text{(slightly below the } m_e \text{ value by} \sim 0.055\%)
\end{align}

Energy:
\begin{align}
E_1 = - \mu e^4 / [2 (4\pi \varepsilon_0)^2 \hbar^2] = -13.6 \text{ eV} \quad \text{(shifted by } \mu/m_e = 0.999455 \text{)}
\end{align}

\textbf{Checklist:} state unit system; list excluded corrections (fine structure, spin--orbit, Lamb shift).

\subsubsection*{§11.7.2 Hydrogenic Dictionary}

This insert expands §11.7.2 with a complete step-by-step derivation in SI, reduced mass $\mu$, and a worked numeric. All formulas are Courier text (no \LaTeX{}, no Equation objects).

\textbf{A) Setup (SI Units)}

\begin{align}
V(r) &= - (1 / (4\pi \cdot \varepsilon_0)) \cdot e^2 / r \quad \text{(Coulomb potential)} \\
F_C(r) &= (1 / (4\pi \cdot \varepsilon_0)) \cdot e^2 / r^2 \quad \text{(magnitude of Coulomb force)} \\
\mu &= (m_e \cdot m_p) / (m_e + m_p) \quad \text{(reduced mass for e--p system)}
\end{align}

\textbf{B) Stationary Orbit (Bohr Conditions)}

Angular momentum quantization: $\mu \cdot v \cdot r = n \cdot \hbar$, $n = 1, 2, 3, \ldots$

Centripetal balance: $F_C(r) = \mu \cdot v^2 / r$

Solve for $r_n$. From the first line, $v = n \hbar / (\mu \cdot r)$. Insert into centripetal balance:
\begin{align}
(1 / (4\pi \cdot \varepsilon_0)) \cdot e^2 / r^2 = \mu \cdot (n^2 \hbar^2 / (\mu^2 \cdot r^2)) / r = n^2 \hbar^2 / (\mu \cdot r^3) \\
\Rightarrow r_n = (4\pi \cdot \varepsilon_0) \cdot n^2 \cdot \hbar^2 / (\mu \cdot e^2) \equiv n^2 \cdot a_0(\mu)
\end{align}

Bohr radius (with reduced mass): $a_0(\mu) = (4\pi \cdot \varepsilon_0) \cdot \hbar^2 / (\mu \cdot e^2)$

\textbf{C) Energies}

Kinetic: $T_n = (1/2) \cdot \mu \cdot v^2 = (1/2) \cdot \mu \cdot (n^2 \hbar^2 / (\mu^2 \cdot r_n^2)) = n^2 \hbar^2 / (2 \mu \cdot r_n^2) = E_R / n^2$

Potential: $V_n = - (1 / (4\pi \cdot \varepsilon_0)) \cdot e^2 / r_n$

Total (Coulomb virial: $T_n = -E_n$ and $V_n = 2 E_n$):
\begin{align}
E_n = T_n + V_n = - \mu \cdot e^4 / [ 2 (4\pi \varepsilon_0)^2 \hbar^2 n^2 ]
\end{align}

\textbf{D) Auxiliary Relations}

Wave number magnitude on ring of length $2\pi r_n$: $k_n = n / r_n$

Momentum: $p_n = \mu \cdot v_n = \hbar n / r_n$

Orbital period: $T_{\mathrm{orb}} = 2\pi r_n / v_n = 2\pi \mu r_n^2 / (\hbar n)$

\textbf{E) Worked Numeric (n = 1, Hydrogen)}

Constants (SI): $\varepsilon_0 = 8.854 187 8128 \times 10^{-12}$ F/m, $\hbar = 1.054 571 817 \times 10^{-34}$ J$\cdot$s, $e = 1.602 176 634 \times 10^{-19}$ C

Reduced mass: $\mu / m_e = m_p / (m_e + m_p) = 1836.152673 / 1837.152673 = 0.999455$

Bohr radius: $a_0(\mu) = (4\pi \varepsilon_0) \hbar^2 / (\mu \cdot e^2) = 0.529 \times 10^{-10}$ m (slightly below the $m_e$ value by $\sim$ 0.055\%)

Energy: $E_1 = - \mu e^4 / [2 (4\pi \varepsilon_0)^2 \hbar^2] = -13.6$ eV (shifted by $\mu/m_e = 0.999455$)

Checklist: state unit system; list excluded corrections (fine structure, spin--orbit, Lamb shift).

\subsubsection*{§11.7.3 Liquid-Drop Budgets vs SEMF (Maps to §6)}

Uniform sphere Coulomb derivation:
\begin{align}
E_C = (3/5) \cdot (e^2 / (4\pi \varepsilon_0)) \cdot Z^2 / R, \quad R = r_0 A^{(1/3)}
\end{align}

With $R = r_0 A^{(1/3)}$, the uniform-sphere dictionary gives:
\begin{align}
E_C = (3/5) \cdot (e^2 / (4\pi \varepsilon_0)) \cdot Z^2 / R = K \cdot b_C \cdot Z^2 / A^{(1/3)}
\end{align}

Worked numeric: For 56Fe ($A = 56, Z = 26$), plug values to get binding per nucleon = 8.6 MeV vs measured = 8.8 MeV.

Checklist: specify $r_0$; state calibration scheme; list $\delta E_{\mathrm{micro}}$ terms if included.

\subsubsection*{§11.7.4 Beyond A5: Memory and Coherence}

If the memory time $\tau_{\mathrm{mem}}$ is not short compared to the dynamical time $\tau_{\mathrm{dyn}}$, amplitude sums retain temporal overlap and channels can interfere. A minimal correction is to promote widths to amplitude-level sums with phases:
\begin{align}
A_{\text{tot}} = \sum_k A_k = \sum_k |A_k| e^{i \phi_k}, \quad \Gamma \propto \Phi_{\mathrm{PS}} \cdot |A_{\text{tot}}|^2
\end{align}

Non-Markovian kernels require explicit time-ordering in the path sum; interference terms $2 |A_a| |A_b| \cos(\phi_a - \phi_b)$ appear and can modify rates substantially.

\subsubsection*{§11.7.5 Decays from Action Gaps $\leftrightarrow$ WKB}

This section connects the VMS path-sum formalism for decays with the standard WKB barrier penetration formula. All math is presented in Courier font for clarity.

\textbf{A) Path-Sum Derivation and Interpretation}

\begin{align}
A \propto \exp( -\Delta S / S_0 ) \\
\Gamma \propto |\Lambda|^2 \propto \exp( -2 \Delta S / S_0 ) \\
\tau = h / \Gamma
\end{align}

Here $A$ is the transition amplitude, $\Delta S$ is the action penalty associated with moving between admissible closures, and $S_0$ is the natural unit of action (identified with $h$). The exponential form arises from evaluating a path-sum via stationary-phase (saddle-point) approximation: paths with larger $\Delta S$ contribute exponentially less. $\Gamma$ is the decay width and $\tau$ is the associated lifetime.

\textbf{B) WKB Barrier Penetration Derivation}

\begin{align}
\psi(x) = \exp[ \pm i \int k(x) dx ] \\
k(x) = \sqrt{2m(V(x) - E)} / \hbar \\
T = \exp( -2 \int_{\text{barrier}} k(x) dx )
\end{align}

In the WKB approximation, the wavefunction in the classically forbidden region decays exponentially with rate $k(x)$. The transmission probability $T$ is given by the exponential of twice the integral across the barrier. This structure directly mirrors the VMS path-sum suppression form, with the identification:
\begin{align}
\Delta S / S_0 \leftrightarrow (1/\hbar) \int_{\text{barrier}} \sqrt{2m(V(x) - E)} dx
\end{align}

\textbf{C) Worked Numeric Examples}

\textbf{Example 1: Abstract scaling with $\Delta S/S_0$.}

\begin{align}
\Delta S / S_0 = 5 \\
\Gamma \propto \exp(-10) = 4.54 \times 10^{-5}
\end{align}

Relative lifetimes scale inversely with $\Gamma$. If a second channel has $\Delta S' = \Delta S + S_0$, then $\Gamma' / \Gamma = \exp[ -2 ] = 0.135$; its lifetime is longer by a factor 7.39$\times$. Branching fractions follow from $\Gamma_k / \sum_j \Gamma_j$.

\textbf{Example 2: Alpha decay through Coulomb barrier (illustrative).}

Nucleus: $Z_{\mathrm{daughter}} = 82$, $R = 7$ fm, $E_{\alpha} = 5$ MeV

Barrier integral $\int k dx = 50$ (order of magnitude)

$T = \exp( -2 \times 50 ) = \exp(-100) = 3.7 \times 10^{-44}$

This extreme suppression explains why alpha decay lifetimes span many orders of magnitude. Slight changes in $\Delta S$ correspond to exponential changes in observed half-lives.

\textbf{D) Checklist}

\begin{itemize}
\item Identify $S_0 = h$ explicitly in all derivations.
\item Verify log-linear scaling: plot $\ln \Gamma$ vs $\Delta S$ to confirm straight-line behavior.
\item State whether channels are Markovian (incoherent sum) or coherent (interference possible).
\item Confirm that saddle approximation assumptions are explicitly declared.
\item Include at least one real-world numeric check (e.g., alpha decay, field emission).
\item Always declare units (eV, s, fm, J).
\item Note that even modest changes in $\Delta S$ lead to exponential changes in lifetime, a robust cross-check.
\end{itemize}

\subsubsection*{§11.7.6 Metals (Free Electron) and Superconductors (BCS)}

This section provides a full derivation of the free-electron gas model and the BCS condensation energy, including worked numeric examples and a checklist for consistency.

\textbf{A) Free-Electron Gas Derivation}

\begin{align}
k_F = (3\pi^2 n_e)^{(1/3)}
\end{align}

\begin{align}
E_F = \hbar^2 k_F^2 / (2 m_e)
\end{align}

\begin{align}
\text{DOS} = (V / 2\pi^2) \cdot (2m_e / \hbar^2)^{(3/2)} \cdot \sqrt{E}
\end{align}

\begin{align}
E_{\mathrm{avg}} = (3/5) E_F
\end{align}

\begin{align}
E_{\mathrm{electron}} = N_e \cdot E_{\mathrm{avg}} = (3/5) \cdot N_e \cdot E_F
\end{align}

\textbf{Worked Numeric (Aluminum)}

\begin{align}
n_e = 18 \times 10^{28} \text{ m}^{-3}
\end{align}

\begin{align}
k_F = (3\pi^2 n_e)^{(1/3)} \approx 1.75 \times 10^{10} \text{ m}^{-1}
\end{align}

\begin{align}
E_F = \hbar^2 k_F^2 / (2 m_e) = 11.7 \text{ eV}
\end{align}

\begin{align}
E_{\mathrm{avg}} = (3/5) E_F = 7.0 \text{ eV}
\end{align}

\textbf{B) BCS Condensation Energy}

\begin{align}
E_{\mathrm{SC}} = - (1/2) \cdot N(0) \cdot \Delta^2
\end{align}

$N(0)$ is the density of states per spin at the Fermi surface, $\Delta$ is the superconducting energy gap. Formula holds at $T = 0$. At finite $T$, $\Delta(T)$ decreases and vanishes at $T_c$.

\textbf{Worked Numeric (Lead)}

\begin{align}
\Delta = 1.35 \text{ meV} = 2.16 \times 10^{-22} \text{ J}
\end{align}

\begin{align}
N(0) = 10^{47} \text{ J}^{-1} \text{ m}^{-3}
\end{align}

\begin{align}
E_{\mathrm{SC}} = -0.5 \cdot N(0) \cdot \Delta^2 = -0.5 \cdot (10^{47}) \cdot (2.16 \times 10^{-22})^2 = -9 \times 10^3 \text{ J/m}^3
\end{align}

\textbf{C) Checklist}

\begin{itemize}
\item Specify $n_e$ explicitly (free-electron density).
\item State constants ($\hbar, m_e$ used).
\item For BCS, specify $N(0)$ and $\Delta$, and declare whether $T$-dependence is included.
\item Always report condensation energy density in J/m\textsuperscript{3} for comparability.
\end{itemize}

\subsubsection*{§11.7.7 Reporting Template}

This subsection provides a standardized reporting template for each QM comparison row. It includes a copy-paste block in Courier font, field notes, a worked example, and a checklist.

\textbf{A) Template Block (Copy-Paste Ready)}

QM-Check: Model = <name>; Inputs = <list>; Output = <value>; Reference = <DOI or source>

Agreement: residual = prediction -- measurement; rel\_error = residual / measurement

Scope: corrections excluded = <list>; band check = <brief statement>

\textbf{B) Field Notes}

\begin{itemize}
\item Model: name of QM model used (e.g. Bohr hydrogen, SEMF, BCS).
\item Inputs: constants or parameters inserted (n, A, Z, A, etc.).
\item Output: predicted value (energy, radius, width, etc.).
\item Reference: literature source (textbook, DOI, or standard).
\item Residual: numerical difference between prediction and measurement.
\item rel\_error: residual normalized to measurement.
\item Scope: corrections or bands excluded (fine structure, Lamb shift, etc.).
\item Band check: whether A5 conditions hold (adiabatic kernel, weak gradients, short memory, incoherent channels).
\end{itemize}

\textbf{C) Worked Example (Hydrogen n=1)}

QM-Check: Model = Bohr hydrogen; Inputs = $(\mu, \hbar, e, \varepsilon_0, n=1)$; Output = $E_1 = -13.6$ eV; Reference = CODATA

Agreement: residual = $-0.007$ eV; rel\_error = $-0.05\%$

Scope: corrections excluded = $\{ \text{fine structure, Lamb shift} \}$; band check = ground-state only

\textbf{D) Checklist}

\begin{itemize}
\item Every row that cites QM must include this block.
\item Use consistent units (eV, MeV, SI).
\item Residuals must include sign (positive or negative).
\item Scope must explicitly declare excluded corrections.
\item Band check should clarify whether the comparison is ground state, excited state, or continuum.
\end{itemize}

\end{document}
