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\title{Particles Mechanics Math Walk-Through}
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We pick up directly from the canonical Foundation Math Walk-Through. We carry a single dimensional scale $S_0 = h$ (SI-locked via the electron anchor) and introduce no new dimensional scales here. All particle structure follows from the same route/closure variational framework; base locks are not re-tuned.

\section*{Foundation Bridge (Display-Area $\rightarrow$ Phase $\rightarrow$ Closed Harmonics)}

Display-area counts patches, action sums them, phase labels them. Closed harmonics are the only quantization rule we need.

\begin{align*}
A_d &= \pi r^2 \\
S[\text{path}] &= \int A_d \cdot ds \\
\varphi &= S / S_0 \quad (S_0 = h) \\
S_{\text{loop}} / S_0 &= 2\pi n \quad (n \in \mathbb{Z})
\end{align*}

\section*{Particles Math Walk-Through (Picks up from Canonical Foundation)}

What we build, VMS-first, with short classical/SM dictionaries only:

\begin{itemize}
\item Quantization ladder from closed harmonics (integer $n$ labels the class: charge steps; spin classes).

\item Family structure from route symmetries (stabilizers mirror U(1)/SU(2)/SU(3) without importing gauge furniture).

\item Mass patterns as ratios from closure-tension $\times$ geometry (no new scales):
  \begin{itemize}
  \item $E_k = \int_0^\ell \tau_k(s) \, ds$ (dimensionless)
  \item $m_i/m_j \approx (E_i/E_j) \, G_{ij}$
  \end{itemize}

\item Mixing from overlap (2-state shown; 3-state skeleton in Appendix):
  \[
  \tan(2\theta) = 2\kappa / \Delta
  \]

\item Widths and lifetimes from an action-gap escape map (ratios first):
  \[
  \Gamma_i / \Gamma_j \approx (\Omega_i / \Omega_j) \cdot e^{\{-(\Delta S_i - \Delta S_j)/S_0\}}
  \]

\end{itemize}

No new tunables. Ratios/bands are published first; calibrated absolutes live in the Appendix/Calibration.

\section*{0) Locks, Scope, and Dictionary Discipline}

\begin{itemize}
\item Locks: $S_0 = h$ (single dimensional scale). Speed of advance $c$ is an acceptance lock. No other units-bearing knobs.

\item Ontology: display-area $A_d = \pi r^2$; route action $S[\text{path}] = \int A_d \, ds$; phase label $\varphi = S/S_0$.

\item Scope: particle classes (charge steps, spin classes), family structure, mass ratios, mixing, widths/lifetimes — all as dimensionless relations. Catalog details and calibrations live in the Pillar Math Appendix.

\begin{align*}
A_d &= \pi r^2 \\
S[\text{path}] &= \int A_d \, ds \\
\varphi &= S / S_0 \quad (S_0 = h)
\end{align*}

\end{itemize}

\section*{1) Display-Area $\rightarrow$ Phase $\rightarrow$ Closed Harmonics (with stability checks)}

\textbf{Step 1 --- Action bookkeeping.} Along any route, sum the hidden display-area patches ($A_d$) to get $S[\text{path}]$. When we need a phase we divide by $S_0$ so $\varphi$ is dimensionless.

\[
S[\text{path}] = \int A_d \, ds = \varphi[\text{path}] = S[\text{path}]/S_0
\]

\textbf{Step 2 --- Caustic optionality.} Real geometry creates folds/foci where multiple near-optimal routes exist. If the spread of near-ties is on the same scale as the accumulating display-area, the front oscillates between options instead of averaging out.

\textbf{Step 3 --- Harmonic closure.} If the geometry feeds the front back to its starting facing after one lap, the loop is closed. Stability requires an exact hand-off of facing, i.e., an integer number of phase turns:

\[
S_{\text{loop}}/S_0 = 2\pi n \, S_0. \quad n \text{ is an integer.}
\]

\textit{Interpretation.} The integer $n$ labels the loop class. These integers are the source of charge steps and spin classes. No separate quantization postulate is imported.

\section*{2) Quantization Ladder (Charge Steps; Spin Classes)}

Closed-loop integer $n$ is the class label. At catalog level:

\[
q = n \, q_0 \quad (\text{integer step } q_0)
\]

spin class: $n$ even $\rightarrow$ boson; $n$ odd $\rightarrow$ fermion

Why this is enough. The loop can hand off its facing only at integers; parity of $n$ tracks the symmetry under exchange (class-level statement; anomaly/symmetry constraints are carried in the Appendix).

\section*{3) Route Symmetries $\rightarrow$ Family Structure (Gauge Dictionary)}

We identify the invariances of $S[\text{path}]$ and the boundary rules that preserve closure. The stabilizer of those invariances partitions couplings into families. Coarse-grained, the connected components mirror U(1), SU(2), SU(3). We do not assume gauge furniture a priori.

Let $\Theta = \{g : S[g \cdot \text{path}] = S[\text{path}]$ and boundary rules preserved$\}$.

$\text{Stab}(\Theta) = \{h \in \Theta : h \cdot \text{pattern} = \text{pattern}\}$.

$\text{Stab}(\Theta)$ acts on couplings $\Rightarrow$ families of couplings (labels: U(1), SU(2), SU(3)).

Let $\Theta$ be the set of transformations that leave the action $S$ and the boundary rules unchanged. The stabilizer $\text{Stab}(\Theta)$ (the elements that keep a given closed pattern fixed) acts on couplings and partitions them into families; under coarse-graining those families mirror U(1), SU(2), SU(3).

\textit{Appendix:} explicit generators and admissibility constraints; here we keep the derivation ladder and the dictionary only.

\section*{4) Mass Patterns --- Closure Tension $\times$ Coupling Geometry (Ratios-First)}

\textit{Idea} --- The closed pattern stores a dimensionless ``tension count'' around its loop. Effective mass scales with that stored count and with how the pattern couples to its environment/other loops. Derive the ratio in three steps.

\textbf{Step A} --- Define a dimensionless stored-tension along the loop. Let $\tau(s)$ be the (dimensionless) closure-tension density and $\ell$ the loop length (dimensionless).

\[
E_{\text{loop}} \equiv \int_0^\ell \ell \, \tau(s) \, ds \approx \bar{\tau} \, \ell
\]

\textbf{Step B} --- Relate effective mass to the loop tension count. The SI lock enters only through $S_0$ when phases are read; the proportionality cancels in ratios:

\[
m \propto E_{\text{loop}}
\]

\textbf{Step C} --- Allow dimensionless geometry in couplings (overlaps, background structure) via $G$. Then:

\[
m_i / m_j \approx (\bar{\tau}_i \, \ell_i) / (\bar{\tau}_j \, \ell_j) \times G_{\{ij\}}
\]

\textit{Notes} --- $G_{\{ij\}}$ is a pure number from coupling geometry; no new dimensional constants appear. The lepton ladder can be fit as one $G$ map; details in the Appendix.

\section*{5) Mixing --- Overlap Diagonalization (2-State and 3-State Skeleton)}

Mixing here is derived from overlap of two closed VMS patterns that are nearly the same shape. We coarse-grain the route counting to a $2 \times 2$ effective energy map built from (detuning $\Delta$) and (overlap $\kappa$). This is not an import of quantum postulates; it's the VMS overlap operator written in the smallest non-trivial basis. Numbers are reported as ratios (units absorbed).

\textit{Effective $2 \times 2$ model} --- define detuning and write the matrix:

\[
H = \begin{bmatrix} E_1, \kappa \\ \kappa, E_2 \end{bmatrix}
\]

\[
\Delta \equiv E_1 - E_2
\]

Diagonalize by a rotation through angle $\theta$ (kill the off-diagonal):

\[
R(\theta) = \begin{bmatrix} \cos\theta, -\sin\theta \\ \sin\theta, \cos\theta \end{bmatrix}
\]

\[
\tan(2\theta) = 2\kappa / \Delta
\]

Eigenvalues after rotation:

\[
\lambda_+ = \tfrac{1}{2} \left( E_1 + E_2 + \sqrt{(\Delta^2 + 4\kappa^2)} \right)
\]

\[
\lambda_- = \tfrac{1}{2} \left( E_1 + E_2 - \sqrt{(\Delta^2 + 4\kappa^2)} \right)
\]

3-state skeleton (dictionary only) --- use a unitary with one CP-phase:

\[
U = U(\theta_{12}, \theta_{23}, \theta_{13}, \delta)
\]

\textit{Appendix note} --- The Appendix carries the explicit $3 \times 3$ parameterization and ratio-first maps to CKM/PMNS-style sets.

\section*{6) Decays \& Widths --- Escape Map from Action Gaps (Full Steps)}

\subsection*{Mass Patterns --- Closure Tension $\times$ Coupling Geometry (Ratios-First)}

\textit{Idea} --- A closed VMS pattern stores a dimensionless ``tension count'' around the loop. Effective mass scales with that stored count and with how the pattern couples to its environment/other loops. We publish ratios so all global proportionalities cancel.

\textbf{Step A} --- Define the dimensionless stored-tension along the loop. Let $\tau(s)$ be the closure-tension density and $\ell$ the loop length (both dimensionless under our normalization):

\[
E_{\text{loop}} \equiv \int_0^\ell \ell \, \tau(s) \, ds \approx \bar{\tau} \cdot \ell
\]

\textbf{Step B} --- Relate effective mass to the tension count. The SI lock enters only through $S_0$ when phases are read; the proportionality cancels in ratios:

\[
m \propto E_{\text{loop}}
\]

\textbf{Step C} --- Include dimensionless coupling-geometry differences between patterns $i$ and $j$ via $G_{\{ij\}}$:

\[
m_i / m_j \approx (\bar{\tau}_i \cdot \ell_i) / (\bar{\tau}_j \cdot \ell_j) \times G_{\{ij\}}
\]

\textit{Why this is VMS-first} --- $\bar{\tau}$ and $\ell$ come from the closed-route geometry (no imported constitutive laws), and $G_{\{ij\}}$ is a pure number from overlap/coupling geometry. No new dimensional scales appear; one global calibration (handled elsewhere) is sufficient.

\section*{7) Mixing --- Overlap Diagonalization (VMS-First)}

\textit{VMS framing} --- Two closed patterns that are nearly the same shape can overlap. Coarse-graining the route counting to the smallest non-trivial basis gives a $2 \times 2$ effective map built from detuning $\Delta$ and overlap $\kappa$. This is not an imported quantum axiom; it's the VMS overlap operator in a tiny basis. We report ratios so units do not matter.

\textit{Effective $2 \times 2$ model (define detuning and write the matrix):}

\[
H = \begin{bmatrix} E_1, \kappa \\ \kappa, E_2 \end{bmatrix}
\]

\[
\Delta \equiv E_1 - E_2
\]

Diagonalize by a real rotation $R(\theta)$ that kills the off-diagonal term:

\[
R(\theta) = \begin{bmatrix} \cos\theta, -\sin\theta \\ \sin\theta, \cos\theta \end{bmatrix}
\]

\[
\tan(2\theta) = 2\kappa / \Delta
\]

Eigenvalues after rotation (observed levels):

\[
\lambda_+ = \tfrac{1}{2} \left( E_1 + E_2 + \sqrt{(\Delta^2 + 4\kappa^2)} \right)
\]

\[
\lambda_- = \tfrac{1}{2} \left( E_1 + E_2 - \sqrt{(\Delta^2 + 4\kappa^2)} \right)
\]

\textit{Interpretation} --- Large overlap ($\kappa$) relative to detuning ($\Delta$) gives a big rotation $\theta$ and a large level split; small $\kappa/\Delta$ gives a small rotation and a small split. Branching-ratio patterns and oscillations follow from the same rotation (see Appendix for the amplitude algebra).

3-state skeleton (dictionary only) --- When three near-patterns mix, use a unitary with one CP-phase:

\[
U = U( \theta_{12}, \theta_{23}, \theta_{13}, \delta )
\]

We keep the $3 \times 3$ parameterization and CKM/PMNS-style dictionaries in the Appendix; here we only carry the VMS-first ladder and the minimal formula.

\section*{8) Composites --- Binding Algebra with 2-Loop and 3-Loop Examples}

\textit{Idea} --- A composite is a bound set of closed VMS patterns (loops). Binding means the total route-action decreases when the loops couple admissibly (hand-offs allowed, no sideways pile-up). We stay ratio-first: global proportionalities cancel.

\subsection*{Definitions (what each symbol means)}

\begin{itemize}
\item $S_i$ --- route-action of loop $i$ when isolated (dimensionless under our normalization).

\item $S_{\text{int}(i,j), S_{\text{int}}(1,2,3,\ldots)}$ --- action changes from admissible couplings (geometry-only, dimensionless).

\item $E_{\text{loop}, i} = \int_0^{\ell_i} \ell \, \tau_i(s) \, ds = \bar{\tau}_i \, \ell_i$ --- loop ``tension count'' (dimensionless).

\item $\kappa_{ij}$ --- effective overlap/coupling between loops $i$ and $j$ (dimensionless, sign fixed by admissibility).

\item $\Delta$ --- detuning between uncoupled loop counts ($\Delta = E_1 - E_2$ in 2-loop case).

\end{itemize}

\subsection*{Binding condition and mass deficit (general)}

\[
S_{\text{total}}^{\{{\text{separate}}\}} = \sum_i S_i
\]

\[
S_{\text{total}}^{\{{\text{bound}}\}} = \sum_i S_i + \sum_{\{i<j\}} S_{\text{int}}(ij) + S_{\text{int}}(1,2,3,\ldots)
\]

\[
\Delta S_{\text{bind}} \equiv S_{\text{total}}^{\{{\text{separate}}\}} - S_{\text{total}}^{\{{\text{bound}}\}} > 0 \quad (\text{bound if positive})
\]

In mass ratios (global proportionality cancels), the composite sits below the sum of the parts by a geometry factor $G_{\text{bind}}$:

\[
m_{\{\text{comp}\}} / (\sum_i m_i) \approx 1 - \left( \Delta S_{\text{bind}} / \sum_i E_{\text{loop}, i} \right) \cdot G_{\{\text{bind}\}}
\]

\subsection*{Two-Loop Composite --- Bonding/Antibonding split (worked)}

Coarse-grain the coupled pair to a $2 \times 2$ effective map built from uncoupled counts ($E_1, E_2$) and a coupling $\kappa$ ($>0$ for admissible bonding).

\[
H = \begin{bmatrix} E_1, -\kappa \\ -\kappa, E_2 \end{bmatrix}
\]

\[
\Delta \equiv E_1 - E_2
\]

Diagonalize with a real rotation $R(\theta)$ that kills the off-diagonal. Read both the mixing angle and the energy split directly:

\[
R(\theta) = \begin{bmatrix} \cos\theta, -\sin\theta \\ \sin\theta, \cos\theta \end{bmatrix}
\]

\[
\tan(2\theta) = 2\kappa / \Delta
\]

\[
\lambda_\pm = \tfrac{1}{2} \left( E_1 + E_2 \pm \sqrt{(\Delta^2 + 4\kappa^2)} \right)
\]

Binding shows up as the lower mode sitting below either part:

\[
\lambda_- < \min(E_1, E_2)
\]

The composite's effective loop count tracks $\lambda_-$. The action gain (dimensionless) relative to separate parts is:

\[
\Delta S_{\text{bind}} / S_0 \approx (E_1 + E_2 - 2\lambda_-) / S_0
\]

Lifetime still follows the action-gap law but now from the bound pattern:

\[
\Gamma_{\{\text{comp}\}} \approx \nu \cdot Q \cdot \Omega \cdot \exp\left( -\Delta S_{\text{exit}}/S_0 \right)
\]

\subsection*{Three-Loop Ring/Braid --- Clear $3 \times 3$ form and special case}

Write a symmetric $3 \times 3$ with admissible couplings $\kappa_{ij}$ (sign chosen so that negative off-diagonals produce a bonding drop).

\textit{Matrix form (readable layout):}

\[
H = \begin{bmatrix}
E_1, & -\kappa_{12}, & -\kappa_{13} \\
-\kappa_{12}, & E_2, & -\kappa_{23} \\
-\kappa_{13}, & -\kappa_{23}, & E_3
\end{bmatrix}
\]

\textit{Binding criterion} --- after diagonalization, the lowest eigenvalue lies below all three uncoupled counts:

\[
\lambda_{\min} < \min(E_1, E_2, E_3)
\]

\textit{Special analytic check (equal loops and equal couplings:} $E_1=E_2=E_3=E, \kappa_{12}=\kappa_{23}=\kappa_{13}=\kappa \to 0$). Then the eigen-pattern is closed form:

\[
\lambda_{\text{bond}} = E - 2\kappa \quad (\text{symmetric bonding mode})
\]

\[
\lambda_{\text{anti}} = E + \kappa \quad (\text{twofold degenerate})
\]

A bound composite exists if $E - 2\kappa < E \Rightarrow \kappa > 0$ (with admissible hand-offs). The eigenvector for the bonding mode is proportional to $[1, 1, 1]$ (all loops in-phase).

If couplings are unequal, numerically diagonalize the $3 \times 3$ above; the same binding criterion on $\lambda_{\min}$ applies.

\subsection*{Admissibility and stability (what to check)}

\begin{itemize}
\item Integer locks across the layout --- each loop's hand-off remains integer-clean and the composite symmetry is admissible.

\item Exit hardness --- the cheapest allowed escape from the bound pattern obeys $\Delta S_{\text{exit}}/S_0 \gtrsim \Lambda$ ($\Lambda$ modest).

\item Quasi-static background --- keep the caustic/coupling geometry slow vs.\ the composite cycle time; otherwise expect drift.

\end{itemize}

\subsection*{Worked micro-check (numbers as a sanity test)}

Take $E_1=1.00, E_2=1.10, \kappa=0.08$. Then $\Delta=-0.10, \tan(2\theta)=2 \cdot 0.08 / -0.10 = -1.6 \Rightarrow \theta=-0.688$ rad ($\approx -39.4°$).

Eigenvalues: $\lambda_\pm = \tfrac{1}{2}(2.10 \pm \sqrt{0.01 + 0.0256}) = \tfrac{1}{2}(2.10 \pm 0.0356) = \tfrac{1}{2}(2.10 \pm 0.1887) \Rightarrow \lambda_+ = 0.9557, \lambda_- = 1.1443$.

Binding check: $\lambda_- < \min(1.00, 1.10) = 1.00$ $\checkmark$ (composite is bound).

\section*{Proton, Neutron, and ``Apparent Size'' from Prying (VMS predictions)}

\subsection*{Three-loop composite picture}

Model each baryon (proton or neutron) as a three-loop composite with uncoupled counts $E_i$ and admissible couplings $\kappa_{ij}$ determined by the layout/orientation. In the symmetric baseline ($E_i = E$ and $\kappa_{ij} = \kappa$), the bonding eigenvalue is $\lambda_{\text{bond}} = E - 2\kappa$. Small orientation-driven differences $(\Delta \kappa_{ij})$ shift the bonding level and set the mass split.

\subsection*{Baseline and first-order shift (Rayleigh quotient)}

For small asymmetries around the symmetric bonding vector $v = [1,1,1]/\sqrt{3}$, the first-order shift of the bonding level is the average coupling change seen by $v$:

\[
\lambda_{\text{bond}} \approx E - (2/3) \cdot ( \kappa_{12} + \kappa_{23} + \kappa_{13} )
\]

Therefore the \emph{proton---neutron mass difference} in this picture follows the $**$sum of couplings$**$:

\[
\lambda_{\text{bond}}^{[n]} - \lambda_{\text{bond}}^{[p]} \approx (2/3) \cdot \left[ (\kappa_{12} + \kappa_{23} + \kappa_{13})^{[p]} - (\kappa_{12} + \kappa_{23} + \kappa_{13})^{[n]} \right]
\]

\textit{Interpretation} --- If the neutron's admissible couplings sum to a $**$smaller$**$ value than the proton's (slightly less binding), then $\lambda_{\text{bond}}^{[n]} > \lambda_{\text{bond}}^{[p]}$ and the neutron is heavier, as observed.

\subsection*{Ratio-first prediction band (no new scales)}

Use mass $\propto$ bonding level. With $E$ as a shared scale (drops in ratios):

\[
(m_n - m_p)/m_p \approx \left[ \lambda_{\text{bond}}^{[n]} - \lambda_{\text{bond}}^{[p]} \right] / \lambda_{\text{bond}}^{[p]}
\]

\[
\approx (2/3) \cdot \left[ (\sum \kappa)^{[p]} - (\sum \kappa)^{[n]} \right] / \lambda_{\text{bond}}^{[p]}
\]

A tiny fractional difference in the $**$sum of couplings$**$ (order $10^{-3}$) produces the observed $O(10^{-3})$ neutron---proton mass ratio split. You can fit the single number $\Delta(\Sigma \kappa)$ once and carry it across related observables (decay/width bands).

\subsection*{``Apparent size'' from prying (compliance)}

Define a small pry coordinate $a$ that opens the three-loop layout (e.g., a symmetric radial separation). The bonding level responds as:

\[
\lambda_{\text{bond}}(a) \approx E - (2/3) \cdot \left[ \kappa_{12}(a) + \kappa_{23}(a) + \kappa_{13}(a) \right]
\]

The effective stiffness against pry is the curvature:

\[
k_{\text{eff}} = - d^2\lambda_{\text{bond}}/da^2 = (2/3) \cdot \left[ -\kappa_{12}''(a) - \kappa_{23}''(a) - \kappa_{13}''(a) \right]
\]

Apparent size tracks compliance (larger compliance $\Rightarrow$ larger apparent radius under small-q probes). As a dictionary to scattering form factors at small $q$:

\[
r_{\{\text{app}\}}^2 \propto 1 / k_{\text{eff}} \quad (\text{dictionary only: slope of form factor} \leftrightarrow \text{compliance})
\]

\textit{Prediction} --- If the neutron's couplings are slightly weaker (smaller $\Sigma\kappa$), it should also be \emph{softer} (larger compliance) under the same pry, giving a larger $r_{\text{app}}$ in this sense.

\subsection*{Verification steps (how to test)}

\begin{enumerate}
\item Sign of the mass split --- Verify that any admissible orientation that reduces $\Sigma\kappa$ for the neutron predicts $m_n > m_p$ (correct sign).

\item Magnitude band --- Fit $\Delta(\Sigma\kappa)$ from the mass ratio once; propagate to width-ratio bands where the exit map matches ($\Omega, \nu$ common).

\item Prying compliance --- Measure small-q slopes (dictionary) or direct pry response in a controlled setup; check $C_n > C_p$ at low pry if $\Sigma\kappa^{[n]} < \Sigma\kappa^{[p]}$.

\item Stability --- Ensure integer locks hold for the composite and $\Delta S_{\text{exit}}/S_0$ sits in the long-lived band; otherwise, expect broadened widths.

\end{enumerate}

\subsection*{Stability \& Selection Rules (Quick)}

\begin{itemize}
\item Integer locks must remain admissible across the coupled layout (no forbidden hand-offs).

\item Exit hardness must be significant: $\Delta S_{\text{exit}}/S_0 \gtrsim \Lambda$ ($\Lambda$ modest).

\item Background must be quasi-static on the composite cycle time to avoid wandering of $n$ or $\theta$.

\end{itemize}

\section*{9) Nuclear Binding, Strong/Weak, and Atomic}

Locks and stance --- VMS-first, ratio-first. Single dimensional scale $S_0 = h$ (SI-locked). Binding and forces are gradients of the same route-action bookkeeping. No imported force laws; the shapes come from overlap kernels and admissibility (hand-off rules) on closed patterns.

\subsection*{9.a) A-Body Binding: VMS Liquid-Drop with Derivation Sketch}

\subsubsection*{9.a.1 Smooth binding from geometry/tension budgets}

We split smooth (bulk) contributions from shell oscillations. Smooth terms follow counting of loop patches (volume), their boundary (surface), orientation-charged overlap (Coulomb dictionary from EM), and proton---neutron imbalance (asymmetry). All coefficients are dimensionless composites of tension/geometry integrals; calibrated once.

\begin{align*}
\Delta S_{\text{bind}}^{\{\text{smooth}\}}(A, Z) / S_0 &= \alpha_V A - \alpha_S A^{2/3} - \alpha_C \left[ Z(Z-1) \right] \\
A^2 &[-1/3] - \alpha_A (A - 2Z)^2 / A + \alpha_G G(A, Z)
\end{align*}

\textit{Why the exponents:} in a roughly compact 3D layout, occupied loop patches scale like $A$ (volume), boundary like $A^{2/3}$ (area), and mean separation like $A^{1/3}$, giving $A^{\{-1/3\}}$ in the Coulomb dictionary term. The asymmetry penalty is quadratic in proton---neutron imbalance divided by size. $G(A, Z)$ collects admissible layout effects (pure number).

\[
R(A) \propto A^{\{1/3\}}
\]

\[
\text{Area} \propto A^{\{2/3\}} \quad ; \quad \text{Mean spacing} \propto A^{\{1/3\}}
\]

\subsubsection*{9.a.2 Shell correction from discrete closure (magic numbers)}

Closed integers in a mean field add extra binding near specific counts (magic numbers) for protons and neutrons. We use a smooth dip near each magic value; nearest shell dominates:

\[
\Delta S_{\text{shell}}(A, Z)/S_0 = - \sum_{\{x \in \{p, n\}\}} \beta_x \cdot \max_k \exp\left( - (N_x - N_{\{\text{magic}, x\}})^{\wedge 2} / (2 \sigma_x^{\wedge 2}) \right)
\]

Nearest-shell approximation: the $\max_k$ picks the closest magic number; \emph{for smooth gradients in fitting, replace max with a log-sum-exp over shells.}

Here $N_p = Z, N_n = A - Z$. Amplitudes $\beta_x$ and widths $\sigma_x$ are dimensionless, set once.

\subsubsection*{9.a.3 Total binding and mass dictionary (ratios)}

Calibration set (fit once, carry across predictions): $\{\alpha_V, \alpha_S, \alpha_C, \alpha_A, \alpha_G, \beta_p, \beta_n, \sigma_p, \sigma_n\}$

\[
\Delta S_{\text{bind}}(A, Z)/S_0 = \Delta S_{\text{bind}}^{\{\text{smooth}\}}/S_0 + \Delta S_{\text{shell}}/S_0
\]

\[
m_{\{A, Z\}} / (\sum_i m_i) \approx 1 - \left[ \Delta S_{\text{bind}}(A, Z) / \sum_i E_{\text{loop}, i} \right] \cdot G_{\{\text{bind}\}}(A, Z)
\]

We publish ratios (per-constituent or to a reference nucleus). Absolutes appear only after a one-time calibration.

\subsection*{9.b) How ``Strong'' Falls Out (Short-Range Attraction with Core)}

In VMS, attraction between nucleons is a drop in total action from admissible overlap of closed loops; range is limited by caustic spread, and a very-short-range penalty prevents sideways pile-up (hard core). Write a two-body action kernel in separation $r$:

\[
\Delta S_{\{\text{NN}\}}(r)/S_0 = - K_{\{\text{att}\}} \cdot \exp( - r / \ell_s ) + K_{\{\text{core}\}} \cdot \exp\left( -(r/r_c)^p \right)
\]

\begin{itemize}
\item $\ell_s$ is a short range set by caustic size; $r_c$ is a core scale; $p \geq 2$ is steep. $K_{\text{att}}, K_{\text{core}}$ are dimensionless strengths from overlap geometry. The ``force'' profile is the gradient of this action ($w$~r.t. $r$ (dictionary to conventional force via acceptance locks):

\[
F_s(r) \propto - d[\Delta S_{\{\text{NN}\}}(r)/S_0] / dr = - (K_{\text{att}}/\ell_s) \exp( - r/\ell_s ) + (K_{\text{core}} \, p / r_c) (r/r_c)^{p-1} \exp\left( -(r/r_c)^p \right)
\]

\end{itemize}

This produces an attractive well at 1--2 short-range units with a repulsive core at very small $r$, matching qualitative nucleon---nucleon phenomenology.

\subsection*{Many-body saturation}

follows because each loop only overlaps a finite neighborhood at short range; the net per-particle gain saturates, giving the volume term with a surface subtraction ($\alpha_V, \alpha_S$).

\subsection*{9.c) How ``Weak'' Falls Out (Class-Changing, Short-Range, Chiral Gate)}

Amplitude-level transition probability (per attempt), with a short-range kernel and an action gap:

\[
P_{\{\text{trans}\}} \approx \Omega_w \cdot \int K_w(r) \, dr \cdot e^{\{ - \Delta S_{\{\text{trans}\}} / S_0 \}}
\]

Width (rate) scales as amplitude squared times phase-space:

\[
\Gamma \propto G_w^{\wedge 2} \cdot \Phi_{\{PS\}} \cdot e^{\{ - 2 \Delta S_{\{\text{trans}\}} / S_0 \}}
\]

Contact strength (short-range integral):

\[
G_w \equiv \int K_w(r) \, dr
\]

\subsection*{Symbols \& Notes}

\begin{itemize}
\item $\Omega_w$ --- orientation/handedness acceptance (chiral gate; dimensionless).

\item $K_w(r)$ --- short-range overlap kernel (dimensionless; rapidly decays with $r$).

\item $\Delta S_{\text{trans}}$ --- action gap to the allowed class-changing exit (dimensionless).

\item $S_0$ --- the single dimensional scale; we lock $S_0 = h$ (SI).

\item $G_w$ --- contact strength from the short-range kernel (dimensionless in our units).

\item $\Phi_{\{PS\}}$ --- phase-space factor (dimensionless).

\item $\Gamma$ --- partial width (rate); totals add as $\Gamma_{\text{total}} = \sum_i \Gamma_i$ across channels $i$.

\end{itemize}

Narrative link: amplitude $\propto e^{\{-\Delta S/S_0\}}$; squaring the amplitude yields the $e^{\{-2\Delta S/S_0\}}$ in widths. Chirality enters via $\Omega_w$ (selection rule), not as an added force. Ratio-first reporting is preferred; absolute calibration is optional.

\subsection*{9.d) Admissibility + Pairing (Many-Loop Rules)}

Admissible composites keep integer hand-offs per constituent and a composite symmetry that doesn't force sideways pile-up. Pairing is the even---even advantage from making hand-offs come in balanced pairs (dimensionless add-on):

\[
\delta_{\text{pair}}(A, Z) = + \alpha_P \cdot A^{\{-3/4\}} \quad (\text{even } Z, \text{ even } N)
\]

\[
\delta_{\text{pair}}(A, Z) = 0 \quad (\text{odd } A)
\]

\[
\delta_{\text{pair}}(A, Z) = - \alpha_P \cdot A^{\{-3/4\}} \quad (\text{odd } Z, \text{ odd } N)
\]

Add $\delta_{\text{pair}}$ to $\Delta S_{\text{bind}}^{\{\text{smooth}\}}$ before the shell term. $\alpha_P$ is dimensionless and calibrated once.

\section*{9.e) Worked Nuclear Examples (Ratios-First)}

\subsection*{9.e.1 Deuteron ($A = 2, Z = 1$) --- shallow bind}

Use the two-body kernel $\Delta S_{\text{NN}}(r)/S_0$ above. There is a shallow bound if the attraction overwhelms the core at some $r = r_0$:

\[
d[\Delta S_{\{\text{NN}\}}(r)/S_0] / dr \big|_{r=r_0} = 0, \quad \Delta S_{\{\text{NN}\}}(r_0)/S_0 > 0
\]

\[
(d/dr) \Delta S_{\{\text{NN}\}}(r) = 0 \Rightarrow (K_{\text{att}}/\ell_s) e^{\{-r/\ell_s\}} = (2 K_{\text{core}} / r_c^{\{2\}}) \, r e^{\{-(r/r_c)^{\{2\}}\}}
\]

Solve this for $r = r_0$ (e.g., numerically for $p = 2$), then check that $\Delta S_{\{\text{NN}\}}(r_0) > 0$ to confirm a shallow bound.

Triplet admissibility (aligned facings) opens the gate ($\kappa_{\text{att}} > 0$); singlet can be near-threshold (small effective $\kappa_{\text{att}}$). This reproduces the pattern: one shallow bound channel and one unbound/marginal.

\subsection*{9.e.2 Helium-4 ($A = 4, Z = 2$) --- tight bind, even---even pairing}

Plug $A = 4, Z = 2$ into the smooth binding with pairing (and a small shell bump near closed sub-structures). Even---even pairing adds binding; short range and saturation explain why extra loops beyond the first shell gain little per particle.

\subsection*{9.g) Atomic Attachment (EM Bridge) --- with Hydrogen Example}

Electrons attach via the EM orientation/Coulomb dictionary. Bound routes follow closure, fermion parity fills shells; no new scales are introduced. The circular closure for Hydrogen:

\[
S_{\text{loop}} = \oint \mathbf{p} \cdot d\mathbf{l} = 2\pi n S_0, \quad n \in \mathbb{Z}^+
\]

\[
\mathbf{p} \cdot \mathbf{r} = n S_0
\]

\[
p^{\wedge 2} / \mu = k_C \, q_0^{\wedge 2} / r \quad (\mu = \text{reduced mass})
\]

\[
r_n = n^{\wedge 2 S_0^{\wedge 2} / (\mu k_C q_0^{\wedge 2})}
\]

\[
E_n = - \tfrac{1}{2} \mu ( k_C q_0^{\wedge 2} )^{\wedge 2} / (n^{\wedge 2 S_0^{\wedge 2}}) \quad \Rightarrow \quad E_n / E_1 = 1 / n^{\wedge 2}
\]

This is the same closure story; the EM pillar carries the full periodicity via screening + shell parity.

\subsection*{9.f) Validity \& Falsifiers}

Holds when composites are integer-locked and backgrounds vary slowly. Strong kernel short-range must be small vs. composite size; weak transitions must be dominated by a single $\Delta S_{\text{trans}}$ channel.

\textit{Falsifiers:} width or mass-ratio patterns that cannot be fitted with a single calibration of $\{\alpha_V, \alpha_S, \alpha_C, \alpha_A, \alpha_G, \alpha_P, \beta_x, \sigma_x\}$; nucleon---nucleon shapes inconsistent with any short-range attractive + core grammar; weak decays demanding non-chiral gates at leading order.

\section*{10) Elemental Table (VMS Taxonomy Schema)}

We catalog **primes** (single closed patterns) and **composites** (admissibly bound sets). Each row is a **pattern class**, not a classical species. The table is ratio-first and dimensionless; calibrated anchors appear in a side band.

\textit{Recommended columns (minimal set):}

\begin{itemize}
\item Class label (integer $n$ and parity) $\mid$ Charge step ($q = n q_0$) $\mid$ Spin class (even$\to$boson, odd$\to$fermion)

\item Family tag (by stabilizer dictionary: U(1)/SU(2)/SU(3)) $\mid$ Loop count $E_{\text{loop}}$ ($\bar{\tau} \ell$) $\mid$ Geometry tag $G$

\item Mixing links (neighbors, angles) $\mid$ Width band from $\Delta S_{\text{exit}}$ $\mid$ Validity strip (integer lock; $\Lambda$)

\item For composites: constituent set $\{i\}$, coupling map $\kappa_{\{ij\}}$, predicted binding $\Delta S_{\text{bind}}/S_0$, admissibility notes (hand-off rules).

\end{itemize}

\textit{Usage} --- Fill **primes** first ($n, q, \text{spin}, E_{\text{loop}}, G$). Then add **composites** with $\kappa_{\{ij\}}$ and $\Delta S_{\text{bind}}$ estimates. Worked ratios go in the rightmost columns; absolutes are added only after Calibration locks are applied.

\subsection*{Build Checklist (fast)}

\begin{enumerate}
\item Prime rows --- fit ($n, q, \text{spin}, E_{\text{loop}}, G$) from clean single-pattern data.

\item Candidate composites --- propose $\kappa_{\{ij\}}$ from overlap geometry; check $\lambda_{\text{min}} < \min(E_i)$.

\item Stability --- verify integer locks and $\Delta S_{\text{exit}}/S_0$ bands.

\item Mixing --- record $\theta$ and (if 3-state) $(\theta_{12}, \theta_{23}, \theta_{13}, \delta)$; ensure unitarity.

\item Ratios first --- publish $m_{\text{comp}} / 2 m_i, \Gamma_i / \Gamma_j$ ratios, and band plots; defer absolutes to Calibration.

\end{enumerate}

\section*{11) Validity, Limits, and Falsifiers}

This walk-through is meant for the regime where closed patterns are well-formed and the environment is gentle enough that the integer hand-off and the coarse-grained overlap picture both make sense. Below are the exact checks we use.

\subsection*{When it holds (green zone)}

\begin{itemize}
\item Clean harmonic closure (integer lock). The loop hands off its facing exactly after a lap:
  \[
  S_{\text{loop}} / S_0 = 2\pi n \pm \varepsilon \text{int} \quad (n \text{ is an integer}, \varepsilon_{\text{int}} \ll 1)
  \]

\item Stationarity at the solution. Small admissible distortions do not change the integer:
  \[
  \delta S_{\text{loop}} = 0 \quad (\text{at fixed } n)
  \]

\item No cheap exits. The minimal action gap to any allowed open route is comfortably above the noise floor:
  \[
  \Delta S_{\min} / S_0 \gtrsim \Lambda \quad (\Lambda \text{ is a modest constant; large values } \Rightarrow \text{ long lifetimes})
  \]

\item Admissible symmetry locks. The pattern's symmetry lies in the stabilizer (no anomaly flags at this level).

\item Quasi-static background. Caustic structure and coupling geometry vary slowly compared to the loop cycle time $\tau_{\text{loop}}$.
  \[
  |d/dt[\mathbf{e}_{\text{nv}}]| \ll 1/\tau_{\text{loop}}
  \]

\end{itemize}

\subsection*{Use caution (yellow zone)}

\begin{itemize}
\item Background energy shreds closure. Strong external coupling lowers $\Delta S_{\min}$; the loop becomes leaky and widths blow up.

\item Rapidly varying caustics. If the fold/focus pattern changes on the loop timescale, the integer lock can wander.
  \[
  \tau_{\text{caus}} \lesssim O(\tau_{\text{loop}}) \Rightarrow \text{expect drift in } n \text{ and effective parameters}
  \]

\item Strong degeneracy. When more than two near-patterns have comparable detuning, the $2 \times 2$ model is insufficient---use the full 3-state (or higher) unitary.
  \[
  \Delta \approx 0 \text{ across } >2 \text{ patterns } \Rightarrow \text{require } U(\theta_{12}, \theta_{23}, \theta_{13}, \delta)
  \]

\end{itemize}

\subsection*{Out-of-scope (red flags)}

\begin{itemize}
\item Non-adiabatic disruptions that break the integer lock ($S_{\text{loop}}/S_0$ ceases to sit near $2\pi n$).

\item Backgrounds that force non-unitary mixing at leading order (lossy, non-reversible channel wiring).

\item Evidence that a new dimensional scale (beyond $S_0 = h$ and acceptance locks) is needed to align independent ratios.

\end{itemize}

\subsection*{What would falsify this treatment (practical tests)}

\begin{itemize}
\item Cross-ratio inconsistency that demands a new scale. Independent mass-ratio bands cannot be reconciled with a single geometry map $G$ (within uncertainty).

\item Width-ratio patterns not following the action-gap law, even after phase-space corrections $\Omega$. A simple linear check over channels $i, j$:
  \[
  \ln(\Gamma_i/\Gamma_j) \approx \ln(\Omega_i/\Omega_j) - (\Delta S_i - \Delta S_j)/S_0
  \]
  If the slope $-1/S_0$ or intercept $\ln(\Omega_i/\Omega_j)$ systematically fail across clean channels, the model is wrong in this regime.

\item Mixing that requires a non-unitary map ($U \dagger U \neq I$) or cannot be captured by a single real rotation for a 2-state sector:
  \[
  \tan(2\theta) \neq 2\kappa/\Delta \quad (\text{after consistent extraction of } \kappa, \Delta \text{ from independent observables})
  \]

\item Integer-closure failure. Direct probes of phase accumulation around the loop do not land at $2\pi n$ within the expected $\varepsilon_{\text{int}}$.

\end{itemize}

\subsection*{How to check in practice (quick checklist)}

\begin{enumerate}
\item Closure lock --- fit $S_{\text{loop}}/S_0$ from the harmonic; verify $|S_{\text{loop}}/S_0 - 2\pi n| \le \varepsilon_{\text{int}}$.

\item Exit hardness --- estimate $\Delta S_{\min}$ from the cheapest allowed channel; check $\Delta S_{\min}/S_0$ against $\Lambda$ (long-lived if large).

\item Mixing sanity --- extract $\kappa$ and $\Delta$ from independent observables; confirm $\tan(2\theta) = 2\kappa/\Delta$ and that $U$ is unitary in the 3-state case.

\item Widths --- compute $\ln(\Gamma_i/\Gamma_j) - \ln(\Omega_i/\Omega_j)$ across channels; residuals should be small and structureless.

\item Stability --- ensure $\tau_{\text{caus}} \gg \tau_{\text{loop}}$ and track drift over time; large drift flags out-of-regime behavior.

\end{enumerate}

\section*{Classical/SM Bridge --- Dictionary Only}

Quantization and spin: integer winding constraints (not imported axioms).

Families: stabilizers of admissible route symmetries (U(1)/SU(2)/SU(3) dictionary).

Mass/mixing/width relations: dimensionless and ratio-first; absolutes anchored in Calibration.

\section*{Appendix --- Cheat Sheet (Particles, VMS)}

\begin{itemize}
\item Lock: $S_0 = h$ (single dimensional scale, SI-locked).

\item Harmonic closure: $S_{\text{loop}}/S_0 = 2\pi n$ ($n \in \mathbb{Z}$).

\item Quantization: $q = n q_0$; spin class: $n$ even $\to$ boson, $n$ odd $\to$ fermion (class-level).

\item Mass ratios: $m_i/m_j \approx [\bar{\tau}_i \ell_i] / [\bar{\tau}_j \ell_j] \times G_{\{ij\}}$.

\item Mixing (2-state): $\tan 2\theta = 2\kappa/\Delta$; eigenvalues: $\lambda_\pm = \tfrac{1}{2}[(E_1+E_2) \pm \sqrt{(\Delta^2 + 4\kappa^2)}]$.

\item Widths: $\Gamma_i / \Gamma_j \approx (\Omega_i/\Omega_j) \exp[{-(\Delta S_i - \Delta S_j)/S_0}]$.

\item Validity strip: clean closure; stable symmetry locks; no cheap $\Delta S$ exits.

\end{itemize}

\end{document}
