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\title{Particles Mechanics Narrative}
\author{}
\date{September 27, 2025 --- Version: Draft (\textbf{DO NOT OVERWRITE} prior versions)}

\begin{document}

\maketitle

Constants: CODATA-2022 (NIST SP 961, May 2024); PDG 2024 for cross-checks (masses/widths).

Locks: import a single dimensional scale $S_0 = \hbar$ (SI-locked). Speed of advance $c$ is an acceptance lock. No new tunables are introduced.

\section*{Locks \& Conventions}

\begin{itemize}
\item Route/closure language: same as EM/Mechanics (display-area $A_d$; action $S[\text{path}] \equiv \int A_d \, ds$ when we need phase).

\item Data anchors: CODATA-2022 and PDG-2024: prefer ratios when tighter than absolutes.

\item Notation sanity: new symbols are dimensionless unless stated; any absolute mass/width is PDG-anchored in Calibration.

\item Single dimensional scale: $S_0 = \hbar$ (SI-locked). No other tunable scales.
\end{itemize}

\section*{Scope}

Build particle classes (charges, spins, families) from closure harmonics and loop topology, derive mass/mixing/decay relations as \textbf{\textbf{dimensionless ratios}} using a single scale $S_0 = \hbar$, and recover standard limits where appropriate. This document sets the picture and the on-ramp to math; the detailed derivations live in the Math Walk-Through and the Pillar Math Appendix.

SM link: width/lifetime ratios compare directly to PDG without adding dimensional dials.

Definitions: $\Delta S \geq 0$ is the minimal action gap to any allowed open route. $Q$ is a dimensionless quality factor (geometry/statistics). $\Omega$ is a dimensionless attempt-frequency ratio (combinatorics/phase-space).

SM link: the 3-state generalization yields a unitary rotation with one CP-phase; mapping to CKM/PMNS parameters appears in the Appendix (structure-level, ratio-first).

\section*{Coverage map}

\begin{itemize}
\item Foundation --- $S_0 = \hbar$; operators $(E, \mathcal{C}, T, S, C)$; route/closure language consistent with other pillars.

\item Quantization --- closed-route integers $\to$ charge steps and spin classes (boson/fermion).

SM limit: integer $n$ labels reproduce charge steps and spin classes (boson/fermion) without importing gauge axioms; anomaly notes live in the Appendix.

\item Gauge structure --- route symmetries $\to$ group structure (U(1), SU(2), SU(3) analogs).

SM limit: stabilizers of the admissible route symmetries mirror U(1), SU(2), and SU(3); we read these from invariances rather than postulating a gauge set.

\item Mass patterns --- closure tension and coupling geometry $\to$ mass ratios (leptons; selected light-baryon bands).

\item Mixing --- overlap of near-patterns $\to$ unitary rotations (2- and 3-state cases).

\item Decay \& widths --- escape map $\Pi_\text{esc}$ with $\Delta S / S_0$; lifetime/width ratios as exponentials with dimensionless prefactors.

\item Predictions \& falsifiers --- ratio-first targets with PDG anchors; no new dimensional knobs.

Example (widths): publish a pairwise width-ratio band using $\Delta S / S_0$ differences and $\Omega$ ratios; compare to PDG medians with no new knobs.

Example (ratios): publish a predicted ordering/band for $m_\mu / m_e$ and $m_\tau / m_\mu$ from one $\mathcal{G}$ geometry; lock absolutes only when calibrated.
\end{itemize}

\section*{In-scope (this pillar)}

\begin{itemize}
\item Charge quantization and spin classes from closed-route integers; anomaly-safe symmetry statements in the Appendix.

\item Lepton mass ratios; selected light-baryon band structure (catalog level).

\item Lifetime/width scaling from an escape map $\Pi_\text{esc}$ with no new dimensional constants.

\item Mixing angles/phases from route-overlap integrals; CKM/PMNS analogs at the level of structure and ratios.
\end{itemize}

\section*{Out-of-scope (here)}

\begin{itemize}
\item Full nuclear structure; heavy-flavor phenomenology (dedicated branches).

\item Material/device microphysics (thermo/material pillars).

\item Beyond-SM numerics; we publish ratios/bands and compare to PDG without adding knobs.
\end{itemize}

\section*{What is a ``particle'' in VMS? (common-sense picture, same ontology)}

We keep the same ontology you've seen in the other pillars: routes and closure. Nothing new is smuggled in. Here's the clean story of how a particle appears, stays around, and eventually ends, using only display-area, caustics, and closed harmonics.

\subsection*{1) Start: display-area expansion (an open route)}

A void advancing at light speed hides a circular patch of space each step: the display-area $A_d = \pi r^2$. Along a route we add those patches up: that running total is the display-area action $S[\text{path}] = \int A_d \, ds$. When we need a phase label, we divide by the single scale $S_0=\hbar$.

\begin{verbatim}
A_d = \pi r^{2}

S[path] = \int A_d   ds

phase label:   \varphi = S / S_{0}    (with S_{0} = \hbar, SI-locked)
\end{verbatim}

At this stage we have an open propagation front. No particles yet---just a clean route with a phase counter we can read if we need it.

\subsection*{2) Optionality: a suitable caustic appears}

Real space isn't flat everywhere. Geometry and background can fold routes together and create a caustic---a place where there are two (or more) nearby, equally clean ways to advance. The front doesn't `pick once and forget'; it oscillates between those near-ties. Whether that oscillation matters is controlled by scale: if the caustic's size and spacing are comparable to the display-area scale being accumulated, the oscillation persists instead of averaging out.

\subsection*{3) Closure: the oscillation locks into a closed harmonic loop}

Now the key step. If the caustic geometry feeds the front back onto itself with the same facing after one lap, the route closes. Closure is only stable when the accumulated action completes an integer number of turns---so the hand-off of facing is exact. That's a closed harmonic:

\begin{verbatim}
harmonic closure condition:   S_loop / S_{0} = 2\pi n    (n \in \mathbb{Z})
\end{verbatim}

Those integers $n$ are not decoration---they label the loop class. In the EM bridge they show up as charge steps $q = n q_0$; in spin language, the parity of $n$ tracks boson/fermion split. The point is: nothing was postulated---we got integers because the loop must hand off perfectly each cycle.

\subsection*{4) Why some loops live (stability) and others don't}

A closed loop persists when three simple checks pass:

\begin{itemize}
\item \textbf{\textbf{Exact hand-off}} (the harmonic above). If $S_\text{loop} / S_0$ drifts off an integer, the facing mis-hands and the loop degrades.

\item \textbf{\textbf{Boundary rules}} (same as everywhere): no sideways pile-up across a boundary; no tearing of closure. These forbid leaks that would kill the loop immediately.

\item \textbf{\textbf{No cheap exit}}: there isn't a nearby open route with a tiny action gap. If there is, the loop will eventually `take it'---that's decay.
\end{itemize}

What varies across classes (and shows up as mass and other labels) is the loop's internal tension and geometry, and how it couples to the background and to other loops. That's why different closed patterns sit at different mass ratios and selection rules without inventing new dials.

\subsection*{5) How lifetimes fall out (no new constants)}

Let $\Delta S$ be the minimal action gap from the loop to any allowed open route. With a dimensionless quality factor $Q$ (geometry/statistics) and a dimensionless attempt ratio $\Omega$ (combinatorics/phase-space), escape per cycle and widths/lifetimes scale as:

\begin{verbatim}
\Pi_{esc}  \approx  Q \cdot \exp( - \Delta S / S_{0} )

\Gamma_{i} / \Gamma_{j}  \approx  (Q_{i} / Q_{j}) \cdot \exp[ - (\Delta S_{i} - \Delta S_{j}) / S_{0} ]
\end{verbatim}

Big $\Delta S$ means long-lived; small $\Delta S$ means short-lived. Ratios are the point---we compare species cleanly without adding a new units-bearing constant.

\subsection*{6) Walking straight into this pillar's scope (what we work out next)}

Once the loop exists, everything we publish in this pillar is a consequence of that same picture:

\begin{itemize}
\item \textbf{\textbf{Quantization}}: integers $n$ from harmonic closure give charge steps and spin classes (no gauge axioms were assumed first).

\item \textbf{\textbf{Gauge structure}}: the allowed symmetries of the route and its couplings pick out the interaction families---the stabilizers mirror the familiar U(1), SU(2), SU(3) sets.

\item \textbf{\textbf{Mass patterns (ratios)}}: loop tension and coupling geometry set the ratio-first mass formula.
\end{itemize}

\begin{verbatim}
m_{i} / m_{j}  \approx  (\tau_{i} \cdot \ell_{i}) / (\tau_{j} \cdot \ell_{j})   \times  G_{ij}
\end{verbatim}

with $\tau \ell$ a geometric tension-length product and $G$ a dimensionless coupling geometry. One geometry can ladder the leptons; the Appendix shows the working.

\begin{itemize}
\item \textbf{\textbf{Mixing}}: near-patterns overlap; diagonalizing a small overlap $\kappa$ against a detuning $\Delta$ gives the rotation angle (2-state sketch):
\end{itemize}

\begin{verbatim}
\tan 2\theta = 2\kappa / \Delta
\end{verbatim}

The 3-state generalization is a unitary rotation with one CP-phase; we map structure-level parameters to CKM/PMNS in the Appendix.

\begin{itemize}
\item \textbf{\textbf{Decay/widths}}: lifetimes and widths are just the $\Delta S / S_0$ story above; we publish width and lifetime ``ratios'' (and bands) first, then calibrated numbers when useful.
\end{itemize}

That's the whole arc: an open route with display-area expansion encounters a suitable caustic, locks into a closed harmonic, and---if three simple checks pass---stays around. Everything else in this pillar is bookkeeping on that loop: symmetries, ratios, and clean comparisons, all on one scale $S_0 = \hbar$.

\section*{7) Quantization from closed routes}

Stable loops carry integer winding numbers. Those give discrete charge steps and spin classes.

\begin{verbatim}
q = n \cdot q_{0}    (n \in \mathbb{Z})

spin class: n even $\to$ boson, n odd $\to$ fermion (class-level statement)
\end{verbatim}

Charge quantization and allowed spins appear as topological integers of closure; the Appendix states symmetry/anomaly constraints explicitly.

SM limit: integer $n$ labels reproduce charge steps and spin classes (boson/fermion) without importing gauge axioms; anomaly notes live in the Appendix.

\section*{8) Gauge structure from route symmetries}

Route invariances generate the interaction families. At coarse-grain the admissible invariances form group structures that mirror U(1), SU(2), and SU(3) analogs. This fixes which couplings exist and which do not; we don't postulate gauge furniture first and explain later---we begin with route symmetries and read off the families.

SM limit: stabilizers of the admissible route symmetries mirror U(1), SU(2), and SU(3); we read these from invariances rather than postulating a gauge set.

\section*{9) Mass from closure tension and coupling geometry (ratios, no new scales)}

Effective mass reflects stored tension in the closed pattern and how patterns couple to the background and to each other. The algebra is ratio-first:

\begin{verbatim}
m_{i} / m_{j}  \approx  (\tau_{i} \cdot \ell_{i}) / (\tau_{j} \cdot \ell_{j})   \times  G_{ij}
\end{verbatim}

Micro-example (in Appendix): show how a single $\mathcal{G}$ map produces a lepton ladder $m_\mu/m_e$ and $m_\tau/m_\mu$ as ratios with one geometry.

Here $\tau \ell$ is a geometric tension-length product for the pattern, and $G_\{ij\}$ is a \textbf{\textbf{dimensionless}} coupling geometry ratio (no new dimensional constants). The pillar keeps $S_0 = \hbar$ as the only units-bearing scale; absolute values are reported only when anchored to PDG references in Calibration.

\section*{10) Mixing as route overlap (2-state sketch)}

When two patterns are similar enough to overlap, the observed states are rotated mixtures. A minimal 2-state model uses a symmetric overlap $\kappa$ and a detuning $\Delta$. Diagonalizing that map is just the usual rotation: a lower `bonding' level and a higher `antibonding' level. If the lower level sits below either part, you've got a bound composite.

\begin{verbatim}
\tan 2\theta = 2\kappa / \Delta
\end{verbatim}

The Appendix carries the 3-state generalization and the mapping to CKM/PMNS-style parameterizations. SM link: the 3-state generalization yields a unitary rotation with one CP-phase; mapping to CKM/PMNS parameters appears in the Appendix (structure-level, ratio-first).

\section*{11) Decay and widths from an escape map (dimensionless, ratio-first)}

Let $\Delta S$ be the minimal action gap to reach any allowed open route from a closed pattern. With an escape quality factor $Q$ (dimensionless geometry/statistics), the escape probability per cycle scales as:

\begin{verbatim}
\Pi_{esc}  \approx  Q \cdot \exp( - \Delta S / S_{0} )
\end{verbatim}

Widths/lifetimes then scale as:

\begin{verbatim}
\Gamma_{i} / \Gamma_{j}  \approx  (Q_{i} / Q_{j}) \cdot \exp[ - (\Delta S_{i} - \Delta S_{j}) / S_{0} ]
\end{verbatim}

where $\Omega$ is a dimensionless attempt-frequency ratio that carries combinatorics and phase-space factors. This form makes clear what is predicted (ratios) without inventing new dimensional dials.

Definitions: $\Delta S \geq 0$ is the minimal action gap to any allowed open route. $Q$ is a dimensionless quality factor (geometry/statistics). $\Omega$ is a dimensionless attempt-frequency ratio (combinatorics/phase-space).

SM link: width/lifetime ratios compare directly to PDG without adding dimensional dials.

\section*{12) Composites \& Apparent Size --- Narrative Insert (VMS-first)}

Plain-English take: a composite particle is just several closed VMS loops that lock together in an admissible layout so the total route-action drops. No new scaling rules, same closure rules, same single scale $S_0 = \hbar$. Binding means the coupled pattern costs less to run per lap than the parts do separately; that's why a composite sits below the sum of its pieces in mass ratios.

\subsection*{What ``binding'' means in our language}

We compare the action of the parts run separately to the action of the coupled layout. If the coupled layout wins (smaller action), it binds:

\begin{verbatim}
S_total^{[separate]} = \sum_{i} S_{i}

S_total^{[bound]} = \sum_{i} S_{i} + \sum_{i<j} S_{int}(ij) + S_{int}(1,2,3,...)

\Delta S_{bind} = S_total^{[separate]} - S_total^{[bound]} > 0
\end{verbatim}

That $\Delta S_\text{bind}$ is dimensionless in our normalization. In ratios, the composite mass comes in lower than the sum of its parts; the exact drop depends on how the loops overlap and hand off (a geometry factor only, no new dimensional knobs).

\section*{13) Two-loop composites (bonding / antibonding)}

For a pair of loops with counts $E_1$ and $E_2$ and an admissible coupling $\kappa$, the smallest faithful model is a $2 \times 2$ map. Diagonalizing that map is just the usual rotation: a lower `bonding' level and a higher `antibonding' level. If the lower level sits below either part, you've got a bound composite.

\begin{verbatim}
H = [[E_{1}, -\kappa], [-\kappa, E_{2}]]

\Delta \equiv E_{1} - E_{2}

R(\theta) = [[\cos \theta, -\sin \theta], [\sin \theta, \cos \theta]]

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

\lambda_{\pm} = \frac{1}{2}(E_{1} + E_{2} \pm \sqrt{\Delta^{2} + 4\kappa^{2}} )
\end{verbatim}

Binding check:

\begin{verbatim}
\lambda_{-} < \min(E_{1}, E_{2})
\end{verbatim}

In words: enough admissible overlap ($\kappa$) relative to the mismatch ($\Delta$) pulls a shared mode down and locks the pair as one object.

\section*{14) Three-loop rings/braids (what changes and what doesn't)}

With three loops the picture is the same in spirit: write the symmetric $3 \times 3$ with admissible couplings and look at the lowest eigenvalue. Equal loops with equal couplings make the point transparent: one fully in-phase `bonding' mode and two higher modes. One crucial point transparent: one fully in-phase `bonding' mode and two higher modes.

\begin{verbatim}
H = [ [ E_{1}, -\kappa_{12}, -\kappa_{13} ],
      [ -\kappa_{12}, E_{2}, -\kappa_{23} ],
      [ -\kappa_{13}, -\kappa_{23}, E_{3} ] ]

Equal-E, equal-\kappa special case (closed form):

\lambda_{bond} = E - 2\kappa    (symmetric bonding mode)

\lambda_{anti} = E + \kappa    (twofold degenerate)

The composite is bound when the lowest level lies below the individual loop counts:

\lambda_{min} < \min(E_{1}, E_{2}, E_{3})

If the \kappa's aren't all equal, you just diagonalize the $3 \times 3$ numerically; the binding criterion is the same.
\end{verbatim}

\subsection*{Proton, neutron: what our 3-loop picture says}

We model light baryons (proton, neutron) as three-loop composites with small orientation-driven differences in the admissible couplings $\kappa_{ij}$. At the symmetric baseline (all $E_i$ equal, all $\kappa_{ij}$ equal) the bonding level is $\lambda_\text{bond} = E - 2\kappa$. Small deviations around that baseline shift the bonding level by the average coupling change seen by the in-phase vector $[1,1,1]/\sqrt{3}$:

\begin{verbatim}
\lambda_{bond} \approx E - (2/3) \cdot (\kappa_{12} + \kappa_{23} + \kappa_{13})
\end{verbatim}

So the neutron---proton mass split tracks the \textbf{\textbf{sum of couplings}}. If the neutron's admissible couplings sum to a slightly smaller number than the proton's (a little less binding), then neutron sits higher in mass---correct sign, with no new binding. A difference of order $10^{-3}$ in that coupling sum produces an order-$10^{-3}$ mass split---right ballpark. We fit that single $\Delta(2\kappa)$ once, then carry it across related observables (width bands where the exit map matches).

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

`Size' here is not an ingredient we put in; it's the composite's compliance under a small, symmetric pry that opens the layout. Let $a$ be the pry coordinate. The bonding level responds as the couplings change with $a$, and the `stiffness' is the curvature:

\begin{verbatim}
\lambda_{bond}(a) = E - (2/3) \cdot [\kappa_{12}(a) + \kappa_{23}(a) + \kappa_{13}(a)]

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

Dictionary to measurement: a larger compliance (smaller $k_\text{eff}$) reads as a larger apparent radius in small-$q$ probes. If the neutron's coupling sum is smaller, we also expect it to be slightly softer to pry---so a larger apparent size in this sense.

\section*{15) What to check (predictions you can actually test)}

\begin{itemize}
\item Sign: from any admissible orientation that makes $(\sum \kappa)_{n} > (\sum \kappa)_{p}$, predict $m_{n} > m_{p}$ (observed sign).

\item Magnitude band: fit $\Delta(\sum \kappa)$ from the mass ratio once; propagate to width-ratio bands via the same exit map ($\Omega, \nu$ common), no new scales.

\item Apparent size: compare low-$q$ slopes (dictionary) or direct pry response; expect the neutron to be slightly more compliant than the proton if its $\sum \kappa$ is smaller.

\item Stability: ensure the integer locks hold for the composite and that $\Delta S_\text{exit} / S_0$ sits in the long-lived band; otherwise widths broaden and the simple picture breaks.

Bottom line: composites and apparent size drop straight out of the same closure-and-geometry story---one ontology, one scale. We keep it ratio-first, publish bands, and let data accept or reject without adding knobs.
\end{itemize}

\section*{16) Nuclear Binding, Strong/Weak, and Atomic}

We don't import nuclear or weak force laws. We keep the same book-keeping we used everywhere else: closed patterns, admissible hand-offs, and one dimensional scale $S_0 = \hbar$. Binding is just the coupled layout runs cheaper per lap than the parts. Forces are gradients of that same action. Everything below mirrors the math insert, step by step, in plain language.

\subsection*{1) A-Body Binding --- What each term means and why the exponents look the way they do}

We split binding into a smooth bulk piece and a shell piece. The smooth piece is geometry/tension counting; the shell piece is the extra dip you get when a count lands near a clean harmonic (a `magic' number). The math wrote this as:

\begin{verbatim}
\Delta S_{[bind]}^{[smooth]}(A, Z) / S_{0} = \alpha_{V} A - \alpha_{S} A^{2/3} - \alpha_{C} [Z(Z - 1)]
                                       - A^{-1/3} - \alpha_{A} (A - 2Z)^{2} / A + \alpha_{G} G(A,Z)
\end{verbatim}

How to read it:

\begin{itemize}
\item Volume term (+): more loops packed coherently $\to$ more binding.

\item Surface term ($-$): boundary costs you; the fraction of loops sitting on a boundary scales like area.

\item Coulomb opposition ($-$): protons repel via the EM dictionary, and the average spacing grows with size.

\item Asymmetry ($-$): too many of one kind (proton vs neutron) frustrates admissible hand-offs.

\item Geometry correction $G(A,Z)$: layout effects (pure number).
\end{itemize}

Why those exponents are not hand-wavy: in a compact 3-D layout radius grows like $A^{1/3}$. That makes area $\sim A^{2/3}$, and average spacing $\sim A^{1/3}$.

\begin{verbatim}
R(A) \propto A^{1/3}

\text{Area} \propto A^{2/3}  ;  \text{Mean spacing} \propto A^{1/3}
\end{verbatim}

So the surface penalty naturally scales with area ($A^{2/3}$), and the Coulomb dictionary brings in an $A^{-1/3}$. Nothing was invented for effect.

\subsection*{2) Shell correction --- Why magic numbers happen (in our language)}

Closed harmonics in the mean field are simply layouts where the hand-off field are simply layouts where the hand-off is integer-clean. You get an extra dip (more binding) when a count lands near one of those integers---both for protons and for neutrons. The compact formula uses a nearest-shell dip; if you want smooth gradients for fitting, swap in a log-sum-exp.

\begin{verbatim}
\Delta S_{[shell]}(A, Z) / S_{0} = - \sum_{x\in\{p,n\}} \beta_{x} \cdot \max_{k} \exp( - (N_{x} -
N_{magic,x}^{(k)})^{2} / (2 \sigma_{x}^{2}) )
\end{verbatim}

\subsection*{3) Total binding and masses --- Why we stay ratio-first}

We add the smooth and shell parts and then talk in ratios so the global proportionality drops out. That's why we can publish bands without adding knobs:

\begin{verbatim}
\Delta S_{[bind]}(A, Z) / S_{0} = \Delta S_{[bind]}^{[smooth]} / S_{0} + \Delta S_{[shell]} / S_{0}

m_{A, Z} / (\sum m_{i}) \approx 1 - [\Delta S_{[bind]}(A, Z) / \sum E_{[loop]}] \cdot G_{[bind]}(A, Z)
\end{verbatim}

Read: positive binding makes the composite lighter than its pieces---exactly what you expect, and it's the same logic we used for simpler composites.

\subsection*{4) Where the ``strong'' comes from --- short-range attraction with a hard core}

Two nucleons `like' each other when the admissible overlap lowers the total action; they `don't like' getting smashed together side-by-side. That grammar is just a short-range attraction with a very-short-range core. You saw it as:

\begin{verbatim}
\Delta S_{[NN]}(r) / S_{0} = - K_{[att]} e^{ - r / \ell_{s} } + K_{[core]} e^{ - (r / r_{c}) ^{p} }
\end{verbatim}

The `force' shape (if you insist on that language) is just the gradient of $\Delta S$ with respect to $r$. That gives an attractive region at $\sim 1$-2 short-range units and a stiff core at tiny $r$.

\begin{verbatim}
F_{s}(r) \propto - d[\Delta S_{[NN]}(r) / S_{0}] / dr = - (K_{[att]} / \ell_{s}) e^{ - r / \ell_{s} } +
(K_{[core]} p / r_{c}) (r / r_{c}) ^{[p-1]} e^{ - (r / r_{c}) ^{p} }
\end{verbatim}

Saturation (why nuclei don't just keep getting more bound per particle) falls straight out: each loop only overlaps a few neighbors at short range. That's your volume term with a surface subtraction in the smooth binding.

\subsection*{5) Where the ``weak'' comes from --- class-changing, short-range, chiral gate}

Weak processes are hand-offs that change the integer/handedness class. They need a special overlap (short-range) and they pay an action gap. That's why the widths fall with an exponential of the gap; the contact-like strength is just the short-range kernel integrated.

\begin{verbatim}
P_{[trans]} \approx \Omega_{w} \cdot \int K_{w}(r) dr \cdot e^{ - \Delta S_{[trans]} / S_{0} } \times G_{w}^{\Lambda 2} \cdot d_{[PS]} \cdot
e^{ - 2 \Delta S_{[trans]} / S_{0} }
\end{verbatim}

One crucial point: the chiral behavior (parity violation) isn't bolted on. It's a selection rule from the orientation gate:

\begin{itemize}
\item Chiral selection rule --- the orientation gate $Q_w$ suppresses one handedness of admissible hand-offs; parity violation shows up as a selection, not as a new force.
\end{itemize}

\subsection*{6) Admissibility + pairing --- why even---even wins a little}

Admissible composites keep everyone's integer locks and avoid sideways pile-up. Pairing is just `make the hand-offs come in balanced pairs.' It's a small dimensionless add-on that favors even $Z$ and even $N$ at fixed $A$ (the effect shrinks with size):

\begin{verbatim}
\delta_{[pair]}(A, Z) = + \alpha_{P} \cdot A^{ - 3 / 4 }    (even Z, even N)

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

\delta_{[pair]}(A, Z) = - \alpha_{P} \cdot A^{ - 3 / 4 }    (odd Z, odd N)
\end{verbatim}

\subsection*{7) Worked nuclear pictures --- what to look for}

Deuteron ($A=2$): one shallow well if the attraction beats the core at some separation $r_0$. In the math that was the derivative-equals-zero line and the ``value positive'' check:

\begin{verbatim}
(d/dr) \Delta S_{[NN]}(r) |_{r=r_{0}} = 0  ,  \Delta S_{[NN]}(r_{0}) / S_{0} > 0
\end{verbatim}

For the $p=2$ case you can even write the well location condition explicitly (solve this for $r_0$):

\begin{verbatim}
(K_{[att]} / \ell_{s}) e^{ - r / \ell_{s} } = (2 K_{[core]} / r_{c}^{2}) \cdot r \cdot e^{ - (r / r_{c})^{2} }
\end{verbatim}

\subsection*{8) Atomic attachment --- why Hydrogen falls into a $1/n^2$ ladder without importing quantum postulates}

We attach electrons with the EM dictionary for the central 1/r well. The same closure condition drives the result; we only change to the reduced mass $\mu$.

\begin{verbatim}
S_{[loop]} = \oint p \cdot dl = 2\pi n S_{0}  ,  n \in \mathbb{Z}^{+}

p \cdot r = n S_{0}

p^{2} / \mu = k_{C} \cdot q_{0}^{2} / r

r_{n} = n^{2} S_{0}^{2} / (\mu k_{C} q_{0}^{2})  ,  E_{n} = - \frac{1}{2} \mu (k_{C} q_{0}^{2})^{2} / (n^{2}
S_{0}^{2})  \Rightarrow  E_{n} / E_{1} = 1 / n^{2}
\end{verbatim}

Same closure story as everywhere else; no new scale, no wavefunction postulate sneaking in the back door.

\section*{17) Bridges to Electromagnetism and Mechanics}

\begin{itemize}
\item Same ontology and lock: $S_0 = \hbar$ once; no retuning by pillar.

\item EM bridge: orientation-gated couplings here act as selection rules seen as polarization/interaction channels in EM's coarse-grain (Maxwell-limit box lives in EM docs).

\item Mechanics/GR bridge: particle clocks obey the same timing kernels (no extra scale); closed-loop inertia/timing align with the cost-map picture.
\end{itemize}

\section*{18) Calibration capsule (how numbers get compared)}

\begin{itemize}
\item Stamp constants (CODATA-2022, PDG 2024), record versions.

\item Prefer \textbf{\textbf{ratios}} when they beat absolute calibration: $m_\pi / m_e$, $m_\mu / m_e$, selected width ratios (where used).

\item Publish both ratio-only and calibrated values; recommend the tighter band and state why.

\item Adopt \textbf{\textbf{falsifiers}}: sustained $2\sigma$ disagreement across independent ratios after systematics control is a fail; no rescue by adding knobs.
\end{itemize}

\section*{19) Predictions \& falsifiers (high-level)}

\begin{itemize}
\item Lepton mass ratio ladder with dimensionless geometry factors $G_\{ij\}$.

\item Selected baryon band tightening (catalog class positions, not per-isotope rows).

\item Lifetime/width ratios from $\Delta S / S_0$ differences; publish predicted ordering and bands.

\item Mixing angle/magnitude patterns constrained by overlap integrals; publish allowed regions.

\item Falsifier: incoherent ratio shifts that would require introducing new dimensional scales (disallowed).
\end{itemize}

\section*{20) Reader map (where to go next)}

\begin{itemize}
\item Math Walk-Through --- proofs and skeletons for the relations above; minimal algebra for overlap models and escape maps.

\item Pillar Math Appendix --- readable, line-by-line derivations; symbol legend up front; anomaly cancellation and symmetry statements.

\item Calibration --- constants stamp, PDG anchors, ratio-vs-calibrated inserts.

\item Verification \& Falsification --- pass/fail tables with DOIs; pre-registered tests where possible.

\item Student Workbook --- pedagogy and worked problems (same notation as Appendix).

\item VMS Laws (Pilot) --- concise value cards for immediate use, with classical/SM limit boxes.
\end{itemize}

\subsection*{Bottom line}

One ontology, one scale ($S_0 = \hbar$), no new tunables. Quantization, gauge structure, mass/mixing/decay relations all drop out of closure and symmetry. We publish ratios and tight bands first, then calibrated numbers when useful. Where data refuse the bands, we call it---no extra knobs.

\end{document}
