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\title{Mechanics --- Narrative}
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\section*{Executive recap (our picture, no fluff)}

Mass is missing space: a closed, light-speed circulation that keeps a fixed slice of space hidden. Bring another spinning loop near it and the clean route leans slightly toward that hidden-space profile and takes a hair longer to pass. That's gravity: route in a shaped background set by another loop's circulation. No ``pulls.'' Same two rules as always: no sideways pile-up across boundaries; no tearing of closure.

\section*{Scope for this pillar (what we do and what we don't)}

Mechanics carries the full gravity story in VMS --- from gentle bends to horizon-scale effects. We keep the same ontology (routes/closure, display-area $A_d = \pi r^2$), the same bookkeeping ($S[\text{path}] = \int A_d \, ds$), and the same single scale $S_0 = \hbar$ (SI-locked).

\begin{itemize}
\item In-scope: weak-field (tiny bend, small delay, clock drift) and strong-field (deep-well lensing, photon-sphere/horizon behavior, periapsis precession in steep wells, frame-dragging around rotators, wave transport across steep gradients).

\item Checks: recover the standard weak-field numbers and the tested strong-field signatures using the same settings; no new tunable scales.

\item Out-of-scope here: cosmology/expansion; quantum corrections beyond $S_0$ (Particle pillar); heavy numerics beyond the kernels shown in the Walk-Through.
\end{itemize}

\section*{What the Math Walk-Through Will Actually Do}

We start with the same bookkeeping you've seen everywhere in this pillar. A Void advancing at light speed hides a circular patch of space as it moves. That patch is the display-area $A_d = \pi r^2$. Along any route, we simply add up how much area gets hidden at each step. That running total is the route `cost' $S[\text{path}] = \int A_d \, ds$. That running total is the route `cost' (display-area action) $S[\text{path}] = \int A_d \, ds$. At first we only compare ratios of $S$---cleaner vs. costlier routes---no new dials, no imported forces.

A stationary mass is a loop that keeps a chunk of space hidden---missing space. From a distance, the detailed swirl blurs into a gentle background that says ``steps here cost a tiny bit more.'' The Walk-Through names that background $n(x)$. Think of $n(x)$ as a smooth cost map: higher $n$ means `this step hides a bit more display-area than average.' It's not a new substance. It's just a smooth way to keep track of the missing-space profile in our own language.

Once you have that background, tiny sideways changes in it do three simple things to passing routes:

\begin{itemize}
\item They tip the route slightly toward the mass (a small bending angle).

\item They make near passes take a hair longer than far passes (a small positive delay).

\item They make parked loops deep in the profile tick a little slower than ones far away (a small clock drift).
\end{itemize}

The Walk-Through shows each of these with one compact expression---little path integrals you can actually plug into. Same knobs as always: one scale $S_0 = \hbar$ (SI-locked), $c$ and $G$ as acceptance locks, everything else is geometry. No extra parameters appear halfway through the derivation.

\section*{How the classical numbers show up (why we're compatible)}

After the story above, we take one smoothing step: blur the rotating loop into a background and keep the leading terms. In that weak-field limit, the three outputs---bend, delay, and clock drift---land on the familiar textbook values (the usual deflection, Shapiro-style travel-time shift, and gravitational redshift/time-dilation). We don't import someone else's field equations as axioms; we just show that our route picture reduces to the same tested numbers when you look from far away. We set the matching coefficients once (those coefficients are the standard weak-field factors used across deflection, delay, and clockshift) and then reuse them across every case.

\section*{Strong-Field Narrative (what changes and why)}

All the pieces are the same --- only the gradients get big. A stationary mass is still a closed circulation that keeps space hidden (missing space). From far away the swirl blurs to a smooth background cost $n(x)$; near compact masses that background gets steep. The same two rules hold: no sideways pile-up across boundaries; no tearing of closure. Those rules plus a steep $n(x)$ profile give you:

\begin{itemize}
\item Routes that wind hard: light and matter bend strongly; near the photon-sphere the clean options wrap before they escape.

\item Passes that get expensive: near-mass paths pay a large time premium (deep-well delay); parked clocks drift heavily (gravitational time-dilation).

\item Background that twists if the mass rotates: the route frame is dragged --- you see it as a small azimuthal shove (frame-dragging/Lense--Thirring in smooth limit).
\end{itemize}

Nothing new was imported. We're still counting hidden display-area along routes. Strong-field just means the cost landscape is steeper, so the same choice rule (cleaner route wins) produces stronger bending, bigger delays, and deeper clock shifts.

\section*{Where the Walk-Through will take it (preview)}

\begin{itemize}
\item Write the strong-field kernels as the same path integrals you already saw in weak-field, now evaluated on steep profiles $n(x)$.
\end{itemize}

\section*{Plain-language checklist (use this as you read the math)}

\begin{itemize}
\item $A_d$ and $S$ are the only moving parts at the start; $n(x)$ is just a smooth summary of the missing-space profile.

\item Three consequences, one cause: tiny sideways gradients in that profile.

\item Outputs are small, positive-defined effects you can integrate along a path (no new dials appear).

\item The weak-field bridge exists to prove we line up with the known numbers---once---then we move on.

\item ``Show how the dictionary to familiar GR results appears in the smooth limit (deflection near compact masses, deep-well timing, frame-dragging terms).''

\item ``Keep the locks: $S_0 = \hbar$ once; $c$ and $G$ as acceptance locks; no mid-derivation knobs.''
\end{itemize}

\section*{Bench protocols (what you can actually run)}

\begin{itemize}
\item Cavity frequency pull near mass: park a high-Q cavity near a stationary loop (mass) and read the frequency shift; compare to the delay kernel.

\item Ring-laser/fiber-gyro bias/null: place mass symmetrically; beyond Sagnac, expect a null. Any bias sets a bound.

\item Bench-top delay: near-mass pass vs far-pass timing difference (Shapiro analog at lab scale).

\item Polarization near mass: expect no rotation/birefringence in this bridge limit; any detection is a bound.
\end{itemize}

\section*{Falsifiers (clean failure modes)}

\begin{itemize}
\item Chromatic gravity in the bridge limit (wavelength-dependent bend/delay) would be a fail or an extensions lead.

\item Polarization rotation near mass in symmetric placement would be a fail or a bound.

\item Non-reciprocity beyond Sagnac in symmetric rings would be a fail or a bound.
\end{itemize}

\section*{Language \& locks}

\begin{itemize}
\item Projected area only: $A_d = \pi r^2$. No $4\pi$ shells. No double-counting.

\item Single scale: $S_0 = \hbar$ (fixed via SI's exact $h$); it's the converter from action to phase when we need phase.

\item No extra dials: we don't retune per experiment; predictions live on the same yardstick.

\item No flux talk: we stay in routes/closure terms the whole time.
\end{itemize}

\section*{Reader map}

\begin{itemize}
\item Walk-Through --- the compact derivations for bend, delay, and clock drift; smooth-limit bridge included.

\item Math Appendix --- the readable equations and validity limits (weak-field, paraxial, symmetry notes).

\item Calibration \& Verification --- procedures, pass/fail tables, ratio-first when tighter than absolute calibration.

\item Student Workbook --- pedagogy and practice problems (separate track).

\item VMS Laws --- concise value cards (one page per law) for quick use.
\end{itemize}

\section*{Bottom line}

We keep the picture simple and the promises tight: one ontology, one scale ($S_0 = \hbar$), three gravity outcomes. The Walk-Through turns that into short formulas you can actually use; the Appendix and tests keep us honest.

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