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\title{Mechanics --- Math Walk-Through}
\author{}
\date{}

\begin{document}

\maketitle

\section*{Purpose}

Derive the working kernels for bending, travel-time delay, and clock drift in the VMS framework, and show the smooth-limit bridge to the standard weak-field results.

\section*{Conventions and Locks}

\subsection*{Clarifications \& Conventions (alignment with Narrative and Appendix)}

\begin{itemize}
\item ``Route action'' is the display-area action $S[\text{path}] \equiv \int A_d \, ds$. We compare routes by $S$; phase appears only when needed via $\varphi_{\text{phase}} = S/S_0$.

\item Background/cost map $n(x)$: think of $n$ as a smooth cost map tied to missing space; higher $n$ means ``this step hides a bit more display-area than average.'' It is not a new substance.

\item Link to potential and sign convention: set $n(x) = 1 + \eta_0 \varphi(x)$ with $\varphi = \Phi/c^2$ and $\Phi < 0$ near mass. With $|\eta_0|$ fixed by deflection ($|\eta_0| = 2$), bending is toward the mass and travel-time shifts are positive.

\item Scope note: weak + strong field live here. For steep profiles $n(x)$ evaluate the same kernels piecewise; avoid turning points with refinement.

\item Single dimensional scale: $S_0 = \hbar$ (SI-locked).

\item Projected display-area: $A_d = \pi r^2$ (orthographic; no $4\pi$ shells).

\item Route action: $S[\text{path}] = \int A_d \, ds$ (ratios first; phase $= S/S_0$ when needed).

\item Speed of advance: $c$; Newton's constant: $G$.
\end{itemize}

\subsection*{Background Model and Ray Equation}

A stationary mass is treated as a closed circulation whose far-field effect is summarized by a smooth cost map $n(x)$. Adopt the WKB/eikonal ansatz $U = A e^{i k_0 S}$. Collecting powers of $k_0$ yields the eikonal and transport equations:
\begin{align}
|\nabla S|^2 &= n(x)^2 \\
2 \nabla S \cdot \nabla A + A \nabla^2 S &= 0
\end{align}

Rays follow the normal to $S$. Fermat form $\delta \int n \, ds = 0$ gives the ray equation:
\begin{equation}
\frac{d}{ds} ( n \mathbf{t} ) = \nabla n \quad \text{($\mathbf{t}$ is the unit tangent)}
\end{equation}

\section*{1. Bending Kernel}

Take the component perpendicular to the ray:
\begin{equation}
\frac{d\theta}{ds} = \frac{1}{n} (\nabla_\perp n) \approx \nabla_\perp \ln n \quad \text{(small angles)}
\end{equation}

Integrate along the unperturbed straight line to get the net deflection vector:
\begin{equation}
\vec{\alpha} \approx \int (\nabla_\perp \ln n) \, ds
\end{equation}

Weak-field model: $n(x) = 1 + \eta_0 \varphi(x)$, with $\varphi = \Phi/c^2$ and $\Phi = -GM/r$. For $|\eta_0 \varphi| \ll 1$, $\ln n \approx \eta_0 \varphi$, so:
\begin{equation}
\vec{\alpha} \approx \eta_0 \int \nabla_\perp \varphi(x) \, ds
\end{equation}

For a grazing pass with impact parameter $b$:
\begin{equation}
|\alpha| \approx 2 |\eta_0| \frac{GM}{b c^2}
\end{equation}

Matching the standard weak-field value $|\alpha| = 4 GM/(b c^2)$ fixes $|\eta_0| = 2$ (sign chosen to match deflection toward mass).

\subsection*{1A. Strong-Field: Photon-Sphere and Capture (dictionary check)}

For steep, spherically symmetric profiles $n(r)$ that grow sharply as $r$ decreases, closed circular null routes appear when transverse curvature balances forward advance. Define the critical impact parameter $b_c$ as the smallest value for which a passing route still escapes; as $b \to b_c^+$, the deflection $|\alpha|$ diverges and multiple ``looped'' images arise. For $b < b_c$, routes are captured.

In the GR-limit dictionary this reproduces the familiar values (photon sphere and critical impact parameter):
\begin{equation}
b_c = \frac{3\sqrt{3} GM}{c^2}
\end{equation}

Evaluation uses the same kernel $\vec{\alpha} = \int \nabla_\perp \ln n \, ds$ on the steep $n(r)$ profile; the divergence near $b_c$ is a geometric property of the logarithmic integral in a rapidly varying cost map.
\begin{equation}
r_{\text{ph}} = \frac{3 GM}{c^2}
\end{equation}

\section*{2. Travel-Time Delay}

Propagation time increment relative to empty space:
\begin{equation}
\Delta t = \frac{1}{c} \int [ n(x) - 1 ] \, ds \approx \frac{\eta_0}{c} \int \varphi(x) \, ds
\end{equation}

For a point mass on a straight-line path with closest approach $b$, evaluation gives the standard logarithmic form after using $|\eta_0| = 2$:
\begin{equation}
\Delta t \approx \frac{2 GM}{c^3} \ln\left( \frac{r_S + r_R + D}{r_S + r_R - D} \right)
\end{equation}

\subsection*{2A. Observer Dictionary --- Lensing Map and Time-Delay Decomposition}

For far-field geometry (thin-lens approximation) with distances $D_l$ (observer$\to$lens), $D_s$ (observer$\to$source), $D_{ls}$ (lens$\to$source), connect the deflection kernel to observed angles:

Here $\alpha(\theta) \approx (1/D_l) \int \nabla_\perp \ln n \, ds$ written along the unperturbed path. Relative arrival time between two images $i$ and $j$ splits into geometric and potential parts:
\begin{equation}
\Delta t_{ij} = \frac{D_\Delta}{c} \left[ \tfrac{1}{2}(\theta_i - \beta)^2 - \tfrac{1}{2}(\theta_j - \beta)^2 - \psi(\theta_i) + \psi(\theta_j) \right]
\end{equation}

\begin{equation}
\beta = \theta - \frac{D_{ls}}{D_s} \alpha(\theta)
\end{equation}

The projected potential $\psi$ is linked to $\ln n$ via the same delay kernel $\Delta t = (1/c) \int (n - 1) \, ds$. This dictionary is only a re-expression of the kernels above for inference use; no new parameters are introduced.

$r_S, r_R$: source/receiver distances from the mass; $D$: separation.

\section*{3. Clock Drift (Redshift / Time-Dilation)}

In a stationary background the local tick rate obeys, to leading order:
\begin{equation}
d\tau \approx dt \cdot \sqrt{1 + 2 \varphi} \quad \Rightarrow \quad \frac{\Delta \nu}{\nu} \approx -\Delta \varphi
\end{equation}

We treat this as the smooth-limit dictionary of the same cost picture (no new parameters).

\section*{4. Bound Orbits: Periapsis Precession (Sketch)}

For slowly moving matter routes in a central profile $n(r) = 1 + \eta_0 \varphi(r)$, transverse curvature from $\nabla_\perp \ln n$ produces a small advance of periapsis per orbit. In the weak-field, low-speed limit the result reduces to:
\begin{equation}
\Delta \varpi \approx \frac{6 \pi GM}{a (1 - e^2) c^2} \quad \text{per orbit}
\end{equation}

Full derivation is provided via an effective potential constructed from the ray equation (Appendix).

\section*{5. Rotators and Frame-Dragging (Preview)}

With angular momentum $J$, the background acquires a small odd-parity correction (azimuthal bias). In the smooth-limit dictionary this corresponds to the Lense--Thirring rate:
\begin{equation}
\Omega_{\text{LT}} = \frac{2 G J}{c^2 r^3} \quad \text{(axis-aligned, far field)}
\end{equation}

We treat these corrections as extensions; in bench contexts they are reported as bounds unless a positive detection is required.

\section*{6. Validity and Limits}

\begin{itemize}
\item Eikonal/WKB: $|\nabla n|/n \ll k_0$. In strong field, integrate piecewise and avoid turning points.

\item Paraxial expressions are used for closed-form deflection; otherwise evaluate $\vec{\alpha} = \int \nabla_\perp \ln n \, ds$ numerically on $n(x)$.

\item No new dimensional scales are introduced beyond $S_0 = \hbar$; $c$ and $G$ are acceptance locks.
\end{itemize}

\section*{7. Replication Examples}

Solar grazing deflection: with $b \approx R_\odot$ and $M_\odot$,
\begin{equation}
|\alpha| \approx \frac{4 G M_\odot}{R_\odot c^2} \approx 1.75 \text{ arcsec}
\end{equation}

Shapiro delay at superior conjunction: logarithmic microsecond-scale shift with the same $\eta_0$.

Near-Earth gravitational redshift: $\Delta\nu/\nu \approx g h / c^2$ at laboratory heights (Pound--Rebka scale).

\section*{Appendix Pointers}

\begin{itemize}
\item Eikonal working: gradients of $U = A e^{i k_0 S}$; transport; derivation of $d/ds(n \mathbf{t}) = \nabla n$.

\item Deflection integral along straight-line parameterization ($x=b$, $z=s$).

\item Delay integral and endpoint geometry for the logarithm.

\item Time-dilation mapping to $\sqrt{1 + 2\varphi}$.

\item Periapsis advance via effective potential.

\item Rotators: odd-parity term and far-field limit.
\end{itemize}

\section*{Bottom Line}

One ontology, one scale ($S_0 = \hbar$), one cost map $n(x)$. The once-set factor that matches solar deflection also fixes delay and clock-shift. No extra knobs appear mid-derivation; kernels are short and SI-locked.

\section*{Appendix Setup (prep for detailed Math Appendix)}

The companion Appendix carries the line-by-line evaluations alluded to above, using the same symbols and sign conventions stated here. Pointers:

\begin{itemize}
\item Rotators: odd-parity correction (far-field Lense--Thirring limit) and bench-style bounds.

\item Periapsis precession: effective potential constructed from $d/ds(n \mathbf{t}) = \nabla n$ and small-eccentricity expansion.

\item Time-dilation: local cost-per-time ratio leading to $\sqrt{1 + 2\varphi}$ and $\Delta\nu/\nu \approx -\Delta\varphi$.

\item Delay kernel: endpoint geometry for the standard log expression; relation to $\psi$ in the thin-lens dictionary.

\item Deflection integral in a $1/r$ potential: straight-line parameterization ($x = b$, $z = s$) and extraction of the logarithmic term.
\end{itemize}

\end{document}
