\documentclass[11pt]{article}
\usepackage[margin=0.78in]{geometry}
\usepackage{amsmath,amssymb,bm}
\usepackage{graphicx}
\usepackage{booktabs}
\usepackage{array}
\usepackage{longtable}
\usepackage{xcolor}
\usepackage{hyperref}
\usepackage{enumitem}
\usepackage{caption}
\usepackage{float}
\usepackage{fancyhdr}
\usepackage{titlesec}
\usepackage[most]{tcolorbox}
\hypersetup{colorlinks=true, linkcolor=blue!40!black, urlcolor=blue!45!black, citecolor=blue!40!black}
\definecolor{rbflblue}{RGB}{24,75,115}
\definecolor{rbflight}{RGB}{237,246,251}
\definecolor{warn}{RGB}{138,57,31}
\pagestyle{fancy}
\fancyhf{}
\lhead{RBFL Gravity Field Law}
\rhead{Speculative technical draft}
\cfoot{\thepage}
\titleformat{\section}{\Large\bfseries\color{rbflblue}}{\thesection}{0.6em}{}
\titleformat{\subsection}{\large\bfseries\color{rbflblue}}{\thesubsection}{0.6em}{}
\setlist[itemize]{topsep=2pt,itemsep=2pt,parsep=0pt}
\setlist[enumerate]{topsep=2pt,itemsep=2pt,parsep=0pt}
\newcommand{\gb}{\mathbf{g}_{b}}
\newcommand{\grbfl}{\mathbf{g}_{\mathrm{RBFL}}}
\newcommand{\ab}{A_b}
\newcommand{\aphi}{A_{\phi}}
\newcommand{\aobs}{A_{\phi,\mathrm{obs}}}
\newcommand{\alock}{A_{\mathrm{lock}}}
\newcommand{\aphiacc}{a_{\phi}}
\newcommand{\atheta}{a_{\Theta}}
\newcommand{\x}{\mathbf{x}}
\newcommand{\grad}{\nabla}
\newcommand{\vect}[1]{\mathbf{#1}}

\begin{document}

\begin{titlepage}
\centering
{\Huge\bfseries Unified RBFL Gravity Field Law\par}
\vspace{0.25cm}
{\Large Locked Acceleration Law, 3D Phase Operator, and a Hypothetical Gravity-Engineering Placeholder\par}
\vspace{0.3cm}
{\large Working technical white paper - no claims of established physics\par}
\vspace{0.45cm}
\includegraphics[width=0.73\textwidth]{figs/rbfl_galaxy_phase_field.png}
\vspace{0.45cm}
\begin{tcolorbox}[colback=rbflight,colframe=rbflblue,width=0.92\textwidth]
\centering
\large
\[
\boxed{\grbfl(\x,t)=\gb(\x,t)+\ab(\x,t)\sqrt{\aphiacc\,|\gb(\x,t)|}\,\widehat{\gb}(\x,t)}
\]
\end{tcolorbox}
\vfill
{Prepared for Levi Haye / RBFL-RBFT development\par}
{\today\par}
\vspace{0.2cm}
\begin{tcolorbox}[colback=orange!5,colframe=warn,width=0.95\textwidth,title=Scientific and engineering disclaimer]
This document is a speculative mathematical formulation. It does not claim that RBFL is established physics, does not claim a working gravity-engineering device, and does not provide construction instructions. The engineering section is a hypothetical response model only: it states what a phase-amplitude change would have to look like mathematically if such control were ever shown to exist. No such control is demonstrated here.
\end{tcolorbox}
\end{titlepage}

\tableofcontents
\newpage

\section{Executive summary}
This paper states an updated RBFL formulation for gravity itself while preserving the separation that has become central to the current framework:
\[
\text{locked gravity law}\quad \neq \quad \text{phase detection and classification unit}.
\]
The proposed core law is the same family as the previously used RBFL acceleration law,
\[
 g_{\mathrm{RBFL}}(r)=g_b(r)+A_{\phi}(r)\sqrt{a_{\phi}g_b(r)},
\]
but written in a cleaner predictive notation:
\[
 g_{\mathrm{RBFL}}(r)=g_b(r)+A_b(r)\sqrt{a_{\phi}g_b(r)}.
\]
The replacement
\[
A_{\phi}\longrightarrow A_b
\]
does not change the physical idea. It clarifies that the amplitude should be predicted or inferred from baryonic structure and phase-detection channels, rather than treated as a free residual-fitting function.

\begin{tcolorbox}[colback=rbflight,colframe=rbflblue,title=Main claim of this draft]
RBFL treats gravity as the sum of the direct baryonic acceleration field and a saturated baryonic phase-response field. The phase response is controlled by a dimensionless phase operator \(A_b(\x,t)\). The law uses \(A_b\); the detector estimates and classifies \(A_b\). The two roles must not be confused.
\end{tcolorbox}

The paper also includes a hypothetical engineering placeholder. It is not a working method. It simply defines what a change in phase response, \(\Delta A_{\mathrm{drive}}\), would imply for the local acceleration field if a controlled phase perturbation were possible.

\section{Status, scope, and no-claim boundary}
This document is a theoretical record and hypothesis-management paper. It should be read as a proposed mathematical structure that can be tested, falsified, revised, or rejected. It is not a peer-reviewed proof and does not replace general relativity, cold dark matter, MOND, or any established theory.

The clean division is:
\begin{enumerate}
\item \textbf{Core law}: the acceleration equation, kept locked.
\item \textbf{Phase operator}: the 3D field-response quantity \(A_b(\x,t)\).
\item \textbf{Phase Detection Unit}: the multi-channel system used to estimate, classify, and validate \(A_b\).
\item \textbf{Hypothetical engineering placeholder}: a mathematical thought experiment for a controlled change in \(A_b\), with no claim that such control is possible.
\end{enumerate}

\section{Baryonic gravity as the base field}
RBFL begins with ordinary baryonic matter. Let the three-dimensional baryonic density field be
\[
\rho_b(\x,t),\qquad \x=(x,y,z).
\]
The standard baryonic potential \(\Phi_b\) obeys
\[
\boxed{\nabla^2\Phi_b(\x,t)=4\pi G\rho_b(\x,t)}
\]
and the baryonic acceleration field is
\[
\boxed{\gb(\x,t)=-\nabla\Phi_b(\x,t).}
\]
Here \(G\) is Newton's gravitational constant. In this draft, \(\gb\) is the baseline vector field produced by the observed baryonic mass distribution.

For a disk galaxy, the radial version is obtained only after projection. If \(r\) is a projected radial coordinate,
\[
 g_b(r)\simeq \frac{G M_b(<r)}{r^2},
\]
where \(M_b(<r)\) is the baryonic mass enclosed within radius \(r\). This radial expression is an observational projection, not the fundamental dimensionality of RBFL.

\section{The 3D saturated phase-field state}
RBFL adds a phase-state description on top of the baryonic source field. The total local phase state is written
\[
\boxed{
\Phi_{\mathrm{total}}(\x,t)=\Phi_{\mathrm{parent}}(\x,t)+\Phi_b(\x,t)+\Phi_{\mathrm{rot}}(\x,t)+\sum_{i,j}\lambda_{ij}\Phi_i(\x,t)\Phi_j(\x,t).
}
\]
The terms are interpreted as follows.

\begin{longtable}{>{\raggedright\arraybackslash}p{0.23\textwidth}>{\raggedright\arraybackslash}p{0.68\textwidth}}
\toprule
Symbol & Meaning \\
\midrule
\(\Phi_{\mathrm{parent}}\) & Background or parent saturated field. In RBFL language, baryons do not create the entire field from zero; they act as local anchors inside a broader phase environment. \\
\(\Phi_b\) & Baryonic compression field associated with visible matter: stars, gas, dust, bulge, disk, and compact baryonic structure. \\
\(\Phi_{\mathrm{rot}}\) & Rotation-organized field representing angular momentum, shear, central-node rotation, disk drag, and organized flow. \\
\(\Phi_i,\Phi_j\) & Local phase components, such as disk, gas, bulge, bar, or environmental contributions. \\
\(\lambda_{ij}\) & Coupling coefficient between components. In a frozen predictive model these coefficients must be fixed or derived, not tuned per object. \\
\bottomrule
\end{longtable}

The total field is bounded through a saturation map:
\[
\boxed{S_{\phi}(\x,t)=\tanh\left(\frac{\Phi_{\mathrm{total}}(\x,t)}{\Phi_c}\right).}
\]
The scale \(\Phi_c\) is the critical phase scale at which the response transitions toward saturation.

\section{Locked gravity law}
The RBFL gravity field is stated as
\[
\boxed{\grbfl(\x,t)=\gb(\x,t)+\mathbf{g}_{\phi}(\x,t),}
\]
where the phase-response acceleration is
\[
\boxed{\mathbf{g}_{\phi}(\x,t)=\ab(\x,t)\sqrt{\aphiacc\,|\gb(\x,t)|}\,\widehat{\gb}(\x,t).}
\]
Thus,
\[
\boxed{
\grbfl(\x,t)=\gb(\x,t)+\ab(\x,t)\sqrt{\aphiacc\,|\gb(\x,t)|}\,\widehat{\gb}(\x,t).
}
\]
The unit direction is
\[
\widehat{\gb}(\x,t)=\frac{\gb(\x,t)}{|\gb(\x,t)|},
\]
where the vector \(\gb\) already points in the local baryonic acceleration direction. In numerical applications, a small regularization \(\epsilon_g\) may be used only to avoid division by zero:
\[
\widehat{\gb}_{\epsilon}=\frac{\gb}{\sqrt{|\gb|^2+\epsilon_g^2}}.
\]
This regularization is not a physical new term.

\begin{tcolorbox}[colback=rbflight,colframe=rbflblue,title=Core interpretation]
The square-root term is the saturated phase-carrier response. The dimensionless multiplier \(A_b(\x,t)\) says how strongly the local baryonic structure couples to that response. The acceleration scale \(a_{\phi}\) sets the phase-carrier scale.
\end{tcolorbox}

\subsection{Radial projection for rotation curves}
For galaxy rotation curves the 3D field is projected into a radial observable:
\[
\boxed{g_{\mathrm{RBFL}}(r)=\left\langle \widehat{\mathbf{r}}\cdot\grbfl(\x)\right\rangle_{\theta,z}.}
\]
In the simplified radial notation this becomes
\[
\boxed{g_{\mathrm{RBFL}}(r)=g_b(r)+A_b(r)\sqrt{a_{\phi}g_b(r)}.}
\]
The predicted circular speed is
\[
\boxed{V_{\mathrm{pred}}(r)=\sqrt{r\,g_{\mathrm{RBFL}}(r)}.}
\]

\section{Outer locked law}
In the outer low-acceleration region,
\[
 g_b(r)\ll \sqrt{a_{\phi}g_b(r)}.
\]
If the field reaches a locked state,
\[
 A_b(r)\rightarrow A_{\mathrm{lock}},
\]
and outside most of the baryonic mass,
\[
 g_b(r)=\frac{GM_b}{r^2}.
\]
Then
\[
 g_{\mathrm{RBFL}}(r)\approx A_{\mathrm{lock}}\sqrt{a_{\phi}\frac{GM_b}{r^2}}
 =\frac{A_{\mathrm{lock}}\sqrt{GM_ba_{\phi}}}{r}.
\]
Since \(v_f^2/r=g_{\mathrm{RBFL}}\),
\[
 v_f^2\approx A_{\mathrm{lock}}\sqrt{GM_ba_{\phi}}.
\]
Squaring gives
\[
\boxed{v_f^4\approx A_{\mathrm{lock}}^2GM_ba_{\phi}.}
\]
Define
\[
\boxed{a_{\Theta}=A_{\mathrm{lock}}^2a_{\phi}.}
\]
Therefore
\[
\boxed{v_f^4=GM_ba_{\Theta}.}
\]
The fixed outer scaling is therefore the locked-state limit of the same law, not a separate empirical add-on.

\section{The phase operator \texorpdfstring{\(A_b\)}{Ab}}
The current 3D detector writes the phase operator as
\[
\boxed{
A_b(\x,t)=A_{\mathrm{lock}}\,S_{\mathrm{phase}}(\x,t)\,D_{\mathrm{phase}}(\x,t)\,H_{\mathrm{phase}}(\x,t)\,G_{\mathrm{rot}}(\x,t)\,B_{\mathrm{switch}}(\x,t).
}
\]
Figure \ref{fig:stack} shows the component logic.

\begin{figure}[H]
\centering
\includegraphics[width=0.95\textwidth]{figs/phase_operator_stack.png}
\caption{Improved 3D phase-operator stack. This stack refines the detector's estimate of \(A_b\); it does not alter the locked acceleration equation.}
\label{fig:stack}
\end{figure}

\begin{longtable}{>{\raggedright\arraybackslash}p{0.23\textwidth}>{\raggedright\arraybackslash}p{0.68\textwidth}}
\toprule
Component & Role \\
\midrule
\(S_{\mathrm{phase}}\) & Phase strength: how strongly the baryonic distribution can produce a saturated response. A radial first approximation is \(\tanh[(\Sigma_b/\Sigma_c)^p]\). \\
\(D_{\mathrm{phase}}\) & Phase degree or coherence: how smooth and coherent the baryonic structure is. Gradients, fragmentation, and sharp transitions reduce coherence. \\
\(H_{\mathrm{phase}}\) & Height or 3D envelope: how far the phase response projects above/below or beyond the disk. This is the term most limited by missing 3D data. \\
\(G_{\mathrm{rot}}\) & Rotational geometry or central-node drag: deformation produced by the rotating central baryonic node, inner disk, bar, bulge, and shear. \\
\(B_{\mathrm{switch}}\) & Switching/boundary factor: flags sharp phase boundaries, event-projected regions, or regions where a smooth radial classification fails. \\
\bottomrule
\end{longtable}

\subsection{Example radial first approximation}
A practical radial first approximation is
\[
S_{\mathrm{phase}}(r)=\tanh\left[\left(\frac{\Sigma_b(r)}{\Sigma_c}\right)^p\right],
\]
\[
D_{\mathrm{phase}}(r)=
\exp\left[-\eta\left|\frac{d\ln\Sigma_b}{d\ln r}\right|\right]
\left[1+\left(\frac{z_d}{R_d}\right)^2\right]^{-1/2}
\left[1+\exp\left(-\frac{r-R_{\phi}}{w_{\phi}}\right)\right]^{-1}.
\]
The outer transition radius is defined from baryons:
\[
\boxed{g_b(R_{\phi})=a_{\phi}}
\]
or, if a frozen softened threshold is declared,
\[
\boxed{g_b(R_{\phi})=\epsilon_{\phi}a_{\phi},\qquad \epsilon_{\phi}\ \text{fixed before testing}.}
\]

\subsection{Rotational geometry}
The rotational term should be computed from baryonic observables. A strict forward version uses
\[
 V_b(R)=\sqrt{R g_b(R)},\qquad \Omega_b(R)=\frac{V_b(R)}{R}=\sqrt{\frac{g_b(R)}{R}}.
\]
A simple shear diagnostic is
\[
\mathcal{S}_{\mathrm{shear}}(R)=\left|\frac{d\ln\Omega_b}{d\ln R}\right|.
\]
A central-node compactness diagnostic can be written schematically as
\[
\mathcal{C}_{\mathrm{node}}=\frac{M_b(<R_c)/R_c^2}{M_b/R_d^2},
\]
where \(R_c\) is a declared core scale. A first rotational geometry term may then be written
\[
G_{\mathrm{rot}}(R,z)=1+\lambda_{\mathrm{rot}}\,\mathcal{C}_{\mathrm{node}}\,\mathcal{S}_{\mathrm{shear}}(R)\,E_z(R,z),
\]
or, in a damping interpretation,
\[
G_{\mathrm{rot,damp}}(R,z)=\left[1+\lambda_{\mathrm{rot}}\,\mathcal{C}_{\mathrm{node}}\,\mathcal{S}_{\mathrm{shear}}(R)\,E_z(R,z)\right]^{-1}.
\]
Both are detector candidates until frozen in an external validation test.

\section{Law versus detector}
The current architecture is shown in Figure \ref{fig:arch}. The formula predicts; the detector estimates, classifies, and diagnoses.

\begin{figure}[H]
\centering
\includegraphics[width=0.92\textwidth]{figs/gravity_architecture_locked_detector.png}
\caption{The locked law remains separated from the detection layer. Lensing/event channels and other diagnostics help estimate the phase operator; they are not inserted as new acceleration terms.}
\label{fig:arch}
\end{figure}

\subsection{Observed phase detector}
After a prediction is made, an observed residual phase amplitude may be reconstructed as
\[
\boxed{A_{\phi,\mathrm{obs}}(r)=\frac{g_{\mathrm{obs}}(r)-g_b(r)}{\sqrt{a_{\phi}g_b(r)}}.}
\]
The acceleration ratio detector is
\[
\Phi(r)=\frac{g_{\mathrm{obs}}(r)}{g_b(r)}=\frac{V_{\mathrm{obs}}^2(r)}{V_{\mathrm{bar}}^2(r)},
\]
and the log detector is
\[
y(r)=\ln\Phi(r).
\]
These quantities are post-test diagnostics. They do not define the prediction in a blind test.

\subsection{Zero-phase reference gauge}
The current detector uses NGC7331 as a zero-phase reference gauge:
\[
\theta_{\mathrm{NGC7331}}=0^{\circ}.
\]
A generic relative phase angle may be written
\[
\theta_{\mathrm{gal}}=\mathrm{wrap}_{180}\left[360^{\circ}\frac{\Delta\ln R_{\mathrm{gal}}-\Delta\ln R_{\mathrm{ref}}}{L_{\mathrm{ref}}}\right].
\]
This gauge helps classify phase families. It is a detector convention, not a new dynamical law.

\subsection{Lensing/event amplification as detection only}
Lensing arcs and Bullet-style projected amplification are used only in the Phase Detection Unit. They do not modify \(g_{\mathrm{RBFL}}\). Define
\[
\Lambda_{\mathrm{lens,obs}}=\frac{\text{observed projected lensing response}}{\text{baryonic projected baseline}}.
\]
The previous approximate event clue \(\Lambda_{\mathrm{event}}\sim4\) is now treated as a detection-stage parameter or band,
\[
\boxed{\lambda_{\mathrm{event}}\in[\lambda_{\min},\lambda_{\max}],}
\]
not as a fixed constant. This is necessary because arc size, convergence, deflection, and apparent magnification are not the same observable. The lensing/event channel helps constrain \(H_{\mathrm{phase}}\) and \(B_{\mathrm{switch}}\), not the core law.

\section{Hypothetical gravity-engineering placeholder}
This section is deliberately framed as a mathematical placeholder. It is not a claim of a working method, not a device design, and not evidence that gravity can be engineered.

\begin{tcolorbox}[colback=orange!5,colframe=warn,title=No engineering claim]
The following equations answer only one question: if a controlled perturbation to the phase operator existed, how would it enter the RBFL acceleration expression? No known mechanism is asserted. No laboratory pathway is provided.
\end{tcolorbox}

Let a hypothetical external process produce a small dimensionless perturbation
\[
\Delta A_{\mathrm{drive}}(\x,t)
\]
to the local phase operator. Then
\[
A_{\mathrm{eff}}(\x,t)=A_b(\x,t)+\Delta A_{\mathrm{drive}}(\x,t).
\]
The hypothetical acceleration change would be
\[
\boxed{\Delta\mathbf{g}_{\mathrm{hyp}}(\x,t)=\Delta A_{\mathrm{drive}}(\x,t)\sqrt{a_{\phi}|\mathbf{g}_b(\x,t)|}\,\widehat{\mathbf{u}}_{\mathrm{proj}}(\x,t).}
\]
The projection direction \(\widehat{\mathbf{u}}_{\mathrm{proj}}\) cannot be arbitrary in a physical model. In the most conservative case it is aligned with the baryonic field direction,
\[
\widehat{\mathbf{u}}_{\mathrm{proj}}=\widehat{\gb}.
\]
If a future theory allowed another projection channel, that channel would have to be specified by a validated 3D phase model.

\subsection{Required phase-amplitude shift for a target acceleration}
If a target incremental acceleration magnitude \(\Delta g_{\mathrm{target}}\) is specified, the required phase-amplitude shift would be
\[
\boxed{\Delta A_{\mathrm{req}}(\x)=\frac{|\Delta g_{\mathrm{target}}(\x)|}{\sqrt{a_{\phi}|\gb(\x)|}}.}
\]
This equation is useful as a dimensional sanity check. It does not say that such \(\Delta A\) can be produced.

\begin{figure}[H]
\centering
\includegraphics[width=0.76\textwidth]{figs/hypothetical_deltaA_required.png}
\caption{Hypothetical required phase-amplitude shift as a function of background baryonic acceleration. This is a mathematical scaling diagram, not an engineering result.}
\end{figure}

\subsection{Hypothetical drive decomposition}
A generic placeholder for a drive-induced perturbation is
\[
\boxed{\Delta A_{\mathrm{drive}}=\chi_{\phi}\,\mathcal{I}_{\mathrm{drive}}\,\mathcal{S}_{\mathrm{mat}}\,\mathcal{C}_{\mathrm{coh}}\,\mathcal{P}_{\mathrm{geom}}\,\mathcal{E}_{\mathrm{bound}}.}
\]
The factors are purely formal:

\begin{longtable}{>{\raggedright\arraybackslash}p{0.23\textwidth}>{\raggedright\arraybackslash}p{0.68\textwidth}}
\toprule
Symbol & Meaning in the hypothetical placeholder \\
\midrule
\(\chi_{\phi}\) & Unknown phase-coupling susceptibility. No measured value is claimed. \\
\(\mathcal{I}_{\mathrm{drive}}\) & Dimensionless normalized drive intensity or perturbation amplitude. No mechanism is specified. \\
\(\mathcal{S}_{\mathrm{mat}}\) & Material/source saturation compatibility, if such a concept exists. \\
\(\mathcal{C}_{\mathrm{coh}}\) & Coherence factor describing whether the perturbation can stay phase-organized. \\
\(\mathcal{P}_{\mathrm{geom}}\) & Projection geometry into the acceleration channel. \\
\(\mathcal{E}_{\mathrm{bound}}\) & Boundary/envelope factor that prevents unbounded response. \\
\bottomrule
\end{longtable}

For a perturbative mathematical regime one would require
\[
|\Delta A_{\mathrm{drive}}|\ll A_{\mathrm{lock}}
\]
so that the locked galaxy-scale law is not being replaced by a different theory.

\subsection{Engineering interpretation}
The hypothetical engineering statement is therefore:
\[
\boxed{\text{To change gravity in RBFL, one would need to change the local phase operator }A_b.}
\]
The placeholder equation says what such a change would imply mathematically. It does not state that the change is physically possible.

\section{Internal benchmark context}
The following values summarize the current internal detector context. They are not external validation. They show why the detector, not the law, is the present focus.

\begin{table}[H]
\centering
\caption{Current internal benchmark summary.}
\begin{tabular}{lcc}
\toprule
Mode & Holdout phase-state match & Holdout velocity pass \\
\midrule
Prior improved 3D detector & 75\% & 70\% \\
Adjustable event-band detector & 95\% & 70\% \\
\bottomrule
\end{tabular}
\end{table}

The interpretation is that the event/lensing band sharpens phase detection but does not change the locked velocity law. Velocity does not improve unless the predictive \(A_b\) recipe improves.

\begin{figure}[H]
\centering
\includegraphics[width=0.78\textwidth]{figs/lambda_event_band_sweep.png}
\caption{Internal detection-stage sweep over the event-band parameter. This parameter is used for phase classification only; it is not inserted into the acceleration law.}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.78\textwidth]{figs/holdout_lambda_detector_states.png}
\caption{Holdout classification plot from the adjustable event-band detector. The improved classification is a detector result, not a new force-law result.}
\end{figure}

\section{Reproducible application recipe}
A researcher attempting to reproduce the current RBFL gravity calculation should separate prediction and detection.

\subsection{Prediction-side calculation}
\begin{enumerate}
\item Choose a baryonic mass prescription and compute \(\rho_b(\x)\), \(\Phi_b(\x)\), and \(\gb(\x)\).
\item Compute or approximate the phase-operator components \(S,D,H,G,B\) from declared baryonic observables.
\item Form
\[
A_b(\x)=A_{\mathrm{lock}}S_{\mathrm{phase}}D_{\mathrm{phase}}H_{\mathrm{phase}}G_{\mathrm{rot}}B_{\mathrm{switch}}.
\]
\item Compute
\[
\grbfl(\x)=\gb(\x)+A_b(\x)\sqrt{a_{\phi}|\gb(\x)|}\widehat{\gb}(\x).
\]
\item Project to the observable of interest, such as a rotation curve:
\[
V_{\mathrm{pred}}(r)=\sqrt{r g_{\mathrm{RBFL}}(r)}.
\]
\end{enumerate}

\subsection{Detection-side calculation}
Only after prediction is scored, calculate residual and phase-detection diagnostics:
\[
A_{\phi,\mathrm{obs}}(r)=\frac{g_{\mathrm{obs}}(r)-g_b(r)}{\sqrt{a_{\phi}g_b(r)}}.
\]
Then compare \(A_b\) and \(A_{\phi,\mathrm{obs}}\), use the NGC7331 reference gauge, and classify suppressed, locked, amplified, switching, or event-projected states. Lensing arcs, wide binaries, and timing/spacing clues may be used as detection channels only if their rules are declared.

\section{Full symbol dictionary}
\begin{longtable}{>{\raggedright\arraybackslash}p{0.22\textwidth}>{\raggedright\arraybackslash}p{0.70\textwidth}}
\toprule
Symbol & Definition \\
\midrule
\(\rho_b(\x,t)\) & Three-dimensional baryonic density. \\
\(\Phi_b\) & Baryonic potential satisfying \(\nabla^2\Phi_b=4\pi G\rho_b\). \\
\(\gb\) & Baryonic acceleration field, \(-\nabla\Phi_b\). \\
\(\grbfl\) & Total RBFL acceleration field. \\
\(A_b\) & Baryon-derived phase operator used by the law. \\
\(A_{\phi,\mathrm{obs}}\) & Observed residual-derived phase amplitude used only after prediction. \\
\(a_{\phi}\) & Phase-carrier acceleration scale. \\
\(A_{\mathrm{lock}}\) & Locked outer-regime amplitude. \\
\(a_{\Theta}\) & Locked outer acceleration scale, \(A_{\mathrm{lock}}^2a_{\phi}\). \\
\(S_{\mathrm{phase}}\) & Phase strength. \\
\(D_{\mathrm{phase}}\) & Phase coherence or degree. \\
\(H_{\mathrm{phase}}\) & 3D height/envelope factor. \\
\(G_{\mathrm{rot}}\) & Rotational geometry/central-node drag factor. \\
\(B_{\mathrm{switch}}\) & Boundary or switching factor. \\
\(\lambda_{\mathrm{event}}\) & Detection-stage event/lensing band parameter. Not a law parameter. \\
\(\Delta A_{\mathrm{drive}}\) & Hypothetical engineered perturbation to the phase operator. No physical realization is claimed. \\
\(\Delta\mathbf{g}_{\mathrm{hyp}}\) & Hypothetical acceleration change implied by \(\Delta A_{\mathrm{drive}}\). \\
\bottomrule
\end{longtable}

\section{Final statements}
The updated RBFL gravity formulation can be summarized as:
\[
\boxed{\grbfl=\gb+A_b\sqrt{a_{\phi}|\gb|}\,\widehat{\gb}.}
\]
This is the locked core law. The phase operator \(A_b\) is the object to be predicted, inferred, and refined.

The detection-stage lesson is:
\[
\boxed{\text{Better gravity predictions require better phase detection, especially better 3D data.}}
\]
The present limitation is not only mathematical. The detector lacks full 3D baryonic density, full lensing convergence maps, complete wide-binary tracer fields, and complete timing/spacing datasets. Current phase shape is therefore estimated from projected observables.

The hypothetical engineering lesson is:
\[
\boxed{\Delta\mathbf{g}_{\mathrm{hyp}}=\Delta A_{\mathrm{drive}}\sqrt{a_{\phi}|\gb|}\,\widehat{\mathbf{u}}_{\mathrm{proj}}.}
\]
This is only a placeholder. No claim is made that \(\Delta A_{\mathrm{drive}}\) can be produced, controlled, or used in practice.

\begin{tcolorbox}[colback=orange!5,colframe=warn,title=Closing no-claim statement]
This white paper is a speculative RBFL theory record. It defines a possible gravity-field equation and a hypothetical response formula. It does not claim experimental verification, does not claim a gravity-engineering method, and does not provide instructions for creating, cancelling, shielding, or manipulating gravity.
\end{tcolorbox}

\end{document}
