# RBFL Equations Dictionary

This dictionary defines the equations used in the reproducibility package.  The package separates the locked RBFL acceleration law from the Phase Detection Unit.  Detection channels are allowed to infer phase from observables, but they do not modify the locked gravity law.

## 1. Locked RBFL gravity law

### 1.1 Three-dimensional vector form

```math
\mathbf{g}_{\rm RBFL}(\mathbf{x},t)
=
\mathbf{g}_{b}(\mathbf{x},t)
+
A_b(\mathbf{x},t)
\sqrt{a_\phi\,|\mathbf{g}_{b}(\mathbf{x},t)|}\,
\widehat{\mathbf{g}}_{b}(\mathbf{x},t)
```

**Meaning.** The total RBFL acceleration is the baryonic acceleration plus a saturated baryonic phase response.  The scalar phase operator `A_b` controls how strongly the local baryonic structure couples to the phase-carrier response.

### 1.2 Radial rotation-curve projection

```math
g_{\rm RBFL}(r)=g_b(r)+A_b(r)\sqrt{a_\phi g_b(r)}
```

```math
V_{\rm pred}(r)=\sqrt{r\,g_{\rm RBFL}(r)}
```

This is the circular-orbit observable used for SPARC-style rotation-curve tests.  It is a projection of the 3D field, not a replacement for the 3D theory.

### 1.3 Locked outer limit

In the low-acceleration outer regime,

```math
g_b(r)\ll \sqrt{a_\phi g_b(r)}
```

and in a locked outer phase,

```math
A_b(r)\rightarrow A_{\rm lock}.
```

For baryonic mass outside most of the visible mass,

```math
g_b(r)=\frac{G M_b}{r^2}.
```

Then,

```math
v_f^4 = G M_b a_\Theta,
\qquad
 a_\Theta=A_{\rm lock}^2 a_\phi.
```

This is the fixed outer scaling law.  It is not a separate law; it is the locked-state limit of the radial projection.

## 2. Three-dimensional phase-field hierarchy

```math
\Phi_{\rm total}(\mathbf{x})
=
\Phi_{\rm parent}(\mathbf{x})
+
\Phi_b(\mathbf{x})
+
\Phi_{\rm rot}(\mathbf{x})
+
\sum_{i,j}\lambda_{ij}\Phi_i(\mathbf{x})\Phi_j(\mathbf{x}).
```

The bounded saturation function is

```math
S_\phi(\mathbf{x})=\tanh\!\left(\frac{\Phi_{\rm total}(\mathbf{x})}{\Phi_c}\right).
```

This field hierarchy motivates the phase operator but is not currently a complete covariant action theory.

## 3. Baryon-derived phase operator

The current detection/prediction paper uses

```math
A_b(\mathbf{x})
=
A_{\rm lock}
S_{\rm phase}(\mathbf{x})
D_{\rm phase}(\mathbf{x})
H_{\rm phase}(\mathbf{x})
G_{\rm rot}(\mathbf{x})
B_{\rm switch}(\mathbf{x}).
```

This is a factorized recipe for the phase-amplitude operator.  The factors are detection-stage estimates of phase strength, coherence, height, rotation/drag geometry, and switching/boundary structure.

### 3.1 Phase strength

```math
S_{\rm phase}(r)=\tanh\!\left[\left(\frac{\Sigma_b(r)}{\Sigma_c}\right)^p\right].
```

- `Sigma_b(r)`: projected baryonic surface density.
- `Sigma_c`: universal critical surface-density scale for saturation.
- `p`: saturation sharpness.

### 3.2 Phase degree / coherence

```math
D_{\rm 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}.
```

- `eta`: gradient penalty.
- `z_d/R_d`: disk thickness ratio.
- `R_phi`: outer transition radius.
- `w_phi`: transition width.

### 3.3 Outer transition radius

```math
g_b(R_\phi)=a_\phi
```

or, if a frozen softened threshold is used,

```math
g_b(R_\phi)=\epsilon_\phi a_\phi.
```

This is the modern form of the original outer-rim insight.  The old detector used the outer rim to infer missing acceleration; the locked prediction version uses baryonic acceleration to define where the phase response becomes externally visible.

### 3.4 Rotational central-node geometry

A prediction-side proxy may be formed from baryonic rotation quantities only:

```math
V_b(R)=\sqrt{R g_b(R)},
\qquad
\Omega_b(R)=\frac{V_b(R)}{R}=\sqrt{\frac{g_b(R)}{R}}.
```

A shear/deformation proxy may use

```math
\mathcal{S}_{\rm shear}(R)=\left|\frac{d\ln\Omega_b}{d\ln R}\right|.
```

The detector interprets the central baryonic node as the bulge, inner disk, bar, gas core, and any compact central mass.  This term estimates how rotation, shear, and core compactness deform the projected phase field.

### 3.5 Switching/boundary term

A simple diagnostic indicator is based on phase gradients:

```math
B_{\rm switch}\sim f\!\left(\left|\nabla A_b\right|,\left|\nabla A_{\phi,\rm obs}\right|,D_{\rm spike}\right).
```

In this package, switching is evaluated through point-level and galaxy-level detector features rather than a single final analytic formula.

### 3.6 Height / 3D envelope term

A conservative envelope placeholder is

```math
H_{\rm phase}(R,z)=\exp\!\left(-\frac{|z|}{h_\phi(R)}\right).
```

Because full 3D baryonic and lensing data are not yet available in the package, the height term is inferred from projected observables and proxy features.  This is a known limitation.

## 4. Observed phase detector

After a prediction has been made, the observed residual channel can be computed:

```math
A_{\phi,\rm obs}(r)
=
\frac{g_{\rm obs}(r)-g_b(r)}{\sqrt{a_\phi g_b(r)}}.
```

with

```math
g_{\rm obs}(r)=\frac{V_{\rm obs}^2(r)}{r}.
```

This detector is diagnostic only.  It must not be used to build the forward prediction in a blind test.

## 5. Acceleration ratio detector

```math
\Phi(r)=\frac{g_{\rm obs}(r)}{g_b(r)}=\frac{V_{\rm obs}^2(r)}{V_{\rm bar}^2(r)}.
```

```math
y(r)=\ln\Phi(r).
```

The radial phase drift statistic is

```math
B_y=y_{\rm outer}-y_{\rm inner}.
```

## 6. NGC7331 zero-phase reference gauge

The original detector uses NGC7331 as the stable reference gauge:

```math
\theta_{\rm NGC7331}=0^\circ.
```

Relative phase offset may be written schematically as

```math
\theta_{\rm gal}
=
{\rm wrap}_{180}\!\left[
360^\circ\frac{\Delta\ln R_{\rm gal}-\Delta\ln R_{\rm ref}}{L_{\rm ref}}
\right].
```

This is a detection-stage gauge.  It does not modify the locked acceleration law.

## 7. Lensing/event projection detector

Lensing arcs and Bullet-style event projection are used only in the Phase Detection Unit.

```math
\Lambda_{\rm lens,obs}
=
\frac{\hbox{observed projected lensing response}}
{\hbox{baryonic projected baseline}}.
```

The adjustable event-band detector uses a center `lambda_event` and width `sigma_event`:

```math
\mathcal{E}_{\rm event}
=
\exp\!\left[-\frac{1}{2}\left(\frac{\ln(\Lambda_{\rm phase}/\lambda_{\rm event})}{\sigma_{\rm event}}\right)^2\right],
```

where

```math
\Lambda_{\rm phase}=\frac{A_{\phi,\rm obs}}{A_b}.
```

In the latest included sweep, the best compact detector band was centered near

```math
\lambda_{\rm event}=2.0,
\qquad
\sigma_{\rm event}=0.25.
```

This is a detection-stage parameter that estimates event-projected 3D phase geometry.  It is not a new force term and not a law parameter.

## 8. Lensing projection bridge

A future lensing test can use

```math
\Sigma_{\rm RBFL}(X,Y)=\int \rho_b(X,Y,Z)A_b(X,Y,Z)\,dZ,
```

```math
\kappa_{\rm RBFL}=\frac{\Sigma_{\rm RBFL}}{\Sigma_{\rm crit}},
```

```math
\nabla^2\psi_{\rm RBFL}=2\kappa_{\rm RBFL},
\qquad
\boldsymbol{\alpha}_{\rm RBFL}=\nabla\psi_{\rm RBFL}.
```

The package contains pilot/proxy lensing tables, not full raw lensing images or full convergence maps.

## 9. Wide-binary and timing channels

These are not run in the current benchmark package because the necessary datasets are not included.  They are reserved for future detection channels.

Wide-binary tracer channel:

```math
\mathcal{W}(\mathbf{x})=
\left[\mathbf{x}_{\rm bin},\mathbf{v}_{\rm COM},\mathbf{s},\mathbf{v}_{\rm rel},M_{\rm pair}\right].
```

Timing/beat channel:

```math
A_{\phi,\rm timing}
=K_\phi\left|\frac{P_{\rm obs}-P_{\rm beat}}{P_{\rm beat}}\right|W_MW_RW_\theta.
```

Both are detection-stage channels only and do not modify the locked galaxy law.

## 10. Phase states

The detector uses the labels below:

- `suppressed`: predicted or observed phase response below the locked reference.
- `locked`: stable reference-like outer phase; tends toward the fixed law.
- `amplified`: response above the locked reference.
- `switching`: phase-gradient, event, boundary, or unresolved-transition state.

## 11. Key reproducibility outputs

Latest included detector benchmark:

- Prior improved detector holdout phase-state match: 75%.
- Adjustable lensing/event-band detector holdout phase-state match: 95%.
- Locked-law holdout velocity pass at 15% MAPE tolerance: 70%.
- Combined velocity plus phase pass after event-band detector: 70%.

These are detector-stage results.  They should not be stated as proof of an established gravity theory.
