Mass scale
\[\displaystyle r_s=\frac{2GM}{c^2},\qquad x=\frac r{r_s}\]
\(r_s\) has units of length; \(x\) is the dimensionless areal radius.
Equation → domain → implementation → evidenceFormula and
Formula and
symbol reference
A formula is shown only with its variables, units, assumptions and failure modes. Display mathematics follows the clear card system of the original research dashboard while preserving accessible source text.
Ξ
Canonical spine
Six equations that define the current static model
Piecewise segment field
\[\displaystyle
\Xi(x)=\begin{cases}
1-e^{-\varphi/x},&x<1.8\\
H_5((x-1.8)/0.4),&1.8\le x\le2.2\\
(2x)^{-1},&x>2.2
\end{cases}\]
The bridge is derivative-matched through second order.
Clock and radial factors
\[\displaystyle D=\frac1{1+\Xi},\qquad A=D^2,\qquad B=D^{-2},\qquad AB=1\]
These are derived from \(\Xi\), not independently fitted functions.
Static spherical line element
\[\displaystyle ds^2=-D^2c^2dt^2+D^{-2}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)\]
The coordinate \(r\) is areal: symmetry spheres have area \(4\pi r^2\).
Static clock comparison
\[\displaystyle d\tau=D(r)\,dt,\qquad 1+z=\frac{D(r_o)}{D(r_e)}\]
Emitter, observer and kinematic terms must be stated.
Current diagonal interior asymptotics
\[\displaystyle A\to\frac14,\qquad R\sim\frac3{2r^2},\qquad K\sim\frac9{4r^4}\]
These terms diagnose the present diagonal extrapolation. They do not decide whether a separately derived SSZ inner solution or boundary geometry can be regular.
Reading discipline
Every formula carries four boundaries
Domain
Coordinate range, branch, approximation order and observer class.
Dimensions
Units of every symbol and a consistent dimensional check.
Provenance
Definition, mathematical consequence, implementation, test or empirical comparison.
Dual velocity closure
\[\displaystyle v_{\rm esc}=c\sqrt{\frac{r_s}{r}},\qquad v_{\rm fall}=c\sqrt{\frac{r}{r_s}},\qquad v_{\rm esc}v_{\rm fall}=c^2\]
vfall is a reciprocal kinematic scale, not a locally measured superluminal 3-velocity. See the interactive Metric tab.
Non-claim
The conclusion that does not follow from the equation or its software test.
Reviewed equations
Complete curated formula browser
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Extraction boundary: 4,000 unique formula candidates were found in ordinary text, Markdown and existing TeX sources. They are available in the machine catalogue but never become canonical without context review. See curation notes.