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.
Formula explanations
Read every canonical equation without hidden assumptions
These authored notes are kept beside the six canonical equations and the dual-velocity closure. They distinguish definition, interpretation, domain and verification so the automatic catalogue cannot silently replace context with a generic sentence.
Explain this formulaMass scale and dimensionless radius
Purpose
\(r_s=2GM/c^2\) fixes the length scale and \(x=r/r_s\) removes units from the radial coordinate.
How to read it
Compute \(r_s\) from the declared mass and constants, then divide \(r\) by \(r_s\).
Domain and meaning
\(r_s\) has length units; \(x\) is dimensionless. This does not by itself define a metric.
Verification
Dimensional and limit checks validate the normalization, not an independent physical claim.
Explain this formulaPiecewise field bridge
Purpose
The inner, quintic bridge and outer branches define one locked field over their stated \(x\)-ranges.
How to read it
Select the branch by \(x\); use the rescaled argument in the middle branch.
Domain and meaning
The two junctions delimit the bridge. Second-order matching means value, first derivative and second derivative agree at both joins.
Verification
Symbolic derivative identities test the bridge; they do not prove a different interior continuation is regular.
Explain this formulaDerived clock and radial factors
Purpose
\(D\), \(A\) and \(B\) are derived from one field \(\Xi\), with \(AB=1\) as an algebraic identity.
How to read it
Evaluate \(D=(1+\Xi)^{-1}\), then square and invert the square.
Domain and meaning
The definition requires \(1+\Xi\ne0\). These are ansatz factors, not separately fitted observables.
Verification
Algebraic, dimensional and regression tests establish the declared chain only.
Explain this formulaStatic spherical line element
Purpose
The line element specifies the current static, spherically symmetric diagonal geometry.
How to read it
The first term is temporal, the second radial, and the final two terms describe the areal two-sphere.
Domain and meaning
\(r\) is areal because spheres have area \(4\pi r^2\); \(D\) is the declared field-derived factor.
Verification
Signature, tensor, curvature and limit checks test this ansatz within its declared scope.
Explain this formulaStatic clock comparison
Purpose
The relation converts coordinate time to proper time and compares static emitter and observer clocks.
How to read it
Evaluate \(D\) at \(r_o\) and \(r_e\); their ratio gives the displayed static redshift factor.
Domain and meaning
Both worldlines are assumed static. Motion and Doppler contributions are outside this equation.
Verification
Units and weak-field limits are checkable; the ratio is not a complete test of the spacetime model.
Explain this formulaInterior asymptotics and scope
Purpose
The limits record the coefficient and curvature scaling of the current diagonal extrapolation.
How to read it
\(A\) tends to a finite value while \(R\) and \(K\) scale as inverse powers of \(r\).
Domain and meaning
The statement concerns the displayed branch and its declared curvature invariants, not every possible interior.
Verification
Symbolic curvature and asymptotic checks validate this expansion only.
Explain this formulaDual velocity closure
Purpose
The reciprocal escape and infall scales multiply to \(c^2\) by construction.
How to read it
Multiply the square roots and cancel \(r_s/r\) against \(r/r_s\).
Domain and meaning
\(v_{\rm fall}\) is a reciprocal kinematic scale, not a locally measured superluminal three-velocity.
Verification
The product identity is exact within this parametrisation and does not assert superluminal propagation.
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
Search the full reviewed catalogue across geometry, regimes, local frames, geodesics, observables, energy conditions, statistics, weak field, strong field and validation. Automatically extracted candidates remain separate.
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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.
Reading compass
How to read Formulas
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Start with the formula or control, identify its domain and inputs, then follow the linked implementation and evidence record.
A passing identity, numerical limit, plot or comparison supports only the stated relation under its recorded assumptions and provenance.
It is not automatically an independent experiment, a complete physical theory, or a proof beyond the explicit claim boundary.
Foundational synthesis
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Canonical source
Use the current P0, JIF or mathematical lock for the formula and domain shown on this page.
Evidence class
Read tests, convergence, reference compatibility, dataset-conditioned comparisons and independent replication as different evidence classes.
Boundary
Every result retains its assumptions, limits and explicit non-claim so a visual or passing assertion is not over-promoted.