{
  "status": "curated current formula reference",
  "count": 98,
  "formulas": [
    {
      "id": "rs",
      "name": "Schwarzschild length scale",
      "latex": "r_s = 2GM/c^2",
      "topic": "geometry",
      "units": "m",
      "domain": "M > 0",
      "caution": "Defines the radial normalization; it does not itself assert a Schwarzschild metric."
    },
    {
      "id": "x",
      "name": "Normalized radius",
      "latex": "x = r/r_s",
      "topic": "geometry",
      "units": "1",
      "domain": "r_s > 0",
      "caution": "Dimensionless coordinate used by the branch functions."
    },
    {
      "id": "phi",
      "name": "Golden ratio",
      "latex": "\\varphi = (1+\\sqrt{5})/2",
      "topic": "foundations",
      "units": "1",
      "domain": "constant",
      "caution": "A declared SSZ model parameter in the strong branch, not an empirical fit result by itself."
    },
    {
      "id": "xi-strong",
      "name": "Strong branch",
      "latex": "\\Xi_s(x)=1-\\exp(-\\varphi/x)",
      "topic": "field",
      "units": "1",
      "domain": "0 < x < 1.8",
      "caution": "Canonical P0 strong branch."
    },
    {
      "id": "xi-weak",
      "name": "Weak branch",
      "latex": "\\Xi_w(x)=1/(2x)",
      "topic": "field",
      "units": "1",
      "domain": "x > 2.2",
      "caution": "Matches GM/(rc²) because r_s=2GM/c²."
    },
    {
      "id": "blend-t",
      "name": "Blend coordinate",
      "latex": "t=(x-1.8)/0.4",
      "topic": "field",
      "units": "1",
      "domain": "1.8 ≤ x ≤ 2.2",
      "caution": "Maps the transition interval to [0,1]."
    },
    {
      "id": "blend",
      "name": "Quintic Hermite bridge",
      "latex": "\\Xi_b(t)=\\sum_{k=0}^{5}a_k t^k",
      "topic": "field",
      "units": "1",
      "domain": "0 ≤ t ≤ 1",
      "caution": "Six coefficients match value, slope and curvature at both endpoints."
    },
    {
      "id": "blend-conditions",
      "name": "C² endpoint conditions",
      "latex": "\\Xi_b^{(k)}(x_i)=\\Xi_i^{(k)}(x_i),\\quad k=0,1,2",
      "topic": "field",
      "units": "varies by derivative",
      "domain": "x_i ∈ {1.8,2.2}",
      "caution": "The derivatives are with respect to the same normalized coordinate."
    },
    {
      "id": "d",
      "name": "Time factor",
      "latex": "D(r)=1/[1+\\Xi(r)]",
      "topic": "metric",
      "units": "1",
      "domain": "declared branch domain",
      "caution": "Maps the segment field to the static clock factor."
    },
    {
      "id": "s",
      "name": "Radial stretch",
      "latex": "s(r)=1+\\Xi(r)=D^{-1}(r)",
      "topic": "metric",
      "units": "1",
      "domain": "declared branch domain",
      "caution": "Reciprocal scaling used in the diagonal ansatz."
    },
    {
      "id": "metric",
      "name": "Static spherical metric",
      "latex": "ds^2=-D^2c^2dt^2+D^{-2}dr^2+r^2d\\Omega^2",
      "topic": "metric",
      "units": "m²",
      "domain": "static spherical effective model",
      "caution": "The angular radius is areal: sphere area is 4πr²."
    },
    {
      "id": "solid-angle",
      "name": "Unit-sphere line element",
      "latex": "d\\Omega^2=d\\theta^2+\\sin^2\\theta\\,d\\phi^2",
      "topic": "geometry",
      "units": "1",
      "domain": "spherical coordinates",
      "caution": "Angular part of the metric."
    },
    {
      "id": "proper-time",
      "name": "Static proper time",
      "latex": "d\\tau=D(r)\\,dt",
      "topic": "clocks",
      "units": "s",
      "domain": "static observer",
      "caution": "Not the proper time of a photon."
    },
    {
      "id": "radial-null",
      "name": "Radial null coordinate slope",
      "latex": "dr/dt=\\pm cD^2(r)",
      "topic": "null paths",
      "units": "m/s",
      "domain": "radial ds²=0",
      "caution": "Coordinate slope, not locally measured light speed."
    },
    {
      "id": "redshift-general",
      "name": "Static emitter-observer redshift",
      "latex": "1+z=D(r_o)/D(r_e)",
      "topic": "observables",
      "units": "1",
      "domain": "static emitter and observer",
      "caution": "Requires both radii; z=Ξ only for an observer at infinity."
    },
    {
      "id": "redshift-infinity",
      "name": "Static redshift to infinity",
      "latex": "z_\\infty=D^{-1}(r_e)-1=\\Xi(r_e)",
      "topic": "observables",
      "units": "1",
      "domain": "D(∞)=1",
      "caution": "Does not include Doppler, plasma or transfer effects."
    },
    {
      "id": "horizon-xi",
      "name": "Horizon segment value",
      "latex": "\\Xi(r_s)=1-e^{-\\varphi}\\approx0.801711847",
      "topic": "limits",
      "units": "1",
      "domain": "x=1",
      "caution": "Finite canonical P0 value."
    },
    {
      "id": "horizon-d",
      "name": "Horizon clock factor",
      "latex": "D(r_s)=[2-e^{-\\varphi}]^{-1}\\approx0.555027709",
      "topic": "limits",
      "units": "1",
      "domain": "x=1",
      "caution": "Finite horizon factor; not proof of a regular centre."
    },
    {
      "id": "centre-a",
      "name": "Central coefficient limit",
      "latex": "A(r)=D^2(r)\\to1/4",
      "topic": "interior",
      "units": "1",
      "domain": "formal r→0 continuation",
      "caution": "Finite A does not imply finite curvature."
    },
    {
      "id": "ricci",
      "name": "Ricci leading asymptotic",
      "latex": "R(r)\\sim3/(2r^2)",
      "topic": "interior",
      "units": "m⁻²",
      "domain": "formal r→0 continuation",
      "caution": "Diverges at the areal centre."
    },
    {
      "id": "kretschmann",
      "name": "Kretschmann leading asymptotic",
      "latex": "K(r)\\sim9/(4r^4)",
      "topic": "interior",
      "units": "m⁻⁴",
      "domain": "formal r→0 continuation",
      "caution": "Coordinate-invariant curvature divergence."
    },
    {
      "id": "ppn-lensing",
      "name": "Leading PPN light deflection",
      "latex": "\\alpha=(1+\\gamma)r_s/b",
      "topic": "weak field",
      "units": "rad",
      "domain": "b ≫ r_s",
      "caution": "With γ=1 this gives 2r_s/b; not a strong-field ray trace."
    },
    {
      "id": "shapiro",
      "name": "Leading Shapiro delay",
      "latex": "\\Delta t=(1+\\gamma)(r_s/c)\\ln(4r_1r_2/d^2)",
      "topic": "weak field",
      "units": "s",
      "domain": "leading conjunction geometry",
      "caution": "Geometry and one-way/two-way conventions must be stated."
    },
    {
      "id": "perihelion",
      "name": "Perihelion advance",
      "latex": "\\Delta\\omega=6\\pi GM/[a(1-e^2)c^2]",
      "topic": "weak field",
      "units": "rad/orbit",
      "domain": "β=γ=1 weak-field limit",
      "caution": "Compatibility here does not validate the interior."
    },
    {
      "id": "sagnac",
      "name": "Leading Sagnac difference",
      "latex": "\\Delta t\\approx4A\\Omega/c^2",
      "topic": "rotation",
      "units": "s",
      "domain": "standard rotating loop",
      "caution": "A is loop area here, not the metric coefficient A(r)."
    },
    {
      "id": "null-potential",
      "name": "Null circular-orbit diagnostic",
      "latex": "V_{\\rm null}(r)\\propto D^2(r)/r^2",
      "topic": "strong field",
      "units": "relative",
      "domain": "declared diagonal metric",
      "caution": "A maximum is a candidate circular null orbit; stability needs full analysis."
    },
    {
      "id": "energy",
      "name": "Stationary geodesic constant",
      "latex": "E=-g_{tt}u^t",
      "topic": "geodesics",
      "units": "convention dependent",
      "domain": "stationary metric",
      "caution": "A conserved geodesic quantity, not automatically locally measured energy."
    },
    {
      "id": "angular-momentum",
      "name": "Azimuthal geodesic constant",
      "latex": "L=g_{\\phi\\phi}u^\\phi",
      "topic": "geodesics",
      "units": "convention dependent",
      "domain": "axisymmetry",
      "caution": "Used with normalization and radial equations."
    },
    {
      "id": "metric-a",
      "name": "Temporal metric coefficient",
      "latex": "A(r)=D^2(r)",
      "topic": "metric",
      "units": "1",
      "domain": "static diagonal ansatz",
      "caution": "A is not an independently fitted field."
    },
    {
      "id": "metric-b",
      "name": "Radial metric coefficient",
      "latex": "B(r)=D^{-2}(r)",
      "topic": "metric",
      "units": "1",
      "domain": "static diagonal ansatz",
      "caution": "The identity A·B=1 is specific to this ansatz."
    },
    {
      "id": "inverse-metric",
      "name": "Inverse metric",
      "latex": "g^{\\mu\\nu}=\\operatorname{diag}[-1/(D^2c^2),D^2,r^{-2},(r^2\\sin^2\\theta)^{-1}]",
      "topic": "metric",
      "units": "component dependent",
      "domain": "D ≠ 0 and spherical chart",
      "caution": "Coordinate components must not be interpreted as local measurements."
    },
    {
      "id": "metric-determinant",
      "name": "Metric determinant",
      "latex": "\\det g=-c^2r^4\\sin^2\\theta",
      "topic": "metric",
      "units": "coordinate dependent",
      "domain": "static diagonal ansatz",
      "caution": "Its simple D cancellation does not establish central regularity."
    },
    {
      "id": "metric-volume",
      "name": "Invariant coordinate-volume density",
      "latex": "\\sqrt{-g}=c\\,r^2|\\sin\\theta|",
      "topic": "metric",
      "units": "coordinate density",
      "domain": "diagonal spherical chart and D·s=1",
      "caution": "A simple volume density does not imply finite curvature."
    },
    {
      "id": "static-coframe",
      "name": "Static orthonormal coframe",
      "latex": "\\vartheta^{\\hat0}=Dc\\,dt,\\ \\vartheta^{\\hat1}=s\\,dr,\\ \\vartheta^{\\hat2}=r\\,d\\theta,\\ \\vartheta^{\\hat3}=r\\sin\\theta\\,d\\phi",
      "topic": "local frames",
      "units": "length",
      "domain": "static diagonal chart",
      "caution": "The static frame is an operational basis, not a freely falling frame."
    },
    {
      "id": "flow-gamma",
      "name": "Flow-form gamma field",
      "latex": "\\gamma(r)=1+\\Xi(r)=D^{-1}(r)",
      "topic": "metric forms",
      "units": "1",
      "domain": "canonical pure metric mapping",
      "caution": "The repository calls this gamma; its physical interpretation must follow the declared coordinate construction."
    },
    {
      "id": "flow-beta",
      "name": "Flow-form beta field",
      "latex": "\\beta(r)=\\sqrt{1-\\gamma^{-2}(r)}=\\sqrt{1-D^2(r)}",
      "topic": "metric forms",
      "units": "1",
      "domain": "γ ≥ 1",
      "caution": "This derived coordinate-flow parameter is not automatically a material velocity."
    },
    {
      "id": "flow-metric",
      "name": "Non-diagonal flow metric",
      "latex": "ds^2=-c^2(1-\\beta^2)dt^2+2\\beta c\\,dt\\,dr+dr^2+r^2d\\Omega^2",
      "topic": "metric forms",
      "units": "length²",
      "domain": "repository flow coordinate form",
      "caution": "A cross term does not by itself establish rotation or a global chart equivalence."
    },
    {
      "id": "flow-null-slopes",
      "name": "Flow-form radial null slopes",
      "latex": "dr/(c\\,dt)=-\\beta\\pm1",
      "topic": "null paths",
      "units": "1",
      "domain": "radial null curve in flow chart",
      "caution": "These are coordinate slopes; local light speed remains c."
    },
    {
      "id": "proper-radius",
      "name": "Proper radial length",
      "latex": "\\ell=\\int_{r_1}^{r_2}dr/D(r)",
      "topic": "measurements",
      "units": "m",
      "domain": "constant-t spatial slice",
      "caution": "This is slice dependent and is not infall proper time."
    },
    {
      "id": "null-time",
      "name": "Radial null travel time",
      "latex": "\\Delta t=c^{-1}\\int_{r_1}^{r_2}dr/D^2(r)",
      "topic": "null paths",
      "units": "s",
      "domain": "radial null curve in static chart",
      "caution": "A coordinate travel time, not a variable local light speed."
    },
    {
      "id": "strong-first",
      "name": "Strong-branch first derivative",
      "latex": "\\Xi_s'(x)=-(\\varphi/x^2)e^{-\\varphi/x}",
      "topic": "field",
      "units": "1",
      "domain": "x > 0",
      "caution": "Derivative is with respect to normalized radius x."
    },
    {
      "id": "strong-second",
      "name": "Strong-branch second derivative",
      "latex": "\\Xi_s''(x)=e^{-\\varphi/x}(2\\varphi/x^3-\\varphi^2/x^4)",
      "topic": "field",
      "units": "1",
      "domain": "x > 0",
      "caution": "Used in the left endpoint C² match."
    },
    {
      "id": "weak-first",
      "name": "Weak-branch first derivative",
      "latex": "\\Xi_w'(x)=-1/(2x^2)",
      "topic": "field",
      "units": "1",
      "domain": "x > 0",
      "caution": "Used in the right endpoint C² match."
    },
    {
      "id": "weak-second",
      "name": "Weak-branch second derivative",
      "latex": "\\Xi_w''(x)=1/x^3",
      "topic": "field",
      "units": "1",
      "domain": "x > 0",
      "caution": "Used in the right endpoint C² match."
    },
    {
      "id": "hermite-left-value",
      "name": "Quintic Hermite left-value basis",
      "latex": "h_{00}=1-10t^3+15t^4-6t^5",
      "topic": "field",
      "units": "1",
      "domain": "0 ≤ t ≤ 1",
      "caution": "One of six endpoint-interpolation basis functions."
    },
    {
      "id": "hermite-left-slope",
      "name": "Quintic Hermite left-slope basis",
      "latex": "h_{10}=t-6t^3+8t^4-3t^5",
      "topic": "field",
      "units": "1",
      "domain": "0 ≤ t ≤ 1",
      "caution": "Multiplied by interval width and left slope."
    },
    {
      "id": "hermite-left-curvature",
      "name": "Quintic Hermite left-curvature basis",
      "latex": "h_{20}=(t^2-3t^3+3t^4-t^5)/2",
      "topic": "field",
      "units": "1",
      "domain": "0 ≤ t ≤ 1",
      "caution": "Multiplied by interval width squared and left curvature."
    },
    {
      "id": "hermite-right-value",
      "name": "Quintic Hermite right-value basis",
      "latex": "h_{01}=10t^3-15t^4+6t^5",
      "topic": "field",
      "units": "1",
      "domain": "0 ≤ t ≤ 1",
      "caution": "One of six endpoint-interpolation basis functions."
    },
    {
      "id": "hermite-right-slope",
      "name": "Quintic Hermite right-slope basis",
      "latex": "h_{11}=-4t^3+7t^4-3t^5",
      "topic": "field",
      "units": "1",
      "domain": "0 ≤ t ≤ 1",
      "caution": "Multiplied by interval width and right slope."
    },
    {
      "id": "hermite-right-curvature",
      "name": "Quintic Hermite right-curvature basis",
      "latex": "h_{21}=(t^3-2t^4+t^5)/2",
      "topic": "field",
      "units": "1",
      "domain": "0 ≤ t ≤ 1",
      "caution": "Multiplied by interval width squared and right curvature."
    },
    {
      "id": "christoffel",
      "name": "Levi-Civita connection",
      "latex": "\\Gamma^\\rho_{\\mu\\nu}=\\tfrac12g^{\\rho\\sigma}(\\partial_\\mu g_{\\sigma\\nu}+\\partial_\\nu g_{\\sigma\\mu}-\\partial_\\sigma g_{\\mu\\nu})",
      "topic": "differential geometry",
      "units": "coordinate dependent",
      "domain": "metric-compatible torsion-free connection",
      "caution": "Connection components can be chart singular even when invariants are finite."
    },
    {
      "id": "riemann",
      "name": "Riemann tensor",
      "latex": "R^\\rho_{\\ \\sigma\\mu\\nu}=\\partial_\\mu\\Gamma^\\rho_{\\nu\\sigma}-\\partial_\\nu\\Gamma^\\rho_{\\mu\\sigma}+\\Gamma^\\rho_{\\mu\\lambda}\\Gamma^\\lambda_{\\nu\\sigma}-\\Gamma^\\rho_{\\nu\\lambda}\\Gamma^\\lambda_{\\mu\\sigma}",
      "topic": "differential geometry",
      "units": "m⁻²",
      "domain": "Levi-Civita connection",
      "caution": "Sign depends on the declared curvature convention."
    },
    {
      "id": "einstein",
      "name": "Einstein tensor",
      "latex": "G_{\\mu\\nu}=R_{\\mu\\nu}-\\tfrac12Rg_{\\mu\\nu}",
      "topic": "differential geometry",
      "units": "m⁻²",
      "domain": "declared metric and sign convention",
      "caution": "A geometric diagnostic here, not a derived SSZ equation of motion."
    },
    {
      "id": "effective-source",
      "name": "Effective stress tensor diagnostic",
      "latex": "T_{\\mu\\nu}^{\\rm eff}=G_{\\mu\\nu}/(8\\pi G)",
      "topic": "dynamics",
      "units": "stress-energy",
      "domain": "GR-style diagnostic convention",
      "caution": "Does not supply a fundamental SSZ matter action."
    },
    {
      "id": "geodesic",
      "name": "Geodesic equation",
      "latex": "d^2x^\\mu/d\\lambda^2+\\Gamma^\\mu_{\\alpha\\beta}(dx^\\alpha/d\\lambda)(dx^\\beta/d\\lambda)=0",
      "topic": "geodesics",
      "units": "coordinate dependent",
      "domain": "affine parameter λ",
      "caution": "Applies to free test trajectories in the effective geometry."
    },
    {
      "id": "four-velocity",
      "name": "Timelike normalization",
      "latex": "g_{\\mu\\nu}u^\\mu u^\\nu=-c^2",
      "topic": "geodesics",
      "units": "m²/s²",
      "domain": "timelike worldline",
      "caution": "Null trajectories instead have zero norm."
    },
    {
      "id": "null-turning",
      "name": "Null turning-point impact parameter",
      "latex": "b^2=r_{\\rm turn}^2/A(r_{\\rm turn})",
      "topic": "strong field",
      "units": "m²",
      "domain": "equatorial null geodesic",
      "caution": "A turning point is not automatically a circular orbit."
    },
    {
      "id": "photon-condition",
      "name": "Circular null-orbit condition",
      "latex": "d[A(r)/r^2]/dr=0",
      "topic": "strong field",
      "units": "m⁻³",
      "domain": "static spherical metric",
      "caution": "Stability follows from the second derivative and global accessibility."
    },
    {
      "id": "clock-ratio",
      "name": "Static clock-rate ratio",
      "latex": "d\\tau_1/d\\tau_2=D(r_1)/D(r_2)",
      "topic": "clocks",
      "units": "1",
      "domain": "static clocks compared in one stationary chart",
      "caution": "Transport and kinematic effects require a fuller protocol."
    },
    {
      "id": "weak-xi",
      "name": "Weak-field potential correspondence",
      "latex": "\\Xi_w=GM/(rc^2)",
      "topic": "weak field",
      "units": "1",
      "domain": "r > 2.2r_s",
      "caution": "Follows algebraically from r_s=2GM/c²."
    },
    {
      "id": "weak-d",
      "name": "Weak-field time-factor expansion",
      "latex": "D=(1+\\Xi)^{-1}=1-\\Xi+\\Xi^2+O(\\Xi^3)",
      "topic": "weak field",
      "units": "1",
      "domain": "|Ξ| < 1",
      "caution": "A local series; observable calculations still need the full metric."
    },
    {
      "id": "residual",
      "name": "Normalized residual",
      "latex": "\\rho_i=[y_i-f(x_i;\\theta)]/\\sigma_i",
      "topic": "data analysis",
      "units": "1",
      "domain": "independent Gaussian uncertainty",
      "caution": "Correlated measurements require a covariance matrix."
    },
    {
      "id": "chi-square",
      "name": "Correlated chi-square",
      "latex": "\\chi^2=(\\mathbf y-\\mathbf f)^T C^{-1}(\\mathbf y-\\mathbf f)",
      "topic": "data analysis",
      "units": "1",
      "domain": "specified covariance C",
      "caution": "Model comparison must also address parameters, priors and selection effects."
    },
    {
      "id": "curvature-dimensions",
      "name": "Curvature dimensional check",
      "latex": "[R]=L^{-2},\\qquad[K]=L^{-4}",
      "topic": "validation",
      "units": "m⁻² and m⁻⁴",
      "domain": "geometric units restored consistently",
      "caution": "A required sanity check for central asymptotics."
    },
    {
      "id": "strong-limit",
      "name": "Strong-branch central field limit",
      "latex": "\\lim_{x\\to0^+}\\Xi_s(x)=1",
      "topic": "limits",
      "units": "1",
      "domain": "formal inner continuation",
      "caution": "This is a field limit, not proof that the centre belongs to a regular manifold."
    },
    {
      "id": "d-centre",
      "name": "Central clock-factor limit",
      "latex": "\\lim_{x\\to0^+}D(x)=1/2",
      "topic": "limits",
      "units": "1",
      "domain": "formal inner continuation",
      "caution": "Finite D does not imply finite curvature."
    },
    {
      "id": "asymptotic-flat",
      "name": "Asymptotic flatness checkpoint",
      "latex": "\\lim_{x\\to\\infty}\\Xi=0,\\quad\\lim_{x\\to\\infty}D=1",
      "topic": "limits",
      "units": "1",
      "domain": "weak branch",
      "caution": "An asymptotic limit does not determine the global interior."
    },
    {
      "id": "strong-weak-intersection",
      "name": "Raw branch intersection",
      "latex": "1-e^{-\\varphi/x_\\times}=1/(2x_\\times)",
      "topic": "regimes",
      "units": "1",
      "domain": "comparison of unblended branch formulas",
      "caution": "The raw intersection is not one of the declared bridge endpoints."
    },
    {
      "id": "hermite-complete",
      "name": "Complete C² bridge",
      "latex": "H_5=h_{00}y_0+h h_{10}y'_0+h^2h_{20}y''_0+h_{01}y_1+h h_{11}y'_1+h^2h_{21}y''_1",
      "topic": "regimes",
      "units": "1",
      "domain": "1.8 ≤ x ≤ 2.2",
      "caution": "Every derivative is taken with respect to the same normalized coordinate x."
    },
    {
      "id": "d-first",
      "name": "Clock-factor derivative",
      "latex": "D'=-\\Xi'/(1+\\Xi)^2",
      "topic": "metric",
      "units": "inverse length or 1 in x",
      "domain": "differentiable branch",
      "caution": "Derivative units depend on whether the independent variable is r or x."
    },
    {
      "id": "a-first",
      "name": "Temporal coefficient derivative",
      "latex": "A'=2DD'",
      "topic": "metric",
      "units": "inverse length or 1 in x",
      "domain": "A=D²",
      "caution": "Used in circular-orbit diagnostics."
    },
    {
      "id": "b-first",
      "name": "Radial coefficient derivative",
      "latex": "B'=-2D^{-3}D'",
      "topic": "metric",
      "units": "inverse length or 1 in x",
      "domain": "B=D⁻²",
      "caution": "Large coordinate coefficients are not by themselves invariant singularities."
    },
    {
      "id": "static-tetrad-time",
      "name": "Static orthonormal time leg",
      "latex": "e_{\\hat 0}=D^{-1}c^{-1}\\partial_t",
      "topic": "local frames",
      "units": "inverse length",
      "domain": "D>0, static observer",
      "caution": "Defines a local frame only where a static observer is physically admissible."
    },
    {
      "id": "static-tetrad-radial",
      "name": "Static orthonormal radial leg",
      "latex": "e_{\\hat r}=D\\,\\partial_r",
      "topic": "local frames",
      "units": "inverse length",
      "domain": "D>0",
      "caution": "Separates local radial measurements from coordinate components."
    },
    {
      "id": "four-acceleration",
      "name": "Static-observer radial acceleration",
      "latex": "a^{\\hat r}=c^2D'(r)",
      "topic": "local frames",
      "units": "m/s²",
      "domain": "static diagonal ansatz and proper radial frame",
      "caution": "Sign and interpretation depend on derivative convention and observer choice."
    },
    {
      "id": "lagrangian",
      "name": "Geodesic Lagrangian",
      "latex": "2\\mathcal L=g_{\\mu\\nu}\\dot x^\\mu\\dot x^\\nu",
      "topic": "geodesics",
      "units": "velocity squared",
      "domain": "affinely parametrised trajectory",
      "caution": "An effective test-particle Lagrangian is not the missing fundamental field action."
    },
    {
      "id": "radial-timelike",
      "name": "Timelike radial equation",
      "latex": "\\dot r^2=E^2/c^2-A(r)\\left(c^2+L^2/r^2\\right)",
      "topic": "geodesics",
      "units": "m²/s²",
      "domain": "equatorial timelike geodesic; convention-dependent E and L",
      "caution": "Must be re-derived if the metric convention changes."
    },
    {
      "id": "radial-null-effective",
      "name": "Null radial equation",
      "latex": "\\dot r^2=E^2/c^2-A(r)L^2/r^2",
      "topic": "geodesics",
      "units": "m²/s²",
      "domain": "equatorial null geodesic",
      "caution": "Turning points and circular orbits are different conditions."
    },
    {
      "id": "photon-stability",
      "name": "Null-orbit stability diagnostic",
      "latex": "d^2[A(r)/r^2]/dr^2\\lessgtr0",
      "topic": "strong field",
      "units": "m⁻⁴",
      "domain": "at a stationary null orbit",
      "caution": "The sign convention must be matched to the chosen effective potential."
    },
    {
      "id": "orbital-frequency",
      "name": "Circular-orbit coordinate frequency",
      "latex": "\\Omega^2=c^2A'(r)/(2r)",
      "topic": "geodesics",
      "units": "s⁻²",
      "domain": "static spherical metric in areal radius",
      "caution": "Coordinate frequency requires conversion before comparison with a local clock."
    },
    {
      "id": "timelike-angular-momentum",
      "name": "Circular timelike angular momentum",
      "latex": "L^2=c^2r^3A'/(2A-rA')",
      "topic": "geodesics",
      "units": "m⁴/s² under specific convention",
      "domain": "circular timelike orbit",
      "caution": "Denominator and normalization conventions must be checked."
    },
    {
      "id": "timelike-energy",
      "name": "Circular timelike energy",
      "latex": "E^2=2c^4A^2/(2A-rA')",
      "topic": "geodesics",
      "units": "energy-per-mass squared",
      "domain": "circular timelike orbit",
      "caution": "Stability requires a separate second-derivative condition."
    },
    {
      "id": "isco-condition",
      "name": "Marginal orbit stability",
      "latex": "d^2V_{\\rm eff}/dr^2=0",
      "topic": "strong field",
      "units": "potential per length²",
      "domain": "on a circular timelike solution",
      "caution": "Solving this condition is metric- and branch-specific."
    },
    {
      "id": "shadow-impact",
      "name": "Critical shadow impact parameter",
      "latex": "b_{\\rm crit}=r_{\\rm ph}/\\sqrt{A(r_{\\rm ph})}",
      "topic": "strong field",
      "units": "m",
      "domain": "accessible unstable circular null orbit",
      "caution": "A shadow observable additionally needs source, inclination, transfer and rotation."
    },
    {
      "id": "redshift-two-static",
      "name": "Two-radius frequency ratio",
      "latex": "\\nu_o/\\nu_e=D(r_e)/D(r_o)",
      "topic": "observables",
      "units": "1",
      "domain": "static emitter and observer",
      "caution": "Inverse placement of emitter and observer changes the reported z convention."
    },
    {
      "id": "doppler-factor",
      "name": "Special-relativistic line-of-sight Doppler factor",
      "latex": "\\delta=[\\gamma(1-\\beta\\cos\\vartheta)]^{-1}",
      "topic": "observables",
      "units": "1",
      "domain": "local inertial comparison",
      "caution": "This kinematic factor is separate from static gravitational scaling."
    },
    {
      "id": "observed-frequency",
      "name": "Combined schematic frequency map",
      "latex": "\\nu_{\\rm obs}=\\nu_{\\rm emit}[D_e/D_o]\\,\\delta\\,\\mathcal T",
      "topic": "observables",
      "units": "Hz",
      "domain": "declared transfer factor T",
      "caution": "The transfer factor is model-dependent; this is a bookkeeping relation, not a universal closed formula."
    },
    {
      "id": "ppn-gtt",
      "name": "PPN temporal metric expansion",
      "latex": "g_{tt}/c^2=-1+2U/c^2-2\\beta U^2/c^4+O(c^{-6})",
      "topic": "weak field",
      "units": "1",
      "domain": "|U|/c² ≪ 1",
      "caution": "The sign of U must follow the declared PPN convention."
    },
    {
      "id": "ppn-gij",
      "name": "PPN spatial metric expansion",
      "latex": "g_{ij}=[1+2\\gamma U/c^2+O(c^{-4})]\\delta_{ij}",
      "topic": "weak field",
      "units": "1",
      "domain": "weak quasi-static field",
      "caution": "A coordinate-gauge statement used for specific observable calculations."
    },
    {
      "id": "energy-density",
      "name": "Effective density projection",
      "latex": "\\rho_{\\rm eff}=T^{\\rm eff}_{\\hat0\\hat0}/c^2",
      "topic": "energy conditions",
      "units": "kg/m³",
      "domain": "chosen orthonormal frame",
      "caution": "Effective-source diagnostics are not fundamental SSZ matter content."
    },
    {
      "id": "wec",
      "name": "Weak energy condition",
      "latex": "\\rho\\ge0,\\qquad \\rho+p_i/c^2\\ge0",
      "topic": "energy conditions",
      "units": "energy density",
      "domain": "orthonormal principal frame",
      "caution": "A diagnostic condition; violations do not alone select or reject an effective geometry."
    },
    {
      "id": "dec",
      "name": "Dominant energy condition",
      "latex": "\\rho\\ge|p_i|/c^2",
      "topic": "energy conditions",
      "units": "energy density",
      "domain": "orthonormal principal frame",
      "caution": "Requires a physically interpreted effective source."
    },
    {
      "id": "sec",
      "name": "Strong energy condition",
      "latex": "\\rho+\\sum_i p_i/c^2\\ge0,\\quad \\rho+p_i/c^2\\ge0",
      "topic": "energy conditions",
      "units": "energy density",
      "domain": "orthonormal principal frame",
      "caution": "Its relevance depends on the underlying dynamical theory."
    },
    {
      "id": "binomial-sign",
      "name": "Two-sided paired sign-test probability",
      "latex": "p=2\\sum_{k=0}^{\\min(w,n-w)}{n\\choose k}2^{-n}",
      "topic": "statistics",
      "units": "1",
      "domain": "independent exchangeable paired signs under the null",
      "caution": "Does not account for model flexibility, selection or correlated pairs."
    },
    {
      "id": "confidence-interval",
      "name": "Bootstrap percentile interval",
      "latex": "CI_{1-\\alpha}=[Q_{\\alpha/2}(\\hat\\theta^*),Q_{1-\\alpha/2}(\\hat\\theta^*)]",
      "topic": "statistics",
      "units": "same as estimator",
      "domain": "representative resampling scheme",
      "caution": "Bootstrap validity depends on sampling structure and independence assumptions."
    },
    {
      "id": "bic",
      "name": "Bayesian information criterion",
      "latex": "\\mathrm{BIC}=k\\ln n-2\\ln\\hat L",
      "topic": "statistics",
      "units": "1",
      "domain": "regular likelihood and sample-size assumptions",
      "caution": "Lower BIC is an asymptotic model-selection heuristic, not proof."
    },
    {
      "id": "aic",
      "name": "Akaike information criterion",
      "latex": "\\mathrm{AIC}=2k-2\\ln\\hat L",
      "topic": "statistics",
      "units": "1",
      "domain": "maximum likelihood comparison",
      "caution": "Relative predictive criterion; absolute fit and data quality remain separate."
    },
    {
      "id": "sagnac-integral",
      "name": "Stationary-spacetime Sagnac integral",
      "latex": "\\Delta t=-2c^{-1}\\oint g_{0i}/g_{00}\\,dx^i",
      "topic": "rotation",
      "units": "s",
      "domain": "stationary metric and declared coordinates",
      "caution": "The leading 4AΩ/c² formula follows only in its controlled rotating-loop limit."
    }
  ]
}