Symbol Glossary

Metric tensor — encodes the geometry (distances and angles) of spacetime at every point

Symmetric 4×4 tensor; μ,ν ∈ {0,1,2,3}. All curvature quantities derived from it.

Einstein tensor — measures spacetime curvature; shorthand for R_μν − ½g_μν R

Left-hand side of the field equations. Contains second-order partial derivatives of g_μν.

Ricci tensor — contraction of the Riemann tensor; encodes how volumes distort

R_μν = R^λ_{μλν}; symmetric.

Ricci scalar — full contraction of the Ricci tensor; single number summarising curvature

R = g^{μν} R_μν. Used in the Lagrangian and in the definition of G_μν.

Christoffel symbols (connection coefficients) — encode how basis vectors change across spacetime

Not a tensor; computed from first derivatives of g_μν.

Spacetime interval — invariant proper distance/time between events

ds² = g_μν dx^μ dx^ν. Negative = timelike; positive = spacelike.

Newtonian-gauge scalar potential — governs non-relativistic gravitational acceleration

Defined by ds² = −(1+2Φ)c²dt² + (1−2Ψ)dx². Non-relativistic dynamics probes ∇Φ.

Spatial curvature potential — second scalar potential in Newtonian gauge

Lensing probes (Φ+Ψ). In standard GR with no anisotropic stress, Φ = Ψ.

Gravitational slip parameter — ratio of the two scalar potentials

η = Ψ/Φ. GR prediction: η = 1 everywhere. Edge-response predicts η ≠ 1 in boundary-layer zone.

Response-sector contribution to the Newtonian potential

Asymptotically Φ_resp ~ v∞² ln(R); gradient gives the 1/R tail in extra acceleration.

Total stress–energy tensor — the full matter/energy source on the right-hand side of the field equation

T_μν = T̄_μν + T^resp_μν in the response-sector decomposition.

Baryonic stress–energy tensor — contribution from ordinary (observable) matter

Directly observed via photometry and HI surveys.

Response-sector stress–energy — the effective source required so that the observed metric satisfies the Einstein equation

Ontologically agnostic: not a new particle. Can be realised as an extra field, EFT correction, or constitutive closure.

Effective energy density of the response sector

For a flat rotation curve, ρ_resp ~ 1/R² — same radial profile as an isothermal dark-matter halo, but derived rather than assumed.

Galactocentric radius — projected distance from galaxy centre in the disc plane

Primary independent variable in rotation-curve analysis. 1 kpc ≈ 3.086 × 10¹⁹ m.

Observed circular speed — measured Doppler-shift rotation velocity

Primary observable from HI 21-cm or Hα spectroscopy.

Baryonic circular speed — prediction from observed baryonic mass alone

v²_bar = V²_gas + Υ_disk V²_disk + Υ_bul V²_bul.

Observed centripetal acceleration — kinematic inference from circular orbit condition

g_obs = v²_obs / R. The fundamental empirical quantity; no dynamical model assumed.

Baryonic acceleration — Newtonian prediction from baryonic mass model

g_bar = v²_bar / R.

Residual (extra) acceleration — empirical excess over baryonic prediction

g_extra = g_obs − g_bar. Framework interprets this as g_resp from the response sector.

Velocity-squared deficit profile — V²_obs − V²_bar at each radius R

The raw signal the framework must explain. Sign changes indicate oscillatory response; 46/175 SPARC galaxies show at least one sign change.

Critical acceleration scale — the MOND / boundary-layer activation threshold

a₀ ≈ 1.2 × 10⁻¹⁰ m s⁻². Empirical scale at which baryonic gravity equals the response activation level. Possibly linked to cH₀.

Transition radius — galactocentric radius where g_bar(R_t) ≈ a₀

Marks the inner edge of the boundary layer; determined deterministically from g_bar.

Edge-response amplitude — the single free parameter per galaxy; asymptotic extra v² contribution

Estimated two ways: Q_best (fitted) and Q_est (robust outer). v²_extra → Q as R → ∞.

Fitted edge amplitude — best-fit Q from χ² minimisation over all data points

Non-negative by construction. Sensitive to inner kinematics.

Robust outer deficit estimator — Huber M-estimator of Δ(R) over the outer data subset (R ≥ 0.6 R_max)

Model-free; can be negative (rarefaction-phase). Spearman ρ(v_best, v_est) = 0.972 across 175 galaxies.

Auxiliary response field — scalar field whose gradient outside the activation zone produces the 1/R acceleration tail

Governed by a boundary-layer differential equation; ∂_R χ ~ 1/R outside activation region.

Source bump width — Gaussian half-width of the activation region

Runner default: 2.0 kpc. Controls how sharply the response is localised near R_t.

Chi-squared statistic — weighted sum of squared residuals between model and data

χ² = Σ_i [(v_model(R_i) − v_obs(R_i)) / e_vobs(R_i)]².

Chi-squared improvement — χ²_bar − χ²_model; measures how much the response sector helps

Z ≈ √(Δχ²) gives approximate Gaussian significance for 1 added parameter.

Approximate Gaussian significance of Δχ²

Classes: weak Z<2, moderate 2–3, strong 3–5, very-strong Z>5.

Bayesian Information Criterion — k ln(n) − 2 ln(L̂); penalises model complexity

ΔBIC = BIC_model − BIC_bar < 0 required to pass. 97% of 175 SPARC galaxies pass.

Spearman rank correlation coefficient — non-parametric measure of monotonic association

Range [−1, +1]. Used throughout to quantify correlations between Q, galaxy properties, and environment.

Oscillation wavelength — spatial period of the boundary-layer standing wave

Median λ ≈ 45.6 kpc across the 46 sign-change galaxies; scales with R_max, not a cosmological constant.

Radial wavenumber of the oscillatory response

κ = 2π/λ. Galaxy-size dependent, not a fixed cosmological scale.

Oscillation amplitude — peak response-potential strength

Related to Q in the asymptotic regime.

Damping / decay scale of the oscillatory response

Characterises how quickly the oscillation envelope falls off with radius.