Appendix A · Kitcey 2026 · v5.0.7

Symbol Reference

Comprehensive symbol glossary for the Intrinsic Response framework — spacetime geometry through the Baryonic Tully–Fisher Relation.

Sections

Symbols

Gμν = 8πG (T̅ + Tᴿᵉˢᵖ)μν

Framework

SPARC-175

Dataset

I

Spacetime Geometry

8 symbols

Symbol	Definition	Units	Notes / Context
gμν	Metric tensor — encodes the geometry (distances and angles) of spacetime at every point	dimensionless	Symmetric 4×4 tensor; μ,ν ∈ {0,1,2,3}. All curvature quantities are derived from it.
Gμν	Einstein tensor — measures spacetime curvature; shorthand for Rμν − ½gμνR	m⁻²	Left-hand side of the field equations. Contains second-order partial derivatives of gμν.
Rμν	Ricci tensor — contraction of the Riemann tensor; encodes how volumes distort	m⁻²	Rμν = Rλμλν; symmetric.
R	Ricci scalar — full contraction of the Ricci tensor; single number summarizing curvature	m⁻²	R = gμνRμν. Used in the Lagrangian and in the definition of Gμν.
Rλμνσ	Riemann curvature tensor — the fundamental curvature object; measures how parallel transport fails to close	m⁻²	4-index tensor; 20 independent components in 4D.
Γλμν	Christoffel symbols (connection coefficients) — encode how basis vectors change across spacetime	m⁻¹	Not a tensor; computed from first derivatives of gμν.
ds²	Spacetime interval — invariant proper distance/time between events	m² (or s²)	ds² = gμν dxμ dxν. Negative = timelike; positive = spacelike.
√−g	Square root of the absolute value of the metric determinant — volume element factor in integrals	dimensionless	Ensures coordinate-invariant integration over spacetime.

II

Weak-Field Metric Potentials

6 symbols

Symbol	Definition	Units	Notes / Context
Φ	Newtonian-gauge scalar potential — governs non-relativistic gravitational acceleration	km² s⁻²	Defined by ds² = −(1+2Φ)c²dt² + (1−2Ψ)dx². Non-relativistic dynamics probes ∇Φ.
Ψ	Spatial curvature potential — second scalar potential in Newtonian gauge	km² s⁻²	Lensing probes (Φ+Ψ). In standard GR with no anisotropic stress, Φ = Ψ.
η	Gravitational slip parameter — ratio of the two scalar potentials η ≡ Ψ/Φ	dimensionless	GR prediction: η = 1 everywhere. Edge-response predicts η ≠ 1 in the boundary-layer zone — a key falsification target.
Φ̅Φ_bar	Baryonic contribution to the Newtonian potential	km² s⁻²	Sourced by T̅μν via the Poisson equation.
Φ_resp	Response-sector contribution to the Newtonian potential	km² s⁻²	Asymptotically Φ_resp ≈ v∞² ln(R); gradient gives the 1/R tail in extra acceleration.
Φ_obs	Effective potential inferred from observed rotation curve by integrating g_obs(R)	km² s⁻²	Defined up to an additive constant. Computed as ∫ g_obs dR from R_min.

III

Stress–Energy Tensors

7 symbols

Symbol	Definition	Units	Notes / Context
Tμν	Total stress–energy tensor — the full matter/energy source on the right-hand side of the field equation	kg m⁻¹ s⁻²	Tμν = T̅μν + Tᴿᵉˢᵖμν in the response-sector decomposition.
T̅μνT_bar	Baryonic stress–energy tensor — contribution from ordinary (observable) matter: protons, electrons, gas, stars	kg m⁻¹ s⁻²	Directly observed via photometry and HI surveys.
TᴿᵉˢᵖμνT^resp	Response-sector stress–energy — the effective source required so that the observed metric satisfies the Einstein equation	kg m⁻¹ s⁻²	Ontologically agnostic: not a new particle. Can be realized as an extra field, EFT correction, or constitutive closure.
ρ_resp	Effective energy density of the response sector	kg m⁻³	For a flat rotation curve, ρ_resp ∝ 1/R² — same radial profile as an isothermal dark-matter halo, but derived rather than assumed.
p_resp	Effective pressure of the response sector	Pa	Cosmological completion requires specifying the equation of state w = p_resp / ρ_resp.
πμν	Anisotropic stress tensor of the response sector	Pa	Determines the slip η. If πμν ≠ 0 then η ≠ 1.
∇μTμν = 0	Bianchi identity / conservation law — the total stress–energy must be covariantly conserved	—	Imposes consistency on Tᴿᵉˢᵖμν. If baryons are minimally coupled, then ∇μTᴿᵉˢᵖμν = 0 independently.

IV

Galactic Kinematics & Observables

12 symbols

Symbol	Definition	Units	Notes / Context
R	Galactocentric radius — projected distance from galaxy center in the disc plane	kpc	Primary independent variable in rotation-curve analysis. 1 kpc ≈ 3.086 × 10¹⁹ m.
R_max	Maximum observed radius — outermost data point for a given galaxy	kpc	Determines the extent of the ‘outer subset’ used by Q_est.
v_obs	Observed circular speed — measured Doppler-shift rotation velocity	km s⁻¹	Primary observable from HI 21-cm or Hα spectroscopy.
e_{v_obs}	1σ observational uncertainty on v_obs	km s⁻¹	Used to construct χ² and propagate uncertainty bands in all panels.
v̅v_bar	Baryonic circular speed — prediction from observed baryonic mass alone	km s⁻¹	v̅² = V²_gas + Υ_disk V²_disk + Υ_bul V²_bul.
v_model	Total model circular speed — baryons plus response sector	km s⁻¹	v²_model = v̅² + Qχ′(R)·R.
v∞	Asymptotic flat rotation velocity — the constant value v_obs approaches at large R	km s⁻¹	Defines the logarithmic potential: Φ_resp ≈ v∞² ln(R).
V_flat	SPARC-catalogue flat velocity — tabulated asymptotic speed from the Lelli+2016 database	km s⁻¹	Used as a proxy for galaxy mass scale; correlated with Q.
g_obs	Observed centripetal acceleration — kinematic inference from circular orbit condition	km² s⁻² kpc⁻¹	g_obs = v²_obs / R. The fundamental empirical quantity; no dynamical model assumed.
g̅g_bar	Baryonic acceleration — Newtonian prediction from baryonic mass model	km² s⁻² kpc⁻¹	g̅ = v̅² / R.
g_extra	Residual (extra) acceleration — empirical excess over baryonic prediction	km² s⁻² kpc⁻¹	g_extra = g_obs − g̅. Framework interprets this as g_resp from the response sector.
g_resp	Response-sector acceleration — theoretical prediction from Tᴿᵉˢᵖμν	km² s⁻² kpc⁻¹	Asymptotically g_resp ≈ Q/R. Matches g_extra in the successful-fit regime.

V

Response-Sector Model Parameters

12 symbols

Symbol	Definition	Units	Notes / Context
a₀	Critical acceleration scale — the MOND / boundary-layer activation threshold	m s⁻² or km² s⁻² kpc⁻¹	a₀ ≈ 1.2 × 10⁻¹⁰ m s⁻². Empirical scale at which baryonic gravity equals the response activation level. Possibly linked to cH₀.
R_t	Transition radius — galactocentric radius where g̅(R_t) ≈ a₀	kpc	Marks the inner edge of the boundary layer; the source bump S(R) is localized near R_t. Determined deterministically from g̅.
χ	Auxiliary response field — scalar field whose gradient outside the activation zone produces the acceleration tail	dimensionless (normalized)	∂_Rχ ≈ 1/R outside activation region.
χ′_unit	Unit-normalized auxiliary response — the shape function, normalized so that χ′_unit → 1/R at large R	kpc⁻¹	Computed from S(R) via the runner’s numerical integration. Multiplied by Q to give the model.
S(R)	Source bump — localized activation function centered near R_t	kpc⁻¹	Gaussian with width σ_kpc (default 2.0 kpc). Drives χ in the boundary-layer equation.
σ_kpc	Source bump width — Gaussian half-width of the activation region	kpc	Runner default: 2.0 kpc. Controls how sharply the response is localized near R_t.
Q	Edge-response amplitude — the single free parameter per galaxy; asymptotic extra v² contribution	km² s⁻²	v²_extra → Q as R → ∞. Estimated two ways: Q_best (fitted) and Q_est (robust outer).
Q_best	Fitted edge amplitude — best-fit Q from χ² minimization over all data points	km² s⁻²	Non-negative by construction. Sensitive to inner kinematics; produced by the SPARC runner.
Q_est	Robust outer deficit estimator — Huber M-estimator of Δ(R) = V²_obs − V̅² over the outer data subset (R ≥ 0.6 R_max)	km² s⁻²	Model-free; can be negative (rarefaction-phase). Spearman ρ(v_best, v_est) = 0.972 across 175 galaxies.
v_best	Fitted asymptotic extra speed — √Q_best	km s⁻¹	Comparable to v_est for ranking purposes.
v_est	Robust asymptotic extra speed — √max(Q_est, 0)	km s⁻¹	Set to ‘neg’ if Q_est < 0 (rarefaction-phase galaxy).
Δ(R)	Velocity-squared deficit profile — V²_obs − V̅² at each radius R	km² s⁻²	The raw signal the framework must explain. Sign changes in Δ(R) indicate oscillatory response; 46/175 SPARC galaxies show at least one sign change.

VI

Baryonic Mass Components

11 symbols

Symbol	Definition	Units	Notes / Context
V_gas	Gas circular-speed template from SPARC rotmod file	km s⁻¹	HI + He contribution; enters v̅² with coefficient 1 (no Υ multiplier).
V_disk	Stellar-disc circular-speed template from SPARC rotmod file	km s⁻¹	Scaled by Υ_disk before adding to v̅².
V_bul	Bulge circular-speed template from SPARC rotmod file	km s⁻¹	Scaled by Υ_bul. Zero for bulgeless galaxies.
Υ_disk	Stellar mass-to-light ratio of the disc — converts surface brightness to surface mass density	M☉/L☉	Runner default: 0.5. Sensitivity sweep over {0.3, 0.4, 0.5, 0.6, 0.7} shows ≤17% change in Q_est for ±40% variation.
Υ_bul	Stellar mass-to-light ratio of the bulge	M☉/L☉	Runner default: 0.7.
M_b	Total baryonic mass — integral of baryonic surface density over the disc	M☉	Appears in the baryonic Tully–Fisher relation: M_b ∝ v⁴_flat.
M_HI	HI (neutral hydrogen) mass	M☉	Correlated with Q_est; HI-rich discs have extended, diffuse edges → shallower boundary-layer gradient → weaker response.
R_disk	Disc scale length — exponential scale radius of the stellar disc	kpc	SPARC metadata. Used as a galaxy size proxy; controls Q base-line.
Q_flag	SPARC photometric quality flag: 1 = best, 2 = moderate, 3 = poorest	—	Spearman ρ(Q_est, outer_rms_z) = −0.035; photometric quality has zero detectable correlation with outer residual scatter.
T	Hubble morphological type index (e.g., 5 = Sc, 9 = Irr, 10 = Im)	—	From RC3 / SPARC Table 1.
D	Distance to galaxy	Mpc	From SPARC catalogue; used to convert angular to physical scales.

VII

Statistical Diagnostics

11 symbols

Symbol	Definition	Units	Notes / Context
χ²	Chi-squared statistic — weighted sum of squared residuals between model and data	dimensionless	χ² = Σ_i [(v_model(R_i) − v_obs(R_i)) / e_{v_obs}(R_i)]².
χ̅²χ²_bar	Baryons-only chi-squared — χ² when v_model = v̅ (no response sector)	dimensionless	Baseline for Δχ² comparison.
χ²_model	Fitted-model chi-squared — χ² with best-fit Q	dimensionless	Always ≤ χ̅² by construction.
Δχ²	Chi-squared improvement — χ̅² − χ²_model; measures how much the response sector helps	dimensionless	Z ≈ √Δχ² gives approximate Gaussian significance for 1 added parameter.
Z	Approximate Gaussian significance of Δχ²	σ	Z = √Δχ². Classes: weak Z < 2, moderate 2–3, strong 3–5, very-strong Z > 5.
rχ²	Reduced chi-squared — χ² / (n − k) where n = data points, k = free parameters	dimensionless	Reported separately for inner (R < 0.6 R_max) and outer (R ≥ 0.6 R_max) regions. Inner/outer ratio = 3.54× confirms shape inadequacy, not Υ bias.
BIC	Bayesian Information Criterion — k ln(n) − 2 ln(L̂); penalizes model complexity	dimensionless	ΔBIC = BIC_model − BIC̅ < 0 required to pass. 97% of 175 SPARC galaxies pass. Guards against overfitting.
ρ_s	Spearman rank correlation coefficient — non-parametric measure of monotonic association	dimensionless	Used throughout to quantify correlations between Q, galaxy properties, and environment. Range [−1, +1].
r	Pearson linear correlation coefficient	dimensionless	Used alongside Spearman ρ for robustness. r(v_best, v_est) = 0.950 across 175 galaxies.
σ_Δv	Propagated uncertainty on velocity deficit Δv	km s⁻¹	σ_Δv ≈ e_{v_obs} when model is treated as exact.
outer_rms_z	Outer-region residual RMS in units of observational uncertainty — (v_obs − v_model) / e_{v_obs} over R ≥ 0.6 R_max	dimensionless	Key diagnostic for fit quality in the outer disc, where the 1/R tail is most active.

VIII

Oscillatory Boundary-Layer Response

8 symbols

Symbol	Definition	Units	Notes / Context
Φ_resp(r)	Oscillatory response potential — the full spatial form of the response-sector potential	km² s⁻²	Φ_resp = A cos(κr + δ) e⁻ʳ/L; encoding amplitude, wavenumber, phase, and decay scale.
A	Oscillation amplitude — peak response-potential strength	km² s⁻²	Related to Q in the asymptotic regime.
κ	Radial wavenumber of the oscillatory response	kpc⁻¹	κ = 2π/λ. Galaxy-size dependent, not a fixed cosmological scale.
λ	Oscillation wavelength — spatial period of the boundary-layer standing wave	kpc	Median λ ≈ 45.6 kpc across the 46 sign-change galaxies; scales with R_max, not a cosmological constant.
δ	Phase offset of the oscillatory response at R_t	rad	Depends on initial conditions.
L	Damping / decay scale of the oscillatory response	kpc	Characterizes how quickly the oscillation envelope falls off with radius.
λ/2	Half-wavelength of oscillation	kpc	Median ≈ 22.8 kpc. Scaling λ/2 ≈ R_max / 2 is the key empirical result distinguishing galaxy-geometry resonance from cosmological scale.
Δ(R) = 0Nodal radius	Galactocentric radius where V²_obs = V̅² exactly	kpc	Stars at this radius orbit on pure baryonic gravity. Framework predicts morphological ring features here — a testable prediction.

IX

Action Principle & Field Theory

8 symbols

Symbol	Definition	Units	Notes / Context
S	Action — integral of the Lagrangian density over spacetime; extremizing S yields the field equations	J·s	S = ∫ d⁴x √−g [R/(16πG) + L̅ + L_χ + L_int].
L̅L_bar	Baryonic Lagrangian density — standard matter fields (ψ)	J m⁻³	Minimally coupled to gravity.
L_χ	Response-field Lagrangian density — kinetic and potential terms for χ	J m⁻³	Choice of L_χ (e.g., f(X) − V(χ)) controls the field equations for χ and the form of Tᴿᵉˢᵖμν.
L_int	Interaction Lagrangian — couples χ to baryonic fields ψ	J m⁻³	Must tie activation to a baryonic invariant so the response is not a free function per galaxy.
X	Kinetic invariant of the response field — X = gμν ∂μχ ∂νχ	kpc⁻²	Used in k-essence / AQUAL-type response fields: f(X) controls the equation of motion.
f(X)	Non-linear kinetic function for the response field	J m⁻³	Choosing f so that the static equation has conserved radial flux outside activation yields ∂_Rχ ∼ 1/R.
G	Newton’s gravitational constant	m³ kg⁻¹ s⁻²	G ≈ 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻². Appears in the coupling 8πG between geometry and stress-energy.
8πG	Gravitational coupling constant in the field equation Gμν = 8πG Tμν	m kg⁻¹ s⁻²	Right-hand side coefficient.

X

Cosmological Sector

7 symbols

Symbol	Definition	Units	Notes / Context
ΛCDM	Standard cosmological model — Lambda (dark energy) + Cold Dark Matter	—	The empirically successful baseline the response sector must be consistent with at large scales.
H₀	Hubble constant — present-day rate of cosmic expansion	km s⁻¹ Mpc⁻¹	H₀ ≈ 70 km s⁻¹ Mpc⁻¹. The product cH₀ motivates a possible cosmological origin for the acceleration scale a₀.
w	Equation-of-state parameter of the response sector — w = p_resp / ρ_resp	dimensionless	Cosmological viability requires w ≈ 0 (dust-like) over structure-formation epochs.
a(t)	Cosmological scale factor — dimensionless ratio of physical to comoving distances	dimensionless	Cosmological completion requires specifying how Tᴿᵉˢᵖμν evolves with a.
ρ_resp(a)	Redshift evolution of response-sector energy density	kg m⁻³	Must behave approximately as ∝ a⁻³ (pressureless) over target epochs to preserve structure formation.
κ_lens	Lensing convergence — integral of (Φ+Ψ) along the line of sight	dimensionless	κ_lens ∝ Φ+Ψ. With η ≠ 1, the lensing signal differs from the kinematic signal — the primary cluster/lensing prediction.
δ_env	Environment overdensity — local large-scale density contrast around a galaxy	dimensionless	Used in SPARC analysis as an environment proxy. Partial correlations with Q_est should become weak once internal (scale) covariates are controlled.

XI

Baryonic Tully–Fisher Relation

3 symbols

Symbol	Definition	Units	Notes / Context
BTFR	Baryonic Tully–Fisher Relation — M_b ∝ v⁴_flat	—	Tight empirical scaling relation across 4 orders of magnitude in baryonic mass. Emerges as a consequence of the response-sector mechanism rather than an imposed prior.
M_b ∝ v⁴_flat	BTFR functional form	M☉ vs km s⁻¹	Derivable when g_extra is the geometric mean of g̅ and a₀, i.e., when the response sector couples to g̅ at the transition scale.
Q ∝ V²_flat	BTF-anchor prediction — the response amplitude Q scales as the square of the asymptotic velocity	km² s⁻²	Mechanism, not a mass-accounting identity. Confirmed empirically across SPARC-175.

Source: Kitcey, R. D. (2026). Intrinsic Response Sector as Dark Gravity: A GR-Compatible Candidate Identity for the Cold Dark Matter Role (SPARC-175) (v5.0.7). Zenodo. https://doi.org/10.5281/zenodo.18778896. Appendix A: Comprehensive Symbol Glossary (Table 5). Lernaean Research™.