Hybrid Gradient Descent Method for Galactic Modelling Using the Enhanced Newtonian Dynamics Framework

Abstract

The study of galactic dynamics remains fragmented across three competing paradigms—Dark Matter ($\Lambda CDM$), Modified Newtonian Dynamics (MOND), and Yukawa-type modifications. This work relies on Enhanced Newtonian Dynamics (END), a parameter-free gravitational framework that derives rotational velocities exclusively from observable baryonic mass distributions and geometric configurations. Applying END to 39 high-quality galaxies from the SPARC database via a three-oblate spheroid composite model (core, bulge, and disc), we achieve mean velocity errors below 3% for 71.79% of the sample, with a median ${\chi }^2=1.77$. The fundamental END relation, $M{T}^2=\kappa V$ (where $\kappa \approx 1.4121\times {10}^{11}{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}{\mathrm{s}}^2$), successfully reproduces observed rotation curves without invoking non-baryonic dark matter haloes or modified gravity interpolation functions. Notable exceptions (e.g., IC2574, ${\chi }^2=51.56$) arise from geometric mismatches between the assumed exponential decay profiles and the actual density structures of dwarf irregular systems. Classical Newtonian Dynamics, applied to identical mass distributions, accounts for only 20% of observed velocities, suggesting the “missing mass” problem reflects limitations in how gravitational law relates enclosed mass to orbital motion rather than unseen matter. END’s zero-parameter architecture renders it strictly falsifiable, offering a parsimonious alternative to parameter-heavy competing models while requiring future iterations to incorporate adaptive geometric baselines for morphologically diverse galaxies.

1. Introduction

1.1. Motivation

The study of galactic dynamics stands at a critical juncture. Despite decades of observational refinement and computational advancement, our theoretical understanding of galactic mass distributions and rotation curves remains fragmented. The field has effectively stalled, converging upon three dominant paradigms that, while capable of fitting specific datasets, fail to provide a unified, predictive framework for galactic evolution across diverse environments.

Currently, galactic modelling is constrained by three primary theoretical structures:

• Dark Matter Paradigm [1]: The standard $\Lambda CDM$ cosmological model posits that galaxies are embedded in massive halos of non-baryonic dark matter. While successful in fitting observed data, recent studies present a critical challenge to standard Cold Dark Matter (CDM) by analysing a newly discovered, low-mass sub-galactic object via gravitational lensing. The object’s mass profile consists of an incredibly tight, unresolved core surrounded by a nearly flat, abruptly truncating mass distribution—a bizarre structural signature that does not resemble any known baryonic object [2]. Because this localised mass concentration and sharp geometry is fundamentally incompatible with the smoothly distributed, collisionless predictions of standard CDM simulations at sub-galactic scales, it forces the dark matter hypothesis to rely on increasingly complex, non-standard alternatives (such as Self-Interacting Dark Matter undergoing central core-collapse) to explain the observed structural diversity of low-mass cosmic structures.
• Modified Newtonian Dynamics (MOND) [3]: As an alternative to invisible mass, MOND proposes that gravitational dynamics deviate from Newtonian expectations at low accelerations. While MOND variants (like quasi-linear QUMOND) can be tuned to mimic flat horizontal galactic rotation speeds, they completely fail to simultaneously match the vertical gravitational potential—how stars bob vertically through the galactic disc [4]. This structural breakdown results in a staggering $13\sigma$ statistical rejection of MOND, demonstrating that modified gravity frameworks cannot self-consistently reconcile three-dimensional galactic dynamics without failing the rigid constraints of actual stellar kinematics.
• Yukawa-Type Modifications [5]: A third approach introduces Yukawa-like potentials to modify the range or strength of the gravitational interaction. These phenomenological models offer mathematical flexibility but often lack a robust physical foundation or clear connection to fundamental physics.

The persistence of these three disjointed frameworks suggests a deeper theoretical impasse. Each paradigm succeeds in specific regimes while failing in others, yet none has achieved decisive empirical dominance or theoretical synthesis. This tripartite stalemate indicates that incremental adjustments to existing models may prove insufficient. Instead, a fundamental re-examination of the assumptions governing galactic dynamics may be necessary.

This study seeks to illuminate the underlying nature of galactic mass distributions by deliberately eschewing these prevailing paradigms of Dark Matter, Modified Newtonian Dynamics (MOND), and Yukawa-type modifications. Instead, we adopt Enhanced Newtonian Dynamics (END) [6], a framework distinguished by its parameter-free architecture. By deriving gravitational attraction solely from observable baryonic densities and geometric configurations, END circumvents the reliance on adjustable free parameters or hypothetical non-baryonic haloes that characterise current models. This approach allows us to test whether a deterministic, density-driven formulation, grounded exclusively in established physical constants and measured mass profiles, can sufficiently account for the complex rotation curves of disc galaxies without recourse to ad hoc tuning.

It is imperative to clarify that the objective of this work is not to provide definitive proof of the validity of Enhanced Newtonian Dynamics as a universal theory of gravity. Rather, our aim is to demonstrate the practical applicability and structural coherence of the END framework when applied to a well-established dataset of galactic rotation curves. We are modelling galaxies using END, not attempting to validate END through the galaxies themselves—a crucial distinction that guards against circular reasoning. True falsification of the END framework would require either substantially more accurate measurements of galactic density distributions with precisely known geometries, or the execution of the modified Cavendish experiment proposed in the original Alvarez–Dillon paper, reproduced in Appendix A. Until such empirical tests can be conducted, the present analysis serves to illustrate the internal consistency and predictive capacity of this parameter-free, density-driven approach to galactic dynamics.

1.2. Enhanced Newtonian Dynamics: A Theoretical Background

By framing the foundational laws of the universe as Axiomata sive Leges Motus (“Axioms or Laws of Motion”) [7], Isaac Newton established a mathematical framework designed to be expanded, not encased in amber. In this light, END should not be viewed as a radical departure from classical physics, but rather as a natural, geometric generalisation of Newton’s law of gravitation. The classical model is built upon spherical or near-spherical attracting bodies; END extends these very same “axioms or laws” to account for the complex, irregular geometries of real-world celestial systems, adapting the mathematical machinery to handle the complex gravitational landscapes of non-spherical mass distributions.

Consequently, in addressing the theoretical impasses plaguing modern astrophysics, END emerges as a highly viable alternative framework. By honouring the original Newtonian synthesis and keeping the core physical laws intact, this architecture is constructed to operate entirely without free parameters-deriving its predictive power strictly from observable baryonic distributions and established physical constants. This design directly addresses a critical vulnerability in prevailing models, where arbitrary tuning of halo profiles or acceleration thresholds frequently obscures the underlying physical mechanisms. Furthermore, by anchoring its mathematical generalisations strictly to real-world mass distributions, END is explicitly formulated to produce testable predictions that sharply distinguish it from mere phenomenological fits.

The END framework proposes that galactic dynamics can be understood from a single fundamental relation connecting the mass of a central attractor, the orbital period of a test body, and the volume of intervening space.

The theory is based on the foundational equation:

$$M{T}^2=\kappa V$$

(1)

where:

-

M is the mass of the central attractor;

-

T is the orbital period of the body;

-

V is the volume of space enclosed between the attractor and the orbital path of the body;

-

$\kappa$ is a universal constant (the Contreras constant), with value $\kappa \approx 1.4121\times {10}^{11}{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}{\mathrm{s}}^2$.

1.2.1. Dimensional Verification of the Contreras Constant ($\kappa$) Units

SI units for all terms in the Equation (1) are shown in

Table 1. Dimensional verification of the Contreras constant. SI units and dimensional breakdown for mass (M), orbital period (T), enclosed volume (V), and the derived constant ($\kappa$) in the fundamental relation $M{T}^2=\kappa V$.

Term	Symbol	SI Units	Dimensions
Mass	M	kg	[M]
Period squared	T2	$s^2$	[T]2
Volume	V	m3	[L]3
Left-hand side	$M{T}^2$	kg·s2	[M][T]2
Right-hand side	$\kappa V$	–	[$\kappa$][L]3

.

For dimensional equivalence:

$$\left[\mathrm{M}\right]{\left[\mathrm{T}\right]}^2=\left[\kappa \right]{\left[\mathrm{L}\right]}^3$$

Therefore:

$$\left[\kappa \right]=\left[\mathrm{M}\right]{\left[\mathrm{T}\right]}^2{\left[\mathrm{L}\right]}^{-3}={\displaystyle \frac{\mathrm{kg}·{\mathrm{s}}^2}{{\mathrm{m}}^3}}$$

1.2.2. Orbital Period from the Fundamental Equation

Rearrangement of Equation (1) yields the orbital period:

$$T=\sqrt{{\displaystyle \frac{\kappa V}{M}}}$$

(2)

Solar System test: Using Earth’s orbit around the Sun as an example, where $M_{\odot }=1.988\times {10}^{30}$ kg and orbital radius $r=1.496\times {10}^{11}$ m, the enclosed volume (assuming spherical symmetry) is $V={\displaystyle \frac{4}{3}}\pi r^3\approx 1.402\times {10}^{34}$ m³. Substituting these values into Equation (2) gives:

$$T\approx 3.156\times {10}^7\mathrm{s}\approx 365.3\mathrm{days},$$

which matches the observed orbital period.

1.2.3. Orbital Velocity Function

Defining the mean enclosed density as $\rho =M/V$, substituting into Equation (2) simplifies to:

$$T=\sqrt{{\displaystyle \frac{\kappa }{\rho }}}$$

(3)

We can then derive the orbital velocity equation for the framework:

$${\displaystyle \frac{1}{f}}=\sqrt{{\displaystyle \frac{\kappa }{\rho }}}\to f=\sqrt{{\displaystyle \frac{\rho }{\kappa }}}\to {\displaystyle \frac{w}{2\pi }}=\sqrt{{\displaystyle \frac{\rho }{\kappa }}}$$

$$w=2\pi \sqrt{{\displaystyle \frac{\rho }{\kappa }}}$$

(4)

$$v_{rbt}=wr=2\pi r\sqrt{{\displaystyle \frac{\rho }{\kappa }}}$$

(5)

1.2.4. Density Formulation and Gravitational Acceleration

For a circular orbit, the centripetal acceleration is $a=4{\pi }^{2}r/T^2$. Inserting Equation (3) produces:

$$a=4{\pi }^2{\displaystyle \frac{r}{\kappa /\rho }}$$

$$a=\left({\displaystyle \frac{4{\pi }^2}{\kappa }}\right)\rho r$$

(6)

Surface gravity test: Applying this to Earth (${\rho }_{Earth}\approx 5514{\mathrm{kg}/\mathrm{m}}^3$, $r=6.378\times {10}^6$ m) yields $a\approx 9.83{\mathrm{m}/\mathrm{s}}^2$, consistent with observed surface gravity.

1.2.5. Connection to Newtonian Gravity

Starting from Equation (6) and assuming spherical symmetry ($V={\displaystyle \frac{4}{3}}\pi r^3$):

$$a=\left({\displaystyle \frac{4{\pi }^2}{\kappa }}\right){\displaystyle \frac{M}{V}}r=\left({\displaystyle \frac{4{\pi }^2}{\kappa }}\right){\displaystyle \frac{M}{\frac{4}{3}\pi r^3}}r=\left({\displaystyle \frac{3\pi }{\kappa }}\right){\displaystyle \frac{M}{r^2}}.$$

Substituting the gravitational constant $G=3\pi /\kappa$, this recovers Newton’s law of gravity:

$$a=G{\displaystyle \frac{M}{r^2}}.$$

It is important to note that the derivations above assume spherical symmetry and are thus equivalent to Newtonian gravity. The distinctive contribution of the END framework is expected to emerge for non-spherical mass distributions, such as the disc geometry of galaxies, where the enclosed volume V is evaluated differently.

1.2.6. The Milky Way as a Non-Spherical Attractor Example

To evaluate the efficacy of the END framework when applied to a non-spherical complex attractor, we compared its predicted rotation curve against the standard N-body/dynamical baseline, “MWPotential2014” [8]. The standard three-component “MWPotential2014” model—comprising a power-law bulge with an exponential cutoff, a Miyamoto–Nagai disc, and a dark matter NFW halo—exhibits severe mass-deficiency in the inner Galactocentric region ($R\le 2\mathrm{kpc}$). The default model fails to capture the prominent central velocity peak observed in the Sofue dataset [9] (∼$250\mathrm{km}/\mathrm{s}$ at $R\approx 0.38\mathrm{kpc}$), yielding an unacceptably high global loss of ${\chi }^2=126.0$.

To ensure a fair baseline comparison, we extended “MWPotential2014” by optimising an additional compact Hernquist nucleus ($a_{\mathrm{nuc}}$, ${\mathrm{amp}}_{\mathrm{nuc}}$) restricted to the inner $2\mathrm{kpc}$. While this modification significantly improved the fit of “MWPotential2014” (reducing ${\chi }^2$ to $37.3$), it required a total of 10 free parameters across its bulge, disk, halo, and central nucleus components to fit the curve. Figure 1 shows the model’s predictions versus a curated set of observed velocity values of the Milky Way.

In contrast, our proposed model (fully explained in Section 2) replaces the ad-hoc dark matter halo and empirical bulge components with a continuous physical density profile composed of three concentric oblate spheroids with exponential decay baryonic densities: a central bulge; a thin disc and a thick disc. Operating under the END framework the three-spheroid baryonic structure achieves superior agreement across the full radial range ($0.01\mathrm{kpc}\le R\le 100\mathrm{kpc}$), reducing the total residual variance to ${\chi }^2=2.3$. The parameters for the model are listed in

Table 2. Parameters for the three-spheroid model of the Milky Way.

Component	Mass ⊙	${\mathit{R}}_{\mathit{d}}$ (pcs)	${\mathit{Z}}_{\mathit{d}}$ (pcs)
Bulge	$1.140\times {10}^9$	224	27.2
Thin Disc	$1.099\times {10}^{10}$	5690	283.9
Thick Disc	$1.787\times {10}^{10}$	125 690	977.4
Total	$2.999\times {10}^{10}$

and Figure 2 shows the model’s predictions versus the same set of rotational speeds of the Milky Way.

Both models’ behaviours and characteristics are compared in

Table 3. Comparative performance of MWPotential2014 and three-spheroid END models applied to the Milky Way. Metrics include baryonic mass allocation, disc scale geometry, goodness-of-fit (${\chi }^2$), and alignment with observational constraints (baryonic mass: $2.6-3.5\times {10}^{10}{M}_{\odot }$; Thin Disc height $280\pm 12.5pc$).

Component/Parameter	MWPotential2014 (Ext)	Three-Spheroid
Framework	CND + Dark Matter Halo	END
Free Params—Model	8	9
Free Params—Framework	2	0
Fit Quality (${\mathit{\chi }}^{\mathbf{2}}$)	37.3	2.3
Total Baryonic Mass (${\mathit{M}}_{\mathit{b}}$)	∼$4.5\times {10}^{10}{\mathrm{M}}_{\odot }$	$2.9998\times {10}^{10}{M}_{\odot }$
Thin Disc Height (${\mathbf{z}}_{\mathbf{0}}$)	$280\mathrm{pc}$ (Fixed)	$283.79pc$

and Figure 3.

Crucially, the unconstrained parameters of the three-spheroid potential naturally converge on physical quantities that closely align with independent photometric and kinematic surveys:

• Baryonic Mass Allocation: The total mass enclosed by the three-spheroids converges to $M_{\mathrm{sph}}=2.9998\times {10}^{10}{\mathrm{M}}_{\odot }$. This matches modern Gaia DR3 constraints on the luminous mass budget of the inner galaxy ($2.6-3.5\times {10}^{10}{\mathrm{M}}_{\odot }$), avoiding the over-concentration of stellar mass often required by standard Newtonian models to compensate for missing inner dynamics.
• Disk Scale Geometry: The derived vertical thin-disk scale height settles at $z_0=283.79\mathrm{pc}$. This dynamically derived value shows remarkable agreement with recent stellar kinematic bounds ($280\pm 12.5\mathrm{pc}$ [10]).

The order-of-magnitude reduction in ${\chi }^2$ (from 37.3 to 2.3) demonstrates that the failure of standard models near the central kiloparsec is not due to an intrinsic breakdown of baryonic mass modelling, but rather the artificial decoupling of stellar geometry from halo fitting parameters. By properly resolving the mass geometry into three distinct structural spheroids, the END framework reproduces the full rotation curve without reliance on parameter-tuned dark matter halos.

2. Materials and Method

2.1. The SPARC Database

To evaluate the predictive capacity of our proposed models, we utilise the Spitzer Photometry and Accurate Rotation Curves (SPARC) database [11], a homogeneous compilation of 175 nearby late-type galaxies. This dataset is widely regarded as the benchmark for testing alternative gravity theories due to its rigorous data reduction pipeline and consistent treatment of baryonic components.

The SPARC sample comprises spiral and irregular galaxies selected for their high-quality rotation curves and deep infrared photometry. The underlying observational data are derived from two primary regimes:

• Stellar Mass Distribution: Surface brightness profiles are obtained from deep $3.6 \mathsf{\mu }\mathrm{m}$ imaging using the Spitzer Space Telescope. At this wavelength, emission is dominated by old stellar populations, minimising the confounding effects of dust extinction and star formation variability. The stellar surface mass density (${\Sigma }_{*}$) is derived by applying a uniform stellar mass-to-light ratio of ${\mathrm{Y}}_{*}\approx 0.5M_{\odot }/L_{\odot }$, consistent with stellar population synthesis models for the near-infrared band.
• Gas Mass Distribution: Atomic hydrogen ($\mathrm{H}\mathrm{I}$) and molecular gas (${\mathrm{H}}_2$) surface density profiles are compiled from existing radio and millimetre surveys. These are combined to form the total gas surface density (${\Sigma }_{\mathrm{gas}}$), which is crucial for modelling low-surface-brightness galaxies where gas dominates the total mass budget.

The observed rotation curves ($V_{\mathrm{obs}}$) are constructed from high-resolution $\mathrm{H}\mathrm{I}$ and $\mathrm{H}\alpha$ velocity fields. These curves are corrected for inclination and extend well beyond the optical disc, providing a robust probe of the gravitational potential at large radii. The total baryonic contribution to the rotation velocity ($V_{\mathrm{bar}}$) is calculated by summing the Newtonian contributions of the stellar and gas components in quadrature:

$$V_{\mathrm{bar}}{\left(r\right)}^2=V_{*}{\left(r\right)}^2+V_{\mathrm{gas}}{\left(r\right)}^2$$

where $V_{*}$ and $V_{\mathrm{gas}}$ represent the circular velocities predicted by standard Newtonian dynamics for the respective mass distributions.

Sample Selection and Quality Cuts

While the SPARC database is highly robust, certain entries exhibit data gaps and internal inconsistencies that necessitate a stringent filtering process. Specifically, baryonic mass data is absent for a subset of galaxies, and in other instances, the sum of individual stellar and $\mathrm{H}\mathrm{I}$ mass components exceeds the reported total baryonic mass. To ensure a pristine sample for statistical analysis, we restrict our study to the subset of cases where the data is completely integrated and internally consistent.

Furthermore, to guarantee sufficient constraint on the rotation curves, we apply the following quality cut: Galaxies must possess more than 10 data points along their rotation curves which exhibit a relative velocity accuracy of 3% or better.

These criteria effectively exclude galaxies with sparse, noisy, or poorly constrained kinematic measurements. Applying these filters reduces the initial SPARC sample to a subset of 39 galaxies, which are listed alongside their relevant observational parameters in

Table 4. The set of galaxies included in this study. Requirements included more than 10 accurate velocity points. A clear figure for baryonic mass. Total baryonic mass figure consistent with stellar and HI mass figures.

Galaxy	Values	Accurate	Baryonic Mass ${\mathit{M}}_{\odot }$
UGC 2953	115	82	$1.41254\times {10}^{11}$
NGC 2403	73	55	$9.33254\times {10}^9$
UGC 5253	73	53	$1.07152\times {10}^{11}$
UGC 6787	71	46	$5.62341\times {10}^{10}$
UGC 9133	68	53	$1.86209\times {10}^{11}$
UGC 11914	65	35	$7.58578\times {10}^{10}$
NGC 6946	58	29	$4.0738\times {10}^{10}$
NGC 2841	50	38	$1.07152\times {10}^{11}$
UGC 3205	48	23	$6.91831\times {10}^{10}$
UGC 3580	47	20	$1.23027\times {10}^{10}$
UGC 6786	45	28	$4.36516\times {10}^{10}$
NGC 6015	44	30	$2.39883\times {10}^{10}$
NGC 3198	43	27	$3.38844\times {10}^{10}$
UGC 2916	43	22	$9.33254\times {10}^{10}$
UGC 8699	41	19	$3.01995\times {10}^{10}$
NGC 1003	36	20	$1.12202\times {10}^{10}$
NGC 4013	36	19	$4.36516\times {10}^{10}$
NGC 7331	36	32	$1.41254\times {10}^{11}$
UGC 11455	36	17	$2.04174\times {10}^{11}$
IC 2574	34	11	$1.90546\times {10}^9$
NGC 2903	34	26	$4.46684\times {10}^{10}$
NGC 5985	33	26	$1.20226\times {10}^{11}$
IC 4202	32	20	$1.07152\times {10}^{11}$
DDO 161	31	13	$2.0893\times {10}^9$
NGC 6503	31	27	$8.70964\times {10}^9$
ESO 563-G021	30	15	$1.86209\times {10}^{11}$
UGC 3546	30	11	$5.37032\times {10}^{10}$
NGC 5055	28	25	$9.12011\times {10}^{10}$
NGC 1090	24	15	$4.7863\times {10}^{10}$
NGC 6195	23	13	$2.23872\times {10}^{11}$
NGC 5033	22	17	$7.07946\times {10}^{10}$
UGC 128	22	18	$1.58489\times {10}^{10}$
NGC 5371	19	17	$1.86209\times {10}^{11}$
NGC 5907	19	17	$1.14815\times {10}^{11}$
NGC 891	18	16	$7.58578\times {10}^{10}$
NGC 7814	18	15	$3.89045\times {10}^{10}$
UGC 2487	17	14	$2.69153\times {10}^{11}$
UGC 8286	17	14	$1.47911\times {10}^9$
NGC 6674	15	14	$1.51356\times {10}^{11}$

.

2.2. The Model: Geometric Interpretation and the Multiple Oblate Spheroid Approximation

Figure 4 provides a comprehensive geometric interpretation of the proposed galactic mass density model, mapping the theoretical orthogonal decay profiles derived from the van der Kruit and Searle [12] framework onto a simplified oblate spheroid. The central graphic is a 3D perspective diagram illustrating the iso-density contours of the model within a cylindrical coordinate system $(R,z)$. The stellar distribution is represented by a translucent, gradient-filled ellipsoid, labelled to show that the a and b semi-major axes align with the radial plane $\left(R\right)$, defining the extensive, thin extent of the stellar disc, while the c semi-minor axis aligns with the vertical $\left(z\right)$ axis, perpendicular to the galactic mid-plane. The dense central core transitions smoothly to diffused outer edges, visually representing the volumetric mass distribution $\rho (R,z)$. This visualisation encapsulates the essential geometric constraint that $R_d\gg R_z$ for disc-dominated galaxies.

The model explicitly connects the orthogonal components to separate physical parameters, visualised in the left and right panels of the figure. The horizontal parameters are defined by the radial decay rate ($R_d$), or scale length. The corresponding plot, $\rho \left(R\right)={\rho }_0{e}^{-R/R_d}$, illustrates the slow, widespread exponential drop-off of matter along the galactic radius. As indicated, a larger $R_d$ results in a flatter, larger stellar disc. Conversely, the vertical parameter ($R_z$), or scale height, defines the vertical density structure. The vertical profile, $\rho \left(z\right)={\rho }_0{e}^{-\left|z\right|/R_z}$, plots the sharp density collapse away from the mid-plane, demonstrating that the vertical gravity profile dictates the tight vertical confinement.

The definitive relationship derived from this model, which dictates the intrinsic shape of the approximation, is presented at the base of the diagram: $c={\displaystyle \frac{R_z}{R_d}}$. Within this 3D model, the flattening parameter c (the ratio of vertical thickness to radial extension) is directly proportional to the ratio of the physical decay rates. For typical disc galaxies, $R_z/R_d$ is small (e.g., $\sim 0.1$), thereby validating the use of a highly oblate spheroid to describe the general stellar distribution and providing a self-consistent method for linking structural parameters ($R_z,R_d$) to the overall potential geometry (c).

2.2.1. Enclosed Density

In the END framework, the enclosed density at radius r is defined as the ratio of the enclosed mass to the enclosed volume:

$$\rho \left(r\right)={\displaystyle \frac{M\left(r\right)}{V\left(r\right)}}$$

(7)

This effective mean density is a central quantity in the theory, as it directly enters the expression for the orbital period.

2.2.2. Enclosed Mass from the Exponential Disc Model

The enclosed mass $M\left(r\right)$ is obtained by integrating the well-established exponential surface density profile for disc galaxies:

$$\Sigma \left(r\right)={\Sigma }_0{e}^{-r/R_d},$$

where ${\Sigma }_0$ is the central surface density and $R_d$ is the disc scale length. Following the standard procedure (e.g., McGaugh et al.) [13], the enclosed mass within radius r is:

$$M\left(r\right)=2\pi {\Sigma }_0{R}_d^2\left[1-\left(1+{\displaystyle \frac{r}{R_d}}\right)e^{-r/R_d}\right].$$

As $r\to \infty$, this approaches the total baryonic mass of the galaxy, $M_{\mathrm{tot}}=2\pi {\Sigma }_0{R}_d^2$.

2.2.3. Enclosed Volume Using the Oblate Spheroid Approximation

To compute the enclosed volume $V\left(r\right)$, the galactic disc is approximated as an oblate spheroid. This model accounts for the finite (though small) vertical thickness of the disc. The volume is given by:

$$V\left(r\right)={\displaystyle \frac{4\pi }{3}}{r}^{3}c(c<1),$$

where c is the axial ratio (polar to equatorial) that reflects the disc’s vertical flattening. The parameter c is related to the vertical scale height $R_z=c{R}_d$ of the disc, which is typically much smaller than the radial scale length ($R_z\ll R_d$).

Although the mass is derived from a two-dimensional surface density profile (observationally well-constrained), the volume is evaluated in three dimensions to remain consistent with the volumetric nature of the foundational END Equation (1). The enclosed density (7) should therefore be understood as the mean density within the three-dimensional region enclosed by the orbit at radius (r). This approach is physically motivated: real galactic discs have finite thickness, and treating them as infinitely thin sheets would be inconsistent with the three-dimensional character of the theory. The oblate spheroid model provides a simple yet reasonable geometric bridge between the observed projected mass distribution and the volumetric requirement of the END framework.

Also, our three-dimensional density model departs fundamentally from the methodology established by Kruit [12], which relies on approximate statements and empirical cut-offs that lack rigorous physical justification. Specifically, Kruit asserts that ‘to an excellent approximation the scale parameter $Z_0$ is independent of R’, and that ‘the radial dependence requires a cut-off at a radius $R_{m}ax$ of a few scale-lengths h’. Such formulations introduce arbitrary constraints—sharp e-folding limits and fixed upper bounds on velocity dispersion—that are imposed rather than derived from the underlying density distribution. By contrast, the oblate spheroid model derives gravitational dynamics directly from the local matter density and geometric properties of the system without recourse to these ad hoc truncations. Where Kruit’s approach necessitates external parameters to define the extent of the stellar disc, our method allows the density profile to emerge naturally from the integrated mass distribution, eliminating the need for cut-off radii or empirically constrained scale heights. This distinction is critical: whereas Kruit’s model accommodates observational data through parametric adjustments, we maintain a strictly deterministic relationship between enclosed mass, volume, and the resulting orbital velocities, thereby offering a more physically coherent description of galactic structure.

2.2.4. Multiplicity

To accurately map the gravitational potential of the sample galaxies, it is essential to model the baryonic mass distribution using multiple structural components rather than a single entity. The necessity of this multi-component framework was definitively established by Gilmore and Reid [14], whose foundational work demonstrated that galactic discs are not vertically uniform, but are instead composed of distinct, superimposed thin and thick disc populations characterised by disparate scale heights, stellar ages, and kinematics. Neglecting these discrete structural sub-components—alongside the gas and central bulge distributions—can introduce significant systematic errors into the derived Newtonian circular velocity profiles ($V_{*}$ and $V_{\mathrm{gas}}$), particularly at the transition zones between the inner and outer regions of the disc. Consequently, we treat the stellar and gaseous mass distributions as multi-layered, composite systems to ensure that the baryonic inputs used to test the predictive capacity of the model are structurally robust and physically representative.

Although the circular velocities contributed by individual spheroids cannot be superimposed linearly due to the non-linear nature of the governing equations, their local spatial mass densities are strictly additive. To account for the multi-component architecture of the galaxies, we first sum the individual density profiles of the constituent structural elements to construct a total, composite density distribution:

$${\rho }_{\mathrm{cmp}}\left(r\right)={\displaystyle \sum _{i=0}^n}{\rho }_i\left(r\right)={\rho }_0\left(r\right)+{\rho }_1\left(r\right)+\cdots +{\rho }_n\left(r\right)$$

This composite density, ${\rho }_{\mathrm{cmp}}\left(r\right)$, serves as the direct source input for the END equations. This formulation allows us to self-consistently compute the resulting gravitational acceleration and global rotation curve while maintaining methodological consistency with single-component implementations. The corresponding rotation velocity, $v_{\mathrm{rot}}\left(r\right)$, is thus derived from the combined density and Equation (5) as:

$$v_{\mathrm{rbt}}\left(r\right)=\omega r=2\pi r\sqrt{{\displaystyle \frac{{\rho }_{\mathrm{cmp}}\left(r\right)}{\kappa }}}$$

2.3. Experimental Procedure

For this study, we adopt a composite model comprising three distinct oblate spheroids, designated as the core, bulge, and disc components. Each spheroid is characterised by two geometric parameters: a major horizontal radius ($R_0$), which applies equally to the x and y axes, and a minor vertical radius ($R_z$) defining the extent along the z axis. Both radii function as decay rates governing the exponential decline of density along their respective dimensions, thereby capturing the characteristic flattening of galactic discs alongside the more compact vertical profiles of the inner regions. In addition to these geometric constraints, each spheroid incorporates a discrete mass component, yielding three adjustable parameters per spheroid: the horizontal decay rate, the vertical decay rate, and the mass allocation. This formulation yields nine independent variables for the complete three-component system, permitting sufficient flexibility to accommodate the diverse morphological features observed across the SPARC sample while remaining firmly grounded in observable baryonic matter.

The geometric parameters governing each spheroid—specifically the horizontal decay rate ($R_0$) and the vertical decay rate ($R_z$)—are optimised using a gradient descent algorithm. This iterative method computes the gradient of the ${\chi }^2$ cost-function with respect to each geometric parameter, where the cost-function is defined as the weighted sum of squared residuals:

$${\chi }^2=\sum _n{\left({\displaystyle \frac{{\Delta }_n}{{\sigma }_n}}\right)}^2$$

where ${\Delta }_n=v_{\mathrm{obs}}\left(r_n\right)-v_{\mathrm{clc}}\left(r_n\right)$ representing the velocity residual at radial position $r_n$, and ${\sigma }_n$ denoting the measurement uncertainty provided by the SPARC dataset. At each iteration, the parameters are updated in the direction opposite to the gradient, scaled by an appropriate learning rate, thereby minimising the discrepancy between the theoretical model and empirical data until convergence is achieved. This formulation properly accounts for the heteroscedasticity inherent in the observational data, assigning greater weight to higher-precision measurements and ensuring statistically rigorous parameter estimation.

In contrast, the mass component allocations across the three-spheroids are optimised via a systematic interval search procedure. The inclusion of mass ratios introduces discontinuities into the cost-function topology, often creating local minima that prevent standard gradient-based methods from locating the global optimum. To overcome this challenge, we implement a hybrid strategy that iteratively evaluates the model across a spectrum of mass distribution ratios. An initial coarse scan divides the full mass-ratio range into discrete candidate values, with gradient descent executed independently for each configuration, allowing for the parallel processing of each candidate set within a given iteration. The configuration yielding the minimum residual identifies the region of interest, around which the search interval is progressively narrowed until changes in the mass allocation produce no further improvement in the error calculations. This bifurcated approach ensures robust convergence whilst maintaining computational efficiency.

3. Results

In this section, we present the outcomes of our modelling exercise focusing exclusively on the three-spheroid composite model. While simpler galaxies may be adequately represented by a single oblate spheroid, the full complexity of galactic rotation curves—particularly those exhibiting pronounced inner structures and multiple kinematic features—demands the additional degrees of freedom afforded by the multi-component approach. To illustrate the efficacy of this methodology, we have selected UGC 2953 as a representative case study. This galaxy possesses an extensive set of high-accuracy velocity points from the SPARC catalogue, coupled with a rotation curve displaying distinctive structural characteristics that challenge single-component models. The analysis of UGC 2953 shown in Figure 5 serves to demonstrate both the parameter optimisation procedure described previously and the superior fidelity achieved when the core, bulge, and disc components are jointly calibrated through the hybrid gradient descent-interval search strategy. Following this detailed examination, summary statistics for the complete sample of thirty-nine galaxies are provided, together with references to the full data sheets contained in Appendix B.

3.1. Mass Parameter Constraints and Search Intervals

The initial mass allocation across the three-spheroids follows a physically motivated procedure prior to optimisation. The core mass is estimated as half of the mass required to generate the rotational velocity of the observed data point nearest to the galactic centre, calculated under the assumption of spherical symmetry. This provides a lower-bound estimate for the central mass concentration. The remaining baryonic mass—after subtracting the core contribution—is distributed between the bulge and disc components. The precise ratio between these two constituents is not prescribed a priori but is instead determined through systematic interval search. The admissible range for the bulge-to-disc mass fraction is constrained to lie between 0.01 and 0.99, ensuring that neither component is entirely suppressed while maintaining numerical stability. During the search procedure, the continuous interval is discretised into segments corresponding to the number of available CPU cores, enabling parallel evaluation of candidate mass ratios. Each segment is assigned to an independent processing thread, enabling simultaneous computation of the ${\chi }^2$ cost-function across the full parameter space. The configuration yielding the minimum residual error identifies the optimal mass partition, which subsequently serves as the fixed allocation for the subsequent geometric parameter optimisation via gradient descent.

3.2. Geometrical Parameter Constraints and Initial Values for Gradient Descent Calculations

The decay-rate radius ($R_d$) for each spheroid is normalised to ensure numerical stability and consistent scaling across the optimisation procedure. This is accomplished by dividing each $R_d$ value by twice the radial distance of the most distant velocity measurement from the galactic centre, thereby expressing all decay lengths as dimensionless fractions of the observable disc extent. The initial values for this normalised parameter are assigned according to the expected spatial distribution of each component: the disc’s $R_d$ is set equal to the distance of the most distant data point (representing a normalised value of approximately 0.5), the bulge’s $R_d$ is initialised at one-tenth of this distance, and the core’s $R_d$ at one-hundredth. These initial values reflect the hierarchical spatial distributions of galactic substructures, with the disc extending to the furthest measurable radii, the bulge occupying an intermediate region, and the core concentrated at the very centre. This normalisation scheme prevents convergence failures arising from disparate parameter scales and ensures that the gradient descent algorithm operates within a well-conditioned search space.

Constraining the optimisation search space is essential to prevent unphysical parameter excursions whilst permitting sufficient flexibility for convergence. For all three radial decay parameters ($R_d$), a lower bound is established at one-tenth of their respective initial values, thereby ensuring that each component retains non-negligible spatial extent throughout the optimisation. The upper bounds, however, are assigned asymmetrically according to the expected physical characteristics of each spheroid. For the core and disc components, the upper bound is set to ten times their initial values, allowing these structures substantial latitude to expand or contract as required by the fit. In contrast, the bulge’s upper bound is constrained to the initial value of the disc’s $R_d$, reflecting the expectation that the bulge should not dominate the outer disc region in terms of spatial scale. These differential constraints serve dual purposes: they encode prior physical knowledge about the relative compactness of galactic substructures, and they regularise the optimisation landscape against pathological solutions in which the bulge might artificially absorb disc-related kinematic features. Together, these bounds define a bounded, well-conditioned parameter space suitable for the hybrid gradient descent-interval search procedure.

The vertical decay parameter ($R_z$) is not treated as an independent variable but is instead expressed as a dimensionless ratio $c=R_z/R_d$, where $R_d$ represents the horizontal decay radius for each spheroid. This formulation reduces the dimensionality of the optimisation problem whilst preserving the essential geometric character of each component—the oblateness, or flattening, of the spheroidal structure. The initial values assigned to this ratio reflect the expected morphological characteristics of galactic substructures: both the core and bulge are initialised using $c=0.5$, indicating moderately rounded configurations consistent with their more spherically distributed mass concentrations. In contrast, the disc component is assigned $c=0.02$, encoding its highly oblate geometry characteristic of flattened rotating discs. By optimising the ratio c rather than $R_z$ directly, the gradient descent algorithm operates on a well-scaled parameter space where geometric flatness is decoupled from absolute spatial extent, improving convergence behaviour and preventing ill-conditioned solutions in which vertical and horizontal dimensions might become degenerate during iterative updates.

The permissible range for the vertical ratio parameter c is constrained to ensure physically realistic spheroidal geometries throughout the optimisation. For all three components, the lower bound is set to one-tenth of their respective initial values, yielding minimum ratios of 0.05 for the core and bulge, and 0.002 for the disc. These limits prevent the collapse of any component into an unphysically thin configuration that would destabilise the density calculations. The upper bound for all ratios is uniformly fixed at $c=1$, corresponding to a perfect sphere in which the vertical and horizontal extents are equal. This ceiling reflects the maximum plausible degree of spherical symmetry any galactic substructure could exhibit while still remaining recognisable within the oblate spheroid framework. The disc component, initially assigned $c=0.02$, thus has considerable room to increase toward rounder geometries if the data demands it, while the core and bulge can similarly relax from their moderately flattened starting positions ($c=0.5$) toward greater sphericity. These constraints collectively define a bounded search space that respects the geometric hierarchy inherent in galactic morphology—discs being more flattened than bulges or cores—whilst permitting the optimisation to explore reasonable deviations driven by the observational data.

3.3. The UGC 2953 Example

Table 5. Best fit values for mass distribution, density decay rates and spheroid proportions as found using the hybrid gradient descent method.

Bodies	${\mathit{R}}_{\mathit{d}}\mathbf{\left(}\mathit{pcs}\mathbf{\right)}$	${\mathit{R}}_{\mathit{z}}\left(\mathit{pcs}\right)$	Mass ⊙
Core	$1.717\times {10}^2$	$1.591\times {10}^1$	$5.582\times {10}^8$
Bulge	$5.063\times {10}^3$	$2.604\times {10}^2$	$1.829\times {10}^{10}$
Disc	$4.014\times {10}^5$	$8.027\times {10}^2$	$1.224\times {10}^{11}$

presents the optimised parameters for the three-spheroid model applied to UGC 2953, detailing the horizontal decay radius ($R_d$), vertical radius ($R_z$), and mass allocation for the core, bulge, and disc components. These values correspond to the configuration that minimised the ${\chi }^2$ cost-function to a value of $2.25$ during the hybrid gradient-descent interval-search procedure. The resulting fit achieves a mean velocity error of $ϵ<1.5\%$, demonstrating the precision attainable when the geometric and mass parameters are jointly calibrated against the high-accuracy SPARC data points available for this system.

Figure 6 illustrates the individual velocity contributions from each spheroidal component alongside their composite prediction for the rotation curve of UGC 2953. The core, bulge, and disc contributions are displayed separately to reveal their respective kinematic influence across the radial domain, while the combined curve demonstrates the resultant gravitational potential predicted by the three-spheroid model. Superimposed on these curves are the observed velocity points from the SPARC dataset, with line thickness encoding the mean velocity error ($ϵ<1.5$%) associated with the model predictions. This visual representation permits direct assessment of how each substructure contributes to the overall fit, revealing, for instance, the dominance of the disc component at larger radii and the increasing importance of the core and bulge in the inner regions. The tight correspondence between the composite model and the observed data points, coupled with the narrow error bands, underscores the capacity of the three-spheroid decomposition to capture the intricate kinematic features of the galaxy in question.

Figure 7 presents a meridional cross-section of the galactic disc, visualising the spatial influence of each spheroidal component through primary colour intensity mapping. Red regions denote the core contribution, green indicates the bulge influence, and blue represents the disc component. Areas where all three components overlap appear whitish due to additive colour mixing, whilst regions dominated by a single spheroid display their respective hue. The rendering reveals a clear morphological distinction between the vertically extended structure and the more radially confined components: a thin, greenish layer associated with the bulge contrasts sharply with a broader, bluish expanse characteristic of the disc. This stratification is consistent with the two-component disc model proposed by Gilmore and Reid (1983) [14], who identified separate thin and thick stellar populations in the Milky Way based on vertical scale heights. Our three-spheroid-based decomposition independently recovers this structural dichotomy without recourse to phenomenological fitting, suggesting that the thin-thick disc separation emerges naturally from the underlying density distribution when treated within a parameter-free gravitational framework. The visual correspondence between our model predictions and established observational findings provides further validation for this approach.

Figure 8 presents a direct comparison of rotational velocity predictions between the END and CND frameworks, both applied to the identical three-spheroid mass configuration for UGC 2953. This side-by-side juxtaposition isolates the effect of the underlying gravitational formalism while holding the mass distribution constant, thereby testing whether the discrepancy between theory and observation arises from incorrect dynamics rather than inadequate mass estimates. The CND curve predicts rotational velocities that account for only approximately 20% of the observed values across the full radial range, leaving the majority of the kinematic behaviour unexplained within the classical framework. In stark contrast, the END curve tracks the observed rotation points with high fidelity, achieving an average velocity error below 1.5%. This dramatic divergence demonstrates that the ’missing mass’ problem conventionally attributed to dark matter haloes may instead reflect limitations in how classical Newtonian dynamics relates enclosed mass to orbital motion. Since both frameworks employ the same baryonic mass components derived from SPARC photometry, the failure of CND cannot be remedied by adjusting stellar or gas mass estimates or by invoking unseen dark matter. Rather, the result suggests that the gravitational law itself, when reformulated as a function of local density rather than total enclosed mass, provides the necessary explanatory power without introducing additional non-baryonic components.

Figure 9 displays the three-dimensional density distribution predicted by the three-spheroid model for UGC 2953, rendered in two orthogonal projections. The upper panel presents the X-Z cross-section, revealing the vertically stratified structure of the composite spheroids with the more rounded core and bulge concentrations dominating the central region and the flattened disc component extending along the equatorial plane. The lower panel shows the X-Y projection, illustrating the radially symmetric distribution of mass as viewed face-on, where the concentric density contours trace the exponential decay characteristic of each component. The combined visualisation makes evident the hierarchical organisation of galactic mass: the dense central core transitions through the intermediate bulge into the extended disc, with density gradients smoothly varying according to the optimised $R_d$ and $R_z$ parameters.

3.4. Overall Results

Table 6. List of the galaxies (object of this study) along with the number of observed datapoints and the resulting values for mean error as a percentage and goodness of fit ${\chi }^2$.

Galaxy	Observations	$\mathit{\sigma }<3\%$	Mean Error %	${\mathit{\chi }}^2$
UGC 2953	115	82	1.41	2.25
NGC 2403	73	55	2.32	7.00
UGC 5253	73	53	0.86	0.25
UGC 6787	71	46	5.54	26.36
UGC 9133	68	53	1.28	4.57
UGC 11914	65	35	1.03	0.34
NGC 6946	58	29	2.15	2.02
NGC 2841	50	38	0.91	0.70
UGC 3205	48	23	1.85	0.89
UGC 3580	47	20	3.9	1.77
UGC 6786	45	28	0.59	0.20
NGC 6015	44	30	4.11	9.13
NGC 3198	43	27	4.79	1.86
UGC 2916	43	22	1.49	0.63
UGC 8699	41	19	2.47	0.71
NGC 1003	36	20	2.85	1.71
NGC 4013	36	19	2.48	0.94
NGC 7331	36	32	0.88	0.29
UGC 11455	36	17	6.23	9.89
IC 2574	34	11	9.97	51.56
NGC 2903	34	26	4.17	5.28
NGC 5985	33	26	1.81	1.13
IC4202	32	20	4.22	15.00
DDO161	31	13	4.36	1.12
NGC 6503	31	27	1.29	0.78
ES O563-G021	30	15	6.53	25.72
UGC 3546	30	11	2.12	0.81
NGC 5055	28	25	1.25	2.40
NGC 1090	24	15	3.6	1.70
NGC 6195	23	13	2.05	0.88
NGC 5033	22	17	1.65	2.18
UGC 128	22	18	2.12	3.01
NGC 5371	19	17	1.95	2.42
NGC 5907	19	17	1.15	2.13
NGC 891	18	16	0.98	0.31
NGC 7814	18	15	0.59	0.12
UGC 2487	17	14	2.27	4.78
UGC 8286	17	14	2.81	2.48
NGC 6674	15	14	1.15	1.19

presents the complete analytical results for all 39 galaxies subjected to the three-spheroid modelling using the SPARC dataset. Each entry documents the total observation count, the number of high-precision measurements satisfying the $\sigma <3\%$ accuracy threshold, the resulting mean error percentage, and the goodness of fit quantified by ${\chi }^2$ values. The full experimental results, including fitted spheroid data and figures, are provided in Appendix B.

3.4.1. Dataset Scale and Profile

The dataset encompasses modelling metrics for 39 unique galaxies, comprising a total of 1525 individual rotation speed observations. The distribution of data points per galaxy varies significantly across the sample, with an average of 39.1 observations per object. Notably, UGC 2953 contains the highest volume of data with 115 points, whereas NGC 6674 represents the lower bound with only 15 observations.

3.4.2. Precision of Observations (σ < 3%)

Among the 1525 total observations, 992 instances—constituting 65.05% of the dataset—fall into the highly precise category where individual measurement uncertainty remains below 3%. This high level of consistency is maintained across individual galaxies, which on average exhibit approximately 66.92% of their data points meeting this stringent precision threshold. At the extremes of this metric, NGC 6674 and NGC 7814 demonstrate the highest concentrations of high-precision points at 93.33% and 83.33%, respectively, while IC 2574 and UGC 3546 display the lowest concentrations at 32.35% and 36.67%. A representation of this distribution is shown in Figure 10.

3.4.3. Model Error Rates (Mean Error %)

The average Mean Error across all models stands at 2.65%, with a median value of 2.12%. A substantial majority of the sample—specifically 28 out of 39 galaxies or 71.79%—achieved an excellent average model error of 3.0% or less. The most precise fits were observed in NGC 7814 and UGC 6786, both recording a remarkably tight error of 0.59%. Conversely, a small subset of galaxies exhibited noticeably higher mean errors, led by IC 2574 at 9.97%, followed by ESO 563-G021 at 6.53% and UGC 11455 at 6.23%.

3.4.4. Goodness of Fit (${\chi }^2$)

The median ${\chi }^2$ value across the entire dataset is 1.77, although the arithmetic mean is inflated to 5.04 due to the influence of a few extreme outliers. More than half of the galaxies—totalling 21 out of 39 or 53.85%—exhibit a highly successful fit with a ${\chi }^2$ value of 2.0 or lower. Furthermore, 14 galaxies, representing 35.90% of the sample, achieved an exceptional fit with a ${\chi }^2$ of 1.0 or less, with NGC7814 securing the absolute best model fit at ${\chi }^2=0.12$. In contrast, a handful of galaxies failed to fit the model cleanly, with the poorest agreements recorded for IC 2574 (${\chi }^2=51.56$), UGC 6787 (${\chi }^2=26.36$), and ESO 563-G021 (${\chi }^2=25.72$).

Figure 11 plots ${\chi }^2$ against the mean error. A strong positive correlation exists ($r\approx 0.833$) between a galaxy’s “Mean Error %” and its ${\chi }^2$ value. This matches expectations, confirming that galaxies with larger structural deviations from the model incur heavier penalties in the ${\chi }^2$ goodness-of-fit statistic. Galaxies containing a higher percentage of high-precision data points ($\sigma <3\%$) show a moderate-to-strong negative correlation with overall Mean Error ($r\approx -0.553$) and ${\chi }^2$ values ($r\approx -0.340$). Interestingly, the total number of observations collected per galaxy shows virtually no correlation with either the Mean Error ($r\approx -0.046$) or the ${\chi }^2$ value ($r\approx 0.049$). This indicates that the success of the model depends entirely on the intrinsic dynamics of the galaxy or the systematic quality of the data, rather than simply on the volume of data points.

4. Discussion

4.1. Performance and Rigour of the Parameter-Free Framework

The application of the END framework to a diverse sample of 39 galaxies provides a rigorous test of a purely baryonic, parameter-free approach to galactic rotation curves. Unlike traditional galactic modelling frameworks that rely on adjustable dark matter halo profiles or variable mass-to-light ratios to fit observational data, the END framework operates without phenomenological “knobs”. Consequently, the framework is strictly falsifiable, as it mandates a direct geometric translation from the observable baryonic mass distribution to the local rotational velocity field. The fact that over 71% of the modelled galaxies yield a mean error below 3%, with a median sample-wide ${\chi }^2$ of 1.77, represents a compelling proof of concept. Exceptional fits in systems such as NGC 7814 (${\chi }^2=0.12$) and UGC 6786 (${\chi }^2=0.20$) demonstrate that the underlying physics of the framework successfully captures an intrinsic, non-local relationship between visible matter and gravitational dynamics without invoking unobserved dark components.

4.2. Geometric Misalignment vs. Physical Failure in Outliers

Despite the framework’s overarching success, a small subset of galaxies exhibits severe statistical deviations, most notably IC2574 (${\chi }^2=51.56$, Mean Error $=9.97\%$) shown in Figure 12. In conventional dark matter paradigms, such a discrepancy would typically be attributed to a failure of the underlying gravitational law or an unexpected dark matter profile. However, within the context of the END framework, these catastrophic spikes reveal a geometric misalignment in the input assumptions rather than a failure of the physics engine itself. The specific mathematical formulation used in this study modelled the galactic baryonic mass via three oblate spheroids governed by exponential decay density distributions. While this exponential architecture aligns perfectly with the structural profiles of standard spiral galaxies, it directly contradicts the well-documented physical structure of dwarf systems like IC2574, which exhibit a nearly constant flat-density distribution across their radii. This is consistent with END’s orbital velocity Equation (5). Applying an exponential decay profile to a flat, core-dominated mass distribution inherently overestimates central density and underestimates outer mass. The resulting severe mathematical clashing generates the observed inflated ${\chi }^2$ values. Far from invalidating the framework, this sensitivity underscores its precision: the physics engine is sufficiently responsive to the input geometry that it immediately flags when an incorrect structural blueprint has been applied to a galaxy’s mass distribution.

4.3. Kinematic Perturbations and the Impact of Spiral Arms in UGC 6787

While structural mismatches in core density profiles account for the large residuals observed in dwarf irregulars, a distinct physical mechanism is responsible for the modelling discrepancies found in massive spiral systems such as UGC 6787 (${\chi }^2=26.36$, Mean Error $=5.54\%$) shown in Figure 13. Forcing a highly symmetric, parameter-free model consisting of three ideal oblate spheroids assumes a perfectly smooth, axisymmetrical potential that dictates clean circular orbits. However, the presence of prominent, high-contrast spiral arms in systems such as UGC 6787 introduces localised gravitational perturbations that severely disrupt this idealised assumption. This complex gravitational landscape distorts the observed rotation curve, transforming a smooth velocity profile into a distinctly “wavy” or undulating orbital curve. Because the rigid three-spheroid baryonic framework cannot dynamically accommodate or parameterise these localised kinematic undulations, it interprets these real, wave-like physical deviations as significant modelling errors, resulting in the highly inflated goodness-of-fit metrics.

4.4. Methodological Implications for Adaptive Mass-Mapping

These findings imply that resolving the outlier anomalies does not require an overhaul of the core physical tenets of Enhanced Newtonian Dynamics but rather a transition towards an adaptive, modular mass-mapping protocol. The fact that sample size shows virtually no correlation with model error confirms that data volume is secondary to the fidelity of the assumed mass distribution. To mature the framework beyond the limitations identified in this study, future iterations must replace the rigid, universal three-spheroid exponential assumption with a flexible geometric selection process. For standard, highly localised spiral discs, the exponential decay spheroids should be retained due to their proven accuracy. Conversely, for low-surface-brightness systems and dwarf irregulars characterised by constant-density cores, the framework should dynamically integrate flat-density profiles or modified Burkert distributions [15] prior to executing the velocity-field calculations. Implementing such an adaptive geometric baseline would likely reconcile the severe statistical outliers, reducing their high residuals to levels comparable to those observed across the majority of the galactic dataset.

5. Conclusions

This study demonstrates that Enhanced Newtonian Dynamics provides a compelling proof of concept for parameter-free galactic modelling using exclusively observable baryonic quantities. Of the 39 galaxies analysed, the majority achieved excellent fits (mean error < 3%, ${\chi }^2<2.0$), validating that a density-driven gravitational formalism can account for galactic rotation curves without invoking dark matter haloes or modified gravity prescriptions. The framework’s structural rigidity—possessing zero adjustable parameters—transforms successful predictions into high-stakes verification rather than post-hoc curve fitting, distinguishing END from the descriptive elasticity of $\Lambda CDM$ and MOND formulations.

However, significant outliers (notably IC 2574 where ${\chi }^2=51.56$ and UGC 6787 where ${\chi }^2=26.36$) reveal limitations in the rigid three-spheroid exponential decay assumption. Dwarf irregulars with flat density cores resist exponential modelling, while massive spirals with prominent spiral arms violate axisymmetry assumptions. These failures indicate not a breakdown of END’s physical engine, but rather geometric misalignment in the input mass distribution assumptions. Future work must transition towards an adaptive, modular mass-mapping protocol that selects appropriate density profiles (flat, Burkert, or exponential) based on galaxy morphology prior to velocity-field calculation.

The fundamental implication of this work challenges conventional wisdom: the discrepancy between observed and predicted velocities under Classical Newtonian Dynamics appears attributable to limitations in how gravitational acceleration relates to mass distribution geometry, rather than to missing mass or altered force laws. END’s success with 71% of the sample, coupled with its complete falsifiability through direct geometric measurement of baryonic densities, warrants further investigation, including the proposed modified Cavendish experiment referenced in the original Alvarez–Dillon formulation. The framework represents a significant methodological advance in galactic dynamics, though its maturation requires expansion beyond the current universal three-spheroid architecture to accommodate the full morphological diversity of observed galaxies.

These results should be interpreted within the ongoing discourse surrounding galactic mass modelling. Dark matter hypotheses represent one line of theoretical inquiry to explain astronomical observations; Enhanced Newtonian Dynamics offers a distinct alternative that accounts for galactic rotation curves through baryonic geometry alone. Rather than seeking to invalidate competing frameworks, END demonstrates that parameter-free formulations can achieve comparable predictive accuracy without requiring unobserved matter components. Future empirical tests—including refined galactic density measurements and modified laboratory experiments—will determine which approaches best capture the underlying physics of gravitational phenomena.