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The Al-Mizan Al-Mutakamil Relational Framework: Discrete Scale Operators, Galactic Kinematics, and Cosmological Expansion

This paper presents the Al-Mizan Al-Mutakamil relational framework, a unified theory that replaces dark matter and the cosmological constant with discrete scale-step operators acting on baryonic flux tensors, achieving high-precision fits to galaxy rotation curves and deriving a revised cosmological expansion model with a transition redshift of z ≈ 2.15 using only three global scalar constants.

Original authors: Davinus Masire Ochanda

Published 2026-10-22

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CC BY 4.0

Original authors: Davinus Masire Ochanda

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For decades, astronomers have faced a stubborn puzzle regarding how galaxies spin. When they measure the speed of stars and gas orbiting the centers of spiral galaxies, the outer edges move far faster than the visible matter alone should allow. In the standard model of the universe, this discrepancy is explained by the existence of invisible "dark matter," a mysterious substance that does not emit light but exerts gravity, wrapping galaxies in massive, unseen halos. While this idea fits many large-scale observations, it struggles to explain the specific details of individual galaxies without constant, fine-tuned adjustments for each one. An alternative view suggests that our understanding of gravity itself might need to change at the vast, low-force scales of the outer galaxy, rather than invoking invisible matter.

A new theoretical framework proposed by Davinus Masire Ochanda offers a third path, one that abandons both invisible halos and simple tweaks to gravity. Instead, this approach suggests that the behavior of galaxies is governed by a hidden, step-by-step relationship between the visible matter and the structure of space itself. The researcher argues that gravity does not act as a smooth, continuous force everywhere, but rather evolves through discrete steps as it moves away from the center of a galaxy. By treating the motion of stars and gas as a series of connected states rather than a single fluid flow, the theory attempts to explain why galaxies spin the way they do using only the matter we can actually see.

In this new study, Ochanda presents a complete mathematical formulation of what he calls the "Al-Mizan Al-Mutakamil" relational framework. The core idea is that the gravitational pull felt by a star is not just a result of the mass nearby, but is enhanced by a non-local connection to the entire galaxy's angular momentum. Imagine the galaxy not as a collection of independent parts, but as a single system where the motion of one part influences the whole through a series of discrete scale steps. The theory proposes that as one moves outward from the galactic center, the gravitational response shifts through these steps, creating an extra boost in acceleration that keeps the outer stars moving fast without needing dark matter.

To test this idea, the researcher applied the theory to a massive database of 175 real galaxies, known as the SPARC catalog. These galaxies range from small, dim dwarf systems to massive, bright spirals, offering a rigorous test for any theory of gravity. The goal was to see if a single, universal set of rules could predict the rotation speeds of all these different galaxies using only the distribution of their visible stars and gas. The study found that it could. By using just three fixed numbers to define how the relational steps work, the model successfully predicted the rotation curves for the entire collection of galaxies. The predictions matched the observed data with a precision that rivals the best-fitting dark matter models, but without needing to invent a unique, invisible halo for every single galaxy.

One of the most significant findings concerns how the theory handles different types of galaxies. Small, dim dwarf galaxies have long been a trouble spot for standard models because they seem to require an unusually high amount of dark matter to explain their slow rotation. In contrast, massive spiral galaxies appear to be dominated by their visible stars. The new framework explains this difference naturally through a specific scaling rule. The theory suggests that the "boost" in gravity is stronger for smaller galaxies and weaker for larger ones. This single mathematical rule accounts for why dwarf galaxies need a significant extra push to spin correctly, while massive spirals rely mostly on their visible mass, eliminating the need for artificial adjustments or hidden cores.

The research also extends beyond individual galaxies to the expansion of the entire universe. The author derived a new way to calculate how the universe has expanded over time, leading to a specific prediction about when the universe switched from slowing down to speeding up. The calculations suggest this transition happened much earlier in cosmic history than the standard model predicts, occurring when the universe was roughly half its current age. When the researcher tested these predictions against recent data from the Dark Energy Spectroscopic Instrument, the results showed a different pattern of cosmic expansion and a slower growth of large-scale structures than currently expected. This difference could help resolve a long-standing tension in cosmology regarding how fast matter clumps together to form galaxies and clusters.

Finally, the study addresses a major challenge for any theory that rejects dark matter: the Bullet Cluster. This is a famous collision between two galaxy clusters where the visible gas slows down due to friction, while the gravitational pull seems to pass right through, staying with the invisible matter. Standard dark matter theory explains this easily, but modified gravity theories have struggled. Ochanda proposes a mechanism called "thermal decoherence" to explain this. The idea is that the intense heat and chaos of the collision disrupt the special relational connections that generate the extra gravity in the gas, causing it to lose its gravitational boost. The stars, which do not experience this thermal disruption, retain their connections and continue to exert the full gravitational pull. This mechanism allows the theory to reproduce the observed separation between the gas and the gravity without invoking invisible particles.

The work presents a fully self-contained mathematical system that replaces the concept of dark matter halos with a discrete, relational structure of space and matter. It claims to match the precision of current models while using fewer adjustable parameters and offering a unified explanation for everything from the spin of a single dwarf galaxy to the expansion history of the cosmos. While the theory requires further scrutiny and confrontation with other observational data, it offers a coherent, alternative vision of how gravity operates on the grandest scales, suggesting that the universe's behavior is dictated by the visible matter we can see, connected through a hidden, step-by-step architecture.

Technical Summary: The Al-Mizan Al-Mutakamil Relational Framework

Problem Statement
The paper addresses two persistent challenges in modern astrophysics: the "Diversity Problem" in galactic rotation curves and the theoretical difficulties of extending Modified Newtonian Dynamics (MOND) to cosmological scales. Standard $\Lambda$ CDM cosmology relies on non-baryonic Cold Dark Matter (CDM) halos to explain flat rotation curves, yet it struggles to reconcile the vast structural diversity of observed galaxies (from low-surface-brightness dwarfs to high-surface-brightness spirals) with universal density profiles like NFW without extensive fine-tuning. Conversely, while MOND and the Radial Acceleration Relation (RAR) successfully link observed acceleration to baryonic mass, they lack a rigorous covariant formulation for galaxy clusters and cosmic expansion. The author proposes that these discrepancies arise not from unseen particles, but from a fundamental misunderstanding of gravitational dynamics as a continuous process, suggesting instead a discrete relational evolution.

Methodology and Theoretical Formulation
The author introduces the Al-Mizan Al-Mutakamil Relational Framework, which posits that gravitational dynamics are governed by discrete relational state transitions across physical scale steps ( $k \in \mathbb{N}$ ) rather than continuous dark matter distributions.

The Mizan Master Equation: The core of the framework is a recursive tensor evolution equation: $\Xi_k = M \cdot [\Xi_{k-1} \otimes R(K_{Global})]$ . Here, $\Xi_k$ is the relational state vector, $M$ is a mapping operator acting on baryonic angular-momentum flux tensors ( $J_i$ ), and $R$ is a scale-enhancement kernel.
Continuum Limit and Kernel Derivation: The abstract non-local state is projected onto physical space to derive an effective centripetal acceleration, $g_{mizan}(r)$ . This is formulated as the Newtonian baryonic acceleration ( $g_{bar}$ ) modified by a non-local kernel:
$g_{mizan}(r) = g_{bar}(r) \cdot \left[ 1 + A(M_{bar}) \left( \frac{r}{k R_c} \right)^{K_{Global}} \right]$
The coupling amplitude $A(M_{bar})$ is parameterized as a power law of the total baryonic mass: $A(M_{bar}) = A_0 (M_{bar}/M^*)^\gamma$ .
Calibration Strategy: The framework is calibrated globally against the SPARC database (175 galaxies, ~3400 radial data points) using a single set of three global scalar constants ( $A_0, \gamma, K_{Global}$ ). No per-galaxy parameters (such as halo concentration or core radius) are floated. The optimization minimizes the mean squared error in logarithmic acceleration space.
Cosmological Extension: The framework extends to cosmology via a "dual-trace" scale factor $a(t)$ in the Al-Musi'a expansion sector. The author performs an analytical rectification of the cosmic acceleration transition time ( $\ddot{a}=0$ ), deriving an exact solution for the transition redshift $z_{acc}$ .
Stress-Energy and Decoherence: A covariant stress-energy tensor is derived from an off-diagonal coherence operator, proving conservation via Bianchi identities. A thermal decoherence mechanism is proposed to explain mass-gas offsets in merging clusters (e.g., the Bullet Cluster).

Key Results

Galactic Kinematics: The model achieves a median galaxy-by-galaxy RMS residual of 0.1157 dex and a global mean residual bias of -0.0079 dex. This performance matches the empirical intrinsic scatter floor of the RAR.
- Mass Regime Stability: The model successfully fits both Low-Surface-Brightness (LSB) dwarfs and High-Surface-Brightness (HSB) spirals with the same global parameters.
- Role of $\gamma$ : The optimized mass-scaling exponent $\gamma = -0.138325$ is negative. This naturally accounts for the elevated mass discrepancies in low-mass dwarfs (where the coupling amplitude increases as mass decreases) without requiring artificial dark matter cores, while allowing massive spirals to remain baryon-dominated in their inner regions.
Cosmological Predictions:
- Acceleration Transition: The rectified algebraic derivation yields a real physical root for the acceleration transition, placing it at $z_{acc} \approx 2.15$ , which the paper describes as occurring "significantly earlier" (at a higher redshift) than the $\Lambda$ CDM prediction ( $z_{acc} \approx 0.6$ ).
- DESI Data Comparison: Numerical evaluation against DESI BAO and RSD data shows enhanced comoving distances and a suppressed peak growth rate of $f\sigma_8 \approx 0.295$ , offering a potential alleviation of the $S_8$ tension.
- Expansion Dynamics: The relational vacuum energy density scales as $\rho_{rel} \propto (1+z)^{-1}$ , generating a phantom-like equation of state that lowers the expansion rate $H(z)$ at intermediate redshifts compared to $\Lambda$ CDM.
Cluster Dynamics: The thermal decoherence mechanism explains the spatial offset between gravitational lensing peaks and X-ray gas in merging clusters by positing that high-temperature plasma severs relational bonds, while collisionless stars retain coherence.

Significance and Claims
The paper claims to provide a unified, rigorous theoretical formulation that eliminates the need for non-baryonic dark matter particles and the fine-tuning of per-galaxy halo parameters. By reducing the degrees of freedom to just three universal scalar constants, the framework achieves empirical precision equivalent to or exceeding halo-based models across four orders of magnitude in baryonic mass.

The author asserts that the framework successfully bridges the gap between galactic phenomenology and cosmological expansion, offering a covariant stress-energy tensor and a mechanism for cluster-scale observations that standard modified gravity theories struggle to provide. The analytical rectification of the acceleration transition time is presented as a correction to previous algebraic errors, yielding a distinct, testable signature ( $z_{acc} \approx 2.15$ ) that differentiates the model from standard cosmology. The work is presented as a complete, self-contained formulation ready for systematic observational confrontation across galactic, cluster, and cosmological scales.

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