The problem in one picture
Measure how fast stars orbit at different distances from a galaxy's centre and you get a rotation curve. Newtonian gravity applied to the matter we can see predicts that the curve should rise and then fall, the way outer planets move more slowly than inner ones. Real spiral galaxies do not do that. Their curves rise and then stay flat far beyond the visible disk.
Vera Rubin and Kent Ford showed this for Andromeda in 1970 (Astrophysical Journal 159, 379), and a decade of measurements made it the rule. Something adds gravity where there is little visible matter.
Answer 1: dark matter halos
The standard answer is invisible matter distributed in a roughly spherical halo around each galaxy. Cosmological simulations predict halo shapes, the best known being the NFW profile of Navarro, Frenk and White (1996). With a halo fitted to each galaxy, rotation curves are reproduced well.
Dark matter also explains things far from galaxies: the pattern of peaks in the cosmic microwave background, the growth of large-scale structure and the Bullet Cluster, where the lensing mass sits apart from the colliding gas. That is why most cosmologists treat it as the default.
Its weak points are at galactic scales. Simulated halos tend to be too dense in the centre (the core–cusp problem), and each galaxy needs its own halo parameters. And after decades of searches, no dark matter particle has been detected in the lab.
Answer 2: MOND
In 1983 Mordehai Milgrom proposed changing the law instead of adding matter. Below an acceleration of about 1.2 × 10⁻¹⁰ m/s², gravity would weaken more slowly than Newton says. With that one constant, MOND predicts many rotation curves from visible matter alone.
McGaugh, Lelli and Schombert (2016) found a tight radial acceleration relation in SPARC galaxies: the observed acceleration is a simple function of the acceleration from visible matter. MOND predicted this. For dark matter it is a coincidence that has to be explained.
MOND's weak points are the reverse of dark matter's: galaxy clusters still need extra mass, and relativistic versions have trouble with the cosmic microwave background.
Answer 3: emergent gravity
Erik Verlinde argued in 2010 that gravity is not fundamental and emerges from how information is stored on surfaces. In 2017 he derived from this an extra gravitational effect on galactic scales that mimics dark matter and reproduces a MOND-like acceleration scale. The model is young, and its tests on real galaxies are mixed.
Where this programme stands
The research programme behind this site does not offer its own rotation-curve model. Its preprint, Pointer Architecture v9.0, works at the level of a formal computational substrate. Its one cosmological statement is a substrate-language reproduction of the standard holographic estimate of dark energy, which concerns the expansion of the universe and says nothing specific about the dynamics of individual galaxies.
Bottom line
Rotation curves can be fitted without dark matter at galactic scales, and a simple law (MOND) predicts them surprisingly well. No alternative yet matches dark matter across clusters and cosmology. The sober position is that the problem is open at the scale of galaxies, and the way forward is models that make falsifiable predictions on public data.