Appendix 4
Relic Dimension Gradients
ΛCDM requirements
The modern cosmological model ΛCDM includes a set of key assumptions:
- there exists an additional gravitating component, “dark matter,” invisible in the electromagnetic spectrum;
- its mass is roughly 5–6 times the mass of all baryonic matter;
- it is distributed throughout the Universe, forms large-scale halos around galaxies and clusters, and is also present in the intergalactic medium;
- it is required to explain:
- flat galaxy rotation curves,
- the stability of galaxy clusters and the retention of satellites,
- weak and strong gravitational lensing,
- the growth of perturbations before and after recombination,
- the heights of acoustic peaks in the CMB and the BAO structure,
- the shape of the fluctuation spectrum and the structure growth rates.
Thus, ΛCDM postulates a global presence of dark mass in order to satisfy, simultaneously, the requirements of both early- and late-time cosmology.
What is actually required
Within the framework of discrete Space made of fundamental cells (SER)—carriers of metric charge—the conclusions are different:
- for local dynamics, local variations of metric charge (dimension gradients) within halos and filaments are sufficient;
- a global “dark matter budget” across the Universe is not required—the additional effective mass does not need to be distributed everywhere;
- in early epochs, the role of the “invisible component” is naturally played by the high-dimensional background, where the cells’ metric charge was much larger, reproducing the effects of a “cold” gravitating component;
- as SER degrades dimensionality, local contrasts persist in the regions where galaxies and clusters form, delivering the observed effects without introducing a separate entity.
I. Late Universe: observational effects
In local zones (halos, filaments, clusters), it is sufficient to have local volume fractions of elevated dimensionality—stepwise gradients of effective dimensionality within the cell composition:
- A shift in the mixture by a few percentage points toward higher-dimensional cells creates an equivalent excess of mass. The gradient structure is regulated by dynamic homeostasis and by the geometric “stitching” rules of cells.
- This local excess reproduces:
- flat rotation-curve tails,
- stabilization of satellites and stellar streams,
- the required lensing of light over tens to hundreds of kiloparsecs.
- Substructure arises due to the granularity of the ΔD distribution, but on average the profile remains smooth—consistent with the observed “halo”.
Thus, the gravitational effects are achieved without a global mass add-on—only through local redistribution of charge.
Baseline setup
The mean dimensionality across the Universe is taken as 4.81 (Model dimensional parameters). The minimal global mixture that yields this mean without introducing extra components is: 0.81 five-dimensional cells and 0.19 four-dimensional cells (others are negligible in the mean background).
Scenario
In the observed picture of flat rotation-curve segments, there is a characteristic level of additional acceleration on the order of 10⁻¹⁰ m/s² (reference a_ref = 1.0000×10⁻¹⁰ m/s²). We use this reference as a scale measure of the effect.
A stable positive deviation of cell composition in the halo region over tens of kiloparsecs (hereafter “delta-dimensionality”) must increase the mean charge per cell enough to produce the additional gravity and match the reference level.
Halo shell (the main halo volume).
To reproduce flat rotation-curve tails at radii of 8–10 kpc and the correct order of weak lensing, the following combinations of stepwise dimensional uplift are approximately sufficient:
- Option A (ceiling up to the 10th step, Dmax): 5D — 24–28%, 6D — 8–10%, 7D — 2.0–2.5%, 8D — 0.50–0.70%, 9D — 0.10–0.15%, 10D — ≤0.03%. Total fraction of “uplifted” volume: 35–41%.
- Option B (ceiling up to the 9th step, Dmax = 9): 5D — 30–32%, 6D — 12–14%, 7D — 3.0–3.5%, 8D — 0.8–1.0%, 9D — 0.20–0.30%. Total fraction of “uplifted” volume: 46–51%.
Such a stable increase of charge across the halo volume is sufficient to reproduce flat rotation-curve tails and the correct order of weak lensing at the observed radii. For the outer halo (~30 kpc), a much thinner ladder is sufficient: 5D — 10–14%, 6D — 2–3%, 7D — ≤0.30%, 8D — ≤0.05% (total 12–18%; ceiling D_max = 9–10 is admissible).
Compact core (optional, no more than 0.1 of the halo volume).
For a steeper central profile—which is natural, as this is the region of the galaxy’s primary center of mass—a thin additional admixture of upper dimensional levels is admissible (fractions are relative to the core volume): 6D — 4–5%, 7D — 1.0–1.5%, 8D — 0.3–0.5%, 9D — 0.05–0.10%, 10D — ≤0.02%.
In this case, we obtain a sufficient gradient for a moderately sharper rise of velocities in the center and a moderately enhanced central weak lensing, without changing the behavior on the flat radii. The core is an addition to the shell, not a replacement; the shares of levels 9–10 in the core are intentionally tiny, to remain compatible with the global budget of higher levels and to avoid “overweighting” the background.
Spatial placement
The positive dimensional deviation is sustained in the perigalactic shell and slowly declines toward outer radii; across the halo volume it is stably maintained above the background level (delta-dimensionality ΔDhalo = +0.3…+0.5 at 8–10 kpc).
In large systems, a moderately compact central core comprising no more than 0.1 of the halo volume is admissible, where a localized enhancement of the effect may occur (around the primary center-of-mass region).
This macro-pattern reproduces what is observed: flat rotation-curve tails, weak lensing of the correct order, and “mass” outside the disk / bulge—without introducing additional dark-matter entities.
The deviation structure is granular on scales of a few kiloparsecs, but in aggregate produces a smooth profile. This explains why, on large scales, the halo appears quasi-continuous, while on small scales substructures are present (satellites, tidal streams).
Cross-check / allocation
The total volume of elevated-dimensionality zones is distributed non-uniformly: the large-scale “cosmic web” (walls, filaments, nodes) occupies 18–24% of the Universe, and within it the levels D≥7 account for 3.0–3.8% of the global volume (working center around 3.5%). Of this share, galaxy halos account for about 15–20%, i.e., ~0.45–0.76% of the cosmic volume. The rest is filaments and nodes. This arrangement is far from “percolation” and does not break background homogeneity and isotropy.
On average across the Universe, the mean dimensionality remains 4.81: higher levels are localized in the web and halos, voids remain close to the background (4–5D), and the global share of D≥7 is narrow (3.0–3.8%). This delivers the same observable set of effects attributed to “dark matter” in ΛCDM—flat rotation-curve tails, gravitational binding in clusters, weak and strong lensing—without introducing a separate substance and without rebuilding the overall cosmological picture (background expansion parameters, Planck parameters, and BAO / CMB are preserved).
The deviation structure remains granular on scales of a few kpc and smooth in aggregate: on small scales substructures are visible (satellites, tidal streams), while on large scales the halo is quasi-continuous with a correctly decaying profile.
Robustness with respect to mean dimensionality
The real mean values of cell dimensionality and their mixtures may differ from the model value 4.81 (in the limit—higher, up to a nominal 15.26, or lower, down to 3.0). In that case, the order of the required local deviation to reproduce observed halo effects remains comparable and tends to decrease as the mean dimensionality increases: for higher-dimensional cells, the “charge step” is significantly larger than for lower-dimensional ones. Therefore, at a higher mean dimensionality, the same observed effect is achieved with a smaller relative gradient of the local composition.
The computed numbers are model reference points, but the deviation order is stable:
- it remains approximately the same across a wide range of mean dimensionalities;
- it decreases as the mean dimensionality increases (due to the larger “charge step” at higher d);
- globally, elevated-dimensionality zones D≥7 occupy 3.0–3.8% of the Universe’s volume (working center around 3.5%). This is sufficient for filaments, nodes, and halos while preserving ⟨D⟩ = 4.81 and the “voidness” of the background.
II. Early epoch: CMB, BAO, and perturbation growth
At recombination, the mean dimensionality of Space was high (D ≈ 15). This implies:
- charge per cell was orders of magnitude larger than today;
- the lattice behaved like a cold, effectively pressureless component;
- the mean free path and pressure of such cells were negligible.
It is precisely this high-dimensional background that behaves, in the linear phase of perturbation growth, like ΛCDM “dark matter”: it supports fluctuation development, delivers the required acoustic peak heights in the CMB, and sets the BAO scale.
Thus, in our picture the “dark-matter equivalent” is present in the background then, but it does not need to remain globally present on average today.
III. Evolution from early epochs to the present
The mechanism is naturally determined by the SER rule:
- each generation of cells transitions to lower dimensionality, creating a stepwise cascade;
- background metric charges decrease, but local contrasts persist in higher-density regions (future halos and clusters);
- part of the “dark-matter equivalent” is gradually reallocated from the background into local structures;
- globally, the mean picture remains isotropic and homogeneous, while local zones are exactly what provides the observed late-time effects.
Conclusions
ΛCDM requires postulating a global additional mass across the Universe in order to explain both early- and late-time effects at once.
In the SER concept, these two task classes are solved separately:
- early-time effects are explained by the high metric charge of the background at large dimensionalities;
- late-time effects are explained by local dimensional contrasts in halos and filaments.
Additional mass as a separate entity is not required: the space lattice already carries the required charge and redistributes it over the course of evolution.
Therefore, all observed dark-matter effects (CMB, BAO, structure growth, galaxy dynamics, lensing) are reproduced without introducing “invisible” matter that no one has found. The classical model relies on a stopgap—a global component—whereas the same reality is explained by the intrinsic properties of Space.
The decay (degradation) of SQ cells proceeds continuously, so local inhomogeneities naturally “dissolve” over time—the dimensional (charge) contrast decreases. The mass of matter does not disappear. To preserve the observed connectivity of structures, the system compensates for degradation within dynamic homeostasis via one of the modes (or a mixture of them):
- Scale mode. The elevated-dimensionality region grows in size while contrast falls moderately: connectivity is maintained by a larger coverage area.
- Contrast mode. The region keeps its size but adapts its contrast (driven by the current mass configuration and by inflows of matter/angular momentum).
- Mixed mode. A small expansion of the region is accompanied by a moderate increase of contrast.
In all cases, the same homeostasis rule applies: restructuring proceeds causally (no faster than c) and brings the system to a new quasi-balance (virialization, angular momentum, feedback). Hence, continuous dimensional degradation does not break connectivity—it is compensated by scale growth, contrast reinforcement, or a combination of both, depending on the local mass history and environment.
Relic Dimension Gradients
Appendix 4