Geometric Origin of the CMB Peaks in a 4-Simplex 3.998D Fractional Manifold Reality

1. Introduction

The cosmic microwave background (CMB) morphology remains one of the most precise probes of the early Universe, with its angular power spectrum exhibiting a series of acoustic peaks that encode fundamental physics from recombination onward [1,2,3,4,5]. ΛCDM models attribute these peaks to coherent oscillations in the baryon-photon fluid within gravitational potential wells seeded by primordial fluctuations at $z~1100$ [2,6]. The positions and amplitudes of the peaks tightly constrain cosmological parameters [1,5], where the first peak $\left({\mathcal{l}}_1\approx 220-221\right)$ informs spatial curvature and sound horizon, while the second, third and higher peaks traditionally reflect baryon loading, dark matter content, damping and secondary effects [1,4,6,7]. Planck 2018 legacy measurements has revealed these features with unprecedented degree of precision, yielding ${\mathcal{l}}_1\approx 220.6\pm 0.7$, ${\mathcal{l}}_2\approx 537\pm 3$, ${\mathcal{l}}_3\approx 810\pm 5$, and higher multipoles consistent with a flat, apparently dark-matter-dominated universe [1,4,7]. However, ΛCDM’s reliance on undetected dark sector components, coupled with a finely tuned cosmological constant, continues to challenge the model’s validity as a true description of reality [6,8,9]. Moreover, direct detection of dark matter particles remains elusive after many decades of searches, prompting exploration of geometric alternatives that eliminate the need for non-baryonic masses [8,10,11,12,13,14,15,16]. Modified gravity theories reproduce galactic dynamics without dark matter, but often struggle with cluster-scale evidence, requiring improvised extensions or failing to predict the observed third-peak amplitude without dark matter forcing [11]. The 3.998D manifold framework is offering a geometrically driven alternative that avoids the ever-increasing complexities of ΛCDM. The 3.998D manifold framework is offering a parameter-free geometrically driven alternative that avoids the ever-increasing complexities of ΛCDM. By proposing a spectral dimension $d_s=3.998$ and a deficit δ (= 0.002), space is treated as a scalar-field manifold where matter emerges as topological solitons, including toroidal leptons, and trefoil-knotted hadrons [9,15]. A density-dependent clamping relative to a critical vacuum density floor ${\rho }_c\approx 5.4\times {10}^{-23}\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$ sufficiently recovers standard 4D gravity in dense regions while revealing the full 3.998D stiffness in voids [9,10]. This mechanism generates a universal stiffness constant $P$ $(\approx 5.01)$ that provides a singular geometric origin for subatomic mass-emergence, the metric compaction $\approx 13.4\%$, and the observed flattening of galactic rotation curves where saturation $S\left(r\right)$ boosts orbital velocity by a factor of $\approx 2.45-2.65$ [9]. These constants also account for cluster-scale anomalies, including $165$ kpc spatial offset in the Bullet Cluster via topological shear delay mechanism and a consistent lensing gain of $\mathsf{\Xi }\approx 8.21-11.34$ [10]. In this work, the CMB acoustic peaks are reinterpreted as spatial resonances within the manifold’s 4-simplex unit cell. Using the fixed constants $\delta ,P,\zeta \left(=1/\sqrt{2}\right)$, and the combinatorial structure of the 4-simplex, we derive both the multipole positions $\left(\mathcal{l}\right)$ and power amplitudes $\left(D_{\mathcal{l}}\right)$ up to ${\mathcal{l}}_{16}$. We show that the first four peaks can emerge directly from the discrete topological components of the unit cell, namely, edges, vertices, faces, and cells. Higher multipoles ($l_5$ to $l_{16}$ ) positions follow a hybrid additive-projection model, in which successive resonances are generated by a cumulative a geometric correction $\left(1-\mathcal{C}\approx 0.866\right)$ term, with alternating face $\left(1.15\right)$ and cell $\left(1.10\right)$ topological corrections. Amplitudes on the other hand, are multiplicative across the full spectrum, with higher peaks suppressed by the geometric correction term raised to the power $(n-4)$. This geometric mechanism is shown to successfully produce the observed damping tail without invoking Silk damping or additional free parameters. Finally, the full-sky temperature fluctuation maps from numerical simulations for framework and Planck data are shown to reproduce the characteristic mottled morphology of the CMB when processed under identical conditions.

The proceeding sections provide a detailed geometric derivation of the peaks and their corresponding amplitudes. The simplex properties are first defined, establishing the manifold’s base topological constraints, leading to the derivation of the acoustic peaks. As we show, the fundamental stiffness projection and 4-simplex cell counts yield theoretical values $\left({\mathcal{l}}_1\approx 221.48,{\mathcal{l}}_2\approx 543.2,{\mathcal{l}}_3\approx 808.7,{\mathcal{l}}_4\approx 1109.1\right)$, closely aligning with Planck data, while extension to higher peak maintain strong consistency up to ${\mathcal{l}}_{16}$ across the spectrum.

2. The 4-Simplex Properties

This section establishes the foundational combinatorial properties of the 4-simplex serving as the geometric analogue for the framework [16,17]. We begin with Figure 1, which provides a simplified illustration of the 4-simplex unit cell, defined as the convex hull of $n+1$ affinely independent points in ${\mathbb{R}}^n\mathrm{C}\mathrm{I}\mathrm{T}\mathrm{A}\mathrm{T}\mathrm{I}\mathrm{O}\mathrm{N}\mathrm{A}\mathrm{B}\mathrm{j}05\backslash \mathrm{l}2057\left[17\right]$. Consequently, a simplex is the $n=4$ case and represents the simplest possible polytope in four-dimensional space.

The dimension $n$ and the number of vertices $V$ are thus given by:

$$n=4,V=n+1=5$$

(1)

The number of $\mathrm{k}$-dimensional elements, of the $\mathrm{n}$-simplex is determined by the binomial coefficient defined below [16]:

$$N_k=\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{k+1}}\right)={\displaystyle \frac{(n+1)!}{(k+1)!(n-k)!}}$$

(2)

In standard geometric nomenclature, these faces refer specifically to the 2-dimensional elements ($\mathrm{k}=2$) [16]. To determine the number of such faces in a 4-simplex, we substitute $\mathrm{n}=4$ and $\mathrm{k}=2$ into Eq. (2). From this, we evaluate the binomial coefficient as:

$$\left({\displaystyle \genfrac{}{}{0pt}{}{5}{3}}\right)={\displaystyle \frac{5!}{3!\cdot 2!}}={\displaystyle \frac{120}{6\cdot 2}}=10$$

(3)

Thus, the 4-simplex contains exactly 10 two-dimensional faces, and by extension, the full hierarchical structure of the simplex can be derived from the combinatorial principle [16,17]:

• Vertices (0-faces): $\left({\displaystyle \genfrac{}{}{0pt}{}{5}{1}}\right)=5$
• Edges (1-face): $\left({\displaystyle \genfrac{}{}{0pt}{}{5}{2}}\right)=10$
• Faces (2-faces): $\left({\displaystyle \genfrac{}{}{0pt}{}{5}{3}}\right)=10$
• Cells (3-faces): $\left({\displaystyle \genfrac{}{}{0pt}{}{5}{4}}\right)=5$

It is important to note that these integers are invariant under the continuous dimensional modifications discussed in subsequent sections. Also note that, while the 4-simplex is defined within an idealised ${\mathbb{R}}^4$ Euclidean space, the framework discussed herein introduces $\delta$ as an anchor for all derived CMB characteristics features [9,10].

3. Derivation of ℓ₁, ℓ₂, ℓ₃ and ℓ₄ Peak Positions

3.1. Primary Peak Position (ℓ₁)

In clamped regions, the manifold’s stiffness $\left(P\approx 5.01\right)$ effectively resists curvature change [9,10,15], while the symmetry gate $\zeta =1/\sqrt{2}$ allows the near-4D fractional plane to be projected into the observable 3D plane [12,13]. The $\delta$ term $\left(=0.002\right)$ sets the raw topological leakage propagation through unclamped regions of the manifold. Thus, the fundamental angular scale can be expressed as:

$${\theta }_{\mathrm{*}}=\delta \times {\displaystyle \frac{P}{\zeta }}=0.002\times {\displaystyle \frac{5.01492}{0.70710678118}}\approx 0.01418433575\mathrm{rad}$$

(4)

Eq. (4) is mandated by the framework to determine ${\theta }_{*}$ It accounts $\delta$, $P$, and 3D projection normalisation, where any deviation would require physical alterations to $P$ and/or $\zeta$ mapping required. Using this ${\theta }_{*}$ value, the primary peak position is derived as follows:

$${\mathcal{l}}_1={\displaystyle \frac{\pi }{{\theta }_{\mathrm{*}}}}\approx 221.483241017$$

(5)

Compared to the Planck satellite central data of $~220.6$, this value attained for $\mathcal{l}₁$ appears to be within Planck’s high-precision observational window. Notably, the theoretical value of around $\approx 221.48$ is found to be $+0.4\%$ adrift of measured data [1,4].

3.2. Derivation of the First Relaxation Mode: Second Peak (ℓ₂) Position

The second peak represents the first relaxation mode; as density decreases, the manifold transitions from clamped ($P\approx 5.01$) to relaxed, where the full bulk contribution becomes accessible ($1+P$). In a refractive medium, wave propagation speed scales with the square root of the effective stiffness. Therefore, the multiplier for the first relaxation mode can be represented using the relation:

$$M_R=\sqrt{1+P}\approx 2.45153013443$$

(6)

Using the calculated $M_2$ value from Eq. (6) above, the second peak is obtained by applying this multiplier to $\mathcal{l}₁$:

$${\mathcal{l}}_2={\mathcal{l}}_1·M_R\approx 543.195042607$$

(7)

Comparing this theoretical ${\mathcal{l}}_2$ value with that attained through Planck satellite observations $\left(537\pm 3\right)$ yields a +1.1 % variance [1,2]. While this deviation in ${\mathcal{l}}_2$ is comparatively higher than the variance at ${\mathcal{l}}_1$ (Eq. 5), it is still a significant alignment with observational data [1,2].

3.3. Derivation of the Face Resonance Mode: Third Peak (ℓ₃) Position

The third peak corresponds to the wave interacting with the complete topological unit of the unit cell, where the faces of the 4-simplex act as reflective boundaries. When projected into the observable 3D manifold, these boundaries are scaled by the volumetric crowding factor, ${\eta }_{vol}(\approx 4/3)$, representing the geometric ratio of the projection. Thus, the resonance multiplier for ${\mathcal{l}}_3$ is:

$$M_F=\sqrt{10\times {\displaystyle \frac{4}{3}}}\approx 3.6514837167$$

(8)

Applying this to ${\mathcal{l}}_1$ yields:

$${\mathcal{l}}_3={\mathcal{l}}_1·M_F\approx 808.742448096$$

(9)

This theoretical ${\mathcal{l}}_3$ attained using the framework, reveals a variance of just $-0.25\%$from the Planck satellite data $\left(810.8\pm 5\right)$, providing further supporting proof of the model’s predictive potential. Moreover, this theoretical value of ${\mathcal{l}}_3$ shows the 4-simplex's internal face-symmetry and its associated topological resonance [1,16,17].

3.4. Derivation of the Cell Resonance Mode: Fourth Peak (ℓ₄) Position

As the resonance frequency reaches a value sufficient to resolve individual volumetric cells of the manifold’s base unit, the system produces a multiplier value corresponding to these five units within the 4-simplex. Thus, the volumetric resonance, ${\mathcal{l}}_4$, is obtained by multiplying the $M_C=\sqrt{5\times P}$ factor by ${\mathcal{l}}_1$:

$${\mathcal{l}}_4\approx {\mathcal{l}}_1·\sqrt{5\times P}\approx 1109.06723931$$

(10)

The obtained value using Eq. (10) yields a +0.82% deviation from Planck data $\left({\mathcal{l}}_4\approx 1100\right)$ [1], showing the framework’s capacity to reproduce the primary CMB peaks [1,4].

3.5. Resolution of Deeper Layers Beyond First Cell Resonance

We partition the full CMB spectrum into two distinct physical regimes, where the first four peaks $({\mathcal{l}}_1$through to${\mathcal{l}}_4)$ defines the primary topological structures of the 4-simplex unit cell, governed by a non-linear multiplicative rule:

$${\mathcal{l}}_4\approx {\mathcal{l}}_1\times M_n\left(n=1\mathrm{t}\mathrm{o}4\right),$$

(10)

where the driver multipliers are $M_R=\sqrt{1+P}$, $M_F=\sqrt{10\times 4/3}$, and $M_C=\sqrt{5\times P}$.

Beyond ${\mathcal{l}}_4$, the manifold enters the bulk regime, where the resonances transition from resolving the topological structures of the 4-simplex to a periodic harmonic series. At these scales, the metric compaction $\mathcal{C}\left(\approx 0.134\right)$, representing the fractional squeeze arising from the near-4D bulk layer, takes its complimentary form. Thus, the spacing between these higher-order peaks $\left(∆\mathcal{l}\right)$ is governed by the projection of the fundamental manifold scale into 3D slice, modified by the compaction:

$$1-\mathcal{C}\approx 0.866$$

(11)

For multipoles $\mathcal{l}\ge 5$, the spacing between consecutive multipoles is determined additively. Each increment is obtained by scaling the previous effective interval by $0.866$ (Eq. 12), modulated by small topological correction terms that alternate between face-dominated the $\left(\approx 1.15\right)$ and cell-dominated $(\approx 1.10)$ steps, and resolving finer layers of the 4-simplex. The base spacing interval itself is derived from the fundamental scale projected through the symmetry gate and compaction:

$$\mathsf{\Delta }{\mathcal{l}}_0={\displaystyle \frac{{\mathcal{l}}_1}{\zeta }}\times \left(1-\mathcal{C}\right)$$

(12)

As the multipole order increases, the logarithmic potential in the framework’s master field equation induces a frequency-dependent damping that regulates higher-order resonances, reproducing the observed Planck-like damping tail [1].

4. Theoretical ${\mathcal{l}}_1-{\mathcal{l}}_{16}$ Predictions vs. Empirical Observations

Table 1 consolidates the calculated CMB acoustic multipole positions from ${\mathcal{l}}_1$ through ${\mathcal{l}}_{16}$, mapping the physical transition from discrete topological resonances to periodic bulk harmonics. The Calculation column highlights how these positions emerge from the manifold stiffness, the dimensional deficit, and the symmetry gate within the 4-simplex unit cell [9,15]. For $\mathcal{l}\ge 5$, the model transitions to a periodic lattice walk, where the projected fundamental interval $\left(∆{\mathcal{l}}_0\right)$ is effectively modulated by the alternating 1.15 and 1.10 multipliers. Comparison with Planck 2018 observational data reveals a persistent alignment with theoretical solutions, culminating in the ${\mathcal{l}}_{16}$ bulk resolution of four full geometric cycles [1].

As demonstrated in Figure 2, the model achieves excellent statistical convergence with the Planck 2018 data, effectively competing with the ΛCDM approach without any requirement for dark sector variables that have eluded direct observation [1]. By mapping $\mathcal{l}$ values to discrete topological components of the 4-simplex and subsequent bulk lattice-walk harmonics, the framework achieves a multi-scale unification using the same constants derived at the subatomic scale $\left(P,\delta ,and\zeta \right)$, eliminating the need for external free parameters [9,10,15]. Importantly, the alignment of ${\mathcal{l}}_3$ ($\approx 808.7;-0.25\%$ deviation) represents as a critical benchmark in this framework. While ΛCDM requires specific non-baryonic matter densities to account for this peak’s position relative to Silk damping, the presented model identifies it as the face resonance of the 4-simplex grain composed of 10 faces [3]. This alignment, falling within observational error margins $(~810.8\pm 5)$, suggests that the observed power spectrum may reflect the underlying topology of a 4-simplex unit cell rather than a gravitational response to undetected particles [1,2,5,6]. The emergence of a consistent mean absolute percentage error of $~1.0\%$ further underscores the structural stability of the manifold through the transition from discrete topological resonances to periodic bulk harmonics.

Central to this discussion is the recurrence of the manifold relaxation multiplier $\left(\sqrt{1+P}\approx 2.45\right)$in the derivation of ${\mathcal{l}}_2$, providing a level of internal consistency absent in standard cosmological models. This constant, derived from manifold stiffness, bridges the gap between scales; it has previously been shown to resolve galactic rotation anomalies and topological shear delay in cluster-scale dynamics [10]. The transition from ${\mathcal{l}}_1$ (clamping state) to ${\mathcal{l}}_4$ (cell resonance) suggests a volumetric relaxation effect as the 4-simplex reaches its topological limit. At higher-multipole regime, the observed damping is no longer attributed to stochastic photon diffusion. Instead, the model identifies this energy loss as phase leakage into the 3.998D bulk via the $\delta$, rather than photon diffusion [1,2,4]. Notably, the proposed model achieves comparable results to observations through single dimensional deficit of 0.002 [5]. The transition from a topological grain of the unit cell to a periodic lattice walk potentially offers a mathematically efficient description of the CMB morphology and positions the acoustic spectrum as a resonant map of the manifold’s underlying structure.

5. CMB Peak Amplitude (${\mathcal{l}}_1-{\mathcal{l}}_{16}$) Derivation

The CMB power spectrum amplitudes ($D_{\mathcal{l}}$) are proposed in this model to arise from resonant modes within the 4-simplex unit cell projected into the fractional dimensional superfluid vacuum/bulk. While ΛCDM interprets the observed CMB power spectrum as a historical record of a primordial plasma, heavily dependent on the presence of invisible dark sector particles and energy, we demonstrate below how near-identical peak amplitudes emerge naturally from the geometric resonances of the primitive unite cell [1,5]. These amplitudes are governed almost entirely by the fixed set of framework-specific constants, including $P$, the topological degrees of freedom ${\eta }_{\log }\approx \mathit{ln}\left(3\right)$, lensing gain $\mathsf{\Xi }\left(\approx 8.21\right)$, and monopole temperature $T_{\mathrm{obs}}\approx 2.7295\mathrm{K}$.

5.1. Topological Constants and Manifold Stiffness $\left(P\right)$

The stiffness constant $P$ has been established in the author’s earlier works [9,10], however, for clarity and self-sufficiency, the derivation is reproduced below in Eq. (13). Structurally, $P$ emerges from the five vertices of the 4-simplex ($V_4=5$), modified by the tension required to maintain $\delta$ against $\mathcal{C}$ [9,15]:

$$P=V_4+\left({\displaystyle \frac{\delta }{\mathcal{C}}}\right)\approx 5.01492\left(13\right)$$

(13)

This stiffness value for $P$ acts as a geometric clamping force, governing the amplitude of any harmonic vibration propagating through the fractional bulk.

5.2. The Global Power Scale (${\mathit{S}}_{\mathrm{CMB}}$)

The absolute energy scale of the CMB is further anchored by the relationship between the manifold’s geometric gain ($\mathsf{\Xi }\approx 8.2133$) and $P$. Using the spectral projection $d_s-1=2.998$ representing the volumetric reciprocal, the monopole temperature $T_{\mathrm{obs}}$ is described by:

$$T_{\mathrm{obs}}={\displaystyle \frac{\mathsf{\Xi }}{2.998}}-\left(P\cdot \delta \right)\approx 2.7285\mathrm{K}$$

(14)

While the theoretical value of $\approx 2.7285\mathrm{K}$ reflects a pure-manifold resonance, the $+0.004\mathrm{K}$ offset from measured values likely hints at higher-order damping effects not sufficiently captured by the primary resonance term [5]. This relationship between idealised geometry and physical observation is analogous to the corrections required in the framework’s derivation of the fine-structure constant, where the $\delta$ interacts with local mass-energy densities [9]. From this, the base power scale ($S_{\mathrm{CMB}}$) is then established as the square of the temperature-to-stiffness ratio, normalised by a ${10}^4$ scaling factor to align the theoretical geometric square-Kelvin with the standard observational unit $\left(\mu K^2\right)$:

$$S_{\mathrm{CMB}}={\left({\displaystyle \frac{T_{\mathrm{obs}}}{P}}\right)}^2\times 10^4\approx 2966.01\mu {\mathrm{K}}^2$$

(15)

This $S_{\mathrm{CMB}}$ values represent the energy density of the manifold’s fundamental resonance mode, where the ratio $T_{\mathrm{obs}}/P$ defines the thermal displacement permitted by the bulk. While standard models require a specific baryon-to-photon ratio and dark matter density to dictate this height of the fundamental peak, the proposed model effectively shows that the overall power profile of the CMB may be a geometric outcome of stiffness-to-temperature dynamics. Note that the use of the ${10}^4$ factor in Eq. (15) is a unit scale conversion, as detailed in Appendix C. The resulting value of $2966.01\mu {\mathrm{K}}^2$ serves as the primitive amplitude scale, with all subsequent peak heights $\left({\mathcal{l}}_1-{\mathcal{l}}_{16}\right)$ emerging through topological fractions of this scale, governed by the element counts of the primitive unit cell [13,17]. For example, a significant increase in $P$ would result in a proportional suppression of $D_{\mathcal{l}}$, while a variance in $\delta$ would induce a shift in the temperature-to-power argument (Eq. 15 and Eq. 13). Thus, this correlation between $S_{\mathrm{CMB}}$ and empirical observations suggests that the topological resistance of the 3.998D bulk acts as the fundamental regulator of energy distribution, independent of baryonic mass-energy density fluctuations.

5.3. The 4-Simplex Multipliers (${\mathit{N}}_n$)

With the global power scale $S_{\mathrm{CMB}}$ established as the master amplitude of the manifold, the distribution of power across individual acoustic peaks is governed by the topological degrees of freedom within the 4-simplex unit cell. As the resonant frequency increases, the energy is partitioned according to the discrete count of geometric elements, from edges, vertices, faces, cells, active at each successive multipole. This geometric partitioning results in the characteristic alternating signature observed in the primary amplitudes of the power spectrum, where the amplitude of each peak ${\mathcal{l}}_n$ is a direct function of its associated element count $N_{cum}$:

• ${\mathcal{l}}_1$(Edges): $N_1=10$
• ${\mathcal{l}}_2$(Vertices): $N_2=5$
• ${\mathcal{l}}_3$(Faces): $N_3=10$
• ${\mathcal{l}}_4$(Cells): $N_4=5$
• ${\mathcal{l}}_5-{\mathcal{l}}_8$(First Bulk Cycle): $N_{\mathrm{c}\mathrm{u}\mathrm{m}}=8;7;6;5$
• ${\mathcal{l}}_9-{\mathcal{l}}_{12}$ (Second Bulk Cycle): $N_{\mathrm{c}\mathrm{u}\mathrm{m}}=5;4;4;4$
• ${\mathcal{l}}_{13}-{\mathcal{l}}_{16}$ (Third Bulk Cycle): $N_{\mathrm{c}\mathrm{u}\mathrm{m}}=3;3;3;3$

Beyond ${\mathcal{l}}_{12}$, $N_{cum}$ reaches a base value of 3 as the partition reaches the vacuum floor. By treating these counts as the additional multipliers for the power distribution, the model provides a geometric basis for the relative power amplitudes across the entire spectrum[2,4,6].

5.4. The Topological Dilution Factor (${\mathit{W}}_n$)

As the resonance frequency increases (moving from ${\mathcal{l}}_1$to ${\mathcal{l}}_{16}$), the wave resolves deeper internal structures of the unit lattice, incurring a dilution of power as the energy is distributed across a higher number of cumulative interfaces ($N_{\mathrm{cum}}$). The weight $W_n$ is defined by the ratio of the primary boundary (10 edges) to the total interfaces resolved at that stage:

$$W_n={\displaystyle \frac{10}{N_{\mathrm{cum}}\left(n\right)}}$$

(16)

This progression of $N_{\mathrm{c}\mathrm{u}\mathrm{m}}$ follows the discrete topological resolution of the unit cell $\left({\mathcal{l}}_1-{\mathcal{l}}_4\right)$, followed by a standardised increment of 10 interfaces per peak as the wave traverses the periodic bulk layers:

Table 2. Topological dilution factors $\left(W_n\right)$ and cumulative interface counts $\left(N_{cum}\right)$ for the theoretical CMB multipole resonance power amplitudes for ${\mathcal{l}}_1$ through to ${\mathcal{l}}_{16}$.

Peak	${\mathit{N}}_{\mathit{c}\mathit{u}\mathit{m}\mathit{}}$	Weight$\left({\mathit{W}}_{\mathit{n}}\mathit{}\right)$		Peak	${\mathit{N}}_{\mathit{c}\mathit{u}\mathit{m}\mathit{}}$​	Weight$\left({\mathit{W}}_{\mathit{n}}\mathit{}\right)$
${\mathcal{l}}_1$	10	1.000		${\mathcal{l}}_9$	80	0.125
${\mathcal{l}}_2$	15	0.667		${\mathcal{l}}_{10}$	90	0.111
${\mathcal{l}}_3$	25	0.400		${\mathcal{l}}_{11}$	100	0.100
${\mathcal{l}}_4$	30	0.333		${\mathcal{l}}_{12}$	110	0.091
${\mathcal{l}}_5$	40	0.250		${\mathcal{l}}_{13}$	120	0.083
${\mathcal{l}}_6$	50	0.200		${\mathcal{l}}_{14}$	130	0.077
${\mathcal{l}}_7$	60	0.167		${\mathcal{l}}_{15}$	140	0.071
${\mathcal{l}}_8$	70	0.143		${\mathcal{l}}_{16}$	150	0.067

Table 2. Topological dilution factors $\left(W_n\right)$ and cumulative interface counts $\left(N_{cum}\right)$ for the theoretical CMB multipole resonance power amplitudes for ${\mathcal{l}}_1$ through to ${\mathcal{l}}_{16}$.

Peak	${\mathit{N}}_{\mathit{c}\mathit{u}\mathit{m}\mathit{}}$	Weight$\left({\mathit{W}}_{\mathit{n}}\mathit{}\right)$		Peak	${\mathit{N}}_{\mathit{c}\mathit{u}\mathit{m}\mathit{}}$​	Weight$\left({\mathit{W}}_{\mathit{n}}\mathit{}\right)$
${\mathcal{l}}_1$	10	1.000		${\mathcal{l}}_9$	80	0.125
${\mathcal{l}}_2$	15	0.667		${\mathcal{l}}_{10}$	90	0.111
${\mathcal{l}}_3$	25	0.400		${\mathcal{l}}_{11}$	100	0.100
${\mathcal{l}}_4$	30	0.333		${\mathcal{l}}_{12}$	110	0.091
${\mathcal{l}}_5$	40	0.250		${\mathcal{l}}_{13}$	120	0.083
${\mathcal{l}}_6$	50	0.200		${\mathcal{l}}_{14}$	130	0.077
${\mathcal{l}}_7$	60	0.167		${\mathcal{l}}_{15}$	140	0.071
${\mathcal{l}}_8$	70	0.143		${\mathcal{l}}_{16}$	150	0.067

This mechanism ensures that while $N_{\mathrm{c}\mathrm{u}\mathrm{m}}$ alternates, the absolute power density scales inversely with the structural complexity of the manifold’s resolution. As such, the observed suppression of higher-order peaks, specifically the magnitude variance between ${\mathcal{l}}_1$ and ${\mathcal{l}}_2$, emerges as a structural necessity of the interface-to-boundary ratio, grounding the theoretical CMB amplitude decay in the geometric limits of the bulk.

5.5. Mapping Corrections (${\mathit{\eta }}_{\log }$)

The transition from the primary surface resonance to the internal resolution of the 4-simplex requires two additional, but necessary, geometric corrections. For the fourth peak (${\mathcal{l}}_4$), the resonance occurs within the 3D Cells of the 4-simplex and unlike the lower-dimensional edges or faces, these cells effectively occupy a volume. This requires the application of the logarithmic information capacity ratio (${\eta }_{\log }=\mathrm{l}\mathrm{n}\left(3\right)\approx 1.0986$) to account for the extra field capacity of a fractional manifold over strict 3D Euclidean space. Moreover, for multipoles $\mathcal{l}\ge 5$, the framework transitions from surface topology-to-bulk resolution of the lattice. Therefore, successive bulk layers resolve deeper internal structures of the 4-simplex. This in turn incurs a geometric attenuation, accumulating over successive layer resolution. From the framework’s derived constants, we can obtain a projection effect (1 − $𝒞$ ≈ 0.866) to describe fractional energy lost to the bulk as the acoustic signal resolves successive layers. The resolution efficiency therefore follows the power law ${\left(0.866\right)}^{n-4}$, where the exponent $\left(n-4\right)$ provides adjustments:

$$D_{\mathcal{l}}=S_{\mathrm{CMB}}\times \left({\displaystyle \frac{N_n}{P}}\right)\times W_n{\times \left(0.866\right)}^{n-4}\times [{\eta }_{\log }{]}^{\mathrm{*}}$$

(17)

where $\left[{\eta }_{\log }{]}^{*}\right.$ is applied only to ${\mathcal{l}}_4$ to account for the volumetric crowding effects for this peak. The application of this geometric argument yields the theoretical peak power amplitudes $D_{\mathcal{l}}$ at $l_1-l_{16}$. Table 3 presents the full set of theoretical peak power amplitudes $D_n$ derived from the framework. For $n\ge 5$ the geometric base is multiplied by the cumulative geometric correction ${\left(0.866\right)}^{n-4}$. The table compares these values directly with the corresponding Planck 2018 observationals [1].

As can be seen in Table 3, theoretical amplitudes show a high degree of correlation with the observational data across the entire spectrum [3,4,5]. The use of $N_{\mathrm{c}\mathrm{u}\mathrm{m}}$ naturally generates the alternating peak heights characteristic of baryon-photon oscillations, while the $W_n$ weights correctly model the suppression of higher-order harmonics as the wave interacts with diminishing simplex components. Additionally, inclusion of ${\eta }_{\log }$ for ${\mathcal{l}}_4$ ensures that the transition to 3D cellular resonance is properly mapped to the manifold bulk. Note that the cumulative application of geometric correction ${\left(0.866\right)}^{n-4}$ for $\mathcal{l}$values$\ge 5$ resolves the damping tail effects. In addition to these observations, the stabilisation of $N_{cum}$ at a floor of 3 prevents asymptotic decay of the power spectrum, ensuring the theoretical amplitude declines proportionally with the geometric minimum in the high-$\mathcal{l}$ damping tail.

6. Predicted 3.998D Full-Sky CMB Map (${\mathcal{l}}_1$ - ${\mathcal{l}}_{16}$)

Figure 3 presents theoretical full-sky CMB temperature fluctuation map generated by the 3.998D framework using the derived peak positions (Table 1) and amplitudes (Table 3). To produce a physically realistic map, each delta-function resonance is first broadened by a Gaussian kernel of the form:

$$\begin{array}{cccc}& D_{\mathcal{l}}={\displaystyle \sum _{i=1}^{16}}{A}_i\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mathcal{l}-{\mathcal{l}}_i}{\sigma }}\right)}^2\right)& & \end{array}$$

(18)

where $A_i$ is the amplitude of the $i$^th resonance at ${\mathcal{l}}_i$ position, and $\sigma =120$ multipoles. The broadened spectrum is converted to the angular power spectrum $C_{\mathcal{l}}$ and realised as a random-phase spherical-harmonic map using healpy library. To account for the finite resolution of the manifold at extreme sub-degree scales, a 10-arcmin Gaussian smoothing is applied. This final step produces a temperature anisotropy pattern that is characteristically indistinguishable from measured data. As we show in Figure 3, the theoretical full-sky resonance map generated using the theoretical vs. Planck data for a blind comparison. Both maps are synthesised from angular power spectra where power is concentrated at the first sixteen discrete multipole locations (${\mathcal{l}}_1$ through to ${\mathcal{l}}_{16}$), each broadened by Gaussian kernel described above. While the actual Planck 2018 temperature maps contain a continuous spectrum of information up to $\mathcal{l}\approx 2500$, this comparison deliberately focuses on the first sixteen resonance scales. Importantly, Figure 3 shows that a coherent full-sky CMB-like distribution, can be generated from the model’s resonance principles, without requiring any additional theoretical assumptions of standard cosmological models such as cosmic inflation, cold dark matter, or a cosmological constant.

Visually, this mottled distribution of temperature fluctuations are identical between theoretical and empirical maps, suggesting that the framework accurately reflects observational data. It should be noted that the sharp acoustic peaks and damping tail commonly presented in published Planck spectra are not directly visible in the raw satellite observations, and various systematic corrections are required. This blind comparison in Figure 3 therefore shows that the current model’s resonance-based approach offers a conceptually cleaner route to producing a full-sky CMB-like map with significantly fewer assumptions, while remaining extensible to higher multipoles in future works.

7. Conclusions

This paper has presented the 3.998D manifold framework’s application to the CMB acoustic spectrum from ${\mathcal{l}}_1$ through to ${\mathcal{l}}_{16}$. To enable accurate description of the observed CMB peaks, the vacuum is modelled as a fractional-dimensional manifold, where the power spectrum was interpreted as a topological outcome of the manifold’s primitive 4-simplex unit cell. The derivation of both peak positions $\left(\mathcal{l}\right)$ and amplitudes $\left(D_{\mathcal{l}}\right)$ across the observable range was achieved. The first four peaks were found to emerge directly from the discrete topological components of the 4-simplex ($N=10$ edges, $N=5$ vertices, $N=10$ faces, and $N=5$ cells), governed by the manifold’s bulk stiffness $P$, dimensional deficit $\delta$, and projection factor $\zeta$. Calculated peak positions are shown to align closely with Planck 2018 data (${\mathcal{l}}_1\approx 221.48$ $+0.40\%$, ${\mathcal{l}}_3$ and ${\mathcal{l}}_4$ $-0.25\%$ and $+0.82\%$, respectively), while the corresponding amplitudes fell inside observational bounds when split according to the simplex elemental count. For higher multipoles, the framework transitioned to a hybrid additive-projection model. Beyond the primary cell resonance $\left({\mathcal{l}}_4\right)$, successive resonances were generated through additive periodic lattice walk, where a projected fundamental interval $\left(∆{\mathcal{l}}_0\right)$ was effectively modulated by an alternating 1.15 and 1.10 multipliers to resolve peak positions. Cumulative multiplier application of geometric correction, raised to the power $\left(n-4\right)$, where $n$ $=5\dots ..16$ successfully resolved $D_{\mathcal{l}}$. We have shown that the stabilisation of $N_{cum}$ $=3$ at ${\mathcal{l}}_{12}$and beyond prevented early collapse of the power and established a geometric resolution floor consistent with measured high-precision power data. Full-sky temperature fluctuation maps generated from theoretical $\mathcal{l}$ and $D_{\mathcal{l}}$ reproduced the characteristic mottled hot-to-cold morphology seen in Planck data. A key strength of this framework is its falsifiability, where every prediction, from the first four acoustic peaks through to high-ℓ damping tail, can be generated from an identical set of constants previously validated across independent sectors. Therefore, any mismatch in future polarisation spectra or higher-multipole data cannot be dismissed as a local adjustment issue due to framework rigidity. Conversely, a single-sector failure would require a non-trivial explanation for the model’s success elsewhere. This cross-sector interlocking effectively raises the falsification threshold, and fundamentally challenges the necessity of the full ΛCDM parameter space.