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 via primordial fluctuations ($\mathrm{a}\mathrm{t}z~1100,\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}$) [2,6]. The positions and amplitudes of the peaks tightly constrain cosmological parameters [1,5], where the first peak ($\mathcal{l}₁~220-221$) informs spatial curvature and sound horizon. The second and third peaks (${\mathcal{l}}_2\mathrm{a}\mathrm{n}\mathrm{d}{\mathcal{l}}_3$, respectively) traditionally reflect baryon loading and dark matter content [1,4,6,7], with higher peaks examining damping and secondary effects. Planck 2018 legacy measurements has subsequently revealed these features with exquisite precision, yielding ${\mathcal{l}}_1~220.6\pm 0.7;{\mathcal{l}}_2~537\pm 3;\mathcal{l}₃~810\pm 5$, and higher multipoles consistent with a flat, apparent dark-matter-dominated universe [1,4,7]. However, ΛCDM’s reliance on undetected dark sector, coupled with a finely tuned cosmological constant is challenging the model’s validity as a true description of reality [6,8,9]. Moreover, direct detection of dark matter particles remains elusive after decades of searches, prompting exploration of geometric alternatives that eliminate non-baryonic mass [8,10,11,12,13,14,15,16]. Modified gravity theories reproduce galactic dynamics without dark matter but struggle with cluster-scale evidence and CMB peaks, 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 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]. By generating a universal stiffness constant $(P\approx 5.01)$, the framework 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 $\approx 2.45-2.65\mathrm{C}\mathrm{I}\mathrm{T}\mathrm{A}\mathrm{T}\mathrm{I}\mathrm{O}\mathrm{N}\mathrm{O}\mathrm{p}\mathrm{o}26\backslash \mathrm{l}2057\left[9\right]$. This mechanism additionally accounts for cluster-scale anomalies, including a $165$kpc spatial offsets 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, CMB acoustic peaks are reinterpreted as spatial resonances of the manifold’s unit cell, navigated by light through unclamped voids [7,15,16]. Crucially, the proposed model avoids any reliance on plasma oscillations, sound horizon, or recombination timing, providing a purely geometric origin to the CMB.

The following sections provide a detailed geometric derivation of the peaks along with their corresponding heights, attained by the framework. We first define the simplex properties to establish the manifold’s base topological constraints, leading to the derivation of the acoustic peaks, where the positions and heights of the first (${\mathcal{l}}_1$), second (${\mathcal{l}}_2$), third (${\mathcal{l}}_3$), and fourth (${\mathcal{l}}_4$) peaks are calculated. As we show in this work, the fundamental stiffness projections to the 4-simplex cell counts yields theoretical values that closely align with Planck data.

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 ${\mathbb{R}}^n$ [17]. 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]:

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

(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

a. 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 angular of the topological leakage propagate through unclamped regions of the manifold. The fundamental angular scale can be expressed as:

$${\theta }_{*}=\delta \times {\displaystyle \frac{P}{\zeta }}=0.002\times {\displaystyle \frac{5.01}{0.70710678118}}\approx 0.0141715728752\mathrm{rad}$$

(4)

Note that Eq. (4) is mandated by the framework to determine ${\theta }_{*}$. It accounts for leakage ($\delta$), deformation resistance, and 3D projection normalisation. Any other arrangement would alter either the stiffness projection or the dimensional mapping required. Using this ${\theta }_{*}$ value, the primary peak position is derived as follows:

$${\mathcal{l}}_1={\displaystyle \frac{\pi }{{\theta }_{*}}}\approx 221.682708141$$

(5)

Compared to the Planck satellite central data of $220.6\pm 0.7$, this value attained for $\mathcal{l}₁$ appears to be within the high-precision observational window. Notably, the theoretical value of around 221.68 is found to be $+0.491\%$ adrift of the measured data [1,4].

a. Derivation of the 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:

$$M_2=\sqrt{1+P}\approx 2.45153013443$$

(6)

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

$${\mathcal{l}}_2={\mathcal{l}}_1·M_2\approx 543.46183929$$

(7)

Comparing this theoretical ${\mathcal{l}}_2$ value with that attained through Planck satellite observations $\left(537\pm 3\right)$ yields a +1.20% variance [1,2]. While this deviation in ${\mathcal{l}}_2$ is comparatively higher than the variance at ${\mathcal{l}}_1$ (Eq. 5), it still represents a significant alignment with observational data [1,2].

a. Derivation of the Third Peak (ℓ₃): Full Topological Resonance

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

$$M_3=\sqrt{10\times {\displaystyle \frac{4}{3}}}\approx 3.6514837167$$

(8)

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

$${\mathcal{l}}_3={\mathcal{l}}_1·M_3\approx 809.470799051$$

(9)

This theoretical ${\mathcal{l}}_3$ attained using the framework, reveals a variance of just $-0.065\%(\pm 0.617\%$) from the Planck satellite data $\left(810\pm 5\right)$, providing further supporting proof of the model’s predictive potential. Moreover, this theoretical ${\mathcal{l}}_3$ hints at a direct manifestation of the 4-simplex’s internal face-symmetry and its associated topological resonance [1,16,17].

a. Fourth Peak: Volumetric Cell Resonance (ℓ₄)

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 $\sqrt{5\times P}$ factor by ${\mathcal{l}}_1$:

$${\mathcal{l}}_4\approx {\mathcal{l}}_1·\sqrt{5\times P}\approx 1109.52140059$$

(10)

This solution from Eq. (10) results in around +0.86% deviation from Planck data $\left({\mathcal{l}}_4\approx 1100\right)$ [1], strengthening the framework’s capacity to reproduce the CMB features without dark matter parameters required by standard inflationary models [1,4].

4. Theoretical ${\mathcal{l}}_1-{\mathcal{l}}_4$ Predictions vs. Empirical Observations

Table 1 present the consolidated solutions for the calculated CMB acoustic peak positions. Of particular relevance is the multiplier relations column in Table 1, which highlights the link between theoretical values and their emergence via the dimensional deficit, manifold stiffness constant, and symmetry gate within the 4-simplex unit cell [9,15]. Comparison with high-precision observational data reveals a clear alignment with these theoretical solutions [1].

As demonstrated further in Figure 2, the model achieves a statistical convergence with Planck 2018 data, competing with ΛCDM approach which relies on dark sector variables that have eluded direct observation to date [1]. By mapping multipole positions $\left(\mathcal{l}\right)$ to discrete components of the 4-simplex projected through $\zeta$, the framework achieves a multi-scale unification using the same constants derived at the subatomic scale, without relying on external free parameters [9,10,15]. Crucially, the precise alignment of ${\mathcal{l}}_3\left(\approx 809.5;-0.065\mathrm{\%}\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}\right)$ serves as a critical benchmark in this framework. While ΛCDM requires specific non-baryonic matter densities to explain this peak against Silk damping, the 3.998D framework identifies it as the face resonance of a triangular grain composed of 10 faces [3]. This convergence, falling well within observational error margins $\left(~810\pm 5\right)$, suggests that ΛCDM reflects the topology of a 4-simplex unit cell rather than a gravitational response to undetected particles [1,2,5,6].

Central to the 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 same constant, derived from manifold stiffness, has been shown to sufficiently 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. Here, higher-order damping is indicative of phase leakage into the 3.998D bulk via the 0.002 deficit, rather than photon diffusion [1,2,4]. Whereas ΛCDM required six free-parameters and invisible dark sector effects, the proposed model achieves comparable precision via a single dimensional deficit of 0.002 deficit, rather than photon diffusion [5]. While ΛCDM requires six free parameters and undetected dark sector effects, this framework achieves comparable precision using its dimensional deficit , offering offering a more mathematically efficient description of the CMB morphology.

5. CMB Peak Amplitude (${\mathcal{l}}_1$ − ${\mathcal{l}}_4$) Derivation

The CMB power spectrum amplitudes ($D_{\mathcal{l}}$) are proposed to arise from resonant modes within the 4-simplex unit cell projected into the 3.998D superfluid vacuum. While ΛCDM interprets the observed CMB power spectrum as a historical record of a primordial plasma, heavily dependent on the presence of invisible dark matter, we demonstrate below how nearly identical peaks can emerge, naturally, from the geometric resonances of the primitive unite cell used throughout this paper [1,5]. We propose that these amplitudes are governed exclusively by a few framework-specific constants, including the manifold stiffness constant $P$, the topological freedom ${\eta }_{\log }\approx \ln \left(3\right)$, the lensing gain $\mathsf{\Xi }\left(\approx 8.21\right)$, and derived monopole temperature $T_{\mathrm{obs}}\approx 2.7295\mathrm{K}$.

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

The manifold’s resistance to displacement (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. (11). Structurally, it emerges from the five vertices of the 4-simplex ($V_4=5$), modified by the tension required to maintain the dimensional deficit ($\delta$) against the metric compaction ($\mathcal{C}$) [9,15]:

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

(11)

This stiffness value acts as a geometric clamping force, governing the amplitude of any harmonic vibration propagating through the 3.998D bulk.

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

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

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

(12)

While the theoretical value of $\approx 2.7285\mathrm{K}$ reflects a pure-manifold resonance, the $+0.004\mathrm{K}$ offset from measured hints at potential the damping effects not fully captured [5]. Thus, this relationship between idealised geometry and physical observation is analogous to the higher-order corrections required in the framework’s derivation of the fine-structure constant, where the $\delta$ interacts with local mass/energy densities [9]. From here, 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 $\left(\mu {\mathrm{K}}^2\right)$:

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

(13)

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 3.998D 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 is a geometric outcome of stiffness-to-temperature dynamics. Note that the use of the ${10}^4$ factor in Eq. (13) 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 $\left({\mathcal{l}}_1-{\mathcal{l}}_4\right)$ emerging through topological fractions of this primitive 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. 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 cosmic energy distribution, independent of baryonic mass-energy density fluctuations.

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

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. In this framework, the peaks are not interpreted as fluid density fluctuations, but as harmonic resonances of the manifold’s structural components. As the resonant frequency increases, the energy is partitioned according to the discrete count of geometric elements, edges, vertices, faces, and cells, active at each successive multipole. This geometric partitioning results in the characteristic “alternating” signature observed in the power spectrum, where the amplitude of each peak ${\mathcal{l}}_n$ is a direct function of its associated element count $N_n$:

• ${\mathcal{l}}_1$(Edges): $N_1=10$
• ${\mathcal{l}}_2$(Vertices): $N_2=5$
• ${\mathcal{l}}_3$(Faces): $N_3=10$
• ${\mathcal{l}}_4$(Cells): $N_4=5$

By treating these counts as the fundamental multipliers for the power distribution, the proposed model provides a geometric basis for the relative power amplitudes at ${\mathcal{l}}_1-{\mathcal{l}}_4$, eliminating the need for independent baryonic or dark matter parameters [2,4,6].

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

As the resonance frequency increases (moving from ${\mathcal{l}}_1$to ${\mathcal{l}}_4$), the wave resolves deeper internal structures of the unit cell. This resolution incurs 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:

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

(14)

• ${\mathcal{l}}_1$: $W_1=10/10=1.0$
• ${\mathcal{l}}_2$: $W_2=10/15\approx 0.667$
• ${\mathcal{l}}_3$: $W_3=10/25=0.4$
• ${\mathcal{l}}_4$: $W_4=10/30\approx 0.333$

$W_n$ characterises the power attenuation resulting from the energy partition across cumulative topological interfaces resolved at higher resonances. This mechanism ensures that while $N_n$ alternates, the absolute power density scales inversely with the structural complexity of the manifold resolution. As such, the observed suppression of higher-order peaks, specifically the magnitude variance between ${\mathcal{l}}_1$ and ${\mathcal{l}}_2$, naturally emerges as structural necessities of the interface-to-boundary ratio.

a. Volumetric Mapping Correction (${\mathit{\eta }}_{\log }$)

For the fourth peak (${\mathcal{l}}_4$), the resonance occurs within the 3D Cells of the 4-simplex. Unlike the lower-dimensional edges or faces, these cells occupy a volume within the 3.998D manifold. This requires the application of the logarithmic information capacity ratio (${\eta }_{\log }=\mathrm{l}\mathrm{n}\left(3\right)\approx 1.0986$), which accounts for the extra field capacity of a fractional manifold over strict 3D Euclidean space. ${\eta }_{\log }$ is applied only to ${\mathcal{l}}_4$. The calculated amplitudes for the first four peaks are given by:

$$D_n=S_{\mathrm{CMB}}\times \left({\displaystyle \frac{N_n}{P}}\right)\times W_n\times [{\eta }_{\log }{]}^{*}$$

(15)

where $\left[{\eta }_{\log }{]}^{*}\right.$ is applied only to ${\mathcal{l}}_4$ to account for the volumetric crowding effects on this peak. The application of this geometric argument yields the theoretical peak power amplitudes $D_n$ at ${\mathcal{l}}_1$-${\mathcal{l}}_4$ against the observational constraints of the Planck 2018 data set, as summarised in Table 2 below.

The derived amplitudes show a high degree of correlation with observational data [3,4,5]. The use of $N_n$ naturally generates the alternating peak heights, while the $W_n$ accounts for the suppression of higher-order harmonics. The inclusion of the volumetric log-ratio for ${\mathcal{l}}_4$ ensures that the transition to 3D cellular resonance is properly mapped from the 3.998D bulk.

6. Conclusions

This paper has presented the 3.998D manifold framework’s application to the CMB acoustic peaks. By modelling the vacuum as a fractional-dimensional manifold, the CMB power spectrum is interpreted as a topological outcome of the manifold’s primitive unit, rather than an empirical imprint of an early-universe dependent on dark matter and baryonic density parameters. The core achievement in this work includes the derivation of both the positions $\left(\mathcal{l}\right)$ and amplitudes $\left(D_{\mathcal{l}}\right)$ of the first four peaks, using the discrete topological components of a 4-simplex unit cell. We have shown that the positions of these peaks emerge from a global manifold stiffness $\left(P\right)$, dimensional deficit $\left(\delta \right)$, and geometric identities of the manifold. The theoretical position of the fundamental ${\mathcal{l}}_1$ peak $(\approx 221.68)$ has been shown to closely align with observational data to within $+0.49\%$, while the higher order ${\mathcal{l}}_3$ and ${\mathcal{l}}_4$ peaks exhibit the strongest precision match with observations, deviating by just $-0.065\%$ and +0.91%, respectively. A primitive base power amplitude is derived and its distribution across the power spectrum mapped to the structural scaffold of the 4-simplex. By partitioning the baseline energy scale according to the boundaries of the unit cell’s edges $(N=10)$, vertices $(N=5)$, faces $(N=10)$, and cells $(N=5)$, derived peak power amplitudes are revealed to fall firmly within the constrained bounds of Planck 2018 data. Specifically, numerical calculations yield an ${\mathcal{l}}_1$ amplitude of $\approx 5920\hspace{0.17em}\mu {\mathrm{K}}^2$ and successfully replicate the characteristic alternating suppression of the higher-order peaks down to the volumetric cell resonance ${\mathcal{l}}_4$ at $\approx 1083\hspace{0.17em}\mu {\mathrm{K}}^2$. This ability of the framework to predict these ΛCDM features traditionally considered proof of both dark matter and the Big Bang, using just a few framework-derived parameters, fundamentally challenges the necessity of the ΛCDM parameter space. A key strength with this framework is its falsifiability, where same fixed constants (δ, P, ζ and 4-simplex counts) that are consistently applied to derive both the positions and amplitudes of the first four CMB $\mathcal{l}$ and $D_{\mathcal{l}}$ values, have also been shown in prior papers to reproduce galactic rotation curves, cluster collision offsets, and the three generations of particle masses while preserving general relativity and Newtonian gravity in dense regions. As such, any valid mismatch in a new dataset, such as polarisation spectra or higher multipoles from future missions, cannot be dismissed as a mere local adjustment. One would then be required to explain why the framework, nevertheless, succeeds across seemingly independent sectors without parameters. This cross-sector interlocking therefore raises the falsification threshold, where a single-sector failure would require a non-trivial account of the agreements in other areas.