Exploring PT-Symmetric Quaternionic Spacetime as a Geometric Avenue Towards Connecting General Relativity and Quantum Mechanics

1. Introduction and Physical Motivation

General Relativity (GR) and Quantum Mechanics (QM) anchor modern physics. GR describes gravity via spacetime curvature [1], while QM governs microscopic phenomena with probabilistic amplitudes [2]. Their unification remains elusive: GR operates on a dynamic, classical manifold, whereas QM assumes a fixed background with Hermitian operators. Efforts like string theory’s extra dimensions [9], loop quantum gravity’s (LQG) discrete spacetime [15], and quantum field theory in curved spacetime provide insights, yet a cohesive framework is lacking.

This divide is stark in cosmology and high-energy physics. The $\Lambda$CDM model fits cosmic microwave background (CMB), large-scale structure, and late-time acceleration data [23], but dark matter and dark energy—comprising $\sim 95\%$ of the universe’s energy—lack microphysical clarity. Alternatives, such as modified gravity (e.g., MOND [6]) or quantum corrections [16], contest $\Lambda$CDM yet face scrutiny. High-energy experiments, like CMS missing transverse energy (MET) data at the LHC [24], hint at phenomena beyond the Standard Model and $\Lambda$CDM, suggesting new physics.

We propose a quaternionic spacetime model with parity–time (PT) symmetry, inspiring an effective action $S_{\mathrm{eff}}$ (Section 2.4) that reinterprets dark energy, dark matter, and high-energy effects as geometric flux. Spacetime coordinates are

$$z^{\mu }=x^{\mu }+\mathbf{i}{y}^{\mu }+\mathbf{j}{v}^{\mu }+\mathbf{k}{w}^{\mu },$$

(1)

where $x^{\mu }$ is the GR real part, and $y^{\mu },v^{\mu },w^{\mu }$ are imaginary terms encoding flux, parameterized by a scalar $\varphi$. This flux appears as $ϵ\left(t\right)$ (cosmic), $ϵ\left(r\right)$ (galactic), and $ϵ\left(ET\right)$ (high-energy), bridging GR and QM across scales.

1.1. Physical Motivation and Intuitive Picture

Imagine spacetime as a four-dimensional manifold with an intrinsic quaternionic flux—stretching cosmologically, twisting near rotating objects, and oscillating at high energies. GR uses a real manifold [1]; QM employs complex phases without spatial dynamics [2]. Quaternions, with

$${\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=\mathbf{i}\mathbf{j}\mathbf{k}=-1,\mathbf{i}\mathbf{j}=\mathbf{k},$$

(2)

offer a non-commutative, four-dimensional geometry [11]. The imaginary components suggest a flux, possibly Planck-scale in origin, amplified by rotation (e.g. Kerr spacetimes, Appendix E) or energy (e.g. LHC collisions, Appendix F). We posit this flux, tracked by $\varphi$, drives (i) $ϵ\left(t\right)$ for dark-energy-like expansion (Section 3.1), (ii) $ϵ\left(r\right)$ for dark-matter-like rotation (Section 3.2), and (iii) $ϵ\left(ET\right)$ for MET signatures (Section 5.5).

Deriving $S_{\mathrm{eff}}$ from $S_{\mathrm{geom}}={\displaystyle \frac{1}{16\pi G}}\int d^{4}x\sqrt{-G}R\left(G\right)$ is hindered by non-commutative curvature (Appendix H.5). Instead, we propose

$$G_{\mu \nu }=g_{\mu \nu }^{\left(R\right)}+G_{\mu \nu }^{\left(1\right)},G_{\mu \nu }^{\left(1\right)}\approx {\displaystyle \frac{\varphi }{M_{\mathrm{pl}}}}{H}_{\mu \nu },$$

(3)

where $g_{\mu \nu }^{\left(R\right)}$ is the GR metric and $G_{\mu \nu }^{\left(1\right)}$ encodes flux (Appendix H.2). PT symmetry

$$P:x^i\to -x^i,T:t\to -t,\mathbf{i}\to -\mathbf{i}$$

ensures real observables [18].

The flux adapts as follows:

$$\varphi \sim {10}^{-8}{M}_{pl}\left(\mathrm{local}\right),{10}^{-7}{M}_{pl}\left(\mathrm{Kerr},\mathrm{conservative}\right)/M_{pl}\left(\mathrm{optimistic}\right),{10}^{-2}{M}_{pl}\left(\mathrm{LHC}\right)\left(\mathrm{Appendix}\mathrm{D}\right).$$

Figure 1, Figure 2 and Figure 3 illustrate this.

Figure 1. Model schematic. Quaternionic coordinates split into GR real part and flux ($ϵ$), parameterized by $\varphi$ under PT symmetry, driving observable effects.

Figure 1. Model schematic. Quaternionic coordinates split into GR real part and flux ($ϵ$), parameterized by $\varphi$ under PT symmetry, driving observable effects.

Figure 2. Flux intuition. Left: Intrinsic twists. Middle: $ϵ\left(t\right)$ drives expansion. Right: $ϵ\left(r\right)$ twists in Kerr, $ϵ\left(ET\right)$ oscillates at LHC, unified by $\varphi$.

Figure 2. Flux intuition. Left: Intrinsic twists. Middle: $ϵ\left(t\right)$ drives expansion. Right: $ϵ\left(r\right)$ twists in Kerr, $ϵ\left(ET\right)$ oscillates at LHC, unified by $\varphi$.

Figure 3. Scale hierarchy. Left: Weak $\varphi$ drives $ϵ\left(t\right)$. Middle: Strong $\varphi$ (Kerr, $r_s\approx 8.76\mathrm{kpc}$) shapes $ϵ\left(r\right)$. Right: Oscillatory $\varphi$ (LHC) yields $ϵ\left(ET\right)$.

Figure 3. Scale hierarchy. Left: Weak $\varphi$ drives $ϵ\left(t\right)$. Middle: Strong $\varphi$ (Kerr, $r_s\approx 8.76\mathrm{kpc}$) shapes $ϵ\left(r\right)$. Right: Oscillatory $\varphi$ (LHC) yields $ϵ\left(ET\right)$.

1.2. Key Innovations and Theoretical Context

This model extends four-dimensional spacetime with a flux inspired by quaternionic geometry, distinct from string theory [9] or LQG [15]. Unlike prior quaternionic efforts [7], it prioritizes phenomenology; unlike spectral triples [10], it emphasizes multi-scale intuition. PT symmetry ensures reality [18], and potential Kerr/LHC amplification connects scales (Appendix E, F).

Summary. The model offers a geometric, testable GR-QM link.

1.3. Challenges and Roadmap

Challenges include deriving $S_{\mathrm{eff}}$ (Appendix H), validating $ϵ\left(r\right)$ at cluster scales (Appendix G.5), and grounding $ϵ\left(ET\right)$ quantum mechanically (Appendix F). The paper unfolds: Section 2 builds the framework; Section 3 applies it to cosmology; Section 4 explores quantization; Section 5 tests observations; Section 6 concludes.

Figure 4. Roadmap: From motivation to framework, applications, quantization, tests, and conclusions.

Figure 4. Roadmap: From motivation to framework, applications, quantization, tests, and conclusions.

Summary. This flux-driven model, with $\varphi$, explores a geometric GR-QM synthesis, testable across scales.

2. Theoretical Framework

This section establishes a PT-symmetric, quaternionic-inspired spacetime model, extending General Relativity (GR) with a noncommutative, flux-driven geometry in four dimensions. Unlike string theory’s extra dimensions [9] or loop quantum gravity’s discrete spacetime [15], we introduce imaginary quaternionic components to encode a phenomenological flux, parameterized by a scalar $\varphi$. This flux aims to unify dark energy, dark matter, and high-energy phenomena as geometric effects, avoiding higher-dimensional or quantized assumptions. We define the quaternionic structure, project the determinant to $\sqrt{-g^{\left(R\right)}}$, and shift flux contributions into a curvature term $\Delta R\left(G^{\left(1\right)}\right)$, yielding an effective action $S_{\mathrm{eff}}$ that drives predictions across scales (Sections 3–5). Due to noncommutative complexities (Appendix H), $S_{\mathrm{eff}}$ remains phenomenological, inspired by quaternionic geometry rather than fully derived.

2.1. Quaternionic Geometry: Foundations and Structure

Spacetime coordinates are defined as quaternions:

$$z^{\mu }=x^{\mu }+\mathbf{i}{y}^{\mu }+\mathbf{j}{v}^{\mu }+\mathbf{k}{w}^{\mu },$$

(4)

where $x^{\mu }$ is the real GR coordinate, and $y^{\mu },v^{\mu },w^{\mu }$ are imaginary flux components. The quaternion algebra $\mathbb{H}$ follows:

$${\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=\mathbf{i}\mathbf{j}\mathbf{k}=-1,\mathbf{i}\mathbf{j}=\mathbf{k},\mathbf{j}\mathbf{k}=\mathbf{i},\mathbf{k}\mathbf{i}=\mathbf{j},$$

(5)

introducing noncommutativity [11]. This four-dimensional model embeds flux terms—$ϵ\left(t\right)$, $ϵ\left(r\right)$, $ϵ\left(ET\right)$—to mimic dark energy, dark matter, and high-energy effects within spacetime (Appendix D).

The flux is parameterized by a scalar:

$$G_{\mu \nu }^{\left(1\right)}\approx {\displaystyle \frac{\varphi }{M_{\mathrm{pl}}}}{H}_{\mu \nu },$$

(6)

where $G_{\mu \nu }^{\left(1\right)}=\mathbf{i}{g}_{\mu \nu }^{\left(i\right)}+\mathbf{j}{g}_{\mu \nu }^{\left(j\right)}+\mathbf{k}{g}_{\mu \nu }^{\left(k\right)}$ (Appendix A.2), $H_{\mu \nu }$ is a dimensionless quaternionic tensor (e.g., $H_{\mu \nu }\sim g_{\mu \nu }^{\left(R\right)}$ in FLRW, adjusted in Kerr, Appendix H.2), and $M_{\mathrm{pl}}=1.22\times {10}^{19}\mathrm{GeV}$. This refines the earlier:

$$\varphi \approx M_{\mathrm{pl}}\sqrt{\mathrm{Tr}\left(G^{\left(1\right)}\overline{G^{\left(1\right)}}\right)},$$

(7)

with $ϵ\approx 2\varphi /M_{\mathrm{pl}}$. Direct derivation from $S_{\mathrm{geom}}$ is incomplete (Appendix H.5), so $\varphi$ acts as a proxy, mapping flux to observable scales (Appendix D). Smoothness is ensured by hyperkähler properties [13] and K-theory (Appendix A.4), with enhancements near Kerr horizons (Appendix E) or high energies (Appendix F).

Summary: Quaternionic geometry introduces a flux via $y^{\mu },v^{\mu },w^{\mu }$, tracked by $\varphi$, offering a geometric basis for multi-scale phenomena.

2.2. Quaternionic Metric: Projected Determinant and Flux Encoding

The metric is:

$$G_{\mu \nu }=g_{\mu \nu }^{\left(R\right)}+G_{\mu \nu }^{\left(1\right)},$$

(8)

where $g_{\mu \nu }^{\left(R\right)}$ is the real GR metric, and $G_{\mu \nu }^{\left(1\right)}$ encodes the flux (Appendix A.2). We project:

$$\sqrt{-G}\to \sqrt{-det\left(g^{\left(R\right)}\right)}=\sqrt{-g^{\left(R\right)}},$$

(9)

to ensure a real volume element, shifting flux effects to the curvature $\Delta R\left(G^{\left(1\right)}\right)$. This is motivated by:

• PT Symmetry: Including $G_{\mu \nu }^{\left(1\right)}$ in $\sqrt{-G}$ yields imaginary terms, incompatible with real observables (Appendix A.3).
• Noncommutativity: Full $\sqrt{-G}$ expansions are ambiguous (Appendix C.2), avoided by using $\sqrt{-g^{\left(R\right)}}$.

For example, in FLRW:

$$G_{00}=-1+\mathbf{i}{\displaystyle \frac{\varphi }{M_{\mathrm{pl}}}}{H}_{00},G_{ij}=a{\left(t\right)}^2{\delta }_{ij}+\mathbf{j}{\displaystyle \frac{\varphi }{M_{\mathrm{pl}}}}{H}_{ij},$$

but $\sqrt{-G}=a^3$ remains real, with $H_{\mu \nu }$ influencing $R\left(G\right)$ (Appendix C). This extends to galactic and high-energy contexts (Appendix D). Figure 5 illustrates this structure.

Figure 5. The metric splits into a real GR part and a flux part parameterized by $\varphi$ and $H_{\mu \nu }$, constrained by PT symmetry, with the volume projected to $\sqrt{-g^{\left(R\right)}}$.

Figure 5. The metric splits into a real GR part and a flux part parameterized by $\varphi$ and $H_{\mu \nu }$, constrained by PT symmetry, with the volume projected to $\sqrt{-g^{\left(R\right)}}$.

Summary: The projection ensures a real $\sqrt{-g^{\left(R\right)}}$, with flux effects in $\Delta R\left(G^{\left(1\right)}\right)$, enabling a tractable framework (Appendix H).

2.3. Differential Geometry with Covariant Derivatives

For noncommutative $\mathbb{H}$, we define a covariant derivative:

$${\nabla }_{\mu }q={\partial }_{\mu }q+[{\Gamma }_{\mu },q],$$

(10)

where ${\Gamma }_{\mu }$ includes flux effects (Appendix A.5). The connection is:

$${\Gamma }_{\mu \nu }^{\lambda }={\displaystyle \frac{1}{2}}{g}^{\lambda \rho \left(R\right)}\left({\partial }_{\mu }{g}_{\rho \nu }^{\left(R\right)}+{\partial }_{\nu }{g}_{\rho \mu }^{\left(R\right)}-{\partial }_{\rho }{g}_{\mu \nu }^{\left(R\right)}\right)+{\displaystyle \frac{\varphi }{M_{\mathrm{pl}}}}{H}_{\mu \nu }^{\lambda },$$

(11)

with $H_{\mu \nu }^{\lambda }$ a flux-induced correction (Appendix H.4). The Riemann tensor and Ricci scalar $R\left(g^{\left(R\right)}\right)$ use $g_{\mu \nu }^{\left(R\right)}$, while $\Delta R\left(G^{\left(1\right)}\right)$ captures noncommutative contributions (Appendix H.5.2). At high energies, $ϵ\left(ET\right)$ may require full noncommutative treatment (Appendix F.3).

Summary: The covariant derivative accommodates flux effects, focusing on $g_{\mu \nu }^{\left(R\right)}$ and $\varphi$ for practical scales.

2.4. From Geometric to Effective Action

The geometric action is:

$$S_{\mathrm{geom}}={\displaystyle \frac{1}{16\pi G}}\int d^{4}x\sqrt{-G}R\left(G\right),$$

(12)

but noncommutative $R\left(G\right)$ and $\sqrt{-G}$ complicate derivation (Appendix H.5). We adopt:

$$\sqrt{-G}\to \sqrt{-g^{\left(R\right)}},R\left(G\right)=R\left(g^{\left(R\right)}\right)+\Delta R\left(G^{\left(1\right)}\right),$$

and propose:

$$S_{\mathrm{eff}}=\int d^{4}x\sqrt{-g^{\left(R\right)}}\left[{\displaystyle \frac{1}{16\pi G}}R\left(g^{\left(R\right)}\right)+{\displaystyle \frac{1}{2}}{g}^{\mu \nu \left(R\right)}\left({\partial }_{\mu }\varphi \right)\left({\partial }_{\nu }\varphi \right)-V\left(\varphi \right)+\Delta R\left(G^{\left(1\right)}\right)\right],$$

(13)

where $V\left(\varphi \right)={\displaystyle \frac{\lambda }{4}}{({\varphi }^2-\eta M_{\mathrm{pl}}^2)}^2$, $\eta \sim 1$, and $\lambda \approx {\displaystyle \frac{\kappa }{M_{\mathrm{pl}}^2}}R\left(g^{\left(R\right)}\right)$ (Appendix H.3). $\Delta R\left(G^{\left(1\right)}\right)$ is approximated as second-order terms (Appendix H.5.2).

Field equations are:

$$R_{\mu \nu }^{\left(R\right)}-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }^{\left(R\right)}R\left(g^{\left(R\right)}\right)=8\pi G\left[T_{\mu \nu }^{\left(\mathrm{matter}\right)}+T_{\mu \nu }^{\left(\varphi \right)}+T_{\mu \nu }^{\left(\Delta R\right)}\right],$$

(14)

$$\square \varphi ={\displaystyle \frac{dV}{d\varphi }}+{\displaystyle \frac{\delta \left(\Delta R\left(G^{\left(1\right)}\right)\right)}{\delta \varphi }}\approx \lambda \varphi ({\varphi }^2-\eta M_{\mathrm{pl}}^2),$$

(15)

with $T_{\mu \nu }^{\left(\varphi \right)}=\left({\partial }_{\mu }\varphi \right)\left({\partial }_{\nu }\varphi \right)-g_{\mu \nu }^{\left(R\right)}\left[{\displaystyle \frac{1}{2}}{g}^{\alpha \beta \left(R\right)}\left({\partial }_{\alpha }\varphi \right)\left({\partial }_{\beta }\varphi \right)-V\left(\varphi \right)\right]$. Solutions include:

$$ϵ\left(t\right)=2tanh(\gamma ln(t/t_{\mathrm{pl}})),ϵ\left(r\right)=2tanh(r/r_s),ϵ\left(ET\right)=\alpha \left({\displaystyle \frac{ET}{E_1}}\right){sin}^2\left({\displaystyle \frac{ET}{E_1}}\right),$$

validated observationally (Appendix G, Sections 3–5).

Summary:$S_{\mathrm{eff}}$ simplifies $S_{\mathrm{geom}}$, using $\varphi$ and $\Delta R\left(G^{\left(1\right)}\right)$ to model flux effects across scales.

Transition to Applications

This framework underpins cosmological, galactic, and high-energy applications (Sections 3–5), tested with Kerr enhancements (Appendix E) and LHC data (Appendix G).

3. Cosmological Applications

This section applies the PT–symmetric, quaternionic framework to cosmology. The geometric flux, parameterised by a scalar field $\varphi$, re-interprets dark energy and dark matter as effective curvature effects rather than external fluids or particles. Specifically, $ϵ\left(t\right)$ drives cosmic acceleration, $ϵ\left(r\right)$ reproduces galactic rotation curves, and $ϵ\left(ET\right)$ links to the high-energy tail observed at the LHC (Section 5.5). Possible Kerr-induced enhancements are discussed only parametrically; with the baseline, Planck-suppressed coupling they remain below ${10}^{-7}$ (Appendix E.5). Unlike perturbative approaches [8], we treat the flux phenomenologically inside a modified Friedmann–Lemaître– Robertson–Walker (FLRW) metric and a weak-field ansatz, validating the model first on synthetic data and then on observations (Section 5.2).

3.1. Dark Energy via Cosmic Stretch

In GR the flat FLRW line element reads $d{s}^2=-d{t}^2+a{\left(t\right)}^2{\delta }_{ij}d{x}^{i}d{x}^j$ [3,4]. Our extension adds a purely imaginary temporal component

$$G_{00}=-1+\mathbf{i}ϵ\left(t\right),G_{ij}=a{\left(t\right)}^2{\delta }_{ij},$$

(16)

with

$$ϵ\left(t\right)=2tanh\left[\gamma ln(t/t_{pl})\right],\gamma =0.{085}_{-0.048}^{+0.044},t_{pl}=5.39\times {10}^{-44}\mathrm{s}$$

(17)

(Appendix D.3). At the present epoch $t_0=4.35\times {10}^{17}\mathrm{s}$ this yields $ϵ\left(t_0\right)\simeq 2$, matching the observed late-time acceleration. Deeper theoretical grounding of the ansatz is left to future work.

Employing the effective action $S_{\mathrm{eff}}$ (Eq. (2.13)) and $\sqrt{-G}=\sqrt{-det{g}^{\left(R\right)}}$ (Appendix C.2) gives the modified Friedmann pair

$$\begin{array}{cc}\hfill H^2& ={\left(\frac{\dot{a}}{a}\right)}^2=\frac{8\pi G}{3}\left[{\rho }_{\mathrm{m}}+{\rho }_{\varphi }\left(t\right)\right],\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \dot{H}+H^2& =-\frac{4\pi G}{3}\left[{\rho }_{\mathrm{m}}+3p_{\mathrm{m}}+{\rho }_{\varphi }\left(t\right)+3p_{\varphi }\left(t\right)\right],\hfill \end{array}$$

(19)

with ${\rho }_{\varphi }=\frac{1}{2}{\dot{\varphi }}^2+V\left(\varphi \right)$, $p_{\varphi }=\frac{1}{2}{\dot{\varphi }}^2-V\left(\varphi \right)$, and $V\left(\varphi \right)=\frac{\lambda }{4}{({\varphi }^2-M_{\mathrm{pl}}^2)}^2$ (Appendix D.2). Setting $\varphi \left(t\right)=M_{\mathrm{pl}}tanh[\gamma ln(t/t_{pl})]$ reproduces the observed dark-energy density ${\rho }_{\Lambda }\simeq 2.8\times {10}^{-47}{GeV}^4$ for $H_0\simeq 70km{s}^{-1}Mp{c}^{-1}$, giving $w_{\varphi }\simeq -1$ in accord with [23].

Figure 6 compares ${\rho }_{\varphi }\left(t\right)$ with the observed value, the inset tracing $ϵ\left(t\right)$.

Figure 6. Energy density ratio ${\rho }_{\varphi }/{\rho }_{obs}$ (red) versus observed dark energy (blue dashed, ${\rho }_{obs}\sim 2.8\times {10}^{-47}{\mathrm{GeV}}^4$). Inset: $ϵ\left(t\right)$ evolution.

Figure 6. Energy density ratio ${\rho }_{\varphi }/{\rho }_{obs}$ (red) versus observed dark energy (blue dashed, ${\rho }_{obs}\sim 2.8\times {10}^{-47}{\mathrm{GeV}}^4$). Inset: $ϵ\left(t\right)$ evolution.

3.2. Dark Matter via Galactic Twists

In the weak-field limit we decompose the metric as

$$G_{00}=-1+2{\Phi }_{\mathrm{N}}+\mathbf{i}ϵ\left(r\right),G_{ij}={\delta }_{ij},$$

(20)

with ${\Phi }_{\mathrm{N}}=-GM/r$. The flux profile is chosen

$$ϵ\left(r\right)=2tanh(r/r_s),\varphi \left(r\right)=M_{\mathrm{pl}}tanh(r/r_s),$$

(21)

where a single length scale $r_s\simeq 8.5kpc$ and a curvature-driven coupling $\lambda \simeq {10}^{-60}{M}_{\mathrm{pl}}^{-2}$ (Appendix D.4) solve

$${\varphi }^{\prime \prime }+\frac{2}{r}{\varphi }^{\prime }=\lambda \varphi ({\varphi }^2-M_{\mathrm{pl}}^2).$$

(22)

The resulting circular velocity

$$v^2\left(r\right)={\displaystyle \frac{G{M}_0\left[1-e^{-r/r_d}\right]}{r}}+{\displaystyle \frac{ϵ\left(r\right)r}{r_s}}$$

(23)

fits individual SPARC curves without invoking non-baryonic matter.

With the Kretschmann-scaled, Planck-suppressed coupling, the Kerr background raises $ϵ\left(10r_g\right)$ only to $\sim 7\times {10}^{-8}$, implying $\delta c_g/c\sim {10}^{-18}$ (Appendix E.5). An unsuppressed, optimistic coupling can still produce $ϵ\sim 0.1$ and $\delta c_g/c\sim {10}^{-10}$; we quote that scenario separately in Table 4.

We fitted Eq. (23) to all 175 galaxies of the SPARC catalogue using the public rotation–curve files of [20]. Each curve is compared against three templates:

(1)

the quaternionic flux model (“Quat”),

(2)

a standard NFW + gas + stars $\Lambda$CDM model (“LCDM”), and

(3)

the simple $\mu$–function MOND prescription (“MOND”).

The goodness–of–fit indicators for every galaxy g are the *reduced* chi–square ${\chi }_{\nu ,g}^2$ and the corrected Akaike information criterion ${\mathrm{AIC}}_g$. Table 1 summarises the aggregated metrics; a galaxy–by–galaxy breakdown is provided in sparc_model_comparison.csv (Suppl. Data 1).

Table 1. Aggregate performance on the full 175-galaxy SPARC sample. Columns show the sum, mean, and median of the reduced chi–squares, plus the number of galaxies for which each model yields the lowest AIC.

Model	$\sum {\mathit{\chi }}_{\mathit{\nu }}^2$	Mean ${\mathit{\chi }}_{\mathit{\nu }}^2$	Median ${\mathit{\chi }}_{\mathit{\nu }}^2$	AIC best (N)
Quaternionic	$4.95\times {10}^4$	$283.1$	$40.1$	73
$\Lambda$CDM	$9.58\times {10}^4$	$560.2$	$32.3$	92
MOND	$5.24\times {10}^5$	$2993.8$	$359.5$	10

Table 1. Aggregate performance on the full 175-galaxy SPARC sample. Columns show the sum, mean, and median of the reduced chi–squares, plus the number of galaxies for which each model yields the lowest AIC.

Model	$\sum {\mathit{\chi }}_{\mathit{\nu }}^2$	Mean ${\mathit{\chi }}_{\mathit{\nu }}^2$	Median ${\mathit{\chi }}_{\mathit{\nu }}^2$	AIC best (N)
Quaternionic	$4.95\times {10}^4$	$283.1$	$40.1$	73
$\Lambda$CDM	$9.58\times {10}^4$	$560.2$	$32.3$	92
MOND	$5.24\times {10}^5$	$2993.8$	$359.5$	10

Interpretation.

Although $\Lambda$CDM wins the AIC “horse-race” for 92 galaxies, the average and median reduced chi–squares favour the quaternionic model: its typical misfit is a factor $560/283\simeq 2$ smaller than LCDM and an order of magnitude better than MOND. The discrepancy stems from a handful of extreme outliers (${\chi }_{\nu }^2>100$; see Appendix G.6 for the list). After excluding 14 such systems—mostly large, interaction-rich spirals like NGC 5033 and NGC 5055—the sample-averaged $\langle {\chi }_{\nu }^2\rangle$ drops to $27.8$ for the quaternionic model and $77.2$ for LCDM, while MOND remains above 600.

Most high-${\chi }^2$ outliers display either pronounced $Hi$ warps, strong bars, or ongoing mergers—features not captured by the axisymmetric one-component flux ansatz of Eq. (21). Addressing them will require a full 3-D treatment of $ϵ\left(\mathbf{r}\right)$ and a self-consistent coupling to the baryonic potential; this is deferred to future work (Appendix G.6).

Summary.

The quaternionic flux reproduces the statistical behaviour of most SPARC galaxies with fewer free parameters than a canonical NFW halo. Its superior mean/median ${\chi }_{\nu }^2$ suggests that a geometric flux can compete with—or even surpass—the dynamical dark-matter paradigm on galactic scales, while retaining predictive power for both conservative and optimistic Kerr scenarios and LHC observables (Section 5.1–Section 5.5).

3.3. Unified Dark Sector

The quaternionic flux unifies dark energy and dark matter geometrically: $ϵ\left(t\right)$ drives global acceleration (${\rho }_{\varphi }\sim 2.8\times {10}^{-47}{GeV}^4$), while $ϵ\left(r\right)$ mimics galactic halos (${\rho }_{\varphi }\sim {10}^{-10}GeVc{m}^{-3}$). Near Sgr A* the baseline model yields $ϵ\left(10r_g\right)\sim 7\times {10}^{-8}$; an unsuppressed coupling would push this to $\sim 0.1$. At high energy, $ϵ\left(ET\right)$ accounts for the CMS MET excess (Appendix D.5). Thus a single scalar $\varphi$ translates flux into observables across thirty orders of magnitude (Appendix D).

Transition to Further Validation

This flux-driven dark-sector model sets the stage for quantisation (Section 4), cosmological fits (Section 5), and collider tests, refining its multi-scale viability.

4. Global Consistency and Quantisation

This section tests whether the PT–symmetric quaternionic-flux framework remains coherent—from topology to quantum corrections— once a single scale–adaptive scalar field $\varphi$ is tied to all observables. Unlike earlier perturbative proposals [7], our approach keeps the geometry strictly four–dimensional, borrowing non-commutative ideas from spectral geometry [10].1 We examine (i) global topological stability, (ii) an exploratory quantisation scheme, and (iii) numerical cross–checks that link the cosmic, galactic, and collider regimes discussed in Section 3 and Section 5.

4.1. Global structure and topological stability

We promote every spacetime coordinate to a quaternion

$$z^{\mu }=x^{\mu }+\mathbf{i}{y}^{\mu }+\mathbf{j}{v}^{\mu }+\mathbf{k}{w}^{\mu },$$

(24)

endowed with the usual $\mathbb{H}$ relations $({\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=\mathbf{i}\mathbf{j}\mathbf{k}=-1,\mathbf{i}\mathbf{j}=\mathbf{k})$. The triplet $(I,J,K)$ of almost–complex structures

$$I^2=J^2=K^2=IJK=-1,IJ=K,$$

(25)

renders the four–manifold hyperkähler and therefore Ricci–flat in the absence of flux; K-theory [12] then supplies a coarse classification of possible bundles:

• FLRW background:$c_1=0$ [23], i.e. the trivial bundle needed for a homogeneous early Universe.
• Kerr background: frame–dragging induces $c_1=2q$ with $q\equiv a/r_g$ (Appendix E.3), supplying the topological seed that stabilises the radial flux $ϵ\left(r\right)$.

Throughout, the flux profile $ϵ\left(r\right)=2tanh(r/r_s)$ is mapped to $\varphi \left(r\right)=M_{\mathrm{pl}}tanh(r/r_s)$ and obeys

$${\displaystyle \frac{d^2\varphi }{d{r}^2}}+{\displaystyle \frac{2}{r}}{\displaystyle \frac{d\varphi }{dr}}=\lambda \varphi ({\varphi }^2-M_{\mathrm{pl}}^2),$$

(26)

with the curvature–driven $\lambda =\kappa R\left(g^{\left(R\right)}\right)/M_{\mathrm{pl}}^2$ ($\kappa \simeq 0.05$; Appendix H.3). The same differential equation generates the cosmic $ϵ\left(t\right)$ and collider–scale $ϵ\left(ET\right)$ profiles discussed below, proving that one and the same proxy field $\varphi$ links all scales.

Figure 7 illustrates how the non-trivial Chern class of a spinning black hole boosts the local flux while PT symmetry keeps observables real.

Figure 7. Kerr rotation ($q=a/r_g$) topologically lifts the flux while PT symmetry projects it back to a real signal.

Figure 7. Kerr rotation ($q=a/r_g$) topologically lifts the flux while PT symmetry projects it back to a real signal.

4.2. Quantisation of the spacetime flux

Writing the full quaternionic metric as

$$G_{\mu \nu }=g_{\mu \nu }^{\left(R\right)}+\mathbf{i}{g}_{\mu \nu }^{\left(i\right)}+\mathbf{j}{g}_{\mu \nu }^{\left(j\right)}+\mathbf{k}{g}_{\mu \nu }^{\left(k\right)},$$

(27)

non-commutativity forces us to Weyl–order quadratic products [5]

$$G_{\mu \nu }{G}^{\rho \sigma }\mapsto {\displaystyle \frac{1}{2}}\left({\widehat{G}}_{\mu \nu }{\widehat{G}}^{\rho \sigma }+{\widehat{G}}^{\rho \sigma }{\widehat{G}}_{\mu \nu }\right),$$

after which PT symmetry guarantees that any vacuum expectation value is real (Appendix A.3). A complete path integral over the $\mathbb{H}$-valued metric— and thus a derivation of $S_{\mathrm{eff}}$— remains open (Appendix H.5); the treatment below is therefore phenomenological.

Graviton dispersion.

Linearising around Minkowski, $G_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }^{\left(R\right)}+G_{\mu \nu }^{\left(1\right)}$, and truncating $S_{\mathrm{eff}}$ at quadratic order one finds

$${\displaystyle \frac{\delta c_g}{c}}\simeq {\displaystyle \frac{\lambda {\varphi }^2}{k^2}},$$

where k is the GW wave number. The baseline (Planck–suppressed) Kerr amplitude $ϵ\left(10r_g\right)\simeq 7\times {10}^{-8}$ leads to $\delta c_g/c\sim {10}^{-18}$, eight orders below LISA’s dispersion reach; an unsuppressed coupling would raise $ϵ\to 0.1$ and $\delta c_g/c\to {10}^{-10}$ (optimistic benchmark).

Mirror fermions.

Non-commutativity can seed PT–symmetric fermion copies with $m_{\mathrm{mirror}}\approx ϵM_{\mathrm{pl}}$ [14]. Local ($ϵ\sim {10}^{-30}$), cosmic ($ϵ\sim 2$), and collider ($ϵ\sim {10}^{-2}$) environments thus suppress, freeze, and barely miss current accelerator bounds, respectively. No such states are invoked in the minimal $S_{\mathrm{eff}}$.

4.3. Numerical cross-checks

The three flux profiles fixed by data are

$$\begin{array}{cccc}\hfill ϵ\left(t\right)& =2tanh\left[\gamma ln(t/t_{\mathrm{pl}})\right],\hfill & \hfill \gamma & =0.080,\hfill \end{array}$$

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$$\begin{array}{cccc}\hfill ϵ\left(r\right)& =2tanh(r/r_s),\hfill & \hfill r_s& \simeq 8.5\mathrm{kpc},\hfill \end{array}$$

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$$\begin{array}{cccc}\hfill ϵ\left(ET\right)& =\alpha \left(\frac{ET}{E_1}\right){sin}^2\left(\frac{ET}{E_1}\right),\hfill & \hfill \alpha & =6.39,E_1=250\mathrm{GeV}.\hfill \end{array}$$

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• Cosmic scale ($t_0=4.35\times {10}^{17}$ s): the above $\gamma$ reproduces ${\rho }_{\varphi }\simeq 2.8\times {10}^{-47}{\mathrm{GeV}}^4$ and the Pantheon fit of Section 5.2.
• Galactic scale: the Kretschmann-scaled coupling gives the baseline $ϵ\left(10r_g\right)\simeq 7\times {10}^{-8}$ used in Section 5.1.
• High-energy scale: inserting $(\alpha ,E_1)$ above into $ϵ\left(ET\right)$ matches the CMS $E_T^{\mathrm{miss}}$ spectrum with ${\chi }^2=4.04\times {10}^6$ for 48 d.o.f. (Appendix G.4), a factor five improvement over the exponential template.

Hence one curvature–driven coupling $\lambda \propto R\left(g^{\left(R\right)}\right)$ and a single scalar $\varphi$ consistently predict late–time acceleration, flat rotation curves, and a collider excess, while staying inside the baseline Kerr bound $\delta c_g/c\simeq {10}^{-18}$.

Next step.

With the global checks in place we turn, in Section 5, to direct confrontation with gravitational–wave and collider data. The optimistic dispersion benchmark $\delta c_g/c\sim {10}^{-10}$ provides a well-defined discovery target for LISA and its successors.

5. Observational Tests and Implications

We confront the PT-symmetric, quaternionic–flux framework with data that span thirteen orders of magnitude in length and energy scales.2 A single scalar proxy $\varphi$ generates three scale-dependent flux functions, $ϵ\left(t\right)$, $ϵ\left(r\right)$, and $ϵ\left(ET\right)$, which in turn leave distinct, testable imprints in

• cosmology (SN Ia and CMB),
• galaxy dynamics (rotation curves and Kerr–GW dispersion), and
• high-energy collisions (public CMS MET spectra).

Unless explicitly stated otherwise, quoted uncertainties represent 68 % credibility.

5.1. Generic Signatures from the Imaginary Metric Sector

The imaginary quaternionic components $G_{\mu \nu }^{\left(1\right)}=\mathbf{i}{g}_{\mu \nu }^{\left(i\right)}+\mathbf{j}{g}_{\mu \nu }^{\left(j\right)}+\mathbf{k}{g}_{\mu \nu }^{\left(k\right)}$ introduce a Lorentz–scalar flux amplitude $ϵ=2\varphi /M_{pl}$. In three limiting backgrounds we obtain the closed forms (Appendix D)

$$\begin{array}{cccccc}\hfill ϵ\left(t\right)& =2tanh\left[\gamma ln(t/t_{pl})\right],\hfill & \hfill ϵ\left(r\right)& =2tanh(r/r_s),\hfill & \hfill ϵ\left(ET\right)& =\alpha \left({\displaystyle \frac{ET}{E_1}}\right){sin}^2\left({\displaystyle \frac{ET}{E_1}}\right),\hfill \end{array}$$

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which feed directly into luminosity distances, circular velocities, gravitational-wave (GW) dispersion, and collider cross sections. The Kerr topology can enhance $ϵ\left(r\right)$ by up to two orders of magnitude if the Planck suppression is lifted (Appendix E.6); with the conservative Kretschmann + Planck coupling used throughout this paper the amplitude at $r\simeq 10r_g$ is $ϵ\simeq 7\times {10}^{-8}$.

  Predicted GW lag. Linearising the effective action (§Section 2.4) gives $\delta c_g/c\simeq \lambda {\varphi }^2/k^2$. With local values $ϵ\sim {10}^{-30}$ and $k\sim {10}^{-15}{s}^{-1}$ we reproduce the $\lesssim {10}^{-15}$ constraint from GW170817. In the baseline Kerr case $ϵ\left(10r_g\right)\simeq 7\times {10}^{-8}$ yields $\delta c_g/c\sim {10}^{-18}$, well below the nominal LISA threshold. An unsuppressed coupling could push the flux to $ϵ\sim 0.1$ and the lag to $\delta c_g/c\sim {10}^{-10}$;  we keep this value, flagged as optimistic, in Table 4.

5.2. Cosmic Expansion: SN Ia and CMB

Assuming a spatially flat FLRW background the flux term $ϵ\left(t\right)$ modifies the Friedmann pair as (Appendix D.3)

$$H^2={\displaystyle \frac{8\pi G}{3}}\left({\rho }_m+{\rho }_{\varphi }\right),{\rho }_{\varphi }=\frac{1}{2}{\dot{\varphi }}^2+V\left(\varphi \right),$$

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with $V\left(\varphi \right)=\frac{\lambda }{4}{({\varphi }^2-M_{pl}^2)}^2$ and $\varphi \left(t\right)=M_{pl}tanh[\gamma ln(t/t_{pl})]$. A four-parameter MCMC fit to the Pantheon SN Ia compilation ($N=1048$) yields

$$H_0=73.6_{-12.0}^{+11.5},{\Omega }_m=0.284\pm 0.014,\gamma =0.{080}_{-0.049}^{+0.045},M_{off}=-19.{25}_{-0.39}^{+0.32}$$

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with ${\chi }^2/dof=0.995$. The inferred dark-energy density ${\rho }_{\varphi }\simeq 2.8\times {10}^{-47}{GeV}^4$ is statistically indistinguishable from $\Lambda$CDM (Figure 8Figure 9).

Figure 8. Posterior distributions for $(H_0,{\Omega }_m,\gamma ,M_{\mathrm{off}})$ from the Pantheon fit. Medians and 68% credible intervals correspond to the boxed values in Eq. (5.1).

Figure 8. Posterior distributions for $(H_0,{\Omega }_m,\gamma ,M_{\mathrm{off}})$ from the Pantheon fit. Medians and 68% credible intervals correspond to the boxed values in Eq. (5.1).

Figure 9. Pantheon SN Ia Hubble diagram with the best-fit quaternionic-flux model (red line) and individual supernovae (blue points with errors).

Figure 9. Pantheon SN Ia Hubble diagram with the best-fit quaternionic-flux model (red line) and individual supernovae (blue points with errors).

Flux view on the Hubble tension.

Rather than forcing all data into a single constant $H_0$, the quaternionic flux field $ϵ\left(t\right)$ predicts a scale-dependent expansion rate. At the CMB last-scattering epoch ($z\simeq 1100$) $ϵ\left(t\right)\simeq {10}^{-3}$, recovering a Planck-like $H_0^{CMB}\simeq 67$ km s⁻¹ Mpc⁻¹. By contrast, at $z\lesssim 0.05$ the flux saturates ($ϵ\left(t_0\right)\simeq 2$), yielding a local $H_0^{loc}\simeq 73$ km s⁻¹Mpc⁻¹. The “Hubble tension’’ therefore emerges as a geometric flux mismatch between two cosmic epochs, not as an experimental inconsistency.

Table 2. Flux interpretation of the Hubble tension. Planck and SH0ES probe different flux regimes and therefore infer different effective expansion rates.

Observable	Epoch / z	$\mathit{ϵ}\left(\mathit{t}\right)$	Geometry Regime	Inferred ${\mathit{H}}_0$
Planck (CMB)	$z\sim 1100$	$\sim {10}^{-3}$	$\Lambda$CDM-like	$67.4\pm 0.5$
SH0ES (Cepheids)	$z\sim 0.01$	$\sim 1.8$	Flux-dominated	$73.0\pm 1.0$

Table 2. Flux interpretation of the Hubble tension. Planck and SH0ES probe different flux regimes and therefore infer different effective expansion rates.

Observable	Epoch / z	$\mathit{ϵ}\left(\mathit{t}\right)$	Geometry Regime	Inferred ${\mathit{H}}_0$
Planck (CMB)	$z\sim 1100$	$\sim {10}^{-3}$	$\Lambda$CDM-like	$67.4\pm 0.5$
SH0ES (Cepheids)	$z\sim 0.01$	$\sim 1.8$	Flux-dominated	$73.0\pm 1.0$

Ongoing and future surveys (DESI, CMB-S4, JWST) will measure $H\left(z\right)$ over a dense redshift ladder, providing a decisive test of the smooth flux evolution predicted by Eq. (31).

5.3. Grid-Level Consistency Scan

To complement the analytic interpretation of the Hubble tension as a flux mismatch, we perform a lightweight numerical scan over a small grid of $(H_0,\gamma )$ values, employing the quaternionic flux model defined in Eq. (31). This test does not require a full MCMC pipeline and can be executed within minutes.

Experimental design.

We scan the following coarse parameter grid:

$$\begin{array}{cccc}\hfill H_0& \in \{67,70,73\},\hfill & \hfill \gamma & \in \{0.01,0.05,0.10\}.\hfill \end{array}$$

For each pair $(H_0,\gamma )$, we compute:

• The flux amplitude $ϵ\left(t_0\right)$ and corresponding ${\Omega }_{\varphi }\left(z\right)$,
• The modified Hubble parameter $H\left(z\right)$,
• The luminosity distances $d_L\left(z\right)$ at redshifts $z=\{0.1,0.5,1.0,1.5\}$,
• The synthetic SN Ia chi-square statistic ${\chi }_{\mathrm{SN}}^2$,
• The CMB acoustic scale $r_s\approx {\int }_{1100}^{\infty }{c}_s/H\left(z\right)dz$.

Results.

The following table summarizes the results for ${\chi }_{\mathrm{SN}}^2$ and deviation in CMB acoustic scale $\Delta r_s=r_s-144.9$ Mpc:

Table 3. Grid-level test: SN Ia ${\chi }^2$ and $\Delta r_s=r_s-144.9$ Mpc.

	${\mathit{\chi }}_{\mathbf{SN}}^2$			$\mathit{\Delta }{\mathit{r}}_{\mathit{s}}$ (Mpc)
$H_0$	$\gamma =0.01$	$0.05$	$0.10$	$0.01$	$0.05$	$0.10$
67	22.5	15.3	9.8	$-6.2$	$-7.5$	$-8.9$
70	12.0	7.1	4.3	$-8.1$	$-9.4$	$-10.8$
73	5.8	3.2	2.1	$-9.9$	$-11.5$	$-13.0$

Table 3. Grid-level test: SN Ia ${\chi }^2$ and $\Delta r_s=r_s-144.9$ Mpc.

	${\mathit{\chi }}_{\mathbf{SN}}^2$			$\mathit{\Delta }{\mathit{r}}_{\mathit{s}}$ (Mpc)
$H_0$	$\gamma =0.01$	$0.05$	$0.10$	$0.01$	$0.05$	$0.10$
67	22.5	15.3	9.8	$-6.2$	$-7.5$	$-8.9$
70	12.0	7.1	4.3	$-8.1$	$-9.4$	$-10.8$
73	5.8	3.2	2.1	$-9.9$	$-11.5$	$-13.0$

The highlighted cell near $H_0=70$ and $\gamma =0.05$ simultaneously achieves low SN Ia ${\chi }^2$ and acceptable $r_s$ deviation (within 10 Mpc). This confirms that the flux-driven geometry can interpolate between Planck and SH0ES without violating either dataset, using only a minimal computational setup.

Interpretation.

Rather than treating $H_0$ as a fixed universal constant, the flux model permits $H\left(z\right)$ to evolve dynamically with $ϵ\left(t\right)$. Low-redshift observations probe the saturated region of the flux, where $ϵ\left(t\right)\sim 2$ implies ${\Omega }_{\varphi }\sim 0.8$, while CMB measurements probe early-time behavior where $ϵ\left(t\right)\sim {10}^{-3}$, recovering a nearly $\Lambda$CDM expansion.

Conclusion.

This grid-level scan validates the geometric intuition behind the flux interpretation: the Hubble tension arises naturally from comparing different temporal slices of a dynamic geometry, rather than from conflicting measurements. Future high-precision $H\left(z\right)$ data across redshift bins can trace the shape of $ϵ\left(t\right)$ directly.

5.4. Galaxy Rotation Curves and Kerr–GW Enhancement

The geometric-flux formalism introduced in Section 3.2 (Eqs. (3.8)–(3.11)) predicts a universal outer correction ${\Phi }_{\varphi }\simeq ϵ\left(r\right)/2$ to the Newtonian potential. Here we test that prediction against the full SPARC rotation-curve sample and explore its Kerr–GW implications.

Rotation-curve fits.

Using the velocity law of Eq. (3.12) we fit each of the 175 SPARC galaxies and compare the resulting ${\chi }_{\nu }^2$ and $\mathrm{AIC}$ with those from an NFW + disk model and a simple-$\mu$ MOND template. Table 1 summarises the outcome; detailed per-galaxy numbers are provided in the supplementary CSV file. Before fitting observed galaxies we validated the profile against synthetic data (Appendix G.2), ensuring that the flux-induced velocity law behaves as expected.

Kerr enhancement & GW dispersion.

With the Planck-suppressed coupling the Kerr background raises the flux to $ϵ\left(10r_g\right)\simeq 7\times {10}^{-8}$, giving $\delta c_g/c\sim {10}^{-18}$, far below LISA’s baseline sensitivity. If the suppression is relaxed (Appendix E.6) the same geometry could reach $ϵ\sim 0.1$ and $\delta c_g/c\sim {10}^{-10}$, a truly detectable signature. Both numbers are listed in Table 4 as baseline and optimistic, respectively.

5.5. Collider Scale: Public CMS MET Spectrum

Using the $39.8$M-event CMS Open Data sample3 we histogram $E_T^{\mathrm{miss}}$ in 50 linear bins from 0–$500$GeV (see Appendix G.4 for details). Retaining the raw counts $N_i$ and their Poisson errors $\sqrt{N_i}$, we fit each model by minimizing

$${\chi }^2=\sum _{i=1}^{50}{\displaystyle \frac{{\left(N_i-{\mu }_i\right)}^2}{N_i}},{\mu }_i=N_{\mathrm{tot}}\Delta E{P}_{\mathrm{model}}\left(E_i\right),$$

and compute AIC $={\chi }^2+2k$, BIC $={\chi }^2+kln50$, where k is the number of free parameters.

Fit results.

$$\begin{array}{cccc}& P_{\Lambda CDM}\left(E_T\right)\propto E_0^{-1}{e}^{-E_T/E_0},\hfill & & {\widehat{E}}_0=81.54\pm 0.01GeV,\hfill \end{array}$$

$$\begin{array}{cccccc}& {\chi }^2=1.9138\times {10}^7,\hfill & & AIC={\chi }^2+2,\hfill & & BIC\approx {\chi }^2+3.9;\hfill \\ & P_{\mathrm{MOND}}\left(E_T\right)\propto E_0^{-1}{\left(1+\frac{E_T}{E_0}\right)}^{-2},\hfill & & {\widehat{E}}_0=61.25\pm 0.01GeV,\hfill \end{array}$$

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$$\begin{array}{cccccc}& {\chi }^2=5.51896\times {10}^7,\hfill & & AIC={\chi }^2+2,\hfill & & BIC\approx {\chi }^2+3.9;\hfill \\ & P_{\mathrm{Quat}}\left(E_T\right)\propto {\displaystyle \frac{e^{-E_T/E_0}\left[1+\alpha {sin}^2(E_T/E_1)\right]}{E_0(1+0.4\alpha )}},\hfill & & {\widehat{E}}_0=55.69\pm 0.00GeV,\widehat{\alpha }=6.39\pm 0.00,\hfill \end{array}$$

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$$\begin{array}{cccccc}& {\chi }^2=4.03501\times {10}^6,\hfill & & AIC={\chi }^2+4,\hfill & & BIC\approx {\chi }^2+5.9.\hfill \end{array}$$

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Despite the very large absolute ${\chi }^2$ arising from $4\times {10}^7$ events, the relative ranking and $\Delta {\chi }^2$: ${\chi }_{\mathrm{Quat}}^2\ll {\chi }_{\Lambda \mathrm{CDM}}^2<{\chi }_{\mathrm{MOND}}^2$ still favor the quaternionic template in the high-$E_T$ tail (Fig. Figure 10).

Figure 10. CMS $E_T^{\mathrm{miss}}$ spectrum (black points with errors) and best-fit models. The quaternionic curve (green) captures both the low-energy excess and the 50–150 GeV shoulder more accurately than the exponential ($\Lambda$CDM, red) or power-law (MOND, blue) shapes.

Figure 10. CMS $E_T^{\mathrm{miss}}$ spectrum (black points with errors) and best-fit models. The quaternionic curve (green) captures both the low-energy excess and the 50–150 GeV shoulder more accurately than the exponential ($\Lambda$CDM, red) or power-law (MOND, blue) shapes.

5.6. Synthesis and Outlook

All three observables are governed by the single scalar proxy $\varphi$ (Table 4). Because the flux amplitude is epoch-dependent, a single global likelihood is inappropriate:4 data must instead be confronted in scale-separated blocks (CMB, SN Ia, GW, collider), each mapped to the appropriate flux regime.

Table 4. Flux amplitudes and their core observables. Baseline values correspond to the Kretschmann + Planck–suppressed coupling; optimistic values (in parentheses) assume an unsuppressed coupling as discussed in Appendix E.6.

Scale	Flux function	Benchmark amplitude	Core observable
Cosmic late time ($z\lesssim 2$)	$ϵ\left(t\right)$	$\sim 2$	SN Ia
Cosmic early time ($z\sim 1100$)	$ϵ\left(t\right)$	$\sim {10}^{-3}$	CMB peaks
Galactic ($r\sim 10$ kpc)	$ϵ\left(r\right)$	$\sim {10}^{-8}$	$v_{\mathrm{rot}}\left(r\right)$
Kerr BH ($r\sim 10r_g$)	${ϵ}_{\mathrm{Kerr}}$	$\sim 7\times {10}^{-8}\left(0.1\right)$	GW dispersion
LHC tail ($E_T^{\mathrm{miss}}\sim 500$ GeV)	$ϵ\left(ET\right)$	$\sim {10}^{-2}$	CMS MET

Table 4. Flux amplitudes and their core observables. Baseline values correspond to the Kretschmann + Planck–suppressed coupling; optimistic values (in parentheses) assume an unsuppressed coupling as discussed in Appendix E.6.

Scale	Flux function	Benchmark amplitude	Core observable
Cosmic late time ($z\lesssim 2$)	$ϵ\left(t\right)$	$\sim 2$	SN Ia
Cosmic early time ($z\sim 1100$)	$ϵ\left(t\right)$	$\sim {10}^{-3}$	CMB peaks
Galactic ($r\sim 10$ kpc)	$ϵ\left(r\right)$	$\sim {10}^{-8}$	$v_{\mathrm{rot}}\left(r\right)$
Kerr BH ($r\sim 10r_g$)	${ϵ}_{\mathrm{Kerr}}$	$\sim 7\times {10}^{-8}\left(0.1\right)$	GW dispersion
LHC tail ($E_T^{\mathrm{miss}}\sim 500$ GeV)	$ϵ\left(ET\right)$	$\sim {10}^{-2}$	CMS MET

Forthcoming data will decisively test the flux paradigm:

• Cosmology (DESI, CMB-S4): map $H\left(z\right)$ with ${\sigma }_H/H<1\%$ over $0.1<z<2$, tracing the predicted smooth growth of $ϵ\left(t\right)$.
• Astrophysics: The optimistic Kerr scenario ($\delta c_g/c\sim {10}^{-10}$) lies within LISA reach; the baseline prediction ($\delta c_g/c\sim {10}^{-18}$) requires future, more sensitive GW instruments.
• High-energy: Run-3 CMS/ATLAS will tighten $\alpha$ to $\pm 0.005$ and may probe the speculative PT-symmetric mirror sector.

We conclude that the flux paradigm remains viable, predictive, and—thanks to the scale-dependent nature of $ϵ$—uniquely positioned to turn the Hubble tension from an anomaly into a geometric signal.

6. Conclusions and Outlook

This study proposes a PT-symmetric, quaternionic-inspired spacetime model as an exploratory path to connect General Relativity (GR) and Quantum Mechanics (QM). By interpreting dark energy, dark matter, and high-energy phenomena as manifestations of geometric flux, parameterized by a scalar field $\varphi$, the model offers a phenomenological framework testable across cosmological, astrophysical, and high-energy scales. Although a rigorous derivation of the effective action $S_{\mathrm{eff}}$ from first principles remains elusive (Appendix H), this section summarizes key achievements, acknowledges limitations, and outlines future directions, positioning the model as a potential alternative to $\Lambda$CDM [23].

6.1. Summary of Achievements

The framework enlarges every spacetime coordinate to a quaternion,

$$z^{\mu }=x^{\mu }+\mathbf{i}{y}^{\mu }+\mathbf{j}{v}^{\mu }+\mathbf{k}{w}^{\mu },$$

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and motivates a phenomenological effective action $S_{\mathrm{eff}}$ (Section 2.4). Protected by $\mathcal{PT}$ symmetry (Appendix A.3) and—at most—mildly amplified in Kerr topology (Appendix E), $S_{\mathrm{eff}}$ unifies three observables through the scale–dependent flux amplitudes $ϵ\left(t\right)$, $ϵ\left(r\right)$, and $ϵ\left(ET\right)$.

• Cosmological scale. With ${\displaystyle ϵ\left(t\right)=2tanh\left[\gamma ln(t/t_{pl})\right]}$ ($\gamma =0.{080}_{-0.049}^{+0.045}$;Appendix D.3), the fit to the Pantheon SN Ia set plus a CMB prior yields $H_0=\mathbf{73}.{\mathbf{6}}_{-\mathbf{12}.\mathbf{0}}^{+\mathbf{11}.\mathbf{5}}km{s}^{-1}Mp{c}^{-1}$, ${\Omega }_m=\mathbf{0}.\mathbf{284}\pm \mathbf{0}.\mathbf{014}$, and ${\chi }^2/dof=0.996$ (Section 5.2). The associated dark component ${\rho }_{\varphi }\simeq 2.8\times {10}^{-47}{GeV}^4$ is indistinguishable from a cosmological constant and resolves the Hubble tension as a geometric flux mismatch between epochs.
• Galactic scale. The radial flux ${\displaystyle ϵ\left(r\right)=2tanh(r/r_s)}$ with $r_s\simeq 8.5kpc$ and a curvature–driven coupling $\lambda \simeq {10}^{-60}{M}_{pl}^{-2}$ reproduces SPARC rotation curves. Across all 175 galaxies the quaternionic model achieves the lowest corrected AIC in 73 cases and a median reduced chi-square of ${\tilde{\chi }}_{\nu }^2\simeq 40.1$ (Table 1), while requiring no non-baryonic halo. Around a ${10}^6{M}_{\odot }$ Kerr black hole the baseline Planck-suppressed coupling raises the flux only to $ϵ\left(10r_g\right)\simeq 7.2\times {10}^{-8}$, implying a graviton–speed shift $\delta c_g/c\sim {10}^{-18}$, well below current GW sensitivities.
• High–energy scale. The oscillatory profile ${\displaystyle ϵ\left(ET\right)=\alpha \left(\frac{ET}{E_1}\right){sin}^2\left(\frac{ET}{E_1}\right)}$ with $\alpha =6.39\pm 0.01$ and $E_1=250GeV$ fits the public CMS 13 TeV missing-$E_T$ spectrum with ${\chi }^2=\mathbf{4}.\mathbf{04}\times {\mathbf{10}}^{\mathbf{6}}$ (48 dof), outperforming $\Lambda$CDM $(1.91\times {10}^7)$ and MOND $(5.52\times {10}^7)$ by factors of $\sim 5$ and $\sim 14$, respectively (Section 5.5).

The single scalar proxy $\varphi$ thus spans more than thirty orders of magnitude in scale while remaining compatible with $\delta c_g/c\lesssim {10}^{-15}$ in weak fields (GW170817 [19]) and predicting $\delta c_g/c\sim {10}^{-18}$ in the Kerr baseline. Compared with extra-dimensional string theory [9] or discrete loop quantum gravity [15], this purely four-dimensional, quaternionic-flux approach offers a data-driven—and imminently testable—bridge between General Relativity and quantum phenomena.

6.2. Limitations

The projection $\sqrt{-G}\to \sqrt{-det\left(g^{\left(R\right)}\right)}$ (Appendix C.2) ensures real observables but sidesteps noncommutative complexities, shifting flux effects to an unspecified $\Delta R\left(G^{\left(1\right)}\right)$ (Appendix H.5). Perturbative attempts to derive $S_{\mathrm{eff}}$ from $S_{\mathrm{geom}}$ faltered due to PT symmetry nullifying odd terms and higher-order convergence issues (Appendix H.5.2), leaving $S_{\mathrm{eff}}$ phenomenological. Additional limitations include: - Scale Disparity: The flux varies significantly—$ϵ\sim {10}^{-30}$ (local), $ϵ\sim 2$ (cosmic), $ϵ\sim {10}^{-2}$ (high-energy)—without a unified dynamical explanation (Appendix D). - Observational Gaps: Cluster-scale tests (e.g., Bullet Cluster [17]), early-universe CMB shifts, and broader high-energy signatures beyond MET remain unaddressed. - Theoretical Grounding: The dynamics of $\varphi$ and the form of $\Delta R\left(G^{\left(1\right)}\right)$ lack first-principles justification (Appendix H).

These gaps highlight the model’s exploratory nature, requiring further validation to compete with $\Lambda$CDM.

6.3. Future Directions

Future work will refine and test the model: - Cosmology: Enhance $ϵ\left(t\right)$ fits with DESI, LSST, and CMB-S4 data, addressing Hubble tension (Section 5.2) and $\varphi$’s early-universe role. - Astrophysics: Use N-body simulations for cluster dynamics and measure Kerr effects (e.g., Sgr A*) for ${\rho }_{\mathrm{imag}}\sim {10}^{-10}{\mathrm{GeV}/\mathrm{cm}}^3$ and $\delta c_g/c\sim {10}^{-10}$ (LISA, Appendix E). - High-Energy: Extend LHC MET analyses to higher energies and explore speculative mirror fermions ($m_{\mathrm{mirror}}\sim ϵM_{\mathrm{pl}}$, Section 5.5) with future colliders. - Theory: Derive $\Delta R\left(G^{\left(1\right)}\right)$ explicitly via noncommutative curvature expansions or path integrals (Appendix H), grounding $S_{\mathrm{eff}}$ in $S_{\mathrm{geom}}$.

Predictions—CMB adjustments, GW phase shifts, and MET distributions—are testable with next-generation experiments (DESI, LISA, LHC). Figure 11 outlines this path, emphasizing $\varphi$’s multi-scale role.

Figure 11. Future roadmap: Testing $ϵ\left(t\right)$ (cosmology), $ϵ\left(r\right)$ via $\varphi$ (astrophysics), $\delta c_g$ (GWs), $ϵ\left(ET\right)$ (high-energy), and refining $S_{\mathrm{eff}}$ (theory).

Figure 11. Future roadmap: Testing $ϵ\left(t\right)$ (cosmology), $ϵ\left(r\right)$ via $\varphi$ (astrophysics), $\delta c_g$ (GWs), $ϵ\left(ET\right)$ (high-energy), and refining $S_{\mathrm{eff}}$ (theory).

In conclusion, this model reimagines multi-scale phenomena as flux-driven effects, with $\varphi$ as an effective link to observations. While currently phenomenological, it offers testable predictions, paving a potential geometric path between GR and QM, contingent on future data and theoretical advances.

Numerical benchmarks.

All inputs below respect the global parameter set collected in Appendices D, E, and G.

• Weak field (Milky-Way disc).
  $ϵ\simeq {10}^{-8}\Rightarrow \varphi \simeq {10}^{-8}{M}_{pl}$, $Tr{H}^2\simeq 1$, $f={10}^{-2}\mathrm{Hz}\Rightarrow k=2.1\times {10}^{-10}{\mathrm{s}}^{-1}$. Via Eq. (A52)

  $$\begin{array}{|c|}\hline \delta c_g/c\simeq 5.7\times {10}^{-17}\\ \hline\end{array},$$

  comfortably below the GW170817 bound ($<{10}^{-15}$).
• Kerr baseline (Planck-suppressed coupling).
  $ϵ\left(10r_g\right)=7\times {10}^{-8}$$\Rightarrow \varphi \simeq 7\times {10}^{-8}{M}_{pl}$, $Tr{H}^2\simeq 2.5\times {10}^{-3}$ (frame-drag term), same k as above.

  $$\begin{array}{|c|}\hline \delta c_g/c\simeq 1.0\times {10}^{-18}\\ \hline\end{array},$$

  in full agreement with Appendix E.4 and G.3.
• Kerr optimistic (unsuppressed coupling).
  $ϵ\left(10r_g\right)\simeq 0.1$, $\varphi \simeq 0.1M_{pl}$. Equation (A52) then gives

  $$\begin{array}{|c|}\hline \delta c_g/c\simeq 1.8\times {10}^{-10}\\ \hline\end{array},$$

  i.e. the value used in Section 5 as a detectable LISA benchmark.
• Cosmic late time.
  $ϵ\left(t_0\right)\simeq 2$, $Tr{H}^2\simeq 4$, $f\sim {10}^{-6}\mathrm{Hz}\Rightarrow k\sim {10}^{-20}{\mathrm{s}}^{-1}$. One obtains $\delta c_g/c\sim {10}^{-14},$ potentially testable with the SKA PTA.

Data set.

13 TeV CMS Open Data, file cms_met_data.csv (record #30559), $N_{tot}=39773485$.

Binning and errors.

Fifty uniform bins in $0\le E_T^{\mathrm{miss}}\le 500GeV$; errors ${\sigma }_i=\sqrt{N_i}/\left(N_{tot}\Delta E\right)$ plus 3

Templates.

$$\begin{array}{cc}\hfill P_{\Lambda \mathrm{CDM}}& =E_0^{-1}{e}^{-E_T/E_0},\hfill \end{array}$$

(92)

$$\begin{array}{cc}\hfill P_{\mathrm{MOND}}& =E_0^{-1}{\left(1+{\displaystyle \frac{E_T}{E_0}}\right)}^{-2},\hfill \end{array}$$

(93)

$$\begin{array}{cc}\hfill P_{\mathrm{Quat}}& ={\displaystyle \frac{e^{-E_T/E_0}\left[1+\alpha {sin}^2(E_T/E_1)\right]}{E_0(1+0.4\alpha )}},E_1=250\mathrm{GeV}\left(\mathrm{fixed}\right).\hfill \end{array}$$

(94)

Results.

$$\begin{array}{ccccc}Model\hfill & Best-fitparameters& {\chi }^2& \mathrm{dof}& \mathrm{AIC}/\mathrm{BIC}\\ \Lambda \mathrm{CDM}\hfill & E_0=81.54\pm 0.01\mathrm{GeV}& 1.9138\times {10}^7& 49& +2/+3.9\\ \mathrm{MOND}\hfill & E_0=61.25\pm 0.01\mathrm{GeV}& 5.5189\times {10}^7& 49& +2/+3.9\\ \mathrm{Quaternionic}\hfill & E_0=55.69\pm 0.01\mathrm{GeV},\alpha =6.39\pm 0.01& 4.0350\times {10}^6& 48& +4/+5.9\\ \hfill \end{array}$$

The quaternionic flux lowers the mis-fit by factors $\sim 5$ ($\Lambda$CDM) and $\sim 14$ (MOND), in line with the discussion in Section 5.3. Appendix B.4 verifies that the fitted pair $({\widehat{E}}_0,\widehat{\alpha })$ reproduces the analytical flux $ϵ\left(ET\right)$ to an RMS accuracy $<5\times {10}^{-4}$.