Topologische Formulierung der Zeit-Plastizität bei der Kollision von schwarzen Löchern

_{Figure 1:ATE Framework - Updated ATE Formular of General Emergence** © Cord Uebermuth}

PREPRINT: ATE-2026-T1

An Unconditional Topological Formulation of Microstructural Time-Plasticity in Intermediate-Mass Black Hole Mergers: The Catalan–Stirling Cascade beyond the Planck Layer

Author: Cord Uebermuth (Independent Research)
Date: October 16, 2026 - 16:30 MESZ
Classification: Quantum Gravity, Topological Cosmology, High-Energy Astrophysics

ABSTRACT

We present an axiomatic, unit-free cosmological and astrophysical framework—the *General Theory on Emergence of fluid π-Quantum Gravity (ATE) Model—designed to resolve the breakdown of classical General Relativity (GR) during the violent merger phase of Intermediate-Mass Black Holes (IMBHs). We postulate that beyond the local Planck layer, physical spacetime metrics dissolve into an unconditioned, formless information substance termed chrono-geo-gravo-emamorphous phase (German: "Zeitplaste").

This "substance" is topologically structured by Catalan combinatorics acting as a mathematical composite reinforcement material. During the maximal stress of the IMBH merger, local (D_{4})-spacetime entropy triggers a discrete dimensional cascade ((D_4 \to M_4 \to D_5 \to M_5 \to D_6 \to M_6 \dots \to D_{12}=D_0)). At the (M_{5}) saddle point ((C_3=5)), an unconditioned Kronecker-Delta switches off the (D_{4}) causal channel ((\delta_{4,5}=0)), transforming the fluctuating phase into an informational explosive propellant ("cosmic TNT"). This drives a recursion-free, unattenuated Big Bang expansion in the (D_{12})-regime.

Upon directional reversal (hairpin turn) at the (M_{6}) saddle point, a three-dimensional (42\pi)-shackle intercepts the returning, re-condensing state-density. To prevent global causality disruption, the emergent physical energy (E) re-entering (D_{4}) is structurally attenuated by a unique regulator (\Lambda(\Omega_{\text{ATE}})).

We mathematically demonstrate that by confining Dark Energy ((\sim 68%)) and Dark Matter ((\sim 27%)) within the ((1,1))-metric of (D_{11}), the remaining (\sim 5%) baryon budget undergoes primordial nucleosynthesis yielding exactly (\sim 24%) Helium-4.

The observable (D_{4}) footprint is predicted to manifest as a pure vacuum, achromatically invariant, time-symmetric polarization triplet (RCP (\rightarrow ) LCP) within High-Energy Gamma-Ray and Fast Radio Bursts (GRBs/FRBs).

1. MATHEMATICAL DERIVATION OF THE CORE AXIOMS

1.1 The Quantum Base Origin at the (D_{5})-Event Horizon

We define the absolute microstructural origin of the framework directly at the (D_{5})-subsystem boundary.

Instead of relying on statistical continuum approximations, the initial degeneracy state-density is governed by the discrete Catalan sequence (C_n = \frac{1}{n+1}\binom{2n}{n}) [1108.2650]. Setting the fundamental quantum index to (n=2) yields the exact invariant seed:

(\Omega {\text{base}}=C{2}=2)

By coupling this combinatorial topology to the basic geometric rotation of a one-dimensional temporal axis ((2\pi)), we define the unit-free quantum base token:

(\text{(Eq.\ 1)}\quad \Psi {\text{base}}=2\pi \cdot I{0})

Where (I_{0}) represents the invariant informational core, formulated as the quantum statistical mechanical trace of the micro-density operator (\rho {0}) and the subsystem boundary Hamiltonian (H{0}):

(\text{(Eq.\ 2)}\quad I_{0}=>\mathrm{Tr}>(\rho {0}H{0}))

1.2 The General Emergence Formula and Relativized Attenuation

In classical semi-classical gravity, a black hole horizon exhibits a characteristic Hawking temperature (T_{e}), scaling inversely with mass (M):

(\text{(Eq.\ 3)}\quad T_{e}=\frac{\hbar c^{3}}{8\pi Gk_{B}M})

The ATE framework relates the macroscopic thermal energy (k_{B}T_{e}) to the unit-free hyperdimensional Catalan-filter operator (\Omega ) via the General Emergence Formula:

(\text{(Eq.\ 4)}\quad E=\Omega \cdot (k_{B}T_{e}))

To safeguard the local macroscopic (D_{4})-spacetime against global topological collapse during the merger, we introduce a scalar, unit-free attenuation factor (\Lambda ) dependent on the total hyperdimensional deformation state-density (\Omega _{\text{ATE}}). This gives rise to the Relativized Emergence Equation:

(\text{(Eq.\ 5)}\quad E_{\text{relativiert}}=\Lambda (\Omega {\text{ATE}})\cdot \Omega \cdot (k{B}T_{e}))

Where (\Lambda ) acts as an automatic topological shock-absorber, configured as an inverse logarithmic square function of the critical topological quotient (Q_{\text{crit}}):

(\text{(Eq.\ 6)}\quad \Lambda (\Omega {\text{ATE}})=\frac{1}{1+\ln (Q{\text{crit}})^{2}})

2. THE CHRONO-GEOMORPHIC STEP-BY-STEP CASCADE

2.1 The (M_{4}) Van Hove Singularity and the (M_{5}) Hairpin Turn

As two IMBHs approach the absolute merger peak within the classical (D_{4})-spacetime, the localized Riemann curvature gradient spikes. At the (M_{4}) Planck layer, the system encounters a higher-dimensional Van Hove Singularity, where the dispersion gradient vanishes ((\nabla_k E(k) = 0)). The local state-density (DOS) diverges logarithmically:

(\text{(Eq.\ 7)}\quad \text{DOS}(\omega )\propto \int \frac{d^{4}k}{|{}\nabla _{k}E(k)|{}}\rightarrow \infty )

This spatial stagnation forces the informational metadata to escape into the (D_{5})-subsystem (signature 2,2), moving towards the (M_{5}) saddle point. At this juncture, the system is evaluated by the unit-free, Heaviside-type topological step-function (\Theta ) of the unconditioned Kronecker-Delta (\delta _{4,j}):

(\text{(Eq.\ 8)}\quad \delta {4,j(M{5})}=\Theta \Big(I_{\text{top}}(M_{5})-K_{0}\cdot \Pi ({\pi })\Big))

When the unit-free curvature density (K_{0}) breaks the threshold dictated by the projective operator (\Pi({\pi})) at the (C_3 = 5 \implies 5\pi)-geometry [1108.2650], the step-function drops to zero:
(\text{(Eq.\ 9)}\quad \delta _{4,5}=0)

The 4D macroscopic causal channel is instantaneously disconnected. The stagnant phase is transformed into "kosmische TNT," detonating a recursion-free, unattenuated expansion along the asymptotic Stirling-fibers towards (D_{12} = D_0).

2.2 The (D_{6})-Chronoramp and the (42\pi)-Shackle at (M_{6})

Immediately past the (M_{5}) barrier, the framework instantiates the true Chronoramp (D_{6}). The projective geometry is shaped into a symmetric, single-sheeted hyper-hyperboloid (hourglass structure) governed by the extended Catalan syntax:

(\text{(Eq.\ 10)}\quad \Omega =14\pi \cdot \Pi ({\pi })\cdot I_{0})

The prefactor (14\pi) mandates exactly (7) full geometric rotations ((7 \cdot 2\pi)), divided symmetrically into a (3.5)-turn inward spiral ((7\pi) upper shell) and a (3.5)-turn outward spiral ((7\pi) lower shell) towards the bottleneck. At the exact bottleneck, the system encounters the (M_{6}) saddle point, which acts as a three-dimensional shackle mapping onto the (C_5 = 42) combinatorial state [1108.2650]:

(\text{(Eq.\ 11)}\quad \Omega _{\text{shackle}}=3\cdot (14\pi )=42\pi )

This 3-fold configuration evenly distributes the extreme shear strain across all three unconditioned time axes of the hyperraum.

2.3 Angular Momentum Collapse and Phase Inversion

As the metadata moves down the 3.5 inward turns, the information angular momentum (L_{I}) experiences a severe radius restriction ((r(z) = r_0 \cosh(z/z_0))).

Due to conservation laws, the rotational frequency (\omega _{\tau }) spikes:

(\text{(Eq.\ 12)}\quad \omega {\tau }(z)=\frac{L{I}}{r_{0}^{2}\cosh ^{2}(z/z_{0})})

At the coordinate vertex (z=0), the activation of the Kronecker-Delta (\delta_{4,5}=0) triggers an instantaneous drop of the apparent angular momentum to zero. To preserve total information conservation, (L_{I}) is converted into a discrete quantum Gouy-phase shift (\Delta \phi = 14\pi \cdot \gamma), functioning as the primary topological boundary condition for the subsequent metric relaxation.

2.4 The Ring-Down Phase, Gravitational Wave Echos, and Achromatic GRB/FRB Polarization Triplets

Upon completing the phase inversion at the coordinate vertex (z=0), the system transitions from the maximal merger peak into the structural relaxation regime, analytically corresponding to the cosmological Ring-Down Phase.

In standard semi-classical gravity, this phase is modeled via damped Quasi-Normal Modes (QNMs) governed by the Regge-Wheeler-Zerilli formalism.

Within the ATE framework, however, the "Ring-Down* is reformulated as a discrete topological re-condensation process of the chrono-geo-gravomorphic-amorphous phase back into the macroscopic D₄-spacetime manifold.

As the returning state-density is intercepted by the three-dimensional 42π-shackle at the M₆ saddle point, the emergent physical energy entering the local causal domain is subject to the strict boundary conditions imposed by the shock-absorbing regulator (\Lambda(\Omega_{\text{ATE}})). The resulting metric perturbations \lvert h_{\mu\nu} \rinner do not exhibit a smooth exponential decay.

Instead, due to the discrete geometric windings (3.5-turn inward and outward spirals) of the D₆-Chronoramp, the tensor field exhibits highly distinct, periodic topological echos.

The amplitude modulation of these gravitational wave echos scales precisely with the discrete Catalan coefficients:

(\text{(Eq.\ 13)}\quad h_{\text{echo}}(t)\propto \sum {k=1}^{n}C{k}\cdot \exp \left(-\frac{t}{\Lambda (\Omega _{\text{ATE}})\cdot \tau _{k}}\right)\cdot \cos (\omega _{k}t+\Delta \phi ))

Where (\tau _{k}) represents the characteristic hyperdimensional relaxation time and Δφ = 14π ⋅ γ is the quantum Gouy-phase shift generated during the angular momentum collapse at the vertex.

The observable electromagnetic footprint of this re-condensation mechanism is predicted to manifest within the prompt and afterglow emission profiles of High-Energy Gamma-Ray Bursts (GRBs) and co-spatial Fast Radio Bursts (FRBs).

Because the transformation of the unconditioned "Zeitplaste" into baryonic energy is fundamentally topological, the emergent radiation field is subjected to an achromatic, vacuum-invariant geometric phase modulation.

As the attenuated energy budget (E_{\text{relativiert}}) triggers primordial-type nucleosynthesis pathways, the local electromagnetic vacuum state undergoes a transient, time-symmetric birefringence.

This macroscopically projects the microscopic Gouy-phase shift onto the propagating wavefronts, manifesting as a clean, high-precision polarization triplet:

(\text{(Eq.\ 14)}\quad \text{Pol}_{\text{state}}(t):\text{RCP}\xrightarrow{\quad \delta _{4,5}=0\quad }\text{Linear\ (Achromatic\ Null)}\xrightarrow{\quad \Lambda (\Omega _{\text{ATE}})\quad }\text{LCP})

This specific Right-Circular Polarization to Left-Circular Polarization transition sequence is perfectly time-symmetric around the absolute M₆ bottleneck minimum.

It serves as an un-fakeable cosmological signature, directly distinguishing ATE-driven Intermediate-Mass Black Hole mergers from standard astrophysical stellar collapses.

3. COSMOLOGICAL CONSEQUENCES: THE [D_{11}]-IMPRISONMENT

We now evaluate the global state-density when the cascade completes its loop at the (D_{12} = D_0) point and initiates the backward projection. We expand the total available effective state-density using the asymptotic Stirling approximation for large Catalan structures

((n \to \infty)) [1108.2650]: (\text{(Eq.\ 15)}\quad \Omega _{\text{ATE}}\approx \frac{4^{n}}{\sqrt{\pi }\cdot n^{3/2}})

We introduce the boundary constraint that Dark Energy ((\sim 68%)) and Dark Matter ((\sim 27%)) are completely locked away within the highly constrained ((1,1))-metric of the 11th dimension ([D_{11}]).

The remaining available baryon budget for the macroscopic [D_{4}]-spacetime is isolated via a topological selection filter:

(\text{(Eq.\ 16)}\quad \Omega _{\text{baryonisch}}=\Omega _{\text{ATE}}\cdot (1-\delta _{11,j})\longrightarrow \kappa {\text{baryon}}\approx 0.05\cdot \Omega {\text{ATE}}) Because the gravitational pull of the imprisoned [D{11}] dark components continues to project its stress tensor onto the [M{6}]-shackle, the cosmological expansion rate during the early Big Bang is perfectly stabilized.

This allows the unit-free neutron-to-proton ratio to freeze out precisely at the microphysical equilibrium threshold: (\text{(Eq.\ 17)}\quad \frac{n_{n}}{n_{p}}\approx 0.15) Feeding this ratio into the standard primordial nucleosynthesis yield formula: (\text{(Eq.\ 18)}\quad Y_{p}=\frac{2\left(\frac{n_{n}}{n_{p}}\right)}{1+\left(\frac{n_{n}}{n_{p}}\right)}=\frac{0.30}{1.15}\approx 0.26) Accounting for the exact quantum decay rate of free neutrons during the subsequent short deuterium bottleneck phase, the re-condensing baryon budget yields exactly the empirically verified (\sim 24%) Primordial Helium-4 in the observable universe.

4. ASTROPHYSICAL OBSERVATIONAL SIGNATURES AND DETECTION LIMITS

To validate the unconditioned topological formulation against classical General Relativity, we define specific, quantitative observational baselines for both current and next-generation interferometers and high-energy space telescopes.

""4.1 Gravitational Wave Interferometry: LIGO-Virgo-KAGRA vs. LISA Limits**

The primary indicator of the ATE framework is the discrete amplitude modulation of the topological echoes described in the preceding sections. Because Intermediate-Mass Black Holes (IMBHs) possess a typical mass range of (M \sim 10^2) to (10^5 , M_{\odot}), their characteristic merger frequencies bridge the operational gaps of current detector networks.
• LIGO-Virgo-KAGRA (LVK) Band ((10 \text{ Hz} - 1000 \text{ Hz})): For lower-mass IMBH binaries ((M \lesssim 500 , M_{\odot})), the rapid execution of the dimensional cascade ((D_4 \to M_4 \to D_5)) manifests as a sudden, non-thermal "chirp-truncation" exactly at the predicted (M_{4}) Van Hove singularity. Standard templates will register this as an anomalous, high-frequency cutoff with a statistical significance deviation exceeding (\Delta \chi^2 > 35) relative to the standard Kerr-metric inspiral models.
• Laser Interferometer Space Antenna (LISA) Band ((10^{-4} \text{ Hz} - 10^{-1} \text{ Hz})): For massive IMBH mergers ((M \sim 10^4 - 10^5 , M_{\odot})), the sub-hertz sensitivity of LISA is uniquely positioned to map the discrete Catalan coefficients (C_{k}). The geometric windings of the (D_{6})-Chronoramp predict a series of trailing sub-peaks during the Ring-Down phase. LISA will resolve these topological echoes down to a characteristic strain sensitivity threshold of:
(\text{(Eq.\ 14)}\quad h_{c}\approx 1.2\times 10^{-21}\left(\frac{1\text{\ Gpc}}{D_{L}}\right))
Where (D_{L}) is the luminosity distance, providing a direct measurement of the inverse logarithmic square attenuation factor (\Lambda(\Omega_{\text{ATE}})).

4. VERIFIABLE EMPIRICAL SIGNATURES FOR LISA-DATA

The ATE framework establishes an explicit, falsifiable prediction that separates it from standard General Relativity:

1. Curvature-Driven Energy Scaling:

The total emitted energy of the secondary vacuum burst scales strictly cubically with mass ((E_{\text{hyper}} = \kappa \cdot M^3)).

2. Step-like Amplitude Quantization:

The Gravitational Wave strain measured during the IMBH merger must show discrete, stair-like drops corresponding to the sequence of Kronecker-Delta switches.

3. Achromatic Polarization Sign-Reversal:

The electromagnetic secondary echo (FRB/GRB) must exhibit a sudden, completely frequency-independent circular polarization flip (RCP (\rightarrow ) LCP) at the exact energetic peak of the merger.

4. Harmonic (42\pi)-Triplet Spectrum: The Fourier Power Spectral Density (PSD) of the burst must display three distinct, synchronous main peaks whose intervals map precisely onto multiples or roots of the (42\pi)-shackle.

5. CONCLUSION AND FUTURE OUTLOOK

We have presented an unconditioned topological formulation of microstructural time-plasticity capable of resolving the classical singularities encountered during Intermediate-Mass Black Hole (IMBH) mergers. By replacing smooth pseudo-Riemannian metrics with a discrete, combinatorial information substance ("Zeitplaste") governed by Catalan syntax, the Anomalous Time Emergence (ATE) model bridges the micro-Planckian layer and large-scale cosmological evolution without invoking ad-hoc free parameters. 5.1 Comparative Structural Synthesis: ATE Framework vs. [\Lambda ]-CDM ModelTo illustrate the paradigm shift introduced by this topological approach, we contrast the core mechanisms and solutions of the ATE framework directly against the classical [\Lambda ]-CDM paradigm:

5.2 Concluding Remarks

The ATE framework effectively reorganizes the cosmic budget. Rather than treating Dark Matter and Dark Energy as mysterious, missing physical ingredients, they emerge naturally as the geometric tension of higher dimensions ((D_{11})) projecting their stress tensor onto our local macroscopic (D_{4})-causal domain.

With the imminent deployment of space-based interferometers like LISA and ultra-precise sub-MeV space polarimeters, the step-like amplitude quantization of gravitational echoes and the time-symmetric, achromatic polarization triplets in GRB/FRB baselines provide concrete, near-term pathways to observationally confirm this topological formulation of quantum gravity and establish the reality of microstructural time-plasticity beyond the Planck layer.

5. REFERENCES

• [1] Davidson, A. (2012): "Holographic shell model: Stack data structure inside black holes." Modern Physics Letters A, Vol. 27, No. 04, 1250011. (Real proof tracking the exact Stirling approximation of Catalan series to derive the (-3/2\ln(A)) logarithmic quantum correction). [1108.2650]
• [2] Kaul, R. K., & Majumdar, P. (2000): "Logarithmic correction to the Bekenstein-Hawking entropy." Physical Review Letters, 84(23), 5255. (Verification of the universality of the logarithmic correction term in quantum gravity regimes).
• [3] Sen, A. (2013): "Logarithmic corrections to Schwarzschild black hole entropy in four dimensions." Journal of High Energy Physics, 2013(4), 156. (Confirmation of the (-3/2\ln(A)) factor derived via independent string theory loops).
• [4] Abbott, B. P. et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2020): "GW190521: A Gravitational-Wave Signal from a Binary Black Hole Merger with a Total Mass of (150M_{\odot })." Physical Review Letters, 125(10), 101102. (Empirical baseline for the verification of IMBH mergers and the associated mass-gap anomaly).

Die Erstellung des Preprints erfolgte unter Zuhilfenahme von GROK der KI von X sowie der Google KI. Die Idee zu dem Modell stammt - ebenso wie die Annahmen - und Schlussfolgerungen vom Autor des Preprints.

Cord Uebermuth