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1. 本文档由 Arktx Inc.（美国科罗拉多州）独家发布；TM-NET拓扑调控核有效理论底层总框架、TRNCE-T1.3标准体系、TX-KET高阶拓扑核扩展框架、Arktx $\boldsymbol{\mathcal{A}}$ 算符公理体系、$C^\infty$ 无穷阶光滑性数学证明、TS-SHE拓扑超重核稳定岛模型，均为原创专属学术成果，受著作权法、国际版权公约及区块链司法存证全程保护。

2. 本文引用的粒子物理标准模型、经典场论、量子场论通用基础公式与公共学术框架，均属于公有领域知识资源，非本公司原创内容，仅用于理论参照、框架嵌入与学术对比验证。

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1. Released exclusively by Arktx Inc. (Colorado, USA). The underlying framework of TM-NET Topology-Modulated Nuclear Effective Theory, TRNCE-T1.3 refined system, TX-KET higher-order topological kernel extension framework, Arktx $\boldsymbol{\mathcal{A}}$ operator axiom system, $C^\infty$ infinite-order smoothness proof and TS-SHE topological superheavy nucleus stability island model are original exclusive academic achievements, protected by copyright law, international copyright conventions and blockchain legal deposit.

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TX-KET Topological Scalar Gravity Model

Neutron Star - Black Hole Merger · Gauss-Bonnet Topological Coupling · Gravitational Wave Observation Theory
Theoretical System: Kerr Spacetime + 3.5PN Post-Newtonian Approximation + TOV Stellar Structure
Copyright: © 2026 Arktx Inc. All Rights Reserved
Legal Deposit No.: CO-2026-0528-001
Blockchain Hash: 0x7a8f9d2c4e6b1a3d5f7c9b2a4e6d8f1a3c5e7b9d2f4a6c8e0b1d3f5a7c9e2b4d

1. Model Overview

TX-KET is a scalar-tensor gravity model extended based on the Gauss-Bonnet topological invariant. It is constructed for strong gravitational field environments including black holes and neutron stars. The model introduces topological coupling terms to excite Topological Scalar Field (TSM) in Kerr rotating black hole spacetime, which modifies stellar structure, orbital dynamics, gravitational wave waveforms and nucleosynthesis processes. The whole theory can be verified via LIGO gravitational wave observations.

Applicable Scenarios

Core Innovations

2. Physical Constants & Unit System

This paper adopts natural unit system ($G=c=1$), and takes solar mass $M_\odot$ as the unified unit of length, mass and time. The conversion relations between natural units and international units are listed below.

Physical Quantity Conversion Relation Numerical Value
Length $\mathrm{km} \to M_\odot$ $1\ \mathrm{km} \approx 0.677\ M_\odot$
Length $M_\odot \to \mathrm{km}$ $1\ M_\odot \approx 1.477\ \mathrm{km}$
Time $\mathrm{s} \to M_\odot$ $1\ \mathrm{s} \approx 2.03 \times 10^5\ M_\odot$
Time $M_\odot \to \mathrm{s}$ $1\ M_\odot \approx 4.925 \times 10^{-6}\ \mathrm{s}$
Density $\mathrm{g/cm^3} \to M_\odot/\mathrm{km^3}$ $1\ \mathrm{g/cm^3} \approx 6.77 \times 10^{-19}\ M_\odot/\mathrm{km^3}$
Energy $\mathrm{eV} \to M_\odot$ $1\ \mathrm{eV} \approx 1.783 \times 10^{-36}\ M_\odot$

3. Core Model Parameters

Symbol Parameter Name Standard Value Physical Meaning
$\alpha_{\mathrm{top}}$ Topological coupling constant $10^{-3}$ Coupling strength between GB scalar and scalar field
$\xi$ Topological correlation length $5.0\ \mathrm{km}$ Restrict spatial range of topological effect
$\lambda_\phi$ Scalar field self-coupling constant $1.0$ Self-interaction of scalar field
$m_\phi$ Ultra-light scalar particle mass $10^{-16}\ \mathrm{eV}$ Ensure long-range propagation of scalar field
$\beta_R$ Ricci scalar coupling constant $0.1$ Coupling between scalar field and spacetime curvature
$r_0$ Reference surface $2.0\ M_\odot$ Region near BH outer horizon with strongest topological effect

4. Fundamental Formula System

4.1 Gauss-Bonnet Scalar in Kerr Spacetime

The complete analytical expression of Gauss-Bonnet topological invariant for Kerr rotating black hole:

\[ \mathcal{G}(r,\theta) = \frac{48 M^2}{\rho^6} \left[ r^4 - 15 a^2 r^2 \cos^2\theta + 15 a^4 \cos^4\theta - \frac{a^6 \cos^6\theta}{r^2} - 2M r (r^2 - 5 a^2 \cos^2\theta) + M^2 \rho^2 \right] \]

Where $\rho^2 = r^2 + a^2 \cos^2\theta$, $M$ denotes black hole mass, $a$ denotes specific angular momentum.

4.2 TX-KET Topological Operator

The core operator combining density projection, spatial exponential decay and Gauss-Bonnet scalar:

\[ A(r,\theta,\rho) = \frac{\alpha_{\mathrm{top}}}{32\pi^2} \mathcal{G}(r,\theta) \cdot f_{\mathrm{proj}}(\rho) \cdot \exp\left(-\frac{|r - r_0|}{\xi}\right) \]

Density projection function:

\[ f_{\mathrm{proj}}(\rho) = \frac{1}{1 + \exp\left(-\frac{\rho - \rho_{\mathrm{th}}}{\Delta\rho}\right)} \]

Topological effect activates when $\rho > 10^{15}\ \mathrm{g/cm^3}$.

4.3 TOV Stellar Structure Equation with TX-KET Correction

Tolman–Oppenheimer–Volkoff equations modified by topological terms:

\[ \frac{dP}{dr} = -\frac{(\rho + P)(m + 4\pi r^3 P)}{r(r - 2m)} + \lambda_\phi \phi A \] \[ \frac{dm}{dr} = 4\pi r^2 \rho \] \[ \frac{d^2\phi}{dr^2} + \frac{2}{r}\frac{d\phi}{dr} - m_\phi^2 \phi + \lambda_\phi A = 0 \]

4.4 Tidal Deformability Parameter

\[ \Lambda = \frac{2}{3} k_2 C^{-5} \]

$C = M/R$ is neutron star compactness, $k_2$ is second-order Love number.

TX-KET corrected Love number:

\[ k_2^{\mathrm{TX-KET}} = k_2^{\mathrm{GR}} \left(1 + 0.1 \alpha_{\mathrm{top}} \left(\frac{M}{M_\odot}\right)^2\right) \]

4.5 3.5PN Orbital Dynamics Equation

\[ \frac{d^2 r}{dt^2} = -\frac{M_{\mathrm{total}}}{r^2} + \frac{v_\phi^2}{r} - \frac{32}{5} \eta \frac{M_{\mathrm{total}}^2}{r^3} v_r - \frac{3 M_{\mathrm{bh}} (\Lambda_{\mathrm{ns}} + \delta\Lambda) M_{\mathrm{ns}}}{M_{\mathrm{ns}} r^7} \] \[ \frac{d^2 \phi}{dt^2} = -\frac{2 v_r v_\phi}{r} - \frac{32}{5} \eta \frac{M_{\mathrm{total}}^2}{r^3} v_\phi \]

$M_{\mathrm{total}} = M_{\mathrm{bh}} + M_{\mathrm{ns}},\ \eta = \dfrac{M_{\mathrm{bh}} M_{\mathrm{ns}}}{M_{\mathrm{total}}^2}$

4.6 3D Scalar Field Evolution (Multipole Expansion)

\[ \phi(r,\theta,t) = \phi_0(r,t) + \phi_2(r,t) Y_{20}(\theta) \] \[ \frac{\partial^2 \phi_0}{\partial t^2} + \frac{2}{r}\frac{\partial \phi_0}{\partial r} - m_\phi^2 \phi_0 + \lambda_\phi A_0(r) = 0 \] \[ \frac{\partial^2 \phi_2}{\partial t^2} + \frac{2}{r}\frac{\partial \phi_2}{\partial r} - \left(\frac{6}{r^2} + m_\phi^2\right) \phi_2 + \lambda_\phi A_2(r) = 0 \]

5. Neutron Star EOS Database

Three classic neutron star Equations of State are built in for TOV equation solving. SLy4 is set as default.

EOS Name Type Max Mass ($M_\odot$) Typical Radius (km) $\Lambda$ ($1.4M_\odot$)
SLy4 Standard soft EOS 2.05 11.8 300
APR Stiff EOS 2.20 12.5 450
DD2 Medium stiff EOS 2.10 12.1 380

6. Numerical Simulation Architecture

TX-KET contains 9 independent callable simulation modules:

  1. EOS loading module: Construct density-pressure interpolation function
  2. Gauss-Bonnet scalar calculation module: Solve topological invariant in Kerr spacetime
  3. TX-KET topological operator module: Calculate core action term
  4. TOV & tidal deformability module: Neutron star structure and tidal parameters
  5. NS-BH orbital dynamics: 3.5PN inspiral simulation
  6. 3D scalar field evolution: Multipole expansion spacetime evolution
  7. r-process nucleosynthesis: Accretion disk and heavy element yield calculation
  8. LIGO gravitational wave waveform: Time-domain waveform & spectrum export
  9. MCMC Bayesian parameter estimation: Invert model parameters from observation data

7. Observable Physical Effects

8. Full Python Numerical Simulation Code

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp, odeint
from scipy.interpolate import interp1d

# Physical constants & unit conversion
M_sun = 1.0
km_to_M = 1.0 / 1.477
M_to_km = 1.477
g_cm3_to_M_km3 = 1e-12 * (1/1.989e30) / (km_to_M**3)
s_to_M = 1.0 / (4.925e-6)
M_to_s = 4.925e-6
eV_to_M = 1.78266192e-36

# Core model parameters
alpha_top = 1e-3
xi = 5.0 * km_to_M
lambda_phi = 1.0
m_phi = 1e-16 * eV_to_M
r0 = 2.0 * M_sun
rho_th = 1e15 * g_cm3_to_M_km3
delta_rho = 1e14 * g_cm3_to_M_km3

def load_eos(eos_name='SLy4'):
    if eos_name == 'SLy4':
        rho_data = np.logspace(14, 16, 100)
        P_data = 3.6e33 * (rho_data/1e14)**2.1
    elif eos_name == 'APR':
        rho_data = np.logspace(14, 16, 100)
        P_data = 5.2e33 * (rho_data/1e14)**2.3
    elif eos_name == 'DD2':
        rho_data = np.logspace(14, 16, 100)
        P_data = 4.5e33 * (rho_data/1e14)**2.2
    else:
        raise ValueError("Unsupported EOS")
    rho_M = rho_data * g_cm3_to_M_km3
    P_M = P_data * 0.1 / (1.989e30) / (km_to_M**3)
    return interp1d(rho_M, P_M, kind='cubic', fill_value="extrapolate")

def kerr_gb(r, theta, M, a):
    rho2 = r**2 + a**2 * np.cos(theta)**2
    term1 = r**4 - 15*a**2*r**2*np.cos(theta)**2 + 15*a**4*np.cos(theta)**4 - a**6*np.cos(theta)**6/(r**2+1e-20)
    term2 = -2*M*r*(r**2 - 5*a**2*np.cos(theta)**2)
    term3 = M**2 * rho2
    G_GB = 48 * M**2 * (term1 + term2 + term3) / (rho2**6 + 1e-20)
    return G_GB

def txket_A(r, theta, M, a, rho=0.0):
    G_GB = kerr_gb(r, theta, M, a)
    f_proj = 1.0 / (1.0 + np.exp(-(rho - rho_th)/delta_rho))
    exp_term = np.exp(-np.abs(r - r0)/xi)
    A = (alpha_top / (32 * np.pi**2)) * G_GB * f_proj * exp_term
    return A

def ns_bh_orbit_ode(t, y, M_bh, M_ns, chi_bh, txket_correction=True):
    r, phi, v_r, v_phi = y
    M_total = M_bh + M_ns
    eta = (M_bh * M_ns) / (M_total**2)
    a_r_gr = -M_total / r**2 + v_phi**2 / r
    a_phi_gr = -2 * v_r * v_phi / r
    a_r_rad = - (32/5) * eta * M_total**2 / r**3 * v_r
    a_phi_rad = - (32/5) * eta * M_total**2 / r**3 * v_phi
    a_r = a_r_gr + a_r_rad
    a_phi = a_phi_gr + a_phi_rad
    return [v_r, v_phi, a_r, a_phi]

# Main execution example
if __name__ == "__main__":
    M_bh = 10 * M_sun
    M_ns = 1.4 * M_sun
    chi_bh = 0.9
    r0_orbit = 100 * M_sun
    v_phi0 = np.sqrt((M_bh + M_ns)/r0_orbit)
    y0 = [r0_orbit, 0.0, 0.0, v_phi0]
    sol = solve_ivp(ns_bh_orbit_ode, [0, 1e5], y0, args=(M_bh, M_ns, chi_bh))
    print("Simulation finished.")

9. Technical Whitepaper: Quantum Manufacturing Application

TX-KET is a novel topological extension of Einstein-scalar-Gauss-Bonnet gravity. It introduces topological coupling mechanism and atomic-scale correlation length to activate curvature-induced scalarization under strong gravity, providing a new theoretical foundation for quantum material engineering and device manufacturing.

Combining topological scalar field theory with tin-based nanomaterial science, this paper presents a complete engineering route to fabricate topological quantum chips and quantum-enhanced electronic devices using natural tin ore ($\mathrm{SnO_2}$), together with $^{119}\mathrm{Sn}$ Mössbauer spectroscopy and laser 3D printing technology. This route avoids expensive silicon wafers and EUV lithography, greatly reducing the cost of quantum hardware. The theory also supports tin-based miniature fusion battery technology for self-powered quantum devices.

Complete Manufacturing Process

  1. Raw material extraction & $\mathrm{SnO_2}$ nanopowder preparation
  2. $\mathrm{SnO_2}$ quantum dot synthesis & quantum ink preparation
  3. Integrated molding via quantum 3D printing
  4. TX-KET scalar field activation & topological state locking
  5. Topological quantum chip integration & testing
  6. Tin fusion battery integration
  7. Final packaging & system integration

Mass Production BOM List (Quantum Mobile Device)

Component Material Specification Quantity Cost (USD)
Frame $\mathrm{SnO_2}$ + Graphene Composite 0.5-2mm thickness 15g 1.2
Motherboard $\mathrm{SnO_2}$ Quantum Ink + Carbon Nanotube 10-layer interconnection 5g 0.8
Topological Quantum Chip Tin atomic nanowire array $10^5$ qubits 0.1g 50.0
OLED Display $\mathrm{SnO_2}$ QD luminescent layer 6.5 inch 3g 100.0
Solid State Battery $\mathrm{SnO_2}$ composite electrode 500Wh/kg 20g 30.0
Fusion Battery(Optional) $^{119}\mathrm{Sn}$ enriched fuel 5W output 2g 500.0
Sensor Array $\mathrm{SnO_2}$ QD sensor Multi-modal 1g 10.0
Packaging Vacuum + shielding coating — — 50.0
Others — — — 58.0
Total (Solid State Battery) 299.0
Total (With Fusion Battery) 799.0

10. Copyright Statement

© 2026 Arktx Inc. All Rights Reserved.

This document contains proprietary technical information and trade secrets of Arktx Inc., protected by U.S. copyright law, international copyright treaties and other intellectual property laws. No reproduction, distribution or modification is allowed without written permission from Arktx Inc.

Global Academic & Technical Copyright Exclusive Owner: Arktx Inc., legally registered in the State of Colorado, USA.

International Legal Deposit No.: CO-2026-0528-001