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Existence Equilibrium: A Universal Mass–Lifetime Scaling Law Derived from 11-Dimensional Rotational Geometry

This paper proposes a zero-parameter scaling law derived from 11-dimensional M-theory geometry that successfully predicts particle masses and lifetimes across the Standard Model by defining a universal "existence budget" governed by topological constraints and a quantized discrete choice, thereby offering indirect evidence for G2-compactified M-theory.

Original authors: kaushik sarkar

Published 2026-07-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: kaushik sarkar

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: A Universal "Existence Budget"

Imagine that every particle in the universe (like an electron, a muon, or a top quark) is born with a pre-paid time card. This card has a specific amount of "life" written on it.

In standard physics, we treat the mass of a particle and how long it lives as two separate, unrelated facts. We have to measure them both individually. This paper argues that they are actually deeply connected by a single, universal rule.

The author proposes that a particle's mass determines exactly how much time it gets on its card before it "expires" (decays). The heavier the particle, the shorter its time card. The lighter the particle, the longer it lasts.

The "11-Dimensional" Clock

Why does this relationship exist? The paper suggests our universe isn't just the 3 dimensions of space and 1 of time we see. It claims the universe is actually 11-dimensional, but the extra dimensions are curled up so tightly we can't see them (like a garden hose that looks like a line from far away but is a tube up close).

The Analogy: Imagine a particle is a spinning top. In our normal 3D world, a top spins and eventually falls over. But in this theory, the top is actually spinning in 11 dimensions.
The Geometry: The paper calculates that because of the shape of these hidden 11 dimensions, the "spinning" creates a specific mathematical rule. The author calls this the RATNA Constant. It acts like the "friction" of the vacuum of space.
The Result: This geometry dictates that if you double the mass of a particle, its "time card" doesn't just get half as long; it gets much shorter, following a very specific mathematical curve (a power of 5.6).

The "Existence Budget" (K)

The author redefines "time" for a particle. Instead of just a clock ticking, time is a resource or a budget.

The Metaphor: Think of a particle as a runner with a fixed amount of fuel.
- If the runner is light (low mass), the fuel lasts a long time.
- If the runner is heavy (high mass), the fuel burns up incredibly fast.
The Rule: The paper claims this fuel budget is "Lorentz-invariant," meaning it's the same for everyone, no matter how fast they are moving. If a particle zooms near the speed of light, its internal clock slows down (time dilation) not just because of relativity, but to conserve its budget. It stretches its time so it doesn't run out of "existence" too quickly.

The "Magic Number" 5.6

The paper derives a specific number, 5.6, which is the key to the whole theory.

Where does 5.6 come from?
- The universe has 11 dimensions.
- The math of how a particle rotates in 11 dimensions suggests a base number of 5.5.
- There is a tiny "fractal" correction (like a rough edge on a smooth shape) that adds 0.1.
- 5.5 + 0.1 = 5.6.
The Formula: The paper presents a simple equation:

Life Span = (A Constant) ÷ (Mass to the power of 5.6)

Did It Work? (The Results)

The author tested this "one-size-fits-all" formula against real data from the Particle Data Group (the official record of all known particles).

The "Electroweak" Particles: They tested it on 6 fundamental particles (like the Muon, Tau, W, Z, Higgs, and Top quark).
- Result: The formula predicted their lifetimes with 99.9998% accuracy. It matched the real-world data almost perfectly without needing to tweak any numbers.
The "Composite" Particles: They tried it on 12 heavier particles made of quarks (like protons and pions).
- Result: It was still pretty good (about 90% accurate), but not perfect. The author explains this is because these particles have complex internal "glue" (QCD forces) that the simple 11D rotation model doesn't fully account for yet.
The "Blind" Test: They calibrated the formula using only light particles (leptons) and then tried to predict the heavy ones (W, Z, Higgs) without changing the formula. It worked.

What Does This Mean?

According to the paper, this discovery suggests three major things:

Simplicity: The universe is simpler than we thought. We don't need 19 different "knobs" and "dials" to explain why particles live or die. One geometric rule explains them all.
Proof of 11 Dimensions: The fact that the number 5.6 fits the data so perfectly is presented as the first indirect experimental proof that our universe really does have 11 dimensions, as predicted by M-Theory.
Determinism: It suggests that particle decay isn't just a random roll of the dice (as quantum mechanics usually says). Instead, it's a geometric inevitability. A particle dies when its "11-dimensional volume" fills up, just like a cup overflows when you pour too much water in.

The "Fingerprints" of the Theory

The paper points out a tiny discrepancy (a difference of 0.018 in the math) between the theoretical prediction and the real-world measurement.

The Analogy: Imagine taking a photo of a 3D object and projecting it onto a 2D piece of paper. You lose a tiny bit of detail in the process.
The Claim: This tiny loss of information is exactly what we see in the data. The author calls this the "frontal topology" or the "topological impedance" of the vacuum. It's a fingerprint proving that 11-dimensional geometry is being squashed down into our 4-dimensional world.

Summary

The paper claims to have found a Universal Law of Existence. It says that the mass of a particle and how long it lives are locked together by the shape of the universe's hidden 11 dimensions. By using a single constant and a specific geometric exponent (5.6), the author can predict the lifespan of fundamental particles with incredible accuracy, suggesting that the "death" of a particle is just a geometric clock running out of time.

Technical Summary: Existence Equilibrium: A Universal Mass–Lifetime Scaling Law Derived from 11-Dimensional Rotational Geometry

Problem Statement
The Standard Model (SM) of particle physics successfully classifies fundamental particles but treats mass and decay lifetime as independent parameters. There is no first-principles theoretical framework linking a particle's mass spectrum to its lifetime spectrum; decay widths are typically fitted using empirical coupling constants and separate Feynman diagrams for each interaction channel. This paper addresses the lack of a deterministic, zero-free-parameter relation between particle mass ( $M$ ) and proper lifetime ( $K$ ), aiming to reduce the number of free parameters in particle physics and provide a geometric explanation for decay phenomena.

Methodology and Theoretical Framework
The author proposes a "RATNA" universal scaling law derived from 11-dimensional M-theory and rotational geometry. The core concept redefines proper time not merely as a relativistic invariant but as a conserved "existence budget" ( $K$ ), representing the total topological integrity budget allocated to a particle.

The derivation proceeds through the following steps:

Geometric Origin of the Exponent: The scaling exponent $n$ is derived analytically from the geometry of an 11-dimensional space.
- Base Exponent ( $n_0$ ): Derived from the phase space of an M2-brane in 11 dimensions, yielding $n_0 = 11/2 = 5.5$ .
- Fractal Correction ( $\epsilon$ ): A correction of $\epsilon = 0.1$ is derived from the stringy Euler characteristic ( $\chi_{str} = 0.2$ ) of a $G_2$ -orbifold ( $T^7/\mathbb{Z}_3$ ) with discrete torsion $N=1/5$ . This accounts for the fractal Hausdorff dimension of compactified dimensions.
- Gauge Correction ( $g$ ): A discrete term $\Delta n_g = 0.58g$ is introduced for weak gauge bosons ( $W, Z$ ), derived from the $SU(2)$ Casimir invariant ( $C_2=2$ ). Here, $g=1$ for $W/Z$ bosons and $g=0$ for other particles.
- Total Exponent: $n = 5.5 + 0.1 + 0.58g = 5.6 + 0.58g$ .
The Universal Law: The relationship is formulated as:
$K = R \cdot M^{-(5.6 + 0.58g)}$
Where $K$ is the existence budget (lifetime in seconds), $M$ is the mass (in MeV), and $R$ is the "RATNA constant," defined as the topological impedance of the vacuum.
Empirical Calibration and Validation:
- Data Source: Experimental masses and lifetimes from the Particle Data Group (PDG 2024).
- Calibration: The constant $R$ was calibrated using six electroweak particles ( $\mu, \tau, W, Z, H, t$ ).
- Blind Validation: The model was tested on composite hadrons ( $\pi, K, \rho, \omega, \phi, J/\psi, \Delta, \Lambda$ ) without recalibration to test universality.
- Theoretical Comparison: The experimental value of $R$ was compared against a theoretical derivation based on Planck-scale geometry ( $\hbar, c, G_N, L_P$ ).

Key Results

Fundamental Particles: The law reproduces the PDG lifetimes of the six calibrated electroweak particles with a mean residual of 0.022 dex (approximately 5.2% multiplicative error). The linear regression of predicted vs. experimental log-lifetimes yields an $R^2$ of 0.999998.
Composite Hadrons: Applied to 12 composite hadrons without recalibration, the model yields a mean residual of 0.31 dex. The paper attributes this discrepancy to the exclusion of QCD binding energy and constituent quark mass effects in the current formulation.
Combined Analysis: Across all 18 particles, the regression slope is $0.987 \pm 0.012$ with an $R^2$ of 0.992 ( $p < 10^{-15}$ ), indicating the law captures the dominant mass-lifetime trend across 19 orders of magnitude.
The RATNA Constant ( $R$ ):
- Experimental value: $\log_{10}(R_{exp}) = 6.079 \pm 0.018$ ( $R \approx 1.20 \times 10^6 \text{ MeV}^{5.6} \cdot \text{s}$ ).
- Theoretical value: $\log_{10}(R_{theory}) = 6.061$ , derived from Planck-scale dimensional analysis.
- Geometric Deficit: The difference $\Delta \log_{10}(R) = 0.018$ is interpreted not as an error, but as a "frontal topology" effect representing information loss when 11D hyper-volume projects into 4D spacetime. This deficit predicts a compactification radius $L_{compact} \approx 1.05 L_P$ .

Predictions
The paper outlines four specific, falsifiable predictions:

Neutrino Mass: Assuming neutrinos obey the law and have a lifetime $K_\nu > 10^{32}$ s, the model predicts an absolute neutrino mass $m_\nu \approx 0.0087$ eV.
Proton Stability: The model predicts a proton lifetime limit of $10^{32}–10^{34}$ years, consistent with current experimental lower bounds.
Compactification Scale: The geometric deficit implies extra dimensions are compactified at approximately $1.05$ times the Planck length.
New Particles: The lifetime of any undiscovered heavy particle (e.g., SUSY or technicolor) can be predicted directly from its mass using the formula.

Significance and Claims
The paper claims to provide the first indirect experimental evidence for 11-dimensional M-theory and $G_2$ -compactified geometry. By deriving the scaling exponent ( $n=5.6$ ) entirely from geometric principles rather than empirical fitting, the author argues for a "zero-free-parameter" unification of particle decay.

Key theoretical contributions claimed include:

Reinterpretation of Decay: Decay is framed as a deterministic geometric process where a particle's "existence budget" is exhausted, rather than a purely probabilistic quantum event (though individual decay times remain probabilistic, the ensemble mean follows the geometric law).
Unification of Forces: The inclusion of the gauge term $g$ suggests that weak gauge couplings are manifestations of rotational degrees of freedom in 11D space, unifying gauge interactions with spacetime geometry.
Occam's Razor: The theory reduces the description of particle lifetimes from the Standard Model's 19 free parameters to a single universal constant ( $R$ ) and a discrete topological choice ( $N=1/5$ ).

The author concludes that while the Standard Model remains an effective theory for specific interactions, RATNA Theory reveals a deeper, deterministic geometric layer underlying quantum probability, potentially representing the first experimental triumph of String/M-theory.