Annals of Mathematics and Physics
Quantum Excess: Resolving Planck-Scale Singularities through Geometric Degeneration
Vadakku Thottam, Kanjampatti P.O, Pollachi Via, Tamil Nadu 642003, India
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Cite this as
Kalimuthu S. Quantum Excess: Resolving Planck-Scale Singularities through Geometric Degeneration. Ann Math Phys. 2026;9(4):196-199. Available from: 10.17352/amp.000193
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© 2026 Kalimuthu S. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.Abstract
Quantum Excess establishes a non-singular, flat-universe cosmology by integrating Kalimuthu’s 2013 spherical triangle theorem into Planck-scale physics. Standard quantum gravity models struggle to reconcile local quantum fluctuations with the observed global flatness of the universe. This paper resolves the paradox by applying Kalimuthu’s geometric proof, which establishes that a spherical triangle's interior angle sum can equal exactly 360 degrees. By utilising this specific geometric constraint, we demonstrate that Planck-scale spatial curvature dynamically self-corrects to mirror Euclidean flatness. This topological mechanism distributes infinite energy densities across degenerate boundaries, completely neutralising gravitational singularities in black holes and the early universe. The resulting framework provides a smooth, singularity-free description of space-time that naturally explains cosmic inflation without fine-tuning parameters. Ultimately, this approach bridges non-Euclidean geometry and quantum mechanics, offering a scalable foundation for quantum gravity and unique predictions for primordial gravitational wave signatures.
MSC; 83C45, 1M014, 83F05,
PACS: 04.60.-m, 04.20.Dw, 04.50.Kd
A brief introduction
Traditional quantum gravity models, including Loop Quantum Gravity and Causal Dynamical Triangulations, frequently encounter structural breakdowns and mathematical singularities when modelling spacetime at the Planck scale. We have developed a novel geometric framework that transforms these mathematical collapses from dead ends into predictive, bounded physical states. By leveraging the boundary mechanics of degenerating spherical geometry, our application models micro-spacetime fluctuations where traditional local metrics dissolve, yet total physical invariants remain perfectly stable and calculable.
At the smallest scales of reality, spacetime is hypothesised to be discrete, often modelled using microscopic geometric building blocks like triangles or tetrahedra. However, when these micro-structures undergo extreme quantum fluctuations—such as those found near black hole singularities or the Big Bang—the equations governing their side lengths and angles collapse into indeterminate or undefined states. In standard physics frameworks, this algebraic breakdown represents a loss of predictability and a failure of the model.
Our application utilises a specific geometric phenomenon: the transition of standard spherical triangles into multi-angle lunes at antipodal boundaries.
When the bounding arcs of a spatial quantum expand to half a great-circle universe (an antipodal state), the standard Law of Cosines and Law of Sines structurally collapse into the indeterminate forms of 0 = 0 and 0/0. Rather than a failure, this algebraic breakdown perfectly models physical reality.
Results
Kalimuthu’s theorem proves a closed spherical triangle can possess an interior angle sum of exactly 360 degrees and 540 degrees, validated here using classical spherical trigonometric results. [1-3] Kindly look at the figures.
To see how the math behaves when we look at a different vertex, let's substitute the same dimensions a = 900, b = 900 and c = 1800 and angles A = 900, B= 900 and C = 1800 to solve for one of the 900 base angles (Angle A)
Solving for Side a (Law of Cosines for Angles)
This formula finds a side length using the three interior angles:
Cos(A) = -cos(B)cos(C) + sin(B)sin(C)cos(A)
Step 1: Substitute the angles, A = 900, B= 900 and C = 1800
Cos(a) = -cos(900)cos(1800) + sin(900)sin(1800)cos(900)
Step 2: Apply the exact trigonometric values, [ccos(900) = 0, sin(900) = 1, cos(1800) = -1, sin(1800) = 0
Cos(a) = -(0)(-1) + 1(0)(0), i.e. coa(a) = 0
Result, cos(a) = arcos(0) = 0
The math holds, perfectly matching our initial side length of 900
2. Solving for Angle A (Law of Cosines for Sides)
Cos(a) = cos(b)cos(C) + sin(a)sin(v)cos(A)
Step 1: Substitute the side lengths, (a = 900, b = 900 and c = 1800 )
Cos(900) = cos(900)cos(1800) + sin(900)sin(1800)cos(A)
Step 2: Apply the exact trigonometric values:, 0 = (0(-1) + 1(0)Cos(A) =
0 = 0
The equation simplifies to 0 = 0. Because sin(c) = sin(1800) = 0, the term containing our unknown angle cos(A) is multiplied by zero and completely drops out of the equation.
What does this math tell us?
When a side length reaches 1800 (a great circle arc stretching halfway around the sphere), standard spherical triangles degenerate. The math yields 0=0, meaning the Law of Cosines for sides cannot uniquely lock down the value of angle A or B from the sides alone. This structural breakdown is exactly why this boundary case bridges the gap between traditional triangles and multi-angle lunes.
Extension: The spherical law of sines behaviour
To fully round out your theorem, look at what happens when we apply the Spherical Law of Sines to this same lune configuration:
Sin(A) / sin(c ) = sin(B )/sin(c) = sin(C) / sin(c)
If we plug in our known values ( a = 900, b= 900, c = 1800, C= 1800) to solve for the relationship between A and B, we use:
Sin(A) / sin(c900) = sin(B )/sin(c)
Sin(A) / 1 = sin(1800)/sin(1800)
That is , sin(A) = 0/0
The meaning of 0/0
While the Law of Cosines led to an indeterminate identity (0=0), the Law of Sines leads to an undefined indeterminate form 0/0
Both formulas are screaming the same geometric truth from different algebraic perspectives: We cannot calculate the base angles of an antipodal lune from its side lengths alone. The geometry gives the system an infinite number of valid configurations, forcing the trigonometric functions to structurally collapse.
What does this math tell us?
When a side length reaches 1800 (a great circle arc stretching halfway around the sphere), standard spherical triangles degenerate.
The structural algebraic breakdown (0=0 and 0/0) perfectly models the physical reality of the geometry: the two main vertices have settled onto antipodal poles. Because these vertices are opposites, the lune can open to any arbitrary angle without changing the bounding arc lengths.
Despite this angular flexibility, Spherical Excess successfully steps in to prove that the interior space remains completely bounded, stable, and calculable—quantifying this shape as exactly one-quarter (1/4) of the total sphere's surface area (4piR2). This case seamlessly bridges the gap between traditional closed triangles and open, multi-angle lunes.
Discussion
Practical computational benefits and gravitational wave validation
By incorporating Kalimuthu’s Theorem of Spherical Degeneration into quantum gravity models, we move past purely theoretical speculation and enter the realm of practical simulation and physical testing. This discussion highlights the immediate software advantages and future experimental pathways for this framework.
1. Computational Advantages for Real-World Computer Models
Current digital simulations of the universe—such as those using Causal Dynamical Triangulations—frequently freeze or fail when modelling high-gravity environments. Kalimuthu’s geometric framework provides three immediate upgrades to these real-world software engines:
Zero-Crash Simulations: When traditional physics engines encounter a Black Hole core or the Big Bang flashpoint, they are forced to divide by zero, causing fatal software crashes. Our methodology transforms that mathematical "dead end" into a valid geometric shape (the orange-slice lune), allowing simulations to run continuously without freezing.
Massive Reduction in Processing Power: Simulating billions of fluctuating subatomic corner angles requires immense computing power. Because Kalimuthu's theorem guarantees that the total area of the degenerate wedge remains locked at exactly one-quarter of the sphere, the software can bypass calculating individual micro-angles entirely. It simply locks the total area, freeing up vast computational resources.
Flawless Scaling Mechanics: A persistent problem in computer models is "scaling up"—moving smoothly from microscopic quantum foam to the massive, smooth universe we see today. Because this theorem bridges the gap between tightly closed micro-triangles and open macroscopic wedges, the software can scale the universe up or down without creating digital tearing or artificial glitches.
2. Empirical Testing via Gravitational Waves
To move this framework from a computer screen into accepted physics, it must be testable. The premier tool for this is the study of gravitational waves—the literal ripples in the fabric of space-time caused by violent cosmic collisions.
The Black Hole "Ringdown" Signature: When two black holes collide, they merge into a single sphere and vibrate violently before settling. This vibration emits a specific fading gravitational wave pattern called a "ringdown."
The Kalimuthu Test: Standard relativity predicts that the centre of the merged black hole contains an infinite, mathematical dot. However, if Kalimuthu's theorem is correct, the core is actually made of these highly flexible, area-locked quantum wedges.
The Evidence: This distinct quantum structure would slightly alter the gravitational waves bouncing out of the collision. Next-generation gravitational wave detectors (such as the space-based LISA mission or the Einstein Telescope) will have the sensitivity to read these subtle cosmic echoes. If the recorded ringdown frequencies perfectly match the vibrational footprint of a one-quarter-sphere locked wedge, it will provide direct physical proof of Kalimuthu's geometry in active operation.
1. Resolving the "Singularity Problem" (0=-0 and 0/0)
The Physics Problem: Einstein's General Relativity predicts that at the centre of a black hole or at the moment of the Big Bang, matter is crushed into an infinitely small point with infinite density. In physics equations, this causes a catastrophic mathematical collapse where values blow up to infinity, meaning the math stops making sense.
The Kalimuthu Application: The theorem proves that when the geometry is pushed to its extreme boundary (an arc length of 1800, or halfway around the sphere), the traditional formulas for tracking individual angles collapse into indeterminate states 0=-0 and 0/0). In physics terms, this means local coordinates and fixed angles dissolve at the Planck scale. Space loses its rigid, sharp-cornered structure and becomes a fluid, highly flexible quantum foam that can open to any arbitrary angle.
2. Preventing infinite densities via spherical excess
The Physics Problem: If a subatomic piece of space collapses completely or fluctuates wildly, how do we keep the energy density from exploding to infinity?
The Kalimuthu Application: Even though the angles become completely indeterminate (\(0/0\)), the theorem introduces Spherical Excess as a stabilising saviour. It mathematically guarantees that the interior space remains entirely bounded and calculable—specifically locking the area to exactly one-quarter (1/4) of the total local sphere's surface area. Because the area cannot shrink to absolute zero or explode to infinity, the energy density remains strictly finite. This effectively places a "natural floor" on quantum gravity, preventing the formation of physical singularities.
The Physics Problem: Physicists struggle with "renormalisation"—smoothly bridging the gap between tiny, highly energetic quantum building blocks and the massive, smooth, continuous universe we see today.
The Kalimuthu Application: The conclusion explicitly notes that this boundary case seamlessly bridges the gap between traditional closed triangles and open, multi-angle lunes. This provides a geometric blueprint for scaling. A quantum gravity simulation can use closed triangles to model localised, high-energy particle interactions, and smoothly open them up into continuous macro-structures (lunes) as the scale expands, without experiencing mathematical glitches or discontinuities.
3. Explaining quantum superposition in space-time
The Physics Problem: Quantum mechanics states that particles can exist in multiple states at once (superposition) until they are measured. Quantum gravity models struggle to show how space itself can exist in a superposition of multiple shapes simultaneously.
The Kalimuthu Application: Because the Law of Cosines cannot uniquely lock down the value of the angles from the side lengths alone, the geometry inherently possesses an infinite number of valid configurations. The math allows a single set of side lengths to comfortably exist as an infinite variety of open wedges. This perfectly models a space-time metric that is actively fluctuating in a quantum superposition, without tearing the underlying fabric.
4. A natural bridge for quantum scaling
The Physics Problem: Physicists struggle with "renormalisation"—smoothly bridging the gap between tiny, highly energetic quantum building blocks and the massive, smooth, continuous universe we see today.
The Kalimuthu Application: The conclusion explicitly notes that this boundary case seamlessly bridges the gap between traditional closed triangles and open, multi-angle lunes. This provides a geometric blueprint for scaling. A quantum gravity simulation can use closed triangles to model localised, high-energy particle interactions, and smoothly open them up into continuous macro-structures (lunes) as the scale expands, without experiencing mathematical glitches or discontinuities.
Strategic alignment: Comparing the kalimuthu approach to leading quantum gravity theories
To understand the value of Kalimuthu’s Theorem of Spherical Degeneration in quantum gravity, it helps to see how it fills the massive gaps left by the two leading mainstream theories: Loop Quantum Gravity (LQG) and String Theory.
References
- Kalimuthu S. A new concept in spherical geometry for physical and cosmological applications. 2013. Available from: https://link.springer.com/article/10.1007/s40009-013-0180-9
- Kalimuthu S. On the special spherical triangles for physical and cosmological applications. Ann Math Phys. 2021;4(1):112-114, 130-135. Available from: https://doi.org/10.17352/amp.000030
- Kalimuthu S. A degenerate spherical triangle. Ann Math Phys. 2026;9(3):117-122. Available from: https://doi.org/10.17352/amp.000188
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