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The Geometric Nature of Kunal Kishor Verma’s Infinitely Self-Referential Field!
By!
Ian Beardsley!
December 19, 2025-January 2, 2026"
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In a recent paper by Kunal Kishor Verma and myself, it is suggested in Kunal’s Timeless Energy
Principle (TEP) that the universe is an infinitely self-referential field of models that only resolve
when we choose one, and it then collapses into an abstract theory that is incomplete, but
allows the mind to model the Universe from one point of view. It is incomplete in the sense that
all theories are incomplete to date because they are formulated with mathematics which is a
formal system. Formal systems are incomplete because they are self-referential and thus
based on at least one assumption (Godel’s Incompleteness Theorem).!
As such, in our paper, we model this infinite regress to get around the inability of physics to tell
us what reality actually is. The model we collapse from the infinite regress is determined by our
assumptions, like the assumption by relativity of curved spacetime, in the sense that curved
spacetime is inferred by its measured effects. All measurements are really inferences because
they are done with devices based on the same principles we are measuring.!
In light of this we often choose a model based on the assumptions that suit our needs because
any assumption opens particular doors, and closes others. Often the most versatile theory for
the moment is the one that keeps the most doors open (Dr. Richard Allen Miller). The draft of
the paper by Kunal and Ian (the final has been submitted for peer review) is at:!
“Nature as infinite regress: a unified framework of the Timeless Energy Principle nd the collapse
of abstract modeling”!
https://www.academia.edu/145616038/
Nature_as_infinite_regress_a_unified_framework_of_the_Timeless_Energy_Principle_and_the_c
ollapse_of_abstract_modelling!
We see that all theories are incomplete due to the assumptions by the following illustration…If
we want to say what space is, or to keep it simple, what distance is, we say it is given by
velocity over time, or!
x=vt!
We want to say what x is, so we must say what velocity is. It is distance per time:!
v=x/t!
We put this expression for v in the equation for distance and we have!
v=(x/t)t!
This returns!
x=x!
Which is to say distance is distance, or more generally space is space. Thus we see physics
makes assumptions, which does not allow us to say what reality is, here the reality of space
because the language is self-referential.!
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The Geometric Nature of the Infinitely Self-Referential Field!
Here I suggest the field is characterized by the equilateral triangle. If the left corner of its base
is our thesis, then the right base corner is the anti-thesis. The apex is the synthesis of the
thesis and the anti-thesis. But this synthesis becomes a new thesis. It in turn has an anti-
thesis, and these latter two form a new synthesis. This new synthesis has an anti-thesis, and
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this new pair of thesis and anti-thesis has a new synthesis. So we see the infinite self-
referentiality of the field, geometrically. Since the equilateral triangle tessellates, we have an
infinite array of triangles, each node can be taken as a thesis, and has an anti-thesis node, with
this pair of nodes having a synthesis. We can choose any node in the infinite array as a thesis,
find its anti-thesis, and form their synthesis.!
If the tessellations go out to infinity symmetrically, they do so in six-directions, because the
dual of the equilateral triangle is the regular hexagon (as shown in the drawing).!
We see this why in physics , , , are so important. Because the fundamental
unit of the field, the equilateral triangle, if you draw in its altitude, form a 30-60-90 triangle. And
we have!
!
!
!
The regular hexagon is important to characterize the field because its radius is equal to its side
and we can take the side and radius as 1, each. This is why six-fold symmetry is so
fundamental as well. This is analogous to the speed of light in relativity having the property of
infinity and being taken as 1. The speed of light is often taken as c=1 giving one the property of
infinity, because at the speed of light time is zero, so in its reference frame it takes no time to
travel any distance giving v=c=1 is infinite velocity. Similarly the self-referential field is
described by the unit regular hexagon having side to radius equal to 1, describes the field
continuing out to infinity with six directions,!
The square is important because it tessellates as well like the equilateral triangle and the
regular hexagon. It is the basis of all coordinate systems, in that for instance while in it a point
is given by (x,y), in polar coordinates a point is given by ( ) where r and theta are given by!
!
!
This is also the source of appearing so much in physics because!
!
45 degrees is half of 90 degrees. 90 degree angles are formed by bisecting a line into two
equal angles.!
In the hyperbolic geometry of spacetime where clocks are described by hyperbolas, the world
line of light is t=x giving an angle of 45 degrees and this is the asymptote of the hyperbola."
1/2
1
2/2
3/2
cos(30
∘
) =
3
2
cos(60
∘
) =
1
2
sin(90
∘
) = 1
r, θ
x = rcos(θ )
r = x
2
+ y
2
2/2
cos(45
∘
) =
2
2
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## Ordered Sets for the Wave-Particle Dialectic Triangle!
### **1. Left Base Corner (THESIS): Wave Nature of Light**!
**Core Premise:** Light is fundamentally a continuous wave phenomenon in a medium-like
ether.!
**Mathematical Framework:**!
1. **Maxwell's Equations** (1865): Unified description of electricity and magnetism!
2. **Wave Equation:** ∇²E = (1/c²)∂²E/∂t²!
3. **Huygens-Fresnel Principle:** Every point on a wavefront is a source of spherical wavelets!
4. **Superposition Principle:** Waves add linearly; constructive/destructive interference!
5. **Transverse Wave Description:** Electric and magnetic fields oscillating perpendicular to
direction of propagation!
**Empirical Evidence:**!
6. **Young's Double-Slit Experiment** (1801): Clear interference patterns!
7. **Diffraction Patterns:** Single-slit, grating, and edge diffraction!
8. **Polarization Effects:** Malus' law, Brewster's angle!
9. **Doppler Effect for Light:** Frequency shifts consistent with wave theory!
10. **Maxwell's Prediction of EM Waves:** Theoretical derivation of light speed from electric
and magnetic constants!
**Theoretical Successes:**!
11. Perfect explanation of reflection, refraction (Snell's law from wave theory)!
12. Coherent explanation of color as wavelength/frequency!
13. Prediction of entire EM spectrum beyond visible light!
14. Self-consistent mathematical formalism requiring no quantization!
15. Elegant symmetry between electric and magnetic phenomena!
**Internal Consistency:**!
16. Continuous energy distribution in wavefronts!
17. Deterministic evolution via partial differential equations!
18. Clear mechanism for energy transport via Poynting vector!
19. Well-defined phase and amplitude at every point in space!
20. Compatibility with continuum mechanics and field theories!
### **2. Right Base Corner (ANTITHESIS): Particle Nature of Light**!
**Core Premise:** Light consists of discrete packets of energy (quanta) with particle-like
properties.!
**Mathematical Framework:**!
1. **Planck's Radiation Law** (1900): E = hν for blackbody oscillators!
2. **Einstein's Photoelectric Equation** (1905): E_max = hν - φ!
3. **Compton Scattering Formula** (1923): λ' - λ = (h/mₑc)(1 - cosθ)!
4. **Photon Momentum:** p = h/λ = E/c!
5. **Bose-Einstein Statistics** (1924): Describing photon gas statistics!
**Empirical Evidence:**!
6. **Photoelectric Effect:** Electron ejection depends on frequency, not intensity!
7. **Compton Scattering:** Wavelength shift of X-rays scattering off electrons!
8. **Blackbody Radiation Spectrum:** Ultraviolet catastrophe resolved by quantization!
9. **Atomic Spectra:** Discrete emission and absorption lines!
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10. **Photon Counting Experiments:** Individual photon detection (Geiger counters,
photomultipliers)!
**Theoretical Necessities:**!
11. Resolution of ultraviolet catastrophe in classical radiation theory!
12. Explanation of why light energy doesn't distribute continuously in photoelectric effect!
13. Conservation of momentum in light-matter interactions requiring particle-like collisions!
14. Discrete atomic energy levels requiring discrete energy exchanges!
15. Statistical behavior of light requiring countable entities!
**Conceptual Framework:**!
16. Light as discrete "atoms of energy" (Lenard's term, Einstein's photon)!
17. Localized energy transfer at specific points in space and time!
18. Particle-like trajectories in some experimental configurations!
19. Countable interactions with matter (one photon ejects one electron)!
20. Incompatibility with continuous wave description for certain phenomena!
### **3. Apex (SYNTHESIS): Quantum Field Theory / Quantum Electrodynamics**!
**Unifying Framework:** Light as quantized excitations of an underlying quantum field.!
**Mathematical Unification:**!
1. **Field Quantization:** Promotion of classical fields to operator-valued fields: Â(x,t)!
2. **Wave-Particle Synthesis:** Photons as quanta of the electromagnetic field with E = ħω, p =
ħk!
3. **Second Quantization:** Creation (a†) and annihilation (a) operators for photons!
4. **Fock Space:** Hilbert space with variable photon number states |n₁,n₂,...⟩!
5. **Coherent States:** Quantum states that most closely resemble classical waves!
**Resolution of Paradoxes:**!
6. **Interference of Single Photons:** Each photon interferes with itself via probability amplitude!
7. **Wavefunction of the Photon:** Not a wave in physical medium but probability amplitude
field!
8. **Complementarity Principle:** Wave and particle aspects are complementary, not
contradictory!
9. **Measurement Theory:** Collapse of wavefunction explains particle-like detection events!
10. **Quantum Superposition:** Photon exists in superposition of paths until measured!
**New Theoretical Framework:**!
11. **Quantum Electrodynamics (QED):** Most precise physical theory ever developed!
12. **Path Integral Formulation:** Sum over all possible histories explains wave-like behavior!
13. **Gauge Invariance:** U(1) local symmetry requiring massless vector bosons (photons)!
14. **Virtual Particles:** Off-shell photons mediate electromagnetic force!
15. **Renormalization:** Systematic removal of infinities through physical parameter redefinition!
**Empirical Triumphs:**!
16. **Lamb Shift:** Minute energy difference in hydrogen levels precisely predicted!
17. **Anomalous Magnetic Moment of Electron:** g-2 calculated to 10 decimal places!
18. **Casimir Effect:** Force between conducting plates from quantum vacuum fluctuations!
19. **Quantum Optics:** Entangled photons, quantum cryptography, squeezed light!
20. **Unified Description:** Single framework explaining ALL previously contradictory evidence!
**Expanded Conceptual Horizon:**!
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21. **Quantum Vacuum:** Not empty but seething with virtual particle-antiparticle pairs!
22. **Spontaneous Emission:** Explained as interaction with vacuum fluctuations!
23. **All Optical Phenomena:** Refraction, dispersion, nonlinear optics derived from QED!
24. **Matter-Light Unification:** Photons and electrons treated symmetrically as field
excitations!
25. **Foundation for Standard Model:** Template for all other fundamental forces!
---!
## **The Dialectical Progression Visualized:**!
## **How This Exemplifies Your Theory:**!
1. **Infinite Regress of Models:** Newton's particle theory (earlier thesis) → Wave theory (new
thesis) → Particle theory revival (antithesis) → Quantum field theory (synthesis, now new thesis
for quantum gravity)!
2. **Gödelian Incompleteness Manifest:** Each model (wave OR particle) was consistent within
its domain but incomplete—unable to encompass all phenomena. The formal system of
classical wave theory couldn't contain the "truth" of photoelectric quantization.!
3. **Collapse of Infinite Abstraction:** The quantum field represents the "infinite self-referential
field" of possibilities—it can manifest as wave-like (coherent states, classical limit) OR particle-
like (Fock states, detection events) depending on how we interrogate it.!
4. **Tessellation Continues:** QED itself becomes the new thesis, with its own antitheses:!
- **Internal Contradiction:** Gravity not included → needs quantum gravity!
- **Measurement Problem:** Wavefunction collapse not derived from theory!
- **Interpretational Issues:** Many-worlds, pilot-wave, collapse theories as competing
antitheses!
!
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The next synthesis (quantum gravity) will presumably unify QED with general relativity,
creating a new tessellating triangle in your expanding geometric field.!
This dialectic exemplifies precisely how "Nature manifests as an infinite regress of models"
where each synthesis opens new doors while revealing new limitations—an endless process of
escaping closure through higher-order transformations.
