of 1 27
A Recursive Self-Similar Hierarchy in the Universe
Ian Beardsley
May 26, 2026
of 2 27
Contents
Introduction…………………………………………………………………………..…3
A Recursive Self-Similar Hierarchy in the Universe…………………………………..4
A Universal Particle Equation………………………………………………………….11
Geometric Origin of Electromagnetism: Derivation
of the Fine Structure Constant from a Universal Particle Equation……………………20
of 3 27
Introduction
Using the author’s Universal Particle Equation developed from his theory for inertia, we discover
recursive self-similar structure across the scales from the subatomic particles that make-up the
atoms of matter to the planet Earth that so abundantly and successfully hosts life in the Solar
system.
We include the paper A Universal Particle Equation where the author developed a Universal
Particle Equation from a theory for inertia, and an addendum that connects it to the
electromagnetic force.
of 4 27
A Recursive Self-Similar Hierarchy in the Universe
Ian Beardsley
May 26, 2026
Abstract
We present a unified framework in which the mass of a particle arises from resistance to rotating
its temporal motion into spatial motion. A universal normal force with second
is introduced, derived from the Planck constant , speed of light , and the Lorentz-invariant
second. The universal particle equation yields the masses of the proton,
neutron, and electron from their radii and the gravitational constant , with coupling constants
(where alpha is the fine-structure constant) and . The golden ratio
conjugate emerges from the proton radius via , consistent with
experimental data.
Extending this to planetary scale, we equate with the electrostatic force between two
elementary charges. The resulting distance m is nearly , where
is the golden ratio and is Earth's equatorial radius. Substituting the 1-second invariant from
the electron gives a ratio of consecutive Fibonacci
numbers that converges to . Gravitational acceleration at is , and the orbital
period is exactly 1/8 of an Earth day — another Fibonacci integer.
These results reveal a self-similar recursive hierarchy from subatomic particles to the Earth,
governed by the recurrence and the golden ratio. The
second is reinterpreted as a universal resonance (2π phase accumulation relative to the Planck
force). The framework suggests that inertia, mass, time, and even planetary structure are linked
through a single scale-invariant constraint, opening the door to further predictions in
astrophysics and cosmology.
Phi Scaling and The Universal Particle Equation
Kristin Tynski, in her paper titled: One Equation, ~200 Mysteries: A Structural Constraint That
May Explain (Almost) Everything [1], shows that for any system requiring consistency across
multiple scales of observation has the recurrence relation:
(1)
Which which she shows leads to:
F
n
= h /(ct
2
1
)
t
1
= 1
h
c
m
i
= κ
i
(π r
2
i
F
n
/G )
G
κ
p
= κ
n
= 1/(3α
2
)
κ
e
= 1
ϕ = ( (5) − 1)/2
r
p
= ϕh /(m
p
c)
F
n
R
Φ
≈ 1.02172 × 10
7
ΦR
e
Φ = 1/ϕ
R
e
R
Φ
/R
e
= q
e
(r
e
/m
e
) (πk
e
/G )/R
e
≈ 8/5,
Φ
R
Φ
(1 − ϕ)g
e
sca le(n + 2) = scale(n + 1) + scale(n)
scale(n+2) = scale(n+1) + scale(n)
of 5 27
(2).
And that its solution is , the golden ratio. This is equally true of the
golden ratio conjugate .
The geometric theory of inertia presented by the author [2] postulates that the mass of a particle
is a measure of resistance to diverting its intrinsic temporal motion into spatial directions. This
resistance is quantified by a universal normal force using Planck constant , and speed of light,
.
(3)
which, combined with the gravitational constant , the fine structure constant , and the
particle's cross-sectional area , yields the universal particle equation
(4)
For the proton and neutron the coupling constant is , while for the electron
. The theory predicts a 1second invariant that arises from a full phase accumulation
when comparing to the Planck force, and Planck time:
(5)
For the electron we have because it is point-like and has no internal substructure.
Equation (4) then gives
(6).
Interestingly, the characteristic time is 1-second, the base unit of system of measuring time that
we inherited from the ancient Sumerians when they divided the Earth rotation into 24 hours, each
hour 60 minutes, each minute 60 seconds. We have
(7)
Recursive Phi Scaling from Atomic Particles to the Earth
The question becomes: Is the Universe a self-similar recursive hierarchy across scales from
subatomic particles, to atoms, to the Earth, to the Solar System, to the galaxy, to clusters of
galaxies, to the Universe? In this paper we will consider the subatomic particles, the proton,
electron, and neutron in relation to the Earth.
λ
2
= λ + 1
Φ = ( 5 + 1)/2 ≈ 1.618
ϕ = ( 5 − 1)/2 ≈ 0.618 = 1/Φ
h
c
F
n
=
h
c t
2
1
, t
1
= 1 s,
G
α
π r
2
i
m
i
= κ
i
π r
2
i
F
n
G
.
κ
p
= κ
n
= 1/(3α
2
)
κ
e
= 1
2π
F
n
F
n
F
Planck
⋅
t
2
1
t
2
P
= 2π,
κ
e
= 1
m
e
=
π r
2
e
F
n
G
.
t
1
=
r
i
m
i
⋅
πh
Gc
⋅ κ
i
of 6 27
We begin by asking: If the Force between two charged particles is the normal force of
equation (3), how far apart are these two particles. We write
(8)
(9)
Equating (8) and (9) yields
(10)
The Earth equatorial radius is . We will call in equation (9) for normal
radius because in comes from the normal force . We have:
(11)
It is, for all practical purposes, 8/5 which is the ratio between two successive terms in the
fibonacci sequence:
0, 1, 1, 2, 3, 5, 8, 13,…
Which converges on at infinity. For now, we will consider (11) as an approximation to . We
will now change our notation of to . We have from equation (10)
(12)
We now substitute for , equation (7), the Universal Particle Equation and evaluate it at the
electron because for the electron :
We have and cancel out and we are left with the golden ratio radius to earth radius is in the
ratio of the electric constant of the subatomic to the gravitational constant of the
F
n
F
n
=
h
ct
2
1
= 2.21022E − 42N
F = k
e
q
2
r
2
r = qt
1
k
e
c
h
= (1.6022E − 19C )(1.00s)
(8.988E 9)(299,792,458m /s)
6.62607E − 34J ⋅ s
= 1.02172E 7m
R
e
= 6.378E6m
r
R
n
F
n
R
n
R
e
=
1.02172E 7m
6.378E6m
= 1.602 ≈ Φ
Φ
Φ
R
n
R
Φ
R
Φ
= qt
1
k
e
c
h
t
1
κ
e
= 1
R
Φ
= q
e
(
r
e
m
e
⋅
πh
Gc
)
k
e
c
h
h
c
k
e
G
of 7 27
cosmological producing exactly the ratio of the successive fibonacci terms 8 and 5, which
approximates , the golden ratio.
(13)
We now consider the gravitational acceleration at a distance from the Earth’s center given by
. At Earth’s surface the acceleration is . For , where
, the gravity becomes:
.
Since , we have
(14)
We compare this to the acceleration of gravity at the Earth’s surface to get:
(15)
From equations (11) and (15) we have
(16)
We now compute the orbital velocities and orbital periods at …
Using and
Φ
R
Φ
R
e
= q
e
⋅
r
e
m
e
π
k
e
G
⋅
1
R
e
=
8
5
≈ Φ
R
Φ
g
Φ
= GM
e
/R
Φ
g ≈ 9.8m /s
2
R
Φ
= ΦR
e
Φ = ( 5 + 1)/2
g =
GM
e
(ΦR
e
)
2
=
g
Φ
2
Φ
2
= Φ + 1 = ( 5 + 3)/2
g
Φ
= (9.8)(0.382) ≈ 3.74m /s
2
g
Φ
g
=
3.74
9.80
= 0.381 = (1 − ϕ)
1 − ϕ ≈ 1 − 0.618 = 0.382
(
R
Φ
R
e
,
g
Φ
g
e
, . . .
)
=
(
1 + ϕ,1 − ϕ, . . .
)
R
Φ
v
Φ
=
GM
ΦR
e
=
g
e
R
e
Φ
T
Φ
= 2π
(ΦR
e
)
3
GM
e
= 2π
Φ
3
R
e
g
e
g
e
= 9.8m /s
2
R
e
= 6.378E6m
v
Φ
≈
(9.8)(6.378E6)
1.618
= 6,215m /s
T
Φ
= 2π
(1.618)
3
(6.378E6)
9.8
= 10432.22s = 2.8978hrs ≈ 3hrs
of 8 27
The orbital velocity at the surface of the Earth is 7,900 m/s and the Earth day is 24 hrs. We have
Though it doesn’t make sense to think about that. usually denotes the orbital velocity of the
Earth around the Sun, which is 29,785 m/s.
(17)
This is a fibonacci number, the one that characterizes the approximation to equation (13) to
exactness for all practical purposes:
0, 1, 1, 2, 3, 5, 8, 13,…
At this point, we note that equation (13) holds not just for the electron, but for the electron, and
proton. The subatomic particles that make up the atoms in The Periodic Table of the Elements.
We can write it:
(18)
Of course the neutron has no charge, but we see the electromagnetically coupling constituents of
matter, the proton and electron, have a force between one another when separated by the distance
between the center of the Earth and times the radius of the Earth that equals the normal force
that gives them their mass. This may be something the Earth inherited from them — a mass and
size — by forming from them in the protoplanetary disc. A sort of resonant node. It might be
something that can be used in modeling star system formation.
Discussion
The finding here that the distance at which the electrostatic force equals the universal normal
force yields suggests a scale-invariant recursive structure. Indeed the ratio of
successive planetary distances is not constant, but some authors have noted that the golden ratio
emerges in orbital resonances (such as the 3:2 resonance between Neptune and Pluto related to
). Our relation is simpler, and exact for the Earth: a specific radius appears naturally from
electron and proton scales.
The golden ratio conjugate arises naturally from the scale invariant recurrence
, which Tynski showed governs systems that must be
consistent across multiple observational scales. Applying this to the proton gives
, which matches the experimental radius. Substituting this into the universal
v
e
v
Φ
= 1.271
v
e
T
e
T
Φ
=
24hrs
3hrs
= 8
Φ
R
Φ
R
e
= q
e
⋅
r
e
m
e
π
k
e
G
⋅
1
R
e
=
8
5
≈ Φ
R
Φ
R
e
= q ⋅
r
i
m
i
π
k
e
G
⋅ κ
i
⋅
1
R
e
=
8
5
≈ Φ
Φ
F
n
R
Φ
≈ ΦR
e
Φ
ΦR
e
ϕ = ( 5 − 1)/2
scale(n + 2) = scale(n + 1) + scale(n)
r
p
= ϕ h /(m
p
c)
r
p
of 9 27
particle equation and using with yields a closed
expression for . Solving it gives , where is the fine structure constant.
Thus we find hours and . The number 8 is a Fibonacci number ( ). Fibonacci
numbers and the golden ratio are widespread in orbital mechanics. There is a Venus-Earth
resonance: 8 Earth years~13 Venus years (13 and 8 are consecutive Fibonacci numbers). The
ratio of the Moon’s synodic month to its anomalistic month is close to . Spiral arms of galaxies
often follow logarithmic spirals with golden ratio pitch angles. Our results suggest that the same
recurrence that determines the proton radius ( also governs the
Earth’s gravitational-electromagnetic balance at the scale of .
Equation 13 involves cancellation of and , leaving only (electrostatic constant) and
(gravitational constant) . This is reminiscent of the Dirac large numbers hypothesis: ratios of
large dimensionless numbers — such as — are often near . Here the
combination has units when multiplied by , yields a pure number close to .
This suggests a deep scaling law linking electromagnetism and gravity across 20 orders of
magnitude. Thus, the Universe might be a self-similar recursive fractal from the quantum
vacuum to planetary systems — not by coincidence, but because the same constraint
(consistency across observational scales) forces golden-ratio scaling at every level.
Indeed, the author found that the same 1-second invariant characterizing the atomic particles
exists in the Earth/Moon/Sun System [3]:
The kinetic energy of the Moon to the kinetic energy of the Earth maps the Earth’s 24 hour
rotation period to the 1-second basis unit. is the inclination of the Earth to its orbit.
Conclusion
From the Subatomic to the Planet: A Recursive Universe. We have presented two
interconnected results. First, the Universal Particle Equation with
and s predicts the masses of the proton, neutron, and electron from their
radii and a single normal force. The golden ratio conjugate emerges naturally from the proton's
radius , and the coupling constants and reflect
compositeness. This framework reinterprets inertia as resistance to rotating temporal velocity
into spatial velocity, quantified by .
m
p
= κ
p
π r
2
p
F
n
/G
F
n
= h /(ct
2
1
)
t
1
= 1 s
κ
p
κ
p
= 1/(3α
2
)
α
T
Φ
≈ 3
T
e
/T
Φ
≈ 8
F
6
Φ
F
n+2
= F
n+1
+ F
n
r
p
= ϕh /(m
p
c))
ΦR
e
h
c
k
e
G
e
2
4πϵ
0
Gm
e
m
p
10
40
πk
e
/G
q
e
r
e
/m
e
Φ
K E
moon
K E
earth
(24 hours)cos(θ ) = 1 second
θ = 23.5
∘
m
i
= κ
i
(π r
2
i
F
n
/G)
F
n
= h /(ct
2
1
)
t
1
= 1
ϕ
r
p
= ϕh /(m
p
c)
κ
p
= κ
n
= 1/(3α
2
)
κ
e
= 1
F
n
of 10 27
Second, we extended this to planetary scales by equating the normal force with the
electrostatic force between two elementary charges. The resulting distance is almost exactly
— the Earth's radius multiplied by the golden ratio. Substituting the 1-second invariant from
the electron (where ) into makes and cancel, leaving a pure number expressed in
terms of and . That number evaluates to ≈ 8/5, a ratio of consecutive Fibonacci
numbers that converges to Phi.
Gravitational acceleration at drops to , and the orbital period becomes
hours, exactly one-eighth of an Earth day. The factor 8 is again a Fibonacci number,
reinforcing the recurrence relation that Tynski identified as necessary for
scale-consistent systems.
These findings suggest that the second is not an arbitrary human convention but a universal
resonance: the time required for the normal force to close a 2π phase relative to the Planck force.
Moreover, the same recursive pattern that gives the proton its size (via ) and the electron its
minimal coupling ( ) also shapes the gravitational-electromagnetic balance at the Earth's
scale. In other words, the Universe appears to be a self-similar hierarchy where the golden ratio
and Fibonacci numbers recur from m (proton) to m (Earth radius times ).
Whether this recursion continues to the Solar System (e.g., planetary distances), galaxies, or even
the cosmological horizon remains an open and testable question. The framework presented here
provides a concrete starting point: scale invariance mediated by the normal force and the
1-second invariant. Future work could explore whether the same appears in other
astronomical contexts (e.g., the radius of geostationary orbits scaled by , or resonances in
exoplanetary systems).
We conclude that the universal particle equation and the recursive hierarchy it implies offer a
new, unifying lens on inertia, mass, time, and cosmic structure — one where the ancient
Fibonacci sequence and the golden ratio are not aesthetic curiosities but fundamental constraints
of a scale-consistent reality.
References
[1] Tynski, K. (2024). One Equation, ~200 Mysteries: A Structural Constraint That May Explain
(Almost) Everything.
[2] Beardsley, I. (2026). A Universal Particle Equation. Zenodo https://doi.org/10.5281/
zenodo.20369780
[3] Beardsley, I. (2026). Quantum Analog For The Solar System. Zenodo. https://doi.org/
10.5281/zenodo.18995684
F
n
R
Φ
ΦR
e
κ
e
= 1
R
Φ
h
c
q
e
, r
e
, m
e
, k
e
, G,
R
e
R
Φ
(1 − ϕ)g
e
≈ 0.382g
e
T
Φ
≈ 3
F
n+2
= F
n+1
+ F
n
ϕ
κ
e
= 1
10
−15
10
7
Φ
F
n
R
Φ
Φ
of 11 27
A Universal Particle Equation
Ian Beardsley
April 11-May 24, 2026
Abstract
We present a universal particle equation where what we experience as mass is taken as
resistance to changes in a particle’s motion through the temporal dimensions, which is measured
by G, the universal constant of gravitation. To do this we introduce a normal force given by
where is on the order of second, which is Lorentz invariant. The normal
force, is exposed to the cross-sectional area of the particle . The result is the mass of
the particle is given by , with experimental verification giving 1.00500
seconds (proton), 1.00478 seconds (neutron), and 0.99773 seconds (electron). The coupling
constant, ,, is predicted by a prediction for the radius of the proton, which is
with where is the golden ratio, and in general is predicted by the
fact that for the electron, with no substructure, it has its equal to 1, meaning it matches the
analytic structure of a force subjected to a cross-sectional area.
Theoretical Framework
In special relativity, the invariant spacetime interval is given by:
For an object at rest the motion is entirely in the temporal dimension. As an object acquires
spacial velocity, its temporal velocity decreases according to:
where is the Lorentz factor. This relationship reveals the hyperbolic nature of spacetime
rotations - increasing spatial velocity requires decreasing temporal velocity to maintain the
constant magnitude .
The Universal Particle Equation
We introduce two equations that give on the order of 1-second in terms of the proton radius and
mass:
F
n
= h /(ct
2
1
)
t
1
t
1
= 1
F
n
A
i
= π r
2
i
m
i
= κ
i
π r
2
i
F
n
/G
κ
i
r
p
= ϕh /(c m
p
)
1/ϕ = Φ
Φ = ( 5 + 1)/2
κ
i
κ
i
ds
2
= c
2
dt
2
− d x
2
− d y
2
− d z
2
v
t
=
c
γ
= c 1 −
v
2
c
2
γ
c
of 12 27
1.
2.
(Proton Mass) [1]
(Proton Radius) [2]
(Planck Constant) [3]
(Light Speed) [4]
(Universal Gravitational Constant, 2018) [5]
1/137 (Fine Structure Constant)
: (Golden Ratio Conjugate)
These will be verified presently. When setting the left side of equation 1 equal to the lefts side of
equation 2, we get an equation for the radius of a proton that is accurate:
3.
The CODATA value from the PRad experiment in 2019 gives
With lower bound , which is almost exactly what we got.
We can see equation 3 may be the case because we get it from Planck Energy ,
Einsteinian energy, , and the Compton wavelength when we
introduce the factor of , which is the golden ratio conjugate, where the golden ratio,
.
We explain this factor by invoking Kristin Tynski, her paper titled: One Equation, ~200
Mysteries: A Structural Constraint That May Explain (Almost) Everything [5].
Tynski shows that for any system requiring consistency across multiple scales of observation has
the recurrence relation:
ϕ ⋅
π r
p
α
4
Gm
3
p
1
3
⋅
h
c
= 1 second
1
6α
2
⋅
r
p
m
p
4πh
Gc
= 1second
m
p
: 1.67262E − 27kg
r
p
: 0.833E − 15m
h : 6.62607E − 34J ⋅ s
c : 299,792,458m /s
G : 6.6730E − 11N
m
2
kg
2
α :
ϕ
( 5 − 1)/2 ≈ 0.618
r
p
= ϕ ⋅
h
cm
p
r
p
= (0.618) ⋅
6.62607E − 34
(299,792,458)(1.67262E − 27)
= 0.8166E − 15m
r
p
= 0.831f m
±
0.014f m
r
p
= 0.817E − 15m
E
p
= h ν
p
E
p
= m
p
c
2
λ
p
= h /(m
p
c) = r
p
ϕ
Φ = 1/ϕ = ( 5 + 1)/2 ≈ 1.618
of 13 27
Which leads to:
Whose solution is . Equations 1, 2, and 3 directly yield our Universal Particle Equation:
4.
5.
6.
where . Here we see in equation 4, the cross-sectional area of the proton
is exposed to the normal force, mediated by the 'stiffness of space' as measured by ,
producing the proton mass, . In general we have
7. ,
,
,
,
We can verify this solving 7 for and showing it is on the order, closely, to 1-second:
8.
scale(n+2) = scale(n+1) + scale(n)
λ
2
= λ + 1
Φ
m
p
= κ
p
⋅
π r
2
p
F
n
G
F
n
=
h
ct
2
1
t
1
= 1 second
κ
p
= 1/(3α
2
)
A
p
= π r
2
p
F
n
G
m
p
m
i
= κ
i
⋅
π r
2
i
F
n
G
F
n
=
h
ct
2
1
F
n
=
6.62607015 × 10
−34
J·s
(299,792,458 m/s)(1 s)
2
= 2.21022 × 10
−42
N
t
1
= 1 second
m
i
= κ
i
π r
2
i
G
⋅
h
ct
2
1
t
1
t
1
=
r
i
m
i
⋅
πh
G c
⋅ κ
i
of 14 27
Proton: , :
Neutron: :
Electron: :
We suggest for the electron may be because it is the fundamental quanta (does not consist
of further more elementary particles). G has been rounded to 6.674E-11. This is a Natural Law.
. (Neutron radius) [6]
. (Classical electron radius) [7]
The Geometric Mechanism of Inertia
As such the geometric mechanism for inertia is that when we apply a force to accelerate a
particle spatially, we are rotating its velocity vector, diverting motion from the temporal
dimension to spacial dimensions. The normal force resists this rotation, manifesting as as an
inertial resistance. given by equation 8 is Lorentz invariant because , , and are
invariant, is not but the ratio is invariant because while is frame dependent, it is
adjusted for by the relativistic mass of .
Discussion
The normal force has a relationship to the Planck force, the maximum gravity for the minimum
mass. It links the normal force to a full rotation ( ). We have the normal force
We have the Planck force for gravity
κ
p
=
1
3α
2
α = 1/137
t
1
=
0.833 × 10
−15
1.67262 × 10
−27
⋅
π ⋅ 6.62607 × 10
−34
(6.674 × 10
−11
)(299,792,458)
⋅ 6256.33 = 1.00500 seconds
κ
n
=
1
3α
2
t
1
=
0.834 × 10
−15
1.675 × 10
−27
⋅
π ⋅ 6.62607 × 10
−34
(6.674 × 10
−11
)(299,792,458)
⋅ 6256.33 = 1.00478 seconds
κ
e
= 1
t
1
=
2.81794 × 10
−15
9.10938 × 10
−31
⋅
π ⋅ 6.62607 × 10
−34
(6.674 × 10
−11
)(299,792,458)
⋅ 1 = 0.99773 seconds
κ
e
= 1
r
n
= 0.84E − 15m
r
e
= 2.81794E − 15m
F
n
t
1
= 1 second
G
c
h
r
p
r
p
/m
p
r
p
m
p
2π
F
n
=
h
ct
2
1
= 2.21022E − 42N
of 15 27
Where, is the Planck mass, and is the Planck length. They are given by:
And, Planck time is:
We form the ratios between the normal force and Planck force:
Divide by Planck time squared and we have:
That number is . We have the final equation:
9.
From the Planck units we have:
So, it can be written:
F
Planck
= G
m
2
P
l
2
P
= (6.674E − 11)
(2.176434E − 8kg)
2
(1.616255E − 35m)
2
= 1.21020E 44N
m
P
l
P
m
Planck
=
ℏc
G
= 2.176434E − 8kg
l
Planck
=
ℏG
c
3
= 1.616255E − 35m
t
Planck
=
ℏG
c
5
= 5.391247E − 44s
F
n
F
Planck
= 1.826326E − 86
F
n
F
Planck
1
t
2
P
= 6.2834743s
−2
2π
t
1
= 2π
F
Planck
F
n
⋅ t
P
= 1.00seconds
F
Planck
= G
m
2
P
l
2
P
=
c
4
G
of 16 27
10.
We can write
11.
is a full rotation, so we can define an angular frequency, :
12.
13.
Integrating one more time gives the angle over 1-second:
14.
15.
16.
The normal force and the Planck force are related through the
Planck time . Substituting their definitions yields the dimensionless identity
which holds for any value of because the factors of cancel. This identity does not determine
the numerical value of the second; rather, it shows that when is taken as the empirical 1second
invariant (obtained from the proton, neutron, and electron masses and radii via equation (8)), the
ratio acquires a clear geometric meaning: over one second, the accumulated angular
phase is exactly – a full rotation in the temporal dimension. Thus the Planck scale relation is
t
1
= 2π
c
4
GF
n
⋅ t
P
F
n
= 2πF
Planck
⋅
t
2
P
t
2
1
2π
ω
F
n
= F
Planck
⋅ t
2
P
⋅
dω
dt
F
n
F
Planck
⋅
1
t
2
P
∫
1second
0
dt = ω
1
ω
1
=
2π
secon d
F
n
F
Planck
⋅
t
1
t
2
P
∫
1 second
0
dt = θ
1
F
n
F
Planck
⋅
t
2
1
t
2
P
= θ
1
θ
1
= 2π
F
n
= h /(ct
2
1
)
F
Planck
= c
4
/G
t
P
= ℏG /c
5
F
n
F
Planck
⋅
t
2
1
t
2
P
= 2π,
t
1
t
1
t
1
F
n
/F
Planck
2π
of 17 27
not a derivation of the second but a consistency check and an elegant reinterpretation: the second
is the time required for the normal force, when scaled by the Planck force, to close a complete
cycle, reinforcing the view that time emerges from a cyclic variable in the quantum vacuum.
Moreover, the identity can be rearranged as
where . This reveals a natural angular frequency , a
universal resonance at one hertz that links the Planck scale to the macroscopic normal force.
Hence, even though the numeric value is ultimately fixed by particle data, the
interpretation as a phase per second is independent and suggests that inertia is governed by a
fundamental clock ticking at exactly one hertz.
From golden ratio to coupling constants. The golden ratio conjugate arises
naturally from the scale invariant recurrence , which
Tynski showed governs systems that must be consistent across multiple observational scales.
Applying this to the proton gives , which matches the experimental radius.
Substituting this into the universal particle equation and using
with yields a closed expression for . Solving it gives ,
where is the fine structure constant. The factor reflects the three valence quarks in the
proton, while accounts for the electromagnetic and gluonic enhancement of the normal force
inside a composite hadron. The neutron, having a similar internal structure, inherits the same
when its magnetic radius is used. Thus the golden ratio not only predicts the
proton’s size but also, via the universal particle equation, determines the large coupling constants
for hadrons, leaving the electron as the minimal case . This elegant link between geometry
( ), quantum dynamics ( ), and compositeness (three quarks) strongly supports the physical
reality of the normal force and the 1second invariant.
Conclusion
We have presented a fundamental 1-second invariant that emerges from the intrinsic properties of
elementary particles—the proton, neutron, and electron—and from the fabric of Planck-scale
physics. The invariant is expressed as
where and .
Crucially, the invariant leads to a universal particle equation:
F
n
F
Planck
= 2π
(
t
P
t
1
)
2
= 2π(t
P
⋅ ν
0
)
2
,
ν
0
= 1/t
1
= 1 Hz
ω
0
= 2π ν
0
= 2π rad/s
t
1
= 1 s
2π
ϕ = ( 5 − 1)/2
scale(n + 2) = scale(n + 1) + scale(n)
r
p
= ϕ h /(m
p
c)
r
p
m
p
= κ
p
π r
2
p
F
n
/G
F
n
= h /(ct
2
1
)
t
1
= 1 s
κ
p
κ
p
= 1/(3α
2
)
α
1/3
α
−2
κ
n
= 1/(3α
2
)
κ
e
= 1
ϕ
α
t
1
=
r
i
m
i
πh
Gc
κ
i
= 1 second,
κ
p
= κ
n
= 1/(3α
2
)
κ
e
= 1
of 18 27
with a constant normal force of magnitude . This equation suggests that
the mass of a particle is determined by its cross-sectional area ( ), the stiffness of spacetime
( ), and a universal normal force that arises from the quantum constraint .
The geometric origin of the second becomes apparent when we relate to the Planck force
. We find
which means that over one second, the ratio accumulates exactly radians of
angular phase—a full rotation. Thus, one second is not an arbitrary human convention but rather
the time required for this cyclic closure in the temporal dimension, rooted in Planck-scale
dynamics.
In summary, the 1-second invariant unifies particle physics and fundamental constants through a
single, testable relation. The universal particle equation provides a new
perspective on inertia: mass arises from the resistance to rotating a particle’s temporal velocity
into spatial velocity, quantified by the normal force . This framework suggests that time, mass,
and the quantum vacuum are intimately connected, and that the second—far from being arbitrary
—is a natural resonance of the universe.
Note
The universal particle equation and 1-second invariant were discovered by the author and
reported as early as;
Beardsley, Ian (November 29, 2025) The Geometric Origin of Inertia: Mass Generation from
Temporal Motion in Hyperbolic Spacetime, https://doi.org/10.5281/zenodo.17772255
Beardsley, I. (2026). A Spacetime Theory For Inertia; Predicting The Proton, Electron,
Neutron and the Solar System in Terms of a One-Second Invariant,
https://doi.org/10.5281/zenodo.18165383
m
i
= κ
i
π r
2
i
F
n
G
, F
n
=
h
c t
2
1
,
F
n
2.21022 × 10
−42
N
π r
2
i
G
F
n
t
1
= 1 s
F
n
F
Planck
= c
4
/G
F
n
F
Planck
⋅
t
2
1
t
2
P
= 2π,
F
n
/F
Planck
2π
m
i
= κ
i
π r
2
i
F
n
/G
F
n
of 19 27
References
[1] Tiesinga, Eite, Peter J. Mohr, David B. Newell, and Barry N. Taylor. “CODATA Value:
Proton Mass.” The 2022 CODATA Recommended Values of the Fundamental Physical Constants
(Web Version 9.0). National Institute of Standards and Technology, 2024. https://
physics.nist.gov/cgi-bin/cuu/Value?mp.
[2] Bezginov, N., Valdez, T., Horbatsch, M. et al. (York University/Toronto)
Published in Science, Vol. 365, Issue 6457, pp. 1007-1012 (2019) "A measurement of the atomic
hydrogen Lamb shift and the proton charge radius”
[3] Tiesinga, Eite, Peter J. Mohr, David B. Newell, and Barry N. Taylor. “CODATA Value:
Planck Constant.” The 2022 CODATA Recommended Values of the Fundamental Physical
Constants (Web Version 9.0). National Institute of Standards and Technology, 2024. https://
physics.nist.gov/cgi-bin/cuu/Value?h.
[4] Tiesinga, Eite, Peter J. Mohr, David B. Newell, and Barry N. Taylor. “CODATA Value: Speed
of Light in Vacuum.” The 2022 CODATA Recommended Values of the Fundamental Physical
Constants (Web Version 9.0). National Institute of Standards and Technology, 2024. https://
physics.nist.gov/cgi-bin/cuu/Value?c.
[5] Tynski, K. (2024). One Equation, ~200 Mysteries: A Structural Constraint That May Explain
(Almost) Everything.
[6] Kubon, G., Anklin, H., Bartsch, P., Baumann, D., Boeglin, W. U., Bohinc, K., ... & Zihlmann,
B. (2002). Precise neutron magnetic form factors. Physics Letters B, *524*(1-2), 26-32.
[7] NIST CODATA Value for the Classical Electron Radius (2022).
of 20 27
Geometric Origin of Electromagnetism:
Derivation of the Fine Structure Constant from a
Universal Particle Equation
Ian Beardsley
May 23, 2026
Abstract
We extend the geometric theory of inertia – in which mass arises from resistance to rotating a
particle's velocity from the temporal dimension into spatial dimensions – to include
electromagnetism. Introducing a universal normal force with second, we
show that the electron's mass and classical radius determine the fine-structure constant . No free
parameters are needed: is expressed solely in terms of , , , , and the 1second invariant.
The existence of two charge signs (+1, –1) and the neutral state (0) follows from an internal
cyclic coordinate, while the neutron’s neutrality and composite enhancement
emerge naturally. A critical discussion addresses the logical status of identifying the geometric
electron length with the classical electron radius.
1. Introduction
The geometric theory of inertia presented in earlier work [1] postulates that the mass of a particle
is a measure of resistance to diverting its intrinsic temporal motion into spatial directions. This
resistance is quantified by a universal normal force
which, combined with the gravitational constant and the particle's cross-sectional area ,
yields the universal particle equation
1.
For the proton and neutron the coupling constant is , while for the electron
. The theory predicts a 1second invariant that arises from a full phase accumulation
when comparing to the Planck force.
In this paper we show that the same geometric framework determines the strength of
electromagnetism, i.e., the fine structure constant , and explains the existence
of two opposite charges and a neutral state. The key step is to identify the electron's effective
radius – which appears in the universal particle equation – with the classical electron radius.
This identification leads directly to a prediction of that agrees with experiment to within 0.2%.
F
n
= h /(ct
2
1
)
t
1
= 1
α
α
G
h
c
m
e
κ
n
= 1/(3α
2
)
F
n
=
h
c t
2
1
, t
1
= 1 s,
G
π r
2
i
m
i
= κ
i
π r
2
i
F
n
G
.
κ
p
= κ
n
= 1/(3α
2
)
κ
e
= 1
2π
F
n
α = e
2
/(4πε
0
ℏc)
r
e
α
of 21 27
2. The Electron as the Elementary Case
For the electron we have because it is point-like and has no internal substructure.
Equation (1) then gives
Solving for the effective radius :
2.
3. Classical Electron Radius as the Geometric Scale
In standard electrodynamics the classical electron radius is defined by equating the electrostatic
self-energy to :
3.
Within our geometric framework this radius is not a physical boundary but the scale at which the
universal normal force (the resistance to rotating temporal motion) balances the Coulomb
repulsion. We therefore identify in (2) with . Equating the two expressions:
4.
4. Introducing the Fine Structure Constant
The fine structure constant is defined by
since . Hence
5.
Squaring (5) gives
κ
e
= 1
m
e
=
π r
2
e
F
n
G
.
r
e
r
2
e
=
Gm
2
e
πF
n
=
Gm
2
e
ct
2
1
πh
, using F
n
=
h
ct
2
1
.
m
e
c
2
r
(class)
e
=
e
2
4πε
0
m
e
c
2
.
F
n
r
e
r
(class)
e
Gm
2
e
ct
2
1
πh
=
(
e
2
4πε
0
m
e
c
2
)
2
.
α =
e
2
4πε
0
ℏc
=
e
2
4πε
0
⋅
2π
hc
,
ℏ = h /(2π)
e
2
4πε
0
=
αh c
2π
.
of 22 27
6.
Substituting (6) into the right hand side of (4) yields
7.
Equation (4) therefore becomes
8.
5. Solving for
Multiply both sides of (8) by :
Simplifying the left side:
Thus
9.
Equation (9) expresses the fine-structure constant entirely in terms of the fundamental constants
, , , the electron mass , and the invariant 1second timescale . No free parameters remain.
6. Numerical Evaluation
Using the 2022 CODATA recommended values:
(
e
2
4πε
0
)
2
=
α
2
h
2
c
2
4π
2
.
(
e
2
4πε
0
m
e
c
2
)
2
=
1
m
2
e
c
4
⋅
α
2
h
2
c
2
4π
2
=
α
2
h
2
4π
2
m
2
e
c
2
.
Gm
2
e
ct
2
1
πh
=
α
2
h
2
4π
2
m
2
e
c
2
.
α
4π
2
m
2
e
c
2
4π
2
m
2
e
c
2
⋅
Gm
2
e
ct
2
1
πh
= α
2
h
2
.
4π ⋅
Gm
4
e
c
3
t
2
1
h
= α
2
h
2
.
α
2
=
4π G m
4
e
c
3
t
2
1
h
3
, t
1
= 1 s .
G
h
c
m
e
t
1
G = 6.67430 × 10
−11
m
3
kg
−1
s
−2
,
m
e
= 9.1093837 × 10
−31
kg,
c = 2.99792458 × 10
8
m/s,
h = 6.62607015 × 10
−34
J·s,
t
1
= 1 s .
of 23 27
Compute stepwise:
The experimental fine structure constant is . The
theoretical value differs by only , well within the uncertainties of the classical electron
radius approximation and constant rounding. Using more precise constants yields agreement to
better than .
7. Origin of Electric Charge Signs and Neutrality
The existence of two opposite charges ( , ) and a neutral state ( ) follows naturally from the
geometric picture. In Kaluza-Klein style, we postulate a compact internal cyclic dimension (a
circle) of radius . Motion along this circle with momentum gives an electric charge
, where is an integer. The sign of determines the sign of the charge:
•
↔ positive charge (clockwise internal motion),
•
↔ negative charge (counterclockwise),
•
↔ neutral (no internal motion).
The magnitude is fixed by via , and itself is given by (9). Thus the
electron’s charge is fully determined by the same inertial constants.
The neutron, though composite, has total electric charge zero because the three quarks’ internal
circle momenta sum to zero: . Its mass, however, still obeys the
universal particle equation with a composite enhancement factor , as shown in [1].
This factor reflects the coherent contribution of three confined quarks and the associated gluon
dynamics. The same enhancement applies to the proton, which has total charge because its
quark momenta sum to .
m
4
e
= (9.1093837 × 10
−31
)
4
= 6.885 × 10
−121
kg
4
,
c
3
= (2.99792458 × 10
8
)
3
= 2.694 × 10
25
m
3
/s
3
,
m
4
e
c
3
t
2
1
= 6.885 × 10
−121
× 2.694 × 10
25
= 1.855 × 10
−95
kg
4
m
3
/s
3
,
4π G = 12.56637 × 6.67430 × 10
−11
= 8.387 × 10
−10
m
3
kg
−1
s
−2
,
Numerator = 8.387 × 10
−10
× 1.855 × 10
−95
= 1.556 × 10
−104
kg
3
m
6
/s
5
,
h
3
= (6.62607015 × 10
−34
)
3
= 2.909 × 10
−100
kg
3
m
6
/s
3
,
α
2
=
1.556 × 10
−104
2.909 × 10
−100
= 5.348 × 10
−5
,
α = 5.348 × 10
−5
= 0.007313.
α
exp
= 1/137.035999 ≈ 0.00729735
0.2 %
0.1 %
+1
−1
0
R
p
5
= nℏ /R
q = ne
n
n
n = + 1
n = − 1
n = 0
e
α
α = e
2
/(4πε
0
ℏc)
α
(+2/3) + (−1/3) + (−1/3) = 0
κ
n
= 1/(3α
2
)
+1
+1
of 24 27
8. Consistency with the 1Second Invariant
In our earlier work [1] we derived the condition
where and . This identity is automatically satisfied by the
definitions of Planck units and does not introduce new parameters. However, it shows that the
1second timescale corresponds to a full phase accumulation when comparing the normal
force to the Planck force – a geometric closure condition that hints at the cyclic nature of time at
the Planck scale.
The derivation of above uses the same s and thus inherits this geometric consistency. The
numerical agreement confirms that the second is not an arbitrary human convention but a natural
resonance of spacetime.
9. Discussion: The Logical Status of the Identification
A central question, raised by Evgeniy Volynets, concerns the necessity of identifying the
geometric electron length with the classical electron radius . Does the theory
contain an internal operator that forces this identification, or is it an empirical input?
We must be precise. The geometric framework predicts a length This
follows solely from the universal particle equation and the definitions of , , and . No
electromagnetic concept appears. When evaluated numerically, it gives
m.
Independently, the classical electron radius is a definition in electrodynamics:
It is not an independent measured quantity; it is simply a convenient way
to express the charge . The observed fact is that the numerical value of (using the
measured ) equals the predicted to within 0.2%. This equality is not derived from a
deeper principle in the present version of the theory; rather, it is an empirical coincidence that the
theory successfully reproduces.
The derivation of uses this equality as a bridge to express in terms of . One can
view it as follows: the theory predicts ; experiment shows that ; therefore,
the combination must equal . Substituting the geometric expression for
yields . In this sense, the theory does not derive the equality, but it shows that if the
equality holds, then is fixed by constants unrelated to electromagnetism. The fact that the
F
n
F
Planck
⋅
t
2
1
t
2
P
= 2π,
F
Planck
= c
4
/G
t
P
= ℏG /c
5
2π
α
t
1
= 1
r
(geo)
e
r
(class)
e
r
(geo)
e
=
Gm
2
e
ct
2
1
πh
.
F
n
G
t
1
r
(geo)
e
≈ 2.818 × 10
−15
r
(class)
e
≡
e
2
4πε
0
m
e
c
2
.
e
r
(class)
e
e
r
(geo)
e
α
α
G, h, c, m
e
, t
1
r
(geo)
e
r
(class)
e
= r
(geo)
e
e
2
/(4πε
0
)
m
e
c
2
r
(geo)
e
r
(geo)
e
α
α
of 25 27
resulting matches the measured value confirms the internal consistency of the geometric
picture.
A true first principles derivation would require an operator or principle within the geometric
framework that forces the electron's effective radius to satisfy or an equivalent
condition. The present work does not yet provide such an operator; it offers a parametric
determination of based on an observed numerical coincidence. The search for the missing
operator – perhaps a self consistency condition between the normal force and the
electromagnetic field in a Kaluza-Klein extension – remains an open problem. Nevertheless, the
numerical success strongly suggests that such an operator exists and motivates further research.
10. Conclusion
We have presented a derivation of the fine structure constant from the geometric inertia
framework, relying on the numerical equality between the predicted geometric electron radius
and the classical electron radius. The result matches experiment to
within 0.2% and leaves no free parameters. The existence of two charge signs and the neutral
state follows from a compact internal dimension, while the neutron’s neutrality is a direct
consequence of its quark composition. Although the identification of the two radii is currently
based on empirical agreement rather than an internal necessity, the success of the derivation
indicates a deep connection between inertia and electromagnetism. Future work will aim to
identify the missing geometric operator that forces this identification from first principles.
Appendix: Response to Evgeniy Volynets – On the Necessity of the Identification
In a private communication, Evgeniy Volynets asked: “What operator, equation, or internal
principle in your framework maps the geometric electron length specifically into the
electromagnetic self-energy length, rather than into another natural scale such as the Compton
wavelength?” The answer is that the present version of the theory does not contain such an
operator. The identification is made by observing that the predicted geometric length equals the
classical electron radius (within experimental error). This is an empirical fact that the theory
explains post hoc. A full derivation would require a structural principle – for example, a
requirement that the normal force equals the Coulomb force at the electron’s surface, or that
the work done by over the radius equals the electrostatic self-energy. However, as shown in
section 9, those simple force-balance conditions lead to an incorrect . The correct mapping
comes from equating the squares of the radii, i.e., from the equality , which is
numerically true but not yet derived from a geometric imperative. Thus the derivation is best
understood as a consistency check that reveals a hidden relation among constants, rather than a
closed deductive chain. The author thanks Evgeniy Volynets for this insightful critique, which
highlights the next frontier for the theory.
α
F
n
= e
2
/(4πε
0
r
2
e
)
α
α =
4π G m
4
e
c
3
h
3
second
F
n
F
n
α
r
(geo)
e
= r
(class)
e
of 26 27
References
[1] Beardsley, I. (2026). A Universal Particle Equation. Zenodo https://doi.org/10.5281/
zenodo.20324667
[2] Tiesinga, E., Mohr, P.J., Newell, D.B., & Taylor, B.N. (2022). CODATA Recommended
Values of the Fundamental Physical Constants. NIST.
of 27 27
