of 1 40
A Theory For Inertia, Testing For It With A Torsion Pendulum, And Its
Subatomic Structure Appearing On A Planetary Scale
Ian Beardsley
June 3, 2026
of 2 40
Contents
Introduction……………………………………………………………….3
A Universal Particle Equation…………………………………………….5
Geometric Origin of Electromagnetism: Derivation
of the Fine Structure Constant from a Universal
Particle Equation…………………………………………………………15
On the 1 Hz “Noise” and the Case for a Torsion Pendulum
Test of the Temporal Invariant……………………………………………22
Quantum Analog For The Solar System………………………………….28
of 3 40
Introduction
The hour was first invented in ancient Egypt by dividing the night and day into 24 units, 12 for
the day and 12 for the night. Since the day is longer in the summer, and the night shorter, and in
the winter the day is shorter and the night is longer the length of an hour depends on the season.
The ancient Greek astronomer, Hipparchus, divided the day and night into hours determined by
the length of day and night during spring and fall equinoxes when length of day equals the length
of night, inventing the equinoctial hour used year round. Hipparchus had access to ancient
Babylonian knowledge of celestial motions where they knew the day of 24 hours gave an hour
that could be divided by 60 minutes, and each minute by 60 seconds. The Babylonians got the
base 60 divisions of the hour from the ancient Sumerians. But passage of time wasn't measured
down to the second until Christiaan Huygens invented his pendulum clock, which was demanded
by the astronomical revolution that came about from the work of Copernicus (Earth moves
around the Sun), Galileo (Earth is not at the center of the Universe from looking at Jupiter's
moons with his telescope), Brahe (data for planetary motions), Kepler (explains Brahe's data
introducing elliptical orbits for the planets), and Newton (explains Kepler's laws of planetary
motion with his universal law of gravitation). However, ancient Sumerians, ancient Egyptians,
ancient Babylonians, and 10th century Arabs have reported of dreams and visions come to them
by the Gods that demonstrate knowledge of the second as far back as 3000 BC. They even
connected it to the human heartbeat.
In this paper we present our findings that the second is characteristic of subatomic particles, and
is on the scale of the planets as well. With a theory for inertia that gives us a Universal Particle
Equation
,
,
,
,
the mass of the particle, its radius, a universal normal force, G the universal constant of
gravitation, and a universal time invariant.
On the planetary scale we find
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
r
i
t
1
of 4 40
the Moon, a Solar System Planck-type constant, c the speed of light.
This holds for the evolved Solar System when
The kinetic energy of the Moon and the kinetic energy of the Earth maps the 24 hour day into our
base unit of counting, 1 second. is the inclination of the Earth to its orbit.
In the second paper we present the relationship of the Universal Particle equation to
Electromagnetism.
In the third paper we present a way of testing for the 1-second invariant in a proposed
experiment using a torsion pendulum. The theory for inertia predicts we should find a spike at 1
Hz, and interestingly in torsion pendulum data there is a 1 Hz spike. It has always been treated as
noise, and there are several theories used to account for this noise, but it has always just been
subtracted out. In our experiment we design a torsion pendulum that should account for this
noise based on the theories for it, to see if the 1 Hz spike still appears.
In the fourth paper we present our findings or how the Earth/Moon/Sun system is based on the
same 1-second characteristic time if we formulate it in a fairly direct analog of the hydrogen
atom solution to the Schrodinger wave equation.
The occurrence of 1-second so implicitly in equations from the subatomic to the Solar System is
striking. However, after having tried to solve the Schrodinger wave equation for the Solar
System, I find it becomes so complex that it may be impossible. The wave equation predicts
nodes in an inverse square field for the atom, so while it might predict nodes in the Solar System
from its inverse square field, it can’t be used to figure out their evolution from the protoplanetary
disc in a direct analog of the atom. The strange analogs for the Solar System with the hydrogen
atom seem very striking, but have to be tweaked because of course subatomic particles are very
different than planets, like planets don’t jump form orbit to orbit and emit energy, or all have the
same size. I wish I could go to other star systems and learn of other structures and harmonies in
them, and study the ancient history of them, and how other off-world beings might have
developed their calendars to describe them, and find possible connections between them. That is,
in such a scenario I would become an exoarchaeologist. I feel there is a great mystery here that
pertains to why we are here and where we are going, and I present this in hopes it will be found
by others who may have another piece to the puzzle, because the undertaking of explaining it
would be too immense for one person.
ℏ
2
⊙
GM
3
m
1
c
=
3.0281E8 m
299,792,458 m/s
= 1.010 seconds ≈ 1 second
M
m
ℏ
⊙
K E
moon
K E
earth
(24 hours)cos(θ ) = 1 second
θ = 23.5
∘
of 5 40
A Universal Particle Equation
Ian Beardsley
April 11-June 3, 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, directly.
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 6 40
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
= hν
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 7 40
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
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 8 40
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 .
The dimensionless factor distinguishes elementary particles from composite hadrons.
Remarkably, the same emerges for the electron, proton, and neutron when their
respective are chosen appropriately.
The factor reflects the three valence quarks inside the proton and neutron. The appears
because the proton’s small radius (relative to its mass) is set by the strong interaction, which is
times stronger than electromagnetism. Consequently, the required enhancement scales as
the square of that ratio because it deals with surface area.
The electron: as the baseline!
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
t
1
= 1 second
G
c
h
r
p
r
p
/m
p
r
p
m
p
κ
i
t
1
= 1 s
κ
i
1/3
α
−2
∼ 1/α
κ
e
= 1
of 9 40
For the electron, using its classical radius and mass
, the equation (8) with gives!
!
within 0.23% of 1 second. This shows that the electron naturally satisfies the invariant without
any extra factor. The value is not assumed as a physical radius; rather, the invariant predicts
it. Solving for yields!
!
Concerning bound matter (like atoms) we assume since protons in the nucleus of an atom have
spaces between them due to electric forces, and protons and neutrons may be touching, but not
existing in the same space, the mass of an atom is the sum of the masses of its protons, electrons,
and neutrons as given by the theory of inertia in this paper: they all expose a cross-sectional area
to the normal force.
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
Where, is the Planck mass, and is the Planck length. They are given by:
And, Planck time is:
r
e
= 2.81794 × 10
−15
m
m
e
= 9.10938 × 10
−31
kg
κ
e
= 1
t
1
=
r
e
m
e
πh
Gc
⋅ 1 = 0.99773 s,
r
e
t
1
= 1 s
r
e
r
e
= m
e
Gc
πh
= 2.82 × 10
−15
m,
2π
F
n
=
h
ct
2
1
= 2.21022E − 42N
F
Planck
= G
m
2
P
l
2
P
= (6.674E − 11)
(2.176434E − 8kg)
2
(1.616255E − 35m)
2
= 1.21020E44N
m
P
l
P
m
Planck
=
ℏc
G
= 2.176434E − 8kg
l
Planck
=
ℏG
c
3
= 1.616255E − 35m
of 10 40
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:
10.
We can write
11.
is a full rotation, so we can define an angular frequency, :
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
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
of 11 40
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
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.
F
n
F
Planck
⋅
1
t
2
P
∫
1second
0
dt = ω
1
ω
1
=
2π
second
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
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
of 12 40
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:
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
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
m
i
= κ
i
π r
2
i
F
n
G
, F
n
=
h
c t
2
1
,
2.21022 × 10
−42
N
π r
2
i
G
t
1
= 1 s
F
Planck
= c
4
/G
F
n
F
Planck
⋅
t
2
1
t
2
P
= 2π,
of 13 40
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
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.
F
n
/F
Planck
2π
m
i
= κ
i
π r
2
i
F
n
/G
of 14 40
[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 15 40
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
α = e
2
/(4πε
0
ℏc)
r
e
α
of 16 40
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
.
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
=
αhc
2π
.
of 17 40
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 Gm
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 18 40
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 = n e
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 19 40
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
.
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 20 40
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 Gm
4
e
c
3
h
3
second
α
r
(geo)
e
= r
(class)
e
of 21 40
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 22 40
On the 1 Hz “Noise” and the Case for a Torsion
Pendulum Test of the Temporal Invariant
Ian Beardsley
Hillbilly Research Division (Independent)
(Date: June 2026)
Abstract
The claim of a universal 1second invariant and a concomitant normal force
implies that any dynamical system coupling to the resistance of temporal rotation
should exhibit an anomalous resonant response at exactly ( ). Torsion
pendulums have been used in precision experiments for centuries, but a systematic search for a
sharp, unexplained peak at 1 Hz has never been performed because such a peak is conventionally
dismissed as environmental noise or electronic artifact. This paper reviews the known sources of
1  Hz contamination (Nyquist aliasing, pendulum cross coupling, microseisms, clock
feedthrough) and shows that none of them can account for a persistent, amplitude insensitive,
and drive-phase-locked peak that survives standard control tests. We propose a dedicated torsion
pendulum experiment with oversampling, analog antialiasing filtering, and a set of falsifiable
controls. If the predicted 1  Hz resonance is observed, it would provide the first direct
experimental evidence for the temporal invariant; its absence, after proper artifact elimination,
would falsify the central prediction of the theory.
1. Introduction
In a recent particle scale framework (Beardsley 2026), a universal invariant emerges
from the masses and radii of the proton, neutron and electron when combined with the normal
force The invariant gives rise to a natural angular frequency
( ). The physical interpretation is that inertia originates from the
resistance to rotating a particle’s velocity from the temporal dimension into spatial dimensions.
Consequently, any macroscopic system that involves periodic acceleration – in particular a
driven torsion pendulum – should exhibit a resonant enhancement of its response when driven
exactly at . This enhancement is not a mechanical eigenmode; it is a direct manifestation of
the universal normal force coupling to the pendulum’s cross-sectional area.
Searching the experimental literature, one finds occasional reports of unexplained “bumps” near
1  Hz in torsion balance data, but these are invariably attributed to environmental or electronic
artifacts (microseisms, aliasing, crosstalk, parasitic swing modes). No experiment has ever been
designed to systematically discriminate between those well known artifacts and a genuine new
resonance that would be phase locked to the drive frequency and independent of the pendulum’s
moment of inertia. This paper reviews the physics of 1  Hz noise in torsion pendulums and
τ
0
= 1 s
F
n
= h /(c τ
2
0
)
ω
0
= 2π rad/s
f
0
= 1 Hz
τ
0
= 1 s
F
n
=
h
c τ
2
0
≈ 2.21 × 10
−42
N .
ω
0
= 2π /τ
0
= 2π rad/s
f
0
= 1 Hz
ω
0
of 23 40
outlines a clean, falsifiable experiment that can unambiguously test the temporal invariant
prediction.
2. Why 1 Hz is Dirty – But Not Unambiguously
Precision torsion balances (such as those used in the EötWash experiment or for measuring the
gravitational constant ) are usually operated at much lower frequencies (mHz to tenths of Hz)
to avoid seismic and thermal noise. Nevertheless, when a pendulum is actively driven at 1 Hz,
the following contaminants are known to appear:
2.1 Nyquist aliasing
If the data acquisition samples at a rate , any signal component above the Nyquist frequency
is folded back into the measured band. For a 1  Hz signal of interest, sampling at
would place the Nyquist limit exactly at 1 Hz, leading to severe aliasing (a pure 1 Hz
input can appear as a DC offset or as an arbitrary low frequency). However, this is trivially
avoided by oversampling: with , the Nyquist limit is above 50 Hz, and no aliasing of
a 1 Hz signal occurs. Modern microcontrollers easily achieve 1 kHz sampling, so aliasing is a
solvable problem, not an intrinsic obstacle.
2.2 Parasitic pendular (swinging) modes
A torsion pendulum is suspended by a thin fiber. If the driving force is not perfectly aligned with
the torsional axis, or if the fiber is slightly asymmetric, the drive can couple into translational
swing modes. For a fiber of length , the pendular frequency is For
, . Therefore, a 1  Hz drive can easily excite the swing mode if any
misalignment exists. That swing mode will appear as an anomalous peak in the torsional signal
because the optical readout cannot perfectly distinguish pure rotation from horizontal translation.
This artifact is eliminated by:
•
Balancing the pendulum mass symmetrically and using a fiber with high torsional
stiffness (low swing resonance) or, conversely, by designing the fiber such that the
pendular frequency is far from 1 Hz (e.g., gives ).
•
Using a second, independent sensor (e.g., a lateral position sensor) to monitor and
subtract the swing component.
•
Verifying that the anomaly disappears when the drive amplitude is reduced to zero (no
artificial excitation of the swing mode).
2.3 Environmental microseisms
Building vibrations, HVAC systems, walking on floors, and even computer fans often have sharp
spectral components near 1 Hz. These vibrations act as a direct displacement of the suspension
G
f
s
f
N
= f
s
/2
f
s
= 2 Hz
f
s
≥ 100 Hz
L
f
pend
=
1
2π
g
L
.
L ≈ 0.25 m
f
pend
≈ 1 Hz
L = 1 m
f
pend
≈ 0.5 Hz
of 24 40
point, which is indistinguishable from a torque on the pendulum. This noise is typically reduced
by:
•
Placing the apparatus on a massive concrete block supported by vibration damping foam
or pneumatic legs.
•
Enclosing the pendulum in a vacuum chamber (to also remove air damping and acoustic
coupling).
•
Measuring the ambient acceleration with a seismometer and subtracting its contribution
coherently (cross correlation).
2.4 Electronic clock feedthrough
Many precision instruments, data loggers, and microcontrollers operate internal loops at exactly
1 Hz (e.g., updating a display, polling a sensor, or generating a timing interrupt). Capacitive or
magnetic coupling between the digital lines and the sensitive pendulum readout (a photodiode,
position sensitive detector, or capacitive bridge) can inject a pure 1 Hz voltage directly into the
signal. This artifact is identified by:
•
Disconnecting the drive and the pendulum readout while keeping the electronics
powered; a residual 1 Hz peak indicates clock feedthrough.
•
Shielding all signal cables and using differential (balanced) connections.
•
Changing the microcontroller’s update rate (e.g., from 1 Hz to 1.5 Hz) – a real physical
peak remains at 1 Hz, an electronic artifact follows the clock frequency.
3. Why Previous Null Results Do Not Falsify the Theory
Importantly, the fact that no experiment has ever reported an unexplained 1 Hz peak in a driven
torsion pendulum is exactly what the theory predicts for any experiment not designed to
distinguish the predicted effect from the artifacts listed above. Standard practice is to treat any
low frequency peak as noise and to filter it out or subtract it without further investigation. No
experimental group has had a theoretical reason to perform the controls that would reveal a
genuine new resonance – a resonance that would be:
•
Strictly proportional to the drive amplitude (linear response),
•
Independent of the pendulum’s natural frequency (i.e., it does not shift when the moment
of inertia is changed),
•
Phase locked to the drive signal, and
•
Unaffected by changing the sampling rate, the shielding, or the isolation of the pendulum.
Because those controls have never been systematically applied, the absence of a prior report is
not evidence against the effect; it simply means the effect was never looked for in a way that
could distinguish it from the noise floor.
of 25 40
4. Mathematical Model of the Predicted Resonance
In the temporal invariant theory, a test body of mass and effective cross-sectional area
experiences a normal force when its velocity is rotated from the
temporal to spatial axes. For a torsion pendulum with moment of inertia and torsional stiffness
, the equation of motion in the presence of an external drive torque becomes
where is the torque produced by the coupling of the rotating pendulum mass to the
universal normal force. For a simple geometry (a point mass at distance from the axis), the
invariant contribution is
with . The resulting steady-state amplitude at the drive frequency is given by the
well known driven harmonic oscillator response, but with an additional resonance denominator
that becomes singular when :
Hence, when , the amplitude increases regardless of the pendulum’s natural frequency.
The fractional increase can be estimated from the dimensionless coupling constant
which, for a milligram scale mass and millimeter scale radius, yields a
potentially measurable shift of order rad. Modern capacitive or optical readouts can resolve
better than rad, so the effect is within reach.
5. Experimental Protocol to Unambiguously Test the Prediction
Based on the above analysis, we propose the following minimal experiment that can falsify or
confirm the 1 Hz invariant.
5.1 Apparatus
•
A torsion pendulum with a symmetric crossbar (e.g., a thin aluminium rod, length 20 cm,
with adjustable masses at the ends). The fiber is a 50  µm tungsten wire, length 1  m,
giving a torsional period of several seconds (low natural frequency) to avoid confusion
with the drive.
•
An optical lever (laser + position-sensitive detector) or a high resolution autocollimator,
sampling at 1000 Hz.
m
A
eff
= π r
2
F
n
= h /(c τ
2
0
)
I
k
θ
τ
drive
(t)
I
··
θ + b
·
θ + k
θ
θ = τ
drive
(t) + τ
invariant
(t),
τ
invariant
(t)
m
R
τ
inv
= R F
n
A
eff
sin(ω
0
t + ϕ
0
),
ω
0
= 2π /τ
0
ω
ω = ω
0
θ(ω) =
τ
drive
(ω) +
R A
eff
F
n
I
δ(ω − ω
0
)
k
θ
− Iω
2
+ ibω
.
ω = ω
0
κ =
R A
eff
F
n
I ω
2
0
θ
drive
,
10
−6
10
−8
of 26 40
•
An electromagnetic drive coil and a small permanent magnet attached to the pendulum.
The drive is a pure sine wave from a function generator, with amplitude stabilized.
•
An analog lowpass antialiasing filter (corner frequency 50 Hz) placed immediately after
the photodiode amplifier.
•
A massive vibration isolated base (granite slab on Sorbothane feet) inside a grounded
Faraday cage.
5.2 Control tests
1. Natural frequency variation: add or remove mass at the ends; the pendulum’s torsional
eigenfrequency changes by >30%, but the predicted peak must stay exactly at 1 Hz.
2. Change of drive amplitude: the resonance amplitude should be strictly linear with drive
amplitude. Any nonlinearity (e.g., from magnetic coupling) would indicate an artifact.
3. Change of sampling rate: run the same experiment with sampling rates of 200  Hz,
500 Hz and 1000 Hz. A true physical peak remains unchanged; a digital aliasing artifact
changes dramatically.
4. Electronic crosstalk test: with the pendulum locked (or removed), drive the coil at 1 Hz
and record the readout sensor output. Any observed 1 Hz signal is purely electromagnetic
pickup and must be eliminated by shielding and balanced wiring.
5. Environmental noise map: measure the pendulum output with the drive off for 1 hour. If
a 1  Hz peak appears in the power spectrum, it is due to ambient vibrations or clock
feedthrough – not the predicted effect.
5.3 Falsification criterion
The theory is falsified if, after implementing all the above controls, no statistically significant
excess amplitude is observed at (within the resolution of the frequency generator,
) when compared to neighbouring frequencies (0.9  Hz, 0.95  Hz, 1.05  Hz, 1.1  Hz).
Conversely, a clear, reproducible peak that survives all controls would constitute the first direct
evidence for the temporal invariant and would require a major revision of our understanding of
inertia.
6. Relation to Other Proposed Tests (Plasma Thruster)
The same 1  Hz resonance is also predicted for pulsed plasma thrusters. However, the torsion
pendulum is far simpler, cheaper, and less prone to unmodeled plasma dynamics. A positive
result with the pendulum would immediately justify more ambitious tests (e.g., with a Hall
thruster). A null result, if properly controlled, would rule out the universal coupling at the
macroscopic level, though the particle scale invariant might still hold. Hence the torsion
pendulum test is the ideal first step experiment.
f
0
= 1.000 Hz
±
0.001 Hz
of 27 40
7. Conclusion
The 1 Hz “noise” that appears in all torsion pendulum measurements is a well studied collection
of environmental and instrumental artifacts. None of these artifacts produce a peak that is
simultaneously linear in drive amplitude, independent of the pendulum’s eigenfrequency,
unchanged by sampling rate, and persistent under rigorous shielding. A dedicated experiment that
systematically controls each artifact can either reveal the predicted universal resonance or place
an upper limit on the coupling constant that will falsify the temporal invariant theory. Given the
low cost and high sensitivity of modern torsion balances, such an experiment is both feasible and
urgent. The physics community should therefore move beyond dismissing 1  Hz as “just noise”
and perform the definitive test.
References
[1] Beardsley, I. (2026). “A Universal Particle Equation: Mass, Inertia and the 1Second
Invariant.” Zenodo. DOI: 10.5281/zenodo.19930951 (preprint).
[2] Beardsley, I. & Blackwell, D. E. (2026). “ThreeDimensional Simulation of Informational
WarpBubble Dynamics.” Zenodo.
[3] Newman, R. D. & Bantel, M. K. (1999). “On the status of measurements of Newton’s
gravitational constant.” Meas. Sci. Technol. 10, 445.
[4] Speake, C. C. & Quinn, T. J. (2006). “The gravitational constant: theory and experiment.”
Phys. Today 59, 33.
[5] Matsumura, S. et al. (2015). “Vibration isolation system for a torsion pendulum.” Rev. Sci.
Instrum. 86, 064501.
of 28 40
Quantum Analog For The Solar System
Ian Beardsley
March 7, 2026
ABSTRACT
We find if consider the evolved state of the Solar System, that its quantum analog to the Bohr
atom is based on a characteristic time of one-second and the Earth's Moon as the defining metric.
1.0 The Quantum Solution To The Solar System
The ancient Sumerians (4500 BCE-1900 BCE) used base 60 counting, and divided the Earth day
into 24 hours. The ancient Egyptians (3100 BCE-30 BCE) divided the Earth day into 24 hours as
well. Since they both divided the day into 12 hours, and the night into 12 hours and, in the
winter, the night is longer than the day and in the summer, the day is longer than the night, the
hours in a day, or night, can be longer or shorter depending on the time of the year. The ancient
Greeks took the 24 hour day from the ancient Egyptians (Hipparchus, 190 BCE-120 BCE) and
and used an hour to be represented by the equinoxes when day equals night, inventing the
equinoctial hour. It was Christiaan Huygens (1629-1695) who took the hour that had been
divided up into 60 minutes, with each minute divided into 60 seconds, from the ancient Sumerian
base 60 counting, and built the first pendulum clock that could measure down to the second
accurately. This was fueled by the need of Newton's (1642-1727) world view for gravity and
mechanics that needed to measure time down to a unit as small as a second.
It is an interesting phenomenon that the Moon near perfectly eclipse the Sun. The eclipse ratio
that allow for this is about 400:
where is the radius of the Sun and is the radius of the Moon. is the orbit radius of the
Earth orbit and is the orbital radius of the Moon. The solar radius is about 400 times the lunar
radius; the Earth-Sun distance is about 400 times the Earth-Moon distance.
The number of seconds in a day are given approximately by:
The number of seconds in a day, 86 400, can be factored as:
The factor 400 is the eclipse ratio. The factor (216) relates to sixfold symmetry, hexagonal
tiling, and the approximation used by Archimedes as his starting point for calculating .
The appearance of 86 400 in ancient timekeeping thus incorporates the eclipse ratio, whether by
accident or by design.
Let us suggest that the kinetic energy of the Moon to the kinetic energy of the Earth maps the 24
hour (Earth rotation period) day into 1 second, our basis unit of measuring time:
1.1
R
⊙
R
m
≈ 400 and
r
⊕
r
m
≈ 400
R
⊙
R
m
r
⊕
r
m
1.2 86,400 seconds/day = (24 hours)(60 minutes)(60 seconds)
1.3 86,400 = (6)(6)(6)(400)
6
3
π ≈ 3
π
of 29 40
Where is the inclination of the Earth to its orbit.
Using average orbital velocities. We can get closer to a second using aphelions and perihelions
and perigees and apogees.
The Moon stabilizes Earth's axial tilt:
The Moon stabilizing the Earth's tilt to its orbit prevents extreme hot and cold on Earth and
allows for the seasons. As such the Moon is key to optimizing conditions for life on the planet.
Perhaps making it possible for intelligent life to evolve.
We form a Planck-type constant for the Solar System:
We take to be given by:
Equation 6 is an approximately 1-second expression for the radius and mass of a proton that uses
a 2/3 fibonacci approximation for $\phi$, discovered by the author. Thus we see we can see a
possible 1-second invariant that may exist across vast scales from atoms to the Solar System. We
have
Using Earth's orbital velocity at perihelion.
The ground state energy for a hydrogen atom (One electron orbiting a proton) is:
For the planetary system we would replace (Coulombs's constant) with (Newton's universal
constant of gravity). The product of (the charge of an electron squared) and (the mass of an
electron) become a mass cubed. We will choose the mass of the Moon, . We have the ground
state equation is:
1.4
K E
moon
K E
earth
(24 hours)cos(θ ) = 1 second
θ = 23.5
∘
K E
earth
= (5.9722E 24 kg)(29,800 m/s)
2
= 5.30355E 33 J
7.6745E28 J
5.30355E33 J
(86,400 s)cos(23.5
∘
) = 1.1466 seconds ≈ 1 second
θ = 23.5
∘
±
1.3
∘
(with Moon)
θ = 0
∘
to 85
∘
(without Moon, chaotic)
1.5 ℏ
⊙
= (1 second) KE
earth
ℏ
⊙
1.6 1.03351 s =
1
3
⋅
h
α
2
c
2
3
⋅
π r
p
Gm
3
p
1.7 ℏ
⊙
= (1.03351 s)(2.7396E 33 J) = 2.8314E 33 J ⋅ s
K E
Earth
=
1
2
(5.972E 24 kg)(30,290 m/s)
2
= 2.7396E 33 J
1.8 r
1
= −
ℏ
2
k
e
e
2
m
e
k
e
G
e
2
m
e
M
m
of 30 40
Where we have converted meters to seconds by measuring distance in terms of time with the
speed of light ( ). We see the mass of the Moon maps the kinetic energy of the Earth over one
second to 1 second. The Moon is the metric.
The solution for the orbit of the Earth around Sun with the Schrödinger wave equation can be
inferred from the solution for an electron around a proton in the a hydrogen atom with the
Schrödinger wave equation. The Schrödinger wave equation is, in spherical coordinates
Its solution for the atom is as guessed by Niels Bohr before the wave equation existed:
is the energy for an electron orbiting protons and is the orbital shell for an electron with
protons, the orbital number. I find the solution for the Earth around the Sun utilizes the Moon
around the Earth. This is different than with the atom because planets and moons are not all the
same size and mass like electrons and protons are, and they don't jump from orbit to orbit like
electrons do. I find that for the Earth around the Sun
is the energy of the Earth, and is the planet's orbit. is the radius of the Sun, is the
radius of the Moon's orbit, is the mass of the Earth, is the mass of the Moon, is the orbit
number of the Earth which is 3 and is the Planck constant for the solar system. Instead of
having protons, we have the radius of the Sun normalized by the radius of the Moon.
We see that the Moon is indeed the metric, as we said before.
The kinetic energy of the Earth is (using orbital velocity at perihelion):
1.9
ℏ
2
⊙
GM
3
m
=
(2.8314E 33)
2
(6.67408E − 11)(7.34763E 22 kg)
3
= 3.0281E8 m
1.10
ℏ
2
⊙
GM
3
m
1
c
=
3.0281E8 m
299,792,458 m/s
= 1.010 seconds ≈ 1 second
c
−
ℏ
2
2m
[
1
r
2
∂
∂r
(
r
2
∂
∂r
)
+
1
r
2
sin θ
∂
∂θ
(
sin θ
∂
∂θ
)
+
1
r
2
sin
2
θ
∂
2
∂ϕ
2
]
ψ + V(r)ψ = E ψ
1.11 E
n
= −
Z
2
(k
e
e
2
)
2
m
e
2ℏ
2
n
2
1.12 r
n
=
n
2
ℏ
2
Zk
e
e
2
m
e
E
n
Z
r
n
Z
Z
n
1.13 E
n
= n
R
⊙
R
m
⋅
G
2
M
2
e
M
3
m
2ℏ
2
⊙
1.14 r
n
=
2ℏ
2
⊙
GM
3
m
⋅
R
⊙
R
m
⋅
1
n
E
3
r
n
R
⊙
r
m
M
e
M
m
n
ℏ
⊙
Z
R
⊙
/R
m
R
⊙
R
m
=
6.96E8 m
1737400 m
= 400.5986
E
3
= (1.732)(400.5986)
(6.67408E − 11)
2
(5.972E24 kg)
2
(7.347673E22 kg)
3
2(2.8314E33)
2
=
= 2.727E 33 J
of 31 40
The kinetic energy of the Earth is about equal to the energy of the system, because the orbit of
the Earth is nearly circular. That is
The whole object of developing a theory for the way planetary systems form is that they meet the
following criterion: They predict the Titius-Bode rule for the distribution of the planets; the
distribution gives the planetary orbital periods from Newton's Universal Law of Gravitation. The
distribution of the planets is chiefly predicted by three factors: The inward forces of gravity from
the parent star, the outward pressure gradient from the stellar production of radiation, and the
outward inertial forces as a cloud collapses into a flat disc around the central star. These forces
separate the flat disc into rings, agglomerations of material, each ring from which a different
planet forms at its central distance from the star. In a theory of planetary formation from a
primordial disc, it should predict the Titius-Bode rule for the distribution of planets today, which
was the distribution of the rings from which the planets formed.
Also, the Earth has been in the habitable zone since 4 billion years ago when it was at 0.9 AU.
Today it is at 1AU, and that habitable zone can continue to 1.2 AU. So we can speak of the
distance to the Earth over much time. The Earth and Sun formed about 4.6 billion years ago. As
the Sun very slowly loses mass over millions of years as it burns fuel doing fusion, the Earth
slips minimally further out in its orbit over long periods of time. The Earth orbit increases by
about 0.015 meters per year. The Sun only loses 0.00007% of its mass annually. The Earth is at
1AU=1.496E11m. We have 0.015m/1.496E11m/AU=1.00267E-13AU. So,
The Earth will only move out one ten thousandth of an AU in a billion years. Anatomically
modern humans have only been around for about three hundred thousand years. Civilization
began only about six thousand years ago.
The Moon slows the Earth rotation and this in turn expands the Moon's orbit, so it is getting
larger, the Earth loses energy to the Moon. The Earth day gets longer by 0.0067 hours per million
years, and the Moon's orbit gets 3.78 cm larger per year.
We suggest the Solar system comes into phase with a possible one second invariant when the
Earth-Sun separation, and Earth-Moon separation, have kinetic energies whose ratio maps the 24
hour day into the 1-second base unit as given by equation 4:
That is is when equations 5 and 10 hold:
K E
earth
=
1
2
(5.972E 24 kg)(30,290 m/s)
2
= 2.7396E 33 J
2.727E33 J
2.7396E33 J
100 = 99.5 %
E
3
∼ K E
earth
(1.00267E − 13 AU/year)(1E 9 years) = 0.0001 AU
1.4
K E
moon
K E
earth
(24 hours)cos(θ ) = 1 second
1.5 ℏ
⊙
= (1 second)K E
earth
of 32 40
Something remains to be done. Is there something about the Sun that is common to other types of
stars; stars that are perhaps larger and hotter than the Sun, or perhaps smaller and cooler, or a
different color, like blue or red, instead of yellow? The answer is yes. I actually found something
in ancient Vedic knowledge, in the Hindu traditions. Apparently, in Hindu yoga the number 108
is an important number. I read that yogis today noticed that the diameter of the Sun is about 108
times the diameter of the Earth and that the average distance from the Sun to the Earth is about
108 solar diameters, with 108 being a significant number in yoga. So I wrote the equivalent:
or for any star and habitable planet:
the radius of the star. the orbital radius of the habitable planet. We consider the HR
diagram that plots temperature versus luminosity of stars. We see the O, B, A stars are the more
luminous stars, which is because they are bigger and more massive and the the F, G stars are
medium luminosity, mass, and size (radius). Our Sun is a G star, particularly G2V, the two
because the spectral classes are divided up in to 10 sizes, V for five meaning main sequence, that
it is part of the S shaped curve and is in the phase where the star is burning hydrogen fuel, its
original fuel, not the by products. And the K and M stars are the coolest, least massive, least
luminous.
Let us consider the habitable zones of different kinds of stars. In order to get , the
distance of the habitable planet from the star, we use the inverse square law for luminosity of the
star. If the Earth is in the habitable zone, and if the star is one hundred times brighter than the
Sun, then by the inverse square law the distance to the habitable zone of the planet is 10 times
that of what the Earth is from the Sun. Thus we have in astronomical units the habitable zone of
a star is given by:
the luminosity of the star, the luminosity of the Sun. AU the average Earth-Sun separation,
which is 1. The surprising result I found was, after applying equation 4, hypothetically predicting
the size of a habitable planet, to the stars of all spectral types from F through K, with their
different radii and luminosities (the luminosities determine , the distances to the
habitable zones), that the radius of the planet always came out about the same, about the radius
of the Earth. This may suggest optimally habitable planets are not just a function of their distance
from the star, which is a big factor in determining their temperature, but are functions of their
size and mass meaning the size of the Earth could be good for life chemistry and atmospheric
1.10
ℏ
2
⊙
GM
3
m
1
c
= 1.010 seconds ≈ 1 second
1.15 R
⊕
= 2
R
2
⊙
r
⊕
,
1.16 R
planet
= 2
R
2
⋆
r
habitable
R
⋆
r
habitable
r
habitable
1.17 r
habitable
=
L
⋆
L
⊙
AU
L
⋆
L
⊙
r
habitable
of 33 40
composition, and gravity. Stars of the same particular luminosities, temperatures and colors have
about the same mass and size (radius). Here are some examples of such calculations of stars of
different sizes, colors, and luminosities using equation 4:
F8V Star
Mass: 1.18
Radius: 1.221
Luminosity: 1.95
F9V Star
Mass: 1.13
Radius: 1.167
Luminosity: 1.66
G0V Star
Mass: 1.06
Radius: 1.100
Luminosity: 1.35
As you can see we consistently get about 1 Earth radius for the radius of every planet in the
habitable zone of each type of star. I have gone through all stars from spectral class A stars to
spectral class M stars and consistency got this result. It may be this radius for a planet is optimal
for life, in particular intelligent life, because given we might, for that, need a material
M
⋆
= 1.18(1.9891E 30 kg) = 2.347E 30 kg
R
⋆
= 1.221(6.9634E8 m) = 8.5023E8 m
r
p
= 1.95L
⊙
AU = 1.3964 AU (1.496E11 m/AU) = 2.08905E11 m
R
p
=
2R
2
⋆
r
p
= 2
(8.5023E8 m)
2
2.08905E11 m
=
6.92076E6 m
6.378E6 m
= 1.0851 EarthRadii
M
⋆
= 1.13(1.9891E 30 kg) = 2.247683E 30 kg
R
⋆
= 1.167(6.9634E8 m) = 8.1262878E8 m
r
p
= 1.66 AU = 1.28841 AU (1.496E11 m/AU) = 1.92746E11 m
R
p
=
2R
2
⋆
r
p
= 2
(8.1262878E8 m)
2
1.92746E11 m
=
6.852184E6 m
6.378E6 m
= 1.0743468 EarthRadii
M
⋆
= 1.06(1.9891E 30 kg) = 2.108446E 30 kg
R
⋆
= 1.100(6.9634E8 m) = 7.65974E8 m
r
p
= 1.35 AU = 1.161895 AU (1.496E11 m/AU) = 1.7382E11 m
R
p
=
2R
2
⋆
r
p
= 2
(7.65974E8 m)
2
1.7382E11 m
=
6.751E6 m
6.378E6 m
= 1.05848 EarthRadii
of 34 40
composition similar to that of Earth, and, in turn, an Earth-like gravity for the right atmosphere,
including atmospheric composition, or planetary mass, the planet might need to be around this
size.
2.0 The Solar Solution
Our solution of the wave equation for the planets gives the kinetic energy of the Earth from the
mass of the Moon orbiting the Earth, but you could formulate based on the Earth orbiting the
Sun. In our lunar formulation we had:
We remember the Moon perfectly eclipses the Sun which is to say
Thus equation 2.1 becomes
The kinetic energy of the Earth is
Putting this in equation 2.3 gives the mass of the Sun:
We recognize that the orbital velocity of the Moon is
So equation 2.5 becomes
This gives the mass of the Moon is
Putting this in equation 2.1 yields
2.1 K E
e
= 3
R
⊙
R
m
⋅
G
2
M
2
e
M
3
m
2ℏ
2
⊙
2.2
R
⊙
R
m
=
r
e
r
m
2.3 K E
e
= 3
r
e
r
m
⋅
G
2
M
2
e
M
3
m
2ℏ
2
⊙
2.4 K E
e
=
1
2
⋅
GM
⊙
M
e
r
e
2.5 M
⊙
= 3 r
2
e
⋅
GM
e
r
m
⋅
M
3
m
ℏ
2
⊙
2.6 v
2
m
=
GM
e
r
m
2.7 M
⊙
= 3 r
2
e
⋅ v
2
m
⋅
M
3
m
ℏ
2
⊙
2.8 M
3
m
=
M
⊙
ℏ
2
⊙
3 r
2
e
v
2
m
2.9 K E
e
=
R
⊙
R
m
⋅
G
2
M
2
e
M
⊙
2 r
2
e
v
2
m
of 35 40
We now multiply through by and we have
The Planck constant for the Sun, , we will call , the subscript for Planck. We have
We write for the solution of the Earth/Sun system:
We can write 2.11 as
Where we say
Let us see how accurate our equation is:
We have that the kinetic energy of the Earth is
Our equation has an accuracy of
Which is very good.
Let us equate the lunar and solar formulations:
M
2
e
/M
2
e
2.10 K E
e
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2 r
2
e
v
2
m
M
2
e
ℏ
⊙
L
p
p
L
p
= r
e
v
m
M
e
= (1.496E11 m)(1022 m/s)(5.972E 24 kg) = 9.13E 38 kg ⋅
m
2
s
L
2
p
= r
2
e
v
2
m
M
2
e
= 7.4483E 77 J ⋅ m
2
⋅ kg = 8.3367E 77 kg
2
⋅
m
4
s
2
2.11 K E
e
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2L
2
p
2.12 KE
e
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2ℏ
2
⊙
ℏ
⊙
= 9.13E 38 J ⋅ s
h
⊙
= 2π ℏ
⊙
= 5.7365E 39 J ⋅ s
K E
e
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2L
2
p
=
R
⊙
R
m
(6.67408E − 11)
2
(5.972E24 kg)
4
(1.9891E30 kg)
2(8.3367E 77 kg
2
⋅
m
4
s
2
)
=
R
⊙
R
m
(6.759E 30 J)
R
⊙
R
m
=
6.957E8 m
1737400 m
= 400.426
K E
e
= 2.70655E 33 J
K E
earth
=
1
2
(5.972E 24 kg)(30,290 m/s)
2
= 2.7396E 33 J
2.70655E33 J
2.7396E33 J
= 98.79 %
of 36 40
This gives:
We remember that
And since,
Equation 2.14 becomes
The condition of a perfect eclipse gives us another expression for the base unit of a second. is
another version of the Planck Constant, which is intrinsic to the solar formulation as opposed to
the lunar formulation. We want to see what the ground state looks like and what its characteristic
time is, if it is 1 second like it is for the lunar formulation. Looking at the equation for energy:
We see the ground state should be:
And, it is equal to 1 second. You will notice where in the derivation for the energy we lost
, we have to put it in the ground state equation. The computation is:
K E
e
= n
R
⊙
R
m
⋅
G
2
M
2
e
M
3
m
2ℏ
2
⊙
K E
e
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2ℏ
2
⊙
3
R
⊙
R
m
⋅
G
2
M
2
e
M
3
m
2ℏ
2
⊙
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2L
2
p
2.13 L
p
=
M
2
e
M
⊙
M
3
m
3
⋅ ℏ
⊙
ℏ
⊙
= (hC ) K E
e
hC = 1 second
K E
e
=
1
2
M
e
v
2
e
2.14 2v
m
=
v
2
e
r
e
(1 second)
M
2
e
M
⊙
M
3
m
3
M
2
e
M
⊙
M
3
m
3
=
(5.972E24 kg)
2
(1.9891E30 kg)
(7.34763E22 kg)
3
(1.732)
= 321,331.459 ≈ 321,331
2.15 1 second = 2r
e
v
m
v
2
e
M
3
m
3
M
2
e
M
⊙
L
p
K E
e
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2L
2
p
2.16
L
2
p
GM
2
e
M
⊙
⋅
3
c
= 1 second
n = 3
of 37 40
3.0 Jupiter and Saturn
We want to find what the wave equation solutions are for Jupiter and Saturn because they
significantly carry the majority of the mass of the solar system and thus should embody most
clearly the dynamics of the wave solution to the Solar System. We also show here how well the
solution for the Earth works, which is 99.5% accuracy.
I find that as we cross the asteroid belt leaving behind the terrestrial planets, which are solid, and
go to the gas giants and ice giants, the atomic number is no longer squared and the square root of
the orbital number moves from the numerator to the denominator. I believe this is because the
solar system here should be modeled in two parts, just as it is in theories of solar system
formation because there is a force other than just gravity of the Sun at work, which is the
radiation pressure of the Sun, which is what separates it into two parts, the terrestrial planets on
this side of the asteroid belt and the gas giants on the other side of the asteroid belt. The effect
the radiation pressure has is to blow the lighter elements out beyond the asteroid belt when the
solar system forms, which are gases such as hydrogen and helium, while the heavier elements are
too heavy to be blown out from the inside of the asteroid belt, allowing for the formation of the
terrestrial planets Venus, Earth, and Mars. The result is that our equation has the atomic number
of the heavier metals such as calcium for the Earth, while the equation for the gas giants has the
atomic numbers of the gasses. We write for these planets
So, for Jupiter we have (And again using the maximum orbital velocity which is at perihelion):
Jupiter is mostly composed of hydrogen gas, and secondly helium gas, so it is appropriate that
.
Our equation for Jupiter is
Where is the atomic number of hydrogen which is 1 proton, and for the orbital
number of Jupiter, $n=5$. Now we move on to Saturn...
(9.13E38 J ⋅ s)
2
(6.674E − 11)(5.972E24 kg)
2
(1.989E30 kg)
⋅
3
c
= 1.0172 seconds
E =
Z
n
G
2
M
2
m
3
2ℏ
2
⊙
K E
j
=
1
2
(1.89813E 27 kg)(13720 m/s)
2
= 1.7865E 35 J
E =
Z
H
5
(6.67408E − 11)
2
(1.89813E27 kg)
2
(7.347673E22 kg)
3
2(2.8314E33)
2
E =
Z
H
5
(3.971E 35 J) = Z
H
(1.776E 35 J)
Z
H
=
1.7865E35 J
1.776E35 J
= 1.006 protons ≈ 1.0 protons = hydrogen (H)
Z = Z
H
E
5
=
Z
H
5
G
2
M
2
j
M
3
m
2ℏ
2
⊙
Z
H
n = 5
of 38 40
The equation for Saturn is then
It is nice that Saturn would use Helium in the equation because Saturn is the next planet after
Jupiter and Jupiter uses hydrogen, and helium is the next element after hydrogen. As well, just
like Jupiter, Saturn is primarily composed of hydrogen and helium gas.
The accuracy for Earth orbit is
The kinetic energy of the Earth is
Which is very good, about 100% accuracy for all practical purposes. The elemental expression of
the solution for the Earth would be
Where
In this case the element associated with the Earth is calcium which is protons.
K E
S
=
1
2
(5.683E 26 kg)(10140 m/s)
2
= 2.92E 34 J
E =
Z
6
(6.67408E − 11)
2
(5.683E26 kg)
2
(7.347673E22)
3
2(2.8314E33)
2
=
Z
2.45
(3.5588E 34 J) = Z(1.45259E34 J)
Z(1.45259E 34 J) = (2.92E 34 J)
Z = 2 protons = Helium (He)
E
6
=
Z
He
6
G
2
M
2
s
M
3
m
2ℏ
2
⊙
E
n
= n
R
⊙
R
m
G
2
M
2
e
M
3
m
2ℏ
2
⊙
R
⊙
R
m
=
6.96E8 m
1737400 m
= 400.5986
E
3
= (1.732)(400.5986)
(6.67408E − 11)
2
(5.972E24 kg)
2
(7.347673E22 kg)
3
2(2.8314E33)
2
=
= 2.727E 33 J
K E
e
=
1
2
(5.972E 24 kg)(30,290 m/s)
2
= 2.7396E 33 J
2.727E33 J
2.7396E33 J
100 = 99.5 %
E
3
= 3
Z
2
Ca
G
2
M
2
e
M
3
m
2ℏ
2
⊙
R
⊙
R
m
→ Z
2
Z = 20
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References
Beardsley, I. (2025) Theory For The Solar System And The Atom's Proton; Linking Microscales
To Macroscales, DOI: 10.13140/RG.2.2.19296.34561
Beardsley, I. (2026) How Physics and Archaeology Point to a Natural Constant of 1-Second,
https://doi.org/10.5281/zenodo.18829259
Beardsley, I. (2026) The Sublime and Mysterious Place of Humans in the Cosmos; A Work in
Exoarchaeology, https://doi.org/10.5281/zenodo.18715148
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