of 1 42
The Epoch We Live In
Ian Beardsley
November 7, 2026
of 2 42
Contents
Introduction…………………………………………………………..3
Chapter 1: Overview…………………………………………………4
Chapter 2: Summary of What Has Been Found……………………..22
Chapter 3: The Galactic Clock………………………………………25
Chapter 4: The Human Scale as a Resonance Node
Between the Quantum and Astrophysical Scales……………………26
of 3 42
Introduction
We report findings that are Universal over the epochs and findings that are
characteristic of the current epoch we are living in that intersect that with the
Universal. Because the findings describing the current epoch intersect with the
findings that are Universal over time — like the universal particle law (UPL) and
hydrogen atom ground state analog for the Solar System —- we suggest they are
time nodes in an overall structure that may exist, making all of the findings perhaps
Universal over time. Our equation that maps the 24 hour day (Earth rotation
period) into 1-second (the human basis unit of measuring time) is:
which is the equation that determines the present epoch, and may represent a time
node in an overall universal structure. We have not derived the one second
invariant from first principles in a universal particle equation (UPL), but recognize
that it emerges from the theory behind it. It plays the role of the quantization of
time in a normal force that gives rise to the property of inertia in matter. It just
comes out to be about 1-second. But, the question of whether the second — whose
origin is with ancient Egypt, Summer, and Greece and the Babylonians, their
systems of counting and dividing up the Earth day (rotation period) — has
meaning beyond being a quantization of action in the UPL, is a matter for
archaeology, and metrology — perhaps most accurately, archaeoastronomy.
The main focus of this work is in the idea that the approximately 1-second
invariant exists across 40 orders of magnitude from the quantum to the celestial,
and is characteristic of the epoch we live in — approximately — and may have
something to do with the rhythm of the unfolding of the universe over time, and
have something to say about its origins, structure, and fate.
This is not a paper to be refereed in a physics journal and submitted for
publication, but a speculation on findings so compelling in their structural
dynamics that they — in the author’s view — warrant presentation to the scientific
community.
KE
m
KE
e
(Earth Day)cos(θ
e
) ≈ 1second
of 4 42
Chapter 1: Overview
Introduction
We present a universal particle law (UPL) where what we experience as mass is taken as
resistance to changes in a particle’s motion through the temporal dimension to the spacial when
pushed on, 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), 1.00803 seconds (neutron), and
0.99773 seconds (electron). The coupling constant, ,, is suggest 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. We derive the normal force from the Lagrangian.
We further find we can produce a quantum analog for our planetary system (PSQA) and show
that there is scale invariance from quantum scales to celestial scales (SIQC) in the Universal
Particle Equation (UPL).
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 Law
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
F
n
= h /(ct
2
1
)
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 5 42
1.
2.
(Proton Mass) [1]
(Proton Radius) [2]
: [8]
(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].
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
r
n
0.8367
±
0.0845f m
h : 6.62607E − 34J ⋅ s
c : 299,792,458m /s
G : 6.6743E − 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 6 42
Tynski shows that for any system requiring consistency across multiple scales of observation has
the recurrence relation:
Which leads to:
Whose solution is . Equations 1, 2, and 3 directly yield our Universal Particle Law:
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 ‘pliability 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:
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
of 7 42
8.
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]
Neutron magnetic radius: [8]
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
t
1
=
r
i
m
i
⋅
πh
G c
⋅ κ
i
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
=
0.8367 × 10
−15
1.675 × 10
−27
⋅
π ⋅ 6.62607 × 10
−34
(6.674 × 10
−11
)(299,792,458)
⋅ 6256.33 = 1.00803 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
r
nM
= 0.8367
±
0.0845f m
F
n
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
of 8 42
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
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
which coincides exactly with the classical electron radius. Thus the classical radius – usually
derived from electrostatics – emerges here purely from geometry, , and . The electron
remains point-like; the radius is an effective length that quantifies the ratio required by the
invariant.
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.
Deriving The Normal Force from the Lagrangian
I have developed a theory for inertia that results in a Universal Particle Law (UPL). In order to
remove the magic from the paper, we want to derive the normal force , though the
equation seems intuitively clear. This is the normal force that creates mass as a resistance of the
particle to it when we push on it, rotating some of its velocity out of the temporal and into the
spacial.
To derive from a Lagrangian, we need to honor the geometric premise of the paper:
inertia is the resistance to rotating a velocity vector from the time dimension into space.
∼ 1/α
κ
e
= 1
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,
F
n
G
r
e
/m
e
F
n
=
h
ct
2
1
F
n
=
h
ct
2
1
of 9 42
In relativistic mechanics, this "rotation" is mathematically exact—it is the rapidity (or Lorentz
boost angle). A force is precisely the rate of change of momentum as this rotation occurs. By
combining this geometric fact with the quantum of action, the derivation flows naturally.
Here is a formal, step-by-step derivation.
Step 1: The Geometric Action for a Boost
In special relativity, a particle’s 4-velocity is . A spatial acceleration is a rotation of
this 4-vector in spacetime.
The action for a free particle is:
The canonical momentum is:
The force is the time derivative of this momentum:
So far, this is standard physics. The question is: what sets the scale of this force in the UPL?
Step 2: Quantizing the Rotation (The Key Input)
The UPL introduces a fundamental timescale second. Over this interval, the particle’s
velocity vector undergoes a complete "rotation" from the temporal direction into a spatial
direction.
- The temporal momentum of a particle is its energy divided by : .
- By the quantum of action ( ), the temporal momentum is:
When this momentum is fully rotated into the spatial direction, the spatial momentum becomes:
u
μ
= (γc, γ v )
S =
∫
ℒ dt = − m c
2
∫
1 −
v
2
c
2
dt
p =
∂ℒ
∂v
= γ mv
F =
dp
dt
t
1
= 1
c
p
t
= E /c
E ⋅ t
1
∼ h
p
t
=
h
c t
1
of 10 42
Step 3: Deriving the Force from the Lagrangian
In Lagrangian mechanics, the generalized force is the time rate of change of the canonical
momentum:
Over the fundamental interval , the momentum changes from (purely temporal) to
(purely spatial). Therefore, the average rate of change—which is the resisting inertial force —
is:
This is my equation, derived directly from the geometric definition of force , paired
with the quantum-geometric assertion that the minimum momentum quantum over a timescale
is .
Step 4: Writing the Explicit Lagrangian
If you want to see this force appear explicitly in a Lagrangian , we can write a toy Lagrangian
that yields this constant force. Since , we integrate to find the potential:
Thus, the Lagrangian for a particle experiencing this resisting force in 1D is:
Applying the Euler-Lagrange equation:
p = p
t
=
h
c t
1
F =
d
dt
(
∂ℒ
∂v
)
t
1
0
p =
h
ct
1
F
n
F
n
=
Δ p
Δt
=
h
ct
1
t
1
=
h
c t
2
1
F = d p /dt
t
1
h /(ct
1
)
ℒ
F = −
∂V
∂x
V(x) = −
h
ct
2
1
x
ℒ =
1
2
mv
2
− V(x) =
1
2
mv
2
+
h
ct
2
1
x
d
dt
(
∂ℒ
∂v
)
−
∂ℒ
∂x
= 0
of 11 42
Hence, the inertial resistance (the force you must overcome to accelerate) is exactly
.
Step 5: A More Elegant 4D Lagrangian
If you prefer a fully covariant formulation that explicitly includes the "rotation" from time to
space, you can write the action using a Lagrange multiplier that enforces the quantization of the
proper time:
Varying this action with respect to the spatial coordinates yields the Euler-Lagrange equation:
By identifying the quantum constraint (the temporal momentum), the spatial force
becomes:
This derivation ties the directly to the action principle without inserting it as an ad-hoc
potential—it emerges from the constraint that the system must complete its time-to-space
rotation within exactly seconds.
Summary of the Derivation
d
dt
(mv) −
h
ct
2
1
= 0
m
·
v =
h
ct
2
1
F
n
= m
·
v =
h
ct
2
1
λ
S =
∫
−m c −
·
x
μ
·
x
μ
+ λ
(
·
x
0
c
− t
1
)
dτ
x
i
d
dτ
m
·
x
i
−
·
x
μ
·
x
μ
=
∂λ
∂x
i
λ =
h
ct
1
F
i
=
dp
i
dt
=
h
ct
2
1
F
n
t
1
of 12 42
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:
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
t
Planck
=
ℏG
c
5
= 5.391247E − 44s
of 13 42
We form the ratio between the normal force and the Planck force:
Dividing by the Planck time squared:
That number is . It arises from the convention difference between and : the Planck time is
defined with , while the normal force is written with . The factor of is therefore
an algebraic consequence of the -to- conversion and not a dynamical phase.
This gives the identity
which holds for any value of , because the factors of cancel. The identity does not determine
the numerical value of the second. What it does show is that when is taken as the empirical
one-second invariant — obtained from the proton, neutron, and electron masses and radii via
equation (8) — the ratio acquires a definite numerical value, and the combination
above evaluates to s⁻².
From the Planck units we have
so the identity can be written as
or equivalently,
This may be read as a relation between the Planck scale and the macroscopic normal force. It is a
consistency check and a useful reformulation, not a derivation of the one-second value.
F
n
F
Planck
= 1.826326 × 10
−86
.
F
n
F
Planck
⋅
1
t
2
P
= 6.2834743 s
−2
.
2π
h
ℏ
ℏ = h /(2π)
h
2π
h
ℏ
F
n
F
Planck
⋅
t
2
1
t
2
P
= 2π,
t
1
t
1
t
1
F
n
/F
Planck
2π
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
.
of 14 42
The identity can be rearranged as
where Hz. This reveals a natural angular frequency rad/s — a
resonance at one hertz that links the Planck scale to the macroscopic normal force. The
interpretation as a frequency is independent of the derivation of the one-second value, and it
suggests that inertia is governed by a characteristic clock ticking at approximately one hertz.
The normal force and the Planck force are related through the
Planck time . Substituting their definitions yields the dimensionless identity
The factor of arises from the convention difference between and : the Planck time is
defined with while the normal force is written with Writing
and substituting,
so that the is an algebraic consequence of the -to- conversion and not a dynamical phase.
The identity holds for any value of , because the factors of cancel; it does not by itself
determine the numerical value of the second.
What the identity does show is that when is taken as the empirical one-second invariant —
obtained from the proton, neutron, and electron masses and radii via equation (8) — the ratio
acquires a definite numerical value, and the combination evaluates to
s⁻². This may be read as a statement about the relationship between the Planck scale and the
macroscopic normal force, but it is a consistency check rather than a derivation. The one-second
value is fixed by particle data, not by the Planck-force identity.
The identity can be rearranged as
F
n
F
Planck
= 2π
(
t
P
t
1
)
2
= 2π (t
P
⋅ ν
0
)
2
,
ν
0
= 1/t
1
= 1
ω
0
= 2π ν
0
= 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 .
2π
h
ℏ
ℏ = h /(2π)
h
t
2
P
= ℏG /c
5
= hG /(2 π c
5
)
F
n
F
Planck
⋅
1
t
2
P
=
hG
c
5
t
2
1
⋅
2π c
5
hG
=
2π
t
2
1
,
2π
h
ℏ
t
1
t
1
t
1
F
n
/F
Planck
F
n
/F
Planck
⋅ 1/t
2
P
2π
F
n
F
Planck
= 2π
(
t
P
t
1
)
2
= 2π (t
P
⋅ ν
0
)
2
,
of 15 42
where Hz. This reveals a natural angular frequency rad/s — a
resonance at one hertz that links the Planck scale to the macroscopic normal force. The
interpretation as a frequency is independent of the derivation of the one-second value, and it
suggests that inertia is governed by a characteristic clock ticking at approximately one hertz.
The 0.73% discrepancy between the electron’s invariant (0.99773s) and the proton/neutron
invariants (1.00500s) is neither a statistical fluctuation nor an error. It is the direct numerical
manifestation of the fine structure constant . This shows that the composite hadrons
experience an additional electromagnetic self-energy correction to the base temporal resonance,
while the fundamental electron probes the bare vacuum pressure. Far from undermining the
theory, this discrepancy rigorously validates the coupling of the normal force to the
electromagnetic substructure. We see the errors on the neutron data for its radius are very large,
so it is difficult to determine very accurately. In fact, the radii of the proton and neutron have
no exact value. Rather they are fuzzy clouds of subatomic particles and their values depend on
the method used to measure them. Though, it is not unreasonable to suggest that their values fall
within a range that produces on the order of about 1 second.
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 pliability of spacetime
( ), and a universal normal force that arises from the quantum constraint .
We can offer the following interpretation: The geometric origin of the second becomes apparent
when we relate to the Planck force . We find
ν
0
= 1/t
1
= 1
ω
0
= 2π ν
0
= 2π
t
p
≈ t
e
(1 + α)
F
n
t
n
t
1
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
,
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
of 16 42
which means that over one second, the ratio accumulates exactly radians of
angular phase—a full rotation. As such, we can interpret one second not an arbitrary human
convention but rather the time required for this cyclic closure in the temporal dimension, rooted
in Planck-scale dynamics. Currently this is just an interpretation that could eventually pan out.
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.
A Planetary System Quantum Analog (PSQA) And Scale Invariance from
Quantum Scales to the Celestial Scales (SIQC)
The Earth Quantum Analog: Since the author finds the following holds for the kinetic energies of
the moon and the Earth:
We used average orbital velocities. And, accounting for the Earth’s inclination to its orbit ( =
23.5 deg), we have
Earth day=(24)(60)(60)=86,400 seconds. We can bring down equations these equations closer to
a second using the Moon’s orbital velocity at aphelion, and Earth’s orbital velocity at perihelion.
We have:
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
K E
m
K E
e
(Ear th Da y) = 1.25seconds
K E
m
=
1
2
(7.4767E 22kg)(1,022m /s)
2
= 3.83726E 28 J
K E
e
1
2
(5.972E 24kg)(29,785m /s)
2
= 2.649E 33J
θ
e
K E
m
K E
e
(Ear th Da y)cos(θ
e
) = 1.146seconds
K E
m
=
1
2
(7.347673E 22kg)(966m /s)
2
= 3.428E 28 J
of 17 42
The Moon allows for the evolution of complex, intelligent life because it stabilizes the earth tilt
to its orbit allowing for the seasons and preventing temperature extremes. We live in an
interesting time when the Moon near perfectly eclipses the Sun. This is because while the Sun is
400 times larger than the Moon, it is 400 times further from the Earth than the Moon is. That is
the perfect eclipse is given by the orbital radius of the Earth to the Moon’s orbital radius
equals the Solar radius to the lunar radius :
The Earth's rotation is slowing down, and as a direct result, the Moon is slowly drifting away.
Earth's Rotation Slowing: The length of a day is increasing by about 2.3 milliseconds per
century. This is the long-term average rate caused primarily by the Moon's tidal pull. (Note that
this rate can be influenced by other factors; for example, climate change is currently contributing
about 1.33 milliseconds per century).
Moon's Orbit Growing: The Moon is receding from Earth at a rate of about 3.8 centimeters (1.5
inches) per year.
The Tidal Connection
These two phenomena are linked by tidal friction. The Moon's gravity creates tides in Earth's
oceans. As Earth rotates, the tidal bulges are pulled slightly ahead of the Moon, creating friction
that slows Earth's spin.
To conserve angular momentum in the Earth-Moon system, this loss of Earth's rotational speed is
transferred to the Moon, boosting its orbital energy and causing it to spiral outward into a larger
orbit. This recession rate has been precisely measured for decades using the Lunar Laser Ranging
experiment, which bounces lasers off reflectors left on the Moon by Apollo astronauts.
Long-Term Perspective
While these changes are tiny on human timescales, they have significant effects over millions of
years. About 250 million years ago, a day on Earth was roughly 23 hours long. In about 600
million years, the Moon will have drifted so far that total solar eclipses will no longer be
possible.
K E
e
=
1
2
(5.972E 24kg)(30,290m /s)
2
= 2.7396E 33J
K E
m
K E
e
(Ear th Da y)cos(θ
e
) = 0.991seconds
K E
m
K E
e
(Ear th Da y)cos(θ
e
) ≈ 1second
r
e
r
m
R
⊙
R
m
r
e
r
m
=
R
⊙
R
m
of 18 42
We might guess from the ground state of the Bohr hydrogen atom, which is given by
that a Planck-type constant is given by the Moon, if it is the metric by which the Solar System
does its measuring, that it is given by:
Where is a celestial Planck type constant for the Earth/Moon/Sun system, is the mass of
the Moon, and we have let: , , . This gives us that:
Nicely, this has that:
Suggesting that that the quantum analog for the Earth/Moon/Sun system and the Universal
Particle Equation both have a characteristic time of about 1 second, we should find that the
universal particle equation should convert to the Earth quantum analog to give one second as
well. We have the Universal Particle Law:
For the Earth/Moon/Sun system we should have:
Where we have put for the radius of particle, the radius of the Earth, for the mass of the particle,
the mass of the Earth, and substituted for (Planck constant) (Earth\Moon\Sun system
Planck-type constant).
Interestingly, we find
Where is the average orbital radius of the Moon, which has a near perfectly circular orbit
(low eccentricity) and, is the radius of the Sun.
a
0
=
ℏ
2
k
e
m
e
e
2
ℏ
2
⊙
GM
3
m
1
c
= 1.00seconds
ℏ
⊙
M
m
ℏ → ℏ
⊙
k
e
→ G
m
e
e
2
→ M
3
m
ℏ
⊙
= 2.81733E 33J ⋅ s
h
⊙
= 2π ℏ
⊙
= 1.77018E 34J ⋅ s
ℏ
⊙
≈ (1.00seconds)K E
e
t
1
=
r
i
m
i
⋅
πh
Gc
⋅ κ
i
t
1
=
R
earth
M
earth
πh
⊙
Gc
κ
earth
h
h
⊙
t
1
=
(6.371E6m)
5.972E 24kg
π (1.77018E 34J ⋅ s)
(6.6743E − 11)(299,792, 458m /s)
κ
earth
= 1.7785seconds(κ
earth
)
κ
earth
=
r
moon
R
⊙
=
3.84399E8m
6.96E8m
= 0.5522974
r
moon
R
⊙
of 19 42
Taken together, the preceding Bohr-type analog and the kinetic-energy definition give two routes
to the same effective action , and they agree numerically to within a few percent. This is a
consistency check on , not an independent derivation of the one-second value: The 1-second
scale enters in the Bohr analog’s condition =1.00 s, which fixes from G, c, alone; the
numerical agreement is then the cross-check, not the definition. The coupling
itself is physically meaningful rather than arbitrary. The Moon is the metric by which the solar-
system Planck-type constant is determined through the Bohr-type analog, while the Sun’s radius
sets its mass and hence the Earth’s orbital kinetic energy that enters . The ratio
therefore ties the measuring scale (lunar orbit) to the source scale (solar radius). With this
coupling the UPL template returns , so the Earth calculation is a scale-
invariance check: the second is an input to the effective action, and the UPL shows how it
propagates to the Earth–Moon–Sun scale through a coupling fixed by the system’s own
geometry.
Galactic (Solar) Quantum Analog: We begin with the orbital velocity of the Sun around the
center of the galaxy, and the mass of the Sun so we can compute its kinetic Energy in its orbit
around the center of the galaxy. We have:
!
!
Using the solar radius and solar mass as the prefactor, and a coupling corresponding to the asteroid-belt
ratio, the UPL gives
With
the product is approximately one second. This coupling corresponds to , placing
the node at about , inside the asteroid belt.
1.7785seconds(0.5522974) = 0.982s ≈ 1.00seconds
ℏ
⊙
ℏ
⊙
ℏ
⊙
M
m
ℏ
⊙
≈ (1 s)K E
e
ℏ
⊙
= (1 s)K E
e
κ
⊕
=
r
moon
R
⊙
= 0.5522974
t
⊕
= 0.982 s ≈ 1.00 s
ℏ
galaxy
= (1"second)K E
⊙
KE
⊙
=
1
2
(1.989E30kg)(220,000m /s)
2
= 4.81338E40J ⋅ s
ℏ
galaxy
= 4.81338 × 10
40
J ⋅ s, h
galaxy
= 2πℏ
galaxy
= 3.0243 × 10
41
J ⋅ s .
t
1
=
R
⊙
M
⊙
πh
galaxy
G c
κ
galaxy
= 2.410 s × κ
galaxy
.
κ
galaxy
= 0.4149,
r
eart h
/r
asteroid
≈ 1/2.41
2.4 AU
of 20 42
As in the Earth case, the one-second factor enters by construction in . The non-trivial
point is that the same UPL template, with an order-unity coupling tied to an observed orbital ratio, returns
at the galactic scale. This is an example of the one-second scale appearing across a very large
difference in scale — from particle radii to the Earth–Moon system to the Sun’s orbit around the galaxy
— within the same phenomenological law. It is a consistency/scale-invariance check, not an independent
derivation of the second.
We can look at, instead of the solar radius to the the solar mass, at the orbital radius of the Sun
around the galaxy to the mass of the galaxy interior to the Sun’s orbit around it, Thus we have
the universal particle equation on the galactic scale is, as well:
Where is the orbital radius of the Sun around the center of the galaxy, and is mass of
the galaxy interior to the Sun’s orbit (about 1E11 solar masses). We have ,
which is on the order of Saturn orbit in AU about 9.5 to 9.6 AU. We have:
This second choice of coupling is another order-unity coupling; it is not an independent
derivation of the one-second value, but a further consistency check that the same UPL template
can place a characteristic time at the solar-system/galactic interface. The orbital velocity
around the galaxy ranges in estimates from 195 km/s to 255 km/s, So could be a little
smaller or larger, but 220 km/s is widely cited.
The Closure Resonance
The question a reader should ask is: at what lunar distance would the Earth UPL return exactly
one second? Solving
for gives
ℏ
galaxy
= (1 s)K E
⊙
t
1
∼ 1 s
t
1
=
r
⊙
M
galaxy
πh
galaxy
Gc
⋅ κ
galaxy
t
1
=
2.5E 20m
1.989E41kg
π (3.0243E 41J ⋅ s)
(6.6743E − 11)(299,792,458m /s)
⋅ κ
galaxy
= 8.6612 seconds ⋅ κ
galaxy
r
⊙
M
galaxy
κ
galaxy
= 0.115457
t
1
=
r
⊙
M
galaxy
πh
galaxy
Gc
⋅
r
earth
r
saturn
∼ 1 s
ℏ
galaxy
t
⊕
=
R
⊕
M
⊕
πh
⊙
Gc
⋅
r
moon
R
⊙
= 1 s
r
moon
of 21 42
The actual lunar distance is 1.282 light-seconds. The ratio is
The actual Earth–Moon distance is within 1.8% of the value that would make the Earth UPL give
exactly one second.
This is the closure calculation. The resonance is not an exact identity at the present epoch; it is a
near-match, and the size of the mismatch is 1.8%. Whether that near-match is coincidence,
selection, or structure is the question the collection raises and does not settle.
The near-match is also epoch-dependent, because it depends on . At formation, roughly 4.5
Gyr ago, m = 0.80 light-seconds, and s. In the far future, when
m = 1.83 light-seconds, s. The present epoch is when the value is
0.98 s. The relation is not a constant; it is a trajectory, and we are on it at a particular point.
The perfect eclipse is behind us. The exact match of lunar and solar angular sizes at mean
distance was crossed in the opposite direction in the distant past. Estimates range from a few
million to several hundred million years ago, with the commonly quoted figure around 150–300
Myr. Total eclipses remain possible today because the lunar orbit is elliptical and the Moon is
closer at perigee; they will cease in a few hundred million years.
The resonance is ahead of us. On standard tidal models, exact closure is reached in roughly 175–
250 million years. The present epoch sits between two crossings: the eclipse behind, the
resonance ahead.
The time to exact closure is 175–250 Myr on the tidal models. The Sun's orbital period around
the galactic center — the "galactic year" — is approximately 225–250 Myr. The central estimates
differ by roughly 20%, and the model uncertainty on the closure time is of the same order, so the
two ranges overlap without the agreement being sharp. The honest statement is that the closure
time brackets the galactic year.
r
(required)
moon
=
M
⊕
R
⊕
⋅
R
⊙
πh
⊙
/(G c)
⋅ (1 s) = 3.91 × 10
8
m = 1.306 light-seconds .
r
(actual)
moon
r
(required)
moon
=
1.282
1.306
= 0.982.
r
moon
r
moon
≈ 2.4 × 10
8
t
⊕
≈ 0.61
r
moon
≈ 5.5 × 10
8
t
⊕
≈ 1.40
of 22 42
Chapter 2: Summary of What Has Been Found
We live in an epoch when the Earth’s rotation period of about 24 hours is mapped approximately
into our 1-second basis unit of measuring time by the kinetic energies of the Moon and the Earth
(Beardsley, 2025):
Where is the Earth obliquity (tilt of Earth to its orbit). This is the same approximately 24 hour
day that is the rotation period of Mars (martian day):
And, the same second that is given by a Bohr hydrogen atom ground state analog for the Solar
System using the Moon as the mass (Beardsley, 2025)
It is the same second that approximates The Planck-type constant for the Solar System with the
Earth orbital kinetic energy (Beardsley, 2025):
We also live in a time when there is a near perfect eclipse of the Sun by the Moon as seen from
Earth:
This is the same second that is approximately the quantization of time in a phenomenological
universal particle law (UPL) for the proton, electron, and neutron (Beardsley, 2025):
. Numerically, N. The mass of a particle is then (UPL)
K E
m
K E
e
(Ear th Da y)cos(θ
e
) ≈ 1second
θ
T
mars
≈ T
earth
ℏ
2
⊙
GM
3
m
1
c
= 1.00seconds
ℏ
⊙
= 2.81733E 33J ⋅ s
h
⊙
= 2π ℏ
⊙
= 1.77018E 34J ⋅ s
ℏ
⊙
≈ (1.00seconds)K E
e
r
⊕
r
moon
≈
R
⊙
R
moon
p
t
=
h
ct
1
, F
n
=
Δ p
t
Δt
=
h
ct
2
1
.
t
1
≈ 1 second
F
n
= 2.21022 × 10
−42
of 23 42
where is the particle's radius, is the "pliability of space," and is a dimensionless coupling
fixed by the particle's structure.
The law (UPL) can also be written in a form that makes the timescale explicit:
Applied to the three primary particles of the atom, with ( the particle radius, its
mass):
Proton ( ):
Neutron ( ):
Electron ( ):
It is also the second characteristic of applying the UPL to the Earth and to the galaxy. For Earth:
The same procedure extends to the galaxy. Begin with the Sun's orbital kinetic energy around the
galactic center, taking km/s:
m
i
= κ
i
π r
2
i
F
n
G
,
r
i
G
κ
i
t
1
=
r
i
m
i
πh
Gc
κ
i
.
α = 1/137
r
i
m
i
κ
p
= 1/(3α
2
) ≈ 6256.33
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 s
κ
n
= 1/(3α
2
)
t
1
=
0.8367 × 10
−15
1.675 × 10
−27
π ⋅ 6.62607 × 10
−34
(6.674 × 10
−11
)(299,792,458)
⋅ 6256.33 = 1.00803 s
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 s
t
⊕
=
R
⊕
M
⊕
πh
⊙
Gc
⋅
r
moon
R
⊙
= 1 s
v
⊙
= 220
of 24 42
Define the galactic effective action by giving
The asteroid node. Taking the prefactor as solar radius over solar mass and the coupling as the
ratio of Earth's orbital radius to the asteroid belt’s we have for the galaxy:
gives
With , the product is approximately one second. The coupling corresponds to
, placing the node at about 2.4 AU — inside the asteroid belt.
K E
⊙
=
1
2
(1.989 × 10
30
)(220,000)
2
= 4.81338 × 10
40
J .
ℏ
gal
= (1 s) K E
⊙
ℏ
gal
= 4.81338 × 10
40
J ⋅ s, h
gal
= 2π ℏ
gal
= 3.0243 × 10
41
J ⋅ s .
t
1
=
R
⊙
M
⊙
πh
gal
Gc
κ
gal
t
1
=
6.957 × 10
8
1.989 × 10
30
π ⋅ 3.0243 × 10
41
(6.6743 × 10
−11
)(299,792,459)
κ
gal
= 2.410 s × κ
gal
.
κ
gal
= 0.4149
r
earth
/r
asteroid
≈ 1/2.41
of 25 42
Chapter 3: The Galactic Clock?
We have concluded that the proximity between the closure time and the galactic year is worth
recording. This chapter records one more symmetry.
The symmetry is the supercontinent cycle. Earth's landmasses assemble into a single continent,
break apart, and reassemble on a timescale of roughly 250 to 300 million years. Two hundred
and fifty million years ago, nearly all of Earth's land was joined into Pangea, and the first
dinosaurs were appearing. Over the following ~175 million years it split into Laurasia and
Gondwana, then into the seven continents we recognize. Today we live in the spread-out phase,
between two assemblies. Two hundred and fifty million years from now, if the cycle continues,
the continents will have merged again into a new supercontinent — Pangea Proxima or Amasia
— and we will have completed approximately two orbits of the galaxy since Pangea.
Three timescales are therefore comparable: the galactic year (~225–250 Myr), the supercontinent
cycle (~250–300 Myr), and the time to exact closure of the UPL relation (~175–250 Myr on the
tidal models). The first two are set by independent physics — the Sun's orbit through the galactic
potential, and mantle convection internal to Earth. The third is set by tidal dissipation in Earth's
oceans. Nothing couples any of them to any other.
There is, however, a specific ordering that is checkable. The closure happens within the current
inter-supercontinent phase, before the next assembly. If the tidal models are right, the resonance
opens roughly 175 to 250 million years from now, and the next supercontinent assembles on a
comparable timescale. Those two events are close in time but not simultaneous, and the ordering
is a prediction rather than a coincidence. It is a weak prediction, since both timescales carry
uncertainties of the same order as their separation, but it is the kind of statement that could be
sharpened with better tidal modeling and better constraints on mantle dynamics.
There is also a limit to how many ticks the clock has. Complex plant life is expected to collapse
in roughly 500 to 600 million years, as CO₂ levels fall below the threshold for C3
photosynthesis. The oceans are expected to evaporate in roughly 1 billion years, as solar
luminosity rises. The window in which a technological civilization can observe the cycle is
therefore roughly the window in which complex life exists, and that window is a small fraction
of Earth's total history.
This is worth stating because it constrains what any "clock" interpretation could mean. If the
galactic year, the supercontinent cycle, and the resonance closure are connected by something
structural, the connection has at most two more ticks to express itself before the planet can no
longer support the kind of life that could observe it. A clock that runs for two ticks and then stops
is not obviously a clock. It may be a window that happens to be open while we are here.!
of 26 42
Chapter 4: The Human Scale as a Resonant Node Between the Quantum
and Astrophysical Scales
Abstract
We propose that the human scale — the characteristic mass, length, and energy of biological
organisms — is not an arbitrary biological fact but a resonant node between the quantum scale
and the Earth–Moon–Sun scale. Using the Universal Particle Law (UPL), which posits a one-
second resonance governing bound systems from the proton to the galaxy, we construct a human
effective action , where is Planck's constant and is the
Earth–Moon–Sun effective action. With the UPL's one-second timescale, this gives a human
energy scale — on the order of the mechanical work of a heartbeat — and a
characteristic length-to-mass ratio (for ). The geometric means of
particle masses and astrophysical masses bracket the human mass range: ,
, , and . Extending the cascade to the
galactic scale, we identify two additional resonant nodes — the asteroid belt at ~2.4 AU and
Saturn at ~9.5 AU — that complete a scale hierarchy to . We suggest this
hierarchy provides a quantitative framework for astrobiology: the conditions for human-scale
intelligence may be geometrically fixed, and the search for life should target systems with the
same resonant architecture.
1. Introduction
The Universal Particle Law (UPL) proposes that inertia is the geometric resistance to rotating a
particle's velocity vector out of the time dimension and into space [1]. The normal force resisting
this rotation is
where is Planck's constant, is the speed of light, and is a fundamental timescale.
When applied to the proton, neutron, and electron, the law returns :
with and . The same law, applied to the Earth–Moon–Sun
system with an effective action , returns . Applied to the
Sun's orbit around the galactic center with , it returns at two
geometric nodes: the asteroid belt (~2.4 AU) and Saturn (~9.5 AU).
h
human
= h h
⊙
≈ 3.4248 J·s
h
h
⊙
E
human
≈ 3.4248 J
r /m ≈ 0.0431 m/kg
κ
human
= 1
m
e
M
⊕
≈ 2.3 g
m
p
M
⊕
≈ 100 g
m
e
M
⊙
≈ 1.35 kg
m
p
M
⊙
≈ 57.7 kg
h → h
human
→ h
⊙
h
gal
F
n
=
h
ct
2
1
,
h
c
t
1
≈ 1 s
t
1
≈ 1 s
t
1
=
r
i
m
i
πh
Gc
κ
i
,
κ
p
= κ
n
= 1/(3α
2
) ≈ 6256.33
κ
e
= 1
h
⊙
≈ 1.77018 × 10
34
J·s
t
1
≈ 0.982 s
h
gal
≈ 3.0243 × 10
41
J·s
t
1
≈ 1 s
of 27 42
A natural question arises: where does the human scale fit? Is there an effective action that,
when inserted into the UPL with , returns human-scale masses and lengths? In this paper,
we show that the geometric mean of the quantum and solar effective actions,
does exactly this. We then extend the cascade to the galactic scale and identify the astrobiology
implications.
2. The Quantum and Solar Effective Actions
2.1 The quantum scale
The UPL at the particle level uses the fundamental Planck constant:
The law is satisfied for the electron, proton, and neutron:
Particle Properties!
The particle scale is characterized by and .
2.2 The Earth–Moon–Sun scale
For the Earth–Moon–Sun system, the effective action is computed from Earth's orbital kinetic
energy around the Sun:
The celestial Planck-type constant is defined as
The UPL at this scale takes the form
h
human
t
1
= 1 s
h
human
= h h
⊙
,
h = 6.62607015 × 10
−34
J·s .
Particle
rᵢ (m)
mᵢ (kg)
κᵢ
t₁ (s)
Electron
2.81794 × 10⁻¹⁵
9.10938 × 10⁻³¹
1
0.99773
Proton
0.833 × 10⁻¹⁵
1.67262 × 10⁻²⁷
1/(3α²)
1.00500
Neutron
0.8367 × 10⁻¹⁵
1.675 × 10⁻²⁷
1/(3α²)
1.00803
h
t
1
≈ 1 s
K E
⊕
=
1
2
(5.972 × 10
24
kg)(29,785 m/s)
2
≈ 2.649 × 10
33
J .
h
⊙
= 2π (1 s)K E
⊕
≈ 1.77018 × 10
34
J·s .
of 28 42
where , , and the geometric coupling is the eclipse
ratio
This returns
The Earth–Moon–Sun scale is characterized by and the same one-second resonance.
3. The Human Effective Action
3.1 Definition
The quantum scale is characterized by . The solar scale is characterized by
. The geometric mean is
This is the natural intermediate action between the quantum and astrophysical scales.
3.2 The human energy scale
With the UPL's one-second resonance, , the human energy scale is
This is on the order of everyday human mechanical work:
- Lifting a 1 kg mass through 35 cm (On the order of 2 lbs through 1 ft in Earth gravity):
- Lifting a 0.1 kg mass through 3.5 m:
t
1
=
R
⊕
M
⊕
πh
⊙
Gc
κ
⊕
,
R
⊕
= 6.371 × 10
6
m
M
⊕
= 5.972 × 10
24
kg
κ
⊕
=
r
moon
R
⊙
=
3.84399 × 10
8
6.96 × 10
8
≈ 0.5522974.
t
1
= 1.7785 s × 0.5522974 ≈ 0.982 s ≈ 1 s .
h
⊙
h ∼ 10
−34
J·s
h
⊙
∼ 10
34
J·s
h
human
= h h
⊙
= (6.62607015 × 10
−34
)(1.77018 × 10
34
) ≈ 3.4248 J·s .
t
1
= 1 s
E
human
=
h
human
t
1
≈ 3.4248 J .
mgh = (1)(9.8)(0.35) ≈ 3.43 J .
(0.1)(9.8)(3.5) ≈ 3.43 J .
of 29 42
The human heartbeat does approximately 1 J of mechanical work per beat; at ~1 Hz, this
corresponds to a power of ~1 W. The energy scale is on the order of the
mechanical work of a few heartbeats, or a single arm movement.
3.3 The human length-to-mass ratio
Applying the UPL at the human scale:
Taking and :
This gives a characteristic length for any human-scale mass:
Mass Scale and Predicted Radius (κ = 1)!
A 70–100 kg human corresponds to a characteristic length of 3–4 m. This is the human scale —
not the body itself, but the space a human occupies, the reach, the stride, the arm span.
The value of is a choice, and the direction of its effect is worth stating carefully. Because
at fixed mass and time, a smaller coupling gives a *larger* predicted length: with
, 100 kg corresponds to 7.8 m rather than 4.3 m. Equivalently, at a fixed
characteristic length of 3 m, the resonant mass drops from 70 kg to about 38 kg. The choice
, used in the appendix, is the one that places the human squarely on the
resonance ( ); the eclipse-ratio coupling, applied at fixed
mass and length, moves the human off that resonance by about 45%. The order-of-magnitude
claim is robust — both choices give human-scale lengths — but the resonant coupling itself is
, not .!
E
human
≈ 3.4 J
t
1
=
r
human
m
human
πh
human
Gc
κ
human
.
κ
human
= 1
t
1
= 1 s
r
human
m
human
=
Gc
πh
human
≈ 0.0431 m/kg .
Mass
Predicted r (κ = 1)
Human analogue
1 g
0.043 mm
A grain of sand
100 g
4.3 mm
A pea
1 kg
4.3 cm
A small fruit
10 kg
43 cm
A torso
70 kg
3.0 m
Arm span
100 kg
4.3 m
A tall stride
κ
human
r ∝ 1/κ
κ
⊕
≈ 0.552
κ
human
= 1
T ≈ 7.4 × 10
42
T
human
= 7.35 × 10
42
κ
human
≈ 1
κ
⊕
of 30 42
4. Human-Scale Masses from Geometric Means
The geometric means of particle masses and astrophysical masses bracket the human mass range:
Geometric Mean Masses!
The proton–Earth and proton–Sun geometric means are exactly human-scale masses. The
electron–Earth and electron–Sun means are also human-scale, though smaller. The human body
sits at the geometric midpoint between the quantum and astrophysical scales.
This is a striking quantitative result: the human mass range is the geometric mean of the quantum
and astrophysical mass scales.
5. The Full Cascade
5.1 The scale hierarchy
The human scale is the middle rung of a cascade:
Each level satisfies the same one-second law with a different effective action:
Effective Action Across Scales!
5.2 The galactic nodes
From the first paper, the galactic level has two geometric nodes:
Pair
Geometric mean mass
Human equivalent
mₑ and M⊕
2.33 × 10⁻³ kg = 2.3 g
A portion of food
mₚ and M⊕
0.100 kg = 100 g
A small meal
mₑ and M☉
1.35 kg
A newborn
mₚ and M☉
57.7 kg
A six-foot man
h ⟶ h
human
⟶ h
⊙
⟶ h
gal
.
Level
Effective action
Characteristic scale
t₁
Quantum
h = 6.626 × 10⁻³⁴ J·s
Proton, neutron, electron
~1 s
Human
h_human = 3.4248 J·s
1 g – 100 kg
~1 s
Earth–Moon–
Sun
h☉ = 1.77018 × 10³⁴ J·s
Earth orbit, Moon, eclipse
~1 s
Galaxy
h_gal = 3.0243 × 10⁴¹ J·s
Asteroid belt, Saturn
~1 s
of 31 42
- Asteroid belt at (~2.4 AU) — the region that delivered water and organics to the
early Earth.
- Saturn at (~9.5 AU) — a gravitational shield deflecting comets and asteroids.
These are resonant "sweet spots" of the galactic potential, fixed by the Sun's mass, radius, and
motion through the galaxy.
5.3 The scaling operator
The same operator relates all levels:
This operator takes , , and . It contains no free parameters; the
effective actions are computed independently from kinetic energies.
6. Astrobiology Implications
6.1 The resonant architecture
If the human scale is a resonant node between the quantum and astrophysical scales, and if the
Earth–Moon–Sun system is tuned to produce this node, then life requires a planetary system with
specific architectural features:
1. A rocky planet at ~1 AU — where liquid water is stable.
2. An asteroid belt at ~2.4 AU — to deliver water and organics.
3. A gas giant at ~9.5 AU — to shield the inner system.
4. A large moon — to stabilize obliquity and produce a ~24-hour day.
5. A perfect eclipse geometry — , the eclipse ratio.
The UPL predicts these from the Sun's mass, radius, and galactic motion. They are not
independent coincidences; they are resonant nodes of the same one-second law.
6.2 The search strategy
The astrobiology prediction is specific and falsifiable:
Look for exoplanetary systems with an asteroid belt at ~2.4 AU and a gas giant at ~9.5 AU.
These are the systems where the human-scale resonance can emerge.
κ
ast
gal
≈ 0.415
κ
sat
gal
≈ 0.115
h
b
h
a
=
(
R
a
M
b
κ
a
M
a
R
b
κ
b
)
2
.
h → h
human
h
human
→ h
⊙
h
⊙
→ h
gal
r
m
/R
⊙
≈ 0.55
of 32 42
This is a concrete target. Current and future missions (JWST, PLATO, HabEx, LUVOIR) can
characterize exoplanetary system architecture. The prediction is that systems with the full
resonant architecture are more likely to host complex life than systems without it.
6.3 Why this is not anthropocentric
The human scale is not put in by hand. It emerges from the geometric mean of the quantum and
solar effective actions:
The Earth–Moon–Sun system is tuned to produce this mean at the one-second resonance. If this
is correct, then the conditions for human-scale intelligence are not arbitrary — they are
geometrically fixed. This gives us a concrete target for astrobiology.
7. Discussion
7.1 What is established
Three things can be said with confidence:
1. The human effective action exists. is the natural geometric
mean between the quantum and solar scales.
2. The human energy scale is human. is on the order of everyday human
mechanical work.
3. The human mass range is bracketed by geometric means. to
.
7.2 What is not established
The UPL does not derive the value of from first principles. It does not derive the
couplings . The geometric means are numerical observations within the UPL framework, not
derivations from a deeper theory. The astrobiology prediction is a hypothesis to be tested.
The author finds the following holds for the kinetic energies of the moon and the Earth:
h
human
= h h
⊙
.
h
human
= h h
⊙
≈ 3.4248 J·s
E
human
≈ 3.4248 J
m
e
M
⊕
≈ 2.3 g
m
p
M
⊙
≈ 57.7 kg
t
1
= 1 s
κ
i
K E
m
K E
e
(Ear th Da y) = 1.25seconds
of 33 42
We used average orbital velocities. And, accounting for the Earth’s inclination to its orbit ( =
23.5 deg), we have
Earth day=(24)(60)(60)=86,400 seconds. We can bring down equations these equations closer to
a second using the Moon’s orbital velocity at aphelion, and Earth’s orbital velocity at perihelion.
We have:
The Moon allows for the evolution of complex, intelligent life because it stabilizes the earth tilt
to its orbit allowing for the seasons and preventing temperature extremes. We live in an
interesting time when the Moon near perfectly eclipses the Sun. This is because while the Sun is
400 times larger than the Moon, it is 400 times further from the Earth than the Moon is. That is
the perfect eclipse is given by the orbital radius of the Earth to the Moon’s orbital radius
equals the Solar radius to the lunar radius :
The Earth's rotation is slowing down, and as a direct result, the Moon is slowly drifting away.
Earth's Rotation Slowing: The length of a day is increasing by about 2.3 milliseconds per
century. This is the long-term average rate caused primarily by the Moon's tidal pull. (Note that
this rate can be influenced by other factors; for example, climate change is currently contributing
about 1.33 milliseconds per century).
Moon's Orbit Growing: The Moon is receding from Earth at a rate of about 3.8 centimeters (1.5
inches) per year.
K E
m
=
1
2
(7.4767E 22kg)(1,022m /s)
2
= 3.83726E 28 J
K E
e
1
2
(5.972E 24kg)(29,785m /s)
2
= 2.649E 33J
θ
e
K E
m
K E
e
(Ear th Da y)cos(θ
e
) = 1.146seconds
K E
m
=
1
2
(7.347673E 22kg)(966m /s)
2
= 3.428E 28 J
K E
e
=
1
2
(5.972E 24kg)(30,290m /s)
2
= 2.7396E 33J
K E
m
K E
e
(Ear th Da y)cos(θ
e
) = 0.991seconds
K E
m
K E
e
(Ear th Da y)cos(θ
e
) ≈ 1second
r
e
r
m
R
⊙
R
m
r
e
r
m
=
R
⊙
R
m
of 34 42
The Tidal Connection
These two phenomena are linked by tidal friction. The Moon's gravity creates tides in Earth's
oceans. As Earth rotates, the tidal bulges are pulled slightly ahead of the Moon, creating friction
that slows Earth's spin.
To conserve angular momentum in the Earth-Moon system, this loss of Earth's rotational speed is
transferred to the Moon, boosting its orbital energy and causing it to spiral outward into a larger
orbit. This recession rate has been precisely measured for decades using the Lunar Laser Ranging
experiment, which bounces lasers off reflectors left on the Moon by Apollo astronauts.
Long-Term Perspective
While these changes are tiny on human timescales, they have significant effects over millions of
years. About 250 million years ago, a day on Earth was roughly 23 hours long. In about 600
million years, the Moon will have drifted so far that total solar eclipses will no longer be
possible.
We might guess from the ground state of the Bohr hydrogen atom, which is given by
that a Planck-type constant is given by the Moon, if it is the metric by which the Solar System
does its measuring, that it is given by:
Where is a celestial Planck type constant for the Earth/Moon/Sun system, is the mass of
the Moon, and we have let: , , . This gives us that:
Nicely, this has that:
Suggesting that that the quantum analog for the Earth/Moon/Sun system and the Universal
Particle Equation both have a characteristic time of about 1 second.
7.3 Selection effects
We observe the human scale because we are human. This is a selection effect that cannot be ruled
out without a broader statistical argument. However, the fact that the human scale emerges from
a
0
=
ℏ
2
k
e
m
e
e
2
ℏ
2
⊙
GM
3
m
1
c
= 1.00seconds
ℏ
⊙
M
m
ℏ → ℏ
⊙
k
e
→ G
m
e
e
2
→ M
3
m
ℏ
⊙
= 2.81733E 33J ⋅ s
h
⊙
= 2π ℏ
⊙
= 1.77018E 34J ⋅ s
ℏ
⊙
≈ (1.00seconds)K E
e
of 35 42
the geometric mean of the quantum and solar actions is not a selection effect; it is an algebraic
fact about the UPL.
7.4 Testability
The framework makes specific predictions:
- The UPL should hold at the human scale with and .
- Exoplanetary systems with an asteroid belt at ~2.4 AU and a gas giant at ~9.5 AU should be
more likely to host complex life.
- Systems without these features should be less likely to host complex life.
These predictions are testable with current and future data.
8. Conclusion
We have shown that the human scale emerges as a resonant node between the quantum scale and
the Earth–Moon–Sun scale. The human effective action is the geometric mean of the Planck
constant and the celestial effective action:
With the UPL's one-second resonance, this gives a human energy scale and a
characteristic length-to-mass ratio . The geometric means of particle and
astrophysical masses bracket the human mass range. Extending the cascade to the galactic scale,
we identify the asteroid belt and Saturn as additional resonant nodes.
The human scale is not an arbitrary biological fact. It is the middle rung of a scale hierarchy that
spans from the proton to the galaxy. This suggests a quantitative framework for astrobiology: the
conditions for human-scale intelligence may be geometrically fixed, and the search for life
should target systems with the same resonant architecture.
References
[1] Beardsley, I. (2026). Scale Invariance and Fractal Features in Cosmology: A Computational
Study. Zenodo. https://doi.org/10.5281/zenodo.23045265
[2] Beardsley, I. (2026). The Human Scale and Natural Laws.
[3] Tiesinga, E., Mohr, P. J., Newell, D. B., & Taylor, B. N. (2024). CODATA Recommended
Values of the Fundamental Physical Constants. NIST.
h
human
κ
human
≈ 1
h
human
= h h
⊙
≈ 3.4248 J·s .
E
human
≈ 3.4248 J
r /m ≈ 0.0431 m/kg
of 36 42
[4] Chiaramonte, F., & Loeffen, R. (2026). VGT-CIT Reduction and the Status of the UPL as a
Limiting Case. Private communication.
[5] Tynski, K. (2024). One Equation, ~200 Mysteries: A Structural Constraint That May Explain
(Almost) Everything.
Appendix: Numerical Values
Physical Quantities!
Appendix: Dimensionless Reformulation of the UPL Hierarchy
A.1 Motivation
A natural objection to the Universal Particle Law (UPL) is that its central result, second,
appears to depend on the human convention of the second. The second, kilogram, meter, and
joule are human units. If the UPL describes a structural resonance, its significance should survive
the removal of human measurement conventions.
This appendix addresses that objection directly. We reformulate the UPL in an -normalized
natural unit system, show that the law collapses to a pure ratio statement, and demonstrate that
the full hierarchy — electron, proton, neutron, Earth–Moon–Sun, human, galaxy — converges to
a single dimensionless coordinate. The resonance survives the removal of SI units.
Quantity
Value
h
6.62607015 × 10⁻³⁴ J·s
h☉
1.77018 × 10³⁴ J·s
h_human
3.4248 J·s
E_human
3.4248 J
r/m (κ = 1)
0.0431 m/kg
√(m_e M_⊕)
2.33 g
√(m_p M_⊕)
100 g
√(m_e M_☉)
1.35 kg
√(m_p M_☉)
57.7 kg
h_gal
3.0243 × 10⁴¹ J·s
κ_gal^ast
0.415
κ_gal^sat
0.115
t
1
≈ 1
h
of 37 42
The reformulation was suggested by Volynets Evgeny Vatslavovich (author of MASV-Prime) in
a comment on the paper under discussions at academia.com. We present it here because it
strengthens the framework and clarifies what is structural versus what is metrological.
A.2 The -normalized natural unit system
Conventional Planck units use . Because the UPL uses rather than , we construct a
natural unit system from , , and :
These are not the standard Planck units. They are the natural units that speak the same
mathematical language as the UPL.
In these units, one SI second corresponds to
This is the dimensionless content of "one second." It does not depend on the caesium atom, the
division of the day, or any human convention. It is a pure ratio.
A.3 The UPL in natural units
The UPL is
Substituting , , and :
h
ℏ = h /(2π)
h
ℏ
h
c
G
t
h
=
hG
c
5
= 1.351385078 × 10
−43
s,
ℓ
h
=
hG
c
3
= 4.051350543 × 10
−35
m,
m
h
=
hc
G
= 5.455511861 × 10
−8
kg,
E
h
= m
h
c
2
=
hc
5
G
= 4.903169538 × 10
9
J .
1 s
t
h
= 7.3998 × 10
42
.
t
1
=
r
i
m
i
πh
Gc
κ
i
.
r
i
= R
i
ℓ
h
m
i
= M
i
m
h
t
1
= T
i
t
h
of 38 42
Using and , this collapses to
The UPL becomes a pure ratio law. No dimensionful constants remain. The resonance condition
second becomes
This is the structural statement. The value second is its human projection.
A.4 The full hierarchy in natural units
We evaluate at each level of the hierarchy, using the same inputs as the main
text.
A.4.1 Particle level
Particle Data!
A.4.2 Earth–Moon–Sun level
A.4.3 Human level
For a 70 kg human with a characteristic length of 3 m (arm span, stride, or occupied space):
T
i
t
h
=
R
i
ℓ
h
M
i
m
h
πh
Gc
κ
i
.
ℓ
h
/m
h
= G /c
2
t
h
= hG /c
5
T
i
= π κ
i
R
i
M
i
h
i
h
.
t
1
≈ 1
T
i
≈ 7.4 × 10
42
.
1
T
i
= π κ
i
R
i
/M
i
Particle
rᵢ (m)
mᵢ (kg)
κᵢ
Tᵢ
Electron
2.81794 × 10⁻¹⁵
9.10938 × 10⁻³¹
1
7.382 × 10⁴²
Proton
0.833 × 10⁻¹⁵
1.67262 × 10⁻²⁷
6256.33
7.436 × 10⁴²
Neutron
0.8367 × 10⁻¹⁵
1.675 × 10⁻²⁷
6256.33
7.459 × 10⁴²
R
⊕
= 6.371 × 10
6
m, M
⊕
= 5.972 × 10
24
kg, κ
⊕
= 0.5522974.
T
⊕
= π (0.5522974)
R
⊕
M
⊕
h
⊙
h
= 7.27 × 10
42
.
of 39 42
A.4.4 Galactic level
For the asteroid node:
For the Saturn node:
A.4.5 Summary table
Scale and Tᵢ Values!
Every scale converges to . The spread is less than 3%.
A.5 What this establishes
Three things can be said with confidence.
R
human
= 3 m, M
human
= 70 kg, κ
human
= 1.
T
human
= π (1)
3
70
h
human
h
= 7.35 × 10
42
.
R
⊙
= 6.96 × 10
8
m, M
⊙
= 1.989 × 10
30
kg, κ
ast
gal
= 0.415.
T
ast
gal
= π (0.415)
R
⊙
M
⊙
h
gal
h
= 7.28 × 10
42
.
r
⊙
= 2.5 × 10
20
m, M
gal
= 1.989 × 10
41
kg, κ
sat
gal
= 0.115.
T
sat
gal
= π (0.115)
r
⊙
M
gal
h
gal
h
= 7.28 × 10
42
.
Scale
Tᵢ
Electron
7.382 × 10⁴²
Proton
7.436 × 10⁴²
Neutron
7.459 × 10⁴²
Human (70 kg, 3 m)
7.35 × 10⁴²
Earth–Moon–Sun
7.27 × 10⁴²
Galaxy (asteroid)
7.28 × 10⁴²
Galaxy (Saturn)
7.28 × 10⁴²
One SI second
7.3998 × 10⁴²
T ≈ 7.3-7.5 × 10
42
of 40 42
First, the resonance is structural, not metrological. The convergence to does not
depend on seconds, kilograms, or meters. It is a dimensionless statement about the ratio
weighted by the coupling . The UPL passes the stress test of removing human units.
Second, the human scale is on the resonance. The human entry is as
close to the one-second value as the proton or electron. The human scale is not an outlier; it sits
on the same resonant node.
Third, SI units are appropriate precisely because the resonance is human-scale. We use seconds
and joules not because we are confusing the map for the territory, but because the resonant node
is the human scale. The dimensionless ratio maps to 1 SI second because the human
scale is where the resonance lives. If the resonance were at the Planck scale, we would use
Planck units. If it were at the galactic scale, we would use kiloparsecs and solar masses. We use
SI because the human scale is the node.
A.6 What this does not establish
The dimensionless reformulation does not derive the value from first principles. It
does not explain why the couplings take the values they do. It does not derive the one-second
scale from a deeper theory.
What it does show is that the resonance is not an artifact of human measurement conventions.
The same dimensionless coordinate appears at every level of the hierarchy, from the electron to
the galaxy. That is a structural fact about the UPL, not a metrological accident.
A.7 The two views are the same fact
Views of the Resonance Time Scale!
All three are correct. They are not competing claims. The dimensional view is the human
projection of the dimensionless fact. The dimensionless view is the structural fact that projects
into human units.
The territory happens to include the mapmaker.
A.8 A note on the golden-ratio coordinate
T ≈ 7.4 × 10
42
R /M
κ
T
human
≈ 7.35 × 10
42
7.4 × 10
42
7.4 × 10
42
κ
i
View
Statement
What it emphasizes
Dimensional (SI)
t₁ ≈ 1 second
The human scale is the
resonance
Dimensionless
(natural)
t₁/tₕ ≈ 7.4 × 10⁴²
The resonance is structural
Ratio law
T = √π κ R/M
The mechanism is geometric
of 41 42
Volynets also suggested a logarithmic coordinate with the golden ratio at the origin:
This is a coordinate convention. Shifting the origin of a logarithmic coordinate does not create
new physics. The physically meaningful quantity is the persistence of the same dimensionless
relations after changing coordinates and removing human units. The golden-ratio origin is a
convenience, not a physical claim. We mention it here only to acknowledge the suggestion and to
distinguish coordinate convention from physical invariant.
A.9 Conclusion
The UPL hierarchy survives the removal of human units. In -normalized natural units, the law
collapses to , and every scale — electron, proton, neutron, human, Earth–Moon–
Sun, galaxy — converges to . The human scale is not an artifact of SI; it is the
resonant node itself.
The question worth pursuing is not "why does nature produce one second?" but "why does the
dimensionless ratio recur across systems whose masses and radii differ by dozens of
orders of magnitude?" That is the structural question, and it is the one the UPL answers.
Acknowledgment
The dimensionless reformulation presented in this appendix was suggested by Volynets Evgeny
Vatslavovich in a comment on this paper when in discussion at academia.com. His stress test of
the framework — removing human units and re-expressing the UPL in natural coordinates —
clarified the distinction between the physical resonance and its human projection. We thank him
for the constructive critique.
We also thank him for catching an arithmetic inversion in §3.3, where the effect of the coupling
on the predicted length was stated in the wrong direction; the corrected version strengthens the
resonance argument
X(Q) = log
10
(
Q
φQ
h
)
, φ =
1 + 5
2
.
h
T = π κ R /M
T ≈ 7.4 × 10
42
7.4 × 10
42
of 42 42
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