of 1 25
The Earth–Moon Distance as a Resonance Condition: The One-Second
Invariant in the Earth–Moon–Sun System
Ian Beardsley
October 25, 2026
of 2 25
Abstract
The Moon is important to life on Earth and we may be at a time in the history of our planet’s
evolution where conditions are optimal for the success of complex life because of where the
Moon is.
1. Introduction
The Universal Particle Law (UPL) identifies a one-second invariant across scales [1]. The law
can be written in two equivalent ways. The first uses Planck's constant :
The second uses the reduced Planck constant . The two formulations are
algebraically equivalent, but they emphasize different aspects of the physics. The -formulation
makes explicit the interaction of the normal force with the particle's cross-sectional area
, which is the geometric core of the inertia mechanism. The -formulation reveals a
direct analogy with the Bohr ground state of hydrogen and leads to a natural definition of
celestial effective actions.
2. The -Formulation: Force, Cross-Section, and Inertia
In [1], the normal force is derived from the quantization of temporal momentum over the one-
second interval:
The mass of a particle is then
where is the cross-sectional area exposed to the normal force and is the "pliability of
space." The -formulation is the natural one for this geometric picture: is the quantum of
action, and the force-area product is the quantity that, when mediated by , generates mass.
h
F
n
=
h
ct
2
1
, m
i
= κ
i
π r
2
i
F
n
G
.
ℏ = h /(2π)
h
A
i
= π r
2
i
ℏ
h
p
t
=
h
ct
1
, F
n
=
Δp
t
Δt
=
h
ct
2
1
.
A
i
= π r
2
i
G
h
h
F
n
A
i
G
of 3 25
Numerically, N. The proton, neutron, and electron all return s with
their respective couplings and [1].
2. A Planetary System Quantum Analog (PSQA) And Scale Invariance from Quantum
Scales to the Celestial Scales
The Earth Quantum Analog: Since the author finds the following holds for the kinetic energies of
the moon and the Earth (See Appendix 1, The 24-Hour Day as a Stable Resonance Plateau):
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 :
F
n
= 2.21022 × 10
−42
t
1
≈ 1
κ
p
= κ
n
= 1/(3α
2
)
κ
e
= 1
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 28J
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 28J
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
of 4 25
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.
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 (Was actually used to derive for the paper):
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:
r
e
r
m
=
R
⊙
R
m
a
0
=
ℏ
2
k
e
m
e
e
2
ℏ
⊙
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
of 5 25
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 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.
is the perfect scaling ratio in that our Planck type constant is determined by the Moon as
the metric and and by the orbital kinetic energy of the Earth, which is determined by the mass of
the Sun, which is determined by its size.
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:
!
!
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
⊙
1.7785seconds(0.5522974) = 0.982s ≈ 1.00seconds
r
moon
R
⊙
ℏ
gala xy
= (1"second)K E
⊙
KE
⊙
=
1
2
(1.989E30kg)(220,000m /s)
2
= 4.81338E40J ⋅ s
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We do like we did in determining the solar Planck-type constant , to determine the galactic
Planck-type constant. We have:
Thus we have the Universal Particle Law for the galaxy takes the form for solar mass to solar
radius if we instead use the reduced galactic Planck-type constant:
, which is on the order of the asteroid belt (2.2 AU to 3.2 AU).
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:
ℏ
⊙
ℏ
galax y
= (1 second)KE
⊙
= 4.81338E 40J ⋅ s
h
galax y
= 2π ℏ
galax y
= 3.0243E41J ⋅ s
t
1
=
R
⊙
M
⊙
πh
galax y
Gc
⋅ κ
galax y
t
1
=
6.957E8m
1.989E 30kg
π (3.0243E41J ⋅ s)
(6.6743E − 11)(299,792,459m /s)
⋅ κ
galax y
= (2.410s)κ
galax y
κ
galax y
= 0.4149
t
1
=
R
⊙
M
⊙
πh
galax y
Gc
⋅
r
earth
r
asteroids
t
1
=
r
⊙
M
galax y
πh
galax y
Gc
⋅ κ
galax y
t
1
=
2.5E 20m
1.989E41kg
π (3.0243E41J ⋅ s)
(6.6743E − 11)(299,792,458m /s)
⋅ κ
galax y
= 8.6612 seconds ⋅ κ
galax y
r
⊙
M
galax y
κ
galax y
= 0.115457
t
1
=
r
⊙
M
galax y
πh
galax y
Gc
⋅
r
earth
r
saturn
of 7 25
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 main relationship for the galaxy scale that follows suit with the direct way the UPL works is
It is also the equation that is very accurate, the Saturn one is ball park.
3. The Moon, The Electron, and The Bohr Radius
The Universal Particle Law is:
It works for the electron, proton, and neutron:
Proton: , :
Neutron: :
Electron: :
But, it also works for the Earth:
ℏ
galax y
t
1
=
R
⊙
M
⊙
πh
galax y
Gc
⋅
r
earth
r
asteroids
t
1
=
r
i
m
i
⋅
πh
Gc
⋅ 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
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Where is given by:
This is a direct analog for the ground state of the Bohr atom:
, , .
This gives us that:
It works nicely in the sense that it has approximately given by 1 second and the Earth’s
orbital kinetic energy:
Thus we want to put
in
Which we will write as:
t
1
=
R
earth
M
earth
πh
⊙
Gc
κ
earth
κ
earth
=
r
moon
R
⊙
=
3.84399E8m
6.96E8m
= 0.5522974
1.7785seconds(0.5522974) = 0.982s ≈ 1.00seconds
h
⊙
ℏ
2
⊙
GM
3
m
1
c
= 1.00seconds
a
0
=
ℏ
2
k
e
m
e
e
2
ℏ → ℏ
⊙
k
e
→ G
m
e
e
2
→ M
3
m
ℏ
⊙
= 2.81733E 33J ⋅ s
h
⊙
= 2π ℏ
⊙
= 1.77018E 34J ⋅ s
h
⊙
ℏ
⊙
≈ (1.00seconds)K E
e
ℏ
2
⊙
GM
3
m
1
c
= 1.00seconds
t
1
=
R
earth
M
earth
πh
⊙
Gc
κ
earth
t
1
=
R
⊕
M
⊕
πh
⊙
Gc
⋅
r
moon
R
⊙
of 9 25
The result is:
We might feel compelled to do the same thing with the electron. We have:
and
This gives:
Using
To eliminate :
This is correct if you set exactly in both the electron and celestial equations. With the actual values
, ), the exact result is
The interesting thing is that
can be written
M
2
⊕
R
2
⊕
⋅
R
2
⊙
r
2
moon
= 2π
2
M
3
m
Gct
4
1
a
0
=
ℏ
2
k
e
m
e
e
2
t
1
=
r
e
m
e
πh
Gc
t
2
1
r
2
e
⋅ G c = 2π
2
e
a
0
k
e
m
3
e
M
2
⊕
R
2
⊕
⋅
R
2
⊙
r
2
moon
= 2π
2
M
3
m
Gct
4
1
2π
2
M
2
⊕
R
2
⊕
⋅
R
2
⊙
r
2
moon
=
Gc
r
2
e
e
m
3
e
a
0
k
e
⋅
M
3
m
Gc
t
1
= 1
t
e
= 0.99773
t
⊕
= 0.9825
M
2
⊕
R
2
⊕
R
2
⊙
r
2
moon
=
t
2
e
t
2
⊕
⋅
G c
r
2
e
e
m
3
e
M
3
m
a
0
k
e
t
2
1
r
2
e
⋅ G c = 2π
2
e
a
0
k
e
m
3
e
of 10 25
Where . A quick check with rounded numbers yields:
=
This is not a derivation of the 2pi resonance or 1-second invariant, 2pi is an artifact of . If
we consider that:
We have:
Or, if we take we have
We could actually label as . Note: The interesting thing is that the distance to the moon
is similar to 1 light-second. It is 1.28 to 1.3 light-seconds.
4. The 1.28 Light-Seconds Observation
This is a genuinely interesting number, and it deserves more than a page break. Here's what it
means in the framework:
- The solar Bohr radius is light-second.
- The Earth–Moon distance is m light-seconds.
2π
ra d
s
2
≈
Gc
π r
2
e
m
3
e
a
0
k
e
⋅
1
e
π r
2
e
= A
e
(6.6743E − 11)(299,792,458)
π (2.81794E − 15)
2
(9.10938E − 31)
3
5.29E − 11)(8.988E 9)
⋅
1
1.602E − 19)
6.31283 ≈ 2π = 6.283
h → ℏ
a
moon
=
ℏ
⊙
GM
3
m
c =
a
moon
t
1
M
2
⊕
R
2
⊕
R
2
⊙
r
2
moon
=
t
2
e
t
2
⊕
⋅
1
r
2
e
e
Gm
3
e
M
3
m
k
e
a
moon
a
0
t
e
≈ t
⊕
M
2
⊕
R
2
⊕
R
2
⊙
r
2
moon
=
1
r
2
e
e
Gm
3
e
M
3
m
k
e
a
moon
a
0
a
moon
a
⊙
a
⊙
= ℏ
2
⊙
/(GM
3
m
) = c ⋅ 1 s = 1
r
moon
= 3.844 × 10
8
= 1.282
of 11 25
- So .
The coupling that appears in the Earth UPL is . If were exactly
, the coupling would be , and the Earth UPL would give s. The
actual is larger than by 28%, which increases the coupling to 0.552 and brings up to
0.982 s.
The near-match
Here is the number that I think is the real prize. Solve the Earth UPL for the value of that
would give exactly s:
Evaluating this with CODATA values gives m light-seconds.
The actual is 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 1 second.
That is the "1.28 to 1.3 light-seconds" range flagged. The actual value (1.282) and the required
value for exact closure (1.306) are both inside that narrow band.
Why this matters
There are three possible interpretations, and the exploration should distinguish them:
1. Coincidence. The Moon's distance is set by tidal evolution (it has been receding at ~3.8 cm/
year for billions of years), and the fact that it is currently near the value that closes the UPL to 1
second is accidental. Over geological time, has varied substantially, and it will continue to
grow. The current epoch happens to be close to the resonance.
2. Selection effect. Complex life requires a moon at approximately this distance (stabilizing
obliquity, creating tides, and producing the perfect eclipse). The one-second invariant holds
because we are observing the system at the epoch when complex life exists. This connects to
r
moon
/a
⊙
= 1.282
κ
⊕
= r
moon
/R
⊙
= 0.552
r
moon
a
⊙
c /R
⊙
= 0.431
t
⊕
= 0.767
r
moon
a
⊙
t
⊕
r
moon
t
⊕
= 1
r
(required)
moon
=
M
⊕
R
⊕
⋅
R
⊙
πh
⊙
/(Gc)
⋅ (1 s)
r
(required)
moon
= 3.91 × 10
8
= 1.306
r
moon
1.282
r
(actual)
moon
r
(required)
moon
= 0.982
r
moon
of 12 25
Appendix 1's "24-hour day as a stable resonance plateau": the plateau is where the invariant
holds, and life evolved on the plateau.
3. Structure. There is a deeper reason why the Earth–Moon system sits near the resonance.
Perhaps the Moon's mass and distance are jointly constrained by the one-second invariant, in the
same way that the electron's mass and radius are. This would require a dynamical mechanism—
something like "tidal evolution drives the system toward the resonance" rather than "the
resonance happens to be where the system currently is."
The distinction matters because it determines what the one-second invariant explains. If it's
coincidence, it explains nothing beyond the numerical observation. If it's selection, it explains
why life exists at this epoch. If it's structure, it explains why the Earth–Moon system has the
configuration it does.
At the time of formation (~4.5 Gyr ago), m = 0.80 light-seconds; in the far
future, it will reach ~ m (1.83 light-seconds). So has varied from ~0.61 s (at
formation) to ~1.15 s (far future), and is currently at 0.98 s. The window in which is within,
say, 10% of 1 second is roughly the last 2 Gyr and the next 2 Gyr—which overlaps with the
window in which complex life has existed and will exist.
If the near-match is structural, other systems with large moons should also sit near the analog of
the resonance. If it is coincidental, they should be scattered. This is a prediction that could be
tested with exoplanet data (once large-moon systems are detectable).
The present Moon distance is 1.8% below the value required for exact closure of the Earth UPL
to one second. On the constant- tidal model, the Moon crosses the required value in roughly
190 Myr, and the window in which extends from about 0.66 Gyr in the past to
1.70 Gyr in the future. Whether the present epoch's proximity to the resonance is coincidence,
selection, or structure is the open question the figure illustrates.
r
moon
≈ 2.4 × 10
8
5.5 × 10
8
t
⊕
t
⊕
Q
|
t
⊕
− 1
|
< 10 %
of 13 25
Earth-Moon UPL analysis (full version)
======================================
Three-panel figure:
(A) Zoom on t_Earth ~ 1 s, showing the 1.8% gap
(B) Full range on log-log scale (formation to far future)
(C) Geological timeline with tidal model and key markers
Requirements: numpy, matplotlib
"""
import numpy as np
import matplotlib.pyplot as plt
# ---------- Constants ----------
G = 6.67430e-11
c = 299792458.0
R_earth = 6.371e6
M_earth = 5.972e24
R_sun = 6.96e8
M_moon = 7.342e22
h_odot = 1.77018e34 # paper value
# ---------- UPL relation ----------
A = (R_earth / M_earth) * np.sqrt(np.pi * h_odot / (G * c))
C = A / R_sun
def t_earth(r_m):
return C * r_m
def r_required_for_t(target_t):
return target_t / C
# ---------- Key values ----------
r_now = 3.844e8 # m, current
r_formation = 4 * R_earth # m, nominal formation estimate
r_future = 5.5e8 # m, ~1.83 lt-s
t_formation = -4.5 # Gyr (independent estimate)
r_req = r_required_for_t(1.0)
lt_s = 1.0 / c # m per light-second
# ---------- Tidal model (valid |t| < 1.5 Gyr) ----------
of 14 25
dr_dt_now = 0.038 # m/yr
K = dr_dt_now * r_now**5.5
def r_of_t(t_gyr):
r_pow = r_now**6.5 + (13.0 / 2.0) * K * t_gyr * 1e9
return np.where(r_pow > 0, r_pow**(2.0 / 13.0), np.nan)
def t_crossing(target_r):
"""Time (Gyr) at which tidal model gives target_r."""
return (target_r**6.5 - r_now**6.5) / ((13.0 / 2.0) * K) / 1e9
# ---------- Print summary ----------
print("=" * 62)
print("EARTH-MOON UPL ANALYSIS")
print("=" * 62)
print(f"Prefactor A : {A:.6f}")
print(f"Coefficient C (s/m) : {C:.6e}")
print(f"Present r_moon : {r_now:.4e} m = {r_now*lt_s:.4f} lt-s")
print(f" t_Earth(present) : {t_earth(r_now):.5f} s")
print(f"Required for t=1 s : {r_req:.4e} m = {r_req*lt_s:.4f} lt-s")
print(f" Gap : {(1 - r_now/r_req)*100:.2f}%")
print(f"Formation (4 R_E) : {r_formation:.4e} m = "
f"{r_formation*lt_s:.4f} lt-s")
print(f" t_Earth(formation) : {t_earth(r_formation):.4f} s")
print(f"Far future (5.5e8 m) : {r_future*lt_s:.3f} lt-s")
print(f" t_Earth(far future) : {t_earth(r_future):.4f} s")
print()
t_cross_1s = t_crossing(r_req)
t_cross_09 = t_crossing(0.9 * r_req)
t_cross_11 = t_crossing(1.1 * r_req)
print(f"Moon crosses t=1 s in : {t_cross_1s*1000:.0f} Myr (future)")
print(f"|t-1| < 10% window : {t_cross_09:+.2f} Gyr to "
f"{t_cross_11:+.2f} Gyr")
print(f" Width : {t_cross_11 - t_cross_09:.2f} Gyr")
print("=" * 62)
# ---------- Figure ----------
fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(18, 6))
# === Panel A: zoom on t~1 s ===
r_zoom = np.linspace(1.10e0 / lt_s * 1.0e0 * c * 0 + 1.10 * c,
1.50 * c, 400)
ax1.plot(r_zoom * lt_s, t_earth(r_zoom), 'b-', lw=2.4,
of 15 25
label=r'$t_\oplus = C\,r_{\rm moon}$')
ax1.axhline(1.0, color='gray', ls=':', lw=1.2)
ax1.axvline(r_now * lt_s, color='green', ls='--', lw=1.6,
label=f'Present ({r_now*lt_s:.3f} lt-s)')
ax1.axvline(r_req * lt_s, color='red', ls='--', lw=1.6,
label=f'Required ({r_req*lt_s:.3f} lt-s)')
ax1.plot(r_now * lt_s, t_earth(r_now), 'go', ms=10, zorder=5)
ax1.plot(r_req * lt_s, 1.0, 'ro', ms=10, zorder=5)
ax1.axvspan(r_now * lt_s, r_req * lt_s, color='red', alpha=0.15)
ax1.annotate(f'Gap: {(r_req-r_now)/r_req*100:.1f}%',
xy=((r_now+r_req)/2 * lt_s, 0.94),
ha='center', fontsize=11, fontweight='bold', color='darkred')
ax1.set_xlabel(r'$r_{\rm moon}$ (light-seconds)', fontsize=12)
ax1.set_ylabel(r'$t_\oplus$ (s)', fontsize=12)
ax1.set_title('(A) Zoom on $t_\\oplus \\approx 1$ s', fontsize=12)
ax1.legend(fontsize=9, loc='upper left')
ax1.grid(alpha=0.3)
# === Panel B: full range, log-log ===
r_full = np.logspace(np.log10(0.05 * c), np.log10(3.0 * c), 500)
ax2.loglog(r_full * lt_s, t_earth(r_full), 'b-', lw=2.4,
label=r'$t_\oplus = C\,r_{\rm moon}$')
ax2.axhline(1.0, color='gray', ls=':', lw=1.2, label=r'$t_\oplus = 1$ s')
# Key markers
ax2.plot(r_formation * lt_s, t_earth(r_formation), 's',
color='saddlebrown', ms=10, zorder=5, label='Formation (est.)')
ax2.plot(r_now * lt_s, t_earth(r_now), 'go', ms=10, zorder=5,
label='Present')
ax2.plot(r_req * lt_s, 1.0, 'ro', ms=10, zorder=5,
label='Required for 1 s')
ax2.plot(r_future * lt_s, t_earth(r_future), '^',
color='purple', ms=10, zorder=5, label='Far future')
ax2.set_xlabel(r'$r_{\rm moon}$ (light-seconds)', fontsize=12)
ax2.set_ylabel(r'$t_\oplus$ (s)', fontsize=12)
ax2.set_title('(B) Full range: formation to far future', fontsize=12)
ax2.legend(fontsize=9, loc='upper left')
ax2.grid(alpha=0.3, which='both')
# === Panel C: geological timeline, log r ===
t_arr = np.linspace(-1.5, 2.0, 500)
r_arr = r_of_t(t_arr)
ax3.plot(t_arr, r_arr * lt_s, 'b-', lw=2.4,
of 16 25
label='Tidal model (constant-Q)')
# Valid-range shading
ax3.axvspan(-1.5, 1.5, color='lightgreen', alpha=0.15,
label='Model-valid range')
# Formation and crossing markers
ax3.plot(t_formation, r_formation * lt_s, 's',
color='saddlebrown', ms=11, zorder=5,
label='Formation (est.)')
ax3.plot(0, r_now * lt_s, 'go', ms=11, zorder=5, label='Present')
ax3.plot(t_cross_1s, r_req * lt_s, 'ro', ms=11, zorder=5,
label=f'$t=1$ crossing ({t_cross_1s*1000:.0f} Myr)')
# Thresholds
ax3.axhline(r_now * lt_s, color='green', ls='--', lw=1.2, alpha=0.7)
ax3.axhline(r_req * lt_s, color='red', ls='--', lw=1.2, alpha=0.7)
ax3.axvline(0, color='black', lw=1.0, alpha=0.3)
# Annotations
ax3.annotate('Moon crosses $t=1$',
xy=(t_cross_1s, r_req * lt_s),
xytext=(t_cross_1s - 0.6, r_req * lt_s * 2.2),
fontsize=9, ha='center', color='darkred',
arrowprops=dict(arrowstyle='->', color='darkred', lw=1))
ax3.annotate('Model breaks down\n(real history differs)',
xy=(t_formation, r_formation * lt_s),
xytext=(-3.8, 0.15),
fontsize=8, ha='center', style='italic', color='gray',
arrowprops=dict(arrowstyle='->', color='gray', lw=1))
ax3.set_yscale('log')
ax3.set_xlabel('Time (Gyr, 0 = present)', fontsize=12)
ax3.set_ylabel(r'$r_{\rm moon}$ (light-seconds, log)', fontsize=12)
ax3.set_title('(C) Geological timeline', fontsize=12)
ax3.legend(fontsize=8, loc='upper left')
ax3.grid(alpha=0.3, which='both')
plt.tight_layout()
plt.savefig('earth_moon_upl_full.png', dpi=160, bbox_inches='tight')
plt.show()
print("\nFigure saved to earth_moon_upl_full.png")
of 17 25
5. Discussion: Planetary Optimization and the Genesis Analogy
The idea that a planetary system might be precisely tuned for the emergence of life has a rich
history in science fiction. Perhaps the most famous example is Star Trek's "Project Genesis," a
fictional technology designed to reorganize matter at the subatomic level, instantly transforming
a dead celestial body into a thriving, life-sustaining world.
While Genesis is a fantasy of instantaneous terraforming, it highlights a profound cosmological
question: What are the necessary physical conditions for a planetary system to optimize itself for
complex life?
The Universal Particle Law suggests that this optimization is not arbitrary, nor does it require
science-fiction technology. It emerges naturally from the geometric resonance of the Earth-
Moon-Sun system. The Moon’s role is not merely gravitational; it is resonant. It stabilizes Earth's
obliquity, preventing chaotic temperature extremes, and it produces the perfect solar eclipse—a
geometric coincidence where the angular sizes of the Moon and Sun match exactly.
In the framework presented here, the "optimization" of a planet for life is written in the geometry
of the one-second invariant. The near-match between the actual Earth-Moon distance (1.282
light-seconds) and the distance required for exact closure (1.306 light-seconds) suggests that the
system is tuned for stability. The Moon’s gravitational influence locks the Earth into a stable
rotation plateau (as detailed in Appendix 1), keeping the day length and tidal forces within a
narrow, life-permitting window for hundreds of millions of years.
Where Star Trek imagined an artificial device to create a paradise, the UPL suggests that the
natural dynamics of hierarchical bound systems may already contain the blueprint. The
"Genesis" of a habitable world may be a geometric resonance, not a technological invention.
5.1 The Role of Earth's Obliquity
To understand how this resonance tunes the system, we must look at the role of Earth's obliquity
angle, . The inclusion of in the resonance equation s is physically
justified because the efficiency of tidal coupling depends on the orientation of Earth's rotation
axis relative to the orbital plane (the ecliptic).
1. Geometric Projection of Tidal Forces: The Moon's tidal force on Earth is not applied to a
perfect sphere but to an oblate spheroid (an equatorial bulge). The tidal torque that acts to slow
Earth's rotation and accelerate the Moon's orbit is proportional to the component of the tidal
force that acts along the equatorial plane. As the obliquity changes, the projection of the tidal
force onto the equatorial plane changes, and this projection is described by a factor of (or
related trigonometric functions). Classic analyses of tidal friction, such as those by G.H. Darwin,
θ
cos θ
KE
m
KE
e
(T
e
)cos θ ≈ 1
θ
cos θ
of 18 25
show that the rate of change of obliquity and the rate of tidal friction depend explicitly on the
obliquity angle.
2. The Moon as a Stabilizer of Obliquity: The Moon's presence is known to stabilize Earth's
obliquity, keeping it from chaotic variations. The torque that the Moon exerts on Earth's
equatorial bulge is what maintains this stability. When we formulate a resonance condition that
involves the orbital kinetic energy of the Moon and the Earth's day length, we are describing a
system where the transfer of angular momentum is mediated by tidal torques that depend on .
The equation s can be seen as a heuristic resonance condition that captures
this geometric dependence. The term ensures that the resonance is "tuned" by the obliquity:
the effective coupling between the Moon's orbital energy and Earth's rotational energy is
maximal when the obliquity is small (i.e., when ). With Earth's present obliquity of
, , which reduces the product slightly, bringing the value closer to
exactly 1 second.
References
[1] Beardsley, I. (2026). A Universal Particle Law (UPL) and a Planetary System Quantum
Analog (PSQA): Scale Invariance from Quantum Scales to the Celestial Scales (SIQC). Zenodo.
https://doi.org/10.5281/zenodo.23571277
[2] Beardsley, I., & Blackwell, D. (2026). Informational Geometry, the 2π-Hz Resonance, and
Warp-Drive Dynamics Without Exotic Matter. Zenodo. https://doi.org/10.5281/zenodo.20028442
[3] Tiesinga, E., Mohr, P. J., Newell, D. B., & Taylor, B. N. (2024). CODATA Recommended
Values of the Fundamental Physical Constants. NIST.
[4] Tynski, K. (2024). One Equation, Mysteries: A Structural Constraint That May
Explain (Almost) Everything.
[5] Beardsley, I. (2026). The Human Scale as a Resonant Node Between the Quantum and
Astrophysical Scales. In [1].
θ
KE
m
KE
e
(T
e
)cos θ ≈ 1
cos θ
cos θ ≈ 1
23.5
∘
cos(23.5
∘
) ≈ 0.917
∼ 200
of 19 25
Appendix 1: The 24-Hour Day as a Stable Resonance Plateau
This appendix consolidates the earlier treatments of the 24-hour day, the perfect-eclipse
resonance, the derivative condition , the ODE anchoring of the day length, the linear
stability analysis, and the final reformulation of the boundary condition as a finite plateau/
window rather than an exact point.
A.1 The resonance function
The resonance condition used throughout the paper is
where is the Moon’s orbital kinetic energy, is Earth’s orbital kinetic energy around the
Sun, is the length of the day, and is Earth’s obliquity. Since
and is effectively constant over the relevant timescales, we may write
Over a century, varies by less than , so it is treated as constant for the instantaneous
derivative.
The UPL boundary condition for the Earth–Moon–Sun system is
This anchors the scale of the solar-system quantum analog such that at the present epoch,
More precisely, as discussed in A.5, the correct statement is that lies within a finite plateau
or window near unity.
A.2 The plateau condition:
·
F ≈ 0
F(t) =
K E
m
(t)
K E
e
T
day
(t)cos θ,
K E
m
K E
e
T
day
θ
K E
m
=
1
2
M
m
v
2
m
=
GM
m
M
e
2r
m
=
𝒢
r
m
, 𝒢 =
1
2
M
m
GM
e
,
K E
e
F(t) = K
T
day
(t)
r
m
(t)
, K =
𝒢 cos θ
K E
e
.
θ
0.01
∘
ℏ
⊙
= (1 s)K E
e
, h
⊙
= 2π ℏ
⊙
≈ 1.77018 × 10
34
J ⋅ s .
F(t
0
) ≈ 1.
F(t
0
)
·
F ≈ 0
of 20 25
The key dynamical statement is that the Earth–Moon–Sun system is not merely passing through
the resonance; it is sitting near a local extremum of the resonance function. Taking the
logarithmic derivative of ,
Thus requires
Using observed values,
the required rate of day lengthening is
Converting to milliseconds per century,
The observed total rate is about . The paper isolates the non-tidal climate
contribution as , leaving the pure lunar tidal component
The residual is
Since the standard uncertainty in the lunar tidal deceleration is typically to
, this residual is within about . Therefore,
within observational error. The system is on a flat plateau: the Moon’s recession and Earth’s
rotational slowdown are dynamically coupled so as to keep the product nearly constant.
F = K T
day
/r
m
·
F
F
=
·
T
day
T
day
−
·
r
m
r
m
.
·
F ≈ 0
·
T
day
T
day
≈
·
r
m
r
m
.
T
day
= 86,400 s, r
m
= 3.844 × 10
8
m ,
·
r
m
= 0.038 m /yr,
·
T
(req)
day
= T
day
·
r
m
r
m
= 86,400
0.038
3.844 × 10
8
= 8.541 × 10
−6
s/yr .
·
T
(req)
day
= 0.854 ms/century .
2.3 ms/century
1.33 ms/century
·
T
(obs)
day
= 2.3 − 1.33 = 0.97 ms/century .
Δ = 0.97 − 0.854 = 0.116 ms/century .
±
0.1
±
0.2 ms/centur y
1σ
·
F(t
0
) ≈ 0
F(t)
of 21 25
A.3 The ODE for the plateau and the anchoring of the day length
If is treated as an attractor condition, then
Integrating,
so
The integration constant is fixed by the UPL boundary condition. In the idealized exact
version,
so
With the observed and the solar-system parameters entering , this gives the observed 24-
hour day. Thus, within the idealized fixed-point treatment, the 24-hour value is the unique
solution selected by the combination of
1. , i.e. plateau stability, and
2. , i.e. the UPL boundary condition.
However, the physically realistic version is that within a finite window, as discussed
below. In that case the ODE still shows that the day length scales linearly with the Moon’s orbital
radius while the system remains on the plateau, and the observed 24-hour day lies inside the
predicted window.
A.4 Linear stability and the attractor eigenvalue
To show that the plateau is a stable attractor rather than a saddle point, linearize the tidal
evolution around . Write
·
F = 0
·
T
day
T
day
=
·
r
m
r
m
.
ln T
day
= ln r
m
+ ln C,
T
day
(t) = C r
m
(t) .
C
F(t
0
) = K
T
day
(t
0
)
r
m
(t
0
)
= 1,
T
day
(t
0
) =
r
m
(t
0
)
K
.
r
m
(t
0
)
K
·
F = 0
F(t
0
) = 1
F(t
0
) ≈ 1
F
*
= 1
F = F
*
+ δF .
of 22 25
The tidal torque depends on the tidal frequency , where
Using
with
we obtain
At the fixed point ,
Linearizing,
and since ,
where
Evaluating gives
Using angular-momentum conservation,
and
,
Γ
ω = Ω − n
Ω =
2π
T
day
, n =
GM
e
r
3
m
.
·
T
day
T
day
= −
·
Ω
Ω
=
Γ
L
rot
,
·
r
m
r
m
=
2Γ
L
orb
,
L
rot
= IΩ, L
orb
= M
m
GM
e
r
m
,
·
F
F
= ϕ(F ) = Γ(F )
(
1
L
rot
(F )
−
2
L
orb
(F )
)
.
F
*
ϕ(F
*
) = 0 ⟹ L
orb,*
= 2L
rot,*
.
ϕ(F ) ≈ ϕ′ (F
*
)δF,
·
F = Fϕ(F )
δ
·
F = F
*
ϕ′ (F
*
)δF ≡ − λ δF,
λ = − F
*
ϕ′ (F
*
) > 0.
ϕ′ (F
*
)
ϕ′ (F
*
) = Γ
*
(
−
L′
rot
L
2
rot,*
+
2L′
orb
L
2
orb,*
)
.
L′
rot
+ L′
orb
= 0 ⟹ L′
orb
= − L′
rot
,
L
orb,*
= 2L
rot,*
of 23 25
we find
For a positive perturbation , the Moon is farther out and the day is longer, so the
rotational angular momentum decreases: . Hence , and since ,
Therefore,
Small perturbations decay exponentially. A numerical estimate using a constant-time-lag tidal
model gives
corresponding to an e-folding time
This is the correct order of magnitude for the plateau to persist over the geological timescale
required for complex life.
A.5 The boundary condition as a finite window
The exact statement should be replaced by the more defensible statement
Equivalently, since
a window in \(F\) maps directly to a window in :
Using the numbers in the paper, with , the mapping is approximately:
ϕ′ (F
*
) = −
3Γ
*
L′
rot
2L
2
rot,*
.
δF > 0
L′
rot
< 0
−L′
rot
> 0
Γ
*
> 0
ϕ′ (F
*
) < 0 ⟹ λ = − F
*
ϕ′ (F
*
) > 0.
δ
·
F = − λ δF, δF(t) = δF(0)e
−λt
.
λ ≈ 3 × 10
−17
s
−1
,
1
λ
≈ 1 Gyr .
F(t
0
) = 1
F(t
0
) ≈ 1 within the window 0.9 ≲ F ≲ 1.25.
F = K
T
day
r
m
,
T
day
T
min
day
≤
Fr
m
K
≤ T
max
day
.
K ≈ 5.07 × 10
3
m /s
of 24 25
F vs T_day!
Thus a window of roughly 22–26 hours corresponds to , centered near .
If one instead keeps as the anchor, the window is more like 19–23 hours. Either way, the
observed 24-hour day sits inside the predicted plateau.
This is the honest formulation: the UPL does not uniquely derive seconds from first
principles. It predicts a stable plateau, and the observed 24-hour day lies within that plateau. The
absolute value of is set by the observed lunar orbital radius and the requirement that
remain near unity. The plateau width is controlled by the sensitivity of the tidal torque to changes
in , i.e. by the Love number and the tidal quality factor ,
A more precise future treatment could define the window width from tidal physics as
or by requiring that the accumulated drift over years be less than some fraction of the
window width.
A.6 Summary
The consolidated picture is:
1. Resonance function.
F
T_day (hours)
0.90
19.0
0.95
20.0
1.00
21.1
1.05
22.2
1.10
23.2
1.14
24.0
1.20
25.3
1.25
26.4
F ∈ [1.05,1.23]
F ≈ 1.14
F ≈ 1
86,400
T
day
F
T
day
k
2
Q
ΔT
day
∼
T
day
λ
·
T
day
T
day
obs
,
5 × 10
8
F(t) = K
T
day
(t)
r
m
(t)
.
of 25 25
2. Plateau condition.
The required tidal rate is ; the observed pure lunar tidal rate is
; the residual is within observational error.
3. ODE solution on the plateau.
The UPL boundary condition fixes the scale. In the idealized exact treatment,
.
4. Linear stability.
with e-folding time Gyr. The plateau is a stable attractor.
5. Window, not point.
The correct boundary condition is within a finite window. The observed 24-hour day
lies inside the predicted plateau, roughly 22–26 hours for , or 19–23 hours if
is retained as the anchor. The exact value s is not uniquely derived from first
principles, but it is a resonant node within a dynamically stable plateau.
Thus the 24-hour day, the perfect-eclipse geometry, and the 1-second UPL resonance are tied
together by a single dynamical statement: the Earth–Moon–Sun system is sitting on a stable
resonance plateau, and the observed day length is the human-scale projection of that plateau.
·
F ≈ 0 ⟺
·
T
day
T
day
≈
·
r
m
r
m
.
0.854 ms/century
0.97 ms/century
0.116 ms/century
T
day
(t) = C r
m
(t) .
F(t
0
) ≈ 1
T
day
(t
0
) = r
m
(t
0
)/K ≈ 86,400 s
δ
·
F = − λ δF, λ > 0, λ ≈ 3 × 10
−17
s
−1
,
∼ 1
F(t
0
) ≈ 1
F ∈ [1.05,1.23]
F ≈ 1
86,400