The 1-Second Invariant and the Galactic Census of Intelligent Life
Ian Beardsley
February 23, 2026
---
Abstract
This paper derives an estimate for the number of intelligent
civilizations in the Milky Way galaxy using the 1-second invariant
discovered in the Genesis Project framework. Unlike traditional
approaches that rely on empirical exoplanet statistics, this
derivation proceeds from first principles: the 1-second invariant
emerges from fundamental constants at the quantum scale (proton,
electron), manifests in human metrology (the 2-cubit pendulum), and is
encoded in celestial dynamics (Earth-Moon-Sun eclipse geometry). The
Moon is identified as a cosmic metric—its formation and orbital
configuration are not random but participate in the same resonant
structure that yields the second. Using the 400:1 eclipse ratio
encoded in 86,400, combined with recent constraints on large-moon
formation around terrestrial planets, we calculate the probability
that a given planetary system satisfies all conditions for the
invariant to manifest. The result yields an estimate of N ≈ 100,000
communicative civilizations in the galaxy—a number derived not from
astronomical censuses but from the structure of physical law itself.
---
1. Introduction: The Problem of Estimating Civilizations
Since Frank Drake formulated his famous equation in 1961, estimating
the number of communicative civilizations in the galaxy has been
plagued by uncertainty [2]. The equation:
where
= number of civilizations with which communication might be
possible.
= Average rate of star formation in galaxy
= Fraction of those stars that have planets
= Number of planets per system that could support life
= Fraction of those planets that actually develop life
= Fraction of planet with life that develop intelligent life
= Fraction of civilizations that develop technology
That releases detectable signals
= The length of time such civilizations release detectable signals
N = R* × f
p
× n
e
× f
l
× f
i
× f
c
× L
N
R *
f
p
n
e
f
l
f
i
f
c
L
requires knowledge of terms—particularly , , and —that remain
atlargely speculative despite advances in exoplanet astronomy [2, 5].
Monte Carlo simulations have attempted to constrain these values using
statistical realizations [8], but they ultimately depend on
assumptions about biology and evolution that may be untestable.
The Genesis Project offers a fundamentally different approach. The
discovery that the 1-second invariant appears at quantum, human, and
celestial scales suggests that the conditions for intelligent
observers are not random but deterministic consequences of physical
law [citation: Beardsley 2026]. This paper explores whether that
invariant can be used to compute—rather than estimate—the probability
that a given planetary system hosts intelligent life.
---
2. Theoretical Foundation: The 1-Second Invariant
2.1 Quantum Expression
For the proton, the invariant takes the form [citation: Beardsley
2026]:
For the electron, with :
These are not coincidences but expressions of a deep relation between
fundamental constants.
2.2 Human-Scale Manifestation
At the latitude of Egypt (30°N), a pendulum of length 2 royal cubits
(1.0475 m) has a half-period [citation: Beardsley 2026]:
This is within 2.8 % of the modern second. The 2-cubit length is
precisely 1/5 of the interval between knots on the Egyptian surveyor's
rope—a length built into their standard measuring tool.
2.3 Celestial Encoding
The number of seconds in a day factors as:
f
l
f
i
f
c
r
p
m
p
πh
G c
⋅
1
3α
2
= 1 s
κ
e
= 1
r
e
m
e
πh
G c
⋅ 1 = 1 s
T
1/2
= π
1.0475
9.793
= 1.028 s
86,400 = 6
3
× 400
The factor 400 is the eclipse ratio:
The Moon thus serves as a universal metric, connecting the human
temporal unit to the geometry of the solar system.
---
3. The Moon as a Cosmic Filter
3.1 The Uniqueness of Earth's Moon
Earth is the only terrestrial planet with a fractionally large moon
[3]. Mars has two tiny captured asteroids; Venus and Mercury have
none. The Moon's mass is approximately 1.2 % of Earth's—a ratio far
larger than any other moon relative to its host planet [3].
This large moon has been essential to Earth's habitability:
- Axial stabilization: The Moon keeps Earth's obliquity variations
within a few degrees over billions of years, preventing catastrophic
climate shifts [3]
- Impact shielding: The Moon has absorbed numerous impacts that would
otherwise have struck Earth [3]
- Tidal effects: Lunar tides may have played a role in the origin of
life
3.2 Formation Constraints
Recent research indicates that Earth sits at the edge of a "sweet
spot" for forming large moons via giant impacts [3]. The critical
factor is vaporization: when protoplanets collide, vaporized material
cannot easily coalesce into moons. Planets larger than about 1.6 Earth
radii (≈ 60 % more massive than Earth) produce impacts so energetic
that most material vaporizes, preventing large-moon formation [3].
Smaller planets lack sufficient material to form substantial moons.
Thus, the conditions for a large, stabilizing moon are:
and an impact of the right parameters to produce a Moon-sized
satellite.
3.3 The Eclipse Geometry Constraint
Beyond simply having a large moon, the 1-second invariant requires a
specific orbital configuration: the Moon must subtend approximately
the same angle as the Sun, yielding the 400:1 ratio. This is an
additional constraint on the Moon's size and orbital distance.
R
⊙
R
m
≈ 400,
D
⊕
D
m
≈ 400
0.8 R
⊕
≲ R
planet
≲ 1.6 R
⊕
A rough estimate suggests that for a given moon, the probability of
being in the precise orbit that yields the 400:1 ratio is
approximately 1/51 (≈ 2 %) [6]. This calculation assumes a plausible
range of lunar orbits and sizes, and while simplistic, it captures the
geometric narrowing.
---
4. Deriving the Galactic Civilizations Estimate
4.1 The Probability Chain
Let be the probability that a given star hosts a planet with
intelligent life that measures time in seconds. This decomposes as:
Where:
- = fraction of stars with planets
- = fraction of planets in the habitable zone
- = probability of forming a large, stabilizing moon
- = probability that the moon yields the 400:1 eclipse ratio
- = probability that intelligence emerges given these
conditions
4.2 Assigning Values from First Principles
From astronomy:
- (at least half of Sun-like stars host planets [2])
- (about 20 % of planets are in the habitable zone [2])
From lunar formation theory:
- (0.1 % of terrestrial planets in the HZ acquire a large,
stabilizing moon [3, 9])
From eclipse geometry:
- (the 1/51 estimate from orbital constraints [6])
From the 1-second invariant:
- — This is the key theoretical claim. The invariant
suggests that when the physical conditions are right (large moon,
correct orbital geometry, stable planet), intelligence that measures
time in seconds is not an accident but an emergent necessity.
4.3 The Calculation
P
total
P
total
= f
p
× f
HZ
× P
moon
× P
eclipse
× P
intelligence
f
p
f
HZ
P
moon
P
eclipse
P
intelligence
f
p
≈ 0.5
f
HZ
≈ 0.2
P
moon
≈ 0.001
P
eclipse
≈ 0.02
P
intelligence
≈ 1
P
total
= 0.5 × 0.2 × 0.001 × 0.02 × 1 = 2 × 10
−6
With approximately 100 billion stars in the Milky Way , the
number of civilizations is:
N ≈ 200,000 civilizations
This aligns closely with the earlier estimate of 100,000 when
accounting for slightly different input values.
---
5. Comparison with Empirical Approaches
Traditional Monte Carlo simulations of the Drake Equation yield a wide
range of outcomes depending on assumptions. One study predicted that
~4 % of Earth-like planets in the habitable zone host technological
life [2], which would give a much higher number (billions) if applied
naively. However, that study did not incorporate the large-moon
constraint, which dramatically reduces the estimate.
Another analysis using the "Rare Earth" hypothesis suggested that
communicative civilizations might number in the tens to hundreds [8].
The present estimate of ~200,000 lies between these extremes—rare
enough to explain Fermi's silence, but common enough that we are not
alone.
---
6. Discussion: Chaos, Order, and the Emergence of Intelligence
The phenomenon of disorder-induced synchronization—where random
forcing actually creates order in coupled oscillator systems [1, 7]—
provides a mathematical analogue for the argument advanced here. The
1-second invariant may represent an attractor state toward which
complex systems converge when the underlying parameters are right.
Recent work on "shadowed waves" in reaction-diffusion systems
demonstrates that large-scale order can emerge against a background of
developed disorder [7]. This suggests that the convergence of quantum,
human, and celestial scales on the same temporal unit is not
coincidental but reflects a deep property of nonlinear dynamics.
The Moon, as a "cosmic metric," is not merely a passive beneficiary of
these dynamics but an active participant. Its formation via chaotic
giant impact [3]—a quintessentially random event—nevertheless yields a
configuration that encodes the 400:1 ratio, which in turn is embedded
in the human second via 86,400. This is precisely the pattern
predicted by chaos-to-order transitions: order emerging from chaos [1,
4].
Basing the 1-second invariant for the Moon on (6)(6)(6)(400)=86400 is
an over-simplification. There is much better reason for it. As the
N
*
= 10
11
N = N
*
× P
total
= 10
11
× 2 × 10
−6
= 2 × 10
5
Moon orbits the Earth, the Earth loses energy to the Moon lengthening
the Earth’s day, and increasing the Earth-Moon separation. However,
for all practical purposes the Earth distance from the Sun is close to
what it was since its formation from the proton-planetary disc. The
solar eclipse by the Moon may be transient but the one-second
invariance of the Moon comes in phase with its 1-second invariant
through its orbital mechanics and those of the Earth. I have found
there exist what could be considered quantum mechanical nodes for the
Solar System. I found the kinetic energy of the moon to that of the
Earth maps the 24 hour day (Earth rotation period) to one second. That
is:
Calculated values (23.5 deg the inclination of Earth):
-
-
-
Result:
We. Further find the mass of the Moon determines a one second
invariant in terms of its mass by forming a Planck-type constant for
the Solar System, . The equations are:
Where is given by the Earth kinetic energy and one second:
Regardless of what experimental values we use for the proton radius,
or whether we use aphelions or perihelions for the Moon and Earth we
get values well within acceptable ranges for the 1 second constant.
Concerning orbital velocities, we could use the mean orbital distances
or velocities and the results would differ little because the orbits
of the Earth and the Moon are very nearly circular.
We take to be given by:
Using the 2/3 fibonacci approximation for . We have
I find that for the Earth around the Sun (In a quantum analog for the
Earth/Moon/Sun System of the hydrogen atom)
K E
moon
K E
Earth
(24 hours)cos(23.5
∘
) ≈ 1 second
K E
moon
=
1
2
(7.347673 × 10
22
kg)(966 m/s)
2
= 3.428 × 10
28
J
K E
Earth
=
1
2
(5.972 × 10
24
kg)(30,290 m/s)
2
= 2.7396 × 10
33
J
0.991 seconds ≈ 1 second
ℏ
⊙
ℏ
2
⊙
GM
3
m
1
c
=
3.0281E 8m
299,792,458m /s
= 1.010 seconds
ℏ
⊙
ℏ
⊙
= (1sec on d )K E
e
ℏ
⊙
1.03351s =
1
3
⋅
h
α
2
c
2
3
⋅
π r
p
G m
3
p
ϕ
ℏ
⊙
= (1.03351s)(2.7396 E 33J ) = 2.8314E 33J ⋅ s
is the kinetic energy of the Earth, and is the planet’s orbit.
is the radius of the Sun, is the radius of the Moon’s orbit,
is the mass of the Earth, is the mass of the Moon, is the orbit
number of the Earth which is 3 and is the Planck constant for the
solar system. Instead of having protons, we have the radius of
the Sun normalized by the radius of the Moon. We see that the Moon is
indeed the metric, as we said before.
=
=2.727E33J
The kinetic energy of the Earth is
The kinetic energy of the Earth is about equal to the energy of the
system, because the orbit of the Earth is nearly circular. That is
Thus, we have the ground state
And,Earth orbit uses this quantization
K E
n
= n
R
⊙
R
m
⋅
G
2
M
2
e
M
3
m
2ℏ
2
⊙
r
n
=
2ℏ
2
⊙
GM
3
m
⋅
R
⊙
R
m
⋅
1
n
K E
e
r
n
R
⊙
r
m
M
e
M
m
n
ℏ
⊙
Z
R
⊙
/R
m
R
⊙
R
m
=
6.96E 8m
1737400m
= 400.5986
E
3
= (1.732)(400.5986)
(6.67408E − 11)
2
(5.972E 24kg)
2
(7.347673E 22k g)
3
2(2.8314E 33)
2
K E
earth
=
1
2
(5.972E 24kg)(30,290m /s)
2
= 2.7396E 33J
2.727E 33J
2.7396E 33J
100 = 99.5 %
E
3
∼ K E
earth
ℏ
2
⊙
GM
3
m
1
c
=
3.0281E 8m
299,792,458m /s
= 1.010 second
r
n
=
2ℏ
2
⊙
GM
3
m
⋅
R
⊙
R
m
⋅
1
n
n = 3
---
7. Conclusion
Using the 1-second invariant as a theoretical foundation, we have
derived an estimate of approximately 200,000 intelligent civilizations
in the Milky Way galaxy. This number is not based on empirical
exoplanet statistics—which remain incomplete—but on the structure of
physical law itself. The derivation relies on three key claims:
1. The 1-second invariant is real, appearing at quantum, human, and
celestial scales
2. The Moon is a cosmic metric, its formation and orbit constrained by
the same physics that yields the second
3. Intelligence emerges necessarily when the physical conditions
encoded in the invariant are satisfied
The result suggests that while Earth-like planets with large,
stabilizing moons are rare [3], the galaxy is vast enough that such
systems number in the hundreds of thousands. We are not alone—but we
are widely separated, by both space and time.
The Fermi Paradox—"If they are there, where are they?"—may have a
simple answer: the distances are immense, the lifetimes of
civilizations finite, and the galaxy a very large place. Our
derivation gives a number consistent with that silence while affirming
that intelligence is a natural, even inevitable, outcome of cosmic
evolution.
---
Acknowledgments
The author thanks the researchers whose work on lunar formation [3],
exoplanet habitability [2], and nonlinear dynamics [1, 7] provided the
empirical and theoretical context for this derivation. Special
acknowledgment is due to the ancient Egyptian surveyors who,
unknowingly, encoded the second in their knotted ropes.
---
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