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The 1-Second Invariant and the Galactic Census of Intelligent Life

Ian Beardsley

February 23, 2026

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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.

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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

× n

× f

× L

R *

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requires knowledge of terms—particularly , , and —that remain

largely 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.

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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:

πh

G c

⋅

3α

= 1 s

= 1

πh

G c

⋅ 1 = 1 s

1/2

= π

1.0475

9.793

= 1.028 s

86,400 = 6

× 400

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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. The factors of six

are convenient factors into which to factor something: Six is the

product of the two smallest primes, 2 and 3, the smallest factors down

to which we can factor anything, one even, one odd. The regular

hexagon has its sides equal to its radius giving geometric advantage.

The mole is defined by by 12 grams of carbon where carbon’s six

protons and six electrons make it the most stable element.

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

⊙

≈ 400,

⊕

≈ 400

0.8 R

⊕

≲ R

planet

≲ 1.6 R

⊕

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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.

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.

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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.

total

= f

× f

× P

moon

× P

eclipse

× P

intelligence

moon

eclipse

intelligence

≈ 0.5

≈ 0.2

moon

≈ 0.001

eclipse

≈ 0.02

intelligence

≈ 1

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4.3 The Calculation

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.

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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.

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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

total

= 0.5 × 0.2 × 0.001 × 0.02 × 1 = 2 × 10

−6

= 10

N = N

× P

total

= 10

× 2 × 10

−6

= 2 × 10

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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

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

near perfect solar eclipse by the Moon may be transient but the one-

second invariance of the Moon comes in phase with it through its

orbital mechanics. 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

K E

moon

K E

Earth

(24 hours)cos(23.5

∘

) ≈ 1 second

K E

moon

(7.347673 × 10

kg)(966 m/s)

= 3.428 × 10

K E

Earth

(5.972 × 10

kg)(30,290 m/s)

= 2.7396 × 10

0.991 seconds ≈ 1 second

ℏ

⊙

ℏ

⊙

3.0281E 8m

299,792,458m /s

= 1.010 seconds

ℏ

⊙

ℏ

⊙

= (1sec on d )K E

ℏ

⊙

1.03351s =

⋅

π r

G m

ℏ

⊙

= (1.03351s)(2.7396 E 33J ) = 2.8314E 33J ⋅ s

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I find that for the Earth around the Sun (In a quantum analog for the

Earth/Moon/Sun System of the hydrogen atom)

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

⊙

⋅

2ℏ

⊙

2ℏ

⊙

⋅

⊙

⋅

K E

⊙

ℏ

⊙

6.96E 8m

1737400m

= 400.5986

= (1.732)(400.5986)

(6.67408E − 11)

(5.972E 24kg)

(7.347673E 22k g)

2(2.8314E 33)

K E

earth

(5.972E 24kg)(30,290m /s)

= 2.7396E 33J

2.727E 33J

2.7396E 33J

100 = 99.5 %

∼ K E

earth

ℏ

⊙

3.0281E 8m

299,792,458m /s

= 1.010 second

2ℏ

⊙

⋅

⊙

⋅

n = 3

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Spectral class M stars are red dwarfs; red and cool and small compared

to the Sun. This puts their habitable zones closer in. So close in

that the planet's rotation is affected by tidal forces from the star

that slow down its rotation (day) until it is the same as its orbital

period (year) thus meaning that one side of the planet is aways facing

the star and thus always hot and daytime, and the other side is always

facing away and cold and always nighttime making the only place good

for life on the boundary between nighttime and daytime where it is

always twilight. Also, these smaller, cooler stars have so much flare

activity that would easily strip away the planet's atmosphere.

Similarly, stars bigger and brighter than the Sun, like blue spectral

class A and F stars, are more short lived not giving intelligence as

much of a chance to evolve. Also they form more gas giant type planets

like Jupiter and Saturn. The Sun is in a perfect compromise between

these two kinds of stars. For these stars the eclipse ratio should be

about 400. The red dwarfs may have a longer life than our Sun but

their flare activity is a problem.

However, while I use the cited probability for stars with planets in

the habitable zone and the probability of those having Moons with the

400 to one eclipse ratio, I am thinking of computing with a

probability that corresponds not necessarily to an eclipse ratio of

400, but for a star simply where there is a perfect eclipse. That

condition is

While the rotation the planet loses energy to its moon, lengthening

its day, and thus the orbital radius of the moon growing, slowly over

millions of years, we suggest this perfect eclipse comes into being

when its kinetic energy to that of the planet it orbits maps a 24 hour

rotation period of the planet into one-second, as it does for our

Solar System. That is when

Would the day be 24 hours for such planets at this phase in its state?

It might be so, because it may mean such planets have Earth size, .

Is there something about the Sun that is common to other types of

stars; stars that are perhaps larger and hotter than the Sun, or

perhaps smaller and cooler, or a different color, like blue or red,

instead of yellow? The answer is yes. I actually found something in

ancient Vedic knowledge, in the Hindu traditions. Apparently, in Hindu

yoga the number 108 is an important number. I read that yogis today

noticed that the diameter of the Sun is about 108 times the diameter

of the Earth and that the average distance from the Sun to the Earth

is about 108 solar diameters, with 108 being a significant number in

yoga. So I wrote the equivalent:

⋆

moon

planet

moon

K E

moon

K E

Earth

(24 hours)cos(23.5

∘

) ≈ 1 second

⊕

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Or, for any star and habitable planet:

radius of the star. the orbital radius of the habitable

planet.

In order to get , the distance of the habitable planet from the

star, we use the inverse square law for luminosity of the star. If the

Earth is in the habitable zone, and if the star is one hundred times

brighter than the Sun, then by the inverse square law the distance to

the habitable zone of the planet is 10 times that of what the Earth is

from the Sun. Thus we have in astronomical units the habitable zone of

a star is given by:

the luminosity of the star, the luminosity of the Sun. AU the

average Earth-Sun separation, which is 1. The surprising result I

found was, after applying equation 4, hypothetically predicting the

size of a habitable planet, to the stars of all spectral types from F

through K, with their different radii and luminosities (the

luminosities determine , the distances to the habitable zones),

that the radius of the planet always came out about the same, about

the radius of the Earth. This may suggest optimally habitable planets

are not just a function of their distance from the star, which is a

big factor in determining their temperature, but are functions of

their size and mass meaning the size of the Earth could be good for

life chemistry atmospheric composition, and gravity.

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

⊕

= 2

⊙

⊕

planet

= 2

⋆

habitable

⋆

habitable

⋆

⊙

⋆

⊙

habitable

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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.

Note: The Galaxy is about 100 to 200 thousand light years across, and

about 1000 light years thick. Doing a thin disc approximation, the

nearest intelligent civilization would be about 200 light years away.

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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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References

1. Ghosh, A. (2020). "Emergence of order from chaos: A

phenomenological model of coupled oscillators." *Chaos, Solitons &

Fractals* 110334.

2. Ramirez, R., Gómez-Muñoz, M.A., Vázquez, R., & Núñez, P.G. (2017).

"New numerical determination of habitability in the Galaxy: the SETI

connection." *International Journal of Astrobiology*.

3. "Earth and the Moon are proportional partners, and that might be

rare." *Astronomy Magazine*.

4. Ghosh, A. (2020). "Emergence of order from chaos: A

phenomenological model of coupled oscillators." *X-MOL*.

5. Ramirez, R., et al. (2016). "New numerical determination of

habitability in the Galaxy: the SETI connection."

IAC-16,A4,2,1,x33791.

6. Stack Exchange Physics (2024). "Revision: A rough and ready

approximation."

7. Yakupov, E., et al. (2025). "Emergence of order within disorder:

Shadowed waves and chimera states in reaction-diffusion systems."

*Chaos* 35(10), 103122.

of 11 11

8. Forgan, D. (2008). "A Numerical Testbed for Hypotheses of

Extraterrestrial Life and Intelligence." *arXiv:0810.2222*.

9. Fuse, Q., & Fuse, C. (2024). "The Variability of Moon Formation

Around Terrestrial and Gas Giant Planets." *Bulletin of the AAS*

56(7).

10. Beardsley, I (2026). “How Physics and Archaeology Point to a

Natural Constant of 1-Second” https://doi.org/10.5281/zenodo.18773959,

DOI: 10.13140/RG.2.2.31824.98560