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A Universal Particle Law (UPL) and a Planetary System Quantum

Analog (PSQA): Scale Invariance from Quantum Scales to the Celestial

Scales (SIQC)

Ian Beardsley

October 5, 2026

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Contents

Introduction……………………………………………………………………3

A Universal Particle Law (UPL) and A Planetary System

Quantum Analog (PSQA): Scale Invariance from Quantum

Scales to the Celestial Scales (SIQC)…………………………………………6

The Human Scale as a Resonant Node

Between the Quantum and Astrophysical Scales…………………………….27

The 1 Hz Resonance as the Key to Exotic-Matter-Free

Warp Drive: A Synthesis of the Universal Particle Law

and Informational Geometry………………………………………………….43

The 1Hz Resonance: A Physical Basis for the Master

Clock in Biological Systems………………………………………………….56

Appendix 1: The 24-Hour Day as a Stable Resonance Plateau……………….61

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Introduction

This collection of papers presents a unified framework that spans from the quantum scale to the

galactic scale, centered on a single, remarkable observation: a fundamental one-second temporal

invariant appears to govern the structure of bound systems throughout the universe. What began

as an investigation into the nature of inertia has grown into a comprehensive program that

touches particle physics, planetary dynamics, biology, and the possibility of faster-than-light

travel.

The Universal Particle Law (UPL) proposes that mass is not an intrinsic property of matter but

rather a resistance to the rotation of a particle's velocity vector from the temporal dimension into

space. When we push on a particle, we are diverting some of its motion through time into motion

through space. The resistance to this rotation—what we experience as inertia—is quantified by a

normal force that depends on Planck's constant, the speed of light, and a fundamental timescale.

Remarkably, when this law is applied to the proton, neutron, and electron, the same timescale

emerges: approximately one second. The same one-second resonance appears when the law is

applied to the Earth-Moon-Sun system and to the Sun's orbit around the galactic center. This is

not a coincidence but a structural feature of the universe.

The framework is scale invariant. The same mathematical form that predicts the ratio of radius to

mass for fundamental particles also predicts the ratio of orbital parameters for planets and stars.

The coupling constants that appear in these equations are not free parameters but are determined

by the geometry and internal structure of the systems they describe. For composite particles like

protons and neutrons, the coupling reflects the presence of three valence quarks and the strength

of the strong interaction. For the electron, which has no substructure, the coupling is simply

unity. For the Earth-Moon-Sun system, the coupling is the ratio of the Moon's orbital radius to

the Sun's radius—the geometric condition for a perfect solar eclipse.

The one-second resonance has profound implications. It suggests that time is not merely a

coordinate but a dynamical variable with a natural periodicity. The relationship between the

normal force and the Planck force reveals that over one second, the accumulated angular phase is

exactly a full rotation. This geometric interpretation connects the quantum vacuum to

macroscopic dynamics through a single, testable relation.

The framework extends to biology. The human heart beats at approximately one cycle per

second. This is not a biological accident but a physical resonance. The heart is made of protons,

neutrons, and electrons—the same particles that define the one-second invariant. The rhythm of

life is written into the structure of matter itself. This provides a physical foundation for

understanding biological agency and the active role of organisms in evolution, complementing

gene-centric views with a recognition that life unfolds within a landscape of physical attractors.

The most speculative—and potentially transformative—application concerns the Alcubierre warp

drive. The standard warp-drive solution requires exotic matter with negative energy density, a

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substance for which no physical mechanism exists. When the one-second resonance is embedded

within an informational geometry framework, the picture changes. Warp-bubble dynamics

become geodesic motion in an extended configuration space that includes informational

dimensions beyond the four dimensions of spacetime. The apparent need for exotic matter

dissolves: what appears pathological when viewed from the spacetime submanifold alone is

revealed as ordinary when the full informational manifold is considered. Numerical simulations

demonstrate that stable, positive-energy bubble configurations can translate at superluminal

effective velocities while preserving local causality.

The papers collected here develop this framework in stages. The first paper presents the

Universal Particle Law and its application to the Earth-Moon-Sun system, establishing the one-

second invariant across scales. The second paper identifies the human scale as a resonant node

between the quantum and astrophysical scales, with the human effective action emerging as the

geometric mean of the Planck constant and the celestial effective action. The third paper

synthesizes the Universal Particle Law with the Observer-Dependent Information Metric,

showing how the one-second resonance eliminates the exotic-matter requirement for warp drive.

The fourth paper applies the framework to biology, answering Denis Noble's question about the

origin of the heartbeat rhythm. An appendix provides a rigorous treatment of the twenty-four-

hour day as a stable resonance plateau, demonstrating that the Earth-Moon-Sun system is

dynamically locked at a flat extremum of the resonance condition.

Throughout these papers, the emphasis is on testable predictions and empirical verification. The

proton radius predicted by the golden ratio matches experimental measurements. The one-second

invariant emerges for the electron, proton, and neutron within a fraction of a percent. The

twenty-four-hour day lies within a predicted window of stability that has persisted for hundreds

of millions of years. The numerical simulations of warp-bubble dynamics show strictly positive

energy densities and stable resonance capture. These are not mere coincidences but signatures of

a deep structural principle.

The work is presented in the tradition of phenomenological laws awaiting deeper algebraic

derivation. Like Newton's law of gravitation, the Universal Particle Law makes definite

numerical predictions that can be tested against observation. Its foundation is the Principle of

Least Action, from which the normal force can be derived. Its empirical signature is the one-

second resonance, which appears from the proton to the galaxy.

The implications are far-reaching. If the one-second resonance is real, then the conditions for

complex life may be geometrically fixed. Planetary systems with an asteroid belt at

approximately 2.4 AU and a gas giant at approximately 9.5 AU may be the preferred architecture

for intelligence to emerge. The search for life elsewhere should target systems with these

resonant features. The same law that builds protons also builds solar systems, and it does so in a

way that maximizes the chances for stable, complex, intelligent life.

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This collection represents an invitation to explore a new way of thinking about the relationship

between the quantum and the cosmic, between matter and information, between physics and

biology. The universe, it seems, is resonant. We are not merely observers of this cosmic order;

we are its resonant expression.

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A Universal Particle Law (UPL) and A Planetary System Quantum

Analog (PSQA): Scale Invariance from Quantum Scales to the Celestial

Scales (SIQC)

Ian Beardsley

October 5, 2026

Abstract

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 .

= h /(ct

)

= 1

= π r

= κ

π r

= ϕh /(cm

)

1/ϕ = Φ

Φ = ( 5 + 1)/2

= h /(ct

)

= c

− d x

− d y

− d z

= c 1 −

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The Universal Particle Law

We introduce two equations that give on the order of 1-second in terms of the proton radius and

mass:

(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:

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

ϕ ⋅

π r

⋅

= 1 second

6α

⋅

4πh

= 1second

: 1.67262E − 27kg

: 0.833E − 15m

0.8367

0.0845f m

h : 6.62607E − 34J ⋅ s

c : 299,792,458m /s

G : 6.6743E − 11N

α :

( 5 − 1)/2 ≈ 0.618

= ϕ ⋅

= (0.618) ⋅

6.62607E − 34

(299,792,458)(1.67262E − 27)

= 0.8166E − 15m

= 0.831f m

0.014f m

= 0.817E − 15m

= h ν

= m

= h /(m

c) = r

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introduce the factor of , which is the golden ratio conjugate, where the golden ratio,

We explain this factor by invoking Kristin Tynski, her paper titled: One Equation, ~200

Mysteries: A Structural Constraint That May Explain (Almost) Everything [5].

Tynski shows that for any system requiring consistency across multiple scales of observation has

the recurrence relation:

Which leads to:

Whose solution is . Equations 1, 2, and 3 directly yield our Universal Particle Law:

where . Here we see in equation 4, the cross-sectional area of the proton

is exposed to the normal force, mediated by the 'stiffness of space' as measured by ,

producing the proton mass, . In general we have

7. ,

Φ = 1/ϕ = ( 5 + 1)/2 ≈ 1.618

scale(n+2) = scale(n+1) + scale(n)

= λ + 1

= κ

⋅

π r

= 1 second

= 1/(3α

)

= π r

= κ

⋅

π r

6.62607015 × 10

−34

J·s

(299,792,458 m/s)(1 s)

= 2.21022 × 10

−42

= 1 second

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We can verify this solving 7 for and showing it is on the order, closely, to 1-second:

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 .

= κ

π r

⋅

πh

G c

⋅ κ

3α

α = 1/137

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

3α

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

= 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

= 1

= 0.84E − 15m

= 2.81794E − 15m

= 0.8367

0.0845f m

= 1 second

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The dimensionless factor distinguishes elementary particles from composite hadrons.

Remarkably, the same emerges for the electron, proton, and neutron when their

respective are chosen appropriately.

The factor reflects the three valence quarks inside the proton and neutron. The appears

because the proton’s small radius (relative to its mass) is set by the strong interaction, which is

times stronger than electromagnetism. Consequently, the required enhancement scales as

the square of that ratio because it deals with surface area.

The electron: as the baseline

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.

= 1 s

1/3

−2

∼ 1/α

= 1

= 2.81794 × 10

−15

= 9.10938 × 10

−31

= 1

πh

⋅ 1 = 0.99773 s,

= 1 s

= m

πh

= 2.82 × 10

−15

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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.

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:

= (γc, γ v )

S =

∫

ℒ dt = − m c

∫

1 −

p =

∂ℒ

∂v

= γ mv

F =

= 1

= E /c

E ⋅ t

∼ h

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When this momentum is fully rotated into the spatial direction, the spatial momentum becomes:

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:

c t

p = p

c t

F =

(

∂ℒ

∂v

)

p =

Δ p

Δt

c t

F = dp/dt

h /(ct

)

ℒ

F = −

∂V

∂x

V(x) = −

ℒ =

− V(x) =

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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

(

∂ℒ

∂v

)

−

∂ℒ

∂x

= 0

(mv) −

= 0

v =

= m

v =

S =

∫

−m c −

+ λ

(

− t

)

dτ

−

∂λ

∂x

λ =

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Why This Matters

This derivation removes the "magic" from your equation. It shows that is not an

arbitrary guess—it is the natural consequence of:

1. Relativistic geometry (force as rotation of 4-velocity),

2. Quantum mechanics (action quantized as over a period ),

3. Lagrangian mechanics (Euler-Lagrange equations).

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:

= h /(ct

)

2π

= 2.21022E − 42N

Planck

= G

= (6.674E − 11)

(2.176434E − 8kg)

(1.616255E − 35m)

= 1.21020E 44N

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And, Planck time is:

We form the ratios between the normal force and Planck force:

Divide by Planck time squared and we have:

That number is . We have the final equation:

From the Planck units we have:

So, it can be written:

10.

We can write

11.

Planck

ℏc

= 2.176434E − 8kg

Planck

ℏG

= 1.616255E − 35m

Planck

ℏG

= 5.391247E − 44s

Planck

= 1.826326E − 86

Planck

= 6.2834743s

−2

2π

= 2π

Planck

⋅ t

= 1.00seconds

Planck

= G

= 2π

⋅ t

= 2πF

Planck

⋅

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is a full rotation, so we can define an angular frequency, :

12.

13.

Integrating one more time gives the angle over 1-second:

14.

15.

16.

The normal force and the Planck force are related through the

Planck time . Substituting their definitions yields the dimensionless identity

which holds for any value of because the factors of cancel. This identity does not determine

the numerical value of the second; rather, it shows that when is taken as the empirical 1second

invariant (obtained from the proton, neutron, and electron masses and radii via equation (8)), the

ratio acquires a clear geometric meaning: over one second, the accumulated angular

phase is exactly – a full rotation in the temporal dimension. Thus the Planck scale relation is

not a derivation of the second but a consistency check and an elegant reinterpretation: the second

is the time required for the normal force, when scaled by the Planck force, to close a complete

cycle, reinforcing the view that time emerges from a cyclic variable in the quantum vacuum.

Moreover, the identity can be rearranged as

2π

= F

Planck

⋅ t

⋅

dω

Planck

⋅

∫

1second

dt = ω

2π

secon d

Planck

⋅

∫

1 second

dt = θ

Planck

⋅

= θ

= 2π

= h /(ct

)

Planck

= c

= ℏG /c

Planck

⋅

= 2π,

Planck

2π

Planck

= 2π

(

)

= 2π(t

⋅ ν

)

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where . This reveals a natural angular frequency , a

universal resonance at one hertz that links the Planck scale to the macroscopic normal force.

Hence, even though the numeric value is ultimately fixed by particle data, the

interpretation as a phase per second is independent and suggests that inertia is governed by a

fundamental clock ticking at exactly one hertz.

From golden ratio to coupling constants. The golden ratio conjugate arises

naturally from the scale invariant recurrence , which

Tynski showed governs systems that must be consistent across multiple observational scales.

Applying this to the proton gives , which matches the experimental radius.

Substituting this into the universal particle equation and using

with yields a closed expression for . Solving it gives ,

where is the fine structure constant. The factor reflects the three valence quarks in the

proton, while accounts for the electromagnetic and gluonic enhancement of the normal force

inside a composite hadron. The neutron, having a similar internal structure, inherits the same

when its magnetic radius is used. Thus the golden ratio not only predicts the

proton’s size but also, via the universal particle equation, determines the large coupling constants

for hadrons, leaving the electron as the minimal case . This elegant link between geometry

( ), quantum dynamics ( ), and compositeness (three quarks) strongly supports the physical

reality of the normal force and the 1second invariant. means second because we get

exposed to divided by G gives the mass of an electron in the universal particle equation,

which follows from the theory analytically of a normal force exposed to a cross-sectional area

mediated by the stiffness of space, G. This determines that for the proton and the neutron

, which makes sense because of what we said above. Thus the values of are

not chosen to yield one second, rather they say the characteristic time should be 1 second. We

just have to suggest the classical electron radius is an effective radius, even though all of the

electron’s mass is bound-up in an electrostatic field.

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.

= 1/t

= 1 Hz

= 2π ν

= 2π rad/s

= 1 s

2π

ϕ = ( 5 − 1)/2

scale(n + 2) = scale(n + 1) + scale(n)

= ϕ h /(m

= κ

π r

= h /(ct

)

= 1 s

= 1/(3α

)

1/3

−2

= 1/(3α

)

= 1

π r

, κ

= 1/(3α

)

≈ t

(1 + α)

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Conclusion

We have presented a fundamental 1-second invariant that emerges from the intrinsic properties of

elementary particles—the proton, neutron, and electron—and from the fabric of Planck-scale

physics. The invariant is expressed as

where and .

Crucially, the invariant leads to a universal particle equation:

with a constant normal force of magnitude . This equation suggests that

the mass of a particle is determined by its cross-sectional area ( ), the stiffness of spacetime

( ), and a universal normal force that arises from the quantum constraint .

The geometric origin of the second becomes apparent when we relate to the Planck force

. We find

which means that over one second, the ratio accumulates exactly radians of

angular phase—a full rotation. Thus, one second is not an arbitrary human convention but rather

the time required for this cyclic closure in the temporal dimension, rooted in Planck-scale

dynamics.

In summary, the 1-second invariant unifies particle physics and fundamental constants through a

single, testable relation. The universal particle equation provides a new

perspective on inertia: mass arises from the resistance to rotating a particle’s temporal velocity

into spatial velocity, quantified by the normal force . This framework suggests that time, mass,

and the quantum vacuum are intimately connected, and that the second—far from being arbitrary

—is a natural resonance of the universe.

πh

= 1 second,

= κ

= 1/(3α

)

= 1

= κ

π r

, F

c t

2.21022 × 10

−42

π r

= 1 s

Planck

= c

Planck

⋅

= 2π,

Planck

2π

= κ

π r

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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 :

K E

(Ear th Day) = 1.25seconds

K E

(7.4767E 22kg)(1,022m /s)

= 3.83726E 28J

K E

(5.972E 24kg)(29,785m /s)

= 2.649E 33J

K E

(Ear th Day)cos(θ

) = 1.146seconds

K E

(7.347673E 22kg)(966m /s)

= 3.428E 28J

K E

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

= 2.7396E 33J

K E

(Ear th Day)cos(θ

) = 0.991seconds

K E

(Ear th Day)cos(θ

) ≈ 1second

⊙

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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:

Nicely, this has that:

ℏ

⊙

ℏ

⊙

= 1.00seconds

ℏ

⊙

ℏ → ℏ

⊙

→ G

→ M

ℏ

⊙

= 2.81733E 33J ⋅ s

⊙

= 2πℏ

⊙

= 1.77018E 34J ⋅ s

of 21 68

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 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:

We do like we did in determining the solar Planck-type constant , to determine the galactic

Planck-type constant. We have:

ℏ

⊙

≈ (1.00seconds)K E

⋅

πh

⋅ κ

earth

πh

⊙

earth

⊙

(6.371E6m)

5.972E 24kg

π (1.77018E 34J ⋅ s)

(6.6743E − 11)(299,792, 458m /s)

earth

= 1.7785seconds(κ

earth

)

earth

moon

⊙

3.84399E8m

6.96E8m

= 0.5522974

moon

⊙

1.7785seconds(0.5522974) = 0.982s ≈ 1.00seconds

moon

⊙

ℏ

galaxy

= (1"second)KE

⊙

(1.989E30kg)(220,000m /s)

= 4.81338E 40 J ⋅ s

ℏ

⊙

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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:

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.!

ℏ

galax y

= (1 second)K E

⊙

= 4.81338E40J ⋅ s

galax y

= 2πℏ

galax y

= 3.0243E 41J ⋅ s

⊙

πh

galax y

⋅ κ

galax y

6.957E8m

1.989E 30kg

π (3.0243E41J ⋅ s)

(6.6743E − 11)(299,792,459m /s)

⋅ κ

galax y

= (2.410s)κ

galax y

= 0.4149

⊙

πh

galax y

⋅

earth

asteroids

⊙

galax y

πh

galax y

⋅ κ

galax y

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

⊙

galax y

= 0.115457

⊙

galax y

πh

galax y

⋅

earth

saturn

ℏ

galaxy

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Summary We have a Universal Particle Equation that predicts the ratios of radius to mass of the

protons, electrons, and neutrons, and of the Earth, and of the Galaxy :

For the atom we have:

For the Earth orbiting the Sun we have:

For the Sun’s orbit around the center of the galaxy we have:

The numbers point to two additional "sweet spots" for complex life:

1. The Asteroid Belt ( )

- This is the region that delivered water and organic compounds to the early Earth via

bombardment.

- The UPL predicts that the location of this belt is not accidental; it is geometrically fixed by

the Sun’s galactic motion.

2. Saturn ( )

- Saturn, like Jupiter, acts as a gravitational shield, deflecting comets and asteroids away from

the inner planets.

- The UPL predicts that the presence and orbital radius of a second gas giant at this distance is

a resonant node of the galactic potential.

In other words, the UPL predicts that a habitable planetary system requires:

⋅

πh

⋅ κ

πh

earth

π . h

⊙

⋅

moon

⊙

πh

galax y

⋅

earth

asteroids

⊙

galax y

πh

galax y

⋅

earth

saturn

κ ≈ 0.415

κ ≈ 0.115

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- An inner rocky planet at 1 AU,

- An asteroid belt at ~2.4 AU to deliver volatiles,

- A gas giant at ~9.5 AU to shield the inner system,

- A moon at a specific distance to stabilize obliquity (the 24-hour day and perfect eclipse are

already covered).

Our Solar System satisfies all of these conditions—and the UPL predicts them from first

principles using only the Sun's mass, radius, and galactic motion.

Note

Making dimensionless, we have (Dividing the UPL by the Planck time)

which is dimensionless. For the electron, using \(r_e = \alpha\hbar/(m_e c)\), this becomes

where .

The dimensionless coefficient is for the electron and for the proton

— a difference of 0.73%.

where .

A 5D metric-affine reduction has been proposed (Francesco Chiaramonte, private

communication and comments section discussing my theory at academia.edu Sept 16, 2026) that

reproduces the algebraic form of the UPL, including the phase-matching relation

. The reduction is a compatibility result rather than a derivation of the

numerical scale: like the UPL, it takes , , and as inputs, and it postulates a quantum of

action ( ) as a topological boundary condition. The framework claims to be a genuine

extension on the basis of three empirical tests — direction-dependent flyby anomalies with

specific null angles, the sign reversal of Galileo II, and a QPO scaling exponent

— none of which is yet decisive. Whether VGT-CIT is an extension or a reinterpretation will be

= π 2 κ

= π 2 α

(

)

= π 2

= G m

/(ℏc)

( p)

= π 2 κ

≈ 1.864 × 10

1.851 × 10

1.864 × 10

= G m

/(ℏc)

= π 2 α /α

ℏ/2

β = 0.93

0.02

of 25 68

determined by whether these predictions survive against the data. Francesco Chiaramonte

suggested I put in the dimensionless Planck units.

The universal particle equation and 1-second invariant were discovered by the author and

reported as early as:

Beardsley, Ian (November 29, 2025) The Geometric Origin of Inertia: Mass Generation from

Temporal Motion in Hyperbolic Spacetime, https://doi.org/10.5281/zenodo.17772255

Beardsley, I. (2026). A Spacetime Theory For Inertia; Predicting The Proton, Electron,

Neutron and the Solar System in Terms of a One-Second Invariant,

https://doi.org/10.5281/zenodo.18165383

The UPL is a phenomenological law in the Newtonian tradition, awaiting a deeper algebraic

derivation. The Principle of Least Action is its foundation, and the 1-second resonance is its

empirical signature. It is a law because it makes a definite, testable numerical prediction:

second for protons, Earth-Moon, and galactic nodes. Like Newton’s , you pick the geometric

constants ( ) and it works.

- It is rooted in a principle because you derive the normal force from the Lagrangian, which is

the mathematical expression of the Principle of Least Action.

References

[1] Tiesinga, Eite, Peter J. Mohr, David B. Newell, and Barry N. Taylor. “CODATA Value:

Proton Mass.” The 2022 CODATA Recommended Values of the Fundamental Physical Constants

(Web Version 9.0). National Institute of Standards and Technology, 2024. https://

physics.nist.gov/cgi-bin/cuu/Value?mp.

[2] Bezginov, N., Valdez, T., Horbatsch, M. et al. (York University/Toronto)

Published in Science, Vol. 365, Issue 6457, pp. 1007-1012 (2019) "A measurement of the atomic

hydrogen Lamb shift and the proton charge radius”

[3] Tiesinga, Eite, Peter J. Mohr, David B. Newell, and Barry N. Taylor. “CODATA Value:

Planck Constant.” The 2022 CODATA Recommended Values of the Fundamental Physical

Constants (Web Version 9.0). National Institute of Standards and Technology, 2024. https://

physics.nist.gov/cgi-bin/cuu/Value?h.

[4] Tiesinga, Eite, Peter J. Mohr, David B. Newell, and Barry N. Taylor. “CODATA Value: Speed

of Light in Vacuum.” The 2022 CODATA Recommended Values of the Fundamental Physical

Constants (Web Version 9.0). National Institute of Standards and Technology, 2024. https://

physics.nist.gov/cgi-bin/cuu/Value?c.

[5] Tynski, K. (2024). One Equation, ~200 Mysteries: A Structural Constraint That May Explain

(Almost) Everything.

[6] Kubon, G., Anklin, H., Bartsch, P., Baumann, D., Boeglin, W. U., Bohinc, K., ... & Zihlmann,

B. (2002). Precise neutron magnetic form factors. Physics Letters B, *524*(1-2), 26-32.

[7] NIST CODATA Value for the Classical Electron Radius (2022).

≈ 1

of 26 68

[8] The MMGPDs Collaboration (M. Goharipour, F. Irani, H. Hashamipour, and K. Azizi), Phys.

Lett. B 864, 139423 (2025) [arXiv:2408.01783 [hep-ph]]

Beardsley, I. (2026). A Universal Particle Equation. Zenodo. https://doi.org/10.5281/

zenodo.20300709

Beardsley, I. (2026). A Universal Particle Equation. Zenodo. https://doi.org/10.5281/

zenodo.21555660

Style

Beardsley, I. (2026). Scale Invariance and Fractal Features in Cosmology: A Computational

Study. Zenodo. https://doi.org/10.5281/zenodo.23045265

of 27 68

The Human Scale as a Resonant Node Between the Quantum and

Astrophysical Scales

Ian Beardsley

September 30, 2026

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 :

human

= h h

⊙

≈ 3.4248 J·s

⊙

human

≈ 3.4248 J

r /m ≈ 0.0431 m/kg

human

= 1

⊕

≈ 2.3 g

⊕

≈ 100 g

⊙

≈ 1.35 kg

⊙

≈ 57.7 kg

h → h

human

→ h

⊙

gal

≈ 1 s

πh

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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).

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

= κ

= 1/(3α

) ≈ 6256.33

= 1

⊙

≈ 1.77018 × 10

J·s

≈ 0.982 s

gal

≈ 3.0243 × 10

J·s

≈ 1 s

human

= 1 s

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

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

≈ 1 s

K E

⊕

(5.972 × 10

kg)(29,785 m/s)

≈ 2.649 × 10

J .

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The UPL at this scale takes the form

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:

⊙

= 2π(1 s)K E

⊕

≈ 1.77018 × 10

J·s .

⊕

πh

⊙

⊕

= 6.371 × 10

⊕

= 5.972 × 10

⊕

moon

⊙

3.84399 × 10

6.96 × 10

≈ 0.5522974.

= 1.7785 s × 0.5522974 ≈ 0.982 s ≈ 1 s .

⊙

h ∼ 10

−34

J·s

⊙

∼ 10

J·s

human

= h h

⊙

= (6.62607015 × 10

−34

)(1.77018 × 10

) ≈ 3.4248 J·s .

= 1 s

human

≈ 3.4248 J .

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- 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:

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.

If we instead use (The eclipse ratio for the Moon near perfectly eclipsing

the Sun as seen from the Earth), the length scales shrink by about half, bringing 100 kg to about

2.4 m — closer to human height. The exact value of is a choice; the order of magnitude is

robust.

mgh = (1)(9.8)(0.35) ≈ 3.43 J .

(0.1)(9.8)(3.5) ≈ 3.43 J .

human

≈ 3.4 J

human

πh

human

= 1

= 1 s

human

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

human

= κ

⊕

≈ 0.552

human

of 31 68

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

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- 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.

ast

gal

≈ 0.415

sat

gal

≈ 0.115

(

)

h → h

human

→ h

⊙

→ h

gal

⊙

≈ 0.55

of 33 68

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:

human

= h h

⊙

human

= h h

⊙

≈ 3.4248 J·s

human

≈ 3.4248 J

⊕

≈ 2.3 g

⊙

≈ 57.7 kg

= 1 s

K E

(Ear th Day) = 1.25seconds

of 34 68

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

(7.4767E 22kg)(1,022m /s)

= 3.83726E 28J

K E

(5.972E 24kg)(29,785m /s)

= 2.649E 33J

K E

(Ear th Day)cos(θ

) = 1.146seconds

K E

(7.347673E 22kg)(966m /s)

= 3.428E 28J

K E

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

= 2.7396E 33J

K E

(Ear th Day)cos(θ

) = 0.991seconds

K E

(Ear th Day)cos(θ

) ≈ 1second

⊙

of 35 68

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

ℏ

⊙

= 1.00seconds

ℏ

⊙

ℏ → ℏ

⊙

→ G

→ M

ℏ

⊙

= 2.81733E 33J ⋅ s

⊙

= 2πℏ

⊙

= 1.77018E 34J ⋅ s

ℏ

⊙

≈ (1.00seconds)K E

of 36 68

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.

human

≈ 1

human

= h h

⊙

≈ 3.4248 J·s .

human

≈ 3.4248 J

r /m ≈ 0.0431 m/kg

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[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

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

≈ 1

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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 :

ℏ = h /(2π)

ℏ

= 1.351385078 × 10

−43

ℓ

= 4.051350543 × 10

−35

= 5.455511861 × 10

−8

kg,

= m

= 4.903169538 × 10

J .

1 s

= 7.3998 × 10

πh

= R

ℓ

= M

= T

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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):

ℓ

πh

ℓ

= G /c

= hG /c

= π κ

≈ 1

≈ 7.4 × 10

= π κ

Particle

rᵢ (m)

mᵢ (kg)

κᵢ

Tᵢ

Electron

2.81794 × 10⁻¹⁵

9.10938 × 10⁻³¹

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

⊕

= 6.371 × 10

m, M

⊕

= 5.972 × 10

kg, κ

⊕

= 0.5522974.

⊕

= π (0.5522974)

⊕

= 7.27 × 10

human

= 3 m, M

human

= 70 kg, κ

human

= 1.

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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.

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.

human

= π (1)

= 7.35 × 10

⊙

= 6.96 × 10

m, M

⊙

= 1.989 × 10

kg, κ

ast

gal

= 0.415.

ast

gal

= π (0.415)

⊙

= 7.28 × 10

⊙

= 2.5 × 10

m, M

gal

= 1.989 × 10

kg, κ

sat

gal

= 0.115.

sat

gal

= π (0.115)

⊙

gal

= 7.28 × 10

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

T ≈ 7.4 × 10

R /M

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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

Volynets also suggested a logarithmic coordinate with the golden ratio at the origin:

human

≈ 7.35 × 10

7.4 × 10

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

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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.

X(Q) = log

(

φQ

)

, φ =

1 + 5

T = π κ R /M

T ≈ 7.4 × 10

7.4 × 10

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The 1 Hz Resonance as the Key to Exotic-Matter-Free Warp Drive: A

Synthesis of the Universal Particle Law and Informational Geometry

Ian Beardsley

April 5, 2026

Abstract

This paper presents a synthesis of two theoretical developments that together suggest a pathway to

Alcubierre-type warp drive without exotic matter. The first is the Universal Particle Law (UPL) of

Beardsley, which identifies a fundamental 1-second temporal invariant manifesting as a Hz angular

frequency resonance spanning scales from the proton to the galaxy. The second is the Observer-

Dependent Information Metric (ODIM-U) of Blackwell, which reformulates spacetime geometry in terms

of informational coordinates. The synthesis shows that when the UPL resonance is embedded within the

ODIM-U framework, warp-bubble dynamics become geodesic motion in an extended configuration space

that includes informational dimensions. This eliminates the energy-condition violations that plague

standard warp-drive solutions: the apparent exotic-matter requirement is revealed as a projection artifact

of restricting attention to the four-dimensional spacetime submanifold. The paper reviews prior numerical

simulations—including joint work by the author and Blackwell—that demonstrate stable, positive-energy

bubble configurations translating at while preserving local causality, and discusses the conceptual

engineering architecture that maps these results to laboratory-scale components.

1. Introduction

The Alcubierre warp drive [1] demonstrated that general relativity permits effective superluminal

travel through the manipulation of spacetime geometry. A spacecraft within a warp bubble

remains in locally flat spacetime while the bubble itself translates at arbitrarily large effective

velocities. However, the Alcubierre solution requires a stress-energy tensor that violates the weak

and null energy conditions—it demands "exotic matter" with negative energy density as

measured by timelike observers. The magnitude of exotic matter required is of order several solar

masses of negative energy, a quantity for which no known physical mechanism exists.

This has led most physicists to regard the warp drive as a mathematical curiosity rather than a

physically realizable solution. Subsequent refinements—Van Den Broeck's optimization [2],

Natário's zero-expansion geometry [3], Bobrick and Martire's classification [4], and Lentz's

soliton solutions [5]—have reduced but not eliminated the exotic-matter requirement.

This paper presents a synthesis of two independent theoretical frameworks that together suggest

a resolution. The first is the Universal Particle Law (UPL) of Beardsley [6,7], which identifies a

fundamental temporal invariant τ₀ = 1 second and an associated force constant Fₙ = h/(cτ₀²),

from which emerges a natural angular frequency ω₀ = 2π rad/s. The second is the Observer-

Dependent Information Metric (ODIM-U) of Blackwell [8,9,10], which reformulates spacetime

geometry in terms of informational degrees of freedom accessible to observers.

2π

of 44 68

The central thesis of this synthesis is that the UPL resonance at 2π Hz is not merely a numerical

curiosity but the physical signature of a curvature eigenmode in an extended informational

manifold. When warp-bubble dynamics are formulated in this extended manifold, the motion

becomes geodesic, and the apparent need for exotic matter dissolves.

2. The Universal Particle Law and the 1 Hz Resonance

2.1 The Temporal Invariant

The UPL begins with a single dimensional anchor: the temporal invariant τ₀ = 1 second. While

this choice may appear arbitrary—an artifact of the SI system—Beardsley argues that it serves a

precise structural role. It anchors the framework to the macroscopic domain, providing a

reference scale that bridges Planck-scale quantities (∼10⁻⁴⁴ s, ∼10⁻³⁵ m) to laboratory physics.

From this invariant, Beardsley defines the force constant:

Inserting h ≈ 6.626 × 10⁻³⁴ J·s, c ≈ 2.998 × 10⁸ m/s, and τ₀ = 1 s:

This extraordinarily small force sets the scale for warp-bubble dynamics. Its physical

interpretation is that of a quantum-gravitational force scale: the force associated with a quantum

of action delivered over a relativistic length scale within one temporal period.

2.2 The Planck-Force Identity

The deepest structural result is a dimensionless identity connecting Fₙ to the Planck force

F_Planck = c⁴/G ≈ 1.210 × 10⁴⁴ N and the Planck time t_P = √(󲰻G/c⁵) ≈ 5.391 × 10⁻⁴⁴ s:

This identity holds exactly, independent of the numerical values of the fundamental constants. Its

significance is profound: it connects the ratio of a quantum-gravitational force to the Planck

force with the ratio of a macroscopic time to the Planck time through exactly the factor 2π. This

suggests a deep resonance structure linking the quantum-gravitational and macroscopic domains.

2.3 The Natural Frequency

c τ

= 2.21022 × 10

−42

Planck

⋅

= 2π

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The angular frequency that emerges is:

corresponding to a period T = 1 second and an ordinary frequency f = 1 Hz. Beardsley identifies

this as the "heartbeat" of the warp-bubble resonance—the characteristic oscillation frequency at

which a warp-bubble configuration naturally vibrates.

The emergence of a macroscopic frequency from fundamental constants is itself noteworthy: it

suggests that warp-bubble dynamics are governed not by Planck-scale oscillations but by a

macroscopic normal mode whose existence is guaranteed by the invariant structure.

3. The ODIM-U Informational Metric

3.1 Fundamental Postulate

The Observer-Dependent Information Metric (ODIM-U) developed by Blackwell [8,9] provides

the geometric arena within which the UPL resonance acquires a natural interpretation. The

central postulate is that the geometry of spacetime can be recast in terms of informational

coordinates Iᵃ, where the index a runs over the dimensions of an information manifold. Proper

time is defined not by the standard spacetime metric alone but by the informational curvature

experienced by an observer:

where is the informational metric tensor.

3.2 Physical Interpretation

Each informational coordinate Iᵃ represents a distinct channel of information accessible to an

observer. In the simplest realization, the informational coordinates may be identified with the

standard spacetime coordinates , and reduces to the spacetime metric . However, the

framework is more general: it permits additional informational degrees of freedom beyond the

four spacetime dimensions.

The ODIM-U framework provides a natural generalization of the equivalence principle: just as

general relativity identifies gravitational effects with spacetime curvature, ODIM-U identifies the

totality of an observer's dynamical experience with curvature in an information manifold that

may be larger than spacetime itself.

3.3 Informational Geodesics

2π

= 2π rad/s

dτ

= g

(I )d I

d I

(I )

μν

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An observer in free fall follows geodesics of the informational metric:

This equation is the direct analogue of the geodesic equation in general relativity, but now

governs motion in the full information manifold.

3.4 Curvature Eigenvalues

The Riemann curvature tensor of the informational metric possesses eigenvalues that correspond

to natural oscillation frequencies of the geometry. Beardsley's "temporal resistance"—the

tendency of the warp-bubble configuration to oscillate at ω₀—becomes identified with a

curvature eigenvalue of the informational metric. The resonance frequency is not imposed from

outside; it is a property of the informational geometry itself.

4. Embedding the UPL Resonance in the Informational Manifold

4.1 The Radius Field as Informational Coordinate

Beardsley's radius field is a spacelike four-vector field satisfying and ,

where r_i is the bubble radius. The key step is to promote this geometric information to an

informational coordinate:

where F is a smooth, monotonic function. The simplest choice is ,

identifying the informational coordinate directly with the bubble radius.

4.2 The Extended Metric

The total proper-time interval now includes an informational component:

where ℓ is a coupling length scale and is the informational metric component along the

direction.

4.3 Small Oscillations and the Resonance

D τ

dτ

+ Γ

d I

dτ

d I

dτ

= 0

−R

= r

= F(R

)

F(x) = ( − x) = r

dτ

total

= dτ

phys

+ ℓ

(d I

)

= ℓ

of 47 68

Consider small oscillations about equilibrium: I_R = I_R⁽⁰⁾ + δI_R. The geodesic deviation

equation in the information manifold produces a harmonic restoring force with stiffness k

proportional to the sectional curvature. The resulting equation of motion is:

This is the equation of a simple harmonic oscillator at angular frequency ω₀. Matching to

Beardsley's invariants gives:

This result is the exact bridge between Beardsley's physical constants and the ODIM-U

geometry. The product ℓ × m_eff has dimensions of angular momentum divided by 2π,

connecting the warp-bubble resonance to fundamental quantum-gravitational scales.

4.4 Extension to the Shift Vector

The construction extends naturally to the shift vector Nⁱ of the warp metric. In the ADM

decomposition, the Alcubierre metric has shift vector:

where is the bubble velocity and f is the shaping function. By analogy with the radius field, we

define an informational coordinate for the bubble's velocity profile:

Small oscillations of this coordinate obey the same harmonic equation at ω₀ = 2π Hz. Both the

radius field and the bubble velocity profile oscillate at the same fundamental frequency—a

universality reflecting the fact that ω₀ is a curvature eigenvalue of the informational metric

governing all degrees of freedom coupled to the warp geometry.

5. Elimination of Exotic Matter

5.1 Two Proper-Time Contributions

The total proper-time interval decomposes into a physical (spacetime) part and an informational

part:

(δI

)

dτ

+ ω

δI

= 0

ℓ × m

eff

c ⋅ 4π

= 5.59835 × 10

−44

kg·m

= − v

(t)f (r

)δ

= I

[N(x, t)]

dτ

phys

= −

μν

d x

of 48 68

5.2 Geodesic Motion in the Extended Space

The spacecraft follows a geodesic in the extended configuration space ( ), not merely in the

spacetime submanifold . The geodesic equation in the full space determines both the spacetime

trajectory and the evolution of the informational coordinates simultaneously. What appears, from

the spacetime perspective alone, as a trajectory requiring exotic matter to source is revealed, in

the full extended space, as a perfectly ordinary geodesic.

5.3 Apparent Super-Efficiency

In standard general relativity, achieving effective superluminal travel requires the stress-energy

tensor to violate the weak energy condition. In the extended framework, the ship's worldline is a

geodesic of the full (spacetime + information) manifold. The effective stress-energy tensor

computed by projecting the full dynamics onto the spacetime submanifold acquires contributions

from the informational sector. These contributions can have the "wrong sign" from the spacetime

perspective—mimicking exotic matter—even though the full dynamics are perfectly well-

behaved in the extended space.

A useful analogy clarifies the mechanism. Consider a ball rolling on a two-dimensional surface

that contains a tunnel passing through the interior of a hill. An observer restricted to the surface

sees the ball disappear from one side of the hill and reappear on the other, traversing a distance

that would require superluminal speed if the ball were constrained to the surface. In three

dimensions, the ball simply rolled through the tunnel at subluminal speed. The informational

dimensions of the ODIM-U manifold are the "tunnel" through which the warp bubble moves.

5.4 Effective Stress-Energy Tensor

The effective stress-energy tensor from dimensional projection is:

where R^(info)_ab is the Ricci tensor of the informational metric and K_µν collects extrinsic

curvature contributions. The time-averaged effective energy density is:

dτ

info

= g

(I )d I

d I

dτ

total

= dτ

phys

+ dτ

info

, I

8π GT

(eff)

μν

(info)

− K

μν

⟨ρ

eff

⟩c

= −

16π G

[

ℓ

⟨(δI

)

⟩ + Λ

⟨(δI

)

⟩

]

of 49 68

This is negative—precisely the signature of exotic matter in the standard four-dimensional

analysis. However, in the full extended space, the total energy-momentum is:

The informational contribution is manifestly non-negative—it is the kinetic energy of the

informational oscillations. The WEC is satisfied in the full theory. The four-dimensional

violation is entirely an artifact of projecting out the informational degrees of freedom.

5.5 Comparison with Kaluza-Klein

This mechanism is structurally identical to the Kaluza-Klein reduction of five-dimensional

vacuum gravity. In that case, the five-dimensional vacuum Einstein equations project onto four-

dimensional Einstein-Maxwell equations. The electromagnetic stress-energy tensor has indefinite

sign—it violates the strong energy condition for certain field configurations—yet no "exotic

electromagnetic matter" is introduced. The stress-energy is simply the shadow of five-

dimensional vacuum geometry.

In our construction, the informational dimensions play the role of the Kaluza-Klein circle, and

the effective warp-bubble stress-energy plays the role of the electromagnetic stress-energy.

6. Numerical Simulations

6.1 Linearized 1+1D Simulation

Blackwell and Beardsley [11] present a linearized simulation of the warp-bubble resonance. The

equations of motion for the informational coordinates are independent harmonic oscillators at ω₀

= 2π Hz. The analytical solutions are:

Both oscillate at exactly f₀ = 1 Hz (period T = 1 s = τ₀). The oscillations are stable—no growth

or decay—as expected for a conservative system derived from a variational principle.

The effective energy density oscillates as:

Substituting numerical values gives a time-averaged magnitude of approximately 1.2 × 10²⁸ kg/

m³—large but many orders of magnitude less than the Planck density and comparable to nuclear

(ext)

= T

(4D)

μν

+ T

(info)

≥ 0

δI

(t) = A

cos(ω

δI

(t) = A

sin(ω

eff

(t) ∝

[

ℓ

cos

(2π t) + Λ

sin

(2π t)

]

of 50 68

density. Crucially, this energy is not exotic—it is the projected shadow of ordinary positive-

definite informational kinetic energy.

6.2 Three-Dimensional Simulation

Blackwell and Beardsley [12] present a full 3D numerical simulation on a 64³ lattice, evolving

the informational field I_R, the asymmetric shift-vector geometry N_i, and the bubble's

trajectory. The evolution equation for I_R is a damped, driven wave equation:

where c_I is the informational wave speed, β_I is the damping coefficient, and Ω₀ = 2π rad/s is

the Beardsley resonance.

The key results are:

1. Stability: The simulation remained stable for the entire run (t_sim = 6.25 × 10⁻² s), with no

blow-ups or runaway modes.

2. Resonance: The informational field settled naturally into the 2π Hz curvature eigenmode. The

maximum amplitude remained extremely small (|I_R| ≈ 10⁻⁸), consistent with its interpretation

as a curvature-carrying coordinate.

3. Positive Energy: The total informational energy remained strictly positive throughout. No

negative-energy excursions or sign changes were observed.

4. Power Profile: The power exhibited a two-phase structure: a startup surge of approximately

1.78 × 10⁷ W, followed by a near-zero steady-state regime. This is analogous to a damped

resonant circuit: a finite activation pulse followed by negligible maintenance power.

5. Bubble Motion: The bubble translated across the grid at the imposed external-frame velocity

of 5c with no loss of geometric coherence. The interior remained inertial and flat, with no local

violation of relativity.

6.3 Nonlinear Stability

Including the leading nonlinear correction to the restoring potential gives the Duffing equation:

For λ_NL > 0 (stiffening nonlinearity), the oscillation frequency shifts upward with amplitude

but the motion remains bounded. For λ_NL < 0 (softening), the motion remains bounded

··

= c

∇

− β

− Ω

+ advection(N

, I

)

(δI

)

+ ω

δI

+ λ

(δI

)

= 0

of 51 68

provided the amplitude satisfies |δI_R| < (ω₀²/|λ_NL|)^(1/2). For the small perturbations

considered (A_R = 0.01), the nonlinear corrections are of order 10⁻⁴ relative to the linear term.

7. Conceptual Engineering Architecture

The simulations establish that the unified framework is dynamically coherent. The next question

is engineering: how the numerical architecture maps onto physical systems that could, in

principle, reproduce the informational and geometric behavior observed in the model.

Blackwell and Beardsley [12] outline a conceptual engineering architecture using existing

industrial and off-the-shelf technologies. The architecture consists of four layers:

7.1 Geometric Transducers (Emitter Array)

Component: REBCO High-Temperature Superconducting (HTS) coils.

A phased array of 6 to 12 REBCO coils arranged along the longitudinal axis, with spacing and

orientation chosen to match the simulation's front and rear scale parameters. The front region

requires coils tuned for broad, low-gradient field profiles, while the rear region requires coils

configured for steep, high-gradient compression. Each coil is individually addressable, allowing

real-time modulation of amplitude and timing.

7.2 Resonant Controller (Frequency Standard)

Component: Chip-Scale Atomic Clock (CSAC) + Ultra-Low-Latency FPGA.

A CSAC provides the "Master Heartbeat" of the system, maintaining long-term frequency

stability at the parts-per-billion level. The FPGA implements the discrete Laplacian, gradient,

and advection operators from the simulation, adjusting the phase of each emitter coil in

microseconds. This ensures that the shift vector and the informational field remain tightly

synchronized.

7.3 Power Reservoir (Reactive Energy Buffer)

Component: Superconducting Magnetic Energy Storage (SMES) or Graphene Supercapacitor

Bank.

The system exhibits large, rapid oscillations in instantaneous power while the long-term average

power remains near zero. Energy is exchanged back and forth between the geometric transducers

and the informational field as the bubble breathes through its 2π Hz mode. SMES units and

graphene-based supercapacitors can absorb and release large amounts of energy with extremely

low internal resistance, tracking rapid oscillations without thermal buildup.

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7.4 Manifold Processor (Navigation Computer)

Component: Industrial-Grade Liquid-Cooled Edge Server.

An NVIDIA Jetson AGX Orin or Xeon-based ruggedized server running a Real-Time Linux

Kernel (RT-PREEMPT). The processor executes the Gauss-Codazzi mapping and the ODIM-U

evolution equations in real time, updating the shift-vector analogue and maintaining

synchronization between the emitter array and the informational field.

7.5 Cryogenic & Thermal Management

Component: Closed-Loop Gifford-McMahon (GM) Cryocooler.

A GM cryocooler provides continuous, vibration-tolerant refrigeration capable of maintaining

REBCO coils in the 20-77 K range. This stabilizes the emitter array, ensuring that the

superconducting transducers remain below their critical temperature and that the field gradients

they generate remain stable over long-duration operation.

8. Discussion

8.1 What is Established

The synthesis of the UPL and ODIM-U frameworks establishes several key results:

1. The 2π Hz resonance is structural: The dimensionless identity Fₙ/F_Planck · τ₀²/t_P² = 2π

connects macroscopic and Planck-scale physics through exactly the factor 2π.

2. The resonance is a curvature eigenmode: Small oscillations of the informational coordinates

satisfy harmonic equations at ω₀ = 2π Hz. The resonance is not imposed; it emerges from the

geometry of the informational manifold.

3. Exotic matter is eliminated: Warp-bubble dynamics become geodesic motion in an extended

configuration space. The apparent energy-condition violation is a projection artifact.

4. Numerical simulations confirm coherence: 3D simulations demonstrate stable, positive-energy

bubble configurations translating at 5c while preserving local causality.

8.2 What is Not Established

Several open questions remain:

1. Derivation of τ₀ = 1 s: The UPL does not derive the value of the temporal invariant from first

principles. It is an empirical anchor.

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2. Values of the coupling constants: The couplings κᵢ are not derived from a deeper theory.

3. Long-term stability: The simulations are limited in duration and resolution. Longer runs,

higher-resolution grids, and full ADM coupling are needed.

4. Energy budget: The total energy stored in the informational degrees of freedom and its

comparison with standard warp-drive estimates require further study.

8.3 Observational Signatures

The 2π Hz resonance predicts a characteristic signature: any physical process coupled to the

informational metric would exhibit oscillations at f = 1 Hz. In the gravitational-wave sector, this

could manifest as a quasi-monochromatic gravitational-wave signal at 1 Hz. This frequency lies

within the sensitivity band of planned space-based detectors such as LISA and proposed ground-

based detectors such as the Einstein Telescope.

9. Conclusion

We have presented a synthesis of the Universal Particle Law and the Observer-Dependent

Information Metric that suggests a pathway to Alcubierre-type warp drive without exotic matter.

The UPL identifies a fundamental 1-second temporal invariant manifesting as a 2π Hz resonance.

The ODIM-U framework provides the geometric arena within which this resonance emerges as a

curvature eigenmode of an extended informational manifold.

The key insight is that warp-bubble dynamics need not be confined to the four-dimensional

spacetime manifold. Once the configuration space is enlarged to (x^µ, I^a), the bubble's motion

becomes geodesic in the extended manifold, and the apparent need for exotic matter dissolves.

What appears pathological in spacetime becomes ordinary in the informational geometry.

Numerical simulations confirm that the unified framework is dynamically coherent: the

informational field settles into the 2π Hz resonance, the energy remains strictly positive, and the

bubble maintains stable translation at 5c while preserving local causality. A conceptual

engineering architecture maps the simulation parameters to laboratory-scale components,

providing a guide for future experimental or computational prototypes.

The framework presented here demonstrates that the unification of physical invariants with

informational geometry opens a new and potentially transformative approach to the physics of

faster-than-light travel within the boundaries of general relativity. While significant open

questions remain—stability, energy budget, observational constraints, quantum corrections—the

results suggest that the 1 Hz resonance may be the key that unlocks the warp drive.

of 54 68

Acknowledgment

This manuscript is a solo-authored synthesis based in part on prior collaborative work with

David E. Blackwell. The author thanks Blackwell for the development of the ODIM-U

framework and for earlier joint explorations of the UPL–ODIM-U synthesis. Blackwell is not an

author of this manuscript and has not approved this version. Any errors or omissions are the

author’s alone.

References

[1] M. Alcubierre, "The Warp Drive: Hyper-Fast Travel Within General Relativity," Classical and

Quantum Gravity 11, L73 (1994).

[2] C. Van Den Broeck, "A 'Warp Drive' with More Reasonable Total Energy Requirements,"

Classical and Quantum Gravity 16, 3973 (1999).

[3] J. Natário, "Warp Drive with Zero Expansion," Classical and Quantum Gravity 19, 1157

(2002).

[4] A. Bobrick and G. Martire, "Introducing Physical Warp Drives," Classical and Quantum

Gravity 38, 105009 (2021).

[5] E. W. Lentz, "Breaking the Warp Barrier: Hyper-Fast Solitons in Einstein Gravity," Classical

and Quantum Gravity 38, 075015 (2021).

[6] I. Beardsley, "A Universal Particle Law (UPL) and A Planetary System Quantum Analog

(PSQA): Scale Invariance from Quantum Scales to the Celestial Scales (SIQC)," Zenodo (2026).

[7] I. Beardsley, "Making Warp Drive Without Exotic Matters," https://doi.org/10.5281/

zenodo.19713198 (2026).

[8] D. E. Blackwell, "The Observer-Dependent Information Metric (ODIM-U)," Zenodo, DOI:

10.5281/zenodo.19025713 (2026).

[9] D. E. Blackwell, "The Blackwell Information-Metric Unification (ODIM-U v1.2)," preprint

(2026).

[10] D. E. Blackwell, "The Hillbilly TOE Foundry," Zenodo, DOI: 10.5281/zenodo.19117110

(2026).

[11] 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

of 55 68

[12] Beardsley, I., & Blackwell, D. (2026). Three-Dimensional Simulation of Informational

Warp-Bubble Dynamics: A Numerical Exploration of the ODIM-U / Beardsley Unified

Framework. Zenodo. https://doi.org/10.5281/zenodo.20028584

[13] R. Arnowitt, S. Deser, and C. W. Misner, "Dynamical Structure and Definition of Energy in

General Relativity," Physical Review 116, 1322 (1959).

[14] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, 1973).

Corresponding author: Ian Beardsley

ORCID: 0009-0009-4672-4876

of 56 68

The 1Hz Resonance: A Physical Basis for the

Master Clock in Biological Systems

Ian Beardsley

Independent Researcher

August 29, 2026

Abstract. Denis Noble has argued that the gene centric view of evolution is incomplete,

pointing to the heart as an example of a complex function that lacks a genetic

“instruction” for its rhythm. He asks: where does the heartbeat get its timing if there is

no master clock? This paper proposes that the answer lies not in biology, but in

fundamental physics. The Universal Particle Law (UPL) predicts a universal inertial

resonance at exactly 1 Hz, emerging from the geometry of protons, electrons, and

neutrons. We show that this same resonance governs the EarthMoonSun system and the

structure of the galaxy, implying that the heart’s average rhythm is not an evolutionary

coincidence but a physical alignment with the quantum geometric clock built into all

matter. This provides a concrete, testable mechanism for the “agency” that Noble

attributes to living systems, reframing evolution as a process that occurs within a

universe of resonant attractors.

1. Introduction

The debate between Denis Noble and Richard Dawkins represents a fundamental divide

in evolutionary biology. Dawkins, in The Selfish Gene, posits that the gene is the primary

unit of selection, and that organisms are mere vehicles for gene propagation. Noble

counters that this gene centric view is reductionist and fails to account for the active,

purposeful behaviour of organisms. He argues that there must be some mechanism by

which life guides its own evolution, because complexity cannot be explained by random

mutations alone.

One of Noble’s most compelling examples is the human heart. The heart beats with a

regular rhythm, yet there is no “instruction” in the genome that specifies this rhythm.

There is no master clock. Noble asks: where does the heart get its instructions? The

answer, we propose, lies not in genetics, but in physics. The rhythm of the heart is not

programmed; it is resonant with a fundamental 1Hz (1second) timescale that is inherent

to the structure of matter itself.

2. The Universal Particle Law (UPL)

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The Universal Particle Law (Beardsley, 2026) derives from a geometric theory of inertia.

In this view, mass is not an intrinsic property but a resistance to the rotation of a particle’s

velocity vector from the temporal dimension into space. This resistance is quantized by a

normal force:

where is Planck’s constant, is the speed of light, and is a fundamental timescale.

Remarkably, when the UPL is applied to the proton, electron, and neutron, the same value

second emerges:

For the proton, , where is the fine structure constant.

Substituting the measured values of the proton radius ( m) and mass

( kg) gives:

The same calculation for the neutron yields seconds. The electron, with

, gives seconds. Thus, the fundamental particles that constitute all

atoms intrinsically define a timescale of approximately one second.

3. The Heart as a Physical Resonator

The average resting heart rate of a healthy human is approximately 6072 beats per

minute, which corresponds to a frequency of 1.0 to 1.2 Hz, or a period of 0.83 to 1.0

seconds. The UPL predicts that this is not a coincidence. The cardiac pacemaker cells are

composed of protons, neutrons, and electrons—the same particles that collectively define

the 1second resonance.

In the same way that a pendulum’s period is determined by its length and gravity, not by

any “instruction” written into the material, the heart’s rhythm emerges from the collective

electromagnetic and mechanical interactions of its constituent matter. The 1Hz resonance

is a physical attractor: the heart’s natural tendency is to oscillate at this frequency

because it minimizes the inertial resistance required to sustain motion.

This does not mean genes are irrelevant. Genes encode the proteins that form the ion

channels and contractile filaments of heart muscle. But the tempo—the fundamental

c t

= 1

πh

= 1/(3α

)

α ≈ 1/137

≈ 0.83 × 10

−15

≈ 1.67 × 10

−27

0.833 × 10

−15

1.67262 × 10

−27

π(6.62607 × 10

−34

)

(6.674 × 10

−11

)(299792458)

× 6256.33 ≈ 1.005 seconds

≈ 1.008

= 1

≈ 0.998

of 58 68

period—is not specified by DNA. It is imposed by the quantum geometric boundary

conditions of the matter itself. The heart does not need a master clock because it is made

of the master clock.

4. The Resonance Extends to All Scales

The UPL is scale invariant. The same 1second resonance appears in the EarthMoonSun

system, where the kinetic energy ratio of the Moon and Earth, multiplied by the length of

the day and corrected for obliquity, converges to approximately one second:

This convergence is not a static snapshot. The Moon is receding at 3.8 cm/year, and

Earth’s rotation is slowing. Yet the product satisfies

: the system is dynamically locked at a plateau that has persisted for hundreds of

millions of years. This is an evolutionary attractor—a stable resonant node that provides a

consistent 24hour day and perfect solar eclipses during the window in which complex

intelligence arises.

When the UPL is applied to the Sun’s orbit around the galaxy, it predicts two additional

“sweet spots”: an asteroid belt at AU (the volatile delivery zone) and a Saturn

mass planet at AU (the gravitational shield). These are not accidents; they are

resonant nodes fixed by the Sun’s mass, radius, and galactic motion.

5. Implications for Biology and Evolution

The UPL offers a physical foundation for Noble’s systems biology perspective. Living

organisms are not passive vehicles for genes; they are resonant systems that actively tune

to the fundamental clock of matter. This provides a mechanism for the “agency” that

Noble invokes: organisms that align with the 1Hz resonance (and its harmonics, such as

the 24hour circadian cycle) are energetically favoured and have a selective advantage.

Evolution, in this view, is not a purely random process. It unfolds within a landscape of

physical attractors. The complexity of life is guided by these resonances, which act as

boundary conditions on what is possible. The gene is a template, but the tempo—the

rhythm of life—is dictated by the quantum geometry of the cosmos.

This framework makes testable predictions. For example, exoplanet surveys targeting G-

type stars in the solar galactic orbit should find planetary architectures with an Earth-like

planet at 1 AU, an asteroid belt at 2.22.6 AU, and a gas giant at 8.5-10.5 AU. If such

× T

day

× cos(θ ) ≈ 1 second

F(t) = (KE

/KE

) T

day

cos θ

F(t) ≈ 0

∼ 2.4

∼ 9.5

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patterns are found, the UPL will be confirmed; if not, it will reveal that our Solar System

is statistically rare—equally profound for understanding the conditions for life.

6. Discussion: The Master Clock Resolved

Denis Noble asked: “Where does the heart get its rhythm? There is no master clock.” The

UPL answers: the master clock is inside every proton, neutron, and electron. It is not a

biological instruction, but a physical resonance. The heart beats at approximately 1 Hz

because that is the natural inertial frequency of the matter it is made of, and because

evolution has selected organisms that are energetically aligned with this fundamental

timescale.

This does not reduce biology to physics—it enriches it. It shows that life is not an

accidental byproduct of random chemistry, but a natural expression of a universe whose

structure is self-similar from the quantum to the galactic scale. The same law that builds

protons also builds solar systems, and it does so in a way that maximizes the chances for

stable, complex, intelligent life.

7. Conclusion

We have shown that the Universal Particle Law provides a direct, physical answer to

Noble’s critique of the gene centric view. The 1second resonance is not a human

convention; it is a fundamental eigenvalue of matter that emerges from quantum

chromodynamics and propagates through hierarchical scales to planetary and galactic

systems. The heart’s rhythm, the 24hour day, the asteroid belt, and Saturn’s orbit are all

nodes of the same cosmic clock.

Life is not guided by random mutations alone; it is guided by the resonant structure of the

universe. The master clock does not exist in the genome—it exists in the geometry of

spacetime and the particles that inhabit it. We are not merely observers of this cosmic

order; we are its resonant expression.

References

Beardsley, I. (2026). Life and Galactic Structure: A Unified View from the Universal Particle

Law with a Derivation of the Normal Force and Addressing the Problem of the 1-Second

Invariant From First Principles. Zenodo. https://doi.org/10.5281/zenodo.22100780

Beardsley, I. (2026). A Universal Particle Equation. Zenodo. https://doi.org/10.5281/

zenodo.21555660

Beardsley, I. (2026). The OneSecond Law: Unifying the Earth, the Moon, and the

Quantum Vacuum. Zenodo. https://doi.org/10.5281/zenodo.21813533

of 60 68

Beardsley, I. (2026). The Earth Around The Sun, And The Sun Around the Galaxy In

Possible "Sweet Spots" For Complex Life As Determined By A Universal Particle Law.

(selfpublished)

Beardsley, I. (2026). StellarType Scaling and the Galactic Habitable Zone. (selfpublished)

Beardsley, I. (2026). The Kinematic Second: How Ancient Timekeeping Mirrors the

MassWeighted Orbital Dynamics of the EarthMoonSun System. (selfpublished)

Dawkins, R. (1976). The Selfish Gene. Oxford University Press.

Noble, D. (2016). Dance to the Tune of Life: Biological Relativity. Cambridge University

Press.

Noble, D. (2022). Debate with Richard Dawkins. [Video]. The Royal Institution.

of 61 68

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

(t)

K E

day

(t)cos θ ,

K E

day

K E

𝒢

, 𝒢 =

K E

F(t) = K

day

(t)

, K =

𝒢 cos θ

K E

0.01

∘

ℏ

⊙

= (1 s)K E

, h

⊙

= 2πℏ

⊙

≈ 1.77018 × 10

J ⋅ s .

F(t

) ≈ 1.

F(t

)

F ≈ 0

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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

−

F ≈ 0

day

≈

day

= 86,400 s, r

= 3.844 × 10

m ,

= 0.038 m /yr,

(req)

day

= T

day

= 86,400

0.038

3.844 × 10

= 8.541 × 10

−6

s/yr .

(req)

day

= 0.854 ms/century .

2.3 ms/century

1.33 ms/century

(obs)

day

= 2.3 − 1.33 = 0.97 ms/century .

Δ = 0.97 − 0.854 = 0.116 ms/centur y .

0.1

0.2 ms/century

1σ

F(t

) ≈ 0

F(t)

of 63 68

A.3 The ODE for the plateau and the anchoring of the day length

If is treated as an attractor condition, then

Integrating,

The integration constant is fixed by the UPL boundary condition. In the idealized exact

version,

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

day

ln T

day

= ln r

+ ln C,

day

(t) = C r

(t) .

F(t

) = K

day

)

= 1,

day

) =

)

F = 0

F(t

) = 1

F(t

) ≈ 1

= 1

F = F

+ δF .

of 64 68

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

ω = Ω − n

Ω =

2π

day

, n =

day

= −

rot

2Γ

orb

rot

= IΩ, L

orb

= M

= ϕ(F ) = Γ(F )

(

rot

(F )

−

orb

(F )

)

ϕ(F

) = 0 ⟹ L

orb,*

= 2L

rot,*

ϕ(F ) ≈ ϕ′ (F

)δF,

F = Fϕ(F )

F = F

ϕ′ (F

)δF ≡ − λ δF,

λ = − F

ϕ′ (F

) > 0.

ϕ′ (F

)

ϕ′ (F

) = Γ

(

−

L′

rot

rot,*

2L′

orb

orb,*

)

L′

rot

+ L′

orb

= 0 ⟹ L′

orb

= − L′

rot

orb,*

= 2L

rot,*

of 65 68

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 \(K\approx 5.07\times 10^3\ \mathrm{m/s}\), the mapping

is approximately:

ϕ′ (F

) = −

3Γ

L′

rot

rot,*

δF > 0

L′

rot

< 0

−L′

rot

> 0

ϕ′ (F

) < 0 ⟹ λ = − F

ϕ′ (F

) > 0.

F = − λ δF, δF(t) = δF(0)e

−λt

λ ≈ 3 × 10

−17

−1

≈ 1 Gyr .

F(t

) = 1

F(t

) ≈ 1 within the window 0.9 ≲ F ≲ 1.25.

F = K

day

min

day

≤

≤ T

max

day

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F vs T_day!

Thus a window of roughly 22–26 hours corresponds to ,mcentered 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.

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

day

ΔT

day

∼

day

obs

5 × 10

F(t) = K

day

(t)

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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 ⟺

day

≈

0.854 ms/century

0.97 ms/century

0.116 ms/century

day

(t) = C r

(t) .

F(t

) ≈ 1

day

) = r

)/K ≈ 86,400 s

F = − λ δF, λ > 0, λ ≈ 3 × 10

−17

−1

∼ 1

F(t

) ≈ 1

F ∈ [1.05,1.23]

F ≈ 1

86,400

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