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The One-Second Resonance: Two Formulations of the Universal Particle

Law and the Scale Hierarchy from the Proton to the Galaxy

Ian Beardsley

October 25, 2026

of 2 63

Table of Contents

Introduction……………………………………………………3

The and Formulation of the Universal

Particle Law: An Energy-Action Hierarchy

from the Proton to the Galaxy…………………………………5

A Universal Particle Law (UPL) and A Planetary

System Quantum Analog (PSQA): Scale Invariance

from Quantum Scales to the Celestial (The Original

Formulation)………………………..…………………………16

The Human Scale as a Resonant Node Between the

Quantum and Astrophysical Scales……………………………41

Appendix 1: The 24-Hour Day as a Stable

Resonance Plateau……………………………………………..57

ℏ

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Introduction

This collection presents the Universal Particle Law in its two complementary formulations,

together with the scale hierarchy that connects the proton to the galaxy and the human scale that

sits between them.

The Universal Particle Law begins from a simple proposal: what we experience as mass is not an

intrinsic property of matter but a resistance to the rotation of a particle's motion out of the

temporal dimension and 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 — inertia — is

quantified by a normal force that depends on Planck's constant, the speed of light, and a

fundamental timescale. Remarkably, when the law is applied to the proton, the neutron, and the

electron, the same timescale emerges: approximately one second.

The first paper in this collection develops the law in two equivalent forms. In one, the normal

force is written in terms of Planck's constant, and the mass of a particle is generated by that force

acting over the particle's cross-sectional area, mediated by the pliability of space as measured by

Newton's constant. This formulation makes the geometric picture of inertia explicit: mass is the

resistance of an extended area to the rotation of temporal velocity into spatial velocity. In the

other formulation, the same law is written in terms of the reduced Planck constant, and it takes a

form directly analogous to the Bohr ground state of hydrogen. The celestial effective action for

the Earth–Moon–Sun system is derived from this analogy, using the accurately known mass of

the Moon as its anchor. Both formulations return the same one-second invariant. They differ only

in which aspect of the physics they emphasize.

The second paper presents the original formulation of the law and its first extension beyond the

particle scale. It shows that the same one-second resonance appears in the Earth–Moon–Sun

system — where the geometric coupling is the ratio of the Moon's orbital radius to the Sun's

radius, the condition for a near-perfect solar eclipse — and in the Sun's orbit around the galactic

center, where two additional resonant nodes appear: the asteroid belt at roughly 2.4 astronomical

units, and the orbit of Saturn at roughly 9.5. These are not independent coincidences. They are

resonant nodes of the same law, fixed by the Sun's mass, radius, and motion through the galaxy.

The third paper identifies the human scale as the middle rung of this hierarchy. The human

effective action is the geometric mean of the Planck constant and the celestial effective action,

and the human energy scale that follows is on the order of the mechanical work of a heartbeat.

The geometric means of particle and astrophysical masses bracket the human mass range, from a

few grams to several tens of kilograms. The human scale is not an arbitrary biological fact; it is

the resonant node between the quantum and the astrophysical.

An appendix provides a careful treatment of the twenty-four-hour day as a stable resonance

plateau. The Earth–Moon–Sun system is dynamically locked near a flat extremum of the

resonance function: the Moon's recession and Earth's rotational slowdown are coupled so as to

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keep the resonance product nearly constant over geological time. The observed day length lies

inside a predicted window of stability, and the plateau has persisted for hundreds of millions of

years — long enough for complex life to emerge.

Two further appendices, included in the broader collection but not reproduced here in full,

examine the relationship between the Universal Particle Law and two independent theoretical

programs: a five-dimensional metric-affine reduction proposed by Chiaramonte and collaborators

within the ANKHOR research group, and the Quantum Relativity framework of Bermanseder. In

both cases the honest assessment is structural complementarity rather than derivation. The

Universal Particle Law supplies an empirical convergence across scales that neither framework

generates on its own; those frameworks supply candidate algebraic foundations for the form of

the force law. The relationship is best described as two views of a common structure, each

awaiting the other's completion.

The thread that runs through all of these papers is the one-second resonance. It appears in the

proton, the neutron, and the electron. It appears in the Earth–Moon–Sun system and in the Sun's

orbit around the galaxy. It appears at the human scale, as the mechanical work of a heartbeat. The

universe, on this account, is resonant — and we are not merely observers of that resonance but

its expression at the human scale.

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

derivation. The normal force can be derived from the Principle of Least Action, and the one-

second invariant has the character of a structural fact rather than a numerical accident. But the

value of the second is not derived from first principles; it is read from particle data. What the law

does is show that the same characteristic time governs bound systems across more than forty

orders of magnitude in mass and radius, and that the couplings which appear in the law are not

free parameters but are determined by the geometry and internal structure of the systems they

describe.

The papers are collected here in the order in which they develop the framework: two

formulations of the law, the celestial extension, the human scale, and the dynamical stability of

the twenty-four-hour plateau. Read together, they present a single, testable proposal: that the

structure of bound systems throughout the universe is governed by a one-second invariant, and

that the human scale is where that invariant becomes visible to itself.

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The ℎ and 󲰻 Formulations of the Universal Particle Law: An Energy–Action

Hierarchy from the Proton to the Galaxy

Ian Beardsley

Independent Researcher

October 22, 2026

Abstract

The Universal Particle Law (UPL) admits two mathematically equivalent formulations. In the

-formulation, the normal force acts on a particle's cross-sectional area to

generate mass, making explicit the geometric picture of inertia as resistance to the rotation of

temporal velocity into space. In the -formulation, the same law takes a form directly analogous

to the Bohr ground state of hydrogen, with the Moon's mass playing the role of the electron's

charge and the gravitational constant replacing the Coulomb constant. This yields a celestial

effective action J·s, derived from and numerically close to .

The energy–action law unifies the scales from the proton to the galaxy. The

human scale emerges as the geometric mean , giving J in the

-formulation and J in the -formulation—both on the order of the mechanical work of a

heartbeat. The relation between and the Planck force is an exact algebraic identity arising

from the convention; it does not fix the one-second value, which comes from particle data.

We argue that both formulations are physically meaningful and complementary, and we identify

open questions for future work.

1. Introduction

The Universal Particle Law (UPL) identifies a one-second invariant across scales [1]. The law

can be written in two equivalent ways. The first uses Planck's constant :

The second uses the reduced Planck constant . The two formulations are

algebraically equivalent, but they emphasize different aspects of the physics. The -formulation

makes explicit the interaction of the normal force with the particle's cross-sectional area

, which is the geometric core of the inertia mechanism. The -formulation reveals a

direct analogy with the Bohr ground state of hydrogen and leads to a natural definition of

celestial effective actions.

= h /(ct

)

= π r

ℏ

⊙

= 2.81733 × 10

(1 s)KE

⊕

ℏ

scale

= (1 s)E

scale

human

= E

⊕

0.545

ℏ

3.4248

2π

h /ℏ

, m

= κ

π r

ℏ = h /(2π)

= π r

ℏ

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This paper develops both formulations and the energy–action hierarchy that connects them. It is

intended as a standalone supplement to [1], and it is written in a spirit of exploration: several of

the connections reported here are numerical observations that await deeper derivation.

2. The -Formulation: Force, Cross-Section, and Inertia

In [1], the normal force is derived from the quantization of temporal momentum over the one-

second interval:

The mass of a particle is then

where is the cross-sectional area exposed to the normal force and is the "pliability of

space." The -formulation is the natural one for this geometric picture: is the quantum of

action, and the force-area product is the quantity that, when mediated by , generates mass.

Numerically, N. The proton, neutron, and electron all return s with

their respective couplings and [1].

The -formulation has a physical advantage that is worth stating explicitly: it shows the normal

force acting over the particle's cross-section. The mass is not a property of a point; it is the

resistance of an extended area to the rotation of temporal velocity into spatial velocity. The

-formulation, while equivalent, obscures this because the factors are absorbed into the

reduced constant. For the geometric picture, is the natural choice.

3. The -Formulation: The Bohr Analog and the Celestial Action

Substituting into the UPL gives

or, solving for ,

, F

Δp

Δt

= κ

π r

= π r

= 2.21022 × 10

−42

≈ 1

= κ

= 1/(3α

)

= 1

ℏ

2π

ℏ

h = 2π ℏ

2π ℏ

, m

= κ

2π

ℏr

Gct

= π 2

ℏ

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This is the -formulation. It has a natural interpretation in terms of the Bohr ground state. In [1],

we observed that the celestial effective action for the Earth–Moon–Sun system satisfies

which is the direct analog of the Bohr ground state

under the substitutions , , . The Moon's mass is known to high

precision from lunar laser ranging, so the equation is a strong anchor. It gives

The equation can be rearranged to read

so the celestial Bohr radius is the light-second. This is a structural fact: the Earth–Moon–Sun

system has a characteristic length equal to the distance light travels in one second.

The numerical coincidence

connects the celestial action to Earth's orbital kinetic energy J. The two

values differ by about 6%, which is within the uncertainty of the Earth's orbital velocity and the

choice of average versus perihelion velocity.

The -formulation is therefore the natural one for the Bohr-like structure of celestial scales. It is

also the natural one for the energy–action law developed in the next section.

4. The Energy–Action Law

At any scale, define the characteristic energy . Then the effective action is

This is dimensionally transparent: action = energy × time, with the time fixed at the one-second

invariant. The three known scales are:

ℏ

⊙

ℏ

⊙

= 1.00 seconds,

ℏ

ℏ → ℏ

⊙

→ G

→ M

ℏ

⊙

= 2.81733 × 10

J·s, h

⊙

= 2π ℏ

⊙

= 1.77018 × 10

J·s .

ℏ

⊙

= c × (1 s) = 2.998 × 10

ℏ

⊙

≈ (1 s) KE

⊕

≈ 2.649 × 10

ℏ

scale

ℏ

scale

= (1 s) E

scale

, h

scale

= 2π (1 s) E

scale

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Scale-dependent energies and Planck constants!

The particle scale is tautological: . The Earth scale is defined by the Bohr analog.

The galactic scale is defined by the Sun's orbital kinetic energy around the galactic center,

J, again using the one-second invariant: .

The energy–action law unifies the hierarchy. The effective action at each scale is the energy at

that scale multiplied by the one-second invariant. The and versions differ by the convention

factor , which is the same at every scale.

4.1 A remark on the 1 rad/s photon

At the particle scale, is the energy of a photon with angular frequency

rad/s. Its ordinary frequency is Hz, and its wavelength is

This is a numerical curiosity: the wavelength of the "1 rad/s photon" is about times the solar

radius. Whether this is structural or coincidental is an open question. What is not coincidental is

that the particle-scale energy is exactly per second—the reduced Planck constant is the energy

quantum of the one-radian-per-second mode.

5. The Human Scale as Geometric Mean

The human scale is defined by the geometric mean of the quantum and solar effective actions

[1]:

In the energy–action language, this is

Numerically,

Scale

E_scale (J)

󲰻_scale (J·s)

h_scale (J·s)

Particle

󲰻/(1 s) = 1.05457×10⁻³⁴

1.05457×10⁻³⁴

6.62607×10⁻³⁴

Earth–Moon–Sun

2.81733×10³³

1.77018×10³⁴

Galaxy

4.81338×10⁴⁰

3.0243×10⁴¹

= ℏ/(1 s)

⊙

≈ 4.81338 × 10

ℏ

gal

= (1 s)KE

⊙

ℏ

2π

= ℏ/(1 s)

ω = E

/ℏ = 1

ν = 1/(2π)

λ =

= 2π c ≈ 1.884 × 10

m ≈ 2.7 R

⊙

≈ e R

⊙

ℏ

human

= hh

⊙

≈ 3.4248 J·s .

human

= E

⊕

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This is on the order of the mechanical work of a heartbeat, which is about 1 J per beat. The

original paper's value J is the -based energy; the -based energy is J.

The factor of is the same convention difference that appears everywhere else. Both are on the

order of joules, and both are compatible with the biological scale.

The heartbeat does about 1 J of mechanical work per beat. The -formulation lands closer to this

value, but the -formulation is also on the correct order of magnitude. The choice of formulation

should be guided by the aspect of the resonance one wishes to emphasize: the -formulation for

the force-area picture, the -formulation for the energy–action hierarchy.

The human mass range is bracketed by the geometric means of particle and astrophysical masses,

as shown in [1]: g to kg. The human scale is the middle rung

of the hierarchy.

6. The Relation: Exact but Conventional

The identity

holds exactly for any . It is sometimes read as a statement that the normal force, scaled by the

Planck force, accumulates a full rotation of angular phase over one second. As noted in [1], this

reading is suggestive but not correct: the arises from the convention difference between and

. The Planck time is defined with , while the normal force is written with . Substituting the

definitions gives

so the is a conversion factor. The algebraic derivation was presented in a joint paper with

David Blackwell [2]. The equals sign in [1] is deliberate: the identity is exact, not approximate. It

is exact because the algebra is exact. But it is a convention artifact, not a dynamical phase.

human

= (1.05457 × 10

−34

)(2.81733 × 10

) ≈ 0.545 J .

human

≈ 3.4248

ℏ

0.545

2π

Formulation

h_human (J·s)

E_human (J)

h-based

3.4248

󲰻-based

0.545

ℏ

⊕

≈ 2.3

⊙

≈ 57.7

2π

Planck

⋅

= 2π

2π

ℏ

Planck

⋅

2π c

2π

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In the -formulation, the same ratio is 1:

This does not diminish either formulation. The -formulation is the natural one for the force-area

picture, where is the quantum of action and is the quantity that generates mass. The

-formulation is the natural one for the Bohr analog and the energy–action law. The is exact,

and it is exact because the algebraic identity is exact; it is not an approximation. But it is a

convention artifact, not a dynamical phase. The one-second value comes from particle data, not

from the identity.

Section 6A: The 2π Resonance and the Cross-Sectional Area Form

X.1 Motivation

The factor appears in the Universal Particle Law in three independent contexts: in the Planck-

force identity ; in the -form of the UPL, ;

and in the Bohr analog. This section shows that the electron UPL, combined with the

independently measured Bohr radius, produces a fourth appearance of the same , and that this

fourth appearance can be written in a form that foregrounds the cross-sectional area on

the left side and a full rotation on the right.

X.2 Derivation

Start from the electron UPL with :

Squaring and using :

X.1.

The Bohr radius gives the equivalent . Substituting into (X.1):

ℏ

(ℏ)

Planck

⋅

= 1, F

(ℏ)

ℏ

2π

Planck

⋅ t

= 2π

ℏ

= π 2 (r

) ℏ/(G c) κ

ℏ

⊙

2π

= πr

= 1

πh

πh = 2π

ℏ

(

)

2π

ℏ

= ℏ

/(k

)

ℏ = e a

⋅

2π

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Rearranging, and dividing one factor of into to form :

X.2.

This is the preferred form. The left side is a combination of fundamental constants and two

independently measured length scales: the classical electron radius (appearing inside ) and

the Bohr radius . The right side is one full rotation, , divided by the square of the one-second

invariant. Both sides have dimension .

Evaluating the left side with CODATA 2022 values gives . The right side with

s gives . The agreement is , which is the same precision as the

electron UPL itself.

Two equivalent forms are worth recording:

- Dimensionless form. Multiplying (X.2) by gives . The left

side is now a pure number, and the right side is the topological constant . This is the cleanest

statement of the identity: a dimensionless geometric quantity constructed from independent

measurements equals one full rotation.

- Angular-frequency form. Multiplying (X.2) by gives .

Both sides are now inverse times. Since s, the right side is rad/s to the accuracy of the

invariant, and the identity reads: the combination on the left is one full rotation per second.

X.3 Interpretation

The left side of (X.2) has a natural reading. The cross-section is exactly the effective

area that appears in the UPL mass formula . The quantity is the

normal force times the area over which it acts, which is the squared mass of the electron

multiplied by the pliability of space. The rest of the left side — — is the

conversion from that force-area product into an inverse squared time, by way of the Bohr radius.

So (X.2) says: the force-area product of the electron, divided by the Bohr radius and by the

fundamental constants, is exactly one full rotation per square second. Or, in the dimensionless

form: the force-area product of the electron, weighted by the independently measured Bohr

radius, evaluates to .

= πr

2π

−2

6.310 s

−2

= 0.99773

6.312 s

−2

0.03 %

⋅ (Gc /(A

e)) m

/(a

) = 2π

2π

⋅ (Gc /(A

e)) m

/(a

) = 2π /t

≈ 1

2π

= πr

= A

= m

(Gc /e) m

/(a

)

2π

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The here is the same that appears in the Planck-force identity and in the -form of the

UPL. It is the conversion factor between and : it enters through the factor in

(X.1), and it survives through the Bohr radius, which itself uses . What is new in (X.2) is that

the is now anchored to the cross-sectional area on the left, rather than to the algebraic

structure of the UPL alone.

X.4 The scope of the cross-check

The identity (X.2) is a genuine cross-check between two independent physical facts: the electron

UPL (which gives s from the classical electron radius) and the Bohr radius (which

gives Å from hydrogen spectroscopy). Neither was used to derive the other, and both

are measured to high precision. The identity says that these two independent facts, combined

with Newton's constant and the speed of light, produce exactly to within the precision of the

one-second invariant.

The same cross-check cannot be extended to the proton or the neutron. There is no hydrogenic

system with a proton orbiting a proton (they repel) and no hydrogenic system with a neutron at

the center at all, so there is no independent "Bohr radius" for either particle. Substituting the

UPL's own golden-ratio prediction into (X.2) gives a tautology: it restates the

UPL rather than checking it.

The celestial case is a parallel construction, not a direct analog. The moon plays the role of the

Bohr anchor through , and the resulting solar-system length is the

light-second. But is defined by the one-second invariant, whereas is measured

independently. So the electron identity is a cross-check against an external anchor, while the

celestial construction is a consistency relation whose anchor is the one-second invariant itself.

X.5 What the identity does and does not establish

The identity (X.2) establishes:

1. The electron UPL, the Bohr radius, Newton's constant, and the speed of light are mutually

consistent, and their combination produces exactly .

2. The of the UPL is the same that appears in the Planck-force identity and the -form,

appearing here in a fourth independent algebraic context.

3. The cross-sectional area is a natural geometric object in the identity; the factor of inside

combines with the remaining to give the on the right.

The identity does not establish:

1. That is a topological invariant. The traces back to , which is a convention

choice (as discussed in Appendix 3 with respect to Bermanseder's framework). The identity

2π

ℏ

πh = 2π

ℏ

2π

= 0.99773

= 0.529

2π

= ϕh /(m

⊙

= ℏ

⊙

/(GM

) = c ⋅ 1 s

⊙

2π

ℏ

2π

h = 2π ℏ

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shows the convention is internally consistent across multiple paths, but it does not show that

is dynamically required by the vacuum geometry.

2. That the identity extends to composite hadrons or to the celestial system as a direct cross-

check. It is specific to the electron because it is specific to hydrogenic systems.

The honest statement in the paper is therefore: the electron UPL and the Bohr radius are mutually

consistent, and their consistency produces in a form that foregrounds the cross-sectional area.

Whether this reflects a deeper topological structure of the vacuum remains an open question, and

is the subject of the frameworks discussed in the appendices.

7. Numerical Summary

Physical Quantities!

8. Open Questions

1. Derivation of . The Bohr analog gives a precise value for , but it is a guess about the

form of the celestial quantum. Can be derived from the Moon's mass and the one-second

2π

Quantity

Value

= h/(c t

²)

2.21022×10⁻⁴² N

(

󲰻

)

= 󲰻/(c t

²)

3.517×10⁻⁴³ N

󲰻

⊙

2.81733×10³³ J·s

⊙

= 2π󲰻

⊙

1.77018×10³⁴ J·s

󲰻

gal

4.81338×10⁴⁰ J·s

gal

= 2π󲰻

gal

3.0243×10⁴¹ J·s

human

= √(h h

⊙

)

3.4248 J·s

󲰻

human

= h

human

/(2π)

0.545 J·s

= 󲰻/(1 s)

1.05457×10⁻³⁴ J

⊕

= 󲰻

⊙

/(1 s)

2.81733×10³³ J

human

= √(E

⊕

)

0.545 J

⊙

2.6715×10⁶⁷

gal

⊙

1.7085×10⁷

gal

4.564×10⁷⁴

human

/h = √(h

⊙

/h)

5.168×10³³

ℏ

⊙

ℏ

⊙

ℏ

⊙

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invariant without the Bohr analog as an input? A derivation from first principles would be

stronger.

2. The galactic scale. The galactic effective action is defined by the Sun's

kinetic energy. Is there a Bohr-like equation for the galaxy, or is the galactic scale defined only

by the energy–action law? The Earth scale has a Bohr analog; the galactic scale does not, at least

not obviously.

3. The 6% difference. The difference between and is about 6%. Is this physically

significant, or is it due to the choice of average versus perihelion velocity? A careful treatment of

the Earth's orbital velocity would settle this.

4. The relation. The algebraic identity is exact, but it might be a shadow of a deeper

geometric fact about the rotation of temporal velocity into space. Is there a dynamical

interpretation of the that survives the convention?

5. The heartbeat. The human energy scale J (in the -formulation) is on the

order of a heartbeat. Can this be connected to a specific biological mechanism, or is it a

numerical coincidence?

6. The 1 rad/s photon. The wavelength of the 1 rad/s photon is about . Is this structural, or is

it a numerical curiosity?

9. Conclusion

The Universal Particle Law admits two equivalent formulations. The -formulation emphasizes

the geometric picture of inertia as a force acting over a cross-section. The -formulation

emphasizes the Bohr-like structure of celestial scales and leads to the energy–action law

. Both formulations give the same one-second invariant, and both are

physically meaningful.

The celestial effective action is anchored by the Bohr analog s, which uses

the accurately known Moon mass and gives J·s. The energy–action law

extends this to the galaxy and to the human scale, where the geometric mean

gives J in the -formulation—on the order of a heartbeat.

The relation between and the Planck force is exact but conventional. It does not fix the

one-second value, which comes from particle data. Both formulations are correct; the choice

between them should be guided by the aspect of the resonance one wishes to emphasize.

ℏ

gal

= (1 s)K E

⊙

ℏ

⊙

(1 s)K E

⊕

2π

human

≈ 0.545

ℏ

⊙

ℏ

scale

= (1 s)E

scale

ℏ

⊙

ℏ

⊙

/(GM

c) = 1

ℏ

⊙

= 2.81733 × 10

human

= E

⊕

0.545

ℏ

2π

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Future work should aim to derive the celestial effective actions from first principles, to test the

predictions of the framework against exoplanetary data, and to explore the connections between

the energy–action hierarchy and biological rhythms.

References

[1] Beardsley, I. (2026). A Universal Particle Law (UPL) and a Planetary System Quantum

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

https://doi.org/10.5281/zenodo.23571277

[2] Beardsley, I., & Blackwell, D. (2026). Informational Geometry, the 2π-Hz Resonance, and

Warp-Drive Dynamics Without Exotic Matter. Zenodo. https://doi.org/10.5281/zenodo.20028442

[3] Tiesinga, E., Mohr, P. J., Newell, D. B., & Taylor, B. N. (2024). CODATA Recommended

Values of the Fundamental Physical Constants. NIST.

[4] Tynski, K. (2024). One Equation, Mysteries: A Structural Constraint That May

Explain (Almost) Everything.

[5] Beardsley, I. (2026). The Human Scale as a Resonant Node Between the Quantum and

Astrophysical Scales. In [1].

∼ 200

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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 - October 25, 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 −

of 17 63

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

= hν

= m

= h /(m

c) = r

of 18 63

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 ‘pliability 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

)

= πr

= κ

⋅

π r

6.62607015 × 10

−34

J·s

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

= 2.21022 × 10

−42

= 1 second

of 19 63

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

of 20 63

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

of 21 63

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

of 22 63

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) =

of 23 63

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 =

∫

−mc −

+ λ

(

− t

)

dτ

−

∂λ

∂x

λ =

of 24 63

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)

= 1.21020E44N

of 25 63

And, Planck time is:

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

Dividing by the Planck time squared:

That number is . It arises from the convention difference between and : the Planck time is

defined with , while the normal force is written with . The factor of is therefore

an algebraic consequence of the -to- conversion and not a dynamical phase.

This gives the identity

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

the numerical value of the second. What it does show is that when is taken as the empirical

one-second invariant — obtained from the proton, neutron, and electron masses and radii via

equation (8) — the ratio acquires a definite numerical value, and the combination

above evaluates to s⁻².

From the Planck units we have

so the identity can be written as

Planck

ℏc

= 2.176434E − 8kg

Planck

ℏG

= 1.616255E − 35m

Planck

ℏG

= 5.391247E − 44s

Planck

= 1.826326 × 10

−86

Planck

⋅

= 6.2834743 s

−2

2π

ℏ

ℏ = h /(2π)

2π

ℏ

Planck

⋅

= 2π,

Planck

2π

Planck

= G

of 26 63

or equivalently,

This may be read as a relation between the Planck scale and the macroscopic normal force. It is a

consistency check and a useful reformulation, not a derivation of the one-second value.

The identity can be rearranged as

where Hz. This reveals a natural angular frequency rad/s — a

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

interpretation as a frequency is independent of the derivation of the one-second value, and it

suggests that inertia is governed by a characteristic clock ticking at approximately one hertz.

The normal force and the Planck force are related through the

Planck time . Substituting their definitions yields the dimensionless identity

The factor of arises from the convention difference between and : the Planck time is

defined with while the normal force is written with Writing

and substituting,

so that the is an algebraic consequence of the -to- conversion and not a dynamical phase.

The identity holds for any value of , because the factors of cancel; it does not by itself

determine the numerical value of the second.

What the identity does show is that when is taken as the empirical one-second invariant —

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

= 2π

⋅ t

= 2πF

Planck

⋅

Planck

= 2π

(

)

= 2π (t

⋅ ν

)

= 1/t

= 1

= 2π ν

= 2π

= h /(ct

)

Planck

= c

= ℏG /c

Planck

⋅

= 2π .

2π

ℏ

ℏ = h /(2π)

= ℏG /c

= hG /(2πc

)

Planck

⋅

2π c

2π

ℏ

of 27 63

acquires a definite numerical value, and the combination evaluates to

s⁻². This may be read as a statement about the relationship between the Planck scale and the

macroscopic normal force, but it is a consistency check rather than a derivation. The one-second

value is fixed by particle data, not by the Planck-force identity.

The identity can be rearranged as

where Hz. This reveals a natural angular frequency rad/s — a

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

interpretation as a frequency is independent of the derivation of the one-second value, and it

suggests that inertia is governed by a characteristic clock ticking at approximately one hertz.

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 pliability 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,

Planck

⋅ 1/t

2π

Planck

= 2π

(

)

= 2π (t

⋅ ν

)

= 1/t

= 1

= 2π ν

= 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)

of 28 63

while the fundamental electron probes the bare vacuum pressure. Far from undermining the

theory, this discrepancy rigorously validates the coupling of the normal force to the

electromagnetic substructure. We see the errors on the neutron data for its radius are very large,

so it is difficult to determine very accurately. In fact, the radii of the proton and neutron have

no exact value. Rather they are fuzzy clouds of subatomic particles and their values depend on

the method used to measure them. Though, it is not unreasonable to suggest that their values fall

within a range that produces on the order of about 1 second.

Conclusion

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

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

physics. The invariant is expressed as

where and .

Crucially, the invariant leads to a universal particle equation:

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

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

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

We can offer the following interpretation: The geometric origin of the second becomes apparent

when we relate to the Planck force . We find

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.

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

of 29 63

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

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

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

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

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

—is a natural resonance of the universe.

A Planetary System Quantum Analog (PSQA) And Scale Invariance from Quantum Scales

to the Celestial Scales

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:

= κ

π r

K E

(Ear th Da y) = 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 Da y)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 Da y)cos(θ

) = 0.991seconds

K E

(Ear th Da y)cos(θ

) ≈ 1second

of 30 63

The Moon allows for the evolution of complex, intelligent life because it stabilizes the earth tilt

to its orbit allowing for the seasons and preventing temperature extremes. We live in an

interesting time when the Moon near perfectly eclipses the Sun. This is because while the Sun is

400 times larger than the Moon, it is 400 times further from the Earth than the Moon is. That is

the perfect eclipse is given by the orbital radius of the Earth to the Moon’s orbital radius

equals the Solar radius to the lunar radius :

The Earth's rotation is slowing down, and as a direct result, the Moon is slowly drifting away.

Earth's Rotation Slowing: The length of a day is increasing by about 2.3 milliseconds per

century. This is the long-term average rate caused primarily by the Moon's tidal pull. (Note that

this rate can be influenced by other factors; for example, climate change is currently contributing

about 1.33 milliseconds per century).

Moon's Orbit Growing: The Moon is receding from Earth at a rate of about 3.8 centimeters (1.5

inches) per year.

The Tidal Connection

These two phenomena are linked by tidal friction. The Moon's gravity creates tides in Earth's

oceans. As Earth rotates, the tidal bulges are pulled slightly ahead of the Moon, creating friction

that slows Earth's spin.

To conserve angular momentum in the Earth-Moon system, this loss of Earth's rotational speed is

transferred to the Moon, boosting its orbital energy and causing it to spiral outward into a larger

orbit. This recession rate has been precisely measured for decades using the Lunar Laser Ranging

experiment, which bounces lasers off reflectors left on the Moon by Apollo astronauts.

Long-Term Perspective

While these changes are tiny on human timescales, they have significant effects over millions of

years. About 250 million years ago, a day on Earth was roughly 23 hours long. In about 600

million years, the Moon will have drifted so far that total solar eclipses will no longer be

possible.

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):

⊙

ℏ

⊙

of 31 63

Where is a celestial Planck type constant for the Earth/Moon/Sun system, is the mass of

the Moon, and we have let: , , . This gives us that:

Nicely, this has that:

Suggesting that that the quantum analog for the Earth/Moon/Sun system and the Universal

Particle Equation both have a characteristic time of about 1 second, we should find 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.

ℏ

⊙

= 1.00seconds

ℏ

⊙

ℏ → ℏ

⊙

→ G

→ M

ℏ

⊙

= 2.81733E 33J ⋅ s

⊙

= 2π ℏ

⊙

= 1.77018E 34J ⋅ s

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

⊙

of 32 63

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:

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:

ℏ

gala xy

= (1"second)K E

⊙

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

= 4.81338E40J ⋅ s

ℏ

⊙

ℏ

galax y

= (1 second)KE

⊙

= 4.81338E 40J ⋅ s

galax y

= 2π ℏ

galax y

= 3.0243E41J ⋅ 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

of 33 63

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.!

Section Y Another Discussion: The Bohr Analog, the Moon as Ground State, and the Scale

Hierarchy

The electron cross-check, reported in the companion paper, shows that the electron UPL and the

Bohr radius together produce exactly; the solar-system construction is parallel in structure but

anchored internally by rather than externally by .

Y.1 The observation that prompted the exploration

The Bohr-analog construction for the solar system was not introduced as a derivation. It was

introduced because the UPL, applied to the Earth with the eclipse ratio as the coupling, gives

s, while the Bohr-analog substitution with the Moon's mass gives s. The

two constructions agree, and the object that makes them agree is the celestial effective action:

This is a strange object. It is not the action of the Moon, or of the Earth, or of the Sun

individually. It is the action of the Earth–Moon–Sun system as a whole, and the quantity that

fixes it is the cube of the Moon's mass. The Moon, not Mercury, not the Earth, and not the Sun, is

the anchor.

Y.2 Why the Moon and not Mercury

If one were constructing a literal Bohr atom for the solar system, the ground state would be the

innermost orbit. Mercury is at 0.387 AU, closer to the Sun than the Earth, and if one substituted

Mercury's mass into the Bohr analog , the resulting length scale would be

different from the light-second. The construction would not close.

The Moon's mass, cubed and divided into , gives exactly one second. That is an

empirical fact about the numerical value of , not a derivation. But it is the fact that makes the

⊙

galax y

= 0.115457

⊙

galax y

πh

galax y

⋅

earth

saturn

ℏ

gala xy

2π

0.982

→ M

1.00

ℏ

⊙

= GM

c ⋅ (1 s) = 2.817 × 10

J·s

⊙

= ℏ

⊙

/(GM

)

(ℏ

⊙

)

/(c)

of 34 63

whole construction work. The Moon is playing the role that the proton plays in the hydrogen

atom: it is the mass whose cube anchors the effective action.

The reason Mercury does not play this role is worth stating clearly. In the hydrogen atom, the

nucleus is the anchor and the electron is the orbiting body, and the ground state is defined by the

innermost stable orbit. In the solar system, the Sun is the anchor (as the proton is the anchor), the

Earth is the orbiting body (as the electron is the orbiting body), and the Moon is... not the

nucleus. The Moon is a secondary body orbiting the Earth.

So the solar-system construction is not a literal atom. It is a structural analog in which the role of

is played by . The choice of the Moon's mass as the anchor is not dictated by the

planetary architecture. It is dictated by the numerical value of that makes the one-second

invariant close.

Y.3 Why solar systems are not atoms

The analog is not literal, and it is worth listing the ways in which it fails, because those failures

are themselves informative.

1. Planets have different masses and sizes. Protons are identical to protons, electrons to electrons.

Planets are not. The Earth and Jupiter differ in mass by a factor of 318, and in radius by a factor

of 11. There is no "planetary constant" analogous to or .

2. Planets do not jump between orbits. When an atom absorbs a photon, the electron jumps from

one orbital to another, and the energy difference is quantized. When a planet absorbs radiation,

its orbit does not change by a discrete amount. The Bohr quantization condition has no

planetary analog.

3. Planets are not probability clouds. The electron in the hydrogen ground state is described by a

wavefunction whose squared modulus gives the probability of finding the electron at a given

position. The Earth is at a definite position (to within the precision of its orbit). There is no

planetary wavefunction.

4. The Sun is not a point charge. The proton is, to excellent approximation, a point charge at the

center of the hydrogen atom. The Sun is an extended body with a radius of m, and its

gravitational field is not that of a point mass at short distances.

5. The "coupling" is different. In the hydrogen atom, the coupling between the proton and the

electron is the Coulomb constant and the elementary charge . In the solar-system analog, the

coupling is the gravitational constant and the mass . The dimensions work out (both

combinations give when written in the Bohr-analog form), but the physical mechanism is

different.

L = nℏ

6.96 × 10

of 35 63

Given these differences, one would not expect the same numerical relation to hold. The fact that

it does—to within a few percent—is the uncanny fact that motivates the exploration.

Y.4 What the construction shows and what it does not

What the construction shows:

- The Earth–Moon–Sun system has a characteristic action that, when inserted into the UPL

with the Earth's mass and radius and the eclipse ratio as coupling, gives s, consistent

with the one-second invariant.

- The quantity is fixed by the Moon's mass through the relation s.

- The Earth's orbital kinetic energy is numerically close to , with a difference of about

6%.

What the construction does not show:

- Why the Moon's mass has the value that makes this work. The Moon's mass is a consequence

of the giant-impact hypothesis and the subsequent dynamical evolution of the Earth–Moon

system. Whether the one-second relation is a coincidence of that history, or whether it is a

selection effect that constrains which planetary systems can host complex life, is an open

question.

- How to extend the construction to other planetary systems. The construction uses the specific

values of , , , and . There is no obvious generalization to a system with two

moons, or with no moon, or with a gas giant in the habitable zone.

- Whether the one-second relation is a consequence of a deeper wave equation. The Schrödinger

equation does not apply to the Earth–Moon–Sun system, and there is no known wave equation

that does. The Bohr analog is a structural similarity, not a derivation.

Y.5 The electron cross-check

The electron cross-check of the previous section is a parallel exploration, not an extension of the

solar-system construction. It asks: if the solar-system construction uses the Bohr analog with ,

what happens if one uses the literal Bohr analog for the electron, with as the anchor?

The answer is that the electron UPL, combined with the independently measured , gives

exactly :

where . This is a genuine cross-check, because is measured independently of the

electron UPL. It cannot be extended to the proton, neutron, or solar system, because there is no

ℏ

⊙

= 0.982

ℏ

⊙

ℏ

⊙

/(GM

c) = 1

ℏ

⊙

/(1 s)

⊕

moon

⊙

2π

⋅

= 2π

= πr

of 36 63

independent "Bohr radius" for those systems: the proton and neutron have no hydrogenic

analogs, and the solar "Bohr radius" s is defined by the one-second invariant rather

than measured independently.

The contrast is instructive:

- Electron: cross-check against an independent anchor ( ). The is confirmed against an

external measurement.

- Solar system: consistency relation with an internal anchor (the Moon's mass). The one-second

invariant is confirmed to be self-consistent, but not against an external measurement.

The two constructions are parallel in structure but different in verification status. The electron

cross-check is a confirmation; the solar-system construction is a consistency relation.

Y.6 The larger project: scale invariance across forty orders of magnitude

The broader motivation for these explorations is the observation that the one-second invariant

appears at every scale from the proton to the galaxy:

Scale Quantities and t₁ Values!

The invariant spans more than forty orders of magnitude in mass and radius. The question is

whether this is a coincidence, a selection effect, or a structural feature of the universe.

The solar-system Bohr analog and the electron cross-check are two attempts to probe this

question. Neither is a derivation from a wave equation. Neither explains why the one-second

invariant appears at every scale. But both make the invariant concrete: they show that it is not

just a numerical pattern, but a relation that can be written in a definite algebraic form, with

definite anchors (the Moon's mass, the Bohr radius), and definite couplings (the eclipse ratio, the

electron’s ).

The hope is that enough explorations of this kind—each of them partial, each of them limited—

will eventually reveal the structure that makes the invariant possible. That structure may not be

⊙

= c ⋅ 1

2π

Scale

Quantity

t₁

Electron

rₑ, mₑ

0.99773 s

Proton

rₚ, mₚ

1.00500 s

Neutron

rₙ, mₙ

1.00803 s

Earth–Moon–Sun

R⊕, M⊕, κ⊕

0.982 s

Asteroid belt

R☉, M☉, κ_gal

≈ 1 s

Saturn

r☉, M_gal, κ_gal

≈ 1 s

= 1

of 37 63

the Schrödinger equation, which applies to atomic and molecular systems, not to planetary ones.

It may be a different wave equation, or a geometric principle that underlies both. The Bohr

analog is one window into that possibility. It is not the answer, but it is a way of looking.

Y.7 A concluding note

The solar-system construction is uncanny. It uses the Moon's mass in a way that has no obvious

physical justification—the Moon is not the ground state of anything, in any conventional sense—

and yet the numerical relation closes to within a few percent. The electron cross-check is cleaner,

because is an independent measurement, but it is specific to the electron and does not

generalize.

The honest state of the exploration is that the one-second invariant is real and spans scales, that

the Bohr analog provides a way of writing it down for both electrons and solar systems, but that

the deeper structure—the wave equation or geometric principle that would explain the invariant

—remains open. The explorations reported here are steps toward that structure, not the structure

itself.

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:

⋅

πh

⋅ κ

πh

earth

π . h

⊙

⋅

moon

⊙

πh

galax y

⋅

earth

asteroids

⊙

galax y

πh

galax y

⋅

earth

saturn

of 38 63

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:

- 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 , this becomes

where .

κ ≈ 0.415

κ ≈ 0.115

= π 2 κ

= αℏ/(m

= π 2 α

(

)

= π 2

= Gm

/(ℏc)

of 39 63

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

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.

( p)

= π 2 κ

≈ 1.864 × 10

1.851 × 10

1.864 × 10

= Gm

/(ℏc)

= π 2 α /α

ℏ/2

β = 0.93

0.02

≈ 1

of 40 63

[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).

[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 41 63

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

of 42 63

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:

=29785m/s - 30,290m/s. 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 .

moon

of 43 63

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 .

of 44 63

- 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 45 63

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 47 63

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 Da y) = 1.25seconds

of 48 63

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 Da y)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 Da y)cos(θ

) = 0.991seconds

K E

(Ear th Da y)cos(θ

) ≈ 1second

⊙

of 49 63

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

⊙

= 2π ℏ

⊙

= 1.77018E 34J ⋅ s

ℏ

⊙

≈ (1.00seconds)K E

of 50 63

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

of 51 63

[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

of 52 63

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

of 53 63

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.

of 54 63

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

of 55 63

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

of 56 63

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

of 57 63

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

⊙

= 2π ℏ

⊙

≈ 1.77018 × 10

J ⋅ s .

F(t

) ≈ 1.

F(t

)

F ≈ 0

of 58 63

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/century .

0.1

0.2 ms/century

1σ

F(t

) ≈ 0

F(t)

of 59 63

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 60 63

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 61 63

we find

For a positive perturbation , the Moon is farther out and the day is longer, so the

rotational angular momentum decreases: . Hence , and since ,

Therefore,

Small perturbations decay exponentially. A numerical estimate using a constant-time-lag tidal

model gives

corresponding to an e-folding time

This is the correct order of magnitude for the plateau to persist over the geological timescale

required for complex life.

A.5 The boundary condition as a finite window

The exact statement should be replaced by the more defensible statement

Equivalently, since

a window in \(F\) maps directly to a window in :

Using the numbers in the paper, with , the mapping is approximately:

ϕ′ (F

) = −

3Γ

L′

rot

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

K ≈ 5.07 × 10

m /s

of 62 63

F vs T_day!

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

If one instead keeps as the anchor, the window is more like 19–23 hours. Either way, the

observed 24-hour day sits inside the predicted plateau.

This is the honest formulation: the UPL does not uniquely derive seconds from first

principles. It predicts a stable plateau, and the observed 24-hour day lies within that plateau. The

absolute value of is set by the observed lunar orbital radius and the requirement that

remain near unity. The plateau width is controlled by the sensitivity of the tidal torque to changes

in , i.e. by the Love number and the tidal quality factor ,

A more precise future treatment could define the window width from tidal physics as

or by requiring that the accumulated drift over years be less than some fraction of the

window width.

A.6 Summary

The consolidated picture is:

1. Resonance function.

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)

of 63 63

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