of 1 35
Setting The Record Straight On The One Second Invariant And The Universal Particle Equation!
By!
Ian Beardsley!
April 4, 2026"
of 2 35
Introduction
I have now become hesitant about collaborating on projects with author’s who have theories that
can use my work. My work, that of Ian Beardsley, has been used in conjunction with work by
Oleg Evdokimov (Олег Евдокимов) in his Ontology of the Fundamental Network theory
(OFN), in three joint papers authored by myself and Evdokimov. The way my work eventually
becomes credited to Evdokimov and Beardsley instead of to Beardsley, is as follows: The first
paper uses my work, and my work is cited in that paper by referring to papers I wrote before the
collaboration. But what happens, is when we collaborate on a second paper together, and my
equations are used, they are now cited by referring to the first collaboration by Evdokimov and
Beardsley, instead of referring to my papers that preceded that collaboration. After having
collaborated on three such papers together, my original work becomes lost to a series of links to
our joint papers where only the first paper directly credits me for my work. This starts to become
worrisome when Evdokimov starts to write papers where I am not coauthor and cites my
equations not by refereeing to my original papers on the subject, but to our later collaborations.
The result is much work that should be credited to Beardsley becomes credited to Evdokimov
and Beardsley. It is the purpose of this paper to set straight that the one second invariant, its
associated universal particle equation, the theory for mass generation, and the normal force
associated with it solely originated with me (in earlier papers by Ian Beardsley). I am a firm
believer that a person should be properly credited for their work.
The good thing is that the first paper Cosmic Knots as Torsional Solitons in the Fundamental
Network:Unifying the One-Second Invariant, Scaling and Baryogenesis, Oleg Evdokimov, Ian
Beardsley, 2026 has an appendix that explains exactly what originated with me. The paper was
submitted to the journal Physical Review A on 21 January, 2026:
The appendix that properly attributes my work to me is as follows…
F
n
ϕ
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Appendix A: The Geometric Theory of Inertia and the One-Second Invariant
(Beardsley, 2025)
A1. Conceptual Foundation
In a complementary line of research, Beardsley (2025) has developed a geometric theory of
inertia that derives the masses of fundamental particles and key solar system relationships from a
single Lorentz-invariant timescale: second. This theory posits that mass (inertia) is the
resistance encountered when diverting an object's motion from the temporal dimension
into the spatial dimensions. When a particle is accelerated, its spacetime velocity vector is
rotated, converting temporal motion ( ) into spatial motion ( ).
A2. The Fundamental Normal Force and Mass Formula
The theory introduces a universal, Lorentz-invariant normal force , which characterizes the
inherent "stiffness" of spacetime at the quantum-gravitational interface:
(A1)
The inertial mass of a particle is then given by:
(A2)
where:
• is the characteristic radius of the particle,
• is the gravitational constant,
• is a dimensionless coupling constant specific to the particle type:
with being the fine-structure constant.
A3. Empirical Verification: Particle Masses
Using the experimentally measured charge radii of the proton ( ), neutron ( ), and electron ( ),
the formula with s yields:
t
1
= 1
v
t
v
s
F
n
m
i
m
i
= κ
i
π r
2
i
F
n
G
,
r
i
G
κ
i
κ
p
= κ
n
=
1
3α
2
, κ
e
= 1,
α
r
p
r
n
r
e
t
1
= 1
Particle
Accuracy
Proton
1.00500
0.5%
Neutron
1.00478
0.5%
Electron
0.99773
0.2%
Predicted (s)
t
1
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This remarkable agreement provides robust, independent empirical evidence for as a
fundamental invariant governing quantum-gravitational interactions at the baryonic scale.
A4. Derivation of the One-Second Scale from Fundamental Constants
The invariant is not postulated but derived from the fundamental constants of nature:
(A3)
where is the Planck time and is the Compton time of the proton. This expression bridges the
Planck scale, quantum mechanics ( ), relativity ( ), and gravitation ( ), demonstrating that the
one-second timescale is a fundamental property of spacetime geometry itself.
A5. Macroscopic Manifestation: The Solar System and
The same one-second invariant governs macroscopic celestial mechanics. By defining a solar
system Planck-type constant:
(A4)
where is the kinetic energy of Earth's orbital motion, key relationships emerge. For
instance, the Moon's orbital ground state is quantized as:
(A5)
where is the mass of the Moon. This directly parallels the quantum mechanical formulation,
suggesting a unified scaling principle from subatomic to planetary scales.
A6. The Golden Ratio and the Proton Radius
The theory predicts an optimal relationship for the proton radius involving the golden ratio
:
(A6)
This result aligns with high-precision measurements and can be understood dynamically: the
proton may fluctuate among quasi-stable states corresponding to Fibonacci approximations
of (e.g., 2/3, 5/8, 13/21), offering a potential resolution to the historical "proton radius puzzle."
A7. Synthesis with the OFN: Cosmic Knots as the Ontological Basis
Within the Ontology of the Fundamental Network (OFN), the geometric theory of inertia finds
its natural foundation:
• The one-second invariant corresponds to the characteristic phase evolution period of a
Cosmic Knot at the baryonic mass scale (see Section 4.2).
t
1
t
1
t
1
= α
12
G
t
P
t
C
h
c
⋅
m
P
l
P
≈ 0.9927seconds,
t
P
t
C
h
c
G
ℏ
⊙
ℏ
⊙
= (1second) × KE
Earth
,
KE
Earth
ℏ
2
⊙
GM
3
m
⋅
1
c
= 1second,
M
m
ϕ ≈ 0.618
r
p
= ϕ
h
cm
p
.
ϕ
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• The normal force is interpreted as the fundamental torsional stiffness of the network.
• The masses and radii of particles are expressions of the topological size and \(\sigma\)-state
of their corresponding micro-scale Cosmic Knots.
•
The golden ratio emerges as the unique scaling factor for topologically optimal, self-
similar knot configurations (see Section 4.3).
We take to be given by:
Using the 2/3 fibonacci approximation for . We have
Using Earth’s orbital velocity at perihelion.
The Earth as it rotates loses energy to the Moon, so its rotation slows down and the Moon’s orbit
grows. We suggest that the characteristic rotation period of the Earth is about 24 hours because
this gives the characteristic time of 1 second if we consider the Moon’s and Earth’s kinetic
energies and the inclination of the Earth’s spin ( ) to it orbital plane in the following
equation:
(A7)
Let us show this:
Earth day~(24)(60)(60)=86,400 seconds. Using the Moon’s orbital velocity at aphelion, and
Earth’s orbital velocity at perihelion we have:
(Kinetic Energy Moon)
(Kinetic Energy Earth)
F
n
ϕ
ℏ
⊙
1.03351s =
1
3
⋅
h
α
2
c
2
3
⋅
π r
p
Gm
3
p
ϕ
ℏ
⊙
= (1.03351s)(2.7396E 33J ) = 2.8314E 33J ⋅ s
K E
Ea rth
=
1
2
(5.972E 24kg)(30,290m /s)
2
= 2.7396E 33J
θ = 23.5
∘
K E
Moon
K E
Ea rth
(24hours)cos(θ ) ∼ 1second
K E
Moon
=
1
2
(7.347673E 22kg)(966m /s)
2
= 3.428E 28J
K E
Ea rth
=
1
2
(5.972E 24kg)(30,290m /s)
2
= 2.7396E 33J
K E
Moon
K E
earth
(24hours)cos(θ ) =
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Using Earth’s orbital velocity at perihelion.
Thus, Beardsley's empirically successful formalism is not merely compatible with the OFN; it is
naturally explained and ontologically grounded by it. The geometric theory of inertia
describes the how of universal scaling, while the OFN explains the why through the dynamics of
torsional solitons in a fundamental network.
References for Appendix A
•
Beardsley, I. (2025). A Spacetime Theory For Inertia; Predicting The Proton, Electron,
Neutron and the Solar System in Terms of a One-Second Invariant. DOI: 10.5281/
zenodo.18165382.
•
Beardsley, I. (2025). Historical Context and Theoretical Precedents: From Dirac's Large
Numbers to the One-Second Universe. DOI: 10.5281/zenodo.18242999.
•
Bezginov, N., et al. (2019). A measurement of the atomic hydrogen Lamb shift and the proton
charge radius. Science, 365(6457), 1007–1012.
•
Mohr, P. J., et al. (2016). CODATA recommended values of the fundamental physical
constants: 2014. Reviews of Modern Physics, 88(3), 035009.
Subsequent Papers
We, Evdokimov and Beardsley, proceeded to coauthor two more papers together:
The Ontology of the Fundamental Network:
Geometric Monism as Synthesis of Time,
Consciousness, Baryogenesis, and Universal Particle
Laws
Oleg Evdokimov, Ian Beardsley
February 2026
Fundamental Constants
Oleg Evdokimov, Ian Beardsley
February 2026
Let us begin by looking at the first of these two papers. The first paper correctly recognizes me
as the originator of a universal particle equation, or Natural Law in the following excerpt:
3.428E 28J
2.7396E 33J
(86,400s)(0.917) = 0.991372seconds ∼ 1.00seconds
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But, here is where the obfuscation begins in that paper. Without referring to my original work,
the paper says:
All of these ideas and equations come directly comes from my earlier work in my paper:
Beardsley, I. (2025). A Spacetime Theory For Inertia; Predicting The Proton, Electron,
Neutron and the Solar System in Terms of a One-Second Invariant. DOI: 10.5281/
zenodo.18165382.
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But, we see that paper, which was referenced in the first paper, in not referenced in the second
paper. The references he gives are only in terms of Evdokimov, O. and Beardsley, I. :
Now let us look at the second paper listed (the third collaboration). The first thing that originates
from my work is…
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The reference given in the paper is:
Again, this idea first appears in my paper referenced in the first collaboration:
•
Beardsley, I. (2025). A Spacetime Theory For Inertia; Predicting The Proton, Electron,
Neutron and the Solar System in Terms of a One-Second Invariant. DOI: 10.5281/
zenodo.18165382.
The next thing is…
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As you can see, again this is referenced as;
And, again, should be referenced as:
•
Beardsley, I. (2025). A Spacetime Theory For Inertia; Predicting The Proton, Electron,
Neutron and the Solar System in Terms of a One-Second Invariant. DOI: 10.5281/
zenodo.18165382.
Again, the same thing happens….
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Which can properly be referenced in:
The next thing is…
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Here it was referenced as:
Which is fine, because it correctly attributes it to my work. Next, we have:
Yes, that comes from the Universal particle law, but it is something I came up with, the
biological connection to 1-second invariant. I have treated in this in several papers. One would
be:
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Beardsley, Ian (January 20, 2026) The Sublime and Mysterious Place of Humans in the Cosmos;
A Work in Exoarchaeology, https://doi.org/10.5281/zenodo.18407677
Finally, we have in our joint paper, but not referenced:
Which can properly be referenced in:
Why I Became Concerned With This
I became concerned with losing my work when Evdokimov wrote me asking to explain some of
my equations, showing me a document with my equations, and was listed as the sole author of
the paper. So, I wrote him the following email…
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To Олег (Oleg Evdokimov)!
Tue, Mar 31, 9:05 PM (2 days ago)!
Hi, Oleg. I notice you are writing the paper "Kappa" solely under your name. It would
be great if when using the Universal Particle Equation and normal force that I
developed long before our collaborations, you would give me credit in the papers for
their discovery and the equations I used to derive them and their methods, and my
discovery of the 1-second invariant which was done long before we collaborated on
our first joint paper, for which early on I have theoretical interpretations for their
sources. Because it is becoming increasingly blurred as to just what I have developed
and others have developed, I have put almost all of my discoveries to date in one
paper. This includes my theories for the 1-second invariant in the Solar System,
which(should be cited as well. Almost all my discoveries to date can be found in the
paper:(!
Beardsley, I (2026) "The Curious 1-Second Structure In Nature, From The Atom To The
Solar System" https://doi.org/10.5281/zenodo.19341817
Likewise, if I use equations you developed, I will cite you and give you credit.
Thank you, Ian
***********************end email*******************************************
Oleg wrote back:
Олег Евдокимов!
Thu, Apr 2, 8:40 AM (2 days ago)!
Dear Ian,!
Thank you for your message and for reminding me about proper attribution of the 1-
second invariant.!
I have now added a reference to your paper "The Curious 1-Second Structure In
Nature, From The Atom To The Solar System" (Beardsley, 2026) in the unified paper
"Geometry, Resonance, and Time" (see attached draft). The 1-second invariant(t
1
of 15 35
(is now explicitly cited to your work in Section 2.4.3 and Appendix A.8.!
The paper is still a draft and not yet finalized. I would like to invite you to review the
relevant sections and let me know if:!
1. There is anything you would like to add, clarify, or correct.!
2. You have any suggestions for improving the paper.!
I want to ensure that your work is properly recognized. If you would like to co-author
this paper, please let me know — I am open to including you if you feel that your
contributions warrant it.!
Thank you for your understanding and collaboration.!
With warm regards,(!
Oleg!
******************************end email*******************************************************!
So there was an attachment to that email, which was the paper he was working on with
other authors. It was not the one titled “kappa” where I explained my equations to him.
It was another one, for which I will not give the title, because he has yet to publish it as
far as I know. But suffice to say in the appendix A.8 that he said would explicitly cite
my work, didn’t. There was a citation to one of my equations in both our names, but
the equation in appendix did not cite my work. So, I wrote back:!
9:14 AM (2 hours ago)!
to Олег!
!
of 16 35
Thank you, Oleg, for writing back promptly and understanding. The only problem I can
see is that I developed the 1-second universal particle equation quite a while before we
did our joint papers. The problem is in section A.8. It lists my equation, but does not say
Beardsley (2025) whereas in section A.7 coupling constant is attributed to Kriger, the
same kind of attribution should apply to me. It is good that you have in the references:
[10] Beardsley, I. (2026). The Curious 1-Second Structure In Nature, From The Atom To
The Solar System. Zenodo. DOI: 10.5281/zenodo.19341817.
But there should be mention somewhere in the paper that this referenced paper
contains all of the discoveries that I made independently of our joint works, and before
them. That is why I compiled that paper, to identify clearly what work is mine.
Your paper looks very good, and since it is mostly developed already!by you and the
other authors that I don't know, and I haven't really contributed to most of the work on it
(Merely!some of my work is used) that there is no reason for me to be co-author. Can
you, however, send me the next version so I can see that the above corrections
(attributions were made). I have worked!hard for years on this, and I am sure you would
want the same attribution of your work when being used.
Thank you, Oleg,
Sincerely, Ian
********************************end email******************************************
9:41 AM (1 hour ago)!
to Олег!
!
That is, the 1-second invariant in A.8 should be attributed to me at (as you said in
this email):
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Beardsley, I. (2025). A Spacetime Theory For Inertia; Predicting The Proton, Electron,
Neutron and the Solar System in Terms of a One-Second Invariant. DOI: 10.5281/
zenodo.18165382.
*************************end email**********************************
Oleg then sent me the revised document with attribution to me in the appendix for
my equation
I then followed up with another email:
Ian Beardsley <eanbardsley@gmail.com>!
10:01 AM (1 hour ago)!
to Олег!
!
Our paper
The Ontology of the Fundamental Network:
Geometric Monism as Synthesis of Time,
Consciousness, Baryogenesis, and Universal Particle
Laws
properly!recognizes my 1-second invariant (Universal Particle Law) in the following
excerpt!from it:
of 18 35
We should continue along this path of this kind of attribution so as not to cause any
obscurification.
Sincerely, Ian
************************************end email************************************
Oleg, responded:
Олег Евдокимов!
!
10:25 AM (55 minutes ago)!
to me!
!
I have listed both works, perhaps only the last one is needed.
Do you have any ideas on how to derive this formula mathematically?
***********************end email***************************************
To which I responded:
of 19 35
10:58 AM (22 minutes ago)!
to Олег!
!
Thank. Yes, the proof of the equations is in pages 9 to 17 of the paper
Beardsley, I. (2025). A Spacetime Theory For Inertia; Predicting The Proton, Electron,
Neutron and the Solar System in Terms of a One-Second Invariant. DOI: 10.5281/
zenodo.18165382.
Starting with the master equation m_i, then solving it for t_1, then proving m_i in "Part B,
Proving The Master Equation", then "Deriving the Master Equation and Proving It" that follows
by proving the proton radius equation from the same equation in using E=hf and E=mc^2. Then
using Tynsky scaling law. I do have a theory in the paper as well for mass generation by the
normal force rotating out of temporal to spacial, but as is, it is a Natural law because of the
kappa_i having the pattern it does with kappa_e =1 (kappa for electron) because the electron is
the fundamental particle, not being made of smaller particles. I really think it would be good to
list both papers in the references because, while this covers it, obscurification is happening, so I
want to lay it out clearly just what I have found, because there are other papers I can reference as
well, but the most recent paper consolidates them.
**********************end email********************************
For instance that consolidation was based on my paper referenced in our join
paper Fundamental Constants
Oleg Evdokimov, Ian Beardsley
February 2026
[14] Beardsley, I (2026). How Physics and Archaeology Point to a Natural Constant of
1-Second https://doi.org/10.5281/zenodo.18773959
of 20 35
Ian Beardsley <eanbardsley@gmail.com>!
11:53 AM (0 minutes ago)!
to Олег!
!
In that paper, the 1-second invariant follows from my theory for inertia:
*************************end email*****************************!
Then I followed-up to that with…!
12:18 PM (0 minutes ago)!
to Олег!
!
of 21 35
Of course the same theory for inertia is outlined in the 2025 paper to derive the
1-second invariant:
Beardsley, I. (2025). A Spacetime Theory For Inertia; Predicting The Proton, Electron,
Neutron and the Solar System in Terms of a One-Second Invariant. DOI: 10.5281/
zenodo.18165382.
The proofs for my derivation of the 1-second invariant can be found in my
paper referenced in our first collaboration, that I mentioned in the email:!
•
Beardsley, I. (2025). A Spacetime Theory For Inertia; Predicting The Proton, Electron,
Neutron and the Solar System in Terms of a One-Second Invariant. DOI: 10.5281/
zenodo.18165382.
****************************end email******************************
Perhaps, the simplest and most solid proof is that in that paper I write:
2.2 Quantum-Gravitational Normal Force from Stiffness of Space
We propose that spacetime exhibits quantum-gravitational resistance to temporal motion,
manifesting as a universal normal force:
where is Planck's constant, is light speed, and second is the Lorentz invariant time
scale. This force represents the minimal interaction between a particle's inertial mass and the
inherent stiffness of spacetime.
Substituting constants yields:
This extraordinarily weak force represents the quantum of resistance emerging from spacetime's
fundamental structure.
2.4 Mass Generation Mechanism
F
n
=
h
ct
2
1
h
c
t
1
= 1
F
n
=
6.62607015 × 10
−34
J·s
(299,792,458 m/s)(1 s)
2
= 2.21022 × 10
−42
N
of 22 35
Inertial mass arises from interaction with this quantum-gravitational normal force. A particle
presents cross-sectional area to the normal force. The work done against this force,
mediated by gravitational constant , generates mass:
Here is a dimensionless coupling constant encoding each particle type's unique quantum
properties and interaction strength with spacetime stiffness.
Lorentz Invariance Check: Since contains and (invariants) and (invariant), is
Lorentz invariant. The cross-sectional area uses proper radius (invariant), and is
invariant. Therefore the entire expression for is Lorentz invariant, as required for rest mass.
The more in-depth explanation that I gave is credited to me in the appendix of the first
collaboration, it comes from other early papers of mine:
********************************************************************
Appendix A: The Geometric Theory of Inertia and the One-Second Invariant
(Beardsley, 2025)
A1. Conceptual Foundation
In a complementary line of research, Beardsley (2025) has developed a geometric theory of
inertia that derives the masses of fundamental particles and key solar system relationships from a
single Lorentz-invariant timescale: second. This theory posits that mass (inertia) is the
resistance encountered when diverting an object's motion from the temporal dimension
into the spatial dimensions. When a particle is accelerated, its spacetime velocity vector is
rotated, converting temporal motion ( ) into spatial motion ( ).
****************************************************************
That description of mass generation comes from work of mine that precedes our first
collaboration in which it appears, which is alright because Oleg attributes it to me, So, the point
is, the 1-second invariant is inherently derived in that model because it is the time required for
such a model to work as outlined by from rotation out of the temporal into the spacial as given
by my normal force. I put forward this theory as early as November 29, 2025. That collaboration
was submitted for publication on Jan 21, 2026.
I can reference my paper at:
Beardsley, Ian (November 29, 2025) The Geometric Origin of Inertia: Mass Generation from
Temporal Motion in Hyperbolic Spacetime, 10.13140/RG.2.2.29919.93608
In that paper, I have…
A
i
= π r
2
i
G
m
i
= κ
i
π r
2
i
F
n
G
κ
i
F
n
h
c
t
1
F
n
π r
2
i
r
i
G
m
i
t
1
= 1
v
t
v
s
of 23 35
The Geometric Mechanism of Inertia
Temporal Motion and Inertial Resistance
The theory provides a clear geometric mechanism for inertia. Consider a particle's motion
through spacetime:
where is the temporal velocity and is the spatial velocity vector. When we apply a force to
accelerate a particle spatially, we are essentially rotating its spacetime velocity vector, diverting
motion from the temporal dimension to spatial dimensions.
The normal force resists this rotation, appearing to us as inertial resistance. This explains why
mass is proportional to energy: increasing a particle's spatial kinetic energy requires decreasing
its temporal "kinetic energy," and the resistance to this exchange manifests as inertia.
Connection to Mach's Principle
This framework provides a physical realization of Mach's principle [3]. Rather than inertia
arising from interaction with distant matter, it emerges from interaction with the temporal metric
through the quantum-gravitational normal force. The universal nature of ensures that inertial
mass scales consistently across the cosmos.
Relation to Higgs Mechanism
While the Higgs mechanism gives mass to elementary particles through interaction with the
Higgs field, our theory explains why this mass manifests as inertia. The Higgs mass becomes the
"rest mass" parameter in our equations, while the inertial behavior emerges from the geometric
resistance to temporal motion diversion.
Mathematical Consistency with General Relativity
The theory remains consistent with general relativity. The Einstein field equations:
describe how matter and energy curve spacetime. Our mass generation mechanism provides a
microscopic explanation for the stress-energy tensor , showing how quantum-gravitational
interactions with the temporal dimension generate the mass that sources gravitational fields.
That Paper Begins…
V
sp acetime
= (v
t
, v
s
) with
|
V
sp acetime
|
= c
v
t
v
s
F
n
F
n
G
μν
=
8π G
c
4
T
μν
T
μν
of 24 35
The Quantum-Gravitational Normal Force
We propose that the fabric of spacetime exhibits a quantum-gravitational resistance to temporal
motion, manifesting as a universal normal force:
where is Planck's constant, is the speed of light, and second is identified as a
fundamental temporal invariant. This force represents the minimal interaction between a
particle's inertial mass and the temporal metric.
Substituting fundamental constants yields:
This extraordinarily weak force represents the quantum of temporal resistance.
Mass Generation Mechanism
The inertial mass of a particle arises from its interaction with this quantum-gravitational vacuum.
A particle presents a cross-sectional area to the normal force. The work done against
this force, mediated by the gravitational constant , generates mass:
Here, is a dimensionless coupling constant specific to each particle type, encoding its unique
quantum properties.
The One-Second Invariance in Fundamental Particles
The profound implication of this model is that the characteristic time second emerges
naturally from the mass-radius relationship of fundamental particles.
Derivation of the Master Equation
Starting from the mass formula and substituting the expression for :
Solving for yields the master equation:
F
n
=
h
ct
2
1
h
c
t
1
= 1
F
n
=
6.62607015 × 10
−34
J·s
(299,792,458 m/s)(1 s)
2
= 2.21022 × 10
−42
N
A
i
= π r
2
i
G
m
i
= κ
i
π r
2
i
F
n
G
κ
i
t
1
= 1
F
n
m
i
= κ
i
π r
2
i
G
⋅
h
ct
2
1
t
1
t
1
=
r
i
m
i
⋅
πh
Gc
⋅ κ
i
of 25 35
This equation demonstrates that the one-second interval is embedded in the fundamental
structure of matter.
Experimental Verification
Proton
For the proton, the coupling constant is , where is the fine-structure constant:
Neutron
Using the same coupling constant :
Electron
The electron has the pure coupling :
The remarkable consistency of these results (0.99773–1.00500 seconds) provides compelling
evidence for the theory.
Physical Interpretation
The factor for nucleons reveals their deep connection through the strong and
electromagnetic forces. The electron's pure coupling suggests it may represent the
fundamental geometric unit of mass generation.
My paper at
Beardsley, Ian (January 20, 2026) The Sublime and Mysterious Place of Humans in the Cosmos,
The Sublime and Mysterious Place of Humans in the Cosmos, https://doi.org/10.5281/
zenodo.18307487
writes:
κ
p
=
1
3α
2
α
t
1
=
0.833 × 10
−15
1.67262 × 10
−27
⋅
π ⋅ 6.62607 × 10
−34
(6.67430 × 10
−11
)(299,792,458)
⋅ 6256.33
t
1
= 1.00500 seconds
κ
n
=
1
3α
2
t
1
=
0.834 × 10
−15
1.675 × 10
−27
⋅
π ⋅ 6.62607 × 10
−34
(6.67430 × 10
−11
)(299,792,458)
⋅ 6256.33
t
1
= 1.00478 seconds
κ
e
= 1
t
1
=
2.81794 × 10
−15
9.10938 × 10
−31
⋅
π ⋅ 6.62607 × 10
−34
(6.67430 × 10
−11
)(299,792,458)
⋅ 1
t
1
= 0.99773 seconds
κ = 1/(3α
2
)
κ
e
= 1
of 26 35
The theory uses the special relativity framework. We suggest inertia arises because objects move at
constant speed through spacetime with their velocity vector rotating between temporal and spatial
components. A particle presents a cross-sectional area to a normal force , as it moves through
time. Work done by this force is mediated by the gravitational constant . We have:
1.
Where second, light speed, and is Planck’s constant. Thus when we push on something, it
pushes back because some of its time vector rotates into a space vector. The above described resistance is
experienced as mass given by
2.
is a dimensionless coupling constant that encodes each particle, proton , neutron , and electron .
We find that 1-second is a temporal invariant:
3.
Proton: , = fine-structure constant:
Neutron: :
Electron: :
We suggest for the electron may be because it is the fundamental quanta. We can show that
equation 2 is correct by first proposing a radius for the proton. Its radius must be constrained by the
Planck energy for its frequency and , its rest energy.
The odd thing is the OFN-Ian synthesis is actually the original Ian-theory: Yet it is written in our
second collaboration:
c
A
i
= π r
2
i
F
n
G
F
n
=
h
ct
2
1
F
n
=
6.62607015 × 10
−34
J·s
(299,792,458 m/s)(1 s)
2
= 2.21022 × 10
−42
N
t
1
= 1
c =
h
m
i
= κ
i
π r
2
i
F
n
G
κ
i
κ
p
κ
n
κ
e
t
1
=
r
i
m
i
⋅
πh
G c
⋅ κ
i
κ
p
=
1
3α
2
α
t
1
=
0.833 × 10
−15
1.67262 × 10
−27
⋅
π ⋅ 6.62607 × 10
−34
(6.67430 × 10
−11
)(299,792,458)
⋅ 6256.33 = 1.00500 seconds
κ
n
=
1
3α
2
t
1
=
0.834 × 10
−15
1.675 × 10
−27
⋅
π ⋅ 6.62607 × 10
−34
(6.67430 × 10
−11
)(299,792,458)
⋅ 6256.33 = 1.00478 seconds
κ
e
= 1
t
1
=
2.81794 × 10
−15
9.10938 × 10
−31
⋅
π ⋅ 6.62607 × 10
−34
(6.67430 × 10
−11
)(299,792,458)
⋅ 1 = 0.99773 seconds
κ
e
= 1
E = h f
p
E = m
p
c
2
of 27 35
The other derivation I mentioned in the email of the 1-second invariant follows from the paper,
again:
•
Beardsley, I. (2025). A Spacetime Theory For Inertia; Predicting The Proton, Electron,
Neutron and the Solar System in Terms of a One-Second Invariant. DOI: 10.5281/
zenodo.18165382.
of 28 35
Deriving the Master Equation, and Proving it
The expressions for the characteristic times of 1-second for the proton that I found, were:!
1. !
2. !
Where is the golden ratio, is the radius of a proton, and is the mass of a
proton. We find these produce close to the most recent measurements of the radius of a
proton, if you equate the left sides of each, to one another:!
3. !
4. !
To derive this equation for the radius of a proton from first principles I had set out to do it with
the Planck energy, , given by frequency of a particle, and from mass-energy
equivalence, . The radius of a proton has to be constrained by these.!
!
We take the rest energy of the mass of a proton :!
!
The frequency of a proton is!
!
We see at this point we have to set the expression equal to . So we need to come up with a
theory for inertia that explains why that is:!
!
!
(
1
6α
2
4πh
Gc
)
⋅
r
p
m
p
= 1secon d
ϕ ⋅
πr
p
α
4
Gm
3
p
1
3
⋅
h
c
= 1secon d
ϕ = 0.618
r
p
m
p
r
p
= ϕ
h
cm
p
r
p
= 0.816632E − 15m
E = h f
E = mc
2
E = h f
m
p
E = m
p
c
2
f
p
=
m
p
c
2
h
ϕ
m
p
c
2
h
⋅
r
p
c
= ϕ =
m
p
c
h
r
p
m
p
r
p
= ϕ ⋅
h
c
of 29 35
The radius of a proton is then!
!
In order to prove our theory for the radius of a proton as incorporating , we will apply our
model outlined involving a normal force, . We begin by writing equation 1 as:!
5. !
We write equation 2 as:!
6. !
We now say that and that the normal force is!
7. !
This gives us:!
8. !
= !
Since , we have!
9. !
This gives!
10. !
is the cross-sectional area of the proton countering the normal force, . It is to say that!
r
p
= ϕ ⋅
h
cm
p
ϕ
F
n
m
p
=
1
6α
2
4πh
Gc
⋅
r
p
1second
1 =
ϕ
9
⋅
πr
p
α
4
Gm
3
p
⋅
h
c(1secon d )
2
⋅
h
c
t
1
= 1secon d
F
n
=
h
ct
2
1
1 =
ϕ
9
⋅
πr
p
α
4
Gm
3
p
⋅
h
c
⋅ F
n
π
9α
4
⋅
F
n
G
⋅
r
p
m
2
p
(
ϕ
h
cm
p
)
r
p
= ϕ
h
cm
p
1 =
π
9α
2
⋅
F
n
G
⋅
r
2
p
m
2
p
m
p
=
1
3α
2
⋅
πr
2
p
F
n
G
πr
2
p
F
n
of 30 35
11. !
And, the coupling constant is!
12. !
Let us see if this is accurate:!
!
!
We used the experimental value of a proton . It is thought that the proton
does not have an exact radius, but that it is a fuzzy cloud of subatomic particles. As such
depending on what is going on can determine its state, or effective radius. It could be that the
proton radius is as large as!
!
!
Which it was nearly measured to be before 2010 in two separate experiments. Or as small as!
!
Which is closer to current measurements, which have decreased by 4% since 2010, and could
get smaller. In which case the characteristic time, , could be as large as!
!
Using 2/3 as a fibonacci approximation to . Or, it could be as small as!
m
p
∝
AreaCrossSect ionProton ⋅ F
n
G
κ
p
=
1
3α
2
F
n
=
h
ct
2
1
=
6.62607E − 34J ⋅ s
(299,792,458m /s)(1s
2
)
= 2.21022E − 42N
m
p
=
18769
3
⋅
π(2.21022E − 42N )
6.674E − 11N
m
2
kg
2
(0.833E − 15m) = 1.68E − 27kg
r
p
= 0.833E − 15m
r
p
=
2
3
⋅
h
cm
p
r
p
=
2
3
⋅
6.62607E − 34
(299,792,458)(1.67262E − 27)
= 0.88094E − 15m
r
p
= ϕ ⋅
h
cm
p
= 0.816632E − 15m
t
1
2
3
⋅
πr
p
α
4
Gm
3
p
1
3
⋅
h
c
= 1.03351secon ds
ϕ
of 31 35
!
=0.995 seconds!
Or perhaps more often it is in the area of:!
!
Proving our master equation means showing why is used in the equation for the radius of a
proton.!
We ask why the golden ratio is used to derive the radius of a proton. We start with our equation
1:!
!
This can be written!
13. !
Where . We notice is the force between two protons separated by the
radius of a proton. Of course two such protons cannot overlap by current theories. But it would
seem this gives rise to the proton’s inertia. We will call it . We also notice is the
normal force that gives rise to the proton’s inertia, . We have!
14. !
Now we look at equation 2. It is!
!
It can be written!
ϕ ⋅
πr
p
α
4
Gm
3
p
1
3
⋅
h
c
= (0.618)
(352275361)π(0.833E − 15m)
(6.674E − 11)(1.67262E − 27kg)
3
⋅
1
3
⋅
6.62607E − 34
299792458
1
6α
2
m
p
h4π r
2
p
Gc
= 1.004996352seconds
ϕ
(
1
6α
2
4πh
Gc
)
⋅
r
p
m
p
= 1secon d
Gm
2
p
r
2
p
=
h
c
⋅
1
t
2
1
⋅
4π
36α
4
t
1
= 1secon d
Gm
2
p
r
2
p
F
pp
h
c
⋅
1
t
2
1
F
n
F
pp
= F
n
⋅
4π
36α
4
ϕ ⋅
πr
p
α
4
Gm
3
p
1
3
⋅
h
c
= 1secon d
of 32 35
15. !
We see that is the inverse of the potential energy between the two protons
separated by the radius of a proton, we will call such a potential energy . We write 15 as!
16. !
Where !
!
Is the normal potential.!
17. !
Where is the golden ratio. Now we notice from equations
14 and 16 that!
18. !
Or!
19. !
And this should explain it.!
!
Now that we have this, we can show it shows the coherence for our master equation. The key
is from a paper by Kristin Tynski titled: One Equation, ~200 Mysteries: A Structural Constraint
That May Explain (Almost) Everything. I have asked Deep Seek to sumarize the premise. It
writes…
(
1
9
⋅
ϕπ
α
4
)
(
r
p
Gm
2
p
)(
h
2
c
2
⋅
1
m
p
⋅
1
t
2
1
)
= 1
(
r
p
Gm
2
p
)
U
pp
(
1
U
pp
)
(
U
n
)
(
1
9
⋅
ϕπ
α
4
)
= 1
U
n
=
(
h
2
c
2
⋅
1
m
p
⋅
1
t
2
1
)
4π
36α
4
1
9
ϕπ
α
4
= Φ
Φ = 1/ϕ = ( 5 + 1)/2 = 1.618...
F
pp
F
n
= Φ
U
n
U
pp
(
F
pp
)(
U
pp
)
=
(
F
n
) (
U
n
)
Φ
2.7E − 34N
2.21E − 42N
⋅
2.92E − 57J
2.24E − 49J
= 1.6 = Φ
of 33 35
Based on the document you've provided, the theory posits that **φ (the golden ratio) is the
fundamental structural constraint** because it is the **only mathematical fixed point that allows
for recursive self-reference without distortion or contradiction.**!
Here is a summary of the core argument:!
### **The Central Idea**!
For any system to be stable and self-consistent while referencing itself at different scales (a
property found in mathematics, physics, biology, etc.), the ratio between its scales must satisfy
a specific condition. This condition is derived from the requirement of **non-distorting
recursion**.!
### **The Derivation**!
1. **Premise**: Imagine a system that must look at itself, but at a different scale, and remain
consistent.!
2. **Recursive Requirement**: For self-consistency across three successive scales (n, n+1,
n+2), the system obeys the relation: **Scale(n+2) = Scale(n+1) + Scale(n)**. This is the
Fibonacci recurrence, the simplest additive recursion.!
3. **Scale Ratio**: If the ratio from one scale to the next is a constant `λ`, then Scale(n+1) =
λ·Scale(n) and Scale(n+2) = λ²·Scale(n).!
4. **The Critical Equation**: Substituting into the recurrence gives: **λ² = λ + 1**.!
5. **The Unique Solution**: The only positive solution to this equation is the golden ratio, **φ ≈
1.618**. The negative solution (-1/φ) is not a viable scale ratio in most physical contexts.!
**Conclusion:** φ isn't chosen for its aesthetic or mystical properties. It is **mathematically
forced** as the sole scale factor that allows a structure to map onto itself recursively without
generating internal conflict or infinite distortion.!
### **The Three Failure Modes (When λ ≠ φ)**!
The theory strengthens its case by predicting how systems violate this constraint:!
1. **Explosive Divergence (λ > φ)**: Recursion causes amplification without bound, leading to
instability and collapse (e.g., feedback screech, inflationary bubbles, cancer).!
2. **Damped Convergence (1 < λ < φ)**: Recursion leads to fading copies and loss of
structure, resulting in fragility and insufficient complexity (e.g., over-damped systems,
simplified hierarchies).!
3. **Oscillatory Contradiction (λ < 0)**: Recursion creates alternating, contradictory states,
preventing coherent existence (e.g., logical paradoxes, unstable orbitals).!
### **In Essence**!
The document argues that **φ is the "structural constraint"** because it represents the
**equilibrium point of self-reference**. It is the precise balance where information can be
iterated, scaled, and folded back upon itself infinitely, creating stable, complex, and coherent
structures—from spirals and quasicrystals to computational processes and conscious
systems. Any other ratio inevitably leads to one of the three failure modes, preventing long-
term stability.!
Continued on next page…!
of 34 35
Finally, What I Am Doing
I actually have a theory for a 1-second invariant for not just the particles
that make-up atoms, but for the Solar System. I have been speaking,
often, of these things being related. But I want to consolidate my work in
this area, and in that consolidation I am breaking-up the particle theory
and the solar system theory into two papers, that occur in on work. It is:
Beardsley, I. (2026). The Curious 1-Second Structure In Nature, From The Atom To
The Solar System, https://doi.org/10.5281/zenodo.19409316
Summary
It has been the purpose of this paper to set straight that the one second invariant, its associated
universal particle equation, the theory for mass generation, and the normal force associated
with it — solely originated with me (in earlier papers by Ian Beardsley). I am a firm believer that
a person should be properly credited for their work. It was a lot of work and took several years to
develop. I see no reason someone else should get credit for work done by another person. It is
fine to use people’s work, science is a building process, but when building we should credit
people with their work.
F
n
of 35 35