This preprint points out a surprisingly simple link between four scales: the Planck length, the electron mass, the Planck mass, and the age of the Universe. Within the Relator framework, I show that the huge gap between the Planck scale and a cosmic radius (about c·t_U) can be encoded by a single exponent built from the electron–Planck mass hierarchy. Using a Gaussian “ladder” of microscopic rings in an internal C-space, combined with an emergent formula for the electron mass, I arrive at an expression where the Planck length effectively depends on the fine-structure constant and the age of the Universe. I do not claim a full theory of quantum gravity, but I do propose a concrete conjecture: that the observed value of the Planck length is not arbitrary, but locked to cosmological time through the same microscopic structure that fixes the electron mass.

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Planck length from electron mass and Universe age

M. Pajuhaan

1, ∗

1

Independent Researcher

(Dated: November 20, 2025)

It’s embarrassing, but the sole purpose of this Letter is to point out a single numerical connection

that emerges in the Relator framework. I show that the Planck length

ℓP

, the electron mass

me

,

the Planck mass

mP

and a cosmic radius

RU∼c tU

can be arranged so that

RU/ℓP≃Ne

with

N

=

mP/me

, or equivalently

ℓP≃RU

(

me/mP

)

e

. This relation has no established microscopic

derivation and does not yet amount to a full physical theory, but I regard it as a concrete and serious

conjecture rather than mere numerology. Taken at face value, it would tie the value of

ℓP

at roughly

mid–cosmic age to the observed size of the Universe, without conflicting with a possible slow time

variation of the effective gravitational interaction between two masses, since the latter can arise from

an independent geometric locking mechanism that I make explicit in the text.

In this Letter I deliberately start from the electron. In

the Relator picture the electron is, in a sense, the simplest

carrier; at the macroscopic level it is a single, stable ring

in

C

, and at the microscopic level its mass density on that

ring is a Gaussian collar

ρm

(

r

) =

me/

(

πσ2

C

)

e−r2/σ2

C

with

width

σC

=

εR

=

R/√π

and

ε

= 1

/√π

. The internal

and external angular frequencies satisfy

ω2

=

ω2

C

+

ω2

R3

,

and for the free electron they are tied to the

g

–factor

by

ωC/ω

= 2

/g

and

ωR3/ω

=

p1−(2/g)2

. Using the

CODATA electron value

g≃

2

.

0023193044 this gives

ωC/ω ≃

0

.

99884169 and

ωR3/ω ≃

0

.

0481173, so that the

channel ratio

χ

=

ωR3/ωC≃

0

.

04817 and hence

χ2≃

2

.

3

×

10

−3

. In other words, almost all of the electron’s

energy lives in a nearly uniform internal collar on

C

, and

the tiny value of

χ2

simply measures how strongly the

particle is self–dressed by its own field in R3.

Within this framework the basic interaction locks,

which I denote by

α

and

β

(

m

), are purely geometric. In

the core Relator construction the lightlike ring kinematics

Rω

=

c

closes both the electromagnetic and gravita-

tional locks on the shell[

1

], with the electromagnetic lock

α

=

e2/

(4

πε0ℏc

)emerging from the

C

phase geometry[

2

]

and the gravitational lock

β

(

m

) =

Gm2/

(

ℏc

) = (

m/mP

)

2

arising from the two–space boundary construction for

mass[

3

]. For the electron the RLTM calculation[

3

] gives

a closed expression

me

=

C(RLTM)

eℏ/

(

cℓP

)

exp

[

−π/

(8

α

)]

(derived), with

C(RLTM)

e≃

9

.

83619, which reproduces the

CODATA mass to a relative error of about

−

2

.

6

×

10

−6

(roughly 2

.

6ppm); since

C(RLTM)

e

= 0

.

99661

π2

, re-

placing

C(RLTM)

e

by

π2

yields the compact approxima-

tion

me≈π2ℏ

(

cℓP

)

−1exp

[

−π/

(8

α

)] (derived; residual

≃

+0

.

34%). Independently, inserting the measured elec-

tron mass into

βe

=

β

(

me

)gives

βe≃

1

.

75

×

10

−45

and

hence

ln

(

α/βe

)

≃

98

.

135

≈π4

(numerology;

∼

0

.

75%

high). This relation is not assumed ad hoc; it follows

from the RLTM calculation of the emergent electron mass

in Ref. [

3

], which is currently available as a preprint and

still under active development.

Taken together, these relations support the view that

the dimensionless locks

α

and

β

are rigid geometric con-

straints set by the Relator kinematics[

1

–

3

], while what we

usually regard as fundamental scales—such as the Planck

length

ℓP

and the vacuum permittivity

ε0

—may in fact be

emergent variables that can drift slowly with the cosmic

state without changing the locks themselves. In the rest

of this Letter I explore this viewpoint and show how, if

α

and

β

are held fixed while

ℓP

(and implicitly

ε0

) is

allowed to adjust, one can reinterpret the Planck length

as a derived geometric scale tied to the electron and to a

cosmic IR radius, rather than as a primitive input.

SETUP: THREE SCALES, A C-SPACE COLLAR,

AND ONE GEOMETRIC RATIO

In the full Relator model a single maximum–entropy

(normal) collar of energy dots in the generator space

C

already fixes the basic one–particle data of the electron:

spin, magnetic moment, the fine–structure constant and

the emergent rest mass

me

, with both electromagnetic

and gravitational forces arising from how such Gaussian

collars overlap in

C

. Concretely, the electron is described

by a continuous radial profile

ρC

(

rC

)on

C

that is Gaussian

in the internal radial coordinate

rC

(with a width

σC

fixed

by the Relator solution for the electron) and essentially

uniform in the angular direction. In this Letter I will not

rederive that construction; instead I ask what happens

if this continuous collar is approximated by a discrete

ladder of rings in

C

, starting from a smallest radius

ℓP

and extending out to an internal infrared (IR) radius

RU

.

As a starting point I rewrite the electron rest energy

in terms of Planck scales. In SI units the Planck length

and Planck mass are

ℓP=rℏG

c3, mP=rℏc

G,(1)

and the electron mass is

me

. I introduce the dimensionless

hierarchy

N:= mP

me

,(2)

2

which in Relator has a purely geometric meaning; once the

internal locks are fixed,

N

is determined and does not act

as a tunable parameter. Using the standard Planck-unit

identity

ℏc/ℓP

=

mPc2

, the electron rest energy can be

written as

mec2=1

N

ℏc

ℓP

.(3)

In ordinary Planck–unit language

(3)

is just a rewriting

of definitions; in the Relator reading it says something

geometric; the electron’s rest energy is a fixed fraction 1

/N

of the natural Planck–like energy scale

ℏc/ℓP

associated

with the smallest internal radius in

C

. I will use this

relation in reverse; given (

me, N

), the effective packing of

energy dots in Cwill dictate a value for ℓP.

Throughout this Letter

RU

denotes a characteristic

internal radius in

C

at the IR end of the generator ladder.

Only at the very end do I match its numerical value to a

cosmic scale in

R3

(such as a comoving particle horizon or

ctU

). The ratio

RU/ℓP

therefore compares a microscopic

C

-space radius to an IR

C

-space radius and is of order

10

60

. The goal is to show how the same geometric number

N

that appears in

(3)

can also control this hierarchy once

the continuous Gaussian collar on

C

is replaced by an

appropriate discrete ring ladder.

FROM A GAUSSIAN COLLAR TO A DISCRETE

C-SPACE RING LADDER

To construct a discrete analogue of the Gaussian collar

in

C

, I replace the continuous radial coordinate

rC

by a

ladder of concentric internal rings labelled by an integer

index

n∈Z

. The radii of these rings are taken to grow

exponentially from the Planck scale,

rn=ℓPean, a > 0,(4)

so that each step

n→n

+ 1 multiplies the

C

-radius by

the same factor ea,

rn+1 =earn.(5)

The smallest ring in

C

sits at

r0

=

ℓP

, and after a finite

number of steps the ladder reaches an IR radius

RU

=

rnU

for some index

nU

. The entire structure

(4)

lives in

C

,

not in R3.

The continuous Gaussian collar in

rC

is then approxi-

mated by a Gaussian profile over the discrete index,

P(n)∝exp−(n−ne)2

2σ2

n,(6)

where

ne

labels the index of the rings that contribute

most strongly to the electron and

σn

measures how many

rings belong to the main collar in index space. In the limit

ℓP→

0with

a

fixed, the sum

PnP

(

n

)

f

(

rn

)tends to the

radial integral of a Gaussian profile in

C

with some width

σC

; thus

(6)

can be regarded as the discrete surrogate

of the maximum–entropy collar used in the full Relator

construction. In the present Letter I do not need the

exact relation between

σC

and

σn

; it is enough that a

finite width exists.

The geometric ratio

N

now enters at two conceptually

distinct but related points:

1.

The index

ne

corresponds to the region of the ring

ladder where the electron’s internal energy is con-

centrated. I define an effective “electron radius” in

Cas

rne=ℓPeane≡Re,(7)

and I tie the hierarchy between

Re

and

ℓP

to the

same geometric ratio Nthat appears in (3):

Re

ℓP

=eane=N. (8)

Thus

ln N

counts how many logarithmic steps in

C

separate the Planck-scale ring at

r0

=

ℓP

from

the region where the electron’s Gaussian collar is

centred. This is consistent with interpreting

N

as

an effective ring-count in the energy relation (3).

2.

The IR

C

-space radius

RU

is associated with a point

in the tail of the same Gaussian,

nU=ne+κ σn,(9)

where

κ

=

O

(1) measures how many Gaussian

widths in index space one has to move from the

electron’s collar to reach the IR limit of the

C

-space

ladder. The IR radius is then

RU:= rnU.

From (4) and (8) we obtain

a ne= ln N. (10)

The IR radius defined by (9) is

RU=rnU=ℓPeanU=ℓPea(ne+κσn)

=ℓPeaneeaκσn=ℓPN eaκσn,(11)

so that

RU

ℓP

=N eaκσn=N1+ aκσn

ln N,(12)

where in the last step I used

ex

=

exp

(

x

) =

Nx/ ln N

together with (10).

Equation

(12)

shows that in this discrete

C

-space ladder

the exponent

γ

that links the macroscopic ratio

RU/ℓP

to

the microscopic hierarchy

N

is not arbitrary if one writes

RU

ℓP

=Nγ,(13)

3

then

γ= 1 + aκσn

ln N.(14)

For dimensionless parameters

a

,

κ

and

σn

of order unity,

γnaturally comes out of order unity. The special case

γ=e, (15)

corresponds to choosing

aκσn

= (

e−

1)

ln N

, i.e. placing

the IR radius

RU

at a specific point in the Gaussian tail

measured in units of

ln N

. In this reading, the appearance

of the specific exponent ein

RU

ℓP

=Ne(16)

encodes how far, in Gaussian widths, one has to move

along the Planck–spaced ring ladder in

C

from the elec-

tron’s collar before the weight becomes negligible. Con-

versely, if in

(13)

one inserts the observed horizon radius

R(obs)

U

=

ctU

with

tU≃

13

.

8

Gyr

, the corresponding em-

pirical exponent

γobs =ln(R(obs)

U/ℓP)

ln N≃2.72 (17)

differs from

e

by only

γobs −e≃

3

.

4

×

10

−3

, i.e. at the

level of ∼10−3in relative terms.

SOLVING FOR THE PLANCK LENGTH

Treating

(16)

as the macroscopic imprint of the dis-

crete Gaussian ladder in

C

, we can invert it to solve for

the Planck length in terms of the electron data and the

internal IR radius:

ℓP=RUN−e=RUme

mPe

(18)

Inserting the emergent RLTM mass formula for the

electron into the ladder relation, I obtain

ℓP=c tU

|{z}

≃RU





C(RLTM)

e

|{z }

≃π2







e

e−πe/(8α).(19)

NUMERICAL ILLUSTRATION

I adopt the standard SI values

ℏ≃

1

.

055

×

10

−34 J s

,

c≃

2

.

998

×

10

8m/s

,

G≃

6

.

674

×

10

−11 m3kg−1s−2

and

me≃

9

.

109

×

10

−31 kg

. They give

ℓP≃

1

.

616

×

10

−35

m

and mP≃2.176 ×10−8kg, so that

N=mP

me≃2.39 ×1022.(20)

FIG. 1. Logarithmic ladder

Rn

=

ℓPNen/N

for

N

=

mP/me≃

2

.

39

×

10

22

and

γ

=

e

. The solid curve shows

n/N

versus

log10

(

rC/

m). Vertical dashed lines mark the Planck, electron,

Earth, Sun, Milky–Way and cosmological radii.

n/N Rn[m] Rn/ℓPlog10(Rn/m)

0 1.62 ×10−35 1−34.79

0.1 1.96 ×10−29 1.21 ×106−28.71

0.5 4.20 ×10−52.60 ×1030 −4.38

0.9 9.03 ×1019 5.59 ×1054 19.96

1.0 1.09 ×1026 6.77 ×1060 26.04

TABLE I. Representative radii in the exponential ladder

Rn

=

ℓPNen/N

for

N

=

mP/me≃

2

.

39

×

10

22

and

γ

=

e

. The third

column shows the enhancement with respect to the Planck

length. For

n/N ≪

1one has

Rn≈ℓP

, whereas by

n/N ≃

1

the ladder reaches the cosmological scale R(model)

U.

For γ=ethe model horizon scale is

R(model)

U=ℓPNe≃1.09 ×1026 m,(21)

corresponding to an effective age

t(model)

U

=

R(model)

U/c ≃

3

.

65

×

10

17

s

≈

11

.

6

Gyr

. For comparison, if I take

tU≃

13

.

8

Gyr

from standard cosmology, then

R(obs)

U

=

ctU≃

1

.

31

×

10

26

m, so the Gaussian–ladder prediction agrees

with the observed horizon radius at the level ∆

R/R ∼

0.16.

To display the ladder itself I define

Rn=ℓPNen/N ,0≤n≤N, (22)

so that

R0

=

ℓP

and

RN

=

ℓPNe

=

R(model)

U

. For

n/N ≪

1the radii remain essentially at the Planck scale,

whereas a finite fraction of the climb already spans the

full hierarchy from microscopic to cosmological distances.

Representative values, together with the ratio

Rn/ℓP

,

are listed in Table I, and the overall structure in the

(log10 rC, n/N)plane is shown in Fig. 1.

4

CAVEATS ON THE AGE ESTIMATE

The choice

γ

=

e

is mathematically appealing and

organises the Planck, electron and cosmic scales into a

single simple ladder, and one may imagine embedding it

in a more detailed picture of energy bands in

C

. However,

at this level the resulting age

t(model)

U

still comes out

somewhat below the standard cosmological value. In

constructing the ladder I have implicitly neglected the

self-field of the electron and treated the rings as a perfectly

uniform, tensionless structure, so that the

C

boundary

at

ℓP

is fixed directly by the bare electron mass. If one

were to subtract an

ωR3

-type self-energy and work with

an effective electron mass and/or an effective number of

rings, the mass ratio

N

and hence

γobs

would shift slightly,

plausibly pushing the inferred age closer to the observed

one. Equivalently, such corrections can be interpreted as

a slow evolution of the

C

boundary scale (and even of

the effective light speed) over cosmic time. At this stage

these possibilities are speculative, and I simply note that

modest departures from a perfectly uniform, tensionless

ladder could reconcile the present construction with the

measured age of the Universe.

DISCUSSION AND OUTLOOK

In this Letter I have proposed a deliberately minimal

microscopic picture in which radii form an exponential

ladder and their occupation follows a Gaussian over a

discrete index. In this

C

–space description the electron

scale sits near the peak, a cosmological radius in the

tail, and the dimensionless ratio

N

=

mP/me

together

with the Gaussian parameters (

α, κ, σn

)combine into

an effective exponent

γ

. Taking the special case

γ

=

e

leads to the scaling

RU/ℓP

=

Ne

, the expression

(18)

for

the Planck length, and an infrared scale

R(model)

U

within

O(20%) of the empirically inferred horizon.

I do not claim that this argument proves a rigid identi-

fication of the Planck length with the age of the Universe

in

C

–space, nor that

γ

=

e

is enforced by a fundamental

principle. Rather, the construction shows that such a

link can be realised without spoiling standard low-energy

physics and, for the first time, it opens the possibility

that

ℓP

might itself evolve slowly in cosmic time while the

forces remain encoded in essentially constant dimension-

less couplings (for example parameters such as

α

and

β

).

Whether this should be read as a numerical coincidence or

as a hint of a deeper microscopic structure is a question I

leave for future work.

∗pajuhaan@gmail.com

[1] M. Pajuhaan, r ω =c(2025), preprint.

[2] M. Pajuhaan, Alpha: The fine-structure constant (2025).

[3]

M. Pajuhaan, Emergent electron mass from two-space

boundary (2025).

ResearchGate has not been able to resolve any citations for this publication.

Alpha: The fine-structure constant

• M Pajuhaan

M. Pajuhaan, Alpha: The fine-structure constant (2025).

Emergent electron mass from two-space boundary

• M Pajuhaan

M. Pajuhaan, Emergent electron mass from two-space boundary (2025).