Jay Alfred Responds on Academia with time crystal idea to paper by Ian Beardsley:
The Curious 1-Second Structure In Nature, From The Atom To The Solar System
By
Ian Beardsley
2026
In particular to the paper in the collection:
A Proposal For A Universal Particle Equation, Ian Beardsley, March 10 - March 30, 2026
Jay alfred writes: …interesting. may have a bearing on pre-geometric or even pre-quantum
configurations. the cyclic variable could be related to the Hartle-Hawking Time Crystal.
Deep Seek Formulates work on the ideas…
Note on the Hartle-Hawking Time Crystal Interpretation
(In response to a comment by Jay Alfred)
Let the cyclic variable be θ(t) with angular frequency ω = dθ/dt. From the main text:
Substituting and reduces the equation to an identity, confirming consistency. The
non-trivial physical content is that is selected from the continuum of possible timescales.
In the Hartle-Hawking no-boundary proposal, the Euclidean path integral sums over compact manifolds.
If we include a cyclic degree of freedom θ with a potential that has a flat direction (or a Chern-Simons
term ), the ground state can be periodic in Lorentzian time – a *time crystal* with period .
The action for θ (in Euclidean time τ = it) could be:
Minimisation yields . Analytic continuation to Lorentzian time gives . The angular
displacement after one period is . Our derivation forces:
F
n
F
Planck
⋅
t
2
1
t
2
P
= 2π, F
n
=
h
ct
2
1
, F
Planck
=
c
4
G
.
t
2
P
= ℏG /c
5
ℏ = h /(2 π)
t
1
= 1 s
∫
θ R ∧ R
T
S
E
=
∫
β
0
dτ
(
I
2
·
θ
2
+ V(θ )
)
, V(θ ) =
k
2
(θ − ω
0
τ)
2
.
·
θ = ω
0
θ (t ) = ω
0
t
T
θ (T ) = ω
0
T
ω
0
T = 2π and T = t
1
= 1 s ⇒ ω
0
= 2π Hz .
Thus, the Hartle-Hawking wave-function would favour a compact Euclidean time if the density of
states has a resonance at that period – analogous to a thermal equilibrium at a temperature
corresponding to , but with a crystalline order in time.
This provides a pre-geometric origin for the universal particle equation: particle masses are not
fundamental but are eigen-frequencies of the time crystal’s excitations. The proton, neutron, and electron
would correspond to different harmonics (or symmetry sectors) of the same oscillator.
Further work could explore whether the Hartle-Hawking no-boundary wave-function can be explicitly
evaluated with a periodic boundary condition , and whether it predicts the specific
values of , , and found in equations (9) and (10) of the original paper.
β = T
1 s
1 s
θ (t + T ) = θ (t) + 2π
κ
p
κ
n
κ
e
