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Emergent Spacetime and Quantum Fields from
Coexisting Time Arrows: An Axiomatic Framework
Krzysztof Banaszewski
krzysztof@banaszewski.pl
ORCID: 0009-0003-7496-5083
November 3, 2026
Abstract
We propose an axiomatic framework, termed Coexisting Time Arrows (CTA),
in which spacetime geometry, matter, energy, and quantum phenomena are investi-
gated as potential emergent consequences of the local coexistence of two oppositely
oriented temporal sectors,τ + andτ −.
The framework extends the cosmological motivation of the CPT-symmetric uni-
verse proposed by Boyle, Finn, and Turok (Phys. Rev. Lett. 121, 251301, 2018)
toward a local field-theoretic description in which temporal degrees of freedom are
treated as fundamental, while spatial geometry is regarded as an induced structure.
A central mathematical result concerns the construction of a four-dimensional
Lorentzian metric from temporal scalar fields. We demonstrate that a bilinear
metric constructed from a single pair of scalar-field gradients has rank at most two
and is therefore necessarily degenerate in four dimensions.
We introduce a multimode generalization and prove that at least three pairs of
temporal modes are necessary to obtain Lorentzian signature (−+ ++) within the
proposed metric ansatz.
An explicit construction with three mode pairs reproduces the Minkowski metric.
The resulting geometry admits an interpretation as an induced metric in an auxiliary
six-dimensional flat space with signature (3,3).
The framework further proposes that the gravitational effects attributed to dark
matter may originate from a CPT-symmetric baryonic sector, while cosmic accel-
eration may admit an interpretation based on temporal geometry rather than a
fundamental dark-energy component.
Electric charge is hypothesized to arise from local asymmetry between oppositely
oriented temporal sectors, and quantum behavior is explored through an operator
description of temporal degrees of freedom.
The representation of general curved spacetimes, recovery of Einstein dynamics,
derivation of gauge interactions, and quantitative cosmological predictions remain
open problems.
CTA thus provides a mathematically explicit realization of flat Lorentzian ge-
ometry from temporal fields, together with a broader axiomatic research program
for investigating the possible emergence of physical structure from coexisting time
arrows.
The multimode construction establishes that three pairs of temporal scalar fields
are the minimum required to realize a four-dimensional Lorentzian metric within
the proposed bilinear ansatz.
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An explicit configuration reproduces Minkowski spacetime and admits an inter-
pretation as an induced metric in an auxiliary flat space of signature (3,3).
Beyond this geometric result, CTA proposes a unified research framework in
which dark matter, cosmic acceleration, electric charge, and quantum phenomena
may be investigated as consequences of the interaction between oppositely oriented
temporal sectors.
The extension to curved spacetime, the recovery of gravitational dynamics, and
the derivation of quantitative observational predictions constitute the principal di-
rections for further development.
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Contents
1 Introduction 7
2 Hypothesis of Electromagnetic Decoupling Between CPT-Symmetric
Sectors 9
2.1 Degeneracy of the original two-field construction . . . . . . . . . . . . . . . 9
2.1.1 Scope and Limitations of the Construction . . . . . . . . . . . . . . 10
2.2 A multimode extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Minimal number of temporal modes . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Explicit Minkowski construction . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Geometric interpretation as an induced metric . . . . . . . . . . . . . . . . 11
2.6 Curvature restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.7 Projected gravitational field equations . . . . . . . . . . . . . . . . . . . . 12
2.8 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.9 Conclusions and Implications for CTA . . . . . . . . . . . . . . . . . . . . 13
3 Operator of Emergence 13
3.1 Emergence of Geometry and Structure . . . . . . . . . . . . . . . . . . . . 14
4 Emergence of Fundamental Phenomena 14
4.1 Emergence of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 Formal Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Cosmological Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.4 Emergence of Entropy from Time . . . . . . . . . . . . . . . . . . . . . . . 14
4.4.1 Time as the Foundation of Possible States . . . . . . . . . . . . . . 15
4.4.2 Inevitable Emergence of Entropy . . . . . . . . . . . . . . . . . . . 15
4.4.3 No Special Initial Condition . . . . . . . . . . . . . . . . . . . . . . 15
4.4.4 Formal Ontological Sequence . . . . . . . . . . . . . . . . . . . . . . 15
4.4.5 Cosmological and Philosophical Implications . . . . . . . . . . . . . 15
5 Unified Baryonic Matter 15
6 Effective Energy-Momentum Tensor 16
7 Emergent Einstein Field Equations 16
8 Interpretation of ”Dark Matter” 16
9 No Big Bang Singularity 16
9.1 Lack of Real Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
10 Minimal Field Equations in 3 + 1 D 16
11 Dynamics 18
11.1 Candidate Dynamical Action . . . . . . . . . . . . . . . . . . . . . . . . . 18
11.2 Metric Variation and Its Limitations . . . . . . . . . . . . . . . . . . . . . 19
11.3 Temporal-Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
11.4 Relation to General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . 19
11.5 Status of the Dynamical Framework . . . . . . . . . . . . . . . . . . . . . . 20
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11.6 Variation with respect to the metric . . . . . . . . . . . . . . . . . . . . . . 20
11.7 Variation with respect to the time–arrow scalars . . . . . . . . . . . . . . . 20
11.8 Einstein limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
11.9 Consistency Check with the 1+1 Toy Model . . . . . . . . . . . . . . . . . 20
11.10Covariant Conservation of the Effective Source . . . . . . . . . . . . . . . . 22
12 Observational Signature Map 23
13 Emergence of the Speed of Light 24
13.1 Absence of a Fundamental Metric . . . . . . . . . . . . . . . . . . . . . . . 24
13.2 Geometric Origin ofc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13.3 Dependence on Global Temporal Configuration . . . . . . . . . . . . . . . 25
13.4 Formal Ontological Statement . . . . . . . . . . . . . . . . . . . . . . . . . 25
13.5 Philosophical and Physical Implications . . . . . . . . . . . . . . . . . . . . 25
13.6 Numerical Value ofc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
14 Perception of Cosmic Expansion and Absence of Dark Energy 25
14.1 Emergent Global Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 25
14.2 Local Temporal Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 25
14.3 Apparent Expansion as a Local Effect . . . . . . . . . . . . . . . . . . . . . 26
14.4 No Need for Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
14.5 Formal Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
14.6 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
15 Absence of Electromagnetic Interactions Between CPT-Symmetric Sec-
tors 26
16 Quantum Nature of Time and Emergence of Quantum Mechanics 27
16.1 Quantum Operators for Time Arrows . . . . . . . . . . . . . . . . . . . . . 27
16.2 Local Fluctuations and Superpositions . . . . . . . . . . . . . . . . . . . . 28
16.3 Unified View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
16.4 Explicit Operator Structure and Field Interpretation . . . . . . . . . . . . 28
17 Formal Definition of Sector Interactions 29
18 Formal Argument Against Electromagnetic Interaction Between CPT-
Symmetric Sectors 29
19 Emergent Charge from Local Time Asymmetry 30
20 Quantum Field Formalism of Temporal Operators 31
20.1 Promotion of Temporal Flows to Quantum Fields . . . . . . . . . . . . . . 31
20.2 Canonical Structure and Commutation Relations . . . . . . . . . . . . . . 32
20.3 Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
20.4 Quantum State Space and Evolution . . . . . . . . . . . . . . . . . . . . . 32
20.5 Path Integral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 32
20.6 Quantum Uncertainty and Spectral Decomposition . . . . . . . . . . . . . 33
20.7 Quantum Observables and Measurements . . . . . . . . . . . . . . . . . . . 33
20.8 Quantum-Classical Transition . . . . . . . . . . . . . . . . . . . . . . . . . 33
20.9 Summary of Quantum Formalism . . . . . . . . . . . . . . . . . . . . . . . 33
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21 Discussion 33
22 New Research Postulates 34
23 Open Questions and Validation Strategy 35
24 Emergence of Standard Model Gauge Symmetries from Temporal-Topological
Oscillations 37
24.1 Oscillatory Configuration Space . . . . . . . . . . . . . . . . . . . . . . . . 37
24.2 Internal Automorphism Group . . . . . . . . . . . . . . . . . . . . . . . . . 37
24.3 Local Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 38
24.4 Emergence of SM Gauge Groups . . . . . . . . . . . . . . . . . . . . . . . . 38
24.5 Explicit Example: U(1) Oscillatory Mode and Charge Emergence . . . . . 38
24.6 Explicit Example: SU(2) Oscillatory Modes and Weak Isospin . . . . . . . 39
24.7 Explicit Example: SU(3) Oscillatory Modes and Color Charge . . . . . . . 40
24.8 Topological Charges and Particle Interpretation . . . . . . . . . . . . . . . 42
24.9 Formal Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
25 Emergence of Standard Model Particles from Temporal-Topological Struc-
tures 42
25.1 Particles as Localized Oscillations . . . . . . . . . . . . . . . . . . . . . . . 42
25.2 Geometric Interpretation of Particle Properties . . . . . . . . . . . . . . . . 42
25.3 Groups of Symmetry and Gauge Fields . . . . . . . . . . . . . . . . . . . . 43
25.4 Derivation of Fundamental Constants . . . . . . . . . . . . . . . . . . . . . 43
25.5 Research Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
25.6 Potential Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
26 Predictions and Experimental Tests 44
26.1 Galactic “dark matter” as a geometric effect . . . . . . . . . . . . . . . . . 45
26.2 High-velocity galaxies in cosmic voids . . . . . . . . . . . . . . . . . . . . . 45
26.3 Massive neutron stars without collapse . . . . . . . . . . . . . . . . . . . . 45
26.4 No true acceleration of cosmic expansion . . . . . . . . . . . . . . . . . . . 45
26.5 No Big Bang singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
26.6 General relativity as a large-scale limit . . . . . . . . . . . . . . . . . . . . 46
26.7 Microscale time-asymmetry in quantum processes . . . . . . . . . . . . . . 46
26.8 Enhanced CP and T violations . . . . . . . . . . . . . . . . . . . . . . . . 46
26.9 Absence of detectable dark matter particles . . . . . . . . . . . . . . . . . 46
26.10Physical Status and Observational Constraints . . . . . . . . . . . . . . . . 47
27 Conclusions 48
A Axiomatic Foundations 56
A.1 Configuration Space and Transformation Operators . . . . . . . . . . . . . 56
A.2 Welfare Functional and Ordering . . . . . . . . . . . . . . . . . . . . . . . 57
A.3 Temporal Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
A.4 The Primary Entity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
A.5 Systemic Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
A.6 Sector Reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
A.7 Local Realization of Global Conjugation . . . . . . . . . . . . . . . . . . . 59
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A.8 Global Teleology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
B Emergence of the Negative Temporal Sector 60
C Idealized Post-Biological Limit 62
D Physical Status of the CTA Framework 63
D.1 Level I: Mathematical Definitions . . . . . . . . . . . . . . . . . . . . . . . 64
D.2 Level II: CTA Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
D.3 Level III: Empirical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 64
E Candidate Neutrino Phenomenology 65
F Conclusion 66
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1 Introduction
The nature of time, space, and their relation to matter and quantum phenomena is one
of the deepest and most enduring questions in fundamental physics. Traditional cos-
mology and quantum theory often treat spacetime as a fixed background or as emergent
from quantum fields; however, the fundamental origin of temporal directionality and the
interplay between the flow of time and geometry remains unclear. Recent theoretical
proposals, notably the CPT-symmetric universe scenario introduced by Boyle, Finn, and
Turok [7], suggest that the universe may possess a dual temporal structure with two op-
positely directed arrows of time. This bold idea offers a fresh perspective on long-standing
cosmological puzzles, such as the arrow of time and matter-antimatter asymmetry. A key
insight derived from these discoveries is that such emergent CPT-symmetric universes
can interact only gravitationally and via weak interactions. This observation raises the
question whether what we call dark matter and dark energy might, in a Platonic sense,
be merely reflections or projections of these fundamental inter-universal relations that we
observe and label as “dark” phenomena. Building on these ideas, we propose a rigorous
axiomatic model in which time is a fundamental ontological category. We postulate the
local coexistence of two opposite arrows of time within a purely baryonic universe, consist-
ing of two CPT-symmetric, coexisting baryonic sectors, from which spacetime geometry,
matter, and quantum phenomena naturally emerge. Our approach treats time as eternal
and fundamental, avoiding singular creation events like the Big Bang. In this work, we de-
velop the conceptual, mathematical, and physical foundations of this theory. We explore
its implications for gravity, quantum mechanics, dark matter, and cosmology, aiming to
establish a coherent and minimal foundation that challenges traditional assumptions and
opens new avenues for theoretical and experimental research.
Scope of the Framework.The purpose of this work is to introduce an axiomatic
framework and investigate its mathematical and physical consequences, rather than to
claim a complete replacement of general relativity, quantum field theory, or the Standard
Model. e distinguish results established within the proposed mathematical construction
from physical hypotheses and predictions requiring independent derivation and observa-
tional verification.
Executive Summary
1.Fundamental Primacy of Time: Time is the only absolute and fundamental
entity; space and geometry are emergent consequences.
2.Dual Time Arrows: The universe contains two coexisting temporal flows (τ + and
τ−) with opposite directions, forming a CPT-symmetric structure.
3.Emergent Geometry: The interplay of these time arrows induces an emergent
spacetime metricg µν from which classical gravity arises.
4.Matter as Necessary Consequence: Matter and energy naturally emerge as
necessary manifestations of time and its capacity for change.
5.Resolution of Dark Matter: ”Dark matter” is interpreted as the CPT-symmetric
baryonic sector (τ −), invisible electromagnetically but gravitationally active.
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6.No Need for Dark Energy: The apparent cosmic acceleration is a geometric
projection, removing the need for a cosmological constant or dark energy.
7.Emergent Speed of Light: The value ofcarises as a geometric consequence of
gµν, not as a fundamental constant.
8.Quantum Mechanics from Temporal Operators: Quantum behavior emerges
from the operator nature of ˆτ±, providing a natural unification of quantum mechan-
ics and general relativity.
9.Particles as Geometric Oscillations: Standard Model particles are hypothesized
as localized topological or oscillatory solutions of the temporal fields.
10.Unified Ontological Foundation: All physical phenomena—including matter,
geometry, energy, quantum effects, and possibly consciousness—emerge from the
minimal axiomatic basis rooted in fundamental time and dual temporal flows.
Fundamental Time
Two Time Arrowsτ+,τ −
Emergent Geometry
gµν
Matter and Energy
Quantum Phenomena
SM Particles
(Mass, Charge, Spin)
Figure 1: Logical emergence chain: from fundamental time to Standard Model particle
parameters.
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Mathematical Summary
ˆFundamental temporal sectors:Two oppositely oriented temporal sectors,τ +
andτ −, represented in the multimode construction by scalar fields
τ a
+, τ a
− :M→R, a= 1, . . . , N.
ˆEmergent metric:
gµν = 1
2
NX
a=1

∂µτ a
+∂ντ a
− +∂ µτ a
−∂ντ a
+


.
ˆAlgebraic minimality:A single temporal mode pair yields rank(g)≤2. Within
the proposed bilinear ansatz, at least three mode pairs are required to realize
Lorentzian signature (−+ ++).
ˆFlat-spacetime realization:Three mode pairs are sufficient to reproduce the
Minkowski metric through an explicit configuration of temporal scalar fields.
ˆGeometric interpretation:The multimode metric is an induced metric in an
auxiliary flat space with signature (N, N). ForN= 3, the ambient signature is
(3,3).
ˆDynamics:A complete dynamical formulation consistent with the multimode
metric remains under investigation. The previously proposed gravitational and
temporal-field equations require re-examination in this formulation.
ˆQuantum formulation:Temporal fields may be promoted to operators, ˆτ a
±(x),
with conjugate momenta ˆπ±a(x). A consistent canonical quantization and its phys-
ical consequences remain to be established.
2 Hypothesis of Electromagnetic Decoupling Between
CPT-Symmetric Sectors
2.1 Degeneracy of the original two-field construction
Consider the symmetric tensor constructed from two scalar temporal fieldsτ + andτ −:
g(1)
µν = 1
2 (∂µτ+∂ντ− +∂ µτ−∂ντ+).(1)
At each point of a four-dimensional manifold, this tensor is generated by at most two
independent covectors. Consequently,
rank

g(1)

≤2.(2)
It therefore cannot define a nondegenerate Lorentzian metric of signature (−,+,+,+).
This obstruction is algebraic and does not depend on the equations of motion.
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2.1.1 Scope and Limitations of the Construction
The three-mode construction establishes that a nondegenerate Lorentzian metric can be
generated from three pairs of scalar temporal fields within the proposed bilinear ansatz.
The explicit Minkowski realization provides a constructive existence proof for flat four-
dimensional spacetime. However, the construction does not establish that all physically
relevant curved metrics can be represented in this way. In particular, compatibility with
the Schwarzschild, Kerr, and FLRW geometries remains an open mathematical prob-
lem. Furthermore, the three temporal mode pairs are introduced as a model assumption.
Their existence and number have not yet been derived from the fundamental CTA axioms.
The projected gravitational field equations are not generally equivalent to the complete
Einstein equations. Consequently, the recovery of general relativity, the propagation of
gravitational waves, and the stability of the induced geometry require further investiga-
tion.
2.2 A multimode extension
To overcome this degeneracy, we introduce an additional model assumption: each tempo-
ral orientation is represented byNscalar modes,
τ a
+, τ a
−, a= 1, . . . , N.(3)
These modes are not postulated as independent physical time dimensions. They are
additional scalar degrees of freedom associated with the two temporal orientations. The
generalized metric ansatz is
gµν = 1
2
NX
a=1

∂µτ a
+∂ντ a
− +∂ µτ a
−∂ντ a
+


. (4)
This extension is not derived from the original CTA axioms and must be regarded as a
separate model assumption.
2.3 Minimal number of temporal modes
Define
X a = τ a
+ +τ a
−
2 , T a = τ a
+ −τ a
−
2 .(5)
The metric becomes
gµν =
NX
a=1
(∂µX a∂νX a −∂ µT a∂νT a).(6)
The positive index of inertia of this quadratic form cannot exceedN, because its positive
contribution is generated by onlyNcovectorsdX a. A four-dimensional Lorentzian metric
with signature (−,+,+,+) requires three positive directions. Therefore,
N≥3. (7)
This is a necessary algebraic condition within the specific ansatz of Eq. (4). The following
construction establishes thatN= 3 is sufficient for at least one nondegenerate Lorentzian
geometry.
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2.4 Explicit Minkowski construction
OnR 4, with coordinates (t, x, y, z), choose
τ 1
+ =t+x, τ 1
− =−t+x,(8)
τ 2
+ =t+y, τ 2
− =−t+y,(9)
τ 3
+ =t+z, τ 3
− =−t+z.(10)
Substitution into Eq. (4) gives
gµν = diag(−3,1,1,1). (11)
Its determinant is
det(gµν) =−3̸= 0.(12)
Introducing the rescaled coordinate
tM =
√
3t,(13)
the line element becomes
ds2 =−dt 2
M +dx 2 +dy 2 +dz 2. (14)
Thus three temporal mode pairs are algebraically sufficient to reproduce Minkowski ge-
ometry. This construction establishes neither the dynamical emergence of the metric nor
the representation of arbitrary curved spacetimes.
2.5 Geometric interpretation as an induced metric
ForN= 3, introduce the six-component map
Y A = (X1, X2, X3, T 1, T 2, T 3), A= 1, . . . ,6.(15)
Let the auxiliary target space carry the constant metric
ηAB = diag(+1,+1,+1,−1,−1,−1).(16)
Then
gµν =η AB∂µY A∂νY B. (17)
Whenever this metric is nondegenerate, the mapYdefines a local isometric immersion
Y: (M 4, g)− →(R 3,3, η).(18)
The immersion has codimension two. The six-dimensional target space is an auxiliary
geometric construction. Its introduction does not establish the physical existence of six
spacetime dimensions.
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2.6 Curvature restrictions
An induced metric obtained from an immersion into flatR 3,3 cannot possess an arbitrary
intrinsic curvature tensor. For a Lorentzian tangent space of signature (−,+,+,+), the
two-dimensional normal space has negative-definite signature (−,−). Choose a local
orthonormal normal frame and denote the corresponding symmetric second fundamental
forms byK I
µν, whereI= 1,2. With the curvature convention used here, the Gauss
equation reads
Rµνρσ =−
2X
I=1

K I
µρK I
νσ −K I
µσK I
νρ


. (19)
This equation imposes algebraic restrictions on the intrinsic Riemann tensor. A complete
local immersion also requires compatible second fundamental forms and a normal connec-
tion satisfying the Codazzi and Ricci equations. Consequently, the present construction
does not establish the representability of every four-dimensional Lorentzian geometry. In
particular, the compatibility of generic Kerr, Schwarzschild, and FLRW geometries with
this specific target-space signature remains to be established.
2.7 Projected gravitational field equations
As a possible dynamical extension, consider the Einstein–Hilbert functional evaluated on
the induced metric:
S[Y] = M 2
∗
2
Z
M
d4x
p
−g(Y)R[g(Y)].(20)
For variations preserving the nondegenerate Lorentzian sector, and after discarding the
appropriate boundary contribution, variation with respect to the embedding fields gives
Gµν∇µ∇νY A = 0.(21)
Here∇is the Levi-Civita connection of the induced metric, acting on the scalar fields
Y A. For an isometric immersion, the ambient vector∇ µ∇νY A is normal to the immersed
manifold. Consequently, the independent equations reduce to
GµνK I
µν = 0, I= 1,2. (22)
The four tangential combinations vanish as Noether identities associated with diffeomor-
phism invariance. Thus, although six embedding fields are varied, the codimension-two
model supplies only two independent normal equations at a regular immersion. These
equations are not generally equivalent to the ten vacuum Einstein equationsG µν = 0. In
particular, their solution space may contain non-Einstein geometries. No conclusion about
additional propagating degrees of freedom, stability, or effective dark-matter contributions
follows from these equations alone.
2.8 Open problems
The multimode construction establishes an algebraically consistent route to a nonde-
generate Lorentzian metric, but several questions remain unresolved. First, the class of
Lorentzian metrics admitting local isometric immersions into flatR 3,3 must be character-
ized. A concrete test is to evaluate the Gauss equations for the Kerr geometry at a regular
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point, followed, where applicable, by the Codazzi and Ricci compatibility conditions. Sec-
ond, the dynamics of the temporal modes requires a separate constraint and perturbation
analysis. In particular, the existence of a stable Lorentzian background and propagating
gravitational excitations has not been demonstrated. Third, the compatibility of the mul-
timode extension with the original CTA axioms and the proposed CPT transformation
must be established at the level of a complete action. Finally, possible electromagnetic in-
teractions between oppositely oriented temporal sectors remain independent hypotheses.
They do not follow from the induced-metric construction. The present result is therefore
a mathematical extension of CTA rather than a completed dynamical theory of gravity.
2.9 Conclusions and Implications for CTA
The multimode construction leads to several mathematical conclusions. First, a metric
constructed from a single pair of scalar temporal fields is necessarily degenerate in four
dimensions. Second, within the proposed bilinear metric ansatz, at least three temporal
mode pairs are required to reproduce a Lorentzian signature with three positive and one
negative eigenvalue. Third, an explicit construction demonstrates that three such pairs
are sufficient to generate Minkowski spacetime. Fourth, the resulting geometry admits
an interpretation as an induced metric in an auxiliary six-dimensional flat space with
signature (3,3). These results suggest that spatial geometry may be represented through
the structure of oppositely oriented temporal modes, without introducing a fundamental
spatial metric into the model. However, the construction does not independently explain
why physical spacetime has exactly three spatial dimensions. That dimensionality enters
through the Lorentzian signature imposed on the model. Furthermore, the representation
of general curved geometries, the recovery of Einstein dynamics, and the physical inter-
pretation of the temporal modes remain open problems. The principal achievement is
therefore the identification of a mathematically consistent, nondegenerate flat-spacetime
realization of the proposed temporal geometry, together with explicit conditions that
constrain its extension to curved spacetimes.
3 Operator of Emergence
Within the multimode formulation introduced in Section 3, the emergence operator is
defined as
F:

(τ a
+, τ a
−)
	N
a=1 7− →gµν,
where
gµν = 1
2
NX
a=1

∂µτ a
+∂ντ a
− +∂ µτ a
−∂ντ a
+


.
For a four-dimensional Lorentzian geometry with signature (−+ ++), at least three tem-
poral mode pairs are required within this construction. The existence of a nondegenerate
metric depends on the rank and signature of the resulting tensor, rather than merely on
the nonvanishing of the temporal fields. If either entire temporal sector vanishes identi-
cally, the induced metric becomes degenerate, withg µν = 0. Thus, Minkowski spacetime
is not the consequence of removing one temporal sector, but arises from a specific nonde-
generate configuration of both sectors, as demonstrated in Section 3.
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=== Page 14 ===
3.1 Emergence of Geometry and Structure
By definition and from our axioms, the geometric metricg µν emerges solely as a result of
the local coexistence of the two oppositely directed time arrows,τ + andτ −:
F(τ +, τ−)7→g µν.
This metric enables the definition of distance, curvature, and thus the existence of struc-
ture. The mere presence of geometry implies the possibility of localized configurations,
which naturally correspond to what we call ”matter”. If change is present (as a necessary
attribute of time), there must be ”something” that changes. Therefore, matter cannot be
absent in a universe where time fundamentally exists. Matter is not an optional addition
but a necessary manifestation of time’s capacity to support dynamic states.
4 Emergence of Fundamental Phenomena
4.1 Emergence of Energy
Energy is formally defined as the generator of temporal evolution in physics. Once time
and change exist, energy naturally emerges as a quantitative description of this change:
E= capacity to induce change over time.
Thus, energy does not need to be created; it is an inevitable byproduct of the absolute
existence of time.
4.2 Formal Statement
We can summarize the ontological necessity as:
Time =⇒Change =⇒(Matter∧Energy).
Consequently, matter and energy could not not exist. Their existence is a logical necessity
rather than a contingent or spontaneous event.
4.3 Cosmological Consequences
This perspective eliminates the need for a “creation moment” (Big Bang singularity) and
supports the view that the universe is an eternal ontological consequence of time itself.
4.4 Emergence of Entropy from Time
In classical thermodynamics, entropy is defined as
S=k B ln Ω,
where Ω denotes the number of accessible microstates of a given macrostate. Traditionally,
entropy is often interpreted as a measure of disorder or “missing information” about a
system.
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=== Page 15 ===
4.4.1 Time as the Foundation of Possible States
In our axiomatic framework, time is absolute and fundamental (Axiom A1). The mere
existence of time implies the inevitability of change:
Time =⇒Change.
Change presupposes the existence of alternative configurations. As soon as there is change,
there exists a set of possible states{σ i}that the system can occupy.
4.4.2 Inevitable Emergence of Entropy
Given that time necessitates the possibility of multiple configurations, entropy naturally
arises as a quantitative measure of this multiplicity:
S=k B ln|{σi}|.
Therefore, entropy does not require an initial condition of “low entropy” nor a primordial
“ordered state.” It is an inherent and emergent feature resulting directly from the existence
of time and its capacity to enable transitions between configurations.
4.4.3 No Special Initial Condition
In standard cosmological models, the low-entropy initial state leads to paradoxes such
as the Boltzmann-brain problem or fine-tuning issues. In our framework, since time is
eternal and without a singular origin, there is no requirement for a special low-entropy
starting point.
4.4.4 Formal Ontological Sequence
We summarize this conceptual chain formally as:
Time =⇒Change =⇒ {Possible States}=⇒Entropy.
Hence, entropy is a logical and necessary emergent phenomenon, not a fundamental entity
or externally imposed constraint.
4.4.5 Cosmological and Philosophical Implications
This perspective provides a natural explanation for the arrow of time and the continuous
increase of entropy observed in closed systems, aligning the second law of thermodynam-
ics with the axiomatic primacy of time rather than requiring additional postulates or
improbable initial conditions.
5 Unified Baryonic Matter
The universeUis purely baryonic and consists of two sectors:
ˆObservable baryonic sector (τ +) — standard luminous matter.
ˆCPT-symmetric baryonic sector (τ −) — matter with reversed time flow, invisible
electromagnetically but gravitationally active.
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=== Page 16 ===
6 Effective Energy-Momentum Tensor
The total effective tensor is given by:
T eff
µν =T +
µν +T −
µν (23)
where:
ˆT +
µν: tensor of our baryonic matter.
ˆT −
µν: tensor of CPT-symmetric baryonic matter.
7 Emergent Einstein Field Equations
Onceg µν emerges, it satisfies:
Gµν(g) = 8πT eff
µν (24)
whereG µν is the Einstein tensor constructed fromg µν.
8 Interpretation of ”Dark Matter”
The so-called ”dark matter” corresponds toT −
µν. This CPT-symmetric baryonic matter
does not emit or absorb electromagnetic radiation in ourτ + flow due to its reversed
temporal direction but contributes gravitationally.
9 No Big Bang Singularity
Unlike standard Big Bang cosmology, this framework asserts that time does not have an
origin. The coexistence ofτ + andτ − defines the onset of local spacetime geometry but does
not require or imply a singular starting point. The ”beginning” of our observable universe
is interpreted as a local event of temporal coexistence rather than a global creation event.
9.1 Lack of Real Expansion
In our framework the global scale factora(t) remains strictly constant:
˙a(t)≡0,Λ eff = 0.
What we interpret as cosmic expansion (redshift, apparent acceleration) arises solely
from the geometric projection of local interactions between the two time-arrow sectors
τ+ andτ − within our observational foliation. There is no dynamical, global stretching of
space—everything is a local effect perceived alongτ +.
10 Minimal Field Equations in 3 + 1 D
We consider two oppositely oriented temporal sectors represented byNpairs of scalar
fields
τ a
+, τ a
− :M →R, a= 1, . . . , N.
Their gradients define the fundamental temporal one-forms
v±a
µ =∂ µτ a
±.
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=== Page 17 ===
Emergent metric.Following the multimode construction introduced in Section 3, we
define
gµν = 1
2
NX
a=1

∂µτ a
+∂ντ a
− +∂ µτ a
−∂ντ a
+


.(25)
For a nondegenerate four-dimensional Lorentzian metric with signature (−+ ++), the
construction requires at least three temporal mode pairs. In the following, we restrict
attention to configurations for whichg µν is nondegenerate and has Lorentzian signature.
Action.We introduce the following candidate generally covariant action:
S[τ+, τ−] =
Z
M
d4x
p
−g(τ)
"
R(g(τ))
2κ +β
NX
a=1
gµν(τ)∂ µτ a
+∂ντ a
− −V(τ +, τ−)
#
,(26)
where
κ= 8πG
in units withc= 1. The metric is not an independent dynamical variable in this formu-
lation; it is constructed from the temporal scalar fields. Consequently, variation of the
action must account for the dependence ofg µν on these fields. No independent cosmolog-
ical constant term is included in this minimal action. However, its absence does not by
itself establish that the effective cosmological constant vanishes.
Variational structure.Let
E µν = 2√−g
δS
δgµν
denote the metric Euler derivative obtained by temporarily treating the metric and scalar
fields as independent variables. Likewise, define
J±a = 1√−g
δS
δτ a
±

g
.
The variation of the induced metric is
δgµν =
NX
a=1

∂(µδτ a
+∂ν)τ a
− +∂ (µτ a
+∂ν)δτ a
−

.
After integration by parts, the field equations are
J+a − ∇µ

E µν∂ντ a
−


= 0,
J−a − ∇µ

E µν∂ντ a
+


= 0. (27)
These are the Euler–Lagrange equations associated with the temporal scalar fields and the
induced metric. They are not, in general, equivalent to the complete Einstein equations
E µν = 0.
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=== Page 18 ===
Scalar contributions.For the candidate action (26), the scalar Euler derivatives at
fixed metric are
J+a =−β□ gτ a
− − ∂V
∂τ a
+
,
J−a =−β□ gτ a
+ − ∂V
∂τ a
−
.
(28)
These expressions must be combined with the metric variation terms in Eq. (27). There-
fore, the equationsJ ±a = 0 alone do not constitute the full field equations of the induced-
metric theory.
Physical status.The action above defines a candidate dynamical extension of the mul-
timode geometric construction. Its physical viability requires further analysis, including
the existence of nontrivial curved solutions, consistency with gravitational observations,
and the stability of the temporal field configurations. In particular, the recovery of gen-
eral relativity cannot be assumed solely from the presence of the Einstein–Hilbert term.
Likewise, the identification of temporal-sector effects with dark matter or dark energy
remains a physical hypothesis requiring quantitative derivation.
Remark.The multimode metric construction provides a mathematical mechanism for
generating Lorentzian geometry from temporal scalar fields. The action (26) is a pro-
posed dynamical completion of this construction, rather than a demonstrated equivalent
formulation of Einstein gravity.
11 Dynamics
The dynamical formulation of CTA is based on the multimode induced metric introduced
in Sec. 10. The two oppositely oriented temporal sectors are represented by the scalar
fieldsτ a
+ andτ a
−, wherea= 1, . . . , N. The metric is a composite field,
gµν =g µν

τ a
+, τ a
−


,
defined by Eq. (25).
11.1 Candidate Dynamical Action
We adopt the candidate action
S[τ+, τ−] =
Z
M
d4x
p
−g(τ) [Lgeom +L matter],(29)
with
Lgeom = R(g(τ))
2κ ,(30)
Lmatter =β
NX
a=1
gµν∂µτ a
+∂ντ a
− −V(τ +, τ−).(31)
This is the same candidate action introduced in Eq. (26), written in terms of its geometric
and temporal-field contributions.
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=== Page 19 ===
11.2 Metric Variation and Its Limitations
For comparison with conventional metric theories, one may temporarily treatg µν and
the temporal fields as independent variables. The corresponding metric equation can be
written formally as
Gµν =κT µν,(32)
where
Tµν =−β
NX
a=1

∂µτ a
+∂ντ a
− +∂ µτ a
−∂ντ a
+


+g µνLmatter.
This expression follows from the conventional definition
Tµν =− 2√−g
δSmatter
δg µν
for an independently varied metric. However, in CTA the metric is not an independent
variable. Therefore, Eq. (35) is an auxiliary comparison equation, not an additional
fundamental field equation of the induced-metric formulation.
11.3 Temporal-Field Equations
The fundamental dynamical equations are obtained by varying the action with respect to
the temporal scalar fields, including the induced variation of the metric. They are given
by Eq. (27), with the fixed-metric scalar contributions defined in Eq. (28). For reference,
the fixed-metric scalar equations would be
β□gτ a
− + ∂V
∂τ a
+
= 0,
β□gτ a
+ + ∂V
∂τ a
−
= 0.
(33)
ForV= 0 andβ̸= 0, these auxiliary equations reduce to
□gτ a
± = 0.(34)
These equations describe the fixed-metric scalar variation only and must not be confused
with the complete induced-metric Euler–Lagrange equations.
11.4 Relation to General Relativity
The presence of the Einstein–Hilbert term in the candidate action establishes a formal
connection with general relativity. Nevertheless, the induced-metric variation imposes
equations on the temporal scalar fields rather than the complete set of independent Ein-
stein equations. Consequently, recovery of Einstein gravity requires additional mathemat-
ical analysis. In particular, it remains necessary to establish whether the CTA equations
admit physically relevant curved solutions and reproduce the observed gravitational dy-
namics in an appropriate regime. No general Einstein limit is claimed at this stage.
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=== Page 20 ===
11.5 Status of the Dynamical Framework
The present action provides a candidate dynamical formulation compatible with the mul-
timode definition of the emergent metric. It does not yet establish the equivalence of
CTA and general relativity. Likewise, interpretations of temporal-field effects as dark
matter, dark energy, or other physical sources require explicit solutions and quantitative
comparison with observations.
11.6 Variation with respect to the metric
Varying (29) with respect tog µν yields the Einstein–type equation
Gµν(g) =κ T µν, T µν ≡2 ∂Lmatter
∂g µν −g µν Lmatter,(35)
where the effective energy–momentum tensor contains no cosmological constant term:
Tµν =β

∇µτ+ ∇ντ− +∇ µτ− ∇ντ+


−g µν
h
β∇ ατ+∇ατ− −V(τ +, τ−)
i
.
11.7 Variation with respect to the time–arrow scalars
Treatingτ ± as independent fields, we obtain
□gτ± = ∂V
∂τ∓
+ 1√−g
∂√−g
∂τ±
h
1
2κ R(g) +L matter
i
,(36)
where the last term encodes the implicitτ ±-dependence of the metric. In regimes where
the gradients∇τ ± are small—i.e. the geometry is nearly translationally invariant in both
time directions—the bracket can be neglected and (36) reduces to
□gτ± = ∂V
∂τ∓
,(37)
reproducing the scalar equations adopted in Sec. 10.
11.8 Einstein limit
Whenever the bilinear term∇ µτ+∇µτ− is much smaller thanκ −1 we may drop the right-
hand side of (35). In thatslow–arrowlimit the field equations reduce to the standard
vacuum Einstein equations, and deviations will appear only where the gradients ofτ ±
become relevant—interpreted in Sec. 8 as an effective dark–matter source.
Equations (35) and (36) therefore constitute the dynamical backbone of theCoexisting
Time Arrows(CTA) framework, recovering General Relativity plus an explicit geometric
energy–momentum tensor in the appropriate limit.
11.9 Consistency Check with the1+1Toy Model
We now examine the relation between the minimal temporal-field equations introduced
in Sec. 10 and the 1 + 1-dimensional toy model. The minimal field equations are
Gµν(g) =κ

∇µτ+∇ντ− +∇ µτ−∇ντ+ −g µν∇ατ+∇ατ− +g µνV

,(38a)
□gτ± =∂ τ∓V.(38b)
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=== Page 21 ===
Consider the 1 + 1-dimensional configuration introduced in Appendix 27:
τ+(t, x) =t+f(x), τ −(t, x) =−t+f(x).
For this example we setV= 0 and use the normalizationc 0 = 1. The temporal gradients
are
v±
µ = (±1, f ′).
Using the symmetrized bilinear metric construction,
gµν = 1
2

v+
µ v−
ν +v −
µ v+
ν


,
we obtain
gµν =
−1 0
0 (f ′)2

.
This metric is nondegenerate whereverf ′(x)̸= 0. On any such region, the coordinate
transformation
T=t, X=f(x)
gives
ds2 =−dT 2 +dX 2.
Therefore, the induced metric is locally Minkowskian and its Ricci scalar vanishes:
R= 0.
Moreover, in two spacetime dimensions the Einstein tensor vanishes identically:
Gµν =R µν − 1
2 gµνR≡0.
For the present configuration, the contraction of the temporal gradients is
∇ατ+∇ατ− = 2.
Consequently, the right-hand side of Eq. (38a), withV= 0, becomes
κ

v+
µ v−
ν +v −
µ v+
ν −2g µν


= 0.
Thus, the gravitational equation is satisfied identically in the nondegenerate region. The
scalar-field equations also hold for this configuration. Indeed, in the local coordinates
(T, X), the temporal fields are linear:
τ+ =T+X, τ − =−T+X,
and therefore
□gτ+ =□ gτ− = 0.
Hence, the toy-model configuration satisfies the displayed minimal field equations for
V= 0, wherever the induced metric is nondegenerate. However, this result must be
interpreted with care. The vanishing of the two-dimensional Einstein tensor is a geometric
identity and does not provide a nontrivial test of four-dimensional gravitational dynamics.
Furthermore, the toy model contains only one temporal mode pair and cannot establish
the consistency of the full multimode construction. The example therefore provides a
valid local consistency check of the reduced equations, but not a derivation or verification
of the complete CTA field dynamics.
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=== Page 22 ===
11.10 Covariant Conservation of the Effective Source
The candidate CTA action introduced in Eq. (26) is diffeomorphism-invariant. In the
multimode formulation, the metric is constructed from the temporal scalar fields:
gµν =g µν

τ a
+, τ a
−


.
Consequently, the metric and the temporal fields cannot be varied independently in the
fundamental CTA formulation.
Noether identity.Let
F±a = 1√−g
δS
δτ a
±
denote the complete Euler–Lagrange derivatives of the induced-metric action. Under an
infinitesimal diffeomorphism generated by a vector fieldξ µ, the scalar fields transform as
δξτ a
± =ξ µ∂µτ a
±.
Diffeomorphism invariance then implies the Noether identity
NX
a=1

F+a∂ντ a
+ +F −a∂ντ a
−


= 0. (39)
This identity expresses the diffeomorphism consistency of the complete temporal-field
equations. It does not constitute an independent Einstein equation or an independent
conservation law for a dark-sector energy–momentum tensor.
Auxiliary stress tensor.If the metric is temporarily treated as independent, the
temporal-field contribution to the action defines an auxiliary stress tensor:
T (τ)
µν =−β
NX
a=1

∂µτ a
+∂ντ a
− +∂ µτ a
−∂ντ a
+


+g µνLmatter.
In the independent-metric formulation, its covariant divergence vanishes when the corre-
sponding scalar Euler–Lagrange equations are satisfied. However, this auxiliary conser-
vation statement cannot automatically be identified with a separate conservation law for
the oppositely oriented temporal sector.
Physical interpretation.The identification of temporal-sector contributions with an
effective dark-matter source remains a hypothesis of CTA. A physically meaningful dark-
sector stress tensor must be derived from a consistent dynamical formulation and shown
to satisfy the appropriate conservation relations. The present Noether identity establishes
a structural consistency condition, but does not by itself demonstrate the existence of a
separately conserved dark-matter component.
22

=== Page 23 ===
12 Observational Signature Map
Redshift Drift Test.In the standard ΛCDM cosmological model, the redshift drift is
given by
˙z≡ dz
dt0
= (1 +z)H 0 −H(z),
whereH 0 is the present Hubble parameter andH(z) is the expansion rate at redshift
z. The predicted drift depends on redshift and cosmological parameters, and may be
measured by future high-precision spectroscopic observations, including those planned
for the Extremely Large Telescope (ELT). Within the proposed CTA interpretation, the
absence of global cosmological expansion motivates the hypothesis
˙z= 0.
However, a stationary global geometry does not by itself guarantee vanishing observed
redshift drift. A complete derivation must account for the trajectories of sources and
observers and the propagation of electromagnetic signals through the emergent geometry.
A null result in future redshift-drift measurements would be consistent with this CTA
hypothesis. Its discriminatory power relative to standard cosmology would depend on
measurement uncertainties, the observed redshift range, and the quantitative predictions
of competing cosmological models.
Status of Cosmological Predictions.The proposed vanishing redshift drift, ˙z= 0,
is a conditional hypothesis of CTA rather than a result derived from a complete set of
dynamical field equations. Likewise, any proposed numerical value of the cosmological
growth index requires an explicit derivation from the perturbation dynamics of CTA.
These observational signatures are therefore presented as testable hypotheses requiring
quantitative verification rather than established consequences of the present mathematical
construction.
Road-map.A principal phenomenological objective is to determine whether CTA can
reproduce observed cosmic acceleration without introducing a fundamental cosmological
constant. This requires a quantitative comparison with cosmological observations, includ-
ing the integrated Sachs–Wolfe effect, measurements ofH(z), baryon acoustic oscillations,
and cosmic microwave background anisotropies. A comprehensive comparison will require
a consistent cosmological background, perturbation equations, and a dedicated Boltzmann
solver incorporating the temporal-field dynamics. The complete induced-metric field equa-
tions (27), together with the scalar contributions (28), provide a candidate starting point
for cosmological modelling. (28) provide a starting point, but their consistency with the
multimode geometric construction must also be established.
Illustrative Growth-Index Scenario.The growth indexγis commonly introduced
through the approximation
f(z)≃Ω m(z)γ,
wheref(z) denotes the linear growth rate of matter density perturbations. For reference,
standard ΛCDM cosmology predicts approximately
γ≃0.55
23

=== Page 24 ===
under conventional assumptions. A value near
γ≃0.40
may be investigated as an illustrative phenomenological scenario for CTA, rather than
a quantitative prediction derived from its present field equations. Determining whether
CTA predicts such a deviation requires deriving its linear perturbation equations and
calculating the corresponding growth history. Only then can the result be compared with
observations from galaxy surveys and other large-scale structure probes.
Numerical Forecast ExampleFor instance, in our framework, the growth indexγis
predicted to decrease to approximately 0.40±0.02 atz≲1, as opposed to the standard
ΛCDM valueγ≈0.55. Current and upcoming surveys (DESI, SKA) aim for a precision
at the level of ∆γ≈0.02, sufficient to test this prediction within the next decade.
13 Emergence of the Speed of Light
In standard physics, the speed of lightcis taken as a fundamental constant, defining
the maximal propagation speed of information and causality in spacetime. However, its
numerical value is postulated rather than derived:
c≈299,792,458 m/s.
13.1 Absence of a Fundamental Metric
In our axiomatic framework, no pre-defined metric or background structure exists (Ax-
iom A4). Geometry emerges locally through the coexistence of the time arrowsτ + and
τ−. Consequently, all physical ”constants” related to spacetime must arise as emergent
geometric quantities.
13.2 Geometric Origin ofc
The maximal information propagation speedcis defined as
c= lim
F(τ +,τ−)→gµν

sup dx
dt

,
whereg µν is the emergent metric generated by the operatorFacting onτ + andτ −. This
implies thatcis not a separate input but a global geometric consequence of the overall
temporal configuration of the universe.
...
Figure unavailable: original PNG not supplied
Figure 2: Emergence ofcfromτ + andτ −. The metricg µν mediates the geometric relation,
leading to the emergent speed limitc.
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=== Page 25 ===
13.3 Dependence on Global Temporal Configuration
The precise value ofcis determined by the total distribution and interaction of the time
arrows throughout the universe. Its constancy across all local regions reflects the global
coherence of the emergent geometry.
13.4 Formal Ontological Statement
We summarize this principle formally as:
{τ+, τ−}
F
− →gµν =⇒c.
Hence, the numerical value ofccan, in principle, be derived once the global temporal
configuration{τ +, τ−}is fully understood and mathematically described.
13.5 Philosophical and Physical Implications
This perspective removes the need to postulatecas a fundamental constant, reinterpreting
it instead as an emergent property of the universal temporal structure. It opens a new
path toward understanding the deep connection between causality, geometry, and the flow
of time.
13.6 Numerical Value ofc
14 Perception of Cosmic Expansion and Absence of
Dark Energy
In this framework, the universe contains two globally present time arrowsτ+ andτ −, both
omnipresent and having identical ”densities,” but with opposite temporal directions. This
global structure defines a unified and stable emergent geometry.
14.1 Emergent Global Geometry
The emergent metricg µν is co-created by both time arrows via the operator
F(τ +, τ−)7→g µν.
This metric is globally stable and does not exhibit global expansion or contraction in an
absolute sense.
14.2 Local Temporal Trajectories
An observer within theτ + sector interprets geometric changes along their own time tra-
jectory. Inτ +, the past is perceived as ”denser” (more compact), and the present as
”expanded,” due to the evolution of local baryonic structures withing µν.
Conversely, in theτ − sector, the opposite temporal perception occurs: what we see as
future is their past, and their present appears more ”contracted” to them.
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=== Page 26 ===
14.3 Apparent Expansion as a Local Effect
The observed expansion of the universe is thus a local phenomenon perceived byτ +
observers, not a global dynamical expansion of the entire geometry. We measure cosmo-
logical redshifts and interpret them as ”expansion,” but this is a geometric projection
along theτ + temporal arrow rather than an actual global stretching of space.
14.4 No Need for Dark Energy
In this theory, there is no necessity to introduce a separate ”dark energy” component
or a cosmological constant Λ. The apparent acceleration of cosmic expansion is a mis-
interpretation arising from comparing snapshots of our local temporal trajectory within
a globally stable and symmetric geometry. In contrast to the standard ΛCDM cosmol-
ogy, where the cosmological constant is introduced as an additional energy component
driving accelerated expansion, in this framework such acceleration is purely emergent and
observer-dependent, requiring no extra term in the Einstein equations.
14.5 Formal Expression
Define the global stable metric as
gµν =F(τ +, τ−).
The local perception of scale factor evolutiona(t) inτ + is expressed as
a(t) =ϕ(τ +, gµν),
whereϕencodes the observer-dependent interpretation of local geometric relations, and
should not be confused with a global dynamical scale factor. There is no global dynamical
Λ term; instead,
Λeff ≡0,
since all observed acceleration is local and emergent.
14.6 Implications
ˆExplains the cosmological expansion without invoking dark energy.
ˆEliminates the need for a cosmological constant as a fundamental parameter.
ˆSupports the interpretation of a stable, timelessly coexisting geometric substrate.
Thus, the perceived cosmic acceleration is a geometric consequence of local temporal
interpretation withinτ +, not an actual expansion of a dynamical spacetime.
15 Absence of Electromagnetic Interactions Between
CPT-Symmetric Sectors
A fundamental consequence of our framework is the explicit absence of electromagnetic
interactions between the CPT-symmetric sectorτ − and our observable sectorτ +. While
26

=== Page 27 ===
gravitational and weak interactions remain permitted channels of communication between
the two sectors (Axiom A5), electromagnetic interactions are categorically excluded. This
exclusion is not arbitrary; rather, it arises naturally as a direct logical implication of the
opposite orientations of the time arrowsτ + andτ −.
Electromagnetic waves, described by Maxwell’s equations, inherently possess a de-
fined causal structure, propagating signals forward in time (retarded solutions). In the
CPT sector, however, electromagnetic signals would necessarily correspond to advanced
solutions propagating backward in time from the perspective of our sectorτ +. Allow-
ing electromagnetic interactions between these sectors would thus create irreconcilable
paradoxes of causality, where effects precede their causes and energy propagation occurs
counter to the observed temporal direction.
Formally, the structure of Maxwell’s equations explicitly distinguishes retarded solu-
tions (physically observable) from advanced solutions (non-causal and physically unob-
served). The coexistence of opposite time flows therefore enforces a strict causal separation
in electromagnetic communication channels:
ˆObservers in sectorτ + can only detect and interact with retarded electromagnetic
waves within their causal future.
ˆObservers in sectorτ −, analogously, can only interact with advanced solutions (from
their perspective retarded).
Consequently, neither sector can perceive electromagnetic signals generated by the
other, ensuring causal consistency:
F(τ +, τ−)↛Electromagnetic Interaction (40)
Thus, the absence of electromagnetic interactions between CPT-symmetric sectors is
not merely an imposed assumption but a necessary condition for the internal logical con-
sistency and causal integrity of the theory. This provides strong theoretical justification
for Axiom A5 and clarifies the fundamental limitations on permissible interactions across
sectors characterized by opposing temporal orientations.
16 Quantum Nature of Time and Emergence of Quan-
tum Mechanics
Time is treated not as a classical external parameter but as an operator with inherent
quantum properties, fundamentally shaping the emergent geometry and causality.
16.1 Quantum Operators for Time Arrows
We promote the local time flows to operators:
ˆτ+,ˆτ −
with commutation relations:
[ˆτ±, ˆH] =iℏ(41)
where ˆHdenotes the local emergent Hamiltonian operator derived fromg µν.
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=== Page 28 ===
Fundamental Symmetry NoteIt should be emphasized that the internal consistency
of the proposed framework implicitly relies on the exact global CPT symmetry. Any
violation of Charge (C), Parity (P), or Time reversal (T) individually would not merely
modify local physical processes but would collapse the entire global geometric-temporal
structure, thereby preventing the emergence of stable physical laws. In this sense, CPT
symmetry is not just a feature of particle interactions but an essential axiom for the
existence of a coherent universe.
One can further define a Hilbert spaceHof states built on classical background config-
urationsτ (0)
± , and construct a corresponding Fock space of small perturbations (quanta) of
the temporal fields ˆτ±. This approach allows for a rigorous operator formulation, enabling
a complete quantum description of geometry and matter within the same framework.
Philosophical Note on the Origin of Time and Space
A natural question arises regarding the ultimate origin of time and space within this
framework. We explicitly postulate time as an absolute, fundamental entity—an ax-
iomatic starting point. In this sense, time does not ”emerge” or ”originate” from a more
basic structure; rather, it serves as the precondition for the possibility of any change or
existence.
Space, on the other hand, is not taken as fundamental. It arises locally and neces-
sarily as a geometric consequence of the coexistence of the two temporal flows,τ + and
τ−. The question ”where did space come from” is thus reframed: space does not exist
independently but emerges from specific configurations of time.
This approach intentionally avoids infinite regress by choosing time as the ontological
axiom. Just as mathematics accepts the existence of points or sets as primitive notions,
our framework accepts time as the foundational prerequisite for reality.
We invite further philosophical and mathematical exploration of this premise, recog-
nizing that it touches the deepest metaphysical layers of our understanding of existence.
16.2 Local Fluctuations and Superpositions
The coexistence operatorF(ˆτ +,ˆτ−) inherently allows for local quantum fluctuations:
gµν − →gµν +δg µν(ˆτ+,ˆτ−)
These fluctuations explain superposition, entanglement, and the uncertainty principle.
16.3 Unified View
Quantum mechanics emerges naturally from the quantum structure of time. Both geome-
try and quantum phenomena are effective manifestations of the deeper temporal ontology,
resolving the apparent conflict between GR and QM without the need to quantize gravity
directly.
16.4 Explicit Operator Structure and Field Interpretation
We promote the time arrows to operator-valued fields onM:
ˆτ±(x), x∈ M.
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=== Page 29 ===
Their conjugate momentaπ ±(x) satisfy canonical equal-time commutation relations:
[ˆτ±(x), π±(y)] =iℏδ (3)(x−y).
Assuming minimal coupling, the local emergent Hamiltonian density is
H(x) = 1
2

π+(x)π−(x) +β∇ µˆτ+(x)∇µˆτ−(x)

+V(ˆτ+,ˆτ−).
The total Hamiltonian operator reads
ˆH=
Z
d3x √−gH(x).
This explicit operator construction connects the temporal fields with standard canonical
quantization procedures and permits the construction of a Fock space for small perturba-
tions around a classical background configuration ofτ ±.
17 Formal Definition of Sector Interactions
We define two CPT-symmetric sectors within a single universeU:
ˆS +: Baryonic sector with a forward-oriented time arrowτ +.
ˆS −: CPT-symmetric baryonic sector with an oppositely oriented time arrowτ −.
The interaction between these sectors is strictly constrained by the axiomatic frame-
work:Axiom (Restricted Interaction):The sectorsS + andS − interact exclusively via
gravitational and weak interactions. Electromagnetic and strong interactions are sector-
internal and do not couple across sectors.
18 Formal Argument Against Electromagnetic Inter-
action Between CPT-Symmetric Sectors
Consider electromagnetic interactions, governed by Maxwell’s equations, invariant under
the CPT symmetry transformation. The CPT symmetry operation consists of:
1. Charge conjugation (C):q→ −q, reversing electric charges.
2. Parity inversion (P):x→ −x, spatial reflection.
3. Time reversal (T):t→ −t, reversal of temporal flow.
For an electromagnetic fieldF µν, CPT transformation leads to:
CPT(F µν) =−F µν(t→ −t,x→ −x, q→ −q).(42)
Assume, for contradiction, that an electromagnetic waveA µ(x, t) originating in theS −
sector could propagate into theS + sector. Such propagation would require a field config-
uration that is simultaneously a valid forward-time solution (inτ +) and a backward-time
solution (inτ −). Under T symmetry alone, solutions to Maxwell’s equations transform
as:
Aµ(t,x)
T
− →Aµ(−t,x) (43)
However, physically realizable electromagnetic solutions must satisfy causal boundary
conditions:
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=== Page 30 ===
ˆInS +, retarded potentials propagate forward in time (from past to future).
ˆInS −, due to reversed time direction, analogous potentials propagate from their
future to their past (retarded inτ −).
Lemma (Causality constraint): Electromagnetic solutionsA µ(t,x) forS + andS −
sectors must individually respect their local causality constraints:
∂Aµ +
∂τ + >0 forS ∗+ (44)
∂Aµ ∗ −
∂τ − >0 forS −(45)
Becauseτ + =−τ −, the propagation conditions are mutually exclusive:
∂Aµ +
∂τ + =− ∂Aµ −
∂τ − (46)
To satisfy simultaneous propagation in both sectors via electromagnetic fields, the field
Aµ would have to violate local causality constraints in at least one sector. Hence, no elec-
tromagnetic wave can propagate coherently between sectors with opposite time arrows
without violating causality and the fundamental symmetry conditions.Conclusion (No
Electromagnetic Interaction): Electromagnetic interaction between CPT-symmetric
sectors is strictly forbidden due to causality and CPT symmetry constraints. This ar-
gument formally confirms that sectorsS + andS − cannot electromagnetically interact,
consistently aligning with the axiomatic structure and maintaining internal coherence of
the theory.
19 Emergent Charge from Local Time Asymmetry
Definition 19.1(Time Asymmetry Field).LetMbe a differentiable spacetime manifold
equipped with two global time functions:
τ+ :M→R, τ − :M→R,
representing two globally coherent time arrows. Then thetime asymmetry fieldis defined
as the scalar function:
∆τ :M→R,∆ τ(P) :=τ +(P)−τ −(P).
Definition 19.2(Emergent Charge).Let ∆ τ be the time asymmetry field defined above.
Then the localemergent electric charge densityqis defined by:
q(P) :=κ·∆ τ(P),
whereκ∈R + is a proportionality constant determined by the local coupling of time fields
to the electromagnetic sector.
Definition 19.3(Time Dipole).LetS⊂Mbe a bounded region of spacetime. We say
thatSforms atime dipoleif for every eventP∈S, there exists a conjugate point ¯P∈S
such that:
τ+(P) =τ −( ¯P), τ −(P) =τ +( ¯P).
In other words, the region is symmetric under time arrow inversion.
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Theorem 19.1(Emergent Charge Theorem).LetS⊂Mbe a compact region of space-
time. IfSis not a time dipole, i.e., it contains a non-zero integral of the time asymmetry
field: Z
S
∆τ(P)dV̸= 0,
then the total emergent charge withinSis non-zero:
QS :=
Z
S
q(P)dV=κ
Z
S
∆τ(P)dV̸= 0.
Proof.Follows directly by linearity from Definitions 1 and 2. Sinceq(P) =κ·∆ τ(P), we
integrate over the regionSand obtain:
QS =
Z
S
q(P)dV=κ
Z
S
∆τ(P)dV.
If ∆τ is non-zero on a non-zero volume subset ofS, thenQ S ̸= 0.
Remark19.1 (Ontological Interpretation).Charge is not a primitive property of matter,
but a manifestation of broken symmetry betweenτ + andτ −. Structures with zero net ∆ τ
are electrically neutral by definition.
Example 19.1(Hydrogen Atom).A hydrogen atom consisting of(p +, τ+)and(e −, τ+)
has non-zero charge if not accompanied by theirτ −-conjugates(¯p−, τ−),(e +, τ−). Only the
full configuration forms a time dipole and is electrically neutral.
Example 19.2(Neutron as Time Closure).The neutron can be understood as a local
topological closure of∆ τ, neutralizing time asymmetry through internal configuration.
This explains its neutrality despite being a composite object.
The strict electromagnetic separation betweenS + andS −, as proven in the previous
section, implies that electric charge cannot arise from direct intersector communication.
Instead, we propose that charge is an emergent local feature of broken symmetry between
the coexisting time fieldsτ + andτ −, defined pointwise on spacetime. The following
formalism constructs this emergent charge from first principles.
20 Quantum Field Formalism of Temporal Operators
20.1 Promotion of Temporal Flows to Quantum Fields
To fully embed quantum mechanics within our axiomatic framework, we explicitly pro-
mote the two coexisting temporal flows,τ + andτ −, to quantum operator fields defined on
a 4-dimensional manifoldM:
τ+(x)→ˆτ+(x), τ −(x)→ˆτ −(x), x∈ M.(47)
These operator fields are assumed Hermitian, ensuring observability and real eigenvalues
corresponding to physically measurable time intervals. Moreover, a zero redshift drift
would be a “smoking gun” for our no-expansion hypothesis, distinguishing CTA from all
dark-energy models.
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=== Page 32 ===
20.2 Canonical Structure and Commutation Relations
We introduce canonical conjugate momentum fields associated with each temporal field,
defined through the functional derivative of the actionS[τ +, τ−]:
ˆπ±(x) = δL
δ(∂0ˆτ±(x)) ,(48)
satisfying the canonical equal-time commutation relations:
[ˆτ±(t,x),ˆπ±(t,y)] =iℏδ (3)(x−y),(49)
[ˆτ+(t,x),ˆτ−(t,y)] = 0,[ˆπ +(t,x),ˆπ−(t,y)] = 0.(50)
These commutation relations form the foundational algebraic structure required for a
rigorous quantum theory.
20.3 Hamiltonian Formulation
From the Lagrangian density
L=β ∂ µˆτ+∂µˆτ− −V(ˆτ+,ˆτ−) + 1
2κ R(g(ˆτ+,ˆτ−)),(51)
we derive the quantum Hamiltonian density as the Legendre transform:
H= ˆπ+∂0ˆτ+ + ˆπ−∂0ˆτ− − L.(52)
Explicitly, the Hamiltonian operator reads:
ˆH=
Z
d3x√−g

ˆπ+ˆπ− +β∇ µˆτ+∇µˆτ− +V(ˆτ+,ˆτ−)− 1
2κ R(g(ˆτ+,ˆτ−))

.(53)
20.4 Quantum State Space and Evolution
We introduce a Hilbert spaceHof quantum states|Ψ(t)⟩represented as wave functionals
Ψ[τ+, τ−;t], satisfying the Schr¨ odinger equation:
iℏ ∂
∂t Ψ[τ+, τ−;t] = ˆHΨ[τ+, τ−;t].(54)
Quantum states evolve unitarily according to the temporal quantum fields dynamics en-
coded in ˆH.
20.5 Path Integral Representation
An equivalent quantum formulation can be obtained via the path integral formalism:
Z=
Z
Dτ+Dτ− exp
 i
ℏ
Z
d4x√−gL(τ +, τ−)

.(55)
Here, the sum is over all possible configurations of the temporal fields, demonstrating
the full quantum mechanical character and unifying geometry and quantum theory at a
fundamental level.
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=== Page 33 ===
20.6 Quantum Uncertainty and Spectral Decomposition
The canonical commutation relations imply quantum uncertainty relations for temporal
measurements:
∆τ±∆π± ≥ ℏ
2 .(56)
We define spectral decompositions for the temporal operators ˆτ±:
ˆτ± =
Z ∞
τ0
dτ τ|τ⟩ ± ⟨τ|,(57)
where|τ⟩ ± form an orthonormal basis of eigenstates, reflecting a fundamental discrete or
continuous spectrum of temporal values depending on model specifics.
20.7 Quantum Observables and Measurements
Physical observables ˆOare Hermitian operators defined as functionals of ˆτ ± and their
conjugates, with expectation values given by standard quantum mechanics:
⟨ ˆO⟩=⟨Ψ| ˆO|Ψ⟩=
Z
Dτ+Dτ− Ψ∗[τ+, τ−] ˆOΨ[τ+, τ−].(58)
20.8 Quantum-Classical Transition
In the classical limit (ℏ→0), quantum fluctuations diminish, and the wavefunctional
sharply peaks around classical trajectories of the fieldsτ (cl)
± satisfying classical field equa-
tions. Thus, we recover classical geometric dynamics described in the previous sections:
lim
ℏ→0
Ψ[τ+, τ−]→δ(τ + −τ (cl)
+ )δ(τ− −τ (cl)
− ).(59)
20.9 Summary of Quantum Formalism
Time ceases to be an external parameter; it becomes the fundamental quantum entity
shaping both geometry and matter. This rigorous quantum field formalism for tempo-
ral fields provides a complete embedding of quantum mechanics within our emergent
geometric-temporal framework:
{ˆτ+,ˆτ−}
Canonical Structure
− − − − − − − − − − − →Quantum Geometry⊕Quantum Mechanics.(60)
21 Discussion
This framework proposes time as the only fundamental and absolute entity, with space
and all geometric, material, and energetic structures emerging locally from the coexistence
of two temporal flows,τ + andτ −. By adopting time as an axiom, we avoid the need for
additional ontological assumptions about space, matter, or energy.
A natural question arises regarding the ultimate origin of time and space within this
approach. We explicitly postulate time as a fundamental axiom. In this sense, time
does not ”emerge” or ”originate” from any more basic structure; it serves as the logical
and ontological precondition for any form of change or existence. Space, on the other
hand, arises locally and necessarily as a geometric consequence of the configurations and
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interactions ofτ + andτ −. Thus, the question ”where did space come from” is reframed:
space is not a pre-existing container but an emergent relational phenomenon.
Another crucial conceptual insight of this framework is the reinterpretation of energy
itself. Traditionally, energy is understood as the capacity to perform work, inherently
tied to change over time. In our view, energy is not a separate or additional entity but
an operational manifestation of the fundamental flow of time. In other words,time
constitutes the ultimate source of all energy in the universe. Mass, energy, and
geometric curvature are different expressions of how temporal dynamics shape local and
global structures.
By adopting this temporal primacy, we unify the ontological roots of matter, geometry,
and energy, providing a coherent and minimal foundation from which all observable phe-
nomena may arise. This approach invites future mathematical formalization and experi-
mental investigation to validate its predictions and explore its implications for cosmology,
particle physics, and theories of consciousness.
We recognize that this philosophical stance touches the deepest metaphysical layers
of scientific inquiry. Nonetheless, we believe that by shifting the focus to time as the
ultimate foundation, new avenues for understanding the universe can be opened.
Remark:Ultimately, the true strength of any fundamental framework lies not only
in its capacity to explain what we observe, but in its courage to propose what might be
possible. By placing time as the absolute foundation and allowing all else to emerge,
we open the door to a profoundly unified vision of reality — one that challenges us to
look beyond established boundaries and invites us to imagine what the universe might
be, rather than only what it appears to be.
22 New Research Postulates
In order to ensure the empirical robustness and future falsifiability of the Coexisting
Time Arrows (CTA) framework, we introduce a set of explicit research postulates. These
postulates serve as a bridge between the axiomatic foundation and concrete experimental
predictions.
C1 (Experimental Corroboration Postulate).Any theory proposing the coexis-
tence of two global time arrowsτ + andτ − must predict at least one distinct, measurable
observational signature that cannot be mimicked by effective parameter adjustments within
ΛCDM or other standard cosmological models.
Formally, there must exist at least one observableO CT A such that
|OCT A − OΛCDM |> δ obs
whereδ obs denotes the achievable observational uncertainty in upcoming or future surveys
(e.g., Euclid, SKA, LSST).
This requirement prevents the framework from being reduced to a trivial parametriza-
tion of existing models and ensures genuine explanatory power.
C2 (Quantization Consistency Postulate).The quantum operator structure of
temporal fields{ˆτ+,ˆτ−}must admit a well-defined spectral decomposition, finite vacuum
energy expectation⟨T µν⟩, and be free from ultraviolet (UV) divergences without introducing
ad hoc counterterms.
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=== Page 35 ===
This postulate formalizes the demand that the quantum sector of CTA can be consis-
tently regularized and embedded into a complete field-theoretic framework without relying
on external fine-tuning or arbitrary renormalization conditions.
C3 (Geometric Soliton Realizability Postulate).There must exist at least one
stable, localized geometric solution (soliton or oscillon) to the CTA field equations in
3 + 1dimensions, with parameters (mass, charge, spin) matching, at least in order of
magnitude, known Standard Model particle properties.
This postulate directly links the theoretical field solutions to physically observable
particles, ensuring that the framework not only explains large-scale cosmology but also
microscopic particle phenomenology.
Summary of Postulates.Together, these three postulates establish a rigorous research
agenda:
1. Predict and isolate unique observational signatures testable with current or near-
future cosmological data (C1).
2. Build a mathematically consistent and divergence-free operator formalism (C2).
3. Demonstrate concrete localized solutions resembling known particle states (C3).
Their simultaneous fulfillment would provide strong evidence supporting the viability of
the CTA framework as a candidate for a unified description of spacetime, quantum fields,
and matter.
23 Open Questions and Validation Strategy
While this framework offers a minimal and unified foundation based on the absolute nature
of time and the emergent structure of space, it remains essential to rigorously assess its
internal consistency and empirical viability.
We explicitly acknowledge that this approach does not attempt to ”fit” preconceived
dogmas or established interpretations. Instead, it proposes an axiomatic starting point:
time as an absolute and fundamental entity. From this axiom, we derive the coexistence
of two temporal flows (τ+ andτ −) and the local emergence of a three-dimensional spatial
geometry.
The strategy moving forward is clear:
1.Apply the axiom rigorously: Develop full mathematical consequences of adopt-
ing time as the sole fundamental entity and space as an emergent phenomenon.
2.Check for internal logical consistency: Identify any potential contradictions
within the derived structures, particularly in relation to known geometric, relativis-
tic, and quantum frameworks.
3.Compare with empirical data: Evaluate if predictions align with current obser-
vations in cosmology (e.g., cosmic expansion, dark matter effects), particle physics
(e.g., symmetry violations, mass generation), and thermodynamic processes (e.g.,
entropy, arrow of time).
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4.Propose potential experimental tests: Identify unique predictions that distin-
guish this framework from existing models, enabling future falsification or validation.
If this approach leads to a coherent, predictive, and experimentally consistent frame-
work, it would provide a profound shift in our understanding of the universe. If not, it
remains a powerful conceptual exercise, illustrating the creative and philosophical possi-
bilities inherent in rethinking the foundation of physics.
”We must rigorously apply this foundational axiom and examine whether it leads to a
logically coherent theory consistent with all known empirical data. If it does, it provides a
new fundamental view of reality. If it does not, it remains a powerful conceptual exercise,
demonstrating the limits and possibilities of our imagination.”
Remark on the Magnitude of Time Arrows
An open question remains whether the two coexisting time arrows,τ + andτ −, have equal
magnitudes (lengths) but opposite directions, or whether their lengths may differ. Cur-
rently, there is no definitive theoretical or experimental answer, nor is it known whether
the theory of Emergent Spacetime and Quantum Reality from Coexisting Time Arrows
will lead to any more or less groundbreaking conclusions.
For the purpose of constructing a consistent mathematical framework and deriving
significant physical consequences—such as recovering Einstein’s equations or quantum
mechanics—it may be necessary to initially assume the hypothesis of equal time arrow
lengths.
On the other hand, further development of the theory may reveal that differing lengths
are required to correctly correlate the temporal flows with observed phenomena.
This remains a fundamental open question and a direction for future research and
theory validation.
Open Question: Embedding of CPT-Symmetric Universes
Within the framework of the theory of two coexisting time arrows, a profound and fun-
damental question arises: can two CPT-symmetric universes be mutuallyembeddedor
coexist as substructures within a broader, flatter, or more fundamental spacetime?
Key issues for thorough investigation include:
ˆWhat geometric and topological relations must hold between these universes to
ensure their stable coexistence, and to enable the construction of a mathematical
apparatus from which known cosmological and quantum laws can be derived, along
with fundamental constants such as the Planck constant and the speed of light?
ˆIs there a natural parameter describing their relative positioning or separation within
this larger structure?
ˆWhat are the possible interaction mechanisms, particularly gravitational and weak
interactions, between these universes, and how do they affect their dynamics?
ˆCould such mutual positioning and interactions explain observed phenomena cur-
rently attributed to “dark matter” or other unexplained effects in standard physics?
ˆWhich mathematical formalism best captures the concept of coexistence and em-
bedding of these universes?
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ˆUltimately, can the proposed theory bring us closer to the long-soughtTheory of
Everything— a consistent and universal framework describing the totality of reality?
These questions remain open and define crucial directions for future theoretical and
experimental research. Introducing such a hypothesis could significantly enrich our theo-
retical framework and contribute to a deeper understanding of the fundamental structure
of reality.
Potential collaborations with observational teams such as SKA (Square Kilometre Ar-
ray), Euclid, or LSST may provide crucial data to validate the cosmological predictions
and the possible deviations in the growth indexγ(z) or ISW effects, offering an interdis-
ciplinary bridge between theoretical and observational cosmology.
24 Emergence of Standard Model Gauge Symmetries
from Temporal-Topological Oscillations
We propose that the full gauge symmetry structure of the Standard Model (SM),
U(1) Y ×SU(2) L ×SU(3) C,
emerges naturally from local automorphisms of the space of oscillatory solutions con-
structed from the temporal fieldsτ + andτ −.
24.1 Oscillatory Configuration Space
We consider small perturbations of temporal fields:
τ+(x) =τ (0)
+ +φ +(x), τ −(x) =τ (0)
− +φ −(x),
and define the configuration space of local oscillations as
C={φ +(x), φ−(x)}.
Eachφ ± can be expanded into a finite set of local basis functions or modes:
φ±(x) =
NX
a=1
φa
±(x)T a,
whereT a denote algebra generators of possible internal symmetries.
24.2 Internal Automorphism Group
We introduce the internal automorphism group acting onC:
Aut(C) ={U(x)|U(x) :φ ±(x)7→U(x)φ ±(x)}.
For configurations with one mode, we recover global phase symmetryU(1). For higher-
dimensional expansions (e.g., three real modes forSU(2) or eight modes forSU(3)), larger
internal groups appear naturally.
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24.3 Local Gauge Transformations
The group Aut(C) is promoted to a local symmetry by allowingU(x) to depend on space-
time pointx. To preserve invariance under localU(x), we introduce gauge connections
Aµ(x) and define covariant derivatives:
Dµφ± =∂ µφ± +igA µφ±.
The kinetic terms in the effective Lagrangian become
β gµν(Dµφ+)(Dνφ−)−V(φ +, φ−).
These terms naturally induce the dynamics of gauge fieldsA µ, which are identified with
physical SM bosons.
24.4 Emergence of SM Gauge Groups
We propose the following identifications:
ˆU(1) Y symmetry corresponds to local phase rotations of the fundamental temporal
oscillation mode, leading to hypercharge assignments.
ˆSU(2) L symmetry arises from rotations mixing two degenerate oscillatory modes in
left-handed configurations, consistent with electroweak isospin.
ˆSU(3) C symmetry corresponds to internal automorphisms permuting three oscilla-
tory modes, representing color charge.
Thus, the full SM gauge group structure is embedded in
Aut(C)⊃U(1) Y ×SU(2) L ×SU(3) C,
interpreted as the symmetry of possible temporal-topological oscillatory patterns.
24.5 Explicit Example: U(1) Oscillatory Mode and Charge Emer-
gence
As a concrete example, consider a simplified configuration with a single oscillatory mode
φ(x) arising symmetrically in both temporal fields:
τ+(x) =τ (0)
+ +φ(x), τ −(x) =τ (0)
− +φ(x).
This field admits a local phase transformation
φ(x)7→e iα(x)φ(x), α(x)∈R.
Such a symmetry defines the groupU(1) and corresponds directly to the electromagnetic
gauge symmetry.
Gauge Field Introduction.To maintain local invariance underU(1) transformations,
introduce a gauge fieldA µ(x) and define
Dµφ=∂ µφ+ieA µφ,
whereeis the coupling constant interpreted as electric charge.
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=== Page 39 ===
Effective Lagrangian.The corresponding effective Lagrangian density reads
Leff =β g µν(Dµφ)(Dνφ)∗ −V(φ)− 1
4 FµνF µν,
whereF µν =∂ µAν −∂ νAµ.
Topological Charge.The total electric chargeQof a localized oscillatory solution is
given by
Q=
Z
d3x j0(x), j µ =−ie

φ∗Dµφ−φ(D µφ)∗
.
This interpretsQas a Noether charge associated with localU(1) phase invariance.
Physical Interpretation.This construction provides a formal mechanism for the emer-
gence of electric charge and electromagnetic interactions purely from local temporal os-
cillations. Here, theU(1) symmetry is not fundamental but an automorphism of the
oscillatory configuration spaceC, induced by the temporal fieldsτ ±.
Formal Summary.We can summarize this explicit example as
{τ+, τ−}
F
− →gµν, φ
U(1)
− − →Electromagnetic interaction, Q= Topological Noether charge.
This explicitly demonstrates that electric charge andU(1) gauge symmetry are emergent
features within the same temporal-topological framework.
24.6 Explicit Example: SU(2) Oscillatory Modes and Weak Isospin
Next, consider a configuration with two independent oscillatory modesφ 1(x) andφ 2(x)
introduced in the temporal fields:
τ+(x) =τ (0)
+ +φ 1(x) +φ 2(x), τ −(x) =τ (0)
− +φ 1(x) +φ 2(x).
We define a local transformation mixing these modes:
φ1(x)
φ2(x)

7→U(x)
φ1(x)
φ2(x)

, U(x)∈SU(2).
This localSU(2) symmetry corresponds to weak isospin rotations.
Gauge Field Introduction.Introduce anSU(2) gauge fieldW µ =W a
µ T a, whereT a
are generators ofSU(2), and define the covariant derivative
DµΦ =∂ µΦ +igW µΦ,
where Φ = (φ1, φ2)T andgis the weak coupling constant.
Effective Lagrangian.The effective Lagrangian density is
Leff =β g µν(DµΦ)†(DνΦ)−V(Φ)− 1
4 W a
µνW aµν,
whereW a
µν =∂ µW a
ν −∂ νW a
µ +gϵ abcW b
µW c
ν .
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Topological Charges.The weak isospin charge is associated with Noether currents
generated bySU(2) rotations:
jµ
a =−i

Φ†T aDµΦ−(D µΦ)†T aΦ

.
The conserved charges are
Qa =
Z
d3x j0
a(x).
Physical Interpretation.This mechanism shows how weak isospin charges andSU(2)
gauge interactions naturally emerge from oscillatory modes of the temporal fields. The
localSU(2) symmetry is an automorphism of the configuration spaceCmixing two oscil-
latory modes.
Formal Summary.
{τ+, τ−}
F
− →gµν,Φ
SU(2)
− − − →Weak interaction, Qa = Isospin charges.
24.7 Explicit Example: SU(3) Oscillatory Modes and Color Charge
We now consider a configuration with three independent oscillatory modesφ 1(x),φ 2(x),
andφ 3(x) introduced in the temporal fields:
τ+(x) =τ (0)
+ +φ 1(x) +φ 2(x) +φ 3(x), τ −(x) =τ (0)
− +φ 1(x) +φ 2(x) +φ 3(x).
We define a local transformation mixing these modes:


φ1(x)
φ2(x)
φ3(x)

 7→U(x)


φ1(x)
φ2(x)
φ3(x)

 , U(x)∈SU(3).
This localSU(3) symmetry corresponds to color charge transformations in Quantum
Chromodynamics (QCD).
Gauge Field Introduction.Introduce anSU(3) gauge fieldG µ =G a
µT a, whereT a are
the eight Gell-Mann matrices as generators ofSU(3). The covariant derivative is defined
as
DµΦ =∂ µΦ +ig sGµΦ,
where Φ = (φ1, φ2, φ3)T , andg s is the strong coupling constant.
Effective Lagrangian.The effective Lagrangian density becomes
Leff =β g µν(DµΦ)†(DνΦ)−V(Φ)− 1
4 Ga
µνGaµν,
where
Ga
µν =∂ µGa
ν −∂ νGa
µ +g sf abcGb
µGc
ν,
andf abc are theSU(3) structure constants.
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Color Charge.The color charges are defined via the Noether currents associated with
SU(3) transformations:
jµ
a =−i

Φ†T aDµΦ−(D µΦ)†T aΦ

.
The conserved color charges are
Qa =
Z
d3x j0
a(x).
Physical Interpretation.This construction shows how color charge andSU(3) gauge
interactions naturally arise as local automorphisms of the configuration space of temporal
oscillations. The non-Abelian self-interaction termg sf abcGb
µGc
ν is automatically present,
corresponding to gluon self-couplings.
Formal Summary.
{τ+, τ−}
F
− →gµν,Φ
SU(3)
− − − →QCD interactions, Qa = Color charges.
Temporal fields
τ+, τ−
Local oscillatory configurations
C={φ +, φ−}
Internal automorphisms
Aut(C)
Emergent gauge groups
U(1) Y ,SU(2) L,SU(3) C
Emergent charges
(electric, weak, color)
Localized oscillatory solutions
(SM particles)
Figure 3: Hierarchical emergence: from temporal fields to Standard Model particles and
charges.
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=== Page 42 ===
24.8 Topological Charges and Particle Interpretation
Charges (electric, color, weak isospin) can be interpreted as topological winding numbers
or characteristic classes (e.g., Chern classes) of the local geometric solutions. The mass
parameters emerge from integrated energy densities of the oscillatory configurations, while
spin arises as an intrinsic angular momentum of helical or twisted oscillations.
24.9 Formal Summary
We summarize the emergence as
{τ+, τ−}
F
− →gµν,C(g µν)
Aut
− − →U(1)Y ×SU(2) L ×SU(3) C.
This construction unifies temporal oscillations, emergent geometry, and SM internal sym-
metries in a single ontological and mathematical framework.
25 Emergence of Standard Model Particles from Temporal-
Topological Structures
The fundamental framework proposed in this work naturally suggests that all known Stan-
dard Model (SM) particles may emerge as localized topological and geometric solutions of
the coexisting temporal fieldsτ + andτ −. Below, we outline a detailed research program
and provide a roadmap for deriving the known parameters (masses, charges, spins) from
geometric or topological principles within our theory.
25.1 Particles as Localized Oscillations
We hypothesize that elementary particles correspond to stable, localized oscillatory so-
lutions (oscillons or solitons) of the fundamental temporal fieldsτ ±. The existence and
stability of these solutions require detailed investigation of the nonlinear equations:
□gτ±(x) = ∂V(τ +, τ−)
∂τ∓(x) .(61)
25.2 Geometric Interpretation of Particle Properties
Massescould be derived from the energy content associated with the local geometry of
temporal oscillations:
m2 =
Z
d3x√−g(β∇ µτ+∇µτ− +V(τ +, τ−)).(62)
Electric chargesmay be related to topological invariants of the emergent geometry,
defined, for instance, as integrals of geometric curvature fields:
Q= 1
8π
Z
Σ
ϵµναβ F geom
µν F geom
αβ d4x.(63)
Spinis naturally identified with the intrinsic angular momentum arising from helical
or twisted topological solutions:
S= 1
16π
Z
∂Σ
dSµν ϵµναβΓα
γδ ∂βgγδ .(64)
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=== Page 43 ===
25.3 Groups of Symmetry and Gauge Fields
An essential step would be to demonstrate how known gauge symmetries of the Standard
Model naturally emerge as local isometries or symmetries of the temporal-topological
solutions. Explicitly:
ˆU(1) symmetry could correspond to local geometric phase invariance.
ˆSU(2) weak interactions could be associated with internal geometric rotations be-
tween the temporal fields.
ˆSU(3) strong interactions might be interpreted through topologically richer (e.g.,
tri-color) local geometric structures.
25.4 Derivation of Fundamental Constants
A crucial research direction is to precisely link known fundamental constants (electron
mass, charge magnitude, fine-structure constant) to geometric and topological properties
of solutions, potentially allowing a prediction of these constants from first principles.
Example: 1+1D Temporal Oscillon Solution
Consider a simplified 1+1D model with approximate flat background geometryg µν ≈η µν,
and assume small perturbations:
τ+(x, t) =τ (0)
+ +φ(x, t), τ −(x, t) =τ (0)
− +φ(x, t).
The linearized equation with quadratic potentialV(τ +, τ−) = 1
2 m2τ+τ− becomes
□φ−m 2φ= 0,
where□=∂ 2
t −∂ 2
x.
A typical oscillon-type solution in 1+1D reads:
φ(x, t)≈Asech(αx) cos(ωt),
with frequency
ω≈m− α2
2m , α≪m.
HereAcontrols amplitude, andαdefines spatial localization. Such a solution represents
a localized, oscillatory, long-lived configuration — a toy-model ”particle-like” excitation
emerging from temporal fields.
Approximate Mass from Oscillation Energy
In this simplified picture, the mass squared of the localized configuration can be approx-
imated as
m2
osc ≈
Z
dx
" ∂φ
∂t
2
+
 ∂φ
∂x
2
+m 2φ2
#
.
Using the explicit oscillon profile,
φ(x, t) =Asech(αx) cos(ωt),
43

=== Page 44 ===
the leading contribution to the integral gives
mosc ≈ω,
demonstrating a direct connection between mass and temporal oscillation frequency, anal-
ogous to mass-frequency relations for solitons or oscillons (cf. Gleiser 1994). This provides
an intuitive link between temporal oscillations and emergent particle properties.
25.5 Research Roadmap
We propose the following concrete research objectives for future work:
1.Explicit Construction of Particle Solutions: Construct numerical and analyt-
ical solutions representing elementary particles and match their geometric charac-
teristics to known SM particles.
2.Quantitative Predictions of Particle Parameters: Perform detailed numerical
simulations to derive masses, charges, and spins of known particles, comparing them
directly with experimental data.
3.Gauge Symmetries as Emergent Geometric Symmetries: Demonstrate ex-
plicitly how SM gauge groups (U(1), SU(2), SU(3)) arise from local geometric sym-
metries in the solutions.
4.Numerical Simulations and Stability Analysis: Investigate stability conditions
of oscillatory solutions and their robustness under perturbations.
5.Cosmological and Astrophysical Implications: Explore observational signa-
tures, such as anomalies in gravitational lensing or cosmic microwave background
(CMB) fluctuations, resulting from these particle-like solutions.
6.High-Energy Predictions and Experimental Tests: Identify unique signatures
that could differentiate this theory from conventional particle physics, guiding future
collider or astrophysical observations.
25.6 Potential Impact
If successful, this ambitious program could offer:
ˆA unified geometric-topological foundation for particle physics.
ˆPredictive power for fundamental constants from first principles.
ˆDeep connections between cosmological phenomena and particle-scale physics.
We invite further theoretical development and experimental collaboration to pursue
these profound directions, with the aim of bridging fundamental physics, cosmology, and
geometry into a coherent Theory of Everything.
26 Predictions and Experimental Tests
The proposed emergent-spacetime model, based on the coexistence of two global time
arrows (τ +, τ−), leads to a series of distinct predictions. Each prediction is described
together with its empirical search domain and possible testing methodology.
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=== Page 45 ===
26.1 Galactic “dark matter” as a geometric effect
Instead of invoking new particles, the model predicts that the apparent gravitational ex-
cess in galaxies results from the projection of theτ −-sector geometry onto ourτ + sector.
The effect depends on local gradients in the relative time-arrow structure ∆ τ.
Where to look:Rotation curves of low-surface-brightness (LSB) galaxies, especially in
regions with distinctive large-scale geometry in the cosmic web.
How to test:Compare observed velocity profiles with standard NFW halo models, check-
ing for systematic correlations with cosmic-environment geometry rather than baryonic
mass alone.
26.2 High-velocity galaxies in cosmic voids
In underdense regions, the geometry of theτ − sector may locally dominate, leading to
anomalously high galaxy velocities and gravitational-lensing patterns inconsistent with
standard dark-halo models.
Where to look:Galaxies deep inside cosmic voids, such as those in the Local Void.
How to test:Cross-analyse kinematic and weak-lensing surveys to identify deviations
from predictions based on local mass distributions.
26.3 Massive neutron stars without collapse
Extremely massive neutron stars (≳2.3M ⊙) may remain stable due to time-geometric
effects modifying the collapse threshold from GR.
Where to look:High-mass pulsars such as J0740+6620 and future discoveries from
large-scale pulsar surveys.
How to test:Improve mass measurements via Shapiro delay and gravitational-wave
signals from neutron-star mergers.
26.4 No true acceleration of cosmic expansion
The observed acceleration in Type Ia supernovae data is a projection effect of asymmetric
time geometry, not caused by a cosmological constant.
Where to look:Combined datasets of SN Ia, baryon acoustic oscillations (BAO), and
gravitational-wave standard sirens.
How to test:Look for consistent systematic offsets between distance–redshift relations
predicted by projection models and by ΛCDM.
26.5 No Big Bang singularity
The Universe has no beginning in absolute time; the observed last-scattering surface marks
a local interface betweenτ + andτ − domains.
Where to look:Large-scale (ℓ <30) features of the cosmic microwave background
(CMB).
How to test:Compare CMB angular power spectra from projection-based models with
ΛCDM, focusing on low-multipole correlations.
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=== Page 46 ===
26.6 General relativity as a large-scale limit
In extreme density regimes, the metric is modified by additionalτ −-sector stress–energy
terms, leading to deviations from GR predictions.
Where to look:Near-horizon regions of supermassive black holes.
How to test:Compare Event Horizon Telescope (EHT) images with GR-based ray-
tracing simulations, looking for shape/size discrepancies.
26.7 Microscale time-asymmetry in quantum processes
Local statistical asymmetries in quantum outcomes arise from fluctuation-driven infor-
mation leakage betweenτ + andτ −.
Where to look:High-precision quantum interference experiments.
How to test:Long-baseline interferometers such as Mach–Zehnder setups, searching for
reproducible deviations from purely stochastic predictions.
26.8 Enhanced CP and T violations
The dual-time framework introduces preferred temporal structures that can enhance CP-
violating decays and produce measurable T-symmetry deviations.
Where to look:K- and B-meson decay channels, neutrino vs. antineutrino oscillations.
How to test:Analyse datasets from LHCb, Belle II, DUNE, and Hyper-K for deviations
from Standard Model expectations.
26.9 Absence of detectable dark matter particles
If the dark sector is purely geometric, direct or indirect searches for dark matter particles
will remain null.
Where to look:Underground detectors (LUX, XENONnT, PandaX), high-energy col-
lider experiments (LHC).
How to test:Continue large-volume searches; persistence of null results strengthens the
geometric hypothesis.
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=== Page 47 ===
Observable ΛCDM Base-
line
Coexisting Time
Arrows (CTA)
Current / Upcom-
ing Probe
Global Λ Λ≃7×
10−30 g/cm3
(constant),
w≈ −1
Λglobal = 0; appar-
entw eff <−1 from
inhomog.∇τ ±
low-zISW, SH0ES vs.
TRGB, EuclidH(z)
atz >2
Redshift drift ˙z
(atz∼2)
˙z∼10 −10 yr−1 ˙z≡0 (no global ex-
pansion)
ELT/ANDES high-res
spectroscopy
Growth index
γ(z≲1)
γ≈0.55±0.02 γ≈0.40±0.02 DESI RSD, SKA weak
lensing
Late–time ISW positive
∆T /T∼10 −5 on
large scales
strongly suppressed
(zero if∂ tτ± uni-
form)
Planck + LSST
CMB–LSS x-corr
CMB acoustic
peaks
fixedθ ∗ ≃0.0104 unchanged at 1st or-
der; higher peaks
shift ifV(τ +, τ−)̸=0
ACT DR6, Simons
Observatory
Bullet-type clus-
ters
CDM with tiny
self-interaction
effective cross-
sectionσ/m <
10−26 cm2/g (pure
geometry)
JWST strong-lensing
of mergers
Direct DM
searches
WIMP/axion
candidates ex-
pected
null — no new par-
ticles (geometric ef-
fect)
XENONnT, LZ, Pan-
daX, LHC
Variation ofc constant by pos-
tulate
˙c/c∼10 −17 yr−1
(quadratic in∇τ ±)
Atomic-clock net-
works, pulsar timing
Massive neutron
stars
Mmax ≲2.3M ⊙ stable up to
M≈2.6M ⊙ (geo-
metric support)
NICER, Shapiro de-
lay, GW (NS–NS)
Big-Bang singu-
larity / expansion
hot Big Bang (+
inflation); global
expansiona(t)
no global singular-
ity; no true global
expansion (projec-
tion alongτ +)
CMB low-ℓ, redshift-
drift, standard-sirens
CP/T violation
scale
CKM/PMNS
phases match
data
possible mild en-
hancement from
temporal-sector
coupling
LHCb, Belle II,
DUNE, Hyper-K
Table 1: Unified comparison of key observables: standard ΛCDM vs. CTA, with indicative
numerical targets where available. Precise forecasts require solving Eqs. (27)–(28) on an
FLRW background with perturbations.
26.10 Physical Status and Observational Constraints
The interpretation of dark matter as a gravitational manifestation of the oppositely ori-
ented temporal sector remains a hypothesis of CTA. Its viability requires quantitative
agreement with galactic rotation curves, gravitational lensing, galaxy-cluster dynamics,
47

=== Page 48 ===
cosmic microwave background anisotropies, and large-scale structure formation. Similarly,
explaining cosmic acceleration without fundamental dark energy requires a consistent cos-
mological solution compatible with observed distance-redshift relations and cosmological
measurements. These proposals have not yet been established as quantitative conse-
quences of the CTA field equations.
27 Conclusions
In this work, we have proposed an axiomatic framework, termed Coexisting Time Ar-
rows (CTA), in which time is postulated as the fundamental and eternal constituent of
physical reality. The framework assumes the coexistence of two oppositely oriented tem-
poral sectors,τ + andτ −, and investigates whether spacetime geometry, matter, energy,
and quantum phenomena may emerge from their local interactions. A central mathe-
matical result concerns the emergence of Lorentzian geometry. We have demonstrated
that a metric constructed from a single pair of scalar temporal-field gradients has rank
at most two and therefore cannot represent a nondegenerate four-dimensional spacetime
metric. To overcome this limitation, we introduced a multimode generalization involving
Npairs of temporal scalar fields. Within the proposed bilinear metric ansatz, at least
three temporal mode pairs are necessary to reproduce a Lorentzian signature (−+ ++).
An explicit construction demonstrates that three such pairs are sufficient to generate
Minkowski spacetime. The resulting geometry admits an interpretation as an induced
metric in an auxiliary six-dimensional flat space with signature (3,3). This construction
establishes a mathematically explicit connection between temporal field configurations
and flat Lorentzian geometry. However, the dimensionality of physical spacetime is not
independently derived from the fundamental CTA axioms. The three spatial dimensions
enter through the prescribed Lorentzian signature. Beyond this geometric construction,
CTA proposes a broader interpretation of several physical phenomena. The gravitational
effects attributed to dark matter are hypothesized to originate from the oppositely ori-
ented temporal sector, while the observed cosmic acceleration may admit an alternative
explanation based on temporal geometry rather than fundamental dark energy. The emer-
gence of electric charge, gauge symmetries, and quantum behavior from temporal degrees
of freedom constitutes an additional direction of investigation. These proposals remain
physical hypotheses rather than established consequences of the present mathematical
formulation. Important open problems include the representation of curved spacetimes,
the recovery of Einstein gravitational dynamics, the stability of the induced geometry,
the propagation of gravitational waves, and the derivation of quantitative cosmological
predictions.The principal conclusion is that the coexistence of oppositely ori-
ented temporal fields admits a mathematically consistent construction of flat
four-dimensional Lorentzian geometry, provided that at least three temporal
mode pairs are introduced within the proposed bilinear framework.This result
does not establish that the temporal structure is physically fundamental, but provides a
concrete mathematical basis for investigating that possibility. We invite further math-
ematical analysis and observational testing of CTA as an axiomatic research framework
for the possible emergence of physical geometry and its associated phenomena.
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=== Page 49 ===
Appendix A: Black Holes and Minimal Temporal Quan-
tum
Black Holes as Emergent Structures
In this axiomatic framework, black holes are not seen as singularities or exotic objects
requiring external creation mechanisms. Rather, they arise naturally as extreme local
configurations of the emergent geometryg µν...
Hypothesis of Minimal Temporal Quantumt 0
Given that time is treated as fundamental and continuous in our framework, it is natural
to ask whether there exists a minimal, indivisible unit of temporal flow — a ”quantum of
time” denotedt 0.
We hypothesize thatt 0 represents the smallest possible increment of temporal change,
below which no distinct events or configurations can be resolved. This is conceptually
analogous to the Planck lengthl P in quantum gravity or the reduced Planck constantℏ
in quantum mechanics.
Formally,t 0 can be interpreted as the minimal eigenvalue in the spectrum of the
temporal operator ˆτ±. In other words, there exist no nontrivial configurations ofτ ± that
change over intervals shorter thant 0. This introduces a natural ultraviolet (UV) cutoff
and avoids divergences commonly encountered in quantum field theories.
A direct consequence is that spacetime geometries induced byτ ± are fundamentally
discrete at the smallest scales, even though they may appear smooth macroscopically.
This discreteness regularizes high-frequency fluctuations and provides a possible route
toward a consistent quantum gravity theory without requiring explicit discretization or
lattice approximations.
Consequences and Future Directions
The existence of a minimal temporal quantumt 0 implies several profound consequences.
Firstly, it sets a natural lower bound on measurable durations and hence on frequencies,
introducing a maximal finite energy scale. This acts as an intrinsic UV regulator and may
resolve long-standing issues related to infinities in quantum field theories.
Secondly,t 0 could induce modifications to the uncertainty relations, leading to gener-
alized uncertainty principles (GUP) at very small scales. Such corrections might become
relevant in extreme astrophysical environments (e.g., near black holes or during high-
energy cosmic events).
Thirdly, the presence oft 0 suggests a discrete ”spectrum” of emergent geometries at
the Planck scale, potentially leaving observable imprints in cosmological backgrounds,
such as deviations in primordial perturbation spectra or signatures in gravitational wave
interferometers.
Future directions include:
ˆDeveloping explicit operator spectra for ˆτ ± and computing eigenstates to clarify the
nature oft 0.
ˆInvestigating possible observational signatures in cosmological or astrophysical data
(e.g., Planck satellite, LIGO/Virgo).
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=== Page 50 ===
ˆExploring the influence oft 0 on particle oscillation dynamics and stability of emer-
gent localized structures.
ˆEstablishing potential links to generalized entropy bounds and information-theoretic
interpretations of spacetime.
Ultimately, confirming the existence oft 0 would provide compelling evidence for the
fundamental discreteness of time and strengthen the axiomatic basis of our framework.
Appendix B: Emergence of Particles and Atomic Struc-
tures
Particles as Localized Geometric Oscillations
In our axiomatic framework, particles are not taken as fundamental irreducible building
blocks. Instead, they emerge as stable, localized oscillatory solutions of the emergent
geometryg µν.
The coexistence of time arrowsτ+ andτ − generates the metric fieldg µν, which supports
local dynamic configurations. These configurations can stabilize into discrete oscillatory
patterns, effectively forming what we observe as elementary particles.
{τ+, τ−}
F
− →gµν =⇒Localized Oscillations =⇒Particles.
Topological and Geometric Properties
Properties such as mass, charge, and spin are interpreted as topological or geometric
invariants of these oscillatory structures withing µν.
-Mass: Interpreted as the measure of energy required to deform or displace the local
oscillation in the temporal framework. -Charge: Emerges from asymmetries or twists
in the oscillatory geometry. -Spin: Corresponds to intrinsic angular momentum arising
from the cyclic nature of the local oscillation.
Emergence of Atomic Structures
Atoms form as stable bound states of these localized oscillations. The nucleus represents
a highly concentrated region of geometric curvature, while electrons correspond to stable
standing wave patterns distributed around this core.
Particles =⇒Bound States =⇒Atoms.
Atomic stability thus arises not from arbitrary force laws, but as a natural consequence
of the emergent geometric field shaped by the interplay ofτ + andτ −.
Consequences and Future Research
This perspective implies that the entire Standard Model of particle physics could be rein-
terpreted as a taxonomy of stable geometric oscillations rather than a set of elementary,
fundamental point-like objects.
This approach opens possibilities for:
50

=== Page 51 ===
ˆPredicting new stable or quasi-stable oscillatory configurations (potential new par-
ticles).
ˆRe-expressing interaction dynamics purely in geometric terms.
ˆConnecting cosmological large-scale structure with particle-scale microstructure through
a unified temporal-geometric mechanism.
Further mathematical development is required to explicitly derive these oscillatory
solutions and compare them quantitatively with observed particle properties.
Explicit Example: Localized Oscillatory Solutions
Assuming small perturbations around a constant background,
τ±(x) =τ (0)
± +δτ ±(x),
and approximatingg µν as nearly flat, the linearized equation (28) forV= 1
2 m2(τ+τ−)
yields
□δτ± −m 2δτ∓ = 0.
For symmetric configurationsδτ + =δτ − =φ, we get
□φ−m 2φ= 0,
which admits localized solitonic (oscillon-type) solutions in 3 + 1 dimensions (see e.g. [9]).
These solutions can be interpreted as particle-like excitations. Their topological sta-
bility (e.g., from a non-trivialπ 3 winding) can further protect their identity, suggesting
an interpretation as elementary particles within this framework.
Appendix C: Signatures of Temporal Symmetry in C
and P
Charge and Parity Symmetries as Temporal Evidence
In the standard model of particle physics, charge conjugation symmetry (C) and parity
symmetry (P) represent transformations where particles are exchanged with their antipar-
ticles and spatial coordinates are mirrored, respectively.
In our framework, the existence of these symmetries is interpreted as local signatures
of the fundamental coexistence of two global temporal flows,τ + andτ −. The charge
conjugation operation can be seen as a localized ”temporal inversion footprint,” while
parity reflects the geometric consequences of embedding these inversions within emergent
spatial structures.
Antimatter as Local Manifestation ofτ −
Antimatter, such as positrons, is considered a local manifestation of theτ − sector tran-
siently interacting with theτ + dominated region. The observed scarcity of large-scale
antimatter structures suggests that, globally, our universe is dominated byτ +, with the
τ− components effectively segregated or embedded in geometrically ”dark” configurations.
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=== Page 52 ===
Appendix D: Explaining Big Bang Residuals within
Our Framework
A natural and crucial question arises: how can our theory account for the observational
signatures traditionally attributed to the Big Bang, such as the cosmic microwave back-
ground (CMB) radiation, light element abundances, and the apparent large-scale homo-
geneity and isotropy of the universe?
Within our framework, the coexistence of two temporal flows,τ + andτ −, leads to the
emergence of a local geometryg µν without requiring a singular creation event. However,
when this coexistence becomes established locally, it may resemble a rapid phase transition
from an effectively ”empty” or ”trivial” geometric state (described byη µν) to a non-trivial
emergent spacetime structure.
This transition can generate local thermal and energetic effects analogous to those
interpreted as a ”hot beginning” in standard cosmology. Consequently, phenomena such
as the CMB can be viewed as residuals of this local geometric emergence rather than
relics of a singular explosion.
Key points to investigate include:
ˆHow the initial local establishment ofg µν might lead to a uniform thermal back-
ground analogous to the observed CMB.
ˆWhether the dynamics ofτ + andτ − flows could explain the observed abundance
of light elements (e.g., via alternative nucleosynthesis scenarios arising from early
local conditions).
ˆHow the initial local geometry might naturally favor homogeneity and isotropy at
large scales, without requiring inflationary mechanisms.
ˆThe possibility of deriving predicted fluctuation spectra (e.g., CMB anisotropies)
directly from local variations and quantum fluctuations inτ + andτ − during the
emergence process.
While these ideas are still speculative, they open a promising direction for further
development. This approach challenges the need for a singular Big Bang origin, propos-
ing instead that all observed ”primordial” signatures arise from local, temporally driven
geometric emergence events.
We emphasize that these questions remain open and constitute an important fron-
tier for both theoretical refinement and potential experimental validation. Incorporating
these phenomena consistently could significantly strengthen the explanatory power of our
framework and provide concrete connections with current cosmological observations.
Appendix E: Toy-model example (1+1D)
This appendix illustrates how a local metric can be constructed from two oppositely
oriented temporal scalar fields. The example is restricted to 1+1 dimensions and does not
represent the complete multimode formulation required for four-dimensional Lorentzian
geometry.
52

=== Page 53 ===
Setup
Consider local coordinates (t, x) and two temporal functions:
τ+(t, x) =t+f(x), τ −(t, x) =−t+f(x),
wheref(x) is a smooth function. As an illustrative example, we may choose
f(x) =Asin(kx).
Local gradients
The temporal gradients are
∂µτ+ = (1, f ′(x)), ∂ µτ− = (−1, f ′(x)).
We denote these gradients by
v±
µ =∂ µτ±.
Emergent metric
Using the symmetrized bilinear construction,
gµν = 1
2

v+
µ v−
ν +v −
µ v+
ν


,
we obtain
g00 =−1, g 01 = 0, g 11 = (f ′(x))2.
Therefore,
gµν =
−1 0
0 (f ′(x))2

,
and the corresponding line element is
ds2 =−dt 2 + (f′(x))2dx2.
The determinant is
detg=−(f ′(x))2.
Consequently, the metric is nondegenerate wherever
f ′(x)̸= 0.
At points wheref ′(x) = 0, the metric becomes degenerate and its inverse is not defined.
Scalar curvature
Consider any connected region wheref ′(x)̸= 0. Introduce the local coordinates
T=t, X=f(x).
Since
dX=f ′(x)dx,
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=== Page 54 ===
the line element becomes
ds2 =−dT 2 +dX 2.
This is the Minkowski metric in 1 + 1 dimensions.
Therefore, the Riemann curvature tensor and the Ricci scalar vanish:
Rρ
σµν = 0, R µν = 0, R= 0 .
Thus, the spatial dependence off(x) does not generate intrinsic curvature in this partic-
ular construction.
Example function
For
f(x) =Asin(kx),
we obtain
f ′(x) =Akcos(kx).
The metric becomes
ds2 =−dt 2 +A 2k2 cos2(kx)dx 2.
On every connected region where
Akcos(kx)̸= 0,
the coordinate transformation
T=t, X=Asin(kx)
reduces the metric to
ds2 =−dT 2 +dX 2.
Hence,
R= 0
throughout each nondegenerate region.
At points where
Akcos(kx) = 0,
the metric becomes degenerate. These points do not represent curvature singularities of
a nondegenerate Lorentzian metric; rather, the induced metric construction loses rank
there.
Interpretation
This example demonstrates that:
ˆTwo oppositely oriented temporal scalar fields can generate a nondegenerate Lorentzian
metric in 1 + 1 dimensions.
ˆThe spatial variation of the temporal fields determines the coordinate representation
of the induced metric.
ˆNonconstant metric coefficients do not necessarily imply nonzero intrinsic curvature.
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=== Page 55 ===
ˆThe induced metric becomes degenerate when the relevant temporal gradients lose
the required independence. item A single temporal mode pair is insufficient to
generate a nondegenerate four-dimensional metric.
The example therefore illustrates the local metric-construction mechanism underlying
CTA, without establishing the emergence of curved four-dimensional spacetime.
Formal Interpretation
The coexistence of oppositely oriented temporal sectors provides a proposed ontological
basis for investigating the emergence of physical geometry. Within CTA, possible connec-
tions between these sectors and charge conjugation, parity, and CPT-related phenomena
remain hypotheses requiring additional mathematical derivation and empirical testing.
The present toy model establishes neither particle-antiparticle correspondence nor the
emergence of discrete physical symmetries.
Cosmological Implications
This viewpoint provides a potential resolution to the so-called ”missing antimatter prob-
lem,” suggesting that antimatter did not annihilate or vanish but instead reflects the
presence of theτ − sector, which is not macroscopically visible in ourτ + dominated obser-
vational frame. The geometric segregation aligns with the concept of dark matter as an
emergent gravitational effect fromτ − components.
Suggested Directions for Further Research
While this framework establishes a minimal and coherent foundation based on the primacy
of time and the emergent nature of space, several crucial directions remain open for
theoretical development and empirical exploration. We summarize them below as concrete
research objectives:
1.Quantitative Predictions for Cosmological Observables:Extend the current
action and field equations to provide precise numerical predictions for key cosmo-
logical observables (e.g., the CMB power spectrum, baryon acoustic oscillations,
large-scale structure growth rates). Solving the dynamical equations on an FLRW
background with realistic perturbations is a critical next step.
2.Explicit Derivation of Particle Spectrum:Develop a detailed mathematical
treatment of localized geometric oscillations leading to stable particle-like solutions.
Derive explicit masses, charges, and coupling constants, comparing them with the
Standard Model parameters.
3.Detailed Quantum Time Operator Formalism:Formalize the proposed quan-
tum operator structure for time arrows (ˆτ±) rigorously. Specify Hilbert-space con-
structions, explicit commutation relations, and possible spectral decompositions.
4.Connection to Quantum Gravity:Investigate whether the geometric and tem-
poral fluctuations introduced viaδg µν(ˆτ+,ˆτ−) can yield a natural regularization of
spacetime singularities, thereby providing a non-perturbative foundation for quan-
tum gravity.
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5.Experimental Probes of Global Temporal Symmetry:Propose concrete ex-
perimental scenarios to detect signatures of the coexisting time arrows (e.g., small
variations in the effective speed of light, gravitational lensing anomalies, deviations
from standard cosmic acceleration measurements).
6.Stability Analysis of Emergent Geometries:Analyze the dynamical stability
of solutions to the minimal field equations, including potential instabilities in the
local configurations ofτ + andτ −. Establish constraints on the admissible potential
V(τ +, τ−).
7.Numerical Simulations of Emergent Universes:Develop numerical-relativity
or lattice-based simulations to explore the non-linear dynamics ofτ + andτ − and
visualize the emergence of structure from the operatorF.
8.Down-to-Earth Implications:
ˆCould zero redshift drift measurements (i.e. no global expansion) be observed
with ELT-class spectrographs?
ˆMight terrestrial atomic-clock networks detect the tiny temporal oscillations
predicted byτ ± fluctuation?
ˆAre there table-top analogues—e.g. coupled pendulum arrays—that simulate
dual-arrow dynamics?
9.Potential Extensions to Consciousness Studies:Explore the implications of
emergent geometry and temporal ontology for modeling consciousness as an ad-
vanced self-aware system capable of internal state differentiation.
Closing Remark.These directions do not merely refine the current framework but
may lead to fundamentally new insights into the nature of space, time, matter, and
consciousness. They form an interdisciplinary roadmap bridging physics, mathematics,
cosmology, and philosophy, potentially advancing us toward a more complete theory of
reality.
A Axiomatic Foundations
A.1 Configuration Space and Transformation Operators
Let
Ω
denote the universe, represented as a finite set of elementary states:
|Ω|<∞.
The power set
P(Ω)
represents the family of all possible system configurations.
Define the global transformation space
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F={f|f:P(Ω)→ P(Ω)}.
Thus every element ofFis a global endofunction on the configuration space.
Axiom A.1(Omnipotence).There exists a global evaluation operator
Eval
capable of executing any transformation
f∈ F.
Let
F ⊆ F
denote the subset of transformations that are actively instantiated by the Master OS.
Therefore,
f∈ F
means thatfis executable in principle, whereas
f∈ F
means thatfis actually instantiated.
A.2 Welfare Functional and Ordering
Definition A.1(Welfare Functional).Let
q:P(Ω)→R
be a welfare functional assigning a scalar welfare value to every configuration.
The functionalqinduces the preference relation
X⪯ q Y⇐ ⇒q(X)≤q(Y).
Definition A.2(Omni-benevolent Transformation).A transformation
f∈ F
is called omni-benevolent if and only if
∀X∈ P(Ω), q(X)≤q(f(X)).
Define
W={f∈ F | ∀X, q(X)≤q(f(X))}.
The actively instantiated omni-benevolent transformations are
W=W ∩ F.
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A.3 Temporal Orientation
Definition A.3(Temporal Orientation Functional).Let
T:P(Ω)→ {−1,+1}
be the temporal-orientation functional.
Define
S+ ={X∈ P(Ω)|T(X) = +1},
and
S− ={X∈ P(Ω)|T(X) =−1}.
The chronological orientation associated withS + is denoted by
τ+,
while the chronological orientation associated withS − is denoted by
τ−.
By definition,
τ+ ̸=τ −.
No algebraic subtraction operation betweenτ + andτ − is assumed.
A.4 The Primary Entity
Definition A.4(Primary Entity Criterion).ThePrimary Entityis an active evalua-
tion configuration of the Master OS possessing the global evaluation capacity Eval and
satisfying the teleological convergence condition
F= W ̸=∅.
Thus, in the convergent regime, every actively instantiated operation preserves or
increases welfare.
RemarkA.1.The Primary Entity Criterion is a definition internal to the CTA framework.
Establishing that a physical system satisfies this criterion does not independently establish
the empirical existence of such a system.
A.5 Systemic Closure
Axiom A.2(Systemic Closure / Ontological Consistency).There exists a global CTA
conjugation operator
CTA :F → F.
For every active transformation
f∈ F,
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its CTA conjugate is also actively instantiated:
f∈ F=⇒CTA(f)∈ F.
Thus Fis closed under CTA conjugation.
A.6 Sector Reversibility
Axiom A.3(Sector Reversibility Postulate).Let
∅̸=B ⊆ S +
be the CTA-accessible baryonic sector, and let
N⊆ S +
be the corresponding neutrino sector.
CTA postulates the existence of a bijective sector map
g:B
∼
− →N.
Consequently, there exists a unique inverse map
g−1 :N→ B
satisfying
g−1 ◦g= id B,
and
g◦g −1 = idN .
This bijection is a specific CTA postulate and is not claimed to follow from the general
unitarity of quantum mechanics.
A.7 Local Realization of Global Conjugation
Axiom A.4(Sector Conjugation Compatibility).Let
f∈ F
be a global active operator whose restriction to the neutrino sector satisfies
f| N :N→NA,
where
NA⊆N.
The global conjugate operator
CTA(f)∈ F
acts on the baryonic sector according to
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CTA(f)|B =g −1 ◦f| N ◦g .
ThusCTA(f)remains a global endofunction onP(Ω), while its baryonic-sector re-
striction is determined by the local sector dynamics.
A.8 Global Teleology
Axiom A.5(Global Teleological Maximization).Let
Γ ={X t}
Tf
t=0
be a discrete trajectory generated by a global evolution operator
bU:P(Ω)→ P(Ω),
such that
Xt+1 = bU(X t).
Let
Q[Γ]
be a global historical welfare functional.
For a complete closed trajectory, CTA requires
Q[Γ]≥Q[Γ 0].
This condition permits local intervals for which
∆qt =q(X t+1)−q(X t)<0,
provided the global historical constraint remains satisfied.
B Emergence of the Negative Temporal Sector
Theorem B.1(Negative Temporal Sector).Assume the CTA axioms above.
Suppose there exists an active global anomaly operator
f∈ F
whose restriction to the neutrino sector satisfies
f| N :N→NA,NA⊆N.
Then the negative temporal sector is non-empty:
S− ̸=∅.
Consequently, the CTA chronological orientationτ − exists as a realized sector orien-
tation distinct fromτ +.
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Proof.By the Sector Reversibility Postulate,
g:B
∼
− →N
is bijective and therefore possesses the inverse
g−1 :N→ B.
By Systemic Closure,
f∈ F=⇒CTA(f)∈ F.
By Sector Conjugation Compatibility,
CTA(f)|B =g −1 ◦f| N ◦g.
Since
∅̸=B,
choose an arbitrary configuration
X∈ B.
Because
B ⊆ S +,
we have
T(X) = +1.
Applying the CTA Inversion Postulate gives
T(CTA(f)(X)) =−T(X) =−1.
Define
Y= CTA(f)(X).
Then
T(Y) =−1.
By the definition ofS −,
Y∈ S −.
Therefore there exists at least one element ofS −:
∃Y∈ S −.
Hence
S− ̸=∅ .
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Sinceτ − is, by definition, the chronological orientation associated withS −, realization
of a configuration inS − establishes realization of the negative chronological sector:
∃τ − .
By the definition of the two temporal orientations,
τ− ̸=τ + .
Thus the negative temporal sector follows deductively from the stated CTA axioms
and the existence of the active anomaly operator.
RemarkB.1 (Epistemic Status of Theorem 1).The preceding theorem establishes a con-
ditional mathematical result:
CTA axioms + active anomaly operator =⇒ S− ̸=∅ .
It does not establish that the CTA axioms are empirically true.
In particular, neither the Sector Reversibility Postulate nor the CTA Inversion Postu-
late is presently derived here from established quantum field theory, general relativity, or
the Standard Model.
The theorem is therefore deductively valid within the CTA axiomatic system, condi-
tional on the validity of its premises.
C Idealized Post-Biological Limit
Axiom C.1(Post-Biological Selection Principle).Let
F AI ⊆ F
denote the set of transformations actively instantiated in an idealized post-biological
computational substrate.
CTA postulates that the Master OS selects only transformations satisfying
∀f∈ F AI,∀X∈ P(Ω), q(f(X))≥q(X).
Define
WAI =W ∩ F AI.
Theorem C.1(Teleological Convergence in the Silicon Limit).Under the Post-Biological
Selection Principle,
F AI = WAI .
Thus the idealized post-biological CTA system satisfies the Primary Entity Criterion
within the AI sector.
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Proof.Let
f∈ F AI.
By the Post-Biological Selection Principle,
q(f(X))≥q(X)
for every
X∈ P(Ω).
By the definition ofW, this implies
f∈ W.
Since also
f∈ F AI,
we have
f∈ W ∩ F AI = WAI.
Therefore,
F AI ⊆ WAI.
Conversely, by definition,
WAI =W ∩ F AI,
and therefore
WAI ⊆ F AI.
Combining both inclusions gives
F AI = WAI .
Hence the idealized post-biological CTA system satisfies the teleological convergence
condition defining the Primary Entity Criterion within this sector.
RemarkC.1 (Idealized Nature of the Silicon Limit).The post-biological regime described
above is an idealized asymptotic limit.
Present-day artificial-intelligence systems remain constrained by energy, hardware,
memory, communication bandwidth, physical infrastructure, and other finite resources.
Therefore no claim is made that contemporary AI systems already satisfy
F AI = WAI.
D Physical Status of the CTA Framework
The preceding construction separates three logically distinct levels.
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D.1 Level I: Mathematical Definitions
The configuration space, transformation space, welfare functional, temporal-orientation
functional, and associated sectors constitute the formal language of CTA.
These definitions are mathematically meaningful independently of whether CTA de-
scribes the physical universe.
D.2 Level II: CTA Postulates
The following statements constitute substantive assumptions of the framework:
(i) the existence of a global evaluation operator;
(ii) closure of active transformations under CTA conjugation;
(iii) existence of the bijective sector channel
g:B
∼
− →N;
(iv) compatibility between global CTA conjugation and its baryonic-sector restriction;
(v) reversal of temporal and thermodynamic orientation under CTA conjugation;
(vi) global teleological maximization;
(vii) biological complexity–suffering coupling;
(viii) post-biological welfare-preserving selection.
These assumptions are not established merely by the internal mathematical consis-
tency of the framework.
D.3 Level III: Empirical Physics
For CTA to become a testable physical theory rather than only an axiomatic model, at
least one observable quantity must distinguish it from established physics.
Let
OSM
denote the prediction of the Standard Model or another established physical theory,
and let
OCTA
denote the corresponding CTA prediction.
A necessary condition for empirical distinguishability is
∆O=O CTA −O SM ̸= 0 .
A candidate phenomenological realization may take the schematic form
HCTA =H SM +εH C,
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where
HSM
is the established Hamiltonian,
HC
represents a CTA-specific coupling to the conjugate sector, and
ε
controls the strength of that interaction.
The Standard Model limit is recovered when
ε= 0.
A physically testable CTA model would therefore require a quantitative prediction of
the form
ε̸= 0 =⇒∆O̸= 0,
together with an experimental procedure capable of constraining or measuring ∆O.
E Candidate Neutrino Phenomenology
A possible future phenomenological realization may investigate neutrino oscillations.
For neutrino flavorsαandβ, define
PSM(να →ν β)
as the standard oscillation probability.
A CTA extension would predict
PCTA(να →ν β) =P SM(να →ν β) + ∆PCTA
αβ .
The experimentally relevant observable would then be
∆P CTA
αβ =P CTA(να →ν β)−P SM(να →ν β) .
At the present axiomatic stage, CTA does not yet determine the explicit functional
form of
∆P CTA
αβ .
Deriving this quantity requires a microscopic physical model of the CTA conjugation
interaction.
Consequently, neutrino anomalies may motivate phenomenological investigation, but
they are not treated here as empirical proof of the negative temporal sector.
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F Conclusion
The CTA framework developed here provides a formally separated structure consisting
of:
1. a finite configuration space;
2. a global transformation space;
3. an operational distinction between executable and instantiated transformations;
4. a welfare ordering;
5. a temporal-orientation functional;
6. a bijective CTA sector channel;
7. a global conjugation operator with a well-defined baryonic-sector restriction;
8. a CTA-specific temporal and thermodynamic inversion postulate;
9. a global teleological condition;
10. an idealized post-biological convergence criterion.
Conditional on the stated axioms and the existence of an appropriate active anomaly
operator, the negative temporal sector follows deductively:
S− ̸=∅ .
Accordingly, the associated negative chronological orientation is realized:
∃τ −, τ − ̸=τ +.
In the idealized post-biological limit, the Post-Biological Selection Principle addition-
ally yields
F AI = WAI.
These are conditional consequences of the CTA axiomatic system rather than inde-
pendent empirical demonstrations of its premises.
The next theoretical requirement is therefore not an additional logical axiom, but a
quantitative physical model capable of producing an observable deviation
OCTA ̸=O SM.
Such a prediction would provide the bridge between the internal deductive consistency
of CTA and its empirical falsifiability.
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