Section 9: CPT Symmetry and Relational Time in Temporal Flow Dynamics

Core Claim

In Temporal Flow Physics (TFP), CPT symmetry is not imposed but emerges or breaks dynamically through evolution of local flow polarity, denoted sgn(F_i). This polarity is self-organized, shaped by network interactions.

• Time-reversal (T) symmetry is broken by local asymmetries in phase flow coupling and interaction energetics.
• Charge conjugation (C) symmetry is likely broken by internal potentials.
• Parity (P) symmetry is preserved by symmetric network topology.
• The combination leads to emergent large-scale CPT violation.

Characteristic Units Recap

Quantity	Physical Dimension
Characteristic Length (Lc)	[L]
Characteristic Time (Tc)	[T]
Characteristic Energy (Ec)	[M L² T⁻²]
Characteristic Speed (cchar)	Lc / Tc, [L T⁻¹]
Characteristic Mass (Mc)	Ec / cchar² = Ec Tc² / Lc², [M]
Characteristic Action (ħc)	Ec × Tc, [M L² T⁻¹]

9.1 Temporal Neutrality and Causal Breaking

Fundamental flow variable: F_i(t + Δt), F_i(t), dimensionless [1].
Relational causal step index t dimensionless [1], physical time t_phys = t × T_c [T].
Discrete step Δt dimensionless [1], physical step Δt_phys = Δt × T_c [T].
Interaction and noise terms dimensionless, directly modifying F_i.

9.1.2 Polarity-Induced Asymmetry

Define local polarity at node i:
p_i = Σ_{j ∈ N(i)} sin(θ_ij)
where phase difference θ_ij = θ_i − θ_j
All dimensionless [1].
Polarity p_i quantifies net phase gradient bias driving asymmetric flow updates.

Flow evolution obeys second-order discrete time differential relation:
∂²F_i / ∂t² = −λ × p_i × F_i
where λ is polarity-driven flow alignment strength, dimensionless [1].
Noise modeled as ε × ξ_i(t), with ε noise scale and ξ_i(t) Gaussian noise, dimensionless [1].
Flow attractors ±v are dimensionless stable states.

9.2 Relational Time Arrows

Local time arrow on link A_ij defined as dimensionless sign function.
Entropy over domain D:
S_D = − Σ_{i ∈ D} ρ_i × log(ρ_i), dimensionless [1],
with probability density
ρ_i = |F_i|² / Σ_{j ∈ D} |F_j|², dimensionless [1].
Physical entropy: S_{D_phys} = S_D × k_B, with Boltzmann constant k_B (units [M L² T⁻² K⁻¹]).

9.3 Discrete Symmetry Properties

Symmetry	Status in TFP	Mechanism
Charge (C)	Broken	Reversing sgn(Fi) alters internal potential unless symmetric attractors exist
Parity (P)	Preserved	Network topology symmetric under spatial relabeling
Time (T)	Broken	Polarity-driven updates induce temporal bias
CPT	Broken	Broken T and C combined with preserved P

9.4 Physical Implications

Matter–antimatter asymmetry relates average polarity ⟨p_i⟩ to neutrino–antineutrino mass difference Δm_ν:
Δm_ν ≈ (ħ × ⟨p_i⟩ / t_P) / c_char²
where ħ = Planck constant ([M L² T⁻¹]), t_P = Planck time ([T]), and c_char = characteristic speed ([L T⁻¹]).

Black hole interiors may exhibit polarity reversal, emitting time-inverted Hawking quanta.
Cosmic inflation linked to rapid global p_i alignment, setting cosmic time arrow.

9.5 Stress-Energy and Dimensional Consistency

Dimensionless effective stress-energy tensor:
T_eff(μν) = (∂_μ δF)(∂_ν δF) − ½ G_μν × [ (∂_ρ δF)(∂^ρ δF) + V(δF) ]
where δF and derivatives are dimensionless [1], and G_μν is dimensionless.

Physical stress-energy tensor:
T_phys(μν) = C_T × T_eff(μν)
Conversion factor:
C_T = m_P × c_char² / ℓ_P
with C_T units [M L⁻¹ T⁻²], matching energy density units in General Relativity.

9.6 Figures and Tables

Figure 1: Illustration of relational time arrows between nodes.

Table 1: Summary of discrete symmetry breaking in TFP.

9.7 Forward Connections

Refer to Sections 10, 12, and 14 for discussions on quantum coherence, cosmological evolution, and flow condensation phenomena.

9.8 Emergent Polarity from Network Dynamics

Restate polarity metric p_i (dimensionless).
Flow update equation with polarity and noise:
∂²F_i / ∂t² = −λ × p_i × F_i + ε × ξ_i(t)
Noise represents intrinsic fluctuations.
Stable polarity attractors ±v correspond to preferred flow directions; signs of F_i and p_i define flow stability.

9.9 Interactions Between Flow Bias and Coupling Evolution (α Dynamics)

Recall coupling update from Section 6:
α(i,j,n+1) = α(i,j,n) × exp[−δ × (F_i − F_j)²] × topology_factor(i,j,n)
Now introduce polarity dependence by modulating effective friction:
δ_eff(i,j) = δ × [1 + κ × |p_i − p_j|]
where κ is a dimensionless parameter controlling polarity influence.

Interpretation:
Larger polarity mismatch |p_i − p_j| increases effective informational friction δ_eff, accelerating coupling decay.
Similar polarity biases reduce friction, stabilizing coupling and coherence.
This feedback stabilizes causal time arrows and dynamically shapes emergent geometry (Section 7) via relational polarity patterns embedded in coupling evolution.

9.10 Philosophical and Physical Interpretation

The arrow of time and CPT violation arise naturally from network self-organization, not imposed laws.
Temporal flow polarity encodes causal direction, irreversibility, and matter–antimatter asymmetry.
Interplay of flow coupling α and polarity p forms the substrate for temporal asymmetry and emergent spacetime curvature.
Time and symmetry are relational, dynamic properties of fundamental information flows, not fixed backgrounds.

Summary: Key Dimensionless Quantities

Quantity	Definition / Role	Dimension
Fi	Local flow variable at node i	Dimensionless [1]
pi	Local polarity, net phase gradient bias	Dimensionless [1]
α(i,j,n)	Coupling strength between nodes i and j	Dimensionless [1]
δ	Sensitivity to flow misalignment	Dimensionless [1]
λ	Polarity alignment strength	Dimensionless [1]
ε	Noise scale	Dimensionless [1]
ξi(t)	Gaussian noise term	Dimensionless [1]