Section 12: Emergence of Gauge Forces from Temporal Flow Multiplets and Role of γ_asym

Characteristic Units Recap (prior sections)

• Characteristic Length: L_c, dimension [L]
• Characteristic Time: T_c, dimension [T]
• Characteristic Energy: E_c, dimension [M L² T⁻²]
• Characteristic Speed: c_char = L_c / T_c, dimension [L T⁻¹]
• Characteristic Mass: M_c = E_c / c_char² = E_c × T_c² / L_c², dimension [M]
• Characteristic Action: ħ_c = E_c × T_c, dimension [M L² T⁻¹]

12.1 Flow Multiplets and Internal Structure of Gauge Fields

Each network node i supports an n-component complex flow multiplet:
Ψ_i(t) = [F_i⁽¹⁾(t), F_i⁽²⁾(t), ..., F_i⁽ⁿ⁾(t)]
with components:
F_i^(k)(t) = A_i^(k)(t) × exp(i × θ_i^(k)(t))
Amplitudes A_i^(k)(t): dimensionless and normalized
Phases θ_i^(k)(t): dimensionless, encoding internal flow orientation
Ψ_i(t) overall dimensionless.
Physical mapping: Ψ_i(t) × √E_c with dimension [M^1/2 L T⁻¹], representing internal degrees of freedom of emergent gauge fields.

12.2 Flow Polarity and Local Charge Emergence

Flow multiplet polarity (phase orientation plus amplitude sign) corresponds to local gauge charges, generalizing scalar polarity to vector flow spaces.
Charges emerge from discrete phase windings and recursive topological flow motifs.

12.3 Local Temporal Asymmetry Field and CPT Violation

Introduce a scale-dependent dimensionless temporal asymmetry field γ_asym(l):
Dimension: [1]
Physical mapping: γ_{asym_phys}(l) = γ_asym(l) / T_c, dimension [T⁻¹]
γ_asym(l) quantifies intrinsic time-reversal (T) symmetry breaking and CPT violation within flow dynamics.
Derived from fundamental microscopic temporal asymmetry γ (Section 1.3), recursively amplified or averaged over scales.

12.4 Gauge Group Correspondence via Multiplet Dimension and Recursion

Gauge Group	Multiplet Dimension n	Recursion Level	Interpretation
U(1)	1	1st order	Minimal single-phase alignment
SU(2)	2	2nd order	Paired recursive phase alignment with chirality
SU(3)	3	3rd order	Triplet phase cycling with intrinsic asymmetry

Local unitary transformations on n-component flow multiplets generate emergent gauge symmetries via recursive phase coherence.

12.5 Discrete Gauge Connection Incorporating Temporal Asymmetry

Define discrete gauge connection between nodes i and j:
A_ij(t) = [θ_j(t) − θ_i(t)] / l_{Planck_TFP} + γ_asym(l) × 𝒜_ij(t)
l_{Planck_TFP}: dimensionless Planck length scale in TFP
𝒜_ij(t): dimensionless asymmetric gauge potentials
A_ij(t): dimensionless
Physical mapping: Multiply A_ij(t) by (1 / L_c) → physical dimension [L⁻¹], matching continuum gauge potentials.

12.6 Continuum Limit and Emergent Gauge Fields

Discrete gauge connections approximate smooth fields a_μ(x) in continuum:
a_μ(x): dimensionless [1]
Physical mapping: a_μ(x) × (E_c × T_c / L_c) → [M L T⁻¹]
γ_asym(l) acts as a dimensionless coupling parameter encoding CPT-violating effects.
Scale evolution: γ_asym(l) decays exponentially at large scales, restoring CPT symmetry macroscopically, but remains sharp at UV scales to drive mass and symmetry breaking.

12.7 Gauge Covariance and Invariance

Gauge covariance arises from local unitary transformations:
Ψ_i(t) → U_i(t) Ψ_i(t), where U_i(t) ∈ U(n).
Gauge connections transform to preserve local flow observables and phase coherence.

12.8 Phase Evolution and Flow Dynamics

Discrete phase evolution:
dθ_i^(k)(t)/dt = Generator_L acting on Ψ_i components
dθ_i^(k)(t)/dt: dimensionless (phase derivative w.r.t. dimensionless time)
Physical frequency: (dθ_i^(k)(t)/dt) / T_c, dimension [T⁻¹]
Generator_L: dimensionless operator governing recursive phase locking and emergent gauge bosons as quantized flow misalignments.

12.9 Emergent Gauge Forces: U(1), SU(2), SU(3)

• U(1) (Electromagnetism): n=1 multiplets; coupling proportional to γ_asym(l).
• SU(2) (Weak force): n=2 multiplets; chirality and parity violation arise via asymmetric recursion modulated by γ_asym(l).
• SU(3) (Strong force): n=3 multiplets; stabilized non-Abelian triplet cycles shaped by topology and flow coherence.

12.10 Generalized Discrete Action with Temporal Asymmetry and Mass Generation

Temporal flow action (Section 16.1) includes nonlinear interaction terms L_int(Ψ_k, Ψ_j) encoding causal feedback.
Asymmetric potential term Φ_asym(F_i, F_j) arises from:
  • Biased phase gradients across neighboring nodes
  • Spatially varying recursion depth
  • Accumulated flow polarity p_i from phase differences
Φ_asym ∝ p_i × F_i represents energetic asymmetry favoring directional coherence, breaking flow polarity symmetry.
Role of γ_asym(l):
  – Scale-dependent coupling multiplier for Φ_asym.
  – Evolves as: γ_asym(l) = γ₀ × exp[−γ_scale × (log l − log l₀)]
  – At large scales: γ_asym(l) → 0 (symmetry restoration).
  – At small scales: γ_asym(l) sharpens, driving mass differentiation.
Mass generation:
  Effective mass m_eff arises dynamically as curvature of total potential:
    V_total = V_symmetric + γ_asym(l) × Φ_asym
    m_eff ∝ d²/dF² [V_total] evaluated at stable F
Mass emerges intrinsically from temporal asymmetry without extra scalar fields.

12.11 Continuum Effective Field Theory and Gauge Currents

Background gauge fields:
B_μ(x) = γ_asym(l) × a_μ(x)
Dimensions:
  B_μ(x): dimensionless
  Physical: B_μ(x) × (E_c × T_c / L_c), dimension [M L T⁻¹]
Conserved currents:
J^μ(x): dimensionless
Physical currents: J^μ(x) × (M_c / (T_c L_c²)), dimension [M L⁻² T⁻¹]
Conserved currents arise from coherent flow multiplet charges, consistent with gauge invariance.

12.12 Role of δ and topology_factor in Gauge Force Stability and Range

δ (Informational Friction): Controls decay of phase coherence. High δ damps recursive alignments, shortens coherence loops, effectively reducing gauge symmetry rank (e.g., SU(3) → SU(2) → U(1)) in turbulent regions.
topology_factor: Measures recursive loop density and closure strength, supporting long-range gauge coherence essential for stable SU(3) triplet cycling.
Explicit connection:
  – High δ increases damping of recursive coherence, accelerating decay of γ_asym(l) → 0 at large scales.
  – Strong topology_factor sustains temporal asymmetry γ_asym(l) by reinforcing phase coherence and mitigating frictional decay.
Interpretation: Gauge force range and strength are dynamically modulated by δ (damping) and topology_factor (reinforcement), producing natural emergent symmetry breaking/restoration without external scalar fields.