Section 14: Formal Derivation of Emergent Gauge Symmetries and Lie Algebras from Temporal Flow Dynamics

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Section 14: Formal Derivation of Emergent Gauge Symmetries and Lie Algebras from Temporal Flow Dynamics

Characteristic Units Recap

  • Lc [L]
  • Tc [T]
  • Ec [M L² T⁻²]
  • cchar = Lc / Tc [L T⁻¹]
  • Mc = Ec × Tc² / Lc² [M]
  • ħc = Ec × Tc [M L² T⁻¹]

14.1 Flow Multiplets and Internal State Representation

Each node i supports an n-component complex flow multiplet:
Ψi(t) = [Fi(1)(t), ..., Fi(n)(t)]
Components:
Fi(k)(t) = Ai(k)(t) × exp(i × θi(k)(t))
Amplitudes Ai(k)(t) and phases θi(k)(t) are dimensionless, encoding internal gauge degrees of freedom and flow orientation.
The n-dimensional complex vector Ψi represents internal symmetry states and emergent quantum numbers.

14.2 Deriving Lie Algebra Structures from Discrete Flow Dynamics

Infinitesimal flow realignments yield gauge symmetry generators:
δΨk = i × εa × Ta × Ψk
  – εa: dimensionless infinitesimal parameters
  – Ta: dimensionless Lie algebra generators
Commutation relations (emerging from non-commutative discrete rephasings):
  – U(1): Abelian, [Ta, Tb] = 0
  – SU(2): [Ta, Tb] = i εabc Tc
  – SU(3): [Ta, Tb] = 2i fabc Tc
These reflect the intrinsic ordering and topological constraints of temporal flow motifs.

14.3 Conservation Laws and Emergent Charges

Noether currents Jμ arise from local internal flow multiplet transformations with emergent metric gμν:
∂μ Jμ = 0
Dimensions:
  Jμ: dimensionless [1]
  Physical currents: Jμ(x) × (Mc / (Tc × Lc²)) with dimension [M L⁻² T⁻¹]
Emergent charges correspond to integrals over charge density ρ(x) = J⁰(x):
Q = ∫ d³x × ρ(x)
Mapping between symmetry groups and physical charges:

Symmetry Group Charge Interpretation Multiplet Dimension n
U(1) Electric charge 1
SU(2) Weak isospin 2
SU(3) Color charge 3

14.4 Specific Gauge Force Mechanisms

14.4.1 SU(2) Gauge Symmetry from Flow Osculation

For n=2 multiplets Ψk(t) = [Fk1, Fk2]T, internal interaction terms of form:
Lint ∝ A1 × A2 × cos(θ1 − θ2)
Invariant under global U(1) phase shifts but only preserved in form by SU(2) transformations.
Demonstrates SU(2) as the minimal non-Abelian algebra compatible with two-component internal phase alignment.
This underlies electroweak symmetry structure in emergent gauge theory.

14.4.2 Emergence of SU(3) Algebra and Confinement from Flow Triplets

For n=3 multiplets Φi = [Fir, Fig, Fib], recursive alignment enforces:
[Ta, Tb] = 2i fabc Tc
Confinement arises from topological holonomy constraints minimizing phase misalignment energy:
Emisalignment ∝ ½ × CKE_Action × c² × (∇θ)²
Energy cost grows with separation for color non-singlets → only color-neutral (singlet) states stable.
This confinement emerges directly from recursive flow motif dynamics and holonomy, not imposed externally.

14.5 Cosmological and Structural Phenomena from Recursive Symmetries

Inflation: Horizon-Scale Recursive Cascade

Early universe causal neighborhoods exhibit:
Γform(linf) ≫ Γdecay(linf)
Allows coherent flow domains to expand faster than causal horizons → triggering inflation.
Flatness and horizon uniformity arise from large-scale phase alignment and recursive coherence.
Quantum fluctuations during incomplete saturation encode primordial power spectrum.
Observable parameters (scalar spectral index ns, tensor-to-scalar ratio r) controlled by δ(l) (informational friction) and topology_factor(l).

Structure Formation: Saturation Ridge Breakdown

As recursive coherence saturates, δ increases → local phase collapses along saturation ridges.
These decoherence pockets serve as gravitational wells attracting baryonic matter.
Matter clustering emerges not from initial mass seeds but from topological recursion failures.
This view reframes structure formation as an emergent topological-kinematic phenomenon.

Dark Matter Candidates: Persistent Partial Recursion

Some flow motifs fail full SU(N) closure but remain topologically locked and stable.
These carry energy but minimal phase overlap with baryonic flows.
Examples: Non-singlet SU(3) motifs, high-δ locked recursive states.
Observable properties:
  – Gravitationally active (mass-energy contribution)
  – Electromagnetically neutral (no charge coupling)
Natural emergent dark matter candidates, arising from motif stability constraints without ad hoc assumptions.

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