Section 15: Scale-Dependent Coherence and Physical Cutoffs in Temporal Flow Physics

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Section 15: Scale-Dependent Coherence and Physical Cutoffs in Temporal Flow Physics

Characteristic Units Recap

  • \( L_c \) [L]
  • \( T_c \) [T]
  • \( E_c \) [M L² T⁻²]
  • \( c_{\text{char}} = \frac{L_c}{T_c} \) [L T⁻¹]
  • \( M_c = \frac{E_c \cdot T_c^2}{L_c^2} \) [M]
  • \( \hbar_c = E_c \cdot T_c \) [M L² T⁻¹]

15.1 Scale Dependence as Physical Reorganization

Scale parameter \( l \) (dimensionless) indexes observational/coarse-graining scale in recursive flow networks.

Scale dependence of physical parameters (e.g., gauge couplings \( \alpha(l) \)) represents real statistical reorganization of coherent temporal flow bundles, not mere formal renormalization.

As \( l \) increases, flow bundles reorganize, changing coherence amplitude, interaction strengths, and stability— manifesting distinct dynamical phases in the universe’s flow structure.

This reflects Phase-Tuned Field Theory: transitions between flow-defined dynamical phases with distinct emergent coupling behaviors.

15.2 Physical Cutoffs from Discrete Flow Constraints

The discrete causal network imposes natural physical cutoffs, preventing ultraviolet divergences plaguing continuum QFT and providing intrinsic regularization for quantum gravity.

Minimal length scale:

\[ l_{\text{min}} \propto \frac{1}{\sqrt{\langle C_{\text{max}} \rangle}} \]

where \( C_{\text{max}} \) represents maximal coherent loop complexity or curvature-related quantity.

Minimal time scale:

\[ t_{\text{min}} \propto \frac{1}{\langle \text{update\_rate} \rangle} \]

with update_rate being the local fundamental network update frequency.

These scales define hard cutoffs below which the continuum spacetime approximation and emergent gauge fields break down.

The scale-dependent temporal asymmetry \( \gamma_{\text{asym}}(l) \) (Section 12.3) modulates directional temporal flow bias near these cutoffs, influencing flow orientation and CPT asymmetry at fundamental scales.

15.3 Emergent Running Couplings and Phase Transitions

Gauge couplings \( \alpha(l) \) evolve with scale via the causal recursion beta function:

\[ \frac{d\alpha}{d(\log l)} = \beta_{\text{TFP}}(\alpha, \gamma_{\text{asym}}(l), \rho(l)) \]

\( \beta_{\text{TFP}} \) encodes the interplay of:

  • \( E_{\text{osculation}} \): Energy cost of maintaining recursive phase alignment
  • \( F_{\text{feedback}} \): Second-order feedback effects from coherence changes

Running coupling flow emerges physically from the energetic and coherence constraints of recursive flow multiplets.

Phase transitions in flow coherence cause non-trivial running:

  • Critical points correspond to fixed points or divergences of \( \alpha(l) \)
  • Transitions between free-flow, critical ridge, and saturated basin phases alter effective coupling behavior

This encapsulates the Phase-Tuned Field Theory concept: physics dynamically adapts via flow phase reorganization.

15.4 Scale-Dependent Hierarchy of Gauge Forces

The gauge group hierarchy (U(1), SU(2), SU(3)) arises naturally from recursive flow multiplet complexity and scale-dependent coherence:

  • U(1): 1st order recursion, simplest phase alignment, dominant at largest scales
  • SU(2): 2nd order recursion, paired phase coherence with chirality, manifests at intermediate scales
  • SU(3): 3rd order recursion, triplet cycles with intrinsic asymmetry and topological holonomy, dominant at smallest coherence scales

Depending on \( l \), different gauge forces manifest differently, explaining scale-dependent dominance and symmetry breaking phenomena.

15.5 Testable Predictions Related to Scale Dependence

  • Modified running of gauge couplings near fundamental scales due to flow coherence feedback and asymmetry corrections, potentially observable in high-energy experiments or cosmological data.
  • Granularity signatures: deviations from perfect Lorentz invariance or continuous spacetime behavior due to discrete temporal flow substrate, detectable via precision tests.
  • Cosmological imprints of phase transitions: relic signatures in cosmic microwave background anisotropies, scalar spectral index, and gravitational wave backgrounds tied to transitions between flow dynamical phases.

Summary

In Temporal Flow Physics, scale dependence is a physical reorganization of coherent temporal flow bundles, fundamentally driving emergent running couplings and the hierarchical manifestation of gauge forces. The discrete causal network enforces natural physical UV cutoffs, providing a predictive, dimensionally consistent framework as an alternative to conventional renormalization.

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