RG Flow ACSet Skill

Renormalization Group flow with ACSet categorical semantics, XY model topological defects, and Powers PCT hierarchical control.

Seed

741086072858456200

Triadic Palette (Powers PCT Cascade)

ACSet Schema: RGFlow

@present SchRGFlow(FreeSchema) begin
  # Objects
  Trace::Ob
  EquivalenceClass::Ob
  RGStep::Ob
  FixedPoint::Ob
  
  # Morphisms
  condenses_to::Hom(Trace, EquivalenceClass)
  transforms_via::Hom(EquivalenceClass, RGStep)
  flows_to::Hom(RGStep, FixedPoint)
  
  # Attributes
  tau::Attr(RGStep, Float64)
  net_charge::Attr(RGStep, Int)
  hue::Attr(EquivalenceClass, Float64)
end

# Predicates (as computed attributes)
NetChargeZero(step) = net_charge(step) == 0
Ordered(step) = tau(step) < 0.893  # Below BKT
Converged(step) = abs(tau(step) - 0.5) < 0.01

XY Model Configuration (τ=0.5)

Phase: Ordered (below BKT critical τ_c ≈ 0.893)
Defects: 2 vortex/antivortex pairs
Net topological charge: 0 (conserved)
Phenomenal bisect: τ* ≈ 0.5 (converged)

Hierarchical Control (Powers PCT)

Level 5 (Program): "triadic" goal
    ↓ sets reference for
Level 4 (Transition): hue velocities [172°, 292°, 52°]
    ↓ sets reference for
Level 3 (Configuration): complementary angles
    ↓ sets reference for
Level 2 (Sensation): target hues
    ↓ sets reference for
Level 1 (Intensity): lightness 0.55

RG Flow Semantics

The morphism chain Trace → EquivalenceClass → RGStep → FixedPoint implements:

1. condenses_to: Traces coarse-grain to equivalence classes (irrelevant operators drop)
2. transforms_via: Equivalence classes evolve under RG transformation
3. flows_to: RG steps converge to fixed points (universality)

GF(3) Conservation

Triadic colors sum to 0 (mod 3):

• #23C8B3 → trit 0 (identity)
• #AA22BE → trit +1 (creation)
• #E0CE51 → trit -1 (annihilation)

Net charge: 0 + 1 + (-1) = 0 ✓

Usage

using ACSets

@acset_type RGFlowACSet(SchRGFlow)

# Create instance at BKT transition
rg = @acset RGFlowACSet begin
  Trace = 4
  EquivalenceClass = 2
  RGStep = 1
  FixedPoint = 1
  condenses_to = [1, 1, 2, 2]
  transforms_via = [1, 1]
  flows_to = [1]
  tau = [0.5]
  net_charge = [0]
  hue = [172.0, 292.0]
end

Related Skills

• xy-model: XY spin dynamics and BKT transition
• phenomenal-bisect: Temperature search for critical τ*
• hierarchical-control: Powers PCT cascade
• gay-mcp: Deterministic color generation