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)

Color	Hue	Hex	Role
Cyan	172°	#23C8B3	Ordered phase
Purple	292°	#AA22BE	Critical/BKT
Gold	52°	#E0CE51	Converged fixed point

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