RG Flow ACSet Skill Skill

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:

condenses_to: Traces coarse-grain to equivalence classes (irrelevant operators drop)
transforms_via: Equivalence classes evolve under RG transformation
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

Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

Annotated Data

anndata [○] via bicomodule
- Hub for annotated matrices

Bibliography References

dynamical-systems: 41 citations in bib.duckdb

Cat# Integration

This skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:

Trit: 0 (ERGODIC)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #26D826

GF(3) Naturality

The skill participates in triads satisfying:

(-1) + (0) + (+1) ≡ 0 (mod 3)

This ensures compositional coherence in the Cat# equipment structure.

Agent Skills: RG Flow ACSet Skill

Install this agent skill to your local

Skill Files