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rg-flow-acset skill

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This skill analyzes RG flow ACSet models to reveal fixed points and universality, applying hierarchical Powers PCT guidance for topological transitions.

npx playbooks add skill plurigrid/asi --skill rg-flow-acset

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---
name: rg-flow-acset
description: 'RG Flow ACSet Skill'
version: 1.0.0
---

# 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

```julia
@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

```julia
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

Overview

This skill implements a Renormalization Group (RG) flow model using ACSet categorical semantics tailored to the XY model and Powers PCT hierarchical control. It encodes traces, equivalence classes, RG steps, and fixed points with computed predicates for ordering, convergence, and net topological charge. The package includes a seed configuration and a ready example at the BKT-related τ = 0.5 converged point.

How this skill works

The schema defines objects (Trace, EquivalenceClass, RGStep, FixedPoint) and morphisms (condenses_to, transforms_via, flows_to) to represent coarse-graining and RG evolution. Attributes such as tau, net_charge, and hue capture physical and categorical data, and predicates compute properties like NetChargeZero, Ordered, and Converged. Example constructors provide an ACSet instance that demonstrates conservation rules (GF(3) triadic color charge) and a converged τ* configuration.

When to use it

Modeling RG coarse-graining and fixed-point flows for XY-like systems.
Encoding topological defect charge conservation in categorical form.
Prototyping hierarchical control links between program goals and sensory parameters.
Producing deterministic, labelable color triads tied to physical states.
Integrating with numerical searches for critical τ (phenomenal-bisect workflows).

Best practices

Represent physical microstates as Trace objects and only expose relevant equivalence classes after condensing.
Store RGStep attributes (tau, net_charge) to enable predicate-based filtering (Ordered, Converged).
Keep hue values aligned with the triadic GF(3) mapping to preserve net-charge arithmetic.
Use the Powers PCT cascade to document design intent: program → transition → configuration → sensation → intensity.
Validate instances with small example seeds before scaling to larger trace datasets.

Example use cases

Create an ACSet instance for the XY model at τ = 0.5 to verify convergence and net charge zero.
Annotate simulation outputs with equivalence-class hues to visualize ordered vs. critical regions.
Compose hierarchical control references to drive visualization palettes or experimental setpoints.
Integrate with a τ search routine to label candidate fixed points as Converged or Ordered.
Demonstrate GF(3) conservation by mapping color triads to creation/annihilation events in defect dynamics.

FAQ

What does the hue attribute represent?

Hue maps equivalence classes to palette angles representing ordered, critical, and fixed-point roles; it also participates in triadic GF(3) bookkeeping.

How do I detect a converged RG step?

Use the Converged predicate: abs(tau - 0.5) < 0.01 marks a converged step in the provided schema.