Renormalization Group flow with ACSet categorical semantics, XY model topological defects, and Powers PCT hierarchical control.
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Renormalization Group flow with ACSet categorical semantics, XY model topological defects, and Powers PCT hierarchical control.
741086072858456200
| Color | Hue | Hex | Role |
|---|---|---|---|
| Cyan | 172° | #23C8B3 | Ordered phase |
| Purple | 292° | #AA22BE | Critical/BKT |
| Gold | 52° | #E0CE51 | Converged fixed point |
@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
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)
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
The morphism chain Trace → EquivalenceClass → RGStep → FixedPoint implements:
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 ✓
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
xy-model: XY spin dynamics and BKT transitionphenomenal-bisect: Temperature search for critical τ*hierarchical-control: Powers PCT cascadegay-mcp: Deterministic color generation