plurigrid ACSet Superior Measurement
Quantitative analysis of ACSets beyond what AlgebraicJulia authors provide.
GF(3) Triad
acset-superior-measurement (+1) ⊗ acsets-relational-thinking (0) ⊗ compositional-acset-comparison (-1) = 0 ✓
The Gap We Fill
| Measurement | Authors | Us |
|---|---|---|
| Surjectivity | incident() |
χ² coverage gadget |
| Uniformity | None | Statistical test |
| Topology | None | β₁ Betti number |
| Paths | Enumerate | Möbius classification |
| Growth | None | O(n)/O(n²)/O(n³) |
| Distance | None | P-adic ultrametric |
Core Module
include("ACSetMeasurement.jl")
using .ACSetMeasurement
# Measure an ACSet
db = create_my_acset()
metrics = measure_acset(db)
# Check coverage
gadget = incident_coverage(db, :E, :V, :src)
println(gadget) # Surjectivity(100→50, coverage=0.92, ✓ UNIFORM)
# P-adic distance between instances
d = padic_acset_distance(metrics_a, metrics_b, 3)
@assert verify_ultrametric(d_xy, d_yz, d_xz) # Strong triangle
Measurement Suite
1. Surjectivity Gadget
struct SurjectivityGadget
n_source::Int # |domain|
n_target::Int # |codomain|
hit_counts::Vector{Int} # hits per target
coverage::Float64 # fraction covered
uniform::Bool # χ² < threshold
chi_squared::Float64 # statistic
end
2. Betti Numbers
β₁ = schema_betti_1(n_objects, n_morphisms, n_components)
# Independent cycles in schema graph
| Schema | β₁ | Meaning |
|---|---|---|
| Tree | 0 | No cycles |
| Graph | 1 | src↔tgt cycle |
| INTERACTION | 4 | Highly connected |
3. Möbius Path Classification
classification = classify_paths(adjacency_matrix, max_length=4)
# classification.prime_paths - μ > 0, clean
# classification.tangled_paths - μ ≤ 0, cyclic
# classification.ratio - prime / total
4. Growth Rate Analysis
sizes = [measure_acset(build(n)).total_parts for n in [3, 9, 27]]
exponent, class = growth_rate_analysis(sizes, [3, 9, 27])
# class ∈ {"O(n)", "O(n²)", "O(n³)"}
5. P-Adic Ultrametric Distance
d = padic_acset_distance(metrics_a, metrics_b, p=3)
# Satisfies: d(x,z) ≤ max(d(x,y), d(y,z))
# Enables hierarchical clustering
File Locations
- ACSetMeasurement.jl - Julia implementation
- ACSET_SUPERIOR_MEASUREMENT.md - Theory & rationale
- MaterializationGame.jl - incident_coverage origin
End-of-Skill Interface
Integration with Existing Skills
From acsets-relational-thinking (0)
@present SchGraph(FreeSchema) begin
V::Ob; E::Ob
src::Hom(E, V); tgt::Hom(E, V)
end
@acset_type Graph(SchGraph)
From compositional-acset-comparison (-1)
# DuckDB vs LanceDB schema comparison
geometric_morphism(duckdb_acset, lancedb_acset)
This skill (+1)
# Measure quality of instances
metrics = measure_acset(db)
println(metrics.avg_coverage) # How well morphisms cover
println(metrics.betti_1) # Schema complexity
println(metrics.mobius_class.ratio) # Path quality
References
- Bumpus et al. - Spasm counting for homomorphism enumeration
- AlgebraicJulia - Base ACSet implementation
- P-adic analysis - Ultrametric hierarchical clustering
Autopoietic Marginalia
The interaction IS the skill improving itself.
Every use of this skill is an opportunity for worlding:
- MEMORY (-1): Record what was learned
- REMEMBERING (0): Connect patterns to other skills
- WORLDING (+1): Evolve the skill based on use
Add Interaction Exemplars here as the skill is used.