Compositional ACSet Comparison Skill

Trit: 0 (ERGODIC - Coordinator) Color: #26D826 (Green) Domain: Compositional algorithm/data analysis via algebraic databases

Homoiconic Insight

In self-hosted Lisps, the boundary between data structures and algorithms dissolves:

Code is data, data is code (homoiconicity)
Evaluation time is phase-scoped (RED/BLUE/GREEN gadgets)
Entanglement avoided by leaving phases open until explicitly closed
Compositional structure preserved across algorithm ↔ data boundary

Overview

Compare data structures and their properties (density/sparsity, dynamic/static, versioning strategies) using the richness afforded by ACSets. Uses Gay.jl-aided superrandom walks for deterministic exploration of comparison dimensions.

Canonical Triads

schema-validation (-1) ⊗ compositional-acset-comparison (0) ⊗ gay-mcp (+1) = 0 ✓  [Property Analysis]
three-match (-1) ⊗ compositional-acset-comparison (0) ⊗ koopman-generator (+1) = 0 ✓  [Dynamic Traversal]
temporal-coalgebra (-1) ⊗ compositional-acset-comparison (0) ⊗ oapply-colimit (+1) = 0 ✓  [Versioning]
polyglot-spi (-1) ⊗ compositional-acset-comparison (0) ⊗ gay-mcp (+1) = 0 ✓  [Homoiconic Interop]

Golden Thread Walk Dimensions

Each dimension is explored via φ-angle (137.508°) golden spiral for maximal dispersion:

Step	Dimension	Hex Color	Hue
1	Storage Hierarchy	#EE2B2B	0°
2	Density/Sparsity	#2BEE64	137.51°
3	Dynamic/Static	#9D2BEE	275.02°
4	Versioning Strategy	#EED52B	52.52°
5	Traversal Patterns	#2BCDEE	190.03°
6	Index Structures	#EE2B94	327.54°
7	Compression	#5BEE2B	105.05°
8	Query Model	#332BEE	242.55°
9	Embedding Support	#EE6C2B	20.06°
10	Interoperability	#2BEEA5	157.57°
11	Concurrency	#DE2BEE	295.08°
12	Memory Model	#C5EE2B	72.59°

Comparison Matrix: DuckDB vs LanceDB

Dimension 1: Storage Hierarchy (#EE2B2B)

DuckDB                          LanceDB
──────                          ───────
Table                           Database
  └─RowGroup (122K rows)          └─Table
      └─Column                        └─Manifest (version)
          └─Segment                       └─Fragment
              └─Block                         └─Column
                                                  └─VectorColumn

ACSet Morphism Depth:

DuckDB: 4 levels (Table→RowGroup→Column→Segment)
LanceDB: 5 levels (Database→Table→Manifest→Fragment→Column)

Dimension 2: Density/Sparsity (#2BEE64)

Property	DuckDB	LanceDB
Default	Dense columnar	Dense Arrow arrays
Sparse Support	Via NULL bitmask	Via Arrow validity bitmask
Vector Sparsity	N/A	Sparse via IVF partitioning
Storage Efficiency	ALP, ZSTD compression	Lance columnar format
ACSet Rep	`DenseFinColumn`	`DenseFinColumn` with `VectorColumn` extension

Density Formula:

density(acset, obj) = nparts(acset, obj) / theoretical_max(acset, obj)
# DuckDB Segment: ~2048 rows per vector batch
# LanceDB Fragment: variable, optimized for vector search

Dimension 3: Dynamic/Static (#9D2BEE)

Property	DuckDB	LanceDB
Schema Evolution	ALTER TABLE	Manifest versioning
Row Updates	In-place (TRANSIENT→PERSISTENT)	Append + compaction
Index Updates	Dynamic B-Tree/ART	Rebuild IVF partitions
ACSet Mutation	`set_subpart!`, `rem_part!`	Append-only, version chains

State Machine:

DuckDB Segment: TRANSIENT ⟷ PERSISTENT (bidirectional)
LanceDB Manifest: V1 → V2 → V3 → ... (append-only chain)

Dimension 4: Versioning Strategy (#EED52B) ⭐ Lance SDK 1.0.0

Critical Update (December 15, 2025): Lance SDK adopts SemVer 1.0.0

Component	Versioning	Strategy
Lance SDK	SemVer 1.0.0	MAJOR.MINOR.PATCH
Lance File Format	2.1	Binary compatibility, independent
Lance Table Format	Feature flags	Full backward compat, no linear versions
Lance Namespace Spec	Per-operation	Iceberg REST Catalog style

Key Insight: Breaking SDK changes will NOT invalidate existing Lance data.

# ACSet representation of versioning strategies
@present SchVersioning(FreeSchema) begin
  SDKVersion::Ob      # SemVer (1.0.0)
  FileFormat::Ob      # Binary compat (2.1)
  TableFormat::Ob     # Feature flags
  NamespaceSpec::Ob   # Per-operation
  
  # Morphisms: SDK ≠ Format
  sdk_file::Hom(SDKVersion, FileFormat)      # Many-to-one
  file_table::Hom(FileFormat, TableFormat)   # Independent
  table_ns::Hom(TableFormat, NamespaceSpec)  # Independent
end

DuckDB Versioning:

Temporal tables via VERSION AT
Extension versioning separate from core

Dimension 5: Traversal Patterns (#2BCDEE)

Pattern	DuckDB	LanceDB
Sequential Scan	RowGroup→Column→Segment	Fragment→Column
Index Scan	ART/B-Tree navigation	IVF partition probe
Vector Search	N/A (extension)	Centroid→Partition→Rows
Time Travel	`FOR SYSTEM_TIME AS OF`	`checkout(version)`

ACSet Incident Queries:

# DuckDB: Find all segments in a column
incident(duckdb_acset, col_id, :column)

# LanceDB: Find all centroids for an index
incident(lancedb_acset, idx_id, :partition_index) |>
  flatmap(p -> incident(lancedb_acset, p, :centroid_partition))

Dimension 6: Index Structures (#EE2B94)

Index Type	DuckDB	LanceDB
Primary	None (heap)	None (Lance format)
Secondary	ART (Radix Tree)	Scalar indexes
Vector	Extension (vss)	IVF_PQ, IVF_HNSW_SQ, IVF_HNSW_PQ
Full-Text	Extension (fts)	N/A

ACSet Index Representation:

# LanceDB vector index hierarchy
VectorIndex → Partition → Centroid
    ↓
index_column → VectorColumn → Column

Dimension 7: Compression (#5BEE2B)

Algorithm	DuckDB	LanceDB
Numeric	ALP (Adaptive Lossless)	Arrow encoding
String	Dictionary, FSST	Dictionary
General	ZSTD, LZ4	ZSTD
Vector	N/A	PQ (Product Quantization)

Dimension 8: Query Model (#332BEE)

Aspect	DuckDB	LanceDB
Language	SQL	Python/Rust API + SQL filter
Optimization	Volcano/push-based	Vector-first + filter
Execution	Vectorized (2048 batch)	Arrow RecordBatch
Parallelism	Morsel-driven	Partition-parallel

Dimension 9: Embedding Support (#EE6C2B)

Feature	DuckDB	LanceDB
Native	No	Yes (FixedSizeList<Float>)
Generation	UDF/Extension	EmbeddingFunction registry
Storage	ARRAY type	VectorColumn
Search	Extension (vss)	Native (IVF, HNSW)

Dimension 10: Interoperability (#2BEEA5)

Format	DuckDB	LanceDB
Arrow	Full support	Native (Lance = Arrow extension)
Parquet	Read/Write	Read (convert to Lance)
CSV/JSON	Read/Write	Via Arrow
ACSets	Via Tables.jl	Via Arrow → Tables.jl

Cross-Language (from ACSets Intertypes):

# Generate interoperable types
generate_module(DuckDBACSet, [PydanticTarget, JacksonTarget])
generate_module(LanceDBACSet, [PydanticTarget, JacksonTarget])

Dimension 11: Concurrency (#DE2BEE)

Aspect	DuckDB	LanceDB
Model	MVCC	Optimistic (manifest-based)
Writers	Single (or WAL)	Single (append)
Readers	Unlimited concurrent	Unlimited concurrent
Isolation	Snapshot	Version snapshot

Dimension 12: Memory Model (#C5EE2B)

Aspect	DuckDB	LanceDB
Buffer Pool	BufferManager	Memory-mapped Arrow
Eviction	LRU	OS page cache
Allocation	Unified allocator	Arrow allocator
Out-of-Core	Automatic spill	Lazy loading

Interleaved 3-Stream Comparison

Using GF(3) conservation for balanced parallel analysis:

Stream 1 (Blue, -1): Validation/Constraints
  #31945E → #B3DA86 → #8810F2 → #2F5194 → #2452AA → #245FB4

Stream 2 (Green, 0): Coordination/Transport
  #6D59D2 → #9E2981 → #72E24F → #31C5B4 → #C04DDD → #1C8EEE

Stream 3 (Red, +1): Generation/Composition
  #E22FA7 → #E812C8 → #6F68E6 → #25D840 → #DA387F → #A82358

Crystal Family Analogy

Data structures map to crystal symmetry:

Crystal Family	Symmetry	DuckDB Analog	LanceDB Analog
Cubic (#9E94DD)	Order 48	RowGroup uniformity	Fragment uniformity
Hexagonal (#65F475)	Order 24	Column types	Vector dimensions
Tetragonal (#E764F1)	Order 16	Segment blocking	Partition structure
Orthorhombic (#2ADC56)	Order 8	Type system	Index types
Monoclinic (#CD7B61)	Order 4	Compression	Quantization
Triclinic (#E4338F)	Order 2	Raw storage	Raw Arrow

Hierarchical Control Palette

Powers PCT cascade for harmonious comparison:

Level 5 (Program): "Compare DuckDB vs LanceDB"
    ↓ sets reference for
Level 4 (Transition): Dimension sequence [30° steps]
    ↓ sets reference for
Level 3 (Configuration): Property relationships
    ↓ sets reference for
Level 2 (Sensation): Individual metrics
    ↓ sets reference for
Level 1 (Intensity): Numeric values

Colors: #B322C0 → #D5268C → #DC3946 → #DF884A → #E0D551 → #A3E04E

XY Model Phenomenology

At τ=0.5 (ordered phase, τ < τ_c=0.893):

Smooth field, defects bound in pairs
High valence, disentangled
Antivortex at (4,3): #C33567

Interpretation: Both DuckDB and LanceDB are in "ordered phase" - mature, production-ready systems with well-defined structures.

Usage

using ACSets, Catlab

# Load both schemas
include("DuckDBACSet.jl")
include("LanceDBACSet.jl")

# Compare morphism structures
compare_schemas(SchDuckDB, SchLanceDB)

# Analyze density
density_analysis = map([SchDuckDB, SchLanceDB]) do sch
  Dict(ob => sparsity_metric(sch, ob) for ob in obs(sch))
end

# Traverse with Gay.jl colors
for (i, dimension) in enumerate(DIMENSIONS)
  color = gay_color_at(1000000, i)
  analyze_dimension(dimension, color)
end

Skill Files

File	Purpose	Gay.jl Seed
`DuckDBACSet.jl`	Schema for DuckDB storage layer	1000000
`LanceDBACSet.jl`	Schema for LanceDB vector store	1000000
`IrreversibleMorphisms.jl`	Analysis of lossy morphisms	2000000
`SideBySideComparison.jl`	Visual comparison tables	3000000
`ComparisonUtils.jl`	12-dimension comparison utilities	1000000
`GhristCoverage.jl`	Persistent homology coverage analysis	4000000
`ColoringFunctor.jl`	Schema coloring + GF(3) verification	4000000
`GeometricMorphism.jl`	Presheaf topos translation analysis	4000000

Ghrist Persistent Homology Integration

Based on de Silva & Ghrist "Coverage in Sensor Networks via Persistent Homology":

AM Radio Coverage Analogy:

Radio stations = Schema objects (Table, Column, etc.)
Coverage radius = Morphism composability range
Signal overlap = Translatable concepts between schemas
Dead zones = Irreversible information loss

Betti Numbers for Schemas:

β₀: Connected components (isolated subsystems)
β₁: Coverage holes (information flow gaps)
β₂: Enclosed voids (unreachable regions)

Persistent Holes (never die):

🔴 parent_manifest: Temporal irreversibility (version chain)
🔴 source_column: Semantic irreversibility (embedding loss)

Geometric Morphism Analysis

For presheaf topoi PSh(SchDuckDB) and PSh(SchLanceDB):

Essential Image (lossless translation):

Table ↔ Table ✓
Column ↔ Column ✓

Partial Coverage (lossy translation):

RowGroup ~ Fragment
VectorColumn → Column (loses vector semantics)

Dead Zones (no translation):

Segment → ??? (DuckDB-only)
Manifest ← ??? (LanceDB-only)
VectorIndex ← ??? (LanceDB-only)

compositional-acset-comparison