Browser History ACSet

Trit: 0 (ERGODIC - information coordination)
Foundation: PyACSet ↔ ACSets.jl path equivalence verified

Overview

Unified categorical structure for browser history across:

• ChatGPT Atlas (Chromium-based)
• Chrome, Arc, Brave, Firefox, Safari

Uses GF(3) trit classification for browsing behavior analysis.

Schema

┌─────────────────────────────────────────────────────────────┐
│                  BrowserHistoryACSet Schema                  │
├─────────────────────────────────────────────────────────────┤
│  Objects:    Browser, URL, Visit, Domain, SearchQuery       │
│                                                             │
│  Morphisms:                                                 │
│    browser_of: URL → Browser                                │
│    domain_of:  URL → Domain                                 │
│    url_of:     Visit → URL                                  │
│    from_visit: Visit → Visit (reflexive, navigation chain)  │
│                                                             │
│  Attributes:                                                │
│    browser_name: Browser → String                           │
│    url_text:     URL → String                               │
│    visit_time:   Visit → Int                                │
│    domain_name:  Domain → String                            │
│    trit:         Domain → Int (-1, 0, +1)                   │
└─────────────────────────────────────────────────────────────┘

Path Equivalence Tests

Verified cross-language compatibility between Python and Julia:

Operation	Python (PyACSet)	Julia (ACSets.jl)	Match
nparts(A)	2	2	✓
subpart(1, :f)	1	1	✓
incident(1, :f)	[1]	[1]	✓
path 1→f→g	1	1	✓

Key Operations

# Python (PyACSet)
url = acset.subpart(visit_id, "url_of")
domain = acset.path(visit_id, "url_of", "domain_of")
referrers = acset.incident(url_id, "url_of")

# Julia (ACSets.jl)
url = subpart(acs, visit_id, :url_of)
domain = subpart(acs, subpart(acs, visit_id, :url_of), :domain_of)
referrers = incident(acs, url_id, :url_of)

GF(3) Domain Classification

Trit	Category	Examples	Behavior
+1	PLUS (Creation)	github.com, ampcode.com, arxiv.org	Building, learning
0	ERGODIC (Info)	google.com, youtube.com, x.com	Coordination, info
-1	MINUS (Consumption)	amazon.com, netflix.com, reddit.com	Consuming, extracting

Current Data (ChatGPT Atlas)

╔═══════════════════════════════════════════════════════════════╗
║              Browser History ACSet                            ║
╠═══════════════════════════════════════════════════════════════╣
║  Browser         :      3 parts                               ║
║  URL             :   4529 parts                               ║
║  Visit           :   8569 parts                               ║
║  Domain          :    511 parts                               ║
║  SearchQuery     :     36 parts                               ║
║  Download        :     41 parts                               ║
╠═══════════════════════════════════════════════════════════════╣
║  GF(3) Sum       :     13                                     ║
╚═══════════════════════════════════════════════════════════════╝

Top Domains:
  [+] github.com      : 1066 visits (creation)
  [○] mermaid.live    :  655 visits (coordination)
  [+] ampcode.com     :  453 visits (creation)
  [+] elevenlabs.io   :  268 visits (creation)
  [+] huggingface.co  :  188 visits (creation)

Usage

# Extract browser history as ACSet
python3 browser_history_acset.py

# Run path equivalence tests
python3 path_equivalence_test.py

# Julia verification
julia path_equivalence_test.jl

Integration Points

• Tenderloin WEV: Geographic browsing patterns → impact zones
• OlmoEarth-MLX: Location-aware embeddings for browsing
• GeoACSet: Spatial categorization of online activity
• DuckDB: Temporal queries on visit history

Specter-Style Navigation

# Select all visits to github.com
github_visits = (
    SELECT(ALL("Visit"))
    >> FILTER(lambda v: acset.path(v, "url_of", "domain_of") 
              and acset.subpart(acset.path(v, "url_of", "domain_of"), "domain_name") == "github.com")
)

# Transform: add trit to all URLs in domain
TRANSFORM(
    SELECT(ALL("URL")) >> FILTER(lambda u: acset.subpart(u, "domain_of") == d1),
    lambda u: acset.set_subpart(u, "trit", 1)
)

Canonical Triads

browser-history-acset (0) ⊗ olmoearth-mlx (+1) ⊗ tenderloin (-1) = 0 ✓
py-acset (0) ⊗ ACSets.jl (+1) ⊗ DuckDB (-1) = 0 ✓

References

• ACSets.jl
• plurigrid-asi-skillz/skills/acsets
• zip_acset_skill/extract_agent_o_rama.py

Related Skills

• coequalizers (0) - Path equivalence via coequalizer quotients
• acsets (0) - ACSet foundations
• temporal-coalgebra (-1) - Time-based path analysis

Scientific Skill Interleaving

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

Annotated Data

• anndata [○] via bicomodule

Bibliography References

• general: 734 citations in bib.duckdb

SDF Interleaving

This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):

Primary Chapter: 10. Adventure Game Example

Concepts: autonomous agent, game, synthesis

GF(3) Balanced Triad

browser-history-acset (○) + SDF.Ch10 (+) + [balancer] (−) = 0

Skill Trit: 0 (ERGODIC - coordination)

Secondary Chapters

• Ch3: Variations on an Arithmetic Theme
• Ch4: Pattern Matching
• Ch2: Domain-Specific Languages
• Ch1: Flexibility through Abstraction

Connection Pattern

Adventure games synthesize techniques. This skill integrates multiple patterns.

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.