worldmat-tidar

World Matrices via TiDAR Executions: 3×3×3 Parallel Triadic Computation

Version: 1.0.0 Trit: 0 (ERGODIC – coordinates execution) Color: #55D9A0

Overview

Worldmat is a 3×3×3 matrix of TiDAR executions where:

• Rows: MINUS/ERGODIC/PLUS polarities (GF(3) agents)
• Columns: PAST/PRESENT/FUTURE temporal phases
• Depth: OBSERVATION/ACTION/PREDICTION modalities

Each cell executes the TiDAR pattern:

1. DIFFUSION: Draft tokens in parallel (like SplitRng.split)
2. AR VERIFY: Verify sequentially (autoregressive)

Architecture

                    TEMPORAL AXIS
                 PAST    PRESENT   FUTURE
                  ↓        ↓        ↓
            ┌─────────────────────────────┐
            │  ┌───┐ ┌───┐ ┌───┐         │
     MINUS  │  │-1 │ │ 0 │ │+1 │  ← GF(3)=0
            │  └───┘ └───┘ └───┘         │
POLARITY    │  ┌───┐ ┌───┐ ┌───┐         │
     ERGODIC│  │ 0 │ │+1 │ │-1 │  ← GF(3)=0
            │  └───┘ └───┘ └───┘         │
            │  ┌───┐ ┌───┐ ┌───┐         │
     PLUS   │  │+1 │ │-1 │ │ 0 │  ← GF(3)=0
            │  └───┘ └───┘ └───┘         │
            └─────────────────────────────┘
                  ↑    ↑    ↑
               GF(3)=0 for each column

Key Properties

TiDAR Pattern (arXiv:2511.08923)

# Phase 1: DIFFUSION (parallel drafting)
def diffusion_draft(self, n_tokens: int = 8):
    streams = self.rng.split(n_tokens)
    return [stream.next()[0] for stream in streams]

# Phase 2: AR VERIFY (sequential verification)
def ar_verify(self):
    prev = self.seed
    for token in self.draft_tokens:
        verified = mix64(prev ^ token)
        self.verified_tokens.append(verified)
        prev = verified

Work Stealing

Idle agents steal work from busy agents:

class WorkStealingScheduler:
    def steal_work(self, thief: Polarity) -> Optional[TiDARCell]:
        busiest = max(self.queues.keys(), key=lambda p: len(self.queues[p]))
        if busiest != thief and self.queues[busiest]:
            return self.queues[busiest].pop(0)
        return None

ACSet Export

wm = Worldmat(master_seed=0x87079c9f1d3b0474)
wm.execute_parallel()
acset = wm.to_acset()
# Returns: {schema, parts, subparts, metadata}

Commands

# Run demo
python worldmat.py

# Verify SPI
python worldmat.py verify

# Export ACSet
python worldmat.py acset > worldmat.json

GF(3) Triads

worldmat-tidar (0) forms balanced triads:

three-match (-1) ⊗ worldmat-tidar (0) ⊗ gay-mcp (+1) = 0 ✓
spi-parallel-verify (-1) ⊗ worldmat-tidar (0) ⊗ triad-interleave (+1) = 0 ✓
tidar_streaming (-1) ⊗ worldmat-tidar (0) ⊗ gay_triadic_exo (+1) = 0 ✓

Integration

With OpenAI ACSet

from worldmat import Worldmat
from openai_acset import build_openai_acset

# Process conversations through worldmat
wm = Worldmat(master_seed=conv_fingerprint)
wm.execute_parallel()

# Each message → cell in worldmat
# Role (user/assistant/tool) → polarity
# Time → temporal phase
# Type (obs/action/pred) → modality

With Gay-MCP

from gay import SplitMixTernary

# Worldmat colors from Gay-MCP
gen = SplitMixTernary(seed=worldmat.fingerprint())
palette = gen.palette_hex(n=27)  # One color per cell

Files

References

• TiDAR: arXiv:2511.08923
• Gay.jl/src/spc_repl.jl – Whale synergy matrix
• rio/gayzip/tidar_streaming.py – TiDAR ZIP implementation
• gay_triadic_exo.py – Triadic agent orchestration

Base directory: file:///Users/bob/.claude/skills/worldmat-tidar

Scientific Skill Interleaving

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

Graph Theory

• networkx [○] via bicomodule
  • Universal graph hub

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

worldmat-tidar (+) + SDF.Ch10 (+) + [balancer] (+) = 0

Skill Trit: 1 (PLUS – generation)

Secondary Chapters

• Ch7: Propagators
• Ch6: Layering
• Ch4: Pattern Matching
• 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.