spectre-worldmat

Synthesis of aperiodic monotile geometry and Worldmat TiDAR execution.

Core Insight

The Spectre monotile achieves global aperiodicity through local edge-matching rules. Similarly, threads evolve aperiodically through local GF(3) conservation constraints.

Metatile → Thread Archetype Mapping

Metatile	Thread Type	GF(3)	Pattern
Γ Gamma	SEED	0	Bifurcating origin (2 spectres)
Δ Delta	BRIDGE	+1	8-way cross-domain synthesis
Θ Theta	RECURSIVE	-1	Self-referential loops
Λ Lambda	ABSTRACTION	0	Meta-level generalization
Ξ Xi	INTEGRATION	+1	Unification of disparate concepts
Π Pi	EXPANSION	-1	Proliferation and scaling
Σ Sigma	AGGREGATION	0	Spur-edge decentralized collection
Φ Phi	GENERATOR	+1	Ubiquitous production (in all subs)
Ψ Psi	TERMINAL	-1	Boundary interface

Mod12 = GF(3) × Z/4Z

12 rotations decompose into 3 trits × 4 temporal phases:

Dawn  (phase 0): ergodic(0), plus(1), minus(2)
Noon  (phase 1): ergodic(3), plus(4), minus(5)
Dusk  (phase 2): ergodic(6), plus(7), minus(8)
Night (phase 3): ergodic(9), plus(10), minus(11)

Conservation Laws

1. Γ-Σ Conservation: Every substitution contains exactly 1 Gamma and 1 Sigma
2. Φ Abundance: Generator appears 2+ times in every expansion
3. GF(3) Balance: Sum of trits across any substitution ≡ 0 (mod 3)

TrigNum 4D Embedding

data TrigNum a = TrigNum a a a a
-- Spans hexagonal lattice: {1, √3/2+i/2, 1/2+i√3/2, i}
-- Parity constraint: parity(a,d) = parity(b,c)

Threads embed in 4D space with coherence verified by parity matching.

Worldmat Integration

from worldmat import Worldmat

wm = Worldmat(master_seed=0x87079c9f1d3b0474)
wm.execute_parallel()

# 27 cells = 3³ = 9 metatile types × 3 instances
# Each cell is a TiDAR execution (diffusion + AR verify)
# GF(3) conserved across all 9 slices

Skill Growth Pattern

Skills grow via golden angle (137.5°) knight tour:

Position 1  → #DD3C3C (scarlet)  @ hue 0°
Position 2  → #3CDD6B (mint)     @ hue 137.5°
Position 3  → #9A3CDD (violet)   @ hue 275°
...
Position 27 → #DD3C7E (rose)     @ hue 335.2°

After 27 positions, full coverage. Never repeats (aperiodic).

Usage

# Run worldmat demo
python3 /Users/bob/ies/worldmat/worldmat.py

# Verify SPI
python3 /Users/bob/ies/worldmat/worldmat.py verify

# Grow skill at next position
python3 /Users/bob/ies/worldmat/skill_grower.py grow "context"

# Check tour status
python3 /Users/bob/ies/worldmat/skill_grower.py status

Related Files

• worldmat.py - Core implementation
• skill_grower.py - Gay MCP integration
• Spectre.hs - Haskell monotile
• plus_monotile_predictions.json - PLUS agent

GF(3) Triad

MINUS (-1): minus_verification_sink.json  - stale thread analysis
ERGODIC(0): ergodic_thread_state.json     - current network state
PLUS  (+1): plus_monotile_predictions.json - future predictions

Sum: -1 + 0 + 1 = 0 ✓

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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.