Ordered Locale Fanout

Status: ✅ Production Ready (Updated Dec 24, 2024)
Trit: 0 (ERGODIC - coordinator/synthesizer)
Principle: Cocone over GF(3) ordered locale (Heunen-van der Schaaf)
Platform: macOS with Apple Silicon + MLX

Mathematical Foundation (CORRECTED)

This is now based on true ordered locales per Heunen-van der Schaaf (2024):

An ORDERED LOCALE is:
  1. Frame L (complete Heyting algebra of opens)
  2. Order ≪ on L (between OPENS, not points!)
  3. Open cone condition: ↑U and ↓U are open when U is open

The fanout is a cocone (colimit diagram) over this ordered locale:

   MINUS(-1)  ERGODIC(0)  PLUS(+1)
       \          |          /
        \    direction      /
         \    0 → 1        /
          \       |       /
           ι₋₁   ι₀    ι₊₁
            \     |     /
             ↘    ↓    ↙
                APEX
              (merged)

The ordered locale structure:

• Opens: {MINUS, ERGODIC, PLUS} — no points, just roles
• Order: -1 ≤ 0 ≤ +1 (compatible with meet)
• Direction: flow from validation → coordination → generation

See

ordered-locale

skill for formal definition.

---

Overview

Ordered Locale Fanout splits work into 3 parallel agents, each with:

1. Fixed trit (-1, 0, +1) determining role
2. Fixed voice/locale for auditory distinction
3. Deterministic seed derived from parent
4. Optional MLX generation for local LLM inference

                    ┌─────────────────┐
                    │  PARENT AGENT   │
                    │  seed=0x42D     │
                    └────────┬────────┘
                             │ fork(3)
           ┌─────────────────┼─────────────────┐
           ▼                 ▼                 ▼
   ┌───────────────┐ ┌───────────────┐ ┌───────────────┐
   │ MINUS (-1)    │ │ ERGODIC (0)   │ │ PLUS (+1)     │
   │ Anna (German) │ │ Thomas (French)│ │ Luca (Italian)│
   │ VALIDATE      │ │ COORDINATE    │ │ GENERATE      │
   │ seed₀         │ │ seed₁         │ │ seed₂         │
   └───────────────┘ └───────────────┘ └───────────────┘

Ordered Locale Table

Trit	Role	Voice	Language	Character
-1	VALIDATOR	Anna	German	Precise, constraining
0	COORDINATOR	Thomas	French	Neutral, synthesizing
+1	GENERATOR	Luca	Italian	Creative, producing

Python Implementation

#!/usr/bin/env python3
"""
ordered_locale_fanout.py - Triadic parallel dispatch with MLX + voices
"""
import subprocess
import hashlib
from dataclasses import dataclass
from typing import Callable, Any, List
from concurrent.futures import ThreadPoolExecutor, as_completed

# SplitMix64 constants
GOLDEN = 0x9E3779B97F4A7C15
MIX1 = 0xBF58476D1CE4E5B9
MIX2 = 0x94D049BB133111EB
MASK64 = 0xFFFFFFFFFFFFFFFF

@dataclass
class Locale:
    trit: int
    role: str
    voice: str
    language: str

ORDERED_LOCALES = [
    Locale(trit=-1, role="VALIDATOR", voice="Anna", language="German"),
    Locale(trit=0, role="COORDINATOR", voice="Thomas", language="French"),
    Locale(trit=1, role="GENERATOR", voice="Luca", language="Italian"),
]

def splitmix64(x: int) -> int:
    z = (x + GOLDEN) & MASK64
    z = ((z ^ (z >> 30)) * MIX1) & MASK64
    z = ((z ^ (z >> 27)) * MIX2) & MASK64
    return (z ^ (z >> 31)) & MASK64

def fork_seeds(parent_seed: int) -> List[int]:
    """Fork parent seed into 3 child seeds."""
    return [
        splitmix64(parent_seed ^ (i * GOLDEN)) 
        for i in range(3)
    ]

def announce(locale: Locale, message: str):
    """Announce via macOS say with locale-specific voice."""
    subprocess.run(
        ["say", "-v", locale.voice, message],
        capture_output=True
    )

def fanout(
    seed: int,
    task_fn: Callable[[Locale, int, Any], Any],
    context: Any = None,
    announce_start: bool = True,
    announce_end: bool = True
) -> dict:
    """
    Execute triadic fanout with ordered locales.
    
    Args:
        seed: Parent seed for deterministic forking
        task_fn: Function(locale, child_seed, context) -> result
        context: Shared context passed to all agents
        announce_start: Whether to announce agent start
        announce_end: Whether to announce completion
    
    Returns:
        Dict with results keyed by role
    """
    child_seeds = fork_seeds(seed)
    results = {}
    
    def run_agent(i: int):
        locale = ORDERED_LOCALES[i]
        child_seed = child_seeds[i]
        
        if announce_start:
            announce(locale, f"Agent {locale.role} starting")
        
        result = task_fn(locale, child_seed, context)
        
        if announce_end:
            announce(locale, f"Agent {locale.role} complete")
        
        return locale.role, result
    
    # Parallel execution
    with ThreadPoolExecutor(max_workers=3) as executor:
        futures = [executor.submit(run_agent, i) for i in range(3)]
        for future in as_completed(futures):
            role, result = future.result()
            results[role] = result
    
    # Verify GF(3) conservation
    trit_sum = sum(loc.trit for loc in ORDERED_LOCALES)
    assert trit_sum % 3 == 0, f"GF(3) violation: sum={trit_sum}"
    
    return {
        "seed": hex(seed),
        "gf3_sum": trit_sum,
        "gf3_conserved": True,
        "results": results
    }


# === MLX Integration ===

def mlx_triadic_generate(seed: int, prompts: dict, model: str = None):
    """
    Generate text with 3 parallel MLX streams, each with ordered locale.
    
    Args:
        seed: Parent seed
        prompts: Dict with keys "VALIDATOR", "COORDINATOR", "GENERATOR"
        model: MLX model path (default: mlx-community/Mistral-7B-Instruct-v0.3-4bit)
    """
    try:
        from mlx_lm import load, generate
        from mlx_lm.sample_utils import make_sampler
    except ImportError:
        raise ImportError("pip install mlx mlx-lm")
    
    model_path = model or "mlx-community/Mistral-7B-Instruct-v0.3-4bit"
    model, tokenizer = load(model_path)
    
    def generate_task(locale: Locale, child_seed: int, ctx: dict):
        prompt = ctx.get(locale.role, f"You are a {locale.role.lower()}.")
        
        # Seed-derived sampling parameters
        temp = 0.5 + (child_seed % 100) / 200  # 0.5-1.0
        sampler = make_sampler(temp=temp, top_p=0.9)
        
        text = generate(
            model, tokenizer,
            prompt=prompt,
            sampler=sampler,
            max_tokens=100
        )
        
        # Compute trit signature of output
        trit_sig = sum(ord(c) for c in text) % 3 - 1
        
        return {
            "text": text,
            "temp": temp,
            "trit_signature": trit_sig
        }
    
    return fanout(seed, generate_task, prompts)


# === CLI Entry Point ===

if __name__ == "__main__":
    import sys
    
    seed = int(sys.argv[1], 16) if len(sys.argv) > 1 else 0x42D
    
    # Demo: simple task that returns locale info
    def demo_task(locale, child_seed, ctx):
        return {
            "locale": locale.voice,
            "seed": hex(child_seed),
            "trit": locale.trit
        }
    
    result = fanout(seed, demo_task, announce_start=True, announce_end=True)
    
    print("\n╔═══════════════════════════════════════════════════════════════╗")
    print("║  ORDERED LOCALE FANOUT                                        ║")
    print("╚═══════════════════════════════════════════════════════════════╝")
    print(f"\nSeed: {result['seed']}")
    print(f"GF(3) Sum: {result['gf3_sum']} (conserved: {result['gf3_conserved']})")
    print("\nResults:")
    for role, data in result['results'].items():
        print(f"  {role}: {data}")

Shell Commands

# Run basic fanout demo
python ~/.agents/skills/ordered-locale-fanout/ordered_locale_fanout.py 0x42D

# Run with MLX generation
python -c "
from ordered_locale_fanout import mlx_triadic_generate
result = mlx_triadic_generate(0x42D, {
    'VALIDATOR': 'Check this code for errors',
    'COORDINATOR': 'Summarize the approach',
    'GENERATOR': 'Propose improvements'
})
print(result)
"

# Test voice locales
say -v Anna "Agent Minus validating"
say -v Thomas "Agent Ergodic coordinating"  
say -v Luca "Agent Plus generating"

Babashka Integration

#!/usr/bin/env bb
(require '[babashka.process :refer [shell]])

(def GOLDEN 0x9E3779B97F4A7C15)
(def MASK64 0xFFFFFFFFFFFFFFFF)

(def ordered-locales
  [{:trit -1 :role "VALIDATOR" :voice "Anna"}
   {:trit 0  :role "COORDINATOR" :voice "Thomas"}
   {:trit 1  :role "GENERATOR" :voice "Luca"}])

(defn splitmix64 [x]
  (let [z (bit-and (+ x GOLDEN) MASK64)
        z (bit-and (* (bit-xor z (unsigned-bit-shift-right z 30)) 0xBF58476D1CE4E5B9) MASK64)
        z (bit-and (* (bit-xor z (unsigned-bit-shift-right z 27)) 0x94D049BB133111EB) MASK64)]
    (bit-and (bit-xor z (unsigned-bit-shift-right z 31)) MASK64)))

(defn announce [{:keys [voice role]}]
  (shell "say" "-v" voice (str "Agent " role " reporting")))

(defn fanout [seed task-fn]
  (let [children (map-indexed 
                   (fn [i loc] 
                     (assoc loc :seed (splitmix64 (bit-xor seed (* i GOLDEN)))))
                   ordered-locales)]
    (doseq [child children]
      (announce child)
      (task-fn child))
    {:seed (format "0x%X" seed)
     :gf3-sum (reduce + (map :trit ordered-locales))
     :gf3-conserved true}))

;; Run
(fanout 0x42D (fn [loc] (println (:role loc) "seed:" (format "0x%X" (:seed loc)))))

GF(3) Conservation

Each fanout maintains triadic balance:

Σ trits = (-1) + (0) + (+1) = 0 ≡ 0 (mod 3) ✓

Fused Skills

Skill	Contribution
mlx-apple-silicon	Local LLM generation
say-narration	Voice announcements per locale
gay-mcp	Deterministic seed → color/trit
triad-interleave	Stream interleaving
parallel-fanout	Task dispatch pattern

Related Skills

• triad-interleave

  - Stream scheduling

• spi-parallel-verify

  - Parallelism verification

• say-narration

  - Voice table

• mlx-apple-silicon

  - MLX generation

• gay-mcp

  - Color/trit assignment

• ordered-locale

  - Proper locale theory with cones/cocones (Heunen-van der Schaaf)

---

Connection to Proper Ordered Locales

This skill uses "ordered locale" in a colloquial sense (ordered assignment of locales/voices). For the mathematical definition (Heunen-van der Schaaf 2024):

PROPER ORDERED LOCALE (mathematical):
  Frame L (complete Heyting algebra) + preorder on points
  Open cone condition: ↑U and ↓U must be open
  
THIS SKILL (operational):
  Fixed triadic assignment for parallel dispatch
  "Locale" = voice/language identity, not topology

For proper frame structure, cones, cocones, and Stone duality, see:

• ~/.claude/skills/ordered-locale/SKILL.md

• ~/.claude/skills/ordered-locale/OrderedLocale.jl

• ~/worlds/ordered-locale/Catlab.jl/

  (cloned)

---

Skill Name: ordered-locale-fanout
Type: Parallel Dispatch / Voice Identity
Trit: 0 (ERGODIC - coordinator)
GF(3): Conserved by construction
Platform: macOS Apple Silicon

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

ordered-locale-fanout (−) + SDF.Ch10 (+) + [balancer] (○) = 0

Skill Trit: -1 (MINUS - verification)

Secondary Chapters

• Ch9: Generic Procedures
• Ch3: Variations on an Arithmetic Theme
• Ch4: Pattern Matching
• Ch6: Layering
• Ch2: Domain-Specific Languages

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.