Transitive Weep

---
name: transitive-weep
description: 'Each world is an **open game** with:'
---
# Transitive Weep Skill

**Trit**: 0 (ERGODIC/Coordinator)  
**Domain**: open-games, category-theory, flow-networks, self-organization  
**Algebra**: GF(3) conservation via coplay feedback

---

## Overview

**Transitive Weep** = flow propagation through open games network with self-balancing GF(3) conservation. The "weep" metaphor: value flows like tears through the graph, finding its own level.

```
╔════════════════════════════════════════════════════════════════════════╗
║                    TRANSITIVE WEEP ARCHITECTURE                        ║
╠════════════════════════════════════════════════════════════════════════╣
║                                                                        ║
║   Source ──play──▶ Game₁ ──play──▶ Game₂ ──play──▶ ... ──▶ Balance    ║
║      │               │               │                        │        ║
║      │    ◀─coplay───┘    ◀─coplay───┘         ◀─coplay──────┘        ║
║      │                                                                 ║
║      └────────────────── Σ(trit × flow) ≈ 0 ──────────────────────────┘║
║                                                                        ║
║   Play:   Forward flow transformation at each node                    ║
║   Coplay: Backward utility → strategy update for GF(3) balance        ║
║                                                                        ║
╚════════════════════════════════════════════════════════════════════════╝
```

---

## Core Concepts

### 1. Open Games (Play/Coplay)

Each world is an **open game** with:
- **Play**: `(incoming_flow, bias) → observation`
- **Coplay**: `(utility, global_sum, generation) → strategy_update`

```julia
struct OpenGame
    id::Symbol
    trit::Int8           # -1, 0, +1
    play::Function       # Forward pass
    coplay::Function     # Backward pass (self-reinvention)
    state::State         # Mutable: generation, bias, memory
end
```

### 2. Transitive Closure

If A weeps to B, and B weeps to C, then A **transitively** weeps to C:
```
A →weep→ B →weep→ C  ⟹  A →weep*→ C
```

The flow decays exponentially: `flow(C) = flow(A) × w(A,B) × w(B,C) × decay²`

### 3. GF(3) Self-Balancing

The weep stops when global sum is conserved:
```
Σ(trit_i × flow_i) ≈ 0  (mod 3)
```

Coplay feedback adjusts routing to prioritize nodes whose trits would **correct** the running sum.

---

## Triadic World Graph

```
         ┌──────────────────────────────────────────┐
         │           26 ACCESSIBLE WORLDS            │
         └──────────────────────────────────────────┘
         
    MINUS (-1)           ERGODIC (0)          PLUS (+1)
    ──────────           ──────────           ─────────
    A: algebraic         B: bmorphism         C: catcolab
    D: ducklake          E: elevenlabs        F: firecracker
    G: gay-colors        H: hatchery          I: iroh
    J: juvix             K: kubernetes        L: localsend
    M: mcp               N: narya             O: open-games
    P: plurigrid         Q: quantum           R: rama
    S: siggraph          T: topos             U: unison
    V: voice             W: wasm              X: xogot
    Y: yoneda            Z: zig
```

### Adjacency (Triadic Connections)

Each world connects to worlds that form balanced triads:
```clojure
{:A [:B :C]      ;; -1 → {0, +1} = balanced potential
 :B [:A :C :G]   ;; 0 → {-1, +1, -1}
 :C [:T :N :A]   ;; +1 → {0, 0, -1}
 :G [:J :R :B]   ;; -1 → {-1, +1, 0}
 ...}
```

---

## Algorithm

```clojure
(defn transitive-weep [source initial-flow max-depth]
  (loop [frontier [[source initial-flow 0]]
         visited #{}
         global-sum 0.0
         trace []]
    
    ;; Early exit: balanced
    (when (and (> (count trace) 5) (< (abs global-sum) 0.1))
      (return {:status :balanced :trace trace :sum global-sum}))
    
    (let [[node flow depth] (first frontier)
          game (get-game node)
          result (play! game flow global-sum)]
      
      ;; Update global sum
      (swap! global-sum + (:contribution result))
      
      ;; Coplay: which neighbors would balance?
      (let [needed-trit (cond (> global-sum 0.3) -1
                              (< global-sum -0.3) +1
                              :else 0)
            neighbors (sort-by-trit-distance (get-neighbors node) needed-trit)]
        
        ;; Add top neighbors to frontier
        (recur (into (rest frontier) 
                     (map #(vector % (* flow 0.7) (inc depth)) 
                          (take 2 neighbors)))
               (conj visited node)
               global-sum
               (conj trace result))))))
```

---

## Game Composition

### Sequential: G₁ ; G₂

```
G₁ ; G₂ : trit(G₁;G₂) = trit(G₁) + trit(G₂)
```

Flow passes through G₁, then G₂.

### Parallel: G₁ ⊗ G₂

```
G₁ ⊗ G₂ : trit(G₁⊗G₂) = (trit(G₁) + trit(G₂)) mod 3
```

Flow splits, processes in parallel, recombines.

### Balanced Triad Composition

```julia
A(-1) ; B(0) ; C(+1) → trit = 0 ✓
```

---

## Example: Weep from A

```
A (algebraic-julia) trit:-1 obs:0.905 Δ:-0.905 Σ:-0.905
  C (catcolab) trit:+1 obs:0.573 Δ:+0.573 Σ:-0.332  ← balancing!
  B (bmorphism) trit:+0 obs:0.317 Δ:+0.000 Σ:-0.332
    T (topos) trit:+0 obs:0.401 Δ:+0.000 Σ:-0.332
    N (narya) trit:+0 obs:0.201 Δ:+0.000 Σ:-0.332
      O (open-games) trit:+1 obs:0.254 Δ:+0.254 Σ:-0.078 ✓ BALANCED!
```

Status: **balanced-early** at iteration 6, Σ = -0.078 ≈ 0

---

## DuckDB Schema

```sql
CREATE TABLE weep_traces (
    trace_id VARCHAR PRIMARY KEY,
    source VARCHAR NOT NULL,
    node VARCHAR NOT NULL,
    depth INT NOT NULL,
    observation DOUBLE NOT NULL,
    contribution DOUBLE NOT NULL,
    running_sum DOUBLE NOT NULL,
    utility DOUBLE NOT NULL,
    trit TINYINT NOT NULL
);

CREATE TABLE game_compositions (
    composition_id VARCHAR PRIMARY KEY,
    game_id VARCHAR NOT NULL,
    composition_type VARCHAR NOT NULL,
    components VARCHAR[] NOT NULL,
    combined_trit TINYINT NOT NULL,
    gf3_conserved BOOLEAN NOT NULL
);
```

---

## Integration

| Skill | Trit | Bridge |
|-------|------|--------|
| `open-games` | +1 | Play/Coplay semantics |
| `accessible-worlds` | 0 | 26-world superposition |
| `gay-mcp` | +1 | Color assignment to nodes |
| `bisimulation-game` | 0 | Attacker/Defender structure |
| `world-hopping` | 0 | Badiou triangle navigation |

---

## Self-Reinvention

The weep is **autopoietic**: each game modifies its own bias based on utility feedback:

```julia
function coplay(utility, global_sum, generation)
    correction = if global_sum > 0.5 && trit == -1
        0.1  # Boost MINUS to correct positive imbalance
    elseif global_sum < -0.5 && trit == 1
        0.1  # Boost PLUS to correct negative imbalance
    else
        -0.05  # Default decay
    end
    correction / (1 + 0.1 * generation)  # Diminishing returns
end
```

---

## Commands

```bash
# Run transitive weep demo
bb src/openapi_to_catp.bb --tts-demo

# Query weep traces
duckdb zubyul_interactome.duckdb "SELECT * FROM weep_traces WHERE source='A'"

# Check game compositions
duckdb zubyul_interactome.duckdb "SELECT * FROM game_compositions WHERE gf3_conserved"
```

---

**Skill Name**: transitive-weep  
**Type**: Flow Network / Open Games  
**Trit**: 0 (ERGODIC)  
**GF(3)**: Self-balancing via coplay feedback


---

## 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.*