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

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:

{: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

(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

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

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

open-games

accessible-worlds

gay-mcp

bisimulation-game

world-hopping

Self-Reinvention

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

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

# 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.