cat

# cat Skill: Derivational Pipe Chaining

**Trit**: 0 (ERGODIC - coordinator)
**Color**: #26D826 (Green)
**Principle**: Chain operations via derivational succession, not temporal

---

## Overview

The `cat` skill implements the `|>` pipe operator using **derivational chaining** instead of temporal succession. Each pipe stage advances a seed: `seed_{n+1} = f(seed_n, trit_n)`.

## Core Formula

```
pipe_chain: A |> f |> g |> h
  seed₀ → f(seed₀, trit_f) → seed₁
  seed₁ → g(seed₁, trit_g) → seed₂  
  seed₂ → h(seed₂, trit_h) → seed₃

GF(3) conservation: Σ(trit_f + trit_g + trit_h) ≡ 0 (mod 3)
```

## Babashka Implementation

```clojure
(ns cat.pipe
  (:require [clojure.string :as str]))

(def GAMMA 0x9E3779B97F4A7C15)
(def MIX 0xBF58476D1CE4E5B9)
(def MASK64 0xFFFFFFFFFFFFFFFF)

(defn chain-seed [seed trit]
  (bit-and (unchecked-multiply 
             (bit-xor seed (* trit GAMMA)) 
             MIX) 
           MASK64))

(defmacro |> 
  "Derivational pipe with GF(3) tracking"
  [seed & forms]
  (reduce (fn [acc [f trit]]
            `(let [result# (~f ~acc)
                   new-seed# (chain-seed (:seed ~acc) ~trit)]
               (assoc result# :seed new-seed# :trit ~trit)))
          `{:value ~seed :seed 0x42D :trit 0}
          (partition 2 forms)))
```

## Usage

```bash
# Pipe with GF(3) conservation
bb -e "(require '[cat.pipe :refer [|>]])
       (|> 'input' 
           [read-fn -1]      ; MINUS: validate
           [transform-fn 0]  ; ERGODIC: coordinate
           [write-fn +1])    ; PLUS: generate
       ; => GF(3) sum = 0 ✓"
```

## Commands

```bash
# Run pipe chain
just cat-pipe 'input' -1 0 +1

# Verify GF(3) conservation
just cat-verify-gf3 chain.edn
```

## Integration

Forms triads with temporal-coalgebra (-1) and synthetic-adjunctions (+1):
```
temporal-coalgebra (-1) ⊗ cat (0) ⊗ synthetic-adjunctions (+1) = 0 ✓
```

---

**Skill Name**: cat
**Type**: Pipe Coordinator  
**Trit**: 0 (ERGODIC)
**Replaces**: dypler-mcp (not found in npm)



## 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

- `category-theory`: 139 citations in bib.duckdb

## 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.