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(seedn, 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

(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

# 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

# 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

SDF Interleaving

This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):

Primary Chapter: 3. Variations on an Arithmetic Theme

Concepts: generic arithmetic, coercion, symbolic, numeric

GF(3) Balanced Triad

cat (−) + SDF.Ch3 (○) + [balancer] (+) = 0

Skill Trit: -1 (MINUS - verification)

Secondary Chapters

• Ch2: Domain-Specific Languages

Connection Pattern

Generic arithmetic crosses type boundaries. This skill handles heterogeneous data.

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.