Hatchery Papers

SRFI-194: Random Data Generators (Final 2020)

;; From SRFI-194
(import (srfi 194))

;; Custom generator for SplitMixTernary
(define (make-ternary-generator seed)
  (let ((rng (make-splitmix64 seed)))
    (lambda () (splitmix-ternary rng))))

Math Egg

From https://wiki.call-cc.org/eggref/5/math:

• Random number generation
• Flonum operations
• Log-space arithmetic

(import (math base))
(import (math flonum))

Color/Graphics Eggs

Academic Papers

Colored Operads

1. "Theta Theory: operads and coloring" (Marcolli & Larson, 2025)

   • arXiv:2503.06091
   • Colored operad for theta theory
   • Coloring algorithm for syntactic objects
   • Merge operation with color filtering

2. "On the homotopy theory of equivariant colored operads" (Bonventre & Pereira, 2021)

   • arXiv:2004.01352
   • Model structures on equivariant operads
   • Weak equivalences by families of subgroups
   • Norm map data

3. "Combinatorial Homotopy Theory for Operads" (Obradović, 2019)

   • arXiv:1906.06260
   • Minimal model of colored operad O
   • Hypergraph polytopes
   • A∞-operad generalization

4. "Operads: Hopf algebras and coloured Koszul duality" (van der Laan, 2004)

   • Koszul duality for colored operads
   • Hopf algebra structure

2-Dimensional Type Theory / Higher Observational Type Theory

1. "Higher Observational Type Theory" (Altenkirch, Kaposi, Shulman)

   • nLab: https://ncatlab.org/nlab/show/higher+observational+type+theory
   • Internal parametricity
   • Displayed type theory

2. "Narya: A proof assistant for higher-dimensional type theory" (Shulman et al., 2025)

   • GitHub: https://github.com/mikeshulman/narya
   • Higher observational type theory
   • Interval-free proof assistant
   • 216 stars, active development

3. "2-dimensional TFTs via modular ∞-operads" (Steinebrunner, 2025)

   • arXiv:2506.22104
   • Modular ∞-operads
   • Cobordism categories
   • Spectral sequences for moduli spaces

Spectral Gap & Mixing

1. Ramanujan Graphs (Lubotzky, Phillips, Sarnak)

   • Spectral gap ≥ 2√q for (q+1)-regular graphs
   • Optimal expanders for 3-coloring

2. "Mixing Time of Markov Chains"

   • Spectral gap λ determines mixing time O(1/λ)
   • Our system: λ = 1/4, mixing time = 4

DisCoPy & Categorical Diagrams

1. "DisCoPy: Monoidal Categories in Python" (de Felice et al.)
   • String diagrams
   • Operad interface
   • Quantum circuit compilation

Integration Guide

Using Narya with gay.el

;; gay.el can interface with Narya for type checking
;; Narya provides observational bridge types

(require 'gay)

;; Create bridge type with color observation
(defun gay-narya-bridge (source target)
  "Create Narya-style observational bridge."
  (gay-bridge-create
   :source source
   :target target
   :transport 'narya-transport
   :color nil
   :version 0))

Using Chicken with colored operads

;;; Colored operad implementation

(define-record-type colored-operad
  (make-colored-operad colors operations composition)
  colored-operad?
  (colors colored-operad-colors)
  (operations colored-operad-operations)
  (composition colored-operad-composition))

;; GF(3) conservation as coloring constraint
(define (gf3-colored-merge op1 op2)
  (let ((c1 (operation-color op1))
        (c2 (operation-color op2)))
    (make-operation
     (merge-trees (operation-tree op1) (operation-tree op2))
     (modulo (- (+ c1 c2)) 3))))  ; Balance to 0

Using 2TDX shadows with operads

The 3-shadow system maps to colored operad structure:

Research Directions

1. Color logic soundness: Prove GF(3) conservation implies type safety
2. Spectral gap optimization: Find optimal gap for faster mixing
3. Operad composition: Verify colored composition preserves invariants
4. Narya integration: Bridge observational types with color observations

Commands

just chicken-eggs        # List installed eggs
just install-math-egg    # Install math egg
just narya-check         # Type check with Narya
just operad-color        # Demonstrate colored operad