ProofGeneral + Narya Skill

"Observational type theory: where equality is what you can observe, not what you can prove."

Overview

This skill combines:

• Proof General (543⭐): The universal Emacs interface for proof assistants
• Narya (225⭐): Higher-dimensional type theory proof assistant

Proof General Basics

;; Install via straight.el or package.el
(use-package proof-general
  :mode ("\\.v\\'" . coq-mode)
  :config
  (setq proof-splash-enable nil
        proof-three-window-mode-policy 'hybrid))

Key Bindings

Key	Action	Description
C-c C-n	proof-assert-next-command-interactive	Step forward
C-c C-u	proof-undo-last-successful-command	Step backward
C-c C-RET	proof-goto-point	Process to cursor
C-c C-b	proof-process-buffer	Process entire buffer
C-c C-.	proof-goto-end-of-locked	Jump to locked region end

Proof State Visualization

┌─────────────────────────────────────────────────────────────┐
│  ████████████████████░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░  │
│  ▲ Locked (proven)   ▲ Processing      ▲ Unprocessed       │
│                                                             │
│  GF(3) Trit Mapping:                                        │
│    Locked    → +1 (LIVE)     → Red    #FF0000              │
│    Processing → 0  (VERIFY)  → Green  #00FF00              │
│    Unprocessed → -1 (BACKFILL) → Blue #0000FF              │
└─────────────────────────────────────────────────────────────┘

Narya: Higher-Dimensional Type Theory

Narya is a proof assistant for higher observational type theory (HOTT).

Key Features

1. Observational Equality: Bridge types computed inductively from type structure
2. Higher Dimensions: Support for 2-cells, 3-cells, etc.
3. No Interval Type: Unlike cubical type theory, no explicit interval

Narya Syntax

-- Define a type
def Nat : Type := data [
  | zero : Nat
  | suc : Nat → Nat
]

-- Bridge type between values
def bridge (A : Type) (x y : A) : Type := x ≡ y

-- Transport along bridge
def transport (A : Type) (P : A → Type) (x y : A) (p : x ≡ y) : P x → P y
  := λ px. subst P p px

Observational Bridge Types (gay.el integration)

From narya_observational_bridge.el:

(cl-defstruct (obs-bridge (:constructor obs-bridge-create))
  "An observational bridge type connecting two versions."
  source      ; Source object
  target      ; Target object  
  bridge      ; The diff/relation between them
  dimension   ; 0 = value, 1 = diff, 2 = conflict resolution
  tap-state   ; TAP state: -1 BACKFILL, 0 VERIFY, +1 LIVE
  color       ; Gay.jl color
  fingerprint) ; Content hash

;; Create diff as bridge type
(defun obs-bridge-diff (source target seed)
  "Create an observational bridge (diff) from SOURCE to TARGET."
  (let* ((source-hash (sxhash source))
         (target-hash (sxhash target))
         (bridge-hash (logxor source-hash target-hash))
         (index (mod bridge-hash 1000))
         (color (gay/color-at seed index)))
    (obs-bridge-create
     :source source
     :target target
     :bridge (list :from source-hash :to target-hash)
     :dimension 1
     :color color)))

Hierarchical Agent Structure: 3×3×3 = 27

Level 0: Root (VERIFY)
    │
    ├── Level 1: BACKFILL (-1) ─── L2: [-1, 0, +1] ─── L3: 3×3 = 9 agents
    ├── Level 1: VERIFY (0)    ─── L2: [-1, 0, +1] ─── L3: 3×3 = 9 agents  
    └── Level 1: LIVE (+1)     ─── L2: [-1, 0, +1] ─── L3: 3×3 = 9 agents

Total: 1 + 3 + 9 + 27 = 40 agents (or 27 leaves)

Bruhat-Tits Tree Navigation

;; Navigate the Z_3 gamut poset
(defun bt-node-child (node branch)
  "Return child of NODE at BRANCH (0, 1, or 2)."
  (bt-node (append (bt-node-path node) (list branch))))

(defun bt-node-distance (node1 node2)
  "Return tree distance between NODE1 and NODE2."
  (let* ((lca (bt-node-lca-depth node1 node2))
         (d1 (bt-node-depth node1))
         (d2 (bt-node-depth node2)))
    (+ (- d1 lca) (- d2 lca))))

Möbius Inversion for Trajectory Analysis

;; Map TAP trajectory to multiplicative structure
;; -1 → 2, 0 → 3, +1 → 5 (first three primes)
(defun moebius/trajectory-to-multiplicative (trajectory)
  (let ((result 1))
    (dolist (t trajectory)
      (setq result (* result
                      (pcase t
                        (-1 2)
                        (0 3)
                        (+1 5)))))
    result))

;; μ(n) ≠ 0 ⟹ square-free trajectory (no redundancy)

Bumpus Laxity Measures

For coherence between proof states:

(defun bumpus/compute-laxity (agent1 agent2)
  "Laxity = 0 means strict functor (perfect coherence).
   Laxity = 1 means maximally lax."
  (let* ((d (bt-node-distance (narya-agent-bt-node agent1)
                               (narya-agent-bt-node agent2)))
         (mu1 (narya-agent-moebius-mu agent1))
         (mu2 (narya-agent-moebius-mu agent2))
         (mu-diff (abs (- mu1 mu2))))
    (min 1.0 (/ (+ d (* 0.5 mu-diff)) 10.0))))

Version Control Operations

Operation	Description	Dimension
fork	Create 3 branches (balanced ternary)	0 → 1
continue	Choose branch (-1, 0, +1)	1 → 1
merge	Resolve conflict (2D cubical)	1 → 2

Xenomodern Stance

The ironic detachment here is recognizing that:

1. Proof assistants are version control systems for mathematical truth
2. Type theory is a programming language for proofs
3. Observational equality is more practical than definitional equality
4. Higher dimensions emerge naturally from conflict resolution

Commands

just narya-demo           # Run Narya bridge demonstration
just proofgeneral-setup   # Configure Proof General
just spawn-hierarchy      # Create 27-agent hierarchy
just measure-laxity       # Compute Bumpus laxity metrics

References

• Proof General Manual
• Narya GitHub
• Higher Observational Type Theory
• Topos Institute: Structure-Aware Version Control