Planar Isotopy Screen Mapping

Maps thread states and observations to screen positions using planar isotopy principles.

Trit Value

0 (ERGODIC) - Coordinate between spatial regions

Purpose

Transform abstract thread relationships into concrete screen positions while preserving topological invariants:

• Adjacency: Neighboring trits occupy adjacent screen regions
• Handedness: MINUS→left, ERGODIC→center, PLUS→right
• Conservation: Screen area sum is invariant under isotopy

Screen Region Mapping

┌─────────────────┬─────────────────┬─────────────────┐
│                 │                 │                 │
│     MINUS       │    ERGODIC      │     PLUS        │
│    (left)       │    (center)     │    (right)      │
│                 │                 │                 │
│  Cold hues      │  Neutral hues   │  Warm hues      │
│  180-300°       │  60-180°        │  0-60°,300-360° │
│                 │                 │                 │
│  Validator      │  Coordinator    │  Generator      │
│                 │                 │                 │
└─────────────────┴─────────────────┴─────────────────┘

Position Computation

(defn seed->screen-position [seed trit screen-width screen-height]
  "Map seed deterministically to (x, y) within trit's region.

   Uses SplitMix64 decomposition:
   - High bits → x offset
   - Low bits → y offset"
  (let [region (trit->region trit screen-width)
        [rand1 seed'] (splitmix64 seed)
        [rand2 _] (splitmix64 seed')
        x (+ (:x region) (* (:width region) (/ rand1 MASK64)))
        y (* screen-height (/ rand2 MASK64))]
    {:x x :y y :region trit}))

Observation Lines

When thread A observes thread B, draw a line between their screen positions:

(defn observation-line [observer observed]
  {:from (seed->screen-position (:seed observer) (:trit observer))
   :to (seed->screen-position (:seed observed) (:trit observed))
   :gf3-sum (mod (+ (:trit observer) (:trit observed)) 3)})

Planar Isotopy Invariants

1. No crossings for conserved observations: Lines between threads with GF(3) sum = 0 should not cross
2. Triadic bundling: Triad members form non-crossing triangles
3. Seed progression: Moving a thread moves its position deterministically

Integration with Mutual Thread Observation

from mutual_thread_observation import MutualThreadObservationSystem

system = MutualThreadObservationSystem()
# ... register threads ...

# Get screen positions
for tid, state in system.thread_states.items():
    pos = seed_to_screen_position(state.seed, state.trit)
    print(f"{tid}: ({pos['x']:.0f}, {pos['y']:.0f}) in {pos['region_label']}")

macOS Integration

Use with macos-use MCP for actual screen interaction:

# Get element at thread's screen position
bb -e '(require \'[mutual-thread-demo :as mtd])
       (let [pos (mtd/seed->screen-position seed trit)]
         (println (:x pos) (:y pos)))'

Then use

mcp__macos-use__macos-use_click_and_traverse

at those coordinates.

Related Skills

• mutual-thread-observation

  - Core observation system

• gay-mcp

  - Deterministic color from seed

• acsets-algebraic-databases

  - Schema modeling

• bisimulation-game

  - Observational equivalence

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

planar-isotopy-screen (○) + SDF.Ch3 (○) + [balancer] (○) = 0

Skill Trit: 0 (ERGODIC - coordination)

Secondary Chapters

• Ch10: Adventure Game Example

Connection Pattern

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