Visual diagrams explaining Cyberspace Protocol concepts — Cantor trees, coordinate interleaving, and Hyperspace entry

Visual Guide

Diagrams and visualizations for understanding Cyberspace Protocol's core mathematical concepts

Cantor Pairing Tree

Movement proofs in Cyberspace are structured as Cantor pairing trees. Unlike arbitrary hash grinding, Cantor trees create real mathematical structure — each number represents an actual subtree that must be traversed.

Cantor Tree for Path [42, 43, 44, 45] Root h=45 h=43-44 45 h=42 43 leaf temporal seed 42 43 44 45 Legend Branch (internal node) Leaf (path element) Computational Cost: O(path_length) Cantor pairings

Temporal Seed

First leaf is derived from previous event ID, preventing proof reuse. Each hop requires fresh computation.

Path Leaves

Each block height in the traversal path becomes a leaf. The tree proves you traversed the entire sequence.

Root Hash

The tree root is the proof, published in Nostr event tags. Verifiers can reconstruct and validate.

Coordinate Interleaving

Cyberspace coordinates are 256-bit integers with three 85-bit axes (X, Y, Z) and a plane bit. Bits are interleaved to preserve spatial locality — similar coordinates have similar integer values.

256-bit Coordinate Structure X (bits 0-84) Y (bits 85-169) Z (bits 170-254) P Interleaved Bit Pattern x₀ y₀ z₀ … z₈₄ y₈₄ x₈₄ P plane X: 85 bits Y: 85 bits Z: 85 bits 256 bits total (including plane bit)

Why Interleave?

Interleaving preserves spatial locality. Points that are close in 3D space have similar 256-bit values, enabling efficient range queries and proofs.

Gibsons & Sectors

Each axis is 85 bits: 30 bits for Gibsons (2³⁰ per sector), 55 bits for sector coordinates. Sector extraction uses top 55 bits.

Hyperspace Entry Mechanism

Hyperspace is a 1-dimensional network formed by Bitcoin block Merkle roots. Each Hyperjump has 3 sector planes (X, Y, Z) that serve as entry points from 3D cyberspace.

Hyperjump Sector Planes (Entry Points) X Y Z Hyperjump Bitcoin Block Merkle Root X-plane Y-plane Z-plane Entry Process 1. Find Sector Match 55-bit sector coordinate 2. Compute Cantor Proof LCA height ≈33 feasible 3. Publish enter-hyperspace kind 3333, A=enter-hyperspace Entry Cost Compute time: ~15 minutes Cloud cost: ~$0.09 LCA height: h≈33 Exit: Exact HJ coord

3 Entry Planes

Each Hyperjump has X, Y, and Z sector planes — 1 sector thick (2³⁰ Gibsons). Match the 55-bit sector coordinate to enter.

Feasible Proof

Entry requires Cantor proof to LCA height ~33, achievable in ~15 min on consumer hardware or $0.09 cloud compute.

Exact Exit

You always exit at the exact 3D coordinate (Hx, Hy, Hz) of the destination Hyperjump, preserving spatial meaning.

Hyperspace Traversal

Once on Hyperspace, travel between Hyperjumps is nearly instant. Traversal proofs use incremental Cantor trees with temporal seeds — O(path_length) computation.

Hyperjump Chain (Bitcoin Blocks) HJ₁ Block N HJ₂ Block N+1 HJ₃ Block N+2 HJ₄ Block N+3 HJ₅ Block N+4 hyperjump Traversal Proof: [temporal_seed, N+1, N+2, N+3, N+4] temporal N+1 N+2 N+3 N+4 Performance 1-block jump: ~200 ns 1000-block jump: ~1 μs Throughput: ~1M/day

Key Properties

  • ▸ Temporal seed binds proof to chain position — prevents replay attacks by requiring previous event ID
  • ▸ Linear cost — O(path_length) Cantor pairings, not exponential. 1-block and 1000-block jumps are both instant
  • ▸ Published as Nostr events — kind 3333, A=hyperjump with proof_hex tag containing Cantor tree root

Use These Diagrams

All diagrams are open source (CC0). Use them in presentations, documentation, or educational content about Cyberspace Protocol.

Browse FAQ View Glossary