CECR.md

Convergent Extremal Coordinate Routing (CECR)

A decentralized routing model combining gossip-based membership convergence, extremal coordinate embedding, and XOR-based direct routing.

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Overview

CECR is a hybrid routing system designed for peer-to-peer networks that:

• Avoids full-mesh connectivity
• Maintains stable routing under churn
• Requires no central authority
• Converges using gossip alone

It combines three mechanisms:

1. Membership Convergence via gossip
2. Extremal Coordinate Embedding using global min/max
3. Hybrid Routing using:
   • normalized coordinate proximity
   • XOR distance (Kademlia-style)

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Definitions

Let the global peer set be:

S = { id₁, id₂, ..., idₙ }

Where each id is a fixed-width identifier (e.g. SHA-1 hash).

Define:

m = min(S) M = max(S) N = |S|

Constraints:

M > m
N ≥ 2

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Coordinate System

Each peer is mapped into a normalized 1D coordinate space:

p(i) = (i - m) / (M - m)

Range:

0 ≤ p(i) ≤ 1

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Distance Metric (Coordinate Space)

Distance between peers:

d_ratio(i, j) = |p(i) - p(j)| = |i - j| / (M - m)

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XOR Metric (Routing Backbone)

Define XOR distance:

d_xor(i, j) = i XOR j

Routing decision:

next_hop = argmin_k (k XOR target)

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Gossip-Based Membership Convergence

Peer set:

S = set of peer IDs

Canonical hash:

H(S) = Hash(sort(S))

Peers converge to:

m = min(S)
M = max(S)
N = |S|

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Broadcast Strategy

Fan-out:

f(N) = ceil(log2(N))

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Stability Model

Between membership changes:

p(i) is constant

Only changes when:

• new global min/max joins
• existing min/max leaves

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Bounded Drift

Using stale extrema:

p̃(i) = (i - m̃) / (M̃ - m̃)

Error is bounded and small:

|p̃(i) - p(i)| ≤ ε

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Hybrid Routing Strategy

Local routing:

minimize d_ratio(i, target)

Global routing:

minimize d_xor(i, target)

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Design Properties

• Decentralized
• Convergent
• Stable
• Scalable (O(log N))
• Fault tolerant

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Summary

CECR creates a shared coordinate system over a dynamic peer set while preserving routing correctness via XOR distance.