---
slug: oip-convergence-edge-4
title: Convergence Edge 4: Criticality ↔ Scale Invariance
canonical: https://miscsubjects.com/a/oip-convergence-edge-4
---

**C05** (Criticality) recurs-with **C10** (Scale Invariance)

**Shared pattern:** Power-law statistics; no characteristic scale; self-similarity across magnitudes; universality of exponents

**Domain distance:** Condensed matter physics → Biology → Urban science (large)

**Derivation independence:** HIGH. Bak (physics, Brookhaven, 1987) derived SOC from sandpile models. Wilson (physics, Cornell, 1971) derived universality from renormalization group. Mandelbrot (mathematics, IBM, 1982) derived fractals from noise analysis. WBE (biology, Santa Fe, 1997) derived allometric scaling from transport networks. Four fields, four methods, same statistics.

**Convergence strength (1–10):** 8

**Note:** The edge captures that criticality and scale invariance are two faces of the same phenomenon — no characteristic scale means events of all sizes, which means power laws.

---

## Corpus map
- C05 (Criticality): [C05 in the Encyclopedia](/a/convergence-encyclopedia-c05) · [C05 in the Catalogue](/a/oip-node-c05-criticality-edge-of-chaos-power-laws)
- C10 (Scale Invariance): [C10 in the Encyclopedia](/a/convergence-encyclopedia-c10) · [C10 in the Catalogue](/a/oip-node-c10-scale-invariance-fractals-allometry)
- Convergence edges: [1](/a/oip-convergence-edge-1) · [2](/a/oip-convergence-edge-2) · [3](/a/oip-convergence-edge-3) · [4](/a/oip-convergence-edge-4) · [5](/a/oip-convergence-edge-5) · [6](/a/oip-convergence-edge-6) · [7](/a/oip-convergence-edge-7) · [8](/a/oip-convergence-edge-8) · [9](/a/oip-convergence-edge-9) · [10](/a/oip-convergence-edge-10)
- Catalogue hub: [Convergence Catalogue — Public Article](/a/oip-convergence-public-article) · [The Schema](/a/oip-convergence-schema)
