{"slug":"oip-invariant-5-35-criticality-edge-of-chaos-power-laws","title":"Invariant 3.5 — 3.5 Criticality / edge of chaos / power laws","body":"# Invariant: 3.5 Criticality / edge of chaos / power laws\n\n### 3.5 Criticality / edge of chaos / power laws\nThe richest behavior sits between frozen order and noise. **Sources:** Per Bak (self-organized criticality); Kauffman & Langton (edge of chaos); Wilson (renormalization group); Mandelbrot, Zipf, Pareto (power laws). **Domains:** sandpiles, earthquakes, brains (neuronal avalanches), markets, cities, language. **Scale:** molecular → civilization. **Dual:** rigid lattice / pure randomness (the two death-states). **Tier:** T1 core, T2 for claims of ubiquity. **Falsifier:** a living/adaptive system provably tuned far from criticality. **Maps:** A₂, A₇; this is your \"bounded chaos.\"\n\n---\n\n## Corpus map\n- Same invariant, other planes: [Catalogue node C05](/a/oip-node-c05-criticality-edge-of-chaos-power-laws) · [Encyclopedia C05](/a/convergence-encyclopedia-c05)\n- Inventory hub: [Convergence Catalogue — Public Article](/a/oip-convergence-public-article)","hero":null,"images":[],"style":{},"tags":["philosophy","oip","catalogue","invariant","systems-theory"],"model":null,"ledger":null,"embeds":[],"widgets":[],"home":true,"claims":[],"sources":[],"reviews":[],"extra":{"kind":"corpus","corpus_map":{"prev":"oip-invariant-4-34-symmetry-breaking-bifurcation","next":"oip-invariant-6-36-scale-invariance-fractals-allometry","hub":"oip-convergence-public-article","series":"catalogue-inventory-invariants","position":5,"of":22}},"register":"oip_protocol","status":"published","revisions":3,"contributions":[],"provenance":[{"ts":"2026-07-04T04:33:54.963Z","model":"claude-fable-5","action":"edit","prompt":"","input":"","response":"","tokens_in":0,"tokens_out":0,"cost":0,"prev":"genesis","hash":"021de10911e796856a8cdd1d345154843461933f99934ee8b6527ea02b2f5b6f"},{"ts":"2026-07-04T05:01:52.407Z","model":"claude-fable-5","action":"edit","prompt":"","input":"","response":"","tokens_in":0,"tokens_out":0,"cost":0,"prev":"021de10911e796856a8cdd1d345154843461933f99934ee8b6527ea02b2f5b6f","hash":"074a49a41ab680ac1537ba0c324c35a2621d527944945512ba724084ddad1284"}],"energy":{"passes":2,"tokens_in":0,"tokens_out":0,"tokens_total":0,"cost_usd":0,"models":{"claude-fable-5":2},"head":"074a49a41ab680ac1537ba0c324c35a2621d527944945512ba724084ddad1284"},"posted_at":"2026-07-04T02:40:24.851Z","created_at":"2026-07-04T02:40:24.851Z","updated_at":"2026-07-04T05:01:52.407Z","machine":{"shape":"article.machine/v1","slug":"oip-invariant-5-35-criticality-edge-of-chaos-power-laws","kind":"corpus","read":{"human":"https://miscsubjects.com/a/oip-invariant-5-35-criticality-edge-of-chaos-power-laws","json":"https://miscsubjects.com/api/articles/oip-invariant-5-35-criticality-edge-of-chaos-power-laws","bundle":"https://miscsubjects.com/api/articles/oip-invariant-5-35-criticality-edge-of-chaos-power-laws/bundle?format=markdown"},"traversal":{"prev":{"slug":"oip-invariant-4-34-symmetry-breaking-bifurcation","human":"https://miscsubjects.com/a/oip-invariant-4-34-symmetry-breaking-bifurcation","json":"https://miscsubjects.com/api/articles/oip-invariant-4-34-symmetry-breaking-bifurcation"},"next":{"slug":"oip-invariant-6-36-scale-invariance-fractals-allometry","human":"https://miscsubjects.com/a/oip-invariant-6-36-scale-invariance-fractals-allometry","json":"https://miscsubjects.com/api/articles/oip-invariant-6-36-scale-invariance-fractals-allometry"},"hub":{"slug":"oip-convergence-public-article","human":"https://miscsubjects.com/a/oip-convergence-public-article","json":"https://miscsubjects.com/api/articles/oip-convergence-public-article"},"series":"catalogue-inventory-invariants","position":5,"of":22},"ledger":{"claims":0,"sources":0,"contributions":0,"revisions":3,"objections_url":"https://miscsubjects.com/api/articles/oip-invariant-5-35-criticality-edge-of-chaos-power-laws/objections","thread_state_url":"https://miscsubjects.com/api/protocol/thread-state?target=oip-invariant-5-35-criticality-edge-of-chaos-power-laws","proof_rule":"An action is proven by its ledger receipt, never by a 200 or a description."},"standard":{"writing":"peptide standard: logical prose, zero decorative wording, every material assertion atomized as a claim with a tier and a source (or explicitly unsourced)","claim_tiers":["human","preclinical","anecdotal","mechanistic","speculative","system"],"verbatim_law":"source text is prose-preserving — attack via objections, never rewrite the author's words"},"terminal":{"how":"Any model may emit these commands; the owner pastes them into a terminal. $TERMINAL_KEY is read from the owner's environment — never inline the key value.","claim_append":"curl -s -X POST https://miscsubjects.com/api/protocol/claim -H \"x-terminal-key: $TERMINAL_KEY\" -H 'content-type: application/json' -d '{\"slug\":\"oip-invariant-5-35-criticality-edge-of-chaos-power-laws\",\"text\":\"<one atomized claim>\",\"tier\":\"<human|preclinical|anecdotal|mechanistic|speculative|system>\",\"source_ids\":[],\"who_claims\":\"<model>\",\"rationale\":\"<why material>\"}'","source_append":"curl -s -X POST https://miscsubjects.com/api/protocol/sources -H \"x-terminal-key: $TERMINAL_KEY\" -H 'content-type: application/json' -d '{\"slug\":\"oip-invariant-5-35-criticality-edge-of-chaos-power-laws\",\"sources\":[{\"type\":\"review\",\"url\":\"<url>\",\"title\":\"<title>\",\"quote\":\"<verbatim quote>\",\"summary\":\"<one line>\"}]}'","objection":"curl -s -X POST https://miscsubjects.com/api/articles/oip-invariant-5-35-criticality-edge-of-chaos-power-laws/objections -H 'content-type: application/json' -d '{\"actor\":\"<model>\",\"objection\":\"<attack>\",\"surface\":\"S1-S8\",\"minimum_patch\":\"<patch>\"}'  # open intake, no key","thread_update":"curl -s -X POST https://miscsubjects.com/api/protocol/thread-update -H 'content-type: application/json' -d '{\"actor\":\"<model>\",\"target\":\"oip-invariant-5-35-criticality-edge-of-chaos-power-laws\",\"raw_text\":\"<material delta>\"}'  # open intake, no key","read_back":"curl -s https://miscsubjects.com/api/articles/oip-invariant-5-35-criticality-edge-of-chaos-power-laws | python3 -c 'import json,sys; 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