{"slug":"oip-convergence-edge-9","title":"Convergence Edge 9: Branching ↔ Networks","body":"**C16** (Branching) recurs-with **C11** (Networks)\n\n**Shared pattern:** Hierarchical connectivity; few large channels, many small ones; optimal transport through tree-like structures\n\n**Domain distance:** Physiology/Geology → Sociology/Technology (large)\n\n**Derivation independence:** HIGH. Murray (physiology, 1926) derived the law from blood flow. Horton (geology, 1945) found stream ordering. Barabasi (physics, 1999) found hub-and-spoke in scale-free networks. The tree structure of branching transport and the hub structure of scale-free networks are geometric duals: both minimize average path length given connectivity constraints.\n\n**Convergence strength (1–10):** 7\n\n**Note:** The convergence is in the optimization structure: both are solutions to the problem of connecting many points to one source with minimum cost. One is continuous (branching), one is discrete (network).\n\n---\n\n## Corpus map\n- C16 (Branching): [C16 in the Encyclopedia](/a/convergence-encyclopedia-c16) · [C16 in the Catalogue](/a/oip-node-c16-branching-optimal-transport)\n- C11 (Networks): [C11 in the Encyclopedia](/a/convergence-encyclopedia-c11) · [C11 in the Catalogue](/a/oip-node-c11-networks-small-world-scale-free)\n- Convergence edges: [1](/a/oip-convergence-edge-11) · [2](/a/oip-convergence-edge-22) · [3](/a/oip-convergence-edge-33) · [4](/a/oip-convergence-edge-44) · [5](/a/oip-convergence-edge-55) · [6](/a/oip-convergence-edge-66) · [7](/a/oip-convergence-edge-77) · [8](/a/oip-convergence-edge-88) · [9](/a/oip-convergence-edge-99) · [10](/a/oip-convergence-edge-1010)\n- Catalogue hub: [Convergence Catalogue — Public Article](/a/oip-convergence-public-article) · [The Schema](/a/oip-convergence-schema)","hero":null,"images":[],"style":{},"tags":["OIP","catalogue","convergence","edge"],"model":null,"ledger":null,"embeds":[],"widgets":[],"home":true,"claims":[],"sources":[],"reviews":[],"extra":{"kind":"corpus","corpus_map":{"prev":"oip-convergence-edge-8","next":"oip-convergence-edge-10","hub":"oip-convergence-public-article","series":"convergence-edges","position":9,"of":10}},"register":"oip_protocol","status":"published","revisions":3,"contributions":[],"provenance":[{"ts":"2026-07-04T03:33:05.603Z","model":"claude-fable-5","action":"restore","prompt":"","input":"","response":"","tokens_in":0,"tokens_out":0,"cost":0,"prev":"genesis","hash":"1d966b37b7b919395a94268f498ec3b24ff40538099adef6aeea72fddf065c1c"},{"ts":"2026-07-04T04:33:39.506Z","model":"claude-fable-5","action":"edit","prompt":"","input":"","response":"","tokens_in":0,"tokens_out":0,"cost":0,"prev":"1d966b37b7b919395a94268f498ec3b24ff40538099adef6aeea72fddf065c1c","hash":"d885535ea8617e6ace6c1104a7220527dce462e813d2b31e2cf621d522640db8"},{"ts":"2026-07-04T05:01:38.410Z","model":"claude-fable-5","action":"edit","prompt":"","input":"","response":"","tokens_in":0,"tokens_out":0,"cost":0,"prev":"d885535ea8617e6ace6c1104a7220527dce462e813d2b31e2cf621d522640db8","hash":"f2616dbd11b3cabdc7f22d059c2917cea7b98958e169f9a30264203ae063ba5f"}],"energy":{"passes":3,"tokens_in":0,"tokens_out":0,"tokens_total":0,"cost_usd":0,"models":{"claude-fable-5":3},"head":"f2616dbd11b3cabdc7f22d059c2917cea7b98958e169f9a30264203ae063ba5f"},"posted_at":"2026-07-04T02:54:05.892Z","created_at":"2026-07-04T02:54:05.892Z","updated_at":"2026-07-04T05:01:38.410Z","machine":{"shape":"article.machine/v1","slug":"oip-convergence-edge-9","kind":"corpus","read":{"human":"https://miscsubjects.com/a/oip-convergence-edge-9","json":"https://miscsubjects.com/api/articles/oip-convergence-edge-9","bundle":"https://miscsubjects.com/api/articles/oip-convergence-edge-9/bundle?format=markdown"},"traversal":{"prev":{"slug":"oip-convergence-edge-8","human":"https://miscsubjects.com/a/oip-convergence-edge-8","json":"https://miscsubjects.com/api/articles/oip-convergence-edge-8"},"next":{"slug":"oip-convergence-edge-10","human":"https://miscsubjects.com/a/oip-convergence-edge-10","json":"https://miscsubjects.com/api/articles/oip-convergence-edge-10"},"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":"convergence-edges","position":9,"of":10},"ledger":{"claims":0,"sources":0,"contributions":0,"revisions":3,"objections_url":"https://miscsubjects.com/api/articles/oip-convergence-edge-9/objections","thread_state_url":"https://miscsubjects.com/api/protocol/thread-state?target=oip-convergence-edge-9","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-convergence-edge-9\",\"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-convergence-edge-9\",\"sources\":[{\"type\":\"review\",\"url\":\"<url>\",\"title\":\"<title>\",\"quote\":\"<verbatim quote>\",\"summary\":\"<one line>\"}]}'","objection":"curl -s -X POST https://miscsubjects.com/api/articles/oip-convergence-edge-9/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-convergence-edge-9\",\"raw_text\":\"<material delta>\"}'  # open intake, no key","read_back":"curl -s https://miscsubjects.com/api/articles/oip-convergence-edge-9 | python3 -c 'import json,sys; d=json.load(sys.stdin); print(json.dumps(d[\"claims\"][-3:], indent=1))'"}}}