{"slug":"oip-node-c10-scale-invariance-fractals-allometry","title":"Node C10: Scale Invariance / Fractals / Allometry","body":"# Node C10: Scale Invariance / Fractals / Allometry\n\nC10 — Scale Invariance / Fractals / Allometry\n{\n  \"id\": \"C10\",\n  \"claim\": \"The same quantitative rule governs structure across many orders of magnitude; branching transport networks, coastlines, and metabolic rates follow power-law scaling with characteristic exponents.\",\n  \"domain\": [\"mathematics\", \"condensed matter physics\", \"biology\", \"geography\", \"urban science\", \"cosmology\"],\n  \"pattern\": [\"fractal\", \"scaling_law\", \"allometry\", \"power_law\", \"self_similarity\"],\n  \"mechanism\": \"Fractals: objects with non-integer Hausdorff dimension, exhibiting self-similarity across scales. Allometry: metabolic rate B ~ M^(3/4) (Kleiber's law), explained by West-Brown-Enquist through optimal branching network geometry minimizing energy dissipation. Renormalization group: critical exponents are universal — same values across different microscopic Hamiltonians.\",\n  \"scale\": \"molecular → cosmic\",\n  \"claim_tier\": \"T1\",\n  \"sources\": [\n    \"Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman.\",\n    \"Kleiber, M. (1932). 'Body Size and Metabolism.' Hilgardia, 6(8), 315-353.\",\n    \"West, G.B., Brown, J.H. & Enquist, B.J. (1997). 'A General Model for the Origin of Allometric Scaling Laws in Biology.' Science, 276, 122-126.\",\n    \"Wilson, K.G. (1971). 'Renormalization Group and Critical Phenomena II.' Phys. Rev. B, 4(9), 3184-3205.\"\n  ],\n  \"dual\": \"Characteristic-scale systems — objects with a single intrinsic scale (like a sphere of fixed radius).\",\n  \"falsifier\": \"A branching transport network (circulatory, river, fungal) that violates the 3/4 metabolic scaling exponent or the fractal dimension predictions under controlled conditions.\",\n  \"rival_frame\": \"Scaling laws are dimensional necessity, not deep structure. The 3/4 exponent emerges from geometric constraints (space-filling + minimal energy), not from a 'grain' of nature. Fractals are descriptive tools, not explanations — they say 'it looks similar at different scales,' not why.\",\n  \"independence_check\": \"HIGH. Mandelbrot (mathematics, IBM, 1982) derived fractals from study of noise and cotton prices. Wilson (physics, Cornell, 1971) derived scaling from renormalization group in quantum field theory. WBE (biology, Santa Fe, 1997) derived allometry from optimal transport network theory. Kleiber (agricultural biology, Davis, 1932) found the 3/4 law empirically decades before theory. Four origins, same pattern: scale-independent rules.\",\n  \"pattern_type\": \"mathematical\",\n  \"maps_to_axiom\": [\"A7\"]\n}\n\n---\n\n## Corpus map\n- Same node, other planes: [Encyclopedia C10](/a/convergence-encyclopedia-c10) · [Inventory invariant](/a/oip-invariant-6-36-scale-invariance-fractals-allometry)\n- Edges touching C10: [convergence edge 4](/a/oip-convergence-edge-4) · [convergence edge 8](/a/oip-convergence-edge-8) · [disconfirming edge 5](/a/oip-disconfirming-edge-5)\n- Catalogue hub: [Public Article](/a/oip-convergence-public-article) · [Schema](/a/oip-convergence-schema)","register":"oip_protocol","tags":["philosophy","oip","convergence-catalogue","node","systems-theory"],"category":null,"style":{},"claims":[{"id":"c1","text":"The same quantitative rule governs structure across many orders of magnitude; branching transport networks, coastlines, and metabolic rates follow power-law scaling with characteristic exponents.","section":"# Node C10: Scale Invariance / Fractals / Allometry","tier":"mechanistic","source_ids":[],"source_status":"unsourced","why_material":"Core assertion defining the node's central scaling phenomenon."},{"id":"c2","text":"Fractals are objects with non-integer Hausdorff dimension, exhibiting self-similarity across scales.","section":"# Node C10: Scale Invariance / Fractals / Allometry","tier":"mechanistic","source_ids":[],"source_status":"unsourced","why_material":"Defines the fractal mechanism central to the scaling claim."},{"id":"c3","text":"Metabolic rate B scales as M^(3/4) per Kleiber's law, explained by West-Brown-Enquist optimal branching network geometry minimizing energy dissipation.","section":"# Node C10: Scale Invariance / Fractals / Allometry","tier":"human","source_ids":[],"source_status":"unsourced","why_material":"Specific quantitative allometric law and its proposed mechanism."},{"id":"c4","text":"Critical exponents from the renormalization group are universal, taking the same values across different microscopic Hamiltonians.","section":"# Node C10: Scale Invariance / Fractals / Allometry","tier":"mechanistic","source_ids":[],"source_status":"unsourced","why_material":"States universality property linking scaling across physics domains."},{"id":"c5","text":"The pattern applies from molecular to cosmic scales.","section":"# Node C10: Scale Invariance / Fractals / Allometry","tier":"speculative","source_ids":[],"source_status":"unsourced","why_material":"Specifies the claimed range of applicability."},{"id":"c6","text":"Characteristic-scale systems are objects with a single intrinsic scale, such as a sphere of fixed radius.","section":"# Node C10: Scale Invariance / Fractals / Allometry","tier":"mechanistic","source_ids":[],"source_status":"unsourced","why_material":"Defines the explicit dual contrast to scale-invariant systems."},{"id":"c7","text":"Independence of the scaling pattern is HIGH, derived independently from mathematics (Mandelbrot 1982), physics (Wilson 1971), biology theory (WBE 1997), and empirical observation (Kleiber 1932).","section":"# Node C10: Scale Invariance / Fractals / Allometry","tier":"anecdotal","source_ids":[],"source_status":"unsourced","why_material":"Asserts convergent origins across four distinct fields."},{"id":"c8","text":"The pattern maps to axiom A7.","section":"# Node C10: Scale Invariance / Fractals / Allometry","tier":"speculative","source_ids":[],"source_status":"unsourced","why_material":"Links the node to the broader axiom set."}],"sources":[],"prov":{"model":"Fable 5 (Claude Code)","action":"write"}}