---
slug: scaling_law
name: Scaling Law
one_liner: "A fixed relationship in which one quantity is governed by a ratio-power of another, so that behavior is set by relative proportion and how it changes with size rather than by absolute magnitude."
type: prime
schema_version: 2
status: density_v2_draft
categories: [physical, structural, biological]
origin_domain: mathematics
alternate_origin_domains: [physics, biology, statistics, design]
aliases: [power_law_relation, allometric_scaling, proportional_scaling_relation, ratio_governance, scale_invariant_relationship]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-05 generalization) — the proposed GENUS of `proportion_scale` (lift: strip the *visual/architectural aesthetic* frame, leaving a content-neutral ratio/power relationship governing how quantities relate and change with size). Discovered in the op-05 framed-stratum sweep 2026-07-03. Tier B candidate. STATUS: HOLD pending owner keep/cut. FRONTIER-REFORMULATED flag: this is a frontier-reformulation, not a simple lift — the neighboring genera `power_law`, `scaling`, and `allometry` are ALL absent from the corpus, so owner judgment is needed on how wide to draw the node (aesthetic proportion vs. power-law vs. scale-invariance are candidate framings of the same region). Owner judgment required."
- "Created 2026-07-03; STUB ONLY (frontmatter + Core Idea + Notes). SI/SF/similarity are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT written."
- "Phase-C action: re-similarity-check vs `proportion_scale` (the input) and any power-law / self-similarity / dimensional-analysis / fractal prime; DECIDE the node's width (does it name aesthetic proportion, allometric power-laws, or scale-invariance broadly?); confirm it is a genuine genus (covers design proportion AND musical intervals AND biological allometry AND typographic ratios) and not a restatement of the input; draw a subsumption edge proportion_scale→scaling_law + sibling edges at incorporation; sync one_liner to a v1 draft per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 5
  domain_breadth: 5
  structural_abstraction: 5
  transfer_evidence: 5
  reasoning: 'The bare structure — behavior is governed by *relative proportion* and by how a quantity changes as a function of size (canonically a ratio or power relation y ∝ x^k), independent of absolute magnitude — is among the most substrate-independent forms in the corpus. It is literally a mathematical relation and recurs identically in geometry, physics, biology, information, and design; the same scale-invariant / power-law skeleton governs allometry, frequency ratios in music, fractal self-similarity, and physical scaling arguments. Domain breadth, structural abstraction, and transfer all score 5 because the pattern is value-neutral, medium-neutral, and carries verbatim across disciplines. Lifting the aesthetic frame off `proportion_scale` produces one of the highest-SI genera in the sweep — a textbook case of a framed instance sitting atop a fully substrate-independent structure.'
similarity_to_nearest_existing_prime:
  slug: proportion_scale
  similarity: 0.58
structural_framed:
  label: structural
  aggregate: 0.14
  criteria: {vocab_travels: 0.9, evaluative_weight: 0.1, institutional_origin: 0.1, human_practice_bound: 0.1, import_vs_recognize: 0.1}
  rationale: 'Strongly structural. Once lifted off the design/aesthetic frame of `proportion_scale`, the residue is a bare mathematical relation (ratio/power governance, scale-invariance) with essentially no evaluative content, no institutional origin, and no binding to human practice: it is recognized in the wild across physics and biology, not imported as a doctrine, and its vocabulary travels almost perfectly. This is the clearest structural candidate in the batch — the only frame residue is that the *input* it was lifted from lives in aesthetics.'
phase_b_passes_completed: []
---
# Scaling Law

## Core Idea

A scaling law is, as a bare structure, a *fixed relationship in which one quantity is governed by the ratio or power of another* — behavior is determined by *relative proportion* and by how a quantity changes as size changes, independent of absolute magnitude. Canonically it takes the form of a ratio or a power relation, so that doubling the input scales the output by a fixed factor set by the exponent rather than by the input's absolute value. It is reached from `proportion_scale` by stripping one constraint: the *visual/architectural aesthetic* frame. The input concerns how the ratios among parts of a designed work affect perception and aesthetic impact; strip the aesthetics and what remains is the underlying commitment to *relational sizing* itself — a mathematical relation about ratios and how they govern behavior across scale, with no reference to art, perception, or design.

The load-bearing anatomy is *scale-relativity*: the governing quantity is a ratio (or exponent) rather than an absolute size, so the same relation holds regardless of the units or the absolute scale at which it is read. From this follow the two recurring corollaries. First, *proportion fixes behavior*: what matters is the relationship among quantities, not their individual magnitudes — change all sizes together and the behavior is preserved; change their ratios and the behavior changes. Second, *predictable change under rescaling*: because the relation is a ratio or power, knowing the exponent lets one extrapolate behavior to sizes never directly observed. The characteristic payoff is *transfer and prediction across scale*: a single ratio or exponent compresses an entire family of size-varying behaviors, so measurements at one scale predict outcomes at others and structures at one size can be redesigned at another by holding proportions fixed. The characteristic cost/failure is *breakdown of scale-invariance*: real systems obey a scaling law only within a regime, and extrapolating a ratio or power beyond its valid range — where a different mechanism dominates — produces confident, systematically wrong predictions (the square-cube problem, the failure of naive geometric scaling).

Read structurally, the same ratio-governance recurs across domains that share nothing aesthetically. In *music*, intervals are frequency proportions — consonance and the identity of a note relative to another are fixed by ratios (an octave is 2:1) independent of absolute pitch. In *biology*, allometric scaling laws relate an organism's metabolic rate, bone thickness, or lifespan to its body mass by fixed power exponents, so that traits change predictably with size across orders of magnitude. In *typography and design*, aspect ratios fix the character of a form independent of its absolute dimensions. `proportion_scale` — the aesthetics of relative sizing in a designed work — is one species, the case where the ratios govern *perception*; the siblings show the genus is the bare ratio/power relation, governing whatever it is applied to. That the neighboring names `power_law`, `scaling`, and `allometry` are all absent from the corpus underlines that this whole region is currently unrepresented.

## Notes

**Provenance.** Produced by operator 05 (generalization / lift) as the proposed genus of `proportion_scale` (input one-liner: "Relative size relationships"), by stripping the visual/architectural aesthetic frame to leave a content-neutral ratio/power relationship governing behavior across scale. Surfaced in the op-05 framed-stratum sweep on 2026-07-03. Claimed siblings: musical interval ratios (frequency proportions), allometric scaling laws in biology, typographic aspect ratios. Manifest tier: B, flagged frontier-reformulated — `power_law`, `scaling`, and `allometry` are all absent from the corpus, so this candidate reformulates a whole missing region and needs owner judgment on node width.

**For owner evaluation (keep/cut).** The case FOR: this is the highest-substrate-independence candidate in the batch — a bare mathematical ratio/power relation that recurs verbatim across physics, biology, music, and design, with genuinely unlike siblings, and it exposes a conspicuous corpus gap: the power-law / scaling / allometry / scale-invariance region has *no* representation at all. Keeping it (in some form) closes a real structural hole. The RISKS / decisions Kurt must make: (1) **Node width — the central judgment.** Because the neighboring genera are all absent, "scaling law" could be scoped narrowly (allometric power-laws), broadly (any ratio/power relation, near `self_similarity` and `dimensional_analysis`), or aesthetically (staying close to the input). The frontier-reformulated flag exists precisely because *how wide to draw this node* is the open question — Kurt should decide whether to admit one broad `scaling_law` / `power_law` prime or several (e.g., separate `power_law`, `scale_invariance`, `allometry`). (2) **Restatement risk vs the input is low** (similarity ~0.58) — the lift genuinely adds the non-aesthetic siblings. (3) **Alias/overlap risk** with any existing `self_similarity`, `fractal`, or `dimensional_analysis` prime — check before drawing. Sharp keep/cut question: *admit a single broad ratio/power-governance prime, or open the power-law region as several sibling primes?* This is a scoping decision more than a keep/cut.

**At incorporation.** Re-run similarity against `proportion_scale` (input) and any power-law / self-similarity / fractal / dimensional-analysis / scale-invariance prime to confirm no collapse and to settle node width. If kept: draw a subsumption edge `proportion_scale` → `scaling_law`, sibling edges to the musical-interval / allometry / typographic instances (or the primes carrying them), and — importantly — queue the absent neighbors (`power_law`, `scaling`, `allometry`) as a decision set so the region is opened coherently rather than piecemeal. Add a missing-region note. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
