#SLL — Scaling‑Law Lens (binding)

Pattern C.18.1 · Stable Part C - Kernel Extension Specifications

One‑screen purpose (manager‑first). Make generation/selection scale‑savvy: at the level of conceptual descriptors, declare (a) which monotone knobs we would scale, (b) the ScaleWindow over which we claim behaviour, and (c) the elasticity class we observed—without imposing numeric fits or vendor tools at Core level. This surfaces knees early and keeps comparisons lawful and fair across families. (Parity is handled by G.9; illumination remains a report-only telemetry unless a CAL policy promotes it.)

Builds on. C.16 (MM‑CHR), C.17 (Creativity‑CHR), C.18 (NQD‑CAL); advisory: C.5 (Resrc‑CAL). Coordinates with. C.19 (E/E‑LOG), G.5 (Selector & Registry), G.9 (Parity Harness), G.10 (Shipping), G.11 (Refresh‑Telemetry), C.24 (Agent‑Tools‑CAL). Keywords. scaling law; Scale Variables (S); ScaleWindow; knee; diminishing returns; iso‑scale parity; UNM/NormalizationMethod‑based mapping; scale‑probe; DoE (design‑of‑experiments); segmented regression; knee detection.

Teams often say a method “scales” without disclosing which resources, across what window, and how outcomes respond (convex rise → knee → plateau). Without that, parity is skewed (unequal budgets, unmatched windows), coverage/illumination report-metrics leak into dominance, and “knees” are found late. SLL supplies a notation‑independent lens to make scale behaviour explicit and comparable.

#Keywords

• scaling law
• scale variables (S)
• compute-elasticity
• data-elasticity
• resolution-elasticity
• exponent class
• knee
• diminishing returns.

#Relations

#Content

#Problem frame

Teams often say a method “scales” without disclosing which resources, across what window, and how outcomes respond (convex rise → knee → plateau). Without that, parity is skewed (unequal budgets, unmatched windows), coverage/illumination report-metrics leak into dominance, and “knees” are found late. SLL supplies a notation‑independent lens to make scale behaviour explicit and comparable.

#Problem

Omitting Scale Variables and the comparison window causes: (i) unfair parity (compute/data/FoA mismatched), (ii) illumination/coverage report-metric creep into dominance by default, (iii) late detection of knees and budget waste. G.9 already forbids scalarising mixed scales and mandates equal FreshnessWindows/pinned editions; SLL complements this with ScaleWindow & elasticity.

#Forces

Notation independence vs useful scaling heuristics; local context vs cross‑context generality; telemetry vs objectives (illumination stays report‑only telemetry unless policy promotes it); early exploration vs reproducible policy.

#Solution — binding lens for generator/selector profiles (normative)

#Types (aliases; ΔKernel = 0).

SLL.Profile is an annotation on a MethodFamily/Generator or a Selector profile; no new U.Types are minted (LEX discipline).

#Fields (conceptual descriptors).

• S — Scale Variables. Minimal set of monotone knobs for the Context: compute (steps/tokens/FLOPs/time/energy), data (size/quality), model capacity (params/branches), iteration budget, freedom‑of‑action (FoA)/environment richness, etc. Declare units via Resrc‑CAL and bind to a ScaleWindow. Where training/inference trade, name the phase the claim concerns.
• ScaleWindow. Declared range of S values for which behaviour claims hold (editioned). This is distinct from FreshnessWindow used by parity.
• Scale‑Probe. At least two (preferably ≥ 3) parity‑respecting points in S within the ScaleWindow, recorded with replicates/seeds and CI/error bars to support elasticity classification. Pick points via a small factorial or Latin‑hypercube when multiple knobs vary.
• ElasticityClass χ ∈ {rising, knee, flat, declining} — a qualitative class; numeric exponents/fits live in domain annexes, not Core.
• ParityNotes. iso‑scale parity? flag (and loss notes if not achieved), plus Bridge/Φ/Ψ IDs when crossing contexts (penalties route to R only).

#Norms (SLL).

• SLL‑1 (Declaration). Any profile claiming scale behaviour SHALL declare S and a ScaleWindow for the Context.
• SLL‑2 (Probe). Early investigation SHALL include a scale‑probe (≥ 2 points in S, with replicates/CI) and record χ. Multi‑knob probes SHALL hold unspecified knobs fixed or pinned, and disclose invariants.
• SLL‑3 (Parity). Where S is declared, comparisons SHALL ensure iso‑scale parity and lawful UNM/NormalizationMethod‑based mapping across heterogeneous knobs (e.g., FLOPs↔tokens) before comparing outcomes; FreshnessWindows/editions must be equal/pinned per G.9. Record seeds/replicates, ComparatorSet, and policy‑ids in telemetry/SCR.
• SLL‑4 (Selection lens). Within the same Context and ScaleWindow, if other heads (N/U/C) are tied, selectors MAY use illumination as a tie‑breaker, but it SHALL NOT change default dominance; illumination remains report‑only telemetry unless a CAL policy promotes it.
• SLL‑5 (Knee test). A knee is claimed only where a monotone rise is followed by a statistically significant slope drop across adjacent probe points within the ScaleWindow; thresholds (e.g., Δslope & CI level) are policy‑defined (E/E‑LOG) and must be cited. Absent such evidence, classify as rising.
• SLL‑6 (Telemetry invariants). Probes SHALL export seeds/replicates, edition pins, policy‑ids, and Resrc‑CAL units to G.11.

#Method — minimal SoTA probe recipe (notation‑agnostic; informative).

1. Choose knobs S that are plausibly monotone in the Context (compute/data/capacity/FoA).
2. Pick 3–5 probe points per active knob (edge/mid/edge) under iso‑scale parity; use a fractional factorial if >2 knobs.
3. Run replicates (≥ 3 preferred) and bootstrap 95% CI on the primary objective(s); log seeds.
4. Estimate local slopes on a log‑log grid; apply piecewise/segmented regression or a knee detector (e.g., L‑curve/Kneedle) to support χ.
5. Record invariants (pinned knobs, safety envelope) and publish SLL.Card@Context.
6. If χ changes across the window, split the ScaleWindow and re‑classify per segment.

#Interfaces — minimal I/O (conceptual)

G.9 Plan/Run Parity consumes S/ScaleWindow to align budgets, pin editions, and perform UNM/NormalizationMethod‑based mapping; G.11 carries policy‑id, PathSliceId, seeds/replicates, CI level, and edition pins per parity CC.

#Conformance Checklist (CC‑SLL)

1. S declared or S = N/A with rationale.
2. Scale‑probe performed; χ recorded with replicates/CI; invariants disclosed.
3. iso‑scale parity or loss notes + penalties → R only; editions/seeds pinned; ComparatorSet cited.
4. If used as tie‑breaker, the selector cites χ and lens id in E/E‑LOG provenance.
5. Knee claims cite the policy threshold and CI level used.

#Anti‑patterns & remedies

Hidden budget mismatches; averaging ordinals across families; illumination in dominance by default; unpinned editions; slope claims without replicates/CI; training/inference phase mixing → cure with G.9 parity (equal windows/editions; normalize‑then‑compare; return sets), phase‑label the claim, and record slope uncertainty per Scale‑Audit discipline.

#Archetypal grounding (post‑2015; informative)

• LLM scaling. Kaplan‑style & Chinchilla‑optimal regimes; Mixture‑of‑Experts and retrieval‑augmented families shift effective capacity with different inference budgets; prompt‑policies often transfer better than narrow pipelines.
• RL/Planning. Model‑based optimization & general agents vs hand‑tuned controllers; slopes reported wrt budget/FoA under safety envelopes.
• QD/OEE. MAP‑Elites, CMA‑ME, DQD, QDax; POET/Enhanced‑POET families: coverage/illumination as telemetry metrics; parity uses fixed grids/spaces and edition pins.

#Payload — exports

SLL.Card@Context (UTS row; editioned): ⟨S{knobs, units, phase}, ScaleWindow, Scale‑Probe{points≥2, design=one‑liner, seeds, CI}, ElasticityClass χ, ParityNotes{iso‑scale?|loss, invariants}, BridgeIds?/Φ/Ψ, PolicyIds? (E/E‑LOG), PathSliceId?⟩.

UTS row template (conceptual; pencil‑ready). SLL.Card@Context := S=(COMPUTE|DATA|CAPACITY|FOA; units=…; phase=TRAIN|INFER), ScaleWindow=[LOW…HIGH], Probe=(points=…, design=factorial|LHD, seeds=…, CI=…), χ=rising|knee|flat|declining, ParityNotes=(iso=true|false; invariants=…), Bridge/Φ/Ψ=(…), PolicyIds=(…), PathSliceId=(…).

#Relations

Builds on: C.16/17/18. Coordinates with: C.19 (lenses/policies), G.5 (set‑returning selector), G.9 (parity; ParetoOnly default; UNM/NormalizationMethod‑based mapping), G.10 (shipping).

Pedagogical cue. Say what you would scale, probe it twice, and use the slope‑class to steer.

#C.18.1:End