RPCS-1

The Matching Laws (R-1 family)

Integrate less when the world changes faster — with the exact scaling depending on whether the receiver detects regime changes.

This is the structural insight behind every TI recommendation. Its current foundation is IMM Paper 18 (Observer Requirements), which derives it from the bias–variance trade-off faced by any bounded observer: the qualitative law — shorter integration in faster-changing environments — is forced under any convex loss. The exact TI ∝ 1/H scaling is conditional: it holds for receivers that detect regime changes and reset their window; fixed-window receivers obey a square-root law instead.

History note.Earlier versions of this page presented the Matching Principle as Pred-09-5 ("TI ≈ 1/H") from IMM Paper 9. Paper 18 supersedes that claim: it is vindicated qualitatively and sharpened quantitatively, and its published derivation claim is retired in favor of the derived R-1 law family. The product's recommendations are unchanged in direction; what improved is the honesty and precision of the foundation. In the current product the law is operationalized as an engineering rule, not presented as a completed validation of the AI-agent extension.

What it means in practice

Environmental entropy H measures how frequently and unpredictably an environment changes. (In Paper 18's terms this is an identification with the operational change rate — labeled honestly as a modeling bridge, not a derivation.) The matching law says the Temporal Integration window (TI) of a well-tuned receiver shrinks as H rises:

  • High entropy (chaotic environment) → short TI. The agent should integrate only recent signals; old context is noise. Maps to short context windows, frequent grounding, small max_tokens.
  • Low entropy (stable environment) → long TI. The agent benefits from integrating deep history; old context is still relevant. Maps to long context windows, rolling summaries, large max_tokens.

The lookup table

The SDK implements this as a linear interpolation over a lookup table (from config/matching.json):

H = 0.10 → TI = 90  (long deep integration)
H = 0.25 → TI = 70  (stable integration)
H = 0.50 → TI = 50  (balanced integration)
H = 0.75 → TI = 30  (responsive integration)
H = 0.95 → TI = 10  (rapid response)

For intermediate values, TI is linearly interpolated between the nearest table entries. For example, H = 0.2 (a "stable" environment) interpolates to TI ≈ 77.

Entropy levels

stable   → H = 0.20
moderate → H = 0.50
dynamic  → H = 0.75
chaotic  → H = 0.95

The oscillation threshold

The matching laws also imply a stability boundary: when SG × TI exceeds the oscillation threshold, the receiver is structurally near the oscillation regime — it integrates too long while amplifying too strongly, causing it to revisit decisions.

oscillation threshold: SG × TI > 7000

# Example: SG=80, TI=90 → product=7200 → oscillation warning

The tuner surfaces a warning whenever your configuration approaches this boundary.

Why this matters for LLMs

LLMs don't have a literal "integration window" in the receiver-dynamics sense, but the structural analogy holds:

  • TI maps to how much context the model is given and expected to use — context window size, rolling-summary frequency, grounding injection rate.
  • SG maps inversely to temperature — a high-gain receiver amplifies signal differences, which in LLMs means low temperature (crisper, more deterministic outputs).
  • The oscillation threshold maps to the regime where a model with a very long context and very low temperature will start re-analyzing the same evidence in loops.

The cross-substrate conjecture (C-12-1 from RPCS-1) holds that the same receiver-dynamics framework applies across biological, artificial, and hybrid cognitive systems. Every production deployment of this SDK is a test of that conjecture, so RPCS1 reports deterministic tuning hypotheses that should be measured against production traces.

The full law family

R-1 (integration) is one of five derived law families in Paper 18 — the others govern update elasticity (R-2), sampling rate (R-3), the detection channel's gain and criterion (R-4), and commitment thresholds (R-5). All five move the same direction as the change rate rises. See the framework page for the three-block picture (estimate, detect, commit) and the honest claim ledger, including the pre-registered checks that failed and were reported.