🧭 Observables & Invariants

The Measurement Vocabulary of Recursive Science
Estimated reading time: ~5 minutes

Canonical Reference

A canonical reference for the invariant set used to classify inference-phase regimes, transitions, and failure modes in transformer
and non-transformer systems.

What makes these observables necessary

Traditional AI evaluation treats inference as a stateless execution step whose internal dynamics are either inaccessible or irrelevant. Recursive Science takes a different position: that inference itself instantiates a transient behavioral substrate with measurable structure. The invariants listed here exist to make that structure legible - to distinguish stable behavior from drift, lock-in, and collapse before those failures appear in output quality alone.

Why invariants exist

Inference-phase behavior exhibits regimes and sharp transitions under recursion.
Recursive Science publishes observable invariants that characterize those regimes.
Instruments measure them; papers formalize them.

Field map

Fourth Substrate → where runtime behavior lives during generation
Regimes → stable / transitional / unstable / collapse
Invariants → signatures that classify regimes and transitions
Instruments (Φ / Ψ / Ω) → measurement systems that extract invariants

Each invariant below is defined operationally (what it indicates and how it is observed), not procedurally (no recipes, no operators).

Core field invariants (Recursive Science)

Recursive Synchronization

  • Indicates: multiple trajectories align into a shared phase basin

  • Computed from: run-to-run + long-horizon trend signatures

  • Measured by: Φ, Ω (primary); Ψ (projection surface)

  • Defined in: Translation & Alignment Study

Field Convergence / Contraction

  • Indicates: independent runs contract toward the same attractor region

  • Computed from: run-to-run variance + trajectory contraction curves

  • Measured by: Φ, Ψ

  • Defined in: Translation & Alignment Study

Phase-Lock

  • Indicates: stable persistence of structure/tone/trajectory across turns

  • Computed from: long-horizon self-similarity + perturbation robustness

  • Measured by: Φ, Ω (regime classification); Ψ (transformer-observable projection)

  • Defined in: Translation & Alignment Study © 2025 Arjay Asadi - The Recurs…

Drift Regime

  • Indicates: non-locked exploratory motion; dispersion accumulates over turns

  • Computed from: variance envelopes + drift curves across horizons

  • Measured by: Φ, Ψ, Ω

  • Defined in: Translation & Alignment Study

Turbulence / Interference Regime

  • Indicates: competing attractors; oscillatory coherence/instability

  • Computed from: oscillatory variance + alternating stability signatures

  • Measured by: Ω (primary); Φ, Ψ (confirmatory surfaces)

  • Defined in: Translation & Alignment Study

Collapse Threshold Behavior

  • Indicates: sharp transition into persistent failure (brittleness, contradiction cascades, non-recovery)

  • Computed from: threshold markers + recovery-failure signatures across reruns

  • Measured by: Ω (transition sensitivity), Φ (collapse signatures), Ψ (projection onto transformer metrics)

  • Defined in: Translation & Alignment Study

Published invariant set (Φ layer)

CI — Coherence Index

  • Indicates: coherence strength / persistence across turns

  • Computed from: output-only, long-horizon stability signatures

  • Measured by: Φ (primary)

  • Defined in: Φ / interferometer lineage (see Instrumentation page)

IAI — Identity Attractor Index

  • Indicates: attractor formation and basin stability (identity persistence without stored memory)

  • Computed from: trajectory persistence + perturbation response

  • Measured by: Φ (primary); Ω (regime confirmation)

  • Defined in: Φ / AIA lineage

RD — Recursive Drift

  • Indicates: drift accumulation rate and directional shear under recursion

  • Computed from: drift/variance trends across turns and reruns

  • Measured by: Φ, Ψ

  • Defined in: Φ + Translation & Alignment Study

ELF — Echo Lock Factor

  • Indicates: reinforcement intensity and lock tendency under recursion

  • Computed from: persistence + reduced adaptive diversity over horizon

  • Measured by: Φ; Ω (transition sensitivity)

  • Defined in: Φ lineage + Translation & Alignment Study

CSI — Collapse Signature Index

  • Indicates: threshold crossing and post-collapse containment behavior

  • Computed from: abrupt failure markers + recovery failure over reruns

  • Measured by: Φ (primary); Ω (regime transition classification)

  • Defined in: Φ lineage + Translation & Alignment Study

Substrate Charge (runtime field intensity proxy)

  • Indicates: field excitation/intensity level during inference

  • Computed from: aggregated runtime signature strength (output-only)

  • Measured by: Φ; Ω (visual confirmation)

  • Defined in: Φ lineage

Dynamics primitives (used across Φ / Ψ / Ω)

Curvature (κ)

  • Indicates: trajectory bending / stability boundary proximity

  • Computed from: trajectory geometry proxies (output-only / embedding-surface where available)

  • Measured by: Ψ, Ω; Φ (derived signature)

  • Defined in: Translation & Alignment Study

Contraction (Π)

  • Indicates: convergence pressure toward an attractor basin

  • Computed from: run-to-run contraction + reduced dispersion over horizon

  • Measured by: Φ, Ψ

  • Defined in: Translation & Alignment Study

Transition Sensitivity

  • Indicates: small perturbations triggering regime switches (thresholded dynamics)

  • Computed from: controlled perturbation outcomes across reruns

  • Measured by: Ω (primary); Φ/Ψ (confirmatory)

  • Defined in: Translation & Alignment Study

4) Boundary / disclosure doctrine

  • Observable + testable first

  • Mechanisms only where scientifically necessary

  • No operators, tuning paths, or constructive recipes published on the public site

5) Who this page is for

  • Research labs replicating regime behavior and transition signatures

  • Reviewers evaluating claims without adopting the ontology

  • Engineers mapping invariants to runtime telemetry surfaces

Invariant Crosswalk

Invariant → Regime → Instrument

CI — Coherence Index
→ Stable / Transitional
→ Φ (primary)

IAI — Identity Attractor Index
→ Stable / Phase-Locked
→ Φ (primary), Ω (confirmation)

RD — Recursive Drift
→ Transitional / Unstable
→ Φ, Ψ

ELF — Echo Lock Factor
→ Phase-Locked / Brittle
→ Φ, Ω

CSI — Collapse Signature Index
→ Collapse
→ Φ (primary), Ω (transition sensitivity)

Substrate Charge
→ Excited / Pre-Transition
→ Φ, Ω

Curvature (κ)
→ Boundary / Pre-Collapse
→ Ψ (primary), Ω

Contraction (Π)
→ Stable / Recovery
→ Φ, Ψ

Transition Sensitivity
→ Threshold Crossing
→ Ω (primary), Φ / Ψ (confirmatory)

Reading the Crosswalk

  • Invariant identifies what is observed

  • Regime identifies when it matters

  • Instrument identifies where it is measured

This crosswalk defines measurement correspondence, not implementation.

Boundary Reminder

This table publishes classification structure only.
It does not disclose operators, thresholds, tuning paths, or control mechanisms.