η̂ₜ
estimated adaptive evidence-integration rate
Scientific Engineering · Mathematical Core
Mathematics turns an idea into a testable claim.
BenchEWS separates mathematical definition, estimator, implementation, measurement quality, and empirical validity. A plausible relationship becomes scientific only through explicit conditions, competing models, and possible refutation.
00 / Plain-Language Bridge
Why describe BenchEWS as a mathematical detector?
The phrase describes its function, not the full theory. BenchEWS combines structural and dynamic quantities to study changes in degrees of freedom, feedback, observability, and self-correction mathematically.
01 / Exact Decomposition
Prediction-error persistence is not a universal effect of a declining learning rate
The relationship AC1(PE) ≈ 1 − η must not be read as a general identity or causal law. The noise term is generally coupled to the current prediction error; the direction and strength of persistence depend on the environmental, noise, and updating regime.
PEₜ₊₁ = (Oₜ₊₁ − Oₜ) + (1 − ηₜ)PEₜPEₜ₊₁ = (1 − ηₜ)PEₜ + (εₜ₊₁ − εₜ)
Specified minimal case: For one fully specified minimal case, ρ₁(PE) = −η/2. This result belongs only to that model and is not a general law.
02 / Revised Research Question
From a rigid chain to regime-dependent hypotheses
The research question is no longer whether a declining learning rate necessarily produces increasing prediction-error autocorrelation, but under which environmental, noise, and updating regimes changes in adaptive evidence integration are associated with altered prediction-error persistence and subsequent signal–response decoupling.
Every arrow is a hypothesis: No universal causal chain is assumed.
03 / Minimal Measurement Vector
BI(t) = [ η̂ₜ, Persistence(PE)ₜ, I(S;R)ₜ, Trecovery ]
Persistence(PE)ₜ
abstract persistence slot; AC1 is a primary candidate
I(S;R)ₜ
rolling environment–response mutual information; NMI may supplement it
T_recovery
time or trials to a defined post-perturbation baseline
Empirical status: The measurement vector has not been empirically validated.
04 / Estimator Latency
A measured lead may be an estimator artifact.
Different estimators use different windows and delays. A temporal lead becomes interpretable only after this latency has been quantified, corrected, and tested against alternative estimators.
05 / Studio 3.0 Research Architecture
Six connected domains — still separate evidence obligations
Structural Dynamics
Structural Compression, effective degrees of freedom, Frozen Kernel, and structural lock-in
Observation / Evidence Quality
FCQ, observation degradation, evidence quality, and observer effects
Persistence Dynamics
regime-dependent persistence and prediction-error dynamics
Propagation Dynamics
epistemic quality, visibility, authority bias, and network propagation
Adaptive Reopening
reopening correction pathways and restoring effective degrees of freedom
Individual Adaptive Systems
evidence integration, coupling, estimator latency, lock-in, and recovery
Conceptual research architecture — not an empirically established universal causal chain.
Studio 3.0 · PLANNED →06 / Self-Correction State Space
A measurable state vector, not a universal score
Xₜ = [Oₜ, Fₜ, Dₜ, Rₜ, Vₜ]ᵀSₜ = G(Xₜ; θdomain)
Xₜ separates observation, feedback, effective degrees of freedom, responsiveness, and validation. Sₜ is a research construct; G is not assumed to be universal. Domain adapters must map concrete observables, scales, and evidence boundaries explicitly.
Open the canonical self-correction framework →