Appendix-B-Operationalization.md

Appendix B: Operationalization — Estimation Procedures

Epistemic status: This appendix is a procedures document. It specifies estimation recipes for core TFT quantities. End-to-end worked examples that instantiate the full chain are provided in Appendix C (Kalman, exact) and Appendix D (RL, approximate).

Purpose

This appendix addresses the measurement gap between TFT's formal objects and practical deployment: how to estimate $U_M$, $U_o$, $\rho$, $\alpha$, $R$, and $|\delta_{\text{critical}}|$ from data.

B.1 Measurement Targets

Quantity	Role in TFT	Typical unit	Minimum data needed
$U_M$	Model uncertainty (TF-06)	domain-specific variance/entropy	Predictive posterior or ensemble spread
$U_o$	Observation uncertainty (TF-04/TF-06)	sensor variance/noise scale	Channel calibration or residual variance
$\rho(t)$	Mismatch injection rate (TF-11)	surprise per time	Time-series of mismatch magnitudes
$\rho_{\text{delib}}$	Local mismatch drift during pauses (TF-09)	surprise per time	Deliberation windows with no corrective action
$\alpha$	Lower correction efficiency bound (App. A)	inverse time	Vector mismatch trajectories + correction term
$R$	Radius where local sector condition holds (App. A)	surprise magnitude	Same as $\alpha$, plus breakdown detection
$|\delta_{\text{critical}}|$	Functional adequacy threshold (TF-11)	surprise magnitude	Task-level performance curve vs mismatch

B.2 Estimator Cookbook

B.2.1 Estimating $U_M$ and $U_o$

Use the most native uncertainty representation available in the domain:

Domain	$U_M$ estimator	$U_o$ estimator
Kalman / linear Gaussian	Prior predictive variance $P_{t \vert t-1}$	Measurement-noise variance $R_t$ (known or EM-estimated)
Conjugate Bayes	Posterior variance / inverse effective sample size	Likelihood variance (or precision inverse)
RL with ensembles	Across-head predictive variance $\operatorname{Var}_i[Q_i(s,a)]$	TD-target noise variance over replay batches
Neural net regression	Ensemble or Laplace posterior variance	Aleatoric head output $\sigma^2(x)$
PID / classical control	State-estimation covariance from observer	Sensor noise from calibration + residual PSD

For TFT's scalar gain heuristic (TF-06), normalize to common units and compute:

[Operational Definition] $$\hat{\eta}^*t = \frac{\hat{U}{M,t}}{\hat{U}{M,t} + \hat{U}{o,t}}$$

B.2.2 Estimating $\rho(t)$ and $\rho_{\text{delib}}$

Let $s_t = |\delta_t|$ in surprise units (e.g., negative log-likelihood residual scale).

Global mismatch injection rate:

[Operational Definition] $$\hat{\rho}(t) = \left[\frac{s_{t+\Delta t} - s_t}{\Delta t} + \hat{\mathcal{T}}t , s_t\right]+$$

where $[x]_+ = \max(x, 0)$ and $\hat{\mathcal{T}}_t$ is estimated adaptive tempo.

Note on estimation sequencing. This estimator requires $\hat{\mathcal{T}}_t$, which is itself estimated from $\hat{\nu}$ and $\hat{\eta}^*$ (steps 2-3 in the recommended sequence, B.3). The estimation is therefore sequential, not simultaneous: estimate the gain and event rate first (from the agent's internal statistics and observation timing), then use these to extract $\rho$ from the mismatch trajectory. The gain and event rate are typically estimable independently of $\rho$ (they depend on the agent's own uncertainty and event timing, not on the environment's rate of change). This sequential structure avoids circularity but introduces sensitivity: errors in $\hat{\mathcal{T}}$ propagate linearly into $\hat{\rho}$.

Local pause-window drift for TF-09:

[Operational Definition] $$\hat{\rho}{\text{delib}} = \operatorname*{median}{w \in \mathcal{W}{\text{pause}}} \frac{s{w,\text{end}} - s_{w,\text{start}}}{\Delta\tau_w}$$

using windows where corrective action is suspended or effectively delayed.

B.2.3 Estimating $\alpha$ (sector lower bound)

Appendix A uses:

[Assumption A2'] $$\delta^T F(\mathcal{T}, \delta) \ge \alpha |\delta|^2 \quad \text{for } |\delta| \le R$$

Operationally:

1. Estimate $\dot{\delta}_t$ (finite differences or filtered derivative).
2. Compute $\widehat{F}_t = -\dot{\delta}_t + w_t$ where disturbance proxy $w_t$ is estimated from exogenous perturbation channels or residual balancing.
3. Form ratios $r_t = (\delta_t^T \widehat{F}_t) / |\delta_t|^2$ on bins of $|\delta_t|$.
4. Set conservative lower bound $\hat{\alpha}$ as a low quantile (for example 10th percentile) of $r_t$ in the valid region.

B.2.4 Estimating $R$ (valid-region radius)

Estimate $R$ as the largest radius for which sector inequality violations remain below tolerance:

[Operational Criterion] $$\hat{R} = \sup \left{ r > 0 : \Pr\left(\delta^T \widehat{F} < \hat{\alpha}|\delta|^2 ,\middle|, |\delta| \le r\right) \le \epsilon \right}$$

with a chosen violation tolerance $\epsilon$ (for example 5%).

B.2.5 Estimating $|\delta_{\text{critical}}|$

Define a mission-level performance metric $J$ (reward rate, tracking error, service SLA, etc.). Set the critical mismatch threshold where performance crosses a minimum acceptable level $J_{\min}$:

[Operational Definition] $$|\hat{\delta}{\text{critical}}| = \inf \left{ d : \mathbb{E}[J \mid |\delta| = d] < J{\min} \right}$$

This anchors TF-11's normalized persistence condition to real task outcomes.

B.3 Recommended Estimation Sequence

1. Fix mismatch representation $\delta$ in one consistent unit system (prefer surprise-scale).
2. Estimate $U_o$ from channel physics/calibration; estimate $U_M$ from model uncertainty.
3. Validate gain behavior against TF-06 ($\hat{\eta}^*$ trend checks).
4. Estimate $\rho_{\text{delib}}$ from pause windows (TF-09) and $\rho(t)$ from full traces (TF-11).
5. Estimate $\alpha$ and $R$ from local correction dynamics (Appendix A).
6. Estimate $|\delta_{\text{critical}}|$ from task-performance degradation.
7. Compute derived diagnostics: tempo margin $\hat{\mathcal{T}} - \hat{\rho}/|\hat{\delta}{\text{critical}}|$, reserve $\widehat{\Delta \rho^} = \hat{\alpha}\hat{R} - \hat{\rho}$, and deliberation feasibility $\Delta\eta^(\Delta\tau)|\delta{\text{post}}| - \hat{\rho}_{\text{delib}}\Delta\tau$.

B.4 Minimal Reproducibility Checklist

End-to-end worked examples demonstrating the full TFT chain are provided in Appendix C (Kalman/linear-Gaussian domain, exact mapping) and Appendix D (nonstationary bandit/RL domain, approximate mapping).

For any domain report claiming TFT validation, include:

1. Mismatch definition and units.
2. Estimation method for $U_M$, $U_o$, and uncertainty calibration diagnostics.
3. Estimation method for $\rho_{\text{delib}}$ and $\rho(t)$ with window definitions.
4. Sector-bound estimation method ($\alpha$, $R$) and violation tolerance $\epsilon$.
5. Task-level definition of $|\delta_{\text{critical}}|$.
6. At least one ablation where $\eta$ is intentionally miscalibrated to test TF-06 predictions.
7. At least one induced-shock test to evaluate reserve prediction $\Delta\rho^*$.

B.5 Estimator Uncertainty and Sample-Size Guidance

The estimators in B.2 are point recipes. This section provides guidance on their reliability.

$\hat{\eta}^*$ (gain estimate). Inherits uncertainty from $\hat{U}_M$ and $\hat{U}_o$. The ratio $U_M/(U_M + U_o)$ is most sensitive when $U_M \approx U_o$ (both contribute equally). Approximate variance via the delta method:

[Operational Guidance] $$\text{Var}(\hat{\eta}^*) \approx \hat{\eta}^{2}(1-\hat{\eta}^)^2 \left[\frac{\text{Var}(\hat{U}_M)}{\hat{U}_M^2} + \frac{\text{Var}(\hat{U}_o)}{\hat{U}_o^2}\right]$$

When $\hat{\eta}^*$ is near 0 or 1, the estimate is stable (dominated by one uncertainty source). Near 0.5, both sources matter and the estimate is most volatile.

$\hat{\rho}$ (mismatch injection rate). Finite-difference estimation amplifies noise. Recommend smoothing the mismatch time series (exponential moving average or local regression) before differencing. Minimum sample guidance: ~20 mismatch observations for a stable trend estimate; ~50 for reliable variance characterization. The $[\cdot]_+$ clipping in the estimator removes negative estimates but introduces positive bias at small $\rho$; report the pre-clipped distribution if possible.

$\hat{\alpha}$ (sector lower bound). The conservative lower-quantile approach (B.2.3) inherently accounts for estimation noise. The 10th-percentile choice gives approximately 90% confidence that the true $\alpha$ exceeds the estimate, under stationarity. Precision scales as $\sim 1/\sqrt{N}$ where $N$ is the number of bins. Report the quantile level, bin count, and bin width alongside $\hat{\alpha}$.

$\hat{R}$ (sector-condition radius). The violation-tolerance criterion (B.2.4) is inherently conservative. Report the tolerance $\epsilon$, the sample size per bin, and the number of bins with violations. A sharp drop-off in sector-condition satisfaction at some radius is a strong signal; a gradual decay is less informative and suggests the sector condition may not hold cleanly.

General nonstationarity caveat. All estimators assume approximate stationarity over the estimation window. If the environment is changing during estimation, the estimates characterize the average dynamics over that window, not instantaneous values. Use sliding windows matched to the expected stationarity timescale. When environment regime changes are suspected (TF-10), re-estimate from post-change data only.

B.6 Decision-Theoretic Procedures

The estimators above support three operational decision procedures.

B.6.1 Estimating $\lambda$ (Exploration Price)

The exploration weight $\lambda(M_t)$ in TF-08's unified policy objective prices information in value-equivalent terms. In domains with sufficient structure, $\lambda$ is derivable:

Context	$\lambda$ estimator	Source
Finite bandits	Gittins index from dynamic programming	Exact (Gittins 1979)
Linear-Gaussian	Probing cost in quadratic objective	Exact (dual control)
Discrete MDP	$(\text{VoI})^2 / \text{info gain}$	Information-directed sampling (Russo & Van Roy)
General	$\hat{\lambda} = c \cdot \hat{U}_M / \hat{U}_o$	Heuristic: scale CIY weight by relative uncertainty

For the heuristic: when $U_M \gg U_o$ (highly uncertain model), exploration is cheap relative to exploitation risk, so $\lambda$ should be large. When $U_M \ll U_o$ (confident model, noisy observations), exploitation dominates. The constant $c$ is domain-specific and should be calibrated by held-out performance.

B.6.2 Deliberation Stopping Policy

From Proposition 9.1 (TF-09), deliberation of duration $\Delta\tau$ is warranted when $\Delta\eta^*(\Delta\tau) \cdot |\delta_{\text{post}}| &gt; \rho_{\text{delib}} \cdot \Delta\tau$. Operationally:

1. Estimate $\rho_{\text{delib}}$ from prior pause windows (B.2.2).
2. Before each deliberation episode, estimate $|\delta_{\text{post}}|$ as current mismatch + $\rho_{\text{delib}} \cdot \Delta\tau_{\text{planned}}$.
3. Estimate $\Delta\eta^*(\Delta\tau)$ from the diminishing-returns profile of past deliberation episodes (or from the marginal improvement of the first few candidate actions evaluated).
4. Stop deliberating when the marginal improvement rate $\partial \Delta\eta^* / \partial \Delta\tau$ drops below $\rho_{\text{delib}} / |\delta_{\text{post}}|$ (the first-order optimality condition from TF-09).

B.6.3 Structural-Switch Trigger

From Proposition 10.1 (TF-10), structural adaptation is indicated when parametric convergence leaves a mismatch floor. Operationally, the switching decision balances expected mismatch reduction against transition cost:

1. Estimate the current mismatch floor $|\delta|_{\text{floor}}$ from the converged residual statistics.
2. Estimate the post-switch expected mismatch as $|\delta|_{\text{new}} \approx \rho / \alpha'$ where $\alpha'$ is the sector bound under the candidate new model class (may need pilot estimation).
3. Estimate transition cost $C_{\text{switch}}$: knowledge loss (parameters that don't transfer), retraining time ($\Delta\tau_{\text{switch}}$), and accumulated mismatch during transition ($\rho \cdot \Delta\tau_{\text{switch}}$).
4. Switch when: $(|\delta|{\text{floor}} - |\delta|{\text{new}}) \cdot T_{\text{horizon}} > C_{\text{switch}}$, where $T_{\text{horizon}}$ is the expected time the new model class will remain adequate.