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Appendix-C-Kalman-Example.md

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Appendix C: Worked Example — 1D Active Tracking (Kalman Domain)

Epistemic status: This appendix is a worked instantiation. Every TFT quantity has an exact Kalman-filter counterpart, making this a validation of the formal chain rather than an approximate mapping. All quantities are computable in closed form.

Purpose

This example instantiates the full TFT chain — TF-01 through Appendix A — in a single coherent system with explicit measurable quantities at each step. The system is minimal but includes both action-conditioned sensing and online model updating.

System: 1D active tracking with selectable sensor mode

The agent tracks scalar state $x_t$ and chooses sensor mode $a_t \in {L, H}$:

  • Dynamics: $x_{t+1} = x_t + v_t$, $v_t \sim \mathcal{N}(0, q)$, $q = 0.25$
  • Observation: $y_t = x_t + n_t^{(a_t)}$
  • Low mode noise: $r_L = 9$
  • High mode noise: $r_H = 1$
  • Event rate: $\nu = 5 \text{ Hz}$
  • High mode has higher energy cost

C.1 TF-01 (Scope)

Mapping status: exact.

  • $\Omega_t = x_t$ is partially observed via noisy channel.
  • $\mathcal{A} = {L, H}$ is non-empty and causally affects observation quality (action-dependent observation function, TF-01).
  • Residual uncertainty persists ($H(\Omega_t \mid \mathcal{C}_t) > 0$) due to process and sensor noise.

C.2 TF-02 (Causal Structure) + TF-08 (CIY)

Mapping status: exact.

Action precedes observation and changes $P(y_t \mid do(a_t))$ through $r_{a_t}$. Using low mode as comparator action ($q(a') = \mathbf{1}[a' = L]$) — equivalent to the policy-induced default convention (TF-08) in this binary-action case — the canonical CIY is the interventional KL divergence:

[Worked Quantity] $$\text{CIY}(H) = D_{\mathrm{KL}}!\big(P(y \mid do(H)) ,|, P(y \mid do(L))\big)$$ $$= \frac{1}{2}\left[\log!\left(\frac{P^- + r_L}{P^- + r_H}\right) + \frac{P^- + r_H}{P^- + r_L} - 1\right]$$

With $P^- = 4.25$, this gives: $$\text{CIY}(H) = \frac{1}{2}\left[\log!\left(\frac{13.25}{5.25}\right) + \frac{5.25}{13.25} - 1\right] \approx 0.161 \text{ nats}$$

C.3 TF-03 (Model)

Mapping status: exact.

Model state $M_t = (\hat{x}{t|t}, P{t|t})$ is a compression of interaction history with recursive update.

C.4 TF-04 (Event-Driven Dynamics)

Each sensor read is an observation event; updates occur asynchronously at $\nu = 5 \text{ Hz}$.

C.5 TF-05 (Mismatch)

Mapping status: exact.

Innovation:

[Worked Quantity] $$\delta_t = y_t - \hat{x}_{t|t-1}$$

C.6 TF-06 (Update Gain)

Mapping status: exact.

Scalar Kalman gain:

[Worked Quantity] $$K_t = \frac{P^-_t}{P^-t + r{a_t}}$$

With $P^- = 4.25$:

  • $K(H) = 4.25 / 5.25 \approx 0.810$
  • $K(L) = 4.25 / 13.25 \approx 0.321$

Exact TF-06 uncertainty ratio mapping:

  • $U_M = P^-_t$
  • $U_o = r_{a_t}$
  • $\eta^* = K_t$

C.7 TF-07 and TF-08 (Action Selection + Exploration)

Use policy objective:

[Worked Objective] $$a_t^* = \arg\max_a \left[\mathbb{E}[\text{value}(a)\mid M_t] + \lambda_t , \mathbb{E}[\text{CIY}(a)\mid M_t]\right]$$

When uncertainty is high ($P^-$ large), $\lambda_t$ and the CIY term favor high mode $H$. As uncertainty falls, policy shifts toward low-cost $L$.

C.8 TF-09 (Deliberation Threshold)

Suppose a planning pause of $\Delta\tau = 0.5 \text{ s}$, with measured local pause drift:

[Measured] $$\rho_{\text{delib}} = 0.40 ;\text{surprise/s}$$

Cost during pause: $\rho_{\text{delib}} \cdot \Delta\tau = 0.20$ surprise units. If $|\delta_{\text{post}}| = 0.70$, deliberation is worthwhile when (Proposition 9.1):

[Threshold] $$\Delta\eta^(0.5)\cdot 0.70 > 0.20 ;\Longrightarrow; \Delta\eta^(0.5) > 0.286$$

If expected gain improvement is below $0.286$, act immediately.

C.9 TF-10 (Structural Adaptation Trigger)

Assume maneuvering regime change introduces sustained residual autocorrelation and mismatch floor. If estimated valid radius drops to $R = 0.08$ while current bound radius is $R^* = \rho/\alpha = 0.12$, parametric adaptation is no longer adequate ($R^* > R$), triggering model-class change (for example constant-velocity → constant-acceleration process model).

C.10 TF-11 (Tempo + Persistence)

Using action mix $70% H, 30% L$:

[Worked Quantity] $$\bar{\eta}^* = 0.7(0.810) + 0.3(0.321) = 0.663$$ [Worked Quantity] $$\mathcal{T} = \nu \bar{\eta}^* = 5 \cdot 0.663 = 3.315 ;\text{s}^{-1}$$

With $\rho = 0.18 \text{ surprise/s}$ and $|\delta_{\text{critical}}| = 1$, persistence condition holds:

[Check] $$\mathcal{T} > \frac{\rho}{|\delta_{\text{critical}}|} ;;\Rightarrow;; 3.315 > 0.18$$

C.11 Appendix A (Lyapunov Robustness Mapping)

From data, suppose conservative estimates:

  • $\alpha = 2.6 \text{ s}^{-1}$
  • $R = 1.4$
  • $\rho = 0.18$

Then:

[Worked Bounds] $$R^* = \frac{\rho}{\alpha} = \frac{0.18}{2.6} \approx 0.069 < R$$ [Worked Reserve] $$\Delta\rho^* = \alpha R - \rho = 2.6(1.4) - 0.18 = 3.46$$

Interpretation:

  • The agent is comfortably within its invariant region.
  • It has substantial adaptive reserve to absorb additional disturbance before structural failure.

This completes one explicit path from TF-01 through Appendix A with measurable quantities at each step.

Summary: Mapping Quality

TFT Concept Kalman Mapping Status
Scope Exact Definitional
Causal structure + CIY Exact Closed-form KL
Model ($M_t$ as sufficient statistic) Exact Kalman state + covariance
Mismatch ($\delta_t$ = innovation) Exact Standard Kalman innovation
Gain ($\eta^* = K_t$) Exact Kalman gain IS uncertainty ratio
Tempo ($\mathcal{T} = \nu \bar{\eta}^*$) Exact Closed-form
Persistence condition Exact Linear ODE solution
Lyapunov bounds ($R^$, $\Delta\rho^$) Exact From estimated sector parameters