The shield and control share one technical core: shadow_stream → 𝓐, σ_cross, β. The detective reports; the engineer acts. The same Theorem 10 \(\mathfrak{A}\) score that the shield emits as an alert becomes a quadratic term in the control cost.
Why this composes
- The framework primitives are derivatives (Law I — only rates of change enter framework reasoning), so the cost is well-defined on streaming data without any external scale (Law II).
- The same threshold \(\beta = 1\) governs every domain (Law IV). The cost layer needs no per-domain tuning constant.
- The only configurable weights are \(\alpha, \beta, \gamma, \delta\) — they trade off the three framework objectives against each other (and against actuation effort).
Cost function detail
$$ J(\text{shadows}, u) = \underbrace{\alpha \cdot \max(0, \mathfrak{A} - \tau_{T3})^2}_{\text{T10 anomaly}} + \underbrace{\beta \cdot \sigma_{\text{cross}}^2}_{\text{T11 consensus}} + \underbrace{\gamma \cdot \max(0, \beta_{\text{brake}} - 1)}_{\text{T1 finite-time}} + \underbrace{\delta \cdot \|u\|^2}_{\text{actuation}} $$
- \(\alpha\)-term (T10): quadratic, so the optimiser is most sensitive when \(\mathfrak{A}\) is far above \(\tau_{T3}\).
- \(\beta\)-term (T11): quadratic, so the policy suppresses shadow disagreement aggressively.
- \(\gamma\)-term (T1): linear-rectified, so the cost is zero when \(\beta < 1\) and grows linearly when it crosses 1.
- \(\delta\)-term: small actuation regulariser; resolves ties between equally-good policies.
Full scope: control/SCOPE.md