Environment-aspect behavior–design loss

Read the equation

For one layer and one environment aspect, the loss combines design/behavior mismatch with weighted cost, state regularization and design regularization. This is the aspect-specific term later mixed with λ weights. Its numerical value depends on the chosen norm and the relative units of all penalties.

Terms & definitions

The recovered passage does not supply a separate symbol table. Read the source context for the notation used in this version; familiar letters may carry different meanings in other builds.

How it works

1. Read L_{\ell,a}^\*(S_\ell', D_\ell'; U, E^{(a)}) as the quantity being defined or updated by this relation.

2. Substitute the source-defined variables or recorded values and follow the displayed arithmetic and indexing exactly.

3. Use this version’s variable definitions and update order. The same symbol may be assigned a different meaning in another build.

Assumptions & limits

Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.

Original context

+ R_D(D')

Now, for layer \ell \in \{1,\dots,7\} and aspect a \in \{1,\dots,K\}, define a local “view” of this same equation:

L_{\ell,a}^\*(S_\ell', D_\ell'; U, E^{(a)})
\|D_{\ell,a}'(U,E^{(a)})-B_{\ell,a}(S_\ell',E^{(a)})\|
+ W_{\ell,a}(U,E^{(a)})\cdot \text{Cost}_{\ell,a}(S_\ell',S_\ell)
+ R_{\ell,a}(S_\ell',E^{(a)})
+ R_{D,\ell,a}(D_\ell')

Same structure, but:
•	fixed layer \ell
•	fixed aspect a
•	only that aspect’s slice E^{(a)}.

Now combine all aspects for one layer:

L_{\ell,a}^\*(S_\ell', D_\ell'; U, E^{(a)})
=
\|D_{\ell,a}'(U,E^{(a)})-B_{\ell,a}(S_\ell',E^{(a)})\|
+ W_{\ell,a}(U,E^{(a)})\cdot \text{Cost}_{\ell,a}(S_\ell',S_\ell)
+ R_{\ell,a}(S_\ell',E^{(a)})
+ R_{D,\ell,a}(D_\ell')

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