Layer-neuron joint state–design minimization

Read the equation

For each layer, the selected state and design minimize behavior–design mismatch, weighted operating cost, state regularization and design regularization. The same environment is visible to every layer in this version. A shared objective form does not guarantee identical optima when layer-specific models, metrics or constraints differ.

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 (S_\ell^\, D_\ell^\) as the quantity being defined or updated by this relation.

2. Applies the stated minimum, maximum, or clamp. These bounds are part of the actual operation.

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

✔ PART 2 — The Full Equation Applied to Each Layer-Neuron

For each layer \ell \in \{1,…,7\}:

(S_\ell^\*, D_\ell^\*)
=\arg\min_{S_\ell', D_\ell'}
\Big[
\|D_\ell'(U,\mathcal{E}) - B_\ell(S_\ell',\mathcal{E})\|
+ W_\ell(U,\mathcal{E})\cdot \text{Cost}_\ell(S_\ell',S_\ell)
+ R_\ell(S_\ell',\mathcal{E})
+ R_{D,\ell}(D_\ell')
\Big]

This is the full global equation, injected directly into each layer.
•	No reductions
•	No simplifications
•	No “subset” formula

Each layer is a full optimizer-module defined only by the math.

(S_\ell^\*, D_\ell^\*)
=\arg\min_{S_\ell', D_\ell'}
\Big[
\|D_\ell'(U,\mathcal{E}) - B_\ell(S_\ell',\mathcal{E})\|
+ W_\ell(U,\mathcal{E})\cdot \text{Cost}_\ell(S_\ell',S_\ell)
+ R_\ell(S_\ell',\mathcal{E})
+ R_{D,\ell}(D_\ell')
\Big]

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