Read the equation
This expression selects a state or parameter setting that minimizes the displayed objective. Its terms describe what is penalized or fitted; the feasible domain, term weights and an actual solver determine the resulting solution.
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
Read (S_\ell^\, D_\ell^\) as the quantity being defined or updated by this relation.
Applies the stated minimum, maximum, or clamp. These bounds are part of the actual operation.
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
Read the surrounding source passage
You already have the full per-layer objective (with neurons removed):
For layer \ell \in \{1,\dots,7\}:
(S_\ell^\*, D_\ell^\*)=
\arg\min_{S_\ell', D_\ell'}
\Big[
\|D_\ell'(U,E_\ell)-B_\ell(S_\ell',E_\ell)\|
+ W_\ell(U,E_\ell)\cdot \text{Cost}_\ell(S_\ell',S_\ell)
+ R_\ell(S_\ell',E_\ell)
+ R_{D,\ell}(D_\ell')
\Big]
Now declare:
• This whole object is the internal operator of a neuron N_\ell.
• Each N_\ell looks like a neuron, but internally it runs that full optimization.
1.1 Macro-neuron state and operatorExact original notation
(S_\ell^\*, D_\ell^\*)=
\arg\min_{S_\ell', D_\ell'}
\Big[
\|D_\ell'(U,E_\ell)-B_\ell(S_\ell',E_\ell)\|
+ W_\ell(U,E_\ell)\cdot \text{Cost}_\ell(S_\ell',S_\ell)
+ R_\ell(S_\ell',E_\ell)
+ R_{D,\ell}(D_\ell')
\Big]Related builds
Sources & provenance
(S_\ell^\*, D_\ell^\*)= \arg\min_{S_\ell', D_\ell'} \Big[ \|D_\ell'(U,E_\ell)-B_\ell(S_\ell',E_\ell)\| + W_\ell(U,E_\ell)\cdot \text{Cost}_\ell(S_\ell',S_\ell) + R_\ell(S_\ell',E_\ell) + R_{D,\ell}(D_\ell') \Big]
(S_\ell^\*, D_\ell^\*)= \arg\min_{S_\ell', D_\ell'} \Big[ \|D_\ell'(U,E_\ell)-B_\ell(S_\ell',E_\ell)\| + W_\ell(U,E_\ell)\cdot \text{Cost}_\ell(S_\ell',S_\ell) + R_\ell(S_\ell',E_\ell) + R_{D,\ell}(D_\ell') \Big]
(S_\ell^\*, D_\ell^\*) = \arg\min_{S_\ell', D_\ell'} \Big[ \|D_\ell'(U,E_\ell)-B_\ell(S_\ell',E_\ell)\| + W_\ell(U,E_\ell)\cdot \text{Cost}_\ell(S_\ell',S_\ell) + R_\ell(S_\ell',E_\ell) + R_{D,\ell}(D_\ell') \Big]
(S_\ell^\*, D_\ell^\*)= \arg\min_{S_\ell', D_\ell'} \Big[ \|D_\ell'(U,E_\ell)-B_\ell(S_\ell',E_\ell)\| + W_\ell(U,E_\ell)\cdot \text{Cost}_\ell(S_\ell',S_\ell) + R_\ell(S_\ell',E_\ell) + R_{D,\ell}(D_\ell') \Big]
(S_\ell^\*, D_\ell^\*)= \arg\min_{S_\ell', D_\ell'} \Big[ \|D_\ell'(U,E_\ell)-B_\ell(S_\ell',E_\ell)\| + W_\ell(U,E_\ell)\cdot \text{Cost}_\ell(S_\ell',S_\ell) + R_\ell(S_\ell',E_\ell) + R_{D,\ell}(D_\ell') \Big]
(S_\ell^\*, D_\ell^\*) = \arg\min_{S_\ell', D_\ell'} \Big[ \|D_\ell'(U,E_\ell)-B_\ell(S_\ell',E_\ell)\| + W_\ell(U,E_\ell)\cdot \text{Cost}_\ell(S_\ell',S_\ell) + R_\ell(S_\ell',E_\ell) + R_{D,\ell}(D_\ell') \Big]