Proposition 3I.5.P0320 of 83 in the corpus
A hidden layer is a learned coordinate system, not a summary.
The hidden vector is not a compressed description of the input. It is whatever set of coordinates makes the next layer's job easy, and it has meaning only relative to the layer that consumes it.
Demonstration
A hidden layer produces a vector. It is tempting to read that vector as a description of the input — a compressed one, a set of detected properties, a summary. Nothing in the definition supports that reading, and one calculation removes it.
Take a network with one hidden layer. Its output is
yhat = relu(x W1 + b1) W2 + b2Now insert any permutation P of the hidden units: replace W1 by W1 P, b1 by b1 P, and W2 by P transposed times W2. Every hidden coordinate has moved. The output has not changed by a single digit — the permutation cancels between the two layers. So the numbering of the hidden coordinates carries no information at all, and neither does any other property of the hidden vector that the second layer’s weights can absorb.
What the layer is optimising. The hidden vector is not chosen to describe x. It is chosen to make the composite loss small, which is a statement about what the next layer needs, not about what the input contains. If the head is a single linear map, the hidden layer’s task is to place the classes where a hyperplane can separate them — which is exactly the failure the perceptron of I.2 could not repair on its own, now repaired one layer earlier.
Which is why the reading matters. Two networks trained on the same data to the same accuracy can have hidden representations with no correspondence between them: different units, different scales, different directions. Any claim about “the feature this unit detects” is a claim about one training run, and it does not transfer.
Corollary
Three consequences, all of them practical.
Interpretation must be earned per model. A unit’s meaning is not given by the architecture. Establishing one requires an experiment — an intervention, an ablation, a probe — and the result belongs to that checkpoint.
Comparisons run through behaviour, not weights. Two models agree if they agree on inputs. Comparing them coordinate by coordinate compares an arbitrary labelling of hidden units, which is Proposition 4.
The last layer is the one that fixes the geometry. Whatever a hidden layer learns is shaped by what consumes it. Change the head — a different loss, a different number of classes — and the representation the same architecture learns changes with it. This is the mechanism behind transfer, and it is why Book III can freeze a backbone and replace only the head.