Is it possible to decompose a scalar value to a inter-dependent vector neural network?
Yes, you can do that by interchanging the position of decoder and encoder in an autoencoder.

In an autoencoder, you give a long vector as input - the encoder reduces it to a short length vector (compressed) - the decoder now takes this compressed vector as input and upsamples it to the size of the original vector. The autoencoder is trained by taking the Mean Square Error (MSE) of the output of decoder with respect to the input vector. This enforces the compressed vector representation to contain the information of the input vector.

Now coming to your case. You simply need to pass the single scalar value to a decoder that upsamples, say your 3 layer fully connected neural networks. Let this output be denotes as "latent representation". Now pass this "latent representation" to the encoder which uses this "latent representation" to output just a single scalar value. Use MSE objective to enforce the the above single scalar output to match the input scalar value. Once the training is done of the above reverse autoencoder, the "latent representation" will give you the required vector containing the information about the scalar value you wished to represent as a vector.