This document provides mathematically consistent derivations for the
NeuralCompose pipeline.
$$
\mathbf{X}(t) \in \mathbb{R}^{4 \times N}
$$
with TP9, AF7, AF8, TP10 channels.
2. Welch Spectral Density $$
\hat{S}_{xx}(f) = \frac{1}{K,U} \sum_{k=1}^{K} \left| \mathcal{F}{w_k h}(f) \right|^2
$$
Band power:
$$
P_b = \int_{f_1}^{f_2} \hat{S}_{xx}(f),df \approx \sum_i \hat{S}_{xx}(f_i),\Delta f
$$
$$
r_\alpha = \frac{P_\alpha^{\mathrm{baseline}}}{P_\alpha}, \qquad r_\alpha^{\mathrm{dB}} = 20,\log_{10}(r_\alpha)
$$
$$
p(c \mid W) = \frac{e^{z_c(W)}}{\sum_{c' \in \mathcal{C}} e^{z_{c'}(W)}}
$$
Unit normalization:
$$
\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|_2}
$$
Cosine similarity:
$$
\cos(\hat{\mathbf{v}}_1, \hat{\mathbf{v}}_2) = \hat{\mathbf{v}}_1^\top \hat{\mathbf{v}}_2
$$
$$
\mathbf{y} = R,\mathbf{v}, \qquad R \in \left{\pm\frac{1}{\sqrt{d}}\right}^{3 \times d}
$$
$$
\mathbf{z} = \frac{\operatorname{concat}(w_i,\mathbf{v}_i)}{\left|\operatorname{concat}(w_i,\mathbf{v}_i)\right|_2}
$$
Define
$$
D = \max_n r_n
$$
where $r_n$ is the repeat count of an immediately repeated period-$n$
token sequence.
Recommended metrics:
decoder loop period
decoder loop repeat count
prompt echo detection
stop reason
generation length
9. Statistical Evaluation Recommended evaluation reports include:
bootstrap confidence intervals
Mann–Whitney U
Cohen's d
Pareto frontier
effect sizes
hypothesis preregistration
These sections align with the Stage 3.4 and Stage 3.5 evaluation
framework.