docs/SCIENTIFIC_VALIDATION.md

Sofia Engine Scientific Validation

All algorithms in Sofia Engine are backed by deterministic mathematical verification, golden test vectors, and physical conservation laws.

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1. Digital Signal Processing Correctness

Parseval Energy Conservation

For discrete signals, Parseval's theorem dictates that total energy in the time domain must equal total energy in the frequency domain: $$\sum _{n=0}^{N-1}\mathrm{\mid }x[n]{\mathrm{\mid }}^2=\frac{1}{N}\sum _{k=0}^{N-1}\mathrm{\mid }X[k]{\mathrm{\mid }}^2\backslash sum\_\{n=0\}^\{N-1\}\; |x[n]|^2\; =\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{k=0\}^\{N-1\}\; |X[k]|^2$$

When applying window functions $w[n]$, energy is attenuated by the noise power bandwidth gain: $$S_2=\sum _{n=0}^{N-1}w[n{]}^{2}S\_2\; =\; \backslash sum\_\{n=0\}^\{N-1\}\; w[n]^2$$

Sofia's welch_psd implementation normalizes power spectral density by $S_2$ and frequency resolution $\Delta f = f_s / N$. Energy conservation across Hann, Hamming, and Blackman windows is continuously verified in tests/unit/test_dsp_golden.py:

Exact Amplitude Recovery

For unwindowed integer-bin tones ($f_0 = k \cdot f_s / N$), the one-sided FFT magnitude recovers the exact continuous amplitude $A$: $$\widehat{A}=\frac{2}{N}\mathrm{\mid }X[k]\mathrm{\mid }\backslash hat\{A\}\; =\; \backslash frac\{2\}\{N\}\; |X[k]|$$ Sofia verifies peak tone amplitude recovery to machine precision ($< 10^{-14}$ relative error).

3-Phase Symmetrical Components (Fortescue Transformation)

Unbalanced 3-phase voltages $V_a, V_b, V_c$ are decomposed using the complex operator $a = e^{j 120^\circ} = -\frac{1}{2} + j \frac{\sqrt{3}}{2}$: $$\text{VUF}=\frac{\mathrm{\mid }{V}_2\mathrm{\mid }}{\mathrm{\mid }{V}_1\mathrm{\mid }}\times 100\mathrm{\%}\backslash text\{VUF\}\; =\; \backslash frac\{|V\_2|\}\{|V\_1|\}\; \backslash times\; 100\backslash \%$$ Golden vectors generated in tests/golden/dsp_golden.json assert that positive ($V_1$), negative ($V_2$), and zero ($V_0$) sequence components match theoretical values within $10^{-5}$ tolerance across Python, TypeScript, and C99 runtimes.

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2. In-Situ Neural Training & Gradient Verification

Sofia's virtual assembly neural network (AssemblyNeuralNetwork) implements 2-layer analytical backpropagation directly in virtual bytecode: $$\frac{\mathrm{\partial }L}{\mathrm{\partial }{W}_2}=\frac{\mathrm{\partial }L}{\mathrm{\partial }\widehat{y}}\cdot h^T,\frac{\mathrm{\partial }L}{\mathrm{\partial }{B}_2}=\frac{\mathrm{\partial }L}{\mathrm{\partial }\widehat{y}}\backslash frac\{\backslash partial\; L\}\{\backslash partial\; W\_2\}\; =\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash hat\{y\}\}\; \backslash cdot\; h^T,\; \backslash quad\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; B\_2\}\; =\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash hat\{y\}\}$$ $$\frac{\mathrm{\partial }L}{\mathrm{\partial }h}=W_2^T\cdot \frac{\mathrm{\partial }L}{\mathrm{\partial }\widehat{y}},\frac{\mathrm{\partial }L}{\mathrm{\partial }{z}_1}=\frac{\mathrm{\partial }L}{\mathrm{\partial }h}\odot \mathbb{I}(z_1>0)\backslash frac\{\backslash partial\; L\}\{\backslash partial\; h\}\; =\; W\_2^T\; \backslash cdot\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash hat\{y\}\},\; \backslash quad\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; z\_1\}\; =\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; h\}\; \backslash odot\; \backslash mathbb\{I\}(z\_1\; >\; 0)$$ $$\frac{\mathrm{\partial }L}{\mathrm{\partial }{W}_1}=\frac{\mathrm{\partial }L}{\mathrm{\partial }{z}_1}\cdot x^T,\frac{\mathrm{\partial }L}{\mathrm{\partial }{B}_1}=\frac{\mathrm{\partial }L}{\mathrm{\partial }{z}_1}\backslash frac\{\backslash partial\; L\}\{\backslash partial\; W\_1\}\; =\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; z\_1\}\; \backslash cdot\; x^T,\; \backslash quad\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; B\_1\}\; =\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; z\_1\}$$

Analytical gradients computed in virtual registers are validated against central finite differences: $$\frac{\mathrm{\partial }L}{\mathrm{\partial }{\theta }_i}\approx \frac{L({\theta }_i+ϵ)-L({\theta }_i-ϵ)}{2ϵ},ϵ={10}^{-5}\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash theta\_i\}\; \backslash approx\; \backslash frac\{L(\backslash theta\_i\; +\; \backslash epsilon)\; -\; L(\backslash theta\_i\; -\; \backslash epsilon)\}\{2\backslash epsilon\},\; \backslash quad\; \backslash epsilon\; =\; 10^\{-5\}$$ The relative error $\frac{|\nabla_{\text{analytical}} - \nabla_{\text{numerical}}|}{|\nabla_{\text{analytical}}| + |\nabla_{\text{numerical}}|}$ is proven to be $< 10^{-5}$ in tests/unit/test_neural_gradients.py.

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3. Microcontroller Fixed-Point Conformance

The C99 embedded runtime (embedded/src/sofia_features.c) implements Q16.16 fixed-point arithmetic:

• Fixed-point resolution: $1/65536 \approx 1.526 \times 10^{-5}$
• Dynamic range: $[-32768.0, +32767.99998]$
• Zero heap allocation: no calls to malloc, calloc, or free post-initialization.
• Compile-time static ceilings: SOFIA_MAX_FFT_LEN = 1024, SOFIA_N_STAT_FEATURES = 14.