Syntony Principle: Deep Learning Validation & Geometric Interpretation of Adam

**🔗 Reproducible code: [github.com/jpbronsard/syntonic-optimizer](https://github.com/jpbronsard/syntonic-optimizer)** This work presents the first empirical validation of the Syntony Principle in deep neural network optimization and a geometric interpretation of the Adam optimizer. Using MNIST as a proof-of-concept, it demonstrates performance parity between a syntony-based optimizer (CUPAI) and Adam, while empirically confirming the predicted temporal scaling lawτ* = κ√(σ²/λ) with κ ≈ 1. Rather than proposing a competing heuristic, this work reframes Adam’s empirical robustness as a consequence of fixed characteristic timescales that coincide with common deep learning regimes. In contrast, the syntony-based approach explicitly infers the relevant temporal integration scale from gradient variance and innovation rate, making the coherence structure explicit. The contribution is intentionally positioned as an initial empirical validation and interpretive framework. It clarifies the status of parameters (calibration versus tuning), avoids claims of superiority, and formulates a clear falsification protocol based on regime-shift experiments. Extended empirical evaluations (e.g., CIFAR-10, non-stationary training) are deferred to subsequent versions. This record is intended as a citable foundation for coherence-based optimization and for further empirical testing of regime-invariant adaptive learning laws.