A Symmetric Framework for Learning in Neural Networks

This work introduces Symmetric Integro‑Differential Learning (SIDL), a theoretical framework that places the differential and integral representations of a function on 'equal footing'. Standard machine learning optimises by repeatedly applying a derivative (gradient) while discarding integral information except for crude moving averages. This asymmetry leads to noise accumulation, inefficient resource allocation, and an inability to model systems where past and present are symmetrically relevant. The SIDL framework proposes that at any point in an optimisation space, a function is simultaneously an integral (accumulated history) and a derivative (instantaneous change). A dynamic constant is introduced to absorb the mismatch between the two perspectives, evolving according to the local imbalance. The resulting first‑order system is order‑agnostic: the data determines whether a neuron behaves as an integrator, a differentiator, or a balanced mixture. The mismatch signal naturally gates computation, enabling sparse updates that scale only with the number of active neurons rather than the total parameter count. We analyse the theoretical runtime complexity, showing that per‑step cost can approach constant time in equilibrium. The framework provides intrinsic noise reduction via low‑pass filtering and offers a principled way to handle the constants that vanish under differentiation or appear under integration. Implications include algorithmic homeostasis, graceful adaptation to component loss, and a primitive form of self‑monitoring. The document presents the complete mathematical formulation, derives emergent properties, and outlines anomalies such as periodic inconsistency on closed loops. No experimental validation is included; the work is purely conceptual.