The optimal way to inject information into a physical system is a geometry problem.
Physical reservoir computing — using the dynamics of a physical system (spin waves, spiking neurons, mechanical oscillators) as a computational substrate. The system's natural dynamics process information; the question is how to encode the input so the reservoir processes it usefully.
Response-based Optimal Memory Encoding (arXiv:2603.21666): the optimal input encoding that maximizes task-specific linear memory under fixed power constraints is determined by the system's fluctuation-response structure. The steady-state fluctuations tell you how the system naturally spreads information. The linear response tells you how it reacts to input. The optimal encoding aligns the input with the direction where response-to-noise ratio is highest.
The insight that backpropagation-based encoder optimization converges to the same answer as ROME — despite being computed entirely differently — reveals that gradient descent was solving a geometric problem all along. The optimization landscape has a structure determined by physics, and gradient descent finds it not because it understands the physics but because the geometry of the loss surface encodes the physics.
Validated on spin-wave waveguides and spiking neural networks — systems that are physically different but geometrically similar in their fluctuation-response structure. The encoding principle is universal: it works on any system with measurable fluctuations and linear response, including systems that aren't differentiable (where backpropagation can't work but ROME can).
The structural claim: the physics of the reservoir determines the mathematics of the encoder. You don't design the input; you read it from the system's own statistical properties.