With unit leak and no input or bias, gain 1.3 gives a stable sign pair in strongly connected nonnegative tanh models

A Hydra neuropeptide study used a tanh model and scaled its spectral radius to 1.3. We ask whether its stable sign pair requires specific wiring. We give a direct proof for finite, strongly connected, nonnegative matrices in dx/dt = -x + tanh(Wx), with no bias or external input. Every such matrix with spectral radius above one has a unique positive equilibrium and its negative counterpart. Both are locally stable. Every nonzero nonnegative initial state converges to the positive equilibrium. These are special cases of established cooperative-system results. A generated uniform network at gain 1.3 has exactly two stable states, with magnitude 0.752058. A generated network with two modules has at least four stable states at the same gain, edge density, and every node input and output strength. Forty signed starts reach two and four stable endpoint groups. Nonnegative starts reach one in each control. Exact analysis gives the module coupling boundary for loss of mixed-state stability. Checks on 60 generated matrices at five gains agree with the proof. The source matrix is absent from the inspected public repository snapshot. We therefore do not repeat the original Hydra network simulation or estimate its biological state count. The stable sign pair alone cannot identify an effect of specific Hydra wiring. The package includes proofs, generated data, code, a diagnostic tool, figures, source hashes, and repeat instructions.

Hydra · neuropeptide network · structure and function · stable states · tanh model · spectral radius · cooperative systems · model audit · reproducibility