Large-graph behavior of the physical field family

Characterize the behavior of the Homeostatic Background Processor’s wave, diffusion, advection, dispersion, and KdV-type dynamics on larger module graphs, including regimes in which dispersion and advection have sufficiently rich spectra and solitonic behavior may arise.

Background

The experiments use graphs with only six nodes, so the implemented KdV and dispersive dynamics do not enter a solitonic regime. The authors explicitly limit their null findings about the physics zoo to these small module graphs.

The unresolved question is whether larger graphs, with more modes per layer or attention head, produce qualitatively different effects from the observed small-graph results. The future-work discussion also identifies larger module graphs as a priority because they may provide sufficient spectral structure for dispersion and advection.

References

(4)~$N{=}6$ nodes: no solitonic regime for the dispersion; the zoo of physics at large $N$ remains open.

— Can a Dynamic Internal Field Govern a Transformer's Cognition? Certifiability, not Superiority, in Homeostatic Compute Control  (2608.24319 - Arrabal-Campos et al., 25 Aug 2026) in Section 7, subsection “Limitations”