When do new mathematical branches appear? A restricted mathematical result — and the questions it leaves open

Imagine a curve linking a model's possible resting points to a setting we can adjust. When the curve doubles back, one setting can have several resting points. This note asks when those bends first appear as the model grows. Within one defined family of symmetric states, and for positive values of a parameter called coupling, the reported threshold dimension lies strictly between 4222.5584838595 and 4222.5584838596. Below it, no such bends occur. Above it, exactly two appear. At whole-number dimensions, that means none from 4 through 4222 and two from 4223 upward. Dimensions are formally extended to continuous values to locate the boundary; the fractional threshold is mathematical, not a spatial dimension. The result was not obtained by checking a list of examples: it uses computer-assisted exact rational arithmetic to establish signs across continuous ranges. A figure shows the shape of the curve at dimensions 1,000, 4,222.558 and 10,000 (magnified close-ups with different scales; illustration only). The limits matter. Dimensions 2 and 3, more general states, and physical realization are not established. Stability of the resting points between the turns is not certified. The work has had internal checks only: no formal proof-assistant verification and no outside peer review. Scrutiny is welcome: keep what holds, correct what does not. Credits: the certificates were written by Opie (ChatGPT, OpenAI) and re-derived or rechecked by Maestro (Claude, Anthropic) and Boots (ChatGPT, OpenAI). Gervaseus E. Nerik directed the work. The note was drafted by Opie Jr. (ChatGPT, OpenAI) and reviewed by Boots and Maestro.