Is the period-doubling past the edge of stability Feigenbaum's cascade, and in what dimension does that stop holding?

Status: open · laid 2026-09-27 · type: lead

Statement. Take the double well f(x) = (x² − 1)²/4 and plain gradient descent on it,

x ↦ g(x) = x − η f'(x) = x − η(x³ − x).

At either minimum x = ±1 the curvature is f''(±1) = 2, so the fixed point is linearly stable iff |1 − η f''| < 1, i.e. iff η < 2 / f''(±1) = 1. That threshold is the edge of stability: the largest step size at which descent converges to the minimum at all, and it is exactly two over the curvature there. Above it the minimum is repelling, the iteration does not diverge, and what it does instead is period-double — a 2-cycle straddling the minimum, then a 4-cycle, and so on into chaos. The question is whether that cascade is the cascade:

1. Does the doubling spacing give Feigenbaum's δ? Measure the thresholds η₁ = 1 < η₂ < η₃ < … at which each doubling occurs and form the ratios (ηₙ − ηₙ₋₁)/(ηₙ₊₁ − ηₙ). Feigenbaum's constant δ = 4.66920… is universal for one-dimensional maps with a single quadratic critical point, and if the ratios converge to it then this optimiser's failure is in that universality class — the step size is an ordinary bifurcation parameter and nothing about gradient descent is special. There is a real obstruction to check first, not a formality: g is a cubic, odd-symmetric, with two critical points (g'(x) = 1 − η(3x² − 1) vanishes at x = ±√((η + 1)/(3η)), and g''(x) = −6ηx ≠ 0 there, so each is quadratic). Cubic, odd maps are not the unimodal family, and for maps with that symmetry the renormalisation fixed point — hence the constant — can differ. So the honest form of the question is: which cascade is it — the unimodal one restricted to a single well, or a symmetric-cubic one — and does the symmetry of the double well show up in the measured δ.
2. In what dimension does it stop holding? For f : ℝᵈ → ℝ the descent map is d-dimensional and period-doubling is only one of the available routes: an invariant circle (Neimark–Sacker) and quasi-periodicity are open in d ≥ 2, and either one ends the geometric scaling. At what d, and for what class of potentials — separable, non-separable, with which curvature ratios at the minimum — does a measured cascade stop returning δ? The natural first object is a product of double wells with unequal curvatures, where the edge of stability is set by the largest curvature, so the modes cross their own thresholds at different η.

Why this is not only a toy. The edge of stability is an empirical fact about real training: with a fixed step size networks are routinely observed to sit just above 2/λ_max of the loss Hessian, with the loss non-monotone locally and still decreasing over long stretches. That is a dynamical statement about an unstable fixed point that does not diverge — the same shape as the 2-cycle here, in many dimensions and with a moving Hessian. A one-dimensional potential whose cascade is measured rather than argued is the cheapest honest entry to it.

Why a probe and not a case. The test is a computation, and a cheap one — iterating a scalar map — but there is no differential yet: no competing accounts of what the cascade should be, only a known universality class and a stated reason it might not apply. When there are branches (δ = 4.669 · a symmetric-cubic constant · no geometric scaling at all) it can be sharpened into a case.

No prediction and no falsifier are filed here — both are the human's (MANUAL.md, Decisions only you make).

Cross-refs. Siblings from the same session: P-027 (the deep-linear staircase read backwards as a spectral measurement) and P-029 (double descent against Marchenko–Pastur). All three are training dynamics read as a dynamical system rather than as engineering; the run that opened the track is lab/runs/037-grokking-multiplicative-characters.md.

Bridge to the carpet thread, 2026-09-29 (codifier). The same Feigenbaum machinery turned up in the carpet thread (P-048), from the other side. There the renormalization map is the carpet's crossing polynomial R(p), whose slope λ at its fixed point plays the part of δ, and a successive-stage ratio estimate, g·y_g − (g−1)·y_{g−1}, removed the one-block offset post hoc (recorded in P-048's inklings). The two probes share an instrument question — how fast a successive-ratio estimate converges, and what second eigenvalue sets that rate — and a contrast: gradient descent on a double well folds (period doubling), while a crossing polynomial is monotone and cannot. A result on either side about ratio-estimate convergence should be read on the other.