Damping without tuning: the measurable stability threshold of the optimality-criteria iteration, and a controller that tracks it

Paper, code, data and full research record for a dynamical-systems study of the damped optimality-criteria (OC) update in SIMP topology optimization. The OC update has one free parameter, the damping exponent eta, conventionally fixed at 1/2 without a stated criterion. Exactly projected finite-difference Jacobians of the full update — finite element solve, filter, multiplier bisection and clipping — taken across meshes, boundary conditions, volume fractions, filters and penalizations, establish three empirical laws. (1) Oscillation is a period-doubling instability at eta* = 2/s_max, where s_max tops the spectrum of the log-log stiffness of the filtered sensitivity field; the local multipliers mu = 1 - eta*s predict measured onset and decay rates to three decimals. (2) That spectrum saturates at p+1 on top — for exact gradients provably an upper bound, the log-coordinate form of Svanberg's (1994) convexity result, so that eta < 2/(p+1) is flip-free at every fixed point — while its lower edge sits near zero, which pins the min-max (Richardson) optimal damping onto the flip boundary itself: at p = 3 the conventional value is that optimum. (3) The observed s_max belongs to the fixed point the iteration selects, and standard damping self-limits to fixed points with s_max ≤ 2/eta, separating from the naturally saturated family exactly at p = 3. The threshold is therefore a moving target: under penalization continuation, fixed damping at 1/2 fails in 20 of 32 runs, 8 of them as oscillation masked below the amplitude tolerance. Because the free trajectory estimators read the flip branch precisely when it is active and the near-zero creep edge otherwise — a regime dependence that estimator-driven relaxation schemes do not account for — a four-line additive-increase/multiplicative-decrease controller can track the threshold unaided. With a single frozen parameter set it avoids oscillation, visible or masked, in all 64 continuation runs, matches the best hand tuning across an 11-configuration suite, and composes with Anderson acceleration so that each covers the other's failures. This deposit contains the paper (paper/main.pdf) together with all code, checkpoints and figure scripts that reproduce every table and figure in it, plus the complete research record in notes/ — including a retracted preview result, a withdrawn mechanism attribution and a causal claim corrected by measurement. Dependencies are numpy, scipy and matplotlib only. Version 2.0.0 adds the spectral enclosure theorem for exact-gradient SIMP together with a sensitivity-filter counterexample (1.01(p+1) at the operator level), recounts the continuation suite including masked limit cycles, and adds the cycle-G1 research record.