The mechanics of the inverted-U: an exact linear theory of reproducible dimensionality, and what governs its rise and fall

Abstract

A companion paper (doi:10.5281/zenodo.22168191, "What must a theory of perturbational complexity explain?") consolidated nine constraints and challenged any theory to pass them — including C3: debiased perturbational complexity (R-dim, the reproducible dimensionality of the evoked response) is an inverted-U in network dynamics, maximal near the edge of chaos. Here we answer part of that challenge with eleven sealed protocols (all preregistered publicly before their blind quantities were computed, each run once; the failed protocols are reported with the same prominence as the successes). (1) An exact linear theory. For linear networks, R-dim is computable in closed form from connectivity alone: component amplitudes from the impulse-response SVD, per-component noise floors from the stationary Lyapunov covariance, and a universal averaging formula. Confirmed on 36 virgin networks: rank correlation +0.966, median absolute error 0.33 dimensions with a single frozen constant, per-network growth-with-trials predicted (ρ = +0.59), and no inverted-U anywhere in the stable linear family — the falling branch is not a linear phenomenon. (2) The formula transports to nonlinear networks. Feeding it the measured coherent spectrum (through an estimator validated on a known-truth bench; all bench iterations documented) and the measured trajectory noise reproduces the full U on 48 virgin networks in order (+0.925) and calibration (median error 0.41 dimensions, constant frozen), while both sealed peak-location criteria failed: the U's top is a flat plateau at this resolution, and a dedicated sealed 128-network estimation returned "indeterminate". (3) Time and richness dissociate. The duration of the coherent response obeys a relaxation law on the stable side (T_c ~ 1/|λ|; +0.943 on virgin networks) and is maximal at the edge (37× deep-stable, 6× deep-chaotic) — yet the *maximum of R-dim does not coincide with the maximum of coherent time* (sealed dissociation, P = 0.99): richness needs more than time. (4) What governs the fall is saturation. In a two-axis design sampling networks by measured λ and spectral radius ρ independently, at matched criticality saturation crushes R-dim (Spearman −0.62 to −0.88 across all λ strata; 7/7 sealed criteria on 156 virgin networks), while at matched saturation criticality's effect is bounded (|ρₛ| < 0.19). The falling branch of the U is largely saturation wearing chaos's clothes. A finer "graded law" across three additional activation families died in its own sealed test (0/3); the burial is reported, with the method lesson it taught — and a properly powered family-level replication then confirmed the coarse tendency in all three families (sealed; median per-stratum Spearman −0.50 / −0.61 / −0.41 for erf, hard-clip, and softsign at 30 networks per stratum): saturation hurts reproducible dimensionality whatever the shape of the ceiling. A final sealed protocol then answered *where* saturation takes over: in a parametric family interpolating softsign to hard clip, the saturation-governed regime advances with the tail speed of the activation (stable-strata effect −0.16 at p = 1 vs −0.70 at p = 8; sealed two-point criterion passed at −0.54 against a −0.30 bar). (5) One formula. A final sealed protocol tested the unification all of the above points to: applying the exact linear theory to the gain-shrunk effective system W_eff = diag(⟨φ′⟩)·W — same frozen constant — predicts R-dim across all seven activation families studied (pooled order +0.653 on 280 virgin networks, p = 2 × 10⁻³⁵), and per-unit gain heterogeneity is sealed as the essential carrier (the scalar-gain comparator loses by 0.435). At the order level, a nonlinear network is, for reproducible dimensionality, its gain-shrunk linear self; absolute calibration remains an open refinement. (6) Scope, attacked. A final sealed protocol re-ran both flagship laws in five worlds never touched: sizes ×2 and ×4, biological E/I wiring, and halved/doubled noise. The saturation law survived all five (stratum-median Spearman −0.48 to −0.63): together with the activation-family campaign, it holds across nonlinearity, size, wiring and noise — a law of systems. The effective-gain theory passed fully in two worlds and drew its boundary in the others, degrading with network size (order +0.29 at N = 256): a small-family approximation with a mapped edge. We close with what remains open: the exact shape of the U's top, the quantitative chaotic decay, the analytic form of the tail-crossover law, the level calibration of the effective theory — and why its order degrades with size.