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

Year: 2026
Type: preprint
Abstract: A companion paper (doi:10.5281/zenodo.22168190, "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 twelve sealed protocols that carry its claims (all preregistered publicly before their blind quantities were computed, each run once; the failed protocols are reported with the same prominence as the successes), plus five sealed protocols that died — two in silico and three on human TMS-EEG — reported in §8 because no paper had yet reported them. (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; (v4) a network-blind constant scores 1.29 against the sealed 1.5 bar, so that criterion could not have refuted the formula — the linear-family criterion of (1) could, its best constant failing at 1.84 against 1.0), 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) A duration law; and a time/richness dissociation that is sealed only in its weak form. 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 as sealed; (v3) 19.5× and 3.2× under a pedestal-exclusion criterion frozen in a later protocol of this program and applied retroactively, which does not change the ordering; (v4) the edge-maximum criterion is also passed by a pure relaxation model T_c ∝ 1/|λ| with no chaos physics, and what that model gets wrong is the informative part: the edge lives about 3× shorter than relaxation alone predicts, and deep chaos about 4× longer). The sealed dissociation criterion passed — the stratum where T_c peaks is at least as close to the edge as the stratum where R-dim peaks, bootstrap P = 0.987 (0.912 under the pedestal criterion) — but (v4) the strict statement printed in v1–v3, that T_c peaks at the edge while R-dim does not, has P = 0.83, below the sealed 0.90 bar, and the sealed matrix conditioned the reading of that criterion on two primary criteria that failed. That richness needs more than time is a suggestion of these data, not a sealed result. (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, 95% intervals all excluding zero; the four sealed G1 criteria passed on 156 virgin networks), while at matched saturation criticality's effect is not resolved at this n (point estimates +0.04 / +0.05 / −0.19; the widest 95% interval reaches [−0.45, +0.11]). (v4) The three sealed G2 "bound" criteria (|ρₛ| < 0.30 at n = 48) were passed, but three independent null correlations pass them together 89% of the time; they are reported as passed and uninformative, and v3's "7/7" is withdrawn. (v4) Evoked amplitude, measured for every network, does not fall with saturation (Spearman(ρ, evoked energy) between −0.12 and +0.49 within strata) and the G1 correlations survive conditioning on it (−0.64 to −0.88); what saturation raises is trial-to-trial variance (3.5× across the ρ range), so the law is a statement about reproducibility, not response size — with the share of the fall attributable to estimator under-reading at low SNR not separable here. 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) The effective-gain construction, and what it does not establish. Applying the exact linear theory to the gain-shrunk effective system W_eff = diag(⟨φ′⟩)·W — same frozen constant — orders R-dim across all seven activation families (pooled +0.653 on 280 virgin networks, p = 2 × 10⁻³⁵), and a second frozen constant (0.610) predicts absolute level to a median 1.18 dimensions across six worlds with no per-world fitting. (v3) Both figures are reproducible, and neither is a demonstration that the theory adds information over trivial predictors: on the same networks, mean gain alone orders at +0.620 and spectral radius alone at +0.615, and the level criterion is passed by a constant that ignores the network entirely (1.37 against a 1.5 bar). The v2 claim that per-unit gain heterogeneity is the sealed essential carrier is withdrawn: the comparator that produced it was pinned against a truncation ceiling in 28% of networks, and once that is removed the margin falls below its own bar. (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 construction did not: (v3) it is outranked by spectral radius alone in four of the five worlds, and at N = 256 it orders at +0.286 against ρ's +0.694. v2 described that as a mapped edge; it is more honestly described as the construction failing to earn its complexity outside the home family. 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. (v4) Every sealed prediction this program has made on human data has failed or been underpowered (§8, §10); the laws above are laws of one model family until shown otherwise.
DOI: 10.5281/zenodo.22168299
Language: English
Open Access status: green
OpenAlex ID: W7204714698
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