Across five preprints (doi:10.5281/zenodo.22100546, .22100826, .22101059, .22120069, .22133403), we measured how perturbational complexity behaves once its estimator is debiased, what quantity it tracks — the reproducible dimensionality of the evoked response (R-dim) — and how that quantity depends on dynamics, trial count, species, and arousal state, always under preregistrations sealed before data contact and with failures published. This paper consolidates the empirical residue into nine constraints that any theory of perturbational complexity must satisfy, reports the autopsies of six hypotheses of ours that died against them (including our own formal theory, killed in two sealed prediction cycles), and offers the minimal account we know of that survives: reproducible components with rapidly decaying amplitudes become visible one by one as trial averaging lowers an effective noise floor. With an approximately exponential amplitude spectrum, this account reproduces, with three lines of algebra, five constraints at once: logarithmic growth of R-dim with trial count, absence of a detectable human ceiling at protocol-scale trial counts, finite system-specific ceilings in simulated networks, level scaling with electrode coverage without growth scaling, and — via uniform attenuation of component amplitudes — the downward displacement of the growth curve under anesthesia. (v2) v1 described that displacement as parallel (level, not slope); preprint 6 v2 withdraws the parallel-shift shape, so C9 now constrains the existence of the contrast and the recovery lag rather than the shape. The account is explicitly not a theory of consciousness: it is silent on why rich spectra require edge-of-chaos dynamics, why networks self-organize toward that regime, and why recovery from anesthesia lags the state change. It does, however, make a quantitative, falsifiable prediction — the vertical displacement between two states' curves should equal twice the within-state slope times the log of the evoked-amplitude gain ratio — and we tested it: the test protocol was sealed publicly before the one unpublished quantity it requires (the gain ratio) was computed for any animal, and run once. The prediction held on both sealed criteria at the field level (ordering ρ = +0.41, one-sided p = 0.038, n = 20; median observed/predicted ratio 1.00). It then failed its cross-level replication: sealed identically and run once on spiking populations of the same brains (commit 76f2fea), ordering inverted (ρ = −0.29, p = 0.84) and observed displacements exceeded predictions by a median factor of 3.0 — anesthesia does more to neuronal dimensionality than uniform amplitude gain allows. The visibility account therefore survives as an economical summary of the trial-scaling constraints and of C9's shape, and dies as a mechanistic claim at the neuronal level; we report the refutation of our own model here rather than elsewhere. We then chased the structural change itself with a further sealed protocol: anesthesia rotates the population response subspace to near-orthogonality (median between-state overlap 0.082 against a within-state control of 0.532; lower in 15 of 15 animals, p = 3.1 × 10⁻⁵) — anesthesia does not attenuate a fixed response so much as replace it — yet the extent of rotation does not scale the excess displacement either (ρ = −0.09, p = 0.62), killing the rotation-as-explanation hypothesis in the same run that established the rotation. The last pre-declared candidate, per-component decoherence, is reported as untested for a reason we document in full: our estimator of trial-level coherence at matched amplitude failed its pre-registered known-truth validation bench (three versions, criteria fixed before each run, all archived), and under our rules an uncertified instrument runs no confirmatory test. The theory this field needs must pass through all nine constraints; we mark where every account we tried has failed so others can start further ahead.
Nicolás Federico Galindez· Zenodo (CERN European Organi...· 0 citations
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.
Nicolás Federico Galindez· Zenodo (CERN European Organi...· 0 citations
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.
Nicolás Federico Galindez· Zenodo (CERN European Organi...· 0 citations
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.
Nicolás Federico Galindez· Zenodo (CERN European Organi...· 0 citations