CF produces quantitative predictions that are confirmed by data. The validation program spans synthetic simulations (ground truth known exactly), real-world neuroimaging (human subjects), empirical datasets, and algebraic manifolds (structure provable). Every chart below is drawn directly from laboratory data.
A diagnostic that only raises alarms is useless. CF must also correctly identify when factorization works — when decomposing a system into independent parts loses nothing essential. This is validated directly:
In SVF simulations at low coupling, the MSE ratio is 1.0 within noise. Factorization is free. The mean-field approximation is essentially exact. CF correctly says: proceed, reductionism is adequate here.
In HLM (pooling archetype), high IC means data is plentiful and each group can estimate independently. CF correctly reverses its verdict: factorization succeeds when coupling signals redundancy rather than structural necessity.
The constant functions (TRUE, FALSE) have IC₂ = 0, ICint = 0 — no relational structure at all. CF correctly reports: nothing to preserve, factorization discards nothing.
Negative controls are built into every validation study: the left edge of each chart (low coupling) confirms that CF does not raise false alarms. The framework's value is not in universally detecting problems — it is in precisely distinguishing where relational structure is load-bearing from where it is safely ignorable.
In the Single-Variate Factorization study (8,000 simulations), two correlated Gaussian variables are modeled with mean-field factorization (q(x)q(z) instead of q(x,z)). As coupling strength (κ) increases, the mean-field approximation deteriorates monotonically. The MSE ratio measures how much worse the factorized model performs compared to the oracle that respects the coupling.
8,000 simulations (400 replications × 20 coupling levels). Error bars: ±1 SD. MSE ratio = 1 means factorization costs nothing; higher values indicate increasing information loss.
In Hierarchical Linear Models (8,000 simulations), the dependency archetype is pooling — relationships carry signal from shared structure. Here the IC direction reverses: high IC means data is reliable enough that each group can estimate well independently, making the hierarchical pooling layer redundant. The MSE ratio (no-pooling / partial-pooling) drops as IC increases.
8,000 simulations (400 replications × 20 τ levels). The negative correlation (r = −0.88 aggregated) confirms the pooling archetype: high IC = hierarchy unnecessary. This is the opposite direction from filtering — demonstrating Dependency Asymmetry empirically.
In deep three-level hierarchies (16,000 simulations), which layer's coupling matters more? The answer: the proximal layer (nearest to the data). Proximal IC₂₁ correlates with MSE ratio at r = 0.31; distal IC₃₂ at only r = 0.13 — a 2.3× difference. The coupling closest to the observation point dominates inference quality regardless of distal structure.
16,000 simulations across all combinations of proximal and distal coupling. Solid: proximal IC (layer 2→1, closest to data). Dashed: distal IC (layer 3→2). The proximal layer's coupling dominates.
All 16 possible two-input Boolean functions are classified by their IC structure. The x-axis shows pairwise coupling (IC₂); the y-axis shows interaction strength (ICint). The classification is exhaustive and algebraically exact: XOR sits at IC₂ = 0, ICint = 1 — pure coplexity, invisible to pairwise methods. AND/OR sit at IC₂ = 0.82, ICint = 0.58 — mixed structure visible to both pairwise and interaction methods.
Complete manifold of all 16 two-input Boolean functions (dot size = multiplicity at each position). Source: Walsh-Hadamard spectral decomposition. XOR/XNOR (green) have zero pairwise coupling but maximal interaction — the archetype of coplexity. 16 functions collapse to exactly 4 positions by IC structure.
In random 3-SAT instances (350 problems across 7 clause ratios), spectral frustration (ρf) — a measure of relational constraint density — tracks the satisfiability phase transition. As the clause ratio α increases past the critical threshold (~4.27 for 3-SAT), satisfiability drops and frustration peaks. CF's spectral frustration measure captures this structural transition.
350 random 3-SAT instances (50 per α level). Solid: fraction satisfiable. Dashed: mean spectral frustration (ρf). Frustration peaks near the satisfiability phase transition.
IC was independently validated against Pareto-Smoothed Importance Sampling (PSIS), the standard Bayesian diagnostic for posterior approximation quality. Across 900 simulations with varied coupling strengths, IC predicts the log-likelihood gap between the factorized and true posteriors with r = 0.86. The two diagnostics — one geometric (IC), one sampling-based (PSIS) — converge on the same answer through entirely different computational paths.
900 simulations. Each point is one model configuration. IC (x-axis) predicts the log-likelihood gap (y-axis) with r = 0.86. Points colored by PSIS k̂ diagnostic. This validates IC as a computationally cheaper alternative to PSIS for detecting factorization failure.
CF metrics applied to real-world neuroimaging data from the ABIDE consortium (871 subjects, 20 sites) and the Human Connectome Project (70 subjects, task fMRI). IC computed from functional connectivity matrices distinguishes clinical populations and tracks cognitive load. The CCRP study (heart-brain coupling) demonstrates cardiac phase modulation of neural IC across task states.
ABIDE: ASD vs Control on 3 IC-derived metrics. All significant (p < 0.025). Effect sizes (Cohen's d) are small but consistent across 20 independent acquisition sites.
CCRP: Mean IC profile across 8 cardiac phase bins for 4 HCP task conditions. Neural coupling is modulated by cardiac cycle — IC tracks heart-brain interaction.
Beyond neuroscience: The cognitive science domain also integrates linguistic iconicity data (14,776 words from Winter et al. 2017), measuring sound-meaning coupling — the degree to which a word's phonological form is non-arbitrarily related to its meaning. High iconicity = high IC between phonology and semantics; the Saussurean arbitrariness assumption is the factorization q(sound)·q(meaning).
All validation data, simulation code, and analysis scripts are maintained in the project repository. Each dataset below has been used in at least one published validation result.
| Dataset | N | Type | Key Result |
|---|---|---|---|
| SVF Validation | 8,000 | Synthetic | MSE ratio grows monotonically with coupling |
| HLM Validation | 8,000 | Synthetic | IC predicts pooling benefit (r = −0.88) |
| Three-Layer Validation | 16,000 | Synthetic | Proximal dominance (2.3× stronger than distal) |
| IC-PSIS Comparison | 900 | Synthetic | IC predicts log-lik gap (r = 0.86) |
| Threshold Calibration | 1,050 | Synthetic | Three-zone diagnostic system validated |
| Boolean Manifold | 16 | Algebraic | Exhaustive classification, Walsh-Hadamard exact |
| SAT Phase Transition | 350 | Combinatorial | Spectral frustration tracks satisfiability boundary |
| ABIDE Neuroimaging | 871 | Empirical | ASD vs Control FC: IC distinguishes clinical populations (p = 0.002) |
| HCP Working Memory | 70 | Empirical | IC contrasts under cognitive load: 4 Bonferroni-significant pairs |
| HCP CCRP (Heart-Brain) | 208 | Empirical | Cardiac phase modulation of neural IC across 4 task conditions |
| Iconicity Ratings | 14,776 | Empirical | Sound-meaning IC: words with high iconicity ratings show non-arbitrary phonology-semantics coupling (Winter et al. 2017) |
| CC Diagnostic Ladder | 54 | Real-world | Wine, Framingham, Air Quality: mediation structure |
| Percolation IC | 25 | Simulation | IC tracks percolation threshold on lattice |
| Literary Corpus | 90+ | Annotated | Narrative structural coupling in literature |