Circulatory Fidelity diagnoses where reductionism fails: where breaking a system into independent parts discards the information that matters most. Validated across 30+ scientific domains with 49,000+ empirical data points.
Modern science decomposes systems into independent parts. This works spectacularly — until it doesn't. When the connections between parts carry more information than the parts themselves, factorized analysis silently discards what matters most.
Mean-field variational inference, the workhorse of modern Bayesian computation, approximates joint distributions as products of independent factors: q(z) = ∏i qi(zi). This explicitly discards all relational structure between variables.
When does this approximation fail?There is no free lunch in factorization. Every time we treat coupled variables as independent, we pay an information-theoretic cost, a cost that is invisible to the factorized analysis itself. The error grows silently and can be catastrophic.
How do we measure what we've lost?Even when we check for correlations, standard pairwise analysis misses synergistic structure. XOR-like dependencies, where three variables are tightly coupled but any two appear independent, are invisible to marginal observation.
IC₂ = 0 does not mean independenceBiological observers evolved under severe energy budgets. Pairwise observation is metabolically cheap; higher-order observation is expensive. The apparent rarity of synergistic structure is a selection effect of methodology, not a fact about the world.
Synergy is ubiquitous but invisible to cheap observationCF starts from a single mathematical fact: any precision matrix decomposes into what stays and what gets discarded under factorization.
D (diagonal) — Nodal structure. Self-precision of each variable. Retained by mean-field approximation.
R (off-diagonal) — Relational structure. Conditional dependencies between variables. Discarded by mean-field approximation.
When R is 'load-bearing,' i.e., when relational structure carries essential information, factorization fails. CF quantifies exactly when and how much.
Mutual information, and hence the cost of factorization, depends only on correlation strength, not on marginal variances. The relational structure, not nodal properties, determines information flow.
IC's meaning is context-dependent. The same coupling strength implies opposite recommendations depending on whether the coupling is constitutive (load-bearing structure) or inductive (learnable signal). This Dependency Asymmetry is a core diagnostic insight.
Coupling is structural: it carries model physics. Factorization destroys load-bearing relational structure.
Coupling is signal: it provides learnable regularization. High coupling means data is informative and hierarchy is redundant.
In multi-layer models, proximal interfaces (near data) are constitutive; distal interfaces (far from data) are inductive.
CF predictions have been tested against simulation studies across multiple statistical modeling contexts. The framework produces quantitative predictions that are confirmed by data.
Validation spans the core statistical modeling contexts where factorization assumptions matter. In single-variate factorization (8,000 simulations), the MSE degradation matches the closed-form 1/(1−IC²) prediction. In hierarchical linear models (8,000 simulations), IC correctly predicts where partial pooling outperforms no-pooling and full-pooling estimators. In deep three-level hierarchies (16,000 simulations), the proximal dominance principle holds: the nearest layer's coupling dominates inference quality regardless of distal structure.
Threshold calibration studies (1,050 configurations) validate the three-zone diagnostic system. Comparison against PSIS-&kcirc; diagnostics (900 configurations) shows r = 0.88 correlation, meaning CF anticipates posterior pathology from prior predictive samples alone, before running any inference. The complete manifold of all 16 two-input Boolean functions confirms Walsh-Hadamard synergy classification, and SAT phase transition studies (700 instances) demonstrate spectral frustration tracking the satisfiability boundary.
All validation data, simulation code, and analysis scripts are publicly available in the project repository.
CF's core decomposition Λ = D + R is not a metaphor when applied across domains: it identifies the same mathematical structure in different contexts. Each connection is formally classified using semiotic translation quality assessment (STQA), distinguishing proven mathematical identities (Class 10) from structural correspondences (Class 7) from analogies (Class 5). The network currently spans 37 domains connected by 109 formally classified edges.
Every domain extension requires explicit operationalization: What are the nodes? What implements R? How would IC be measured? What would falsify the extension?
Standard correlation analysis is blind to synergistic structure. CF provides a two-stage protocol that detects pure higher-order dependencies invisible to pairwise observation, grounded in Walsh-Hadamard analysis of Boolean functions.
Compute IC₂ for all variable pairs. High IC₂ indicates standard pairwise coupling signaling that MFVI will fail predictably.
When IC₂ ≈ 0, compute IC₃ via Walsh-Hadamard transform. Non-zero IC₃ detects pure synergy, XOR-like structure invisible to marginals.
Pure synergy ↔ affine over GF(2). The Computational Synergy Principle: synergistic functions are exactly the XOR-like functions in their natural encoding.
This is not an analogy. The algebraic structures are identical: Boolean synergy detection over GF(2) and quantum stabilizer operations share the same group-theoretic foundation. Both detect the boundary between efficiently simulable and genuinely complex structure.
CF possesses an inherent Lorentzian geometry. This is not imposed: it emerges from the metric structure of the precision matrix decomposition and connects CF to information geometry, statistical mechanics, and special relativity.
CF quantities satisfy hyperbolic, not circular, geometry. The symmetric leverage S and asymmetric leverage A decompose observational effects with a Minkowski-signature metric.
IC maps to position in the Poincaré disk. The boundary (IC = 1) is infinitely far away: perfect correlation requires infinite geodesic distance, corresponding to infinite information.
Mutual information is the action integral over the geodesic, with IC as the Lagrangian density. Information accumulates along the path through coupling space.
| Identity | Interpretation |
|---|---|
| S² − A² = IC² | Lorentzian metric signature |
| d = arctanh(IC) | IC as hyperbolic position |
| I = log(cosh d) | MI as geodesic action |
| φ = ½ ln(r) | Variance ratio as rapidity |
| CC₁ × CC₂ = IC² | Product identity |
| Λ = I(μ) | Precision IS Fisher information |
| Constant | Value | Meaning |
|---|---|---|
| ICcrit | 1/√3 ≈ 0.577 | Layer L₁/L₂ boundary |
| K | [−6, −2] | Fisher-Rao curvature range |
| nmin | 3 | Minimal synergistic dimension |
| Geff(∞) | 1/4 | Statistical Newton's constant |
CF diagnostics are available in Python and Julia. MIT licensed. Designed for integration into existing Bayesian workflows.
from circulatory_fidelity import inference_coupling, diagnose # Estimate IC between latent and observed ic, se = inference_coupling(z_samples, x_samples) # Full diagnostic workflow result = diagnose(z, x, model_type='filtering') print(f"IC = {result['ic']:.3f}") print(f"Risk: {result['risk_level']}") print(f"MSE ratio: {result['mse_ratio']:.2f}")
using CirculatoryFidelity # Estimate IC from samples ic, se = inference_coupling(z, x) # Closed-form for Gaussian systems ic = ic_gaussian(ρ) # Two-stage synergy detection result = two_stage_protocol(X, y) println("Pairwise IC: $(result.ic2)") println("Synergy IC: $(result.ic3)")
CF is grounded in an explicit ontological commitment: relations are primary, nodes are emergent. This is not decorative philosophy: it is the load-bearing foundation that generates specific, testable predictions.
Patterns of relation are primary; objects are stable patterns within the relational field, not primitives that relations connect. R maintains D, not the reverse.
Nodal decomposition arose from survival pressure on biological observers, not from the structure of reality. Pairwise observation is cheap; higher-order observation is expensive. Science inherited the selection effect.
When we treat coupled variables as independent, we pay an information-theoretic cost bounded below by mutual information. IC quantifies this cost before inference, from model structure alone.
The apparent rarity of higher-order structure is a selection effect. The two-stage protocol detects synergy invisible to pairwise analysis. This remains a testable claim, not a proven truth; evidence across 30+ domains is accumulating.
Is the relational structure load-bearing?
Every CF analysis addresses this single question. If yes — factorization loses essential information. Preserve R. Pay the observation cost. If no then reductionism is adequate. Factorize freely.
CF does not claim that all systems have high relational structure, nor that reductionism always fails. It provides the diagnostic tools to distinguish the cases, and the vocabulary to characterize what factorized analysis discards.
The framework is complementary, not competitive. Use CF when component-level analysis has failed unexpectedly, when interventions on parts fail to produce expected system-level effects, or when there is suspected coupling structure that might be carrying the information that matters.
CF maintains rigorous classification of all claims. Mathematical derivations (proven), empirical predictions (tested against data), structural correspondences (formally mapped), and analogies (suggestive only) are never conflated. Every domain extension requires explicit operationalization, validation criteria, and falsification conditions.
CF is in active development, preparing v1.1 publication. Collaboration inquiries, technical questions, and critical engagement are welcome.