Circulatory Fidelity rests on one commitment: relations are primary, and nodes emerge from them. That isn't decorative philosophy bolted onto the math, it's the foundation the math emerges from, and every prediction we test follows from it.
The operation CF measures is factorization: replacing a joint distribution with a product of independent marginals.
q(x, z) → q(x) · q(z)
This deletes the relational structure between variables. Whatever the joint distribution knew about how x and z co-vary (their coupling, their conditional dependencies, the off-diagonal structure of their shared precision matrix) is set to zero. Mean-field factorization is one sharp instance of what philosophy calls reductionism: treating the whole as the sum of its parts.
Reductionism isn't the enemy here. It's the dominant strategy in modern science and it works extremely well across most of what we do. But it's a strategy, a choice about how to look, not a fact about how reality is built. CF measures when that choice is cheap and when it costs you. The question is always the same: how much did factorizing cost?
CF's ground is that patterns of relation are primary. Objects (nodes, parts, components) are stable patterns within a relational field, not atomic primitives that relations later connect. The precision matrix Λ encodes this directly: its off-diagonal elements R are the relational structure, its diagonal elements D the nodal self-precision. Mathematically, R maintains D, not the reverse.
This is a strong claim, not just a modeling preference. When we factorize (when we replace the joint with a product of marginals), we aren't simply approximating. We're asserting that the relations are eliminable, that the parts on their own contain all the information. IC measures the error in that assertion.
Every system's precision decomposes into what each part knows alone (D) and what the connections know (R). Factorization retains D and discards R. IC measures the relative magnitude of what was discarded.
What we perceive as objects are stable attractors in the relational field. Their boundaries are observer-relative constructions driven by metabolic constraint, not natural joints at which reality carves.
Information discarded by factorization doesn't vanish. It becomes prediction error, failed interventions, unexplained variance. IC quantifies this cost before inference, from model structure alone.
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, factorize freely. The parts suffice.
IC is the diagnostic, not the verdict. What it means depends on what the relational structure is doing in the system: encoding the physics (filtering), or managing redundancy (pooling). That neutrality is load-bearing. It keeps CF from forcing a relational reading onto systems that don't need one.
Why is factorization the default? Not because reality is fundamentally decomposable, but because observing relational structure is metabolically expensive. Biological observers evolved under severe energy constraints. Pairwise observation (checking whether two things are correlated) scales linearly with the number of pairs. Higher-order observation (three-way dependencies, four-way, the full joint) scales combinatorially.
Under metabolic pressure, organisms developed a factorized perceptual architecture: identify objects (nodes), track pairwise associations (cheap R), ignore higher-order structure (expensive R). It works. It built civilization. But it has a built-in limit: structure that takes expensive observation to detect looks like it isn't there.
Science's default reflex shares this economy: isolate variables, control for confounds, study parts independently. But science is also the apparatus built to push past metabolic shortcuts, with controls, replication, and higher-order statistics. CF adds one more tool to that effort. It measures where decomposition's costs exceed its savings.
The relational structure that factorization discards isn't only a constraint on the system. It's also what the system can do. Coupling between variables is both the cost of observation (you have to account for their joint behavior) and the channel information flows through (their co-variation carries signal that neither carries alone).
This duality runs through every domain CF enters. The off-diagonal precision that makes inference hard is the same structure that makes prediction possible. The coupling that frustrates factorized analysis is the same coupling that enables coordinated function. The metabolic cost of relational observation is real, and so is the information it buys.
IC quantifies the tension: the magnitude of the relational structure, apart from whether it helps or hinders any particular strategy. The framework holds constraint and affordance together, not resolving the tension but measuring it, so each domain can work out its own relationship to the cost.
Factorization discards pairwise coupling. But what about structure that's invisible even to pairwise observation? Coplexity names relational structure that exists only in three-or-more-body interactions: dependencies where any two variables look independent, but the full set is tightly coupled.
XOR is the minimal example: knowing A tells you nothing about C; knowing B tells you nothing about C; but knowing both A and B determines C exactly. The pairwise mutual information is zero. The joint structure is maximal. No amount of pairwise investigation will detect this dependency. You must observe the triple.
The metabolic constraint makes coplexity systematically invisible. Pairwise observation is the cheapest relational test, and science defaults to it. Structures that take higher-order observation are therefore underrepresented in our picture of reality, perhaps because they're genuinely rare, perhaps only because they're expensive to see. The detection protocol is what tells the two apart. CF's two-stage protocol (check IC₂, then if it's near zero check IC₃ via Walsh-Hadamard analysis) is built to get past this observational limit.
The claim that coplexity is ubiquitous is a testable hypothesis, not a proven truth. Evidence across … domains is accumulating. But even if coplex structure turns out to be rare, the detection protocol is still necessary. You can't tell "rare" from "unobserved" without a method that sees what pairwise analysis can't.
A framework that spans … domains risks becoming unfalsifiable, explaining everything by explaining nothing. CF keeps rigorous internal classification to prevent that. Every claim carries an explicit epistemic status, and the framework actively watches for overclaiming.
Mathematical identities (proven), structural correspondences (algebraically exact, interpretation requires caveats), and analogies (suggestive patterns) are never conflated. Each domain extension requires explicit operationalization, validation criteria, and falsification conditions before CF claims applicability.
IC is interpretively neutral: its direction depends on what the relational structure is doing. In filtering regimes (where coupling encodes the mechanism), high IC is a warning. In pooling regimes (where coupling manages redundancy), high IC means simplification is safe. No IC verdict is valid without first classifying the regime.
CF would be falsified by: systems with high IC where factorization works perfectly; systems with low IC where relational interventions dominate; or coplexity detection failures where known coplex structure goes undetected. These aren't hand-waved away. They're actively sought as negative controls.
CF doesn't claim that all systems have high relational structure, or that factorization always fails. Most systems, most of the time, decompose well. The parts suffice. CF is the diagnostic for knowing when they don't, and the vocabulary for what factorized analysis discards when the parts aren't enough.
Use CF when component-level analysis has failed unexpectedly; when interventions on parts fail to produce expected system-level effects; when there is suspected coupling structure carrying the information that matters; or when you need a principled answer to whether the connections between parts can safely be ignored.
What does factorization cost here?
If the answer is "nothing", factorize freely. The parts are enough.
If the answer is "everything", the relational structure is the phenomenon.
IC measures the continuum between.