Circulatory Fidelity rests on an explicit ontological commitment: relations are primary, nodes are emergent. This is not decorative philosophy appended to a mathematical framework — it is the load-bearing foundation from which the mathematics arises, and from which every testable prediction follows.
The central operation that CF measures is factorization — the mathematical act of replacing a joint distribution with a product of independent marginals:
q(x, z) → q(x) · q(z)
This operation explicitly 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. Factorization is the mathematical formalization of what philosophy calls reductionism: treating the whole as nothing more than the sum of its parts.
Reductionism is not a bad word here. It is the overwhelmingly dominant epistemic strategy in modern science, and it works spectacularly well in many regimes. But it is a strategy — a methodological choice — not a fact about the structure of reality. CF provides the diagnostic that distinguishes where factorization is adequate from where it is violent to the phenomenon under study. The question is always: how much did that operation cost?
CF's ontological 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 define the relational structure; its diagonal elements D describe nodal self-precision. Mathematically: R maintains D, not the reverse.
This is a strong metaphysical claim, not merely a modeling preference. It means that when we factorize — when we replace the joint with a product of marginals — we are not simply approximating. We are asserting that the relations are eliminable: that the parts, considered independently, 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 does not 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. The direction of its meaning depends on what the relational structure is doing in the system — encoding the physics (filtering) or managing redundancy (pooling). This interpretive neutrality is itself load-bearing: it prevents CF from becoming a hammer that sees every system as a relational nail.
Why is factorization the default strategy? 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 — detecting three-way dependencies, four-way dependencies, 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). This architecture is extraordinarily successful. It built civilization. But it also creates a systematic blindspot: structure that requires expensive observation to detect appears not to exist.
Science inherited this blindspot. The methodological commitment to decomposition — isolate variables, control for confounds, study parts independently — is a direct descendant of the metabolic constraint. It is a survival heuristic elevated to an epistemology. CF does not reject this strategy; it measures where the strategy's costs exceed its savings.
The relational structure that factorization discards is not merely a constraint on the system — it is simultaneously what the system can do. Coupling between variables is both the cost of observation (you must account for their joint behavior) and the channel through which information flows (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 — but so is the information it purchases.
IC quantifies the tension: the relational structure's magnitude, abstracted from whether it helps or hinders any particular analytic strategy. The framework holds constraint and affordance as a generative pair — not resolving the tension but measuring it, so that each domain can determine its own relationship to the cost.
Factorization discards pairwise coupling — but what about structure that is invisible even to pairwise observation? Coplexity names relational structure that exists only in three-or-more-body interactions: dependencies where any two variables appear 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; science defaults to it. Structures that require higher-order observation to detect are therefore selected against in our picture of reality — not because they are rare in nature, but because they are expensive to see. CF's two-stage detection protocol (check IC₂, then if near zero check IC₃ via Walsh-Hadamard analysis) is designed specifically to penetrate this observational blindspot.
The claim that coplexity is ubiquitous remains a testable hypothesis, not a proven truth. Evidence across 41 domains is accumulating. But even if it turns out that coplex structure is genuinely rare, the detection protocol remains necessary: you cannot distinguish "rare" from "unobserved" without a method that can see what pairwise analysis cannot.
A framework that spans 41 domains risks becoming unfalsifiable — explaining everything by explaining nothing. CF maintains rigorous internal classification to prevent this. Every claim carries an explicit epistemic status, and the framework actively monitors 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 are not hand-waved away — they are actively sought as negative controls.
CF does not claim that all systems have high relational structure, nor that factorization always fails. Most systems, most of the time, decompose well. The parts suffice. CF provides the diagnostic to know when they don't — and the vocabulary to characterize what factorized analysis discards when the parts are not 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 sufficient.
If the answer is "everything" — the relational structure is the phenomenon.
IC measures the continuum between.