Each domain is positioned by its composite validation score: higher-confidence domains orbit closer to the center. Edges represent formally classified connections (STQA Class 7+). The Poincaré ball is the native geometry of inference coupling: IC = tanh(geodesic distance).
These connections aren't analogies. Each edge means the same mathematical object (the same IC, the same geodesic coordinate, the same cost function) appears independently in both domains. A Class 10 edge is a proven identity; Class 7 is an algebraically exact structural correspondence. Resemblances (Class 5) are tracked but not shown here.
An edge isn't a resemblance or a metaphor. It means the same mathematical object turns up in two different domains: the same IC, the same geodesic coordinate, the same cost function, doing the same work in each. CF separates what transfers between domains (that shared structure) from what has to be rebuilt locally in each one (what the nodes are, how IC is measured, what a result means on the ground). An edge records the part that transfers.
Edges are graded by how exact that shared structure is, and only the two strongest grades are drawn here. A weaker resemblance you can't yet pin down is a lead, not a connection, so it stays off the map.
The two sides are the same object, connected by a proven theorem with a complete derivation. There's no interpretive room: where the identity holds, the quantities are equal.
Variational inference and differential geometry. IC isn't analogous to a distance on the Fisher-Rao manifold. It is one. The precision matrix is the Fisher information matrix, the geodesic coordinate is d = arctanh(IC), and the cost of factorizing is the action I(d) = log cosh d. Inference and geometry are two readings of one structure.
Information theory and thermodynamics. The information lost when you factorize a coupling equals the minimum energy it takes to erase that coupling: I(d) = log cosh d = Wdiss/kT = Sgen/k. That's Landauer's bound applied to relational structure. The same number is a quantity of information and a quantity of heat.
The mapping between the two domains is algebraically exact, but the two settings aren't the same thing, so the interpretation carries caveats. The structure transfers; the meaning has to be checked on each side.
Computation and quantum information. CF's coplexity test on classical Boolean data, "is this function affine over GF(2)?", is the same structural test the stabilizer formalism runs on quantum states: XOR maps to CNOT, affine maps to Clifford. The algebra is identical. But classical bits and quantum amplitudes aren't, so the correspondence is exact as algebra and needs care as physics.
Chemistry. Chemistry connects to eight other domains, and every link is Class 7. The same coupling structure recurs exactly across bonds, reaction networks, and spectra, but each reading needs its own caveats, and none has yet been tightened into a proven identity. A domain can be densely connected and still be waiting for its first Class 10 edge.
There's a third grade the map doesn't show. Class 5 is a real resemblance with no exact mapping yet, sometimes quantitative, sometimes only qualitative. The lab tracks these as research leads, but the bar for a drawn edge is an exact correspondence (Class 7) or a proven identity (Class 10), nothing softer. And when a translation has to chain through several steps, its grade is the weakest link in the chain, not the strongest.
Every domain extension requires explicit operationalization: What are the nodes? What implements R? How would IC be measured? What would falsify the extension? Each connection is classified using semiotic translation quality assessment (STQA).
Formal, empirical, and operational soundness all above 0.75. Quantitative predictions confirmed by data.
Structurally sound with clear operationalization. Empirically suggestive but awaiting full Stage 3 validation against independent data.
A composite score grades a single domain's own soundness, and it sets where the domain sits in the network: higher scores orbit closer to the center. It's the minimum across three dimensions, not the average: formal soundness (is the CF operationalization mathematically rigorous?), empirical grounding (are the predictions confirmed by data?), and operational completeness (are all six extension questions answered, including falsifiability and the dependency archetype?). Taking the minimum means a domain is only as strong as its weakest dimension. Tier 1 requires all three above 0.75; a strong formal apparatus with thin empirical support stays in Tier 2.
A score and an edge class measure different things. The score grades a domain on its own; the edge class grades a connection between two domains. A Tier 2 domain can still carry a Class 10 edge, and a Tier 1 domain can connect to others only by Class 7. One is about how well CF is grounded in a field; the other is about how exactly two fields share the same structure.