Definition. A five-part compatibility datum [fdip-009Z]

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A compatibility datum is a tuple \[(G,\mathbb K,\rho ,\alpha ,\sigma ,C).\] Here \(G\) is a group, \(\mathbb K\in \{\mathbb R,\mathbb C\}\) is its coefficient field, and \(\rho :G\to GL_{\mathbb K}(F)\) is a finite-dimensional representation on a fiber \(F\). The map \(\alpha \) is an action of \(G\) on a finite graph substrate, in the sense of a representation datum. The selector \(\sigma \) is an idempotent in a declared algebra of operators on \(F\), or on a finite direct sum of such fibers. The reference comparison \(C\) is a stated homomorphism, quotient, or numerical identification against one named object. The coefficient field is scalar data for \(F\). It is not a residual gauge group. When \(\mathbb K=\mathbb R\), a complex representation enters only after realification, as a real representation on the underlying real vector space. A direct sum of that real fiber with another real fiber is then one real representation.

The datum is tested against 5 criteria.

• Exactness. The group, the field, the representation and the substrate action are all specified. An inclusion into a matrix group is not a substitute for the missing slot.
• Declared maps. Each comparison with a Dirac, Lorentz, gauge or gravitational structure names the homomorphism, kernel, cover or branching rule that is actually used.
• Separation. A spinor or Dirac action is identified with a gravitational action only when a separate compatibility law is proved. Sharing a name, or an inclusion into \(SU(2)\), is not that law.
• Selector effect. The selector is given, and there is a proof of the change it makes to the operator or parameter named in the datum.
• Reference recovery. The comparison names the ordinary object recovered in the chosen reference sector, or it records that the datum claims no such recovery.

For a fixed datum, each criterion is marked proved, conditional or open. Proved means the finite obligation stated in that criterion is established. Conditional means a stated part is established and the complementary part is unresolved. Open means the required map or recovery is not constructed. An open mark records that absence. It is not a theorem that no later construction exists.