Structural Addresses: Continuous Proximity, Discrete Membership, and Compatible Markers in Inheritance Systems

Reproducibility resources: A fully reproducible analysis package, including source code, processing scripts, documentation, and instructions for reproducing the analyses presented in this study, is available in the accompanying GitHub repository (link provided below).

This technical note develops a methodology for locating and classifying objects in analytic, arithmetic, and inherited families by separating two different tasks: continuous navigation and exact structural identification. A structural address is written as

\operatorname{Addr}(x)=\bigl(\Sigma_c(x),\delta(x)\bigr),

where the continuous signature \Sigma_c provides local proximity information and the discrete invariant \delta determines exact stratum membership. The note shows that continuous signatures provide reliable local coordinates only where the signature map is a local embedding, while exact membership is recoverable from continuous data only when the discrete invariant factors through the continuous signature. For inherited systems, a marker supports well-defined reduced dynamics precisely when its fibers are compatible with the inheritance map; closure of the reduced dynamics and separation of structural classes are shown to be independent requirements.

The framework is illustrated with three controls. In the detuned Ramanujan 1/\pi family, the parameter moves through one fixed hypergeometric local system: the observable varies analytically, the convergence rate changes monotonically, and the exact value at D=396 is not signaled by a local analytic extremum. In the beta/gamma family

J(s)=\frac{\Gamma(s+1)^2}{\Gamma(2s+2)},

continuous parameter proximity is shown to differ from exact gamma-period relations, with reflection and multiplication identities providing discrete structural relations that are not captured by raw parameter distance alone. A minimal four-state inheritance model then exhibits markers that close but fail to separate, markers that separate but fail to close, and a composite marker that satisfies both requirements.

The note is methodological rather than theorem-driven: its contribution is the synthesis of classical tools into a structural-address framework for surveying continuous neighborhoods, identifying exact strata, and testing whether reduced markers remain compatible under inheritance.