A Common Edge Family for the (0, 1, s)b Triad

We isolate a common structural mechanism behind the three edges of the $(0,1, s)_b$ triad. The starting point is a blurred multiplicative object assembled from prime-local data inside a completed spectral host. The central claim is that the three edges $$0 \leftrightarrow 1, \quad 1 \leftrightarrow s, \quad 0 \leftrightarrow s$$ can be read as three coefficient languages extracted from the same parent object by three natural procedures: local tower expansion, prime-overlap expansion, and obstruction/Taylor expansion. The theorem proved here is intentionally first-stage and structural. Its purpose is to identify a shared mechanism rather than to build a maximal universal theory. Once a prime-successor multiplicative skeleton, a completed host, a spectral portrait, and a blur family are fixed, the three edge readings belong to one common family. The familiar Riemann case then appears as a benchmark realization of this more general blurred multiplicative architecture.