The Non-Invertibility of Many-to-One and the Rank-One Kernel

A rank-one collapse is exact in dimension and lossy in content. A many-to-one map is not invertible: the fibers of cardinality greater than one are the distinctions the output does not determine. A rank-one linear map φ: V → W, with V finite-dimensional over a field k and dim(V) ≥ 2, factors through a single linear functional, φ(v) = h(v)·w with w a nonzero vector spanning its image, so its kernel is the kernel of that functional, a hyperplane of codimension one, and the map's fibers are the level sets of h. By the rank-nullity theorem the collapse is dimensionally exact, dim(ker φ) = dim(V) − 1, and the domain's accounting closes across the map. The exactness is what rank-nullity gives, the lossiness is what the non-invertibility of a many-to-one map gives, and the theorem measures what was set aside without retrieving it. A conservation of the collapsed content is a different and stronger claim, available from a system the map is embedded in rather than from the map. Landauer's principle supplies one such claim in thermodynamics, where erasing a bit dissipates at least k_B T ln 2 to the environment. It is cited for that claim in its own domain, not as a second route to the dimensional result. The paper operates at the scale of the linear map taken alone, its operands being a map, its rank, its kernel, and a dimension count, so nothing it states requires an instance and no instance is admitted as a premise. The mathematics is stated as standard linear algebra. The generative reading (the leftover is not lost, the hole is the reserve, cycle recovery) is in *The Unsigned Bit Is Generative: Why the Closure That Cannot Sign Itself Is the Source and Not the Limit* (Stewart, 2026aj), which cites this one for the mathematics. No interpretive claims are made here. The paper hands forward the direction of citation: every result reached through this theorem is an instance of it rather than a premise of it, and this paper cites none of the papers carrying those results for its mathematics. **Keywords:** linear algebra, rank-nullity theorem, linear map, kernel, nullity, rank-one operator, dual space, linear functional, hyperplane, fiber, injectivity, non-invertibility, many-to-one map, information erasure, Landauer's principle