Dark Sectors as Relative Accessibility Kernels: Finite-Dimensional Dark Quotients and Response Recognition

Dark-sector inference often begins with a familiar pattern: a declared visible channel fails to separate some source distinction, while another response appears to register what the visible channel misses. In physics, this pattern occurs whenever a source distinction is inferred through a response channel rather than direct visible access. Gravitational response, curvature response, holonomy response, vacuum response, anomaly matching, and scattering or asymptotic response are familiar examples. This paper isolates the finite-dimensional accessibility structure common to such situations. The paper does not model a dark sector as hidden matter, absolute absence, or an intrinsically invisible object. It formalizes dark-sector inference as a relative accessibility problem. A source distinction is treated as a traceless self-adjoint state-difference direction on a declared source carrier. Given a declared source carrier, a visible channel, and a response channel, the source distinctions erased by each channel determine a response-recognizable dark sector: the source distinction space erased by visible access, modulo the part also erased by response access. Thus a distinction is dark only relative to the visible channel, and it becomes response-recognizable only when it is not also erased by the response channel. The main theorem proves that this response-recognizable dark sector is exactly the incremental source distinction space gained by adding response access to visible access. Its dimension is equivalently the drop in compression under joint access, and it is dual to the corresponding response survivor quotient. The recognition theorem separates three claims that are often conflated: visible failure, response recognition, and source identification. A dark contrast is response-recognizable precisely when it factors through the relative dark quotient. A state-level invariant is response-recognizable when it factors through the response accessibility quotient; it identifies the declared source identity class only when it is source-complete for that relation. Therefore, visible failure is not source absence, and response separation is not yet source identification. The theorem supplies a finite-dimensional audit criterion for dark-sector claims. Such a claim requires a declared source carrier, a visible channel, a response channel, and a source identity relation. Once these data are fixed, the theorem determines which source distinctions remain invisible, which become response-recognizable, and which are sufficient for source-complete identification. It thereby separates relative darkness, response recognition, and source identification as distinct structural questions. License note: Distributed under CC BY-NC-ND 4.0.