When dilution resolves a low sandwich-immunoassay signal: robust certificates and the limits of spike controls

A low sandwich-immunoassay signal can mean little analyte, or so much analyte that capture and detection reagents are spread over different molecules (the high-dose hook effect). We ask what a second reading of a diluted aliquot certifies in a finite-reagent, independent-site equilibrium model when capacities, affinities, dilution factor, gain drift and native availability are known only to within intervals. The noise-free ambiguity is described exactly: the response is strictly unimodal, every sub-maximal signal has exactly two preimages (exchanged by an involution for symmetric sites), and the diluted-to-neat ratio is strictly increasing. We then prove robust two-reading certificates: for near-unit parameters, dilution in $[9,11]$ and $5\%$ gain drift, the diluted-minus-neat source contrast is nonpositive for accessible concentration at most $0.4$ and at least $0.04$ on $[20,100]$; a general-box theorem gives analogous margins when capacities exceed dissociation constants a hundredfold, for unequal capacities, and up to $10^4$ with an explicit $1/U$ law. Each continuum inequality reduces to eight positive Bernstein coefficients. We derive error budgets from precision profiles, a reduction of relative error to gain drift, conclusions without a binary promise and extensions to availability drift and incomplete equilibration, with a kinetic theorem giving a sufficient incubation time. Exact indistinguishability results show what a neat reading, accessible spikes and finite dilution panels cannot certify. For sequential assays, ideal washing gives a monotone saturating response, any fixed carry-over restores the high-dose tail, and a finite-range wash specification is given. The source inequalities, decisions, certificates and obstructions are verified in Lean 4; the structural and kinetic theorems have conventional proofs. These are conditional certificates, not clinical performance claims.