When response times identify decision and nondecision processes: identification, its limits and design consequences in the fixed-parameter diffusion model

Unrefereed research candidate. In one sentence: in the fixed-parameter diffusion model with a completely unspecified nondecision-time distribution shared by all conditions and responses, the data determine the split of response time into decision and nondecision time exactly when the response-time distributions given each response are not all identical. Diffusion models split each response time into a decision time (noisy evidence drifting to one of two boundaries) and a nondecision time (encoding and movement). The D*M method of Verdonck and Tuerlinckx (2016) leaves the nondecision distribution unspecified and rests on an uniqueness premise. This paper settles that premise for the fixed-parameter model (fixed boundaries, drift and start in each condition, unit diffusion coefficient, nondecision time independent of the decision). Theorem 1: all parameters and the whole nondecision distribution are identified if and only if the normalised response-time distributions given each response are not all identical; equivalently, one condition with a biased start suffices, or two unbiased conditions differing in boundary separation or drift magnitude. The standard unbiased design with drifts +v and -v under one boundary setting is exactly non-identified. When identification fails, a process with half the boundary distance and a quarter of the mean decision time, plus a suitable independent extra delay, fits the data exactly (Lemma 3); Theorem 2 gives the complete identified set of boundary distances with an upper bound that is attained under a lower-tail condition (Corollary 1). Corollary 2 answers the constant-boundary case of a conjecture of Fudenberg, Newey, Strack and Strzalecki (2020). Three limits on estimation are proved: no uniformly consistent estimator of boundary separation over the identifiable class (Theorem 3), including far from the degenerate designs when delay spread is unbounded (Proposition 1); two unbiased conditions whose response log-odds differ by 2ε need more than 1/(4ε2) trials per condition (Theorem 4); and with response-specific delays no single condition suffices (Proposition 2). A start-perturbation analogue of Theorem 4 is stated as a conjecture (Remark 3). The archive contains the manuscript (PDF and Markdown source), figures, the bibliography, a verification script with 25 explicit checks (each failure exits with an error, also under python -O) and a bounded synthetic recovery study with its raw results and summaries, a Monte Carlo uncertainty script, the dated literature audit, a rating-free internal review summary, CI and SHA-256 sums. Assurance: written proofs; numerical checks of the formulae; the simulation is an illustration with one estimator, not validation; no real data; internal AI-assisted review only (not peer review, independent reproduction or formal verification). Across-trial variability and collapsing boundaries are not covered. Creator: Anonymous. Text, data and figures CC0-1.0; code MIT.