The Fragmentation Ceiling: Dimensionless Groups for Nurse Staffing Adequacy

Nursing capacity is usually treated as headcount times hours. This paper proposes that it is not: task switching consumes a fraction of every nurse-hour exposed to it, and that fraction is the product of two observable quantities, an interruption rate and a per-interruption cost in nurse-time. Writing effective capacity as headcount times hours times one minus that fraction, and saturation as demand over the result, gives a conditional dimensionless capacity model. That model, rather than any single prediction drawn from it, is what the paper asks to be judged on. Its content is that a ward's behaviour is a function of the one dimensionless group rather than of headcount, census, acuity and fragmentation separately. A factorial ward simulation tests the sufficiency claim and it holds: one dimensionless group stands in for four varying quantities, leaving 7 to 11 per cent of outcome variance unexplained where patients per nurse leaves 35 to 40 per cent and fragmentation on its own leaves 87 to 92. The same test scopes the claim, and the scope is reported rather than buried: the fragmentation correction earns its place in proportion to how much fragmentation varies between the units being compared, from essentially nothing across similar wards to nearly half the remaining variance across dissimilar ones. Setting the saturation to unity returns a licensed patients-per-nurse ratio, which on the reconstruction is 11.58, 6.89 and 3.97 at low, mid and high acuity, a factor of three across one mandate. That ratio is a ceiling on a defensible staffing level rather than a recommendation, so the verdict it supports is one-sided: a mandated five patients per nurse is past the knife edge at high acuity on any reading, and sits below the ceiling at low and mid, which bounds the mandate rather than clearing it. A secondary prediction, a cost-efficiency reversal between hiring and attention protection at a dimensionless threshold, is derived and simulated but shown to be inert on ordinary wards. The parameter set is reconstructed from six published settings in five countries and no single ward has been measured end to end, so the numbers are illustrative rather than representative and the paper says so throughout. The deposit includes a reproduction bundle with a SHA-256 manifest over every input, script and output, a pre-registration written before the empirical work it constrains, and a verifier that recomputes every number, checks the hashes printed in the manuscript against the recomputed ones, re-derives forty-five headline quantities from the reconstruction, and checks four further claims the paper states in words.