SUMMARY For a scalar or vector parameter of interest with a regular statistical model, we determine the definitive null density for testing a particular value of the interest parameter: continuity gives uniqueness without reference to sufficiency but the use of full available information is presumed. We start with an exponential family model, that may be either the original model or an approximation to it obtained by ancillary conditioning. If the parameter of interest is linear in the canonical parameter, then the null density is third order equivalent to the conditional density given the nuisance parameter score; and when the parameter of interest is also scalar then this conditional density is the familiar density used to construct unbiased tests. More generally but with scalar parameter of interest, linear or curved, this null density has distribution function that is third order equivalent to the familiar higher-order p-value ![\"\Phi(r^*)\"](\"http:\/\/jsr.isrt.ac.bd\/wp-content\/ql-cache\/quicklatex.com-278dcefdbf3de8d3b0cf598cbf727e7f_l3.png\" "\\"Rendered")
. Connections to the bootstrap are described: the continuity-based ancillary of the null density is the natural invariant of the bootstrap procedure. Also ancillarity provides a widely available general replacement for the sufficiency reduction. Illustrative examples are recorded and various further examples are available in Davison et al. (2014) and Fraser et al. (2016).<\/p>\n