An Exact Null for Many-to-One Strategy Comparison Under Unequal, Estimated Volatilities

A desk comparing several candidate strategies against one benchmark faces a multiple-comparison problem with a specific structure: every comparison shares the benchmark, and the benchmark's volatility is itself estimated, so the shared estimate sits in every denominator. This is Dunnett's many-to-one problem, solved in 1955 for equal variances and unsolved since for unequal ones. The state of the art, published in 2023, approximates the joint law of the denominators and generates it by simulation. This paper computes that law exactly. Both couplings run through the benchmark and nothing else, so conditioning on both of the benchmark's sufficient statistics, rather than on its mean alone as tradition does, makes the comparisons exactly independent. A (2K+2)-dimensional integral becomes a two-dimensional outerrule over K one-dimensional inner rules, and the outer dimension does not grow with the number of candidates. The joint law of the denominators is never formed, so it is never approximated. Six falsifiers pin the answer.The law is then used as an instrument rather than as a procedure. It gives the true familywise error rate of every rule in use: the studentised maximum modulus route realises as little as 58% of the error rate it is allowed, and Dunnett's constant carried across over-rejects by up to 84%. Run head-to-head against the 2023 method on its own scenarios, the exact law with estimated volatility ratios loses, and the reason is instructive: where that method's approximation bias is largest, its own estimation error has the opposite sign and cancels most of it. Below five degrees of freedom on a group that method should be preferred; by seven the exact rule is within 2% of nominal, and a standard prepivoting calibration halves what remains. Separating approximation error from estimation error is what exposed that cancellation, and no single simulation can do it; the same law also supplies a control variation whose true value is known exactly.The guarantee a desk needs can also be stated as a bound rather than as an estimate. Admitting a region of ratio vectors whose coverage is certified by exact quadrature, and taking the exact constant over the region's vertices, gives a rule whose familywise error rate is at most the nominal level plus the region's miss probability. That rule is never liberal on the designs of Section 7, and it raises the critical constant by 1.6 to 23%. Branch and bound with interval arithmetic and a Bernstein-ellipse remainder then certifies the supremum over the continuum of ratios on 12 of 17 designs, at a tolerance declared in advance and a median gap of 9.5e07. The region rule is nonetheless not the one to run. It realises 0.7% of the error rate it is allowed and gives up 72.1 of 80 percentage points of power, because it meets a conditional statement on every draw where the guarantee asks only for an average over draws. Imposing the guarantee unconditionally, on the same exact law and at the same ceiling, gives up 4.5 points instead. A global linear program over the invariant class places the deployed rule 0.21 to 0.24 percentage points above the best such rule on mean power, and 0.21 to 0.43 on the worst-case criterion the invariance argument covers. Serial correlation and factor exposure are then handled as declared nuisances, where an exact transform returns the independent problem and every certificate with it. A serial coefficient fitted on a disjoint calibration period gives 0.0414 against a ceiling of 0.025, so that result is a declared-coefficient guarantee and nothing weaker.