A new arXiv preprint says that objective functions designed to respect ordinal or interval-scaled data can become extremely restrictive when they must preserve their results under every allowed affine transformation. Under the paper’s main rule, an adequate function has one unique relevant component, giving that component a decisive structural role.
The work is identified as arXiv version 1, dated 20 Aug 2026. It is a formal theorem-and-proof study: its objects are mathematical sets, real-valued functions, transformation groups and objective functions, rather than observations collected from participants or an applied dataset.
A demanding test for mathematical rules
The paper uses “adequateness” for a specific test of whether an objective function respects the measurement structure of data. In ordinary terms, the function must preserve its minimizing result after the measurement values have been changed by any transformation allowed by the specified group.
The formal setup begins with a non-empty set Ω, a natural number n with n at least 1, and a non-empty set X of real-valued functions on Ω. The question is not how people respond to a measurement, but how a formally defined function behaves when its inputs are transformed.
For the main interval-scaled analysis, the permitted changes are strictly increasing affine transformations. They preserve order while allowing the values to be shifted and rescaled through an affine rule with a positive slope.
The paper’s Proposition 2.3 gives the central equivalence: adequateness means preserving the strict ordering of objective-function values under every allowed transformation. If one value is strictly below another before the permitted change, the same strict ordering must remain afterward.
One component becomes decisive
The main structural theorem concerns a nontrivial, nonnegative adequate objective function. It says that the function must have one unique relevant component and a strictly increasing function H attached to it. When the relevant component values agree across the arguments, the objective function uses H of their common value; when they do not agree, the function is assigned zero.
The authors describe this as an Arrow-type result. In their interpretation, exactly one component acts as a structural dictator, so requiring a non-dictatorial rule rules out an adequate objective function under the stated transformation group.
Here, “component” is a term from the formal construction, not the name of a factor identified through an empirical comparison. The conclusion describes the allowable shape of an objective function under the paper’s assumptions; it does not establish a measured cause or a pattern in an observed population.
The continuous version keeps the same shape
The continuous counterpart retains the one-component structure but requires H to be both strictly increasing and continuous. Theorem 3.4 connects that representation with continuity and adequateness of the objective function in the bounded-function setting and the natural topology specified by the paper.
The authors say the main theorems are extremely restrictive and cannot be strengthened simply by focusing on ordinal-scaled data. That makes the transformation requirement itself the crucial issue: the more demanding the invariance rule, the narrower the class of functions that can satisfy it.
The paper’s interpretation is that many classical objective functions may be inappropriate for data observed in practice because they lack adequateness. That is a consequence the authors draw from the formal characterization, not a result of testing those functions against a particular dataset.
A narrower path for distance measures
The paper then considers dissimilarity coefficients, numerical rules used to express how different two formal objects are. Definition 4.1 introduces these coefficients for a finite Ω and includes the city block and Euclidean metrics.
The transformation group used for this part is narrower than the group behind the main characterization. It requires the affine transformations applied to the different components to have matching absolute slopes.
Under that restricted affine group, Proposition 4.3 reports that both the city block and Euclidean dissimilarity coefficients are adequate. The result therefore depends on the specific transformation conditions stated for this section.
This second result does not remove the tension identified by the main theorems. It shows that familiar dissimilarity measures can meet the adequacy requirement under a restricted group, while the broader common affine requirement produces the highly constrained one-component characterization.
What the mathematics does—and does not—establish
The evidence in the preprint is entirely formal. There is no sample estimate, observed outcome or empirical comparison from which to calculate sampling uncertainty; the conclusions are conditional on the definitions, transformation groups and proof assumptions used in the paper.
The study does not show that a particular standard method misinterprets a particular dataset, that the characterized functions outperform alternatives, or that the formal restrictions apply unchanged to transformation groups the paper did not study. Its claims concern adequateness within the stated mathematical settings.
That distinction is important for readers who use objective functions in applied measurement work. The paper offers a test for whether a rule preserves the required ordering and minimizing result; it does not supply an applied benchmark for deciding which method works best in practice.
The next question is how much to weaken the rule
The authors acknowledge that allowing all affine order-automorphisms may be too demanding. Their proposed direction for future work is to limit the allowed transformations or consider different transformation types in search of less restrictive characterizations.
That leaves a practical question for later work: whether a weaker transformation requirement can preserve the intended measurement structure while allowing objective functions that are not controlled by a single component. The supplied paper proposes that direction but does not establish the answer.
The front matter lists institutional affiliations and author contact details, but the supplied material reports no funding source or conflict-of-interest disclosure. No journal, DOI or peer-reviewed publication status is supplied; the work remains identified as an arXiv preprint.
Paper data and sources
Original title: Characterizations of continuous adequate objective functions for ordinal or interval scaled data
Authors: Gianni Bosi, Gabriele Sbaiz, Magalì Zuanon
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text