Lawful measurement coordinates

### PDF page 7 9. What kind of realism is being claimed? The paper does not claim that ψ is a hidden substance behind the measured coordinate. Nor does it claim that all admissible representations are metaphysically unequal. The claim is weaker and more operational. If two perturbations combine according to a stable empirical law, then the coordinate that makes that law additive represents an invariant of the operation. This is a mild structural realism about operations, not a heavy realism about coordinates as entities. The view is also not pure conventionalism. Once the operation is fixed, Aczel's theorem restricts the coordinate to an affine family. The remaining freedom is meaningful - choice of zero and unit - but it is not arbitrary. In this sense the composition law performs a coordination role: it tells us which numerical differences are operationally homogeneous. A difficulty remains. Often the operation itself is unknown or contested. Bliss, odds-additive shared-target, ZIP, HSA, and full Loewe models can be treated as rival hypotheses about how effects compose. The framework does not solve that identification problem by fiat. It clarifies what each hypothesis commits us to and how the hypotheses can be compared: by admissibility checks where possible, by held-out prediction under explicit error models, and by mechanistic evidence about the operation. 10. Limitations and repairs left to future work First, the theorem used here is one-dimensional and associative. Many real systems are multivariate, context- sensitive, path-dependent, or only approximately associative. Those cases require product structures, simplex geometry, dynamical systems, or non-associative algebraic tools; they are not solved by this paper. Second, the pharmacological contrast is between Bliss independence and a simplified odds-additive shared-target reference. General Loewe additivity with unequal potencies and non-unit Hill slopes is richer. In such cases the mid- range discrimination point shifts. The diagonal e≈0.618 result should be read as a clean analytic limit case and experimental-design guide, not as a universal constant of synergy. Third, the real-data illustration is intentionally modest. A decisive methods paper would analyse a large public screen such as NCI-ALMANAC or DrugCombDB, compare full pharmacological baselines, include measurement error in single-agent marginals, and test associativity using triple-combination data. Such work is needed, but it would be a different paper. Fourth, the Euclidean error is not always large. Near the identity, or when ψ is approximately affine over the observed range, native-coordinate inference may be harmless. The framework is a certificate for when to ask the coordinate question; it is not a denunciation of every native-scale model. 11. Conclusion The history of measurement teaches that numbers do not carry arithmetic rights by themselves. Addition, averaging, extrapolation, and regression are licensed by empirical structure. For bounded compositional observables, Aczel's theorem gives the corresponding lesson in a precise form: if a continuous, monotone, associative operation with identity is present, then there is a unique additive coordinate up to affine transformation. The Euclidean error is the failure to ask whether the native bounded coordinate is affine in that additive coordinate. Sometimes it is. Often it is not. When it is not, the correct mean is a Kolmogorov-Nagumo mean, the bias of native averaging is a Jensen effect, and linear modelling belongs in the coordinate selected by the operation or in an explicitly compared joint coordinate/error model. The first question for a bounded observable is therefore not whether it can be plotted on a line. It is what operation, if any, makes it lawful.

6. Adjacent coordinate traditions: agreement and conflict The framework does not replace established statistical traditions. It asks a different first question. Generalized linear models select links in relation to likelihoods, variance functions, and linear predictors. Aitchison geometry selects log-ratio coordinates because perturbation and powering are the natural operations on compositions. Fisher's z transform selects artanh(r) because it stabilises the sampling distribution of correlation. Information geometry selects coordinates by invariance and metric structure. Composition-derived coordinates select ψ from an empirical operation. These criteria may coincide. When they do, the coordinate receives convergent justification. They may also diverge. In that case the appropriate choice depends on the scientific question. If the goal is sample-efficient prediction under a known distribution, a distribution-derived link may be primary. If the goal is to represent how interventions combine, the composition-derived coordinate has priority. The comparison is then not transform worship but model comparison between joint coordinate/error hypotheses. Criterion Question answered Example Composition-derived In which coordinate does the operation add? rapidity for velocity addition; log-survival 7. Case study: Bliss and the odds-additive shared-target reference Drug-combination pharmacology is a useful case because the same bounded observable, fractional effect E in [0,1], can be associated with different mechanistic composition hypotheses. The case is used here as a philosophical and mathematical illustration of coordinate choice, not as a full pharmacological theory of synergy. For independent failure or independent inhibition, the unaffected fraction multiplies: 1 - E_AB = (1 - E_A)(1 - E_B). The additive coordinate is therefore psi_B(E) = -log(1-E), and the usual Bliss prediction is B(E_A,E_B) = 1 - (1-E_A)(1-E_B). For a simplified shared-target reference with unit Hill slope and matched potency, the odds coordinate adds: E_AB/(1-E_AB) = E_A/(1-E_A) + E_B/(1-E_B). This is a restricted odds-additive reference, not the full general Loewe theory with arbitrary potency ratios and Hill

with g(e*)=(5√5-11)/2≈0.090. The number is the positive root of the quadratic produced by this particular comparison; its significance is experimental, not numerological. It says that mid-range matched effects discriminate the two coordinate hypotheses far better than saturated effects. Figure 1. Diagonal separation between Bliss independence and the simplified odds-additive shared-target reference. The separation vanishes near zero and saturation and reaches its maximum at e≈0.618. 8. Additive Composition Models as a philosophical experiment in theory choice The empirical question is not whether a transform looks elegant. It is whether a joint coordinate/error hypothesis predicts better and leaves less structured residual error. For a candidate coordinate ψ, the scalar Additive Composition Model used in the illustrative case study is ψ(E_AB) = k[ψ(E_A) + ψ(E_B)] + ε. The scalar k summarises departure from the reference composition in that coordinate. k=1 is the reference law; k>1 indicates stronger-than-reference composition on that coordinate; k<1 indicates weaker-than-reference composition. In the drug-combination illustration, k is fitted per dose-response matrix on the training fold. Likelihood comparisons across coordinates require a common density scale. The reported NLLs below are native- scale Gaussian scores computed after transforming predictions back to the fractional-effect scale. For a held-out point, the score has the form NLL = ½ log(2πσ_train²) + (E_obs - E_pred)²/(2σ_train²),

where σ_train is the native-scale residual standard deviation estimated on the training fold for that coordinate/model. This is a common-scale predictive score, not a Jacobian-corrected ψ-scale likelihood. A full likelihood theory with coordinate-scale error, native-scale error, and measurement error in the single-agent marginals is left for future work. As a small real-data illustration, four public SynergyFinder example matrices were analysed: two Mathews Griner et al. matrices and two O'Neil/Merck matrices, comprising 82 interior dose pairs. The analysis used 200 random 70/30 splits stratified by matrix. These splits measure stability of the illustrative workflow, not population-level uncertainty about all pharmacology. No triple-combination data were used, so associativity was not tested. The case therefore illustrates coordinate-sensitive prediction; it does not certify the full admissibility structure. Coordinate Model Held-out RMSE Held-out NLL native reference k=1 0.613 1.018 Bliss reference k=1 0.555 0.896 odds-additive reference k=1 0.534 0.847 odds-additive ACM per-matrix k 0.318 0.302 Bliss ACM per-matrix k 0.165 -0.354 Table 2. Illustrative held-out prediction on four public SynergyFinder example matrices. Lower is better. The result shows that coordinate choice can matter on real data, but it is not a screen-wide validation and does not test associativity. Figure 2. Held-out RMSE on the native fractional-effect scale in the small public SynergyFinder illustration. Error bars represent variability over random train/test splits of the same four matrices, not uncertainty over a population of drug combinations. The result is philosophically useful because it shows that a coordinate commitment can be operationalised as a predictive commitment. It should not be overread. The four matrices are not representative of all combinations; the simplified odds-additive reference is not full Loewe; and the best predictive coordinate in one small set is not a general law. The point is that coordinate choice can be decided by empirical adequacy once the candidate operations have been made explicit.

composition law performs a coordination role: it tells us which numerical differences are operationally homogeneous. A difficulty remains. Often the operation itself is unknown or contested. Bliss, odds-additive shared-target, ZIP, HSA, and full Loewe models can be treated as rival hypotheses about how effects compose. The framework does not solve that identification problem by fiat. It clarifies what each hypothesis commits us to and how the hypotheses can be compared: by admissibility checks where possible, by held-out prediction under explicit error models, and by mechanistic evidence about the operation. 10. Limitations and repairs left to future work First, the theorem used here is one-dimensional and associative. Many real systems are multivariate, context- sensitive, path-dependent, or only approximately associative. Those cases require product structures, simplex