Mechanism-selected composition

### PDF page 8 Daniel J. Murray Revised September 2026 Gate Claim being tested What a verified failure establishes Aluminium application A specified chemical and effect model predicts unused combinations The named model fails for that assay; the abstract representation theorem is unaffected These gates are logically different. Failure should be assigned to the proposition actually tested. A positive reversal contrast is not a dimension theorem. A non-significant contrast is not an equiv- alence certificate. When multiple histories, doses, endpoints or stages contribute to a decision, confidence bounds and multiplicity control must cover the complete planned decision family. 9. The surviving unification The paper establishes a structural correspondence under explicit mechanisms: multiplication of unaffected fractions, addition of equivalent doses, and competitive dose ratios each supply a useful additive representation. The Aczél uniqueness is conditional on the declared total scalar composition and fixes the generator only up to scale. Bliss and common-shape Loewe are different reference laws. Their 𝑛 = 1 gap and its global optimum are exact. The half-dose example exposes a concrete invalid comparison baseline. None of these statements proves that all pharmacological systems are associative, that all scalar interventions commute, that every bounded curve is logistic, or that one geometry or molecular mechanism underlies all applications. A failed increment law can require a different scalar dynam- ics; a failed matched-present sufficiency test can require additional predictive information. The distinction is experimentally consequential. 10. Connection to predictive closure A dose-response curve depends on preparation, intervention, observation and endpoint. Promoting it to a state law additionally requires that the relevant future kernel factor through the proposed present representation. Different histories that share the measured effect must be challenged by the same future while accounting for matching error and biological margins. This rule supplies a stopping point for scalar-composition modelling. If only the Aczél increment assumption fails, test a more general scalar transition family before claiming hidden dimension. If histories at the same scalar have different future laws, merely changing a scalar synergy score or repa- rameterizing that same input cannot restore the erased information. A candidate additional state variable is then evaluated by held-out prediction and residual-history equivalence, with common support in the augmented representation. The role of this paper in the programme is therefore precise: characterize an important classical scalar branch, derive exact differences among its mechanism-selected members, and separate tests of that branch from tests of scalar state sufficiency. 11. Conclusion Pharmacological formulas can share an additive-generator structure while retaining different mech- anisms and different experimental meanings. Boundedness does not choose the generator. The physical composition rule supplies it, the Aczél hypotheses constrain it, and held-out futures test whether the encoded response is a sufficient state. ---

independent information optimum. A half-dose example demonstrates how an invalid baseline can produce apparent synergy under either lawful null. Endpoint-preserving Möbius transformations yield logit translations only under a separately justified projective hypothesis; their metric differs from Bernoulli Fisher geometry. An aluminium/SOD competition model is retained as a conditional testbed with explicit chemical and inferential limits. Order dependence rejects the commutative Aczél increment model when its encoding assumptions apply, but can occur in a fully closed scalar state. Matched-present, common-future tests are therefore required to diagnose missing state rather than merely noncommuting scalar dynamics. Keywords: Aczél representation; bounded composition; Bliss independence; Loewe additivity; Hill equation; competitive antagonism; drug combinations; aluminium toxicology; predictive sufficiency. 1. The structural problem Hill, Langmuir, Michaelis–Menten and Henderson–Hasselbalch forms describe different physical problems. Bliss and Loewe supply different combination references, while Schild and Cheng–Prusoff relations concern particular competitive mechanisms. Their similar formulas do not establish that

Boundedness alone performs none of these tasks. This revision retains the original paper’s mechanism-conditioned generators, exact Bliss–Loewe comparison, baseline critique and prospective aluminium test. It clarifies generator uniqueness, distinguishes reversal from scalar-state failure, makes the design objective explicit, and separates a conditional chemical bookkeeping model from an established aluminium mechanism. Classical results are cited as such [1–15]; the contribution is their precise combination into an experimental audit. 𝑎 ⊕ 𝑏 = 𝜙−1{𝜙(𝑎) + 𝜙(𝑏)}, 𝜙(𝑒 0) = 0. (1) The operation is total on the stated interval. A fitted local rule on an arbitrary subset of 𝐼 × 𝐼 does not establish these global hypotheses. Saturating endpoints where strict monotonicity fails must be treated as limits, rather than silently included in the strict theorem. In particular, the pharmacological examples below use 𝑒 ∈ [0, 1); complete saturation is a limiting value. For a nontrivial increasing generator, uniqueness is up to a positive multiplicative constant . To see why an arbitrary affine change is not allowed, put 𝜓 = 𝑐𝜙 + 𝑏 in the additive identity. Its semigroup and its identity, [0, ∞)is natural. No continuous finite-valued increasing generator can map a compact response interval with both endpoints included onto either range while preserving the strict hypotheses. These domain restrictions are part of the representation, not biological predictions. Equation (1) implies commutativity. It does not select 𝜙 from boundedness. Distinct increasing transformations can encode distinct operations on the same bounded interval. Thus the represen- tation constrains an already specified operation; it cannot infer independence, dose equivalence or projectivity from a response bound.

Daniel J. Murray Revised September 2026 encoding, associativity, the chosen endpoint model, or predictive sufficiency of the scalar. The experiment alone does not identify which assumption failed. In particular, noncommuting interventions do not prove that the state needs more than one coordi- nate. On the fully observed interval (0, 1), define 𝐴(𝑥) =𝑥 Population pooling must also be specified. Conditional independence within latent classes generally gives 𝑠𝐴𝐵 = 𝐸[𝑠 𝐴(𝑍)𝑠𝐵(𝑍)], which differs from 𝐸[𝑠𝐴(𝑍)]𝐸[𝑠𝐵(𝑍)]by a covariance term. A depar- ture from a pooled Bliss prediction can therefore reflect heterogeneity or a changed observation unit, rather than a direct molecular interaction. 3.2 Loewe: equivalent-dose addition For common response shape and constant relative potency, normalize each agent’s dose by its own potency scale. For the Hill family

𝐴𝐵 = 1 − (2/3)2 = 5/9, 𝑒 𝐿 𝐴𝐵 = 1/2. (15) Both are lawful no-extra-interaction predictions under their respective references. An informal benchmark equal to half the full-dose effect , namely 1/4, is smaller by factors 20/9 and 2. Such a comparison cannot establish synergy: it has confused halving a dose with halving a nonlinear response. This is an algebraic example, not a reanalysis of a particular animal study. Synergy is defined relative to a specified reference surface, endpoint and dose jurisdiction [11–15]. A sound experiment measures adequate single-agent curves, propagates their calibration uncertainty into the combina- tion predictions, and compares held-out combinations with a predeclared reference and material response margin. Selecting whichever reference yields the desired conclusion after seeing the mix- ture is not a valid test.

6. When projective geometry is justified 6.1 Endpoint-preserving Möbius transformations Add a separate hypothesis: admitted transformations are orientation-preserving fractional-linear maps fixing both endpoints of (0, 1). They then have the form 𝑇𝜆(𝑒) = 𝜆𝑒 1 + (𝜆 − 1)𝑒, 𝜆 > 0, (16) so 𝑘SOD[SOD] + 𝑘Al𝐶Al . (20) Here 𝐶Al denotes the activity-equivalent concentration of the aluminium species hypothesized to participate. It is not automatically total tissue aluminium or free Al 3+. The products 𝑘SOD[SOD] and 𝑘Al𝐶Al must both have units of inverse time, and 𝑣prod concentration per time. The formula assumes a valid quasi-steady regime, specified speciation, effective mass-action competition and no omitted feedback large enough to change those rates.

calibration is still required. Saturation can make the two references hard to distinguish in absolute effect. As an illustration, at 𝑒𝐴 = 𝑒 𝐵 = 0.95, Bliss predicts 0.9975 and the 𝑛 = 1 Loewe reference predicts 1.90/1.95 ≈ 0.97436, a gap of about 0.02314. These are model calculations, not extracted measurements from [23]. Whether that difference is resolvable depends on assay uncertainty. A prospective test should therefore include validated mid-range single-agent effects, for example tar- gets spanning 0.3–0.7 and the maximum-gap benchmark near 0.62 when the common-shape 𝑛 = 1 assumption is supported. Estimate the full single-agent response and uncertainty on calibration material; freeze the null predictions; then evaluate unused combinations with biological replication. Include assay interference, viability, speciation and target-engagement controls if the chemical path- way is claimed. Agreement with an effect surface remains a prediction result conditional on these declarations, not proof of a unique aluminium mechanism. 8. A falsification ladder with distinct failure meanings Gate Claim being tested What a verified failure establishes Reversal Interventions act as declared future laws That representation is predictively insufficient in the tested scope Bliss Unaffected fractions compose multiplicatively for the stated unit the full dose-space model Competition Antagonist produces the predicted dose ratios and shape-preserving shifts The simple competitive-translation assumptions fail in that range

establishes Aluminium application A specified chemical and effect model predicts unused combinations The named model fails for that assay; the abstract representation theorem is interventions commute, that every bounded curve is logistic, or that one geometry or molecular mechanism underlies all applications. A failed increment law can require a different scalar dynam- ics; a failed matched-present sufficiency test can require additional predictive information. The distinction is experimentally consequential. 10. Connection to predictive closure A dose-response curve depends on preparation, intervention, observation and endpoint. Promoting it to a state law additionally requires that the relevant future kernel factor through the proposed present representation. Different histories that share the measured effect must be challenged by the same future while accounting for matching error and biological margins. histories at the same scalar have different future laws, merely changing a scalar synergy score or repa- rameterizing that same input cannot restore the erased information. A candidate additional state variable is then evaluated by held-out prediction and residual-history equivalence, with common support in the augmented representation. The role of this paper in the programme is therefore precise: characterize an important classical scalar branch, derive exact differences among its mechanism-selected members, and separate tests of that branch from tests of scalar state sufficiency. Pharmacological formulas can share an additive-generator structure while retaining different mech- anisms and different experimental meanings. Boundedness does not choose the generator. The physical composition rule supplies it, the Aczél hypotheses constrain it, and held-out futures test whether the encoded response is a sufficient state.