Pharmacology predecessor

PDF page 35 Murray — Aczél-Family Structure of Pharmacology (v14) Page 33 scenarios whose realisation in pharmacology may exist but is less common. The exhaustiveness claim is limited to the empirically observed linear fractional class (Section 4.7). The framework does not provide a mechanistic alternative to the Exley pathway. It is an interpretive overlay on Exley's chemistry that connects substrate-competition kinetics to the broader compositional structure of pharmacological combinations. If the [AlO2•]2+ pathway is revised by future computational or experimental work, the structural framing adapts; the Aczél unification of pharmacology persists. 11. Conclusion The universality of aluminium-induced oxidative stress — the same qualitative damage pattern across mammals, fish, plants, and cell-free preparations — is a structural consequence of bounded compositional kinetics, not a coincidence of independent chemistries. SOD- mediated superoxide clearance on a bounded interval, perturbed by Al3+ substrate competition, follows rules that are species-independent by mathematical necessity. Aczél's representation theorem reveals those rules, and they are the same rules that govern Hill kinetics, Langmuir adsorption, Michaelis-Menten enzymology, Henderson-Hasselbalch acid- base equilibrium, Bliss independence, Loewe additivity, Schild antagonism, and the Cheng- Prusoff relationship — independently derived over 63 years and now identifiable as realisations of one structural family. The Bliss-Loewe distinction, long operationalised in practical mixture toxicology, is here given a geometric foundation as a structural classification of mechanism: independent perturbations on a bounded observable produce Bliss composition, shared-target perturbations produce Loewe composition, and the gap between them (maximum (5√5−11)/2 PDF page 36 Murray — Aczél-Family Structure of Pharmacology (v14) Page 34 ≈ 0.090 at e = (√5−1)/2 ≈ 0.618, the reciprocal of the golden ratio) is strictly positive by algebraic necessity. The dominant half-dose-mix synergy demonstration design in natural- products toxicology produces apparent synergy of factor 2 from saturation alone. The framework's load-bearing empirical commitment — the discriminating Bliss-vs-Loewe test in mid-range matched-dose combination data — is implementable in standard Al-toxicity laboratory infrastructure and has not yet been performed. Its outcome will determine whether Al3+ competes with antioxidant interventions on an independent or shared substrate, with direct implications for combination intervention design.

which natural coordinate the physical specification adopts. In Al-Otaibi et al. (2018), quercetin (e = 0.950) combined with α-lipoic acid (e = 0.945) gave observed ecombo = 1.007 against Bliss prediction 0.997 and Loewe prediction 0.973 — consistent with both in the saturated regime. The distributed empirical foundation across 60 years of mixture toxicology and enzyme kinetics literature validates the two compositions in their respective mechanistic domains. The dominant half-dose-mix synergy design in the Al-toxicity literature is shown to produce apparent supra-Bliss effects of factor approximately 2 (2.00–2.22 in the Hill nH=1 The framework resolves the Bliss-Loewe debate, explains the universality of Al-induced oxidative damage, identifies a methodological correction for the combination natural- products literature, and generates six falsifiable predictions (five core, one exploratory) testable in standard laboratory infrastructure.

biological organisation rather than modelling preference. The present paper arrives at the same foundation through functional equations rather than algebraic structure, and extends it to generate specific, experimentally testable predictions for combination antioxidant interventions in Al toxicology. 2. The Kinetic Foundation: Aluminium and Superoxide Dismutase 2.1 Bounded steady-state superoxide The observable of interest is the steady-state superoxide concentration [O2•−]ss in a given

Murray — Aczél-Family Structure of Pharmacology (v14) Page 7 magnitude only; the framework's qualitative predictions do not depend sensitively on the precise value of kAl. 3. Structural Foundation: Aczél's Theorem on Bounded Observables 3.1 The theorem Consider a biological observable u confined to a bounded interval I ⊆ ℝ, and a binary

perturbed (e.g., Al exposure inducing SOD expression on a timescale comparable to the experiment), sequential perturbations do not commute. Plant systems under prolonged Al stress show such induction; the framework's clean predictions are restricted to short exposures relative to the SOD-induction timescale. Irreversible damage. Perturbations that cause permanent changes (cell death, fibrillar accumulation, covalent modification) violate the steady-state assumption underlying associativity. the steady-state framing required for the Aczél construction. Compartment heterogeneity. Heterogeneity across compartments at different rate- limiting steps produces apparent deviations from the framework's clean predictions when aggregate measurements are made. Each failure mode produces a structurally specific signature. The framework is therefore falsifiable through axiom-violation tests as well as through direct prediction tests. 4. The Aczél-Family Unification of Pharmacology

θ(x) = x / (K + x) (6) This is simultaneously: the Hill equation with nH = 1 [12], the Langmuir adsorption isotherm with K replaced by P0 [13], the Michaelis-Menten rate equation with K replaced by Km [14], and the Henderson-Hasselbalch acid dissociation equation with K replaced by [H+]ref [15,16]. The four equations are mathematically identical; their independent derivations each correctly arrived at the same Aczél-family realisation through different physical reasoning. The Hill coefficient nH ≠ 1 is a rescaling of the rapidity coordinate (y → ny), not a different equation.

who use Bliss as the null model and find 'sub-Bliss synergy' for agents acting on the same target are not finding antagonism; they are correctly identifying shared-target mechanism, which the framework predicts gives Loewe (which is sub-Bliss by equation 10). The terminology 'sub-Bliss synergy' and 'Loewe additivity' and 'mechanism overlap' are three names for the same prediction. 4.3 Schild regression as log-dose rapidity translation The Schild equation [19] for competitive antagonism states that the dose ratio DR (agonist EC50 with antagonist / EC50 without) satisfies: log(DR − 1) = log[B] − log(Kᴮ) (11)

For a constant-mechanism intervention that shifts the rapidity coordinate by Δ, the dose at which any fixed effect e is reached shifts by Δ in log-dose. EC50 is the conventional anchor because it is the rapidity zero-point, but the predicted shift is identical at EC10, EC90, or any other ECx. Disagreement in log(ECx) shifts across x indicates either non-translation-group composition or that the intervention is rescaling the rapidity (changing nH) rather than translating it. This is testable in any dose-response dataset that reports multiple ECx values under the same intervention.

Page 16 exhaustive. Systems exhibiting composition laws outside the linear fractional class would extend rather than falsify the framework; they would indicate biological mechanisms not yet captured by the Aczél-family axioms as currently stated. 4.8 From physical commitment to parameter dimension A natural concern with the framework as developed in Sections 4.1–4.7 is that the rapidity coordinate appears to be chosen to match each known equation: log-dose for Hill, log- describe the composition. Once the structural type is fixed by physics, the rapidity is forced uniquely up to affine transformation, and the parameter count of the composition law is bounded by the dimension of the underlying isometry group. The latter is the predictive content of the framework: any bounded-observable pharmacological composition is described by a strictly bounded number of structural parameters, set by the geometry rather than by the equation. 4.8.1 The physical-commitment-to-isometry-class map

the bounded interval (0,1) viewed as a hyperbolic geodesic is fixed by three real parameters. This dimensional fact propagates to a structural bound on the parameter count of any pharmacological composition law on (0,1), which is the framework’s strongest predictive content. Single-agent (k = 1). A hyperbolic-class single-agent dose-response θ(x) = x/(K + x) requires one parameter (K, equivalently log K, the location of the rapidity zero). The Hill coefficient n_H is the affine-scale parameter from Aczél’s theorem, not an independent structural with α = 0 recovering Bliss and α = 1 recovering Loewe. The structural parameter α is determined by the perturbation structure (Section 4.8.1, axiom 3); fitting it to data is a hypothesis test, not a free fit. The total dimension of the two-agent composition family is therefore 3, exhausting the hyperbolic-parabolic content of PSL(2,ℝ). k-agent combinations. For k pharmacological agents acting on a single bounded observable, an Aczél-family composition has structural parameter count bounded by N_params ≤ k(k+1)/2 + ε (16)

couplings. A dataset that is best fit by a parameter count exceeding k(k+1)/2 + (k−1)(k−2)/2 = k² − k + 1 cannot be accommodated within any Aczél-family composition on the bounded interval, even by recourse to elliptic structure, and therefore falsifies the framework regardless of how generously ε is interpreted. For practical pharmacology where pairwise reductions hold, ε = 0 and N_params = k(k+1)/2: 1 for single-agent, 3 for two-agent, 6 for three-agent, 55 for a 10-agent screen, 210 for a 20-agent screen. An unconstrained empirical model of k-agent combinations on a bounded effect can in exponential parameter space to a polynomial one: from 2ᵏ − 1 to k(k+1)/2 + ε. For a 10-agent matched-dose screen, this is a reduction from 1023 free parameters to 55. For a 20-agent screen, from approximately 10⁶ to 210. The framework’s predictive content is precisely this counting reduction: it asserts that bounded compositional pharmacology is structurally low- dimensional, and the dimension is set by the geometry rather than the chemistry. 4.8.3 Falsification by parameter dimension The parameter-counting bound (16) is a falsification criterion. A pharmacological dataset is incompatible with the Aczél-family framework if it requires more than k(k+1)/2 + ε independent structural parameters to describe a bounded compositional response and resists

●Genuine elliptic-class structure: the composition involves bidirectional cross- modulation requiring an elliptic isometry of the multi-agent configuration space. This is the framework’s most ambitious prediction (Section 8, Prediction 6); a bounded dataset best fit by k(k+1)/2 + 1 parameters with the extra parameter manifesting as a closed-orbit signature in the (e_A, e_B, e_combo) surface would constitute confirmation rather than falsification of the framework, and identify a class of biology not currently recognised in synergy taxonomy. Each diagnostic condition has a structurally specific signature that distinguishes it from the others. The framework is therefore falsifiable across three independent directions: parameter dimension exceeds bound (axiom failure or new biology), boundary behaviour inconsistent (bounded-observable failure), and closed-orbit signature in higher-dimensional combination data (elliptic-class confirmation). The mid-range matched-dose discriminating test (Section 5.4, Prediction 1) is the load-bearing two-agent test of whether the framework’s 3-parameter prediction holds on the simplest non-trivial case; the parameter-dimension argument extends this to a falsification map valid across all k. 5. Empirical Grounding 5.1 Structural baseline and the Bliss-Loewe gap Figure 1 shows the three Aczél-family compositions on (0,1) alongside the diagonal ecombo = eA baseline. Two regions are highlighted: the saturated regime (eA = eB > 0.85) where all

effect ecombo as a function of equal individual effects eA = eB = e under Bliss independence (independent mechanisms), Loewe additivity (shared target), and the maximum rule (one agent dominates). The three predictions converge at saturation (e → 1) and at zero effect (e → 0); they are maximally discriminating along the diagonal at the golden-ratio reciprocal e = (√5−1)/2 ≈ 0.618. (B) The Bliss − Loewe gap across the (eA, eB) plane. The gap peaks at eA = eB ≈ 0.618 with magnitude (5√5−11)/2 ≈ 0.090 and vanishes on the boundaries. The yellow star marks the diagonal maximum at the golden-ratio reciprocal; the cyan marker 5.2 Distributed empirical foundation in the mixture toxicology and enzyme kinetics literature The structural taxonomy derived in Sections 3-4 predicts that combined effects must follow either Bliss independence or Loewe additivity, depending on whether agents perturb the system through independent mechanisms or compete for the same rate-limiting step. This is not a novel empirical claim; it is the geometric restatement of a binary classification that has been validated across pharmacology, toxicology, and enzyme kinetics over the past six

addition should be used for independently acting chemicals. This guidance was adopted after exhaustive review of mixture studies across dozens of chemical classes. Kortenkamp et al. [27] reviewed over 90% concordance between mechanism-predicted model and observed mixture effect in well-characterised systems. Backhaus and Faust [28] showed that the independent-action model gives accurate predictions for dissimilarly acting chemicals across hundreds of binary and multi-component mixtures when individual effects lie within the mid- range — the discriminating regime the present framework identifies as optimal. Each mixture study that correctly fits one model and rejects the other is an implicit measurement of the Bliss-Loewe gap.

mechanisms of carcinogenicity in a rat liver foci assay, with individual effect probabilities spanning 0.3–0.7 (their Figure 3 reports the dose-response and combination data). The observed combined effect agreed with Bliss prediction within experimental error, while the Loewe prediction significantly overestimated the response (observed 0.46, Bliss prediction 0.48, Loewe prediction 0.62). Similar independent-action confirmation has been reported for mixtures of dissimilarly acting pesticides on algal growth [32] and for anti-androgenic chemicals with different molecular initiating events [33]. In each case, when individual effect levels were in the discriminating regime, the data clustered tightly around the Bliss surface. The body of evidence summarised in Section 5.2 does not replace the targeted Al-specific discriminating test proposed in Prediction 1 (Section 8); it rather demonstrates that the two composition laws have been individually and repeatedly validated across the entire (0,1) domain, including the mid-range, in their respective mechanistic domains. The present paper's geometric unification explains why these validations succeed and provides the mathematical taxonomy the field has used implicitly. The next step for Al toxicology

Page 24 Loewe testing. Suppression fractions normalised against Al-only versus untreated control are eQ = 0.950, eALA = 0.945, ecombo = 1.007. The framework's predictions, with no fitted parameters: Endpoint Bliss prediction Loewe prediction Observed Lipid peroxidation 0.997 0.973 1.007 Protein carbonyl 0.989 0.948 1.032 Table 2. Saturated-regime test from Al-Otaibi et al. (2018). Both predictions are consistent with observation within experimental noise (gap between Bliss and Loewe is 2.4 and 4.1 percentage points, less than typical assay precision). Computational detail is provided in Supplementary §S4. Both predictions are consistent with the data within experimental noise. The dataset is in the saturated regime where Bliss and Loewe converge; we report the test as consistent with the framework's predictions but non-discriminating between independent and shared-target mechanisms. The observed ecombo > 1.0 in both endpoints reflects the bounded observable's absolute physiological floor sitting below the experimental control baseline — a feature the framework predicts, not a measurement artefact. 5.4 The discriminating mid-range test At eA = eB = 0.5, Bliss predicts ecombo = 0.750 and Loewe predicts 0.667 — a gap of 8.3 percentage points at this conventional EC₅₀ anchor, within the discriminating range of standard assays. The optimal target on the diagonal is at the golden-ratio reciprocal e ≈ 0.618, where the gap reaches its maximum of (5√5−11)/2 ≈ 9.0 percentage points (approximately 8% larger than at e = 0.5); the practical 0.3–0.7 discriminating regime brackets both anchors and either choice supports clean discrimination. Prediction 1 (load-bearing): In an Al-toxicity combination study with two antioxidant agents at matched full doses producing individual suppressions eA and eB each in the range 0.3-0.7, the combined suppression ecombo will lie within ±0.05 of either Bliss prediction (7) or

Murray — Aczél-Family Structure of Pharmacology (v14) Page 25 Loewe prediction (9), and the choice indicates whether the two agents act on independent rate-limiting steps or share a target. Systematic supra-Bliss effects (>0.05 above Bliss) indicate genuine positive interaction. Systematic sub-Loewe effects (<0.05 below Loewe) indicate antagonism beyond simple competition. 6. Reinterpretation of the Al-Toxicity Combination Literature as the dominant interpretive frame and the artefact has been routinely reported as mechanism- level synergy. The geometric framework of the rest of the paper makes the structural baseline against which genuine synergy should be measured rigorous and predictive; the critique below stands on the simpler ground of dose-response curvature alone. 6.1 The dominant study design The Al-toxicity combination literature is large and growing, with hundreds of publications testing whether two natural compounds, antioxidants, or chelators jointly mitigate Al-induced

Two designs allow clean discrimination: Matched full-dose design. Test agents A and B at full doses individually and at full- dose A combined with full-dose B. The prediction is direct: combined ecombo against Bliss (7) or Loewe (9). Discrimination requires individual effects in the mid-range (0.3-0.7); studies should choose dose levels that achieve this. Full dose-response design. Measure full Hill curves under each individual treatment at multiple doses and combination effects at multiple matched dose ratios. Deviations from the Aczél predictions at multiple dose pairs constitute genuine mechanism-level synergy or antagonism. This is standard practice in oncology combination drug studies [35] but is rarely applied in natural-products toxicology. The half-dose-mix design supports neither test and should be retired as primary evidence for combination interaction.

statistical inference is not powered, and the trend should be reported as preliminary. The qualitative observation does survive and is sign-conclusive: the cross-kingdom slope is opposite in sign to a free-Al3+-only speciation rescue (which predicts +2.24 from hydrolysis alone [36]). This rules out a pure speciation account of cross-kingdom universality. The quantitative magnitude is dominated by a kingdom-specific [SOD] confound (plant cytoplasmic [SOD] ≈ 1 μM vs mammalian ≈ 10-40 μM contributes −0.34 to −0.55 per pH unit) that, once removed, leaves a residual consistent with the Gouy-Chapman-Stern model of reported in Supplementary §S1; we treat it as motivating rather than validating the architecture. 8. Predictions and Falsification The framework yields six testable predictions.

Murray — Aczél-Family Structure of Pharmacology (v14) Page 29 Prediction 1 (load-bearing, repeated from Section 5.4). Bliss-vs-Loewe discrimination at matched mid-range doses. Confirmed by: matched-dose combination data with individual effects in 0.3-0.7 range falling within ±0.05 of one of the two predictions across multiple dose pairs. Falsified by: systematic deviation (>0.10) at multiple matched dose pairs that cannot be attributed to mechanism overlap, axiom-violation modes (Section 3.3), or measurement error. Prediction 2. log(ECx) shifts are identical for all x under constant-mechanism intervention. Any intervention that shifts the rapidity of an Al-toxicity dose-response by Δ should produce equal shifts in log(EC10), log(EC50), log(EC90). Falsified by: systematic differences in shifts across x. Prediction 3. Schild slope = 1 for true competitive Al-binding antagonists. Compounds that bind Al3+ competitively with its superoxide-binding site should show Schild plots with slope 1.0 ± 0.1. Falsified by: systematic Schild slope deviation indicating non-competitive mechanism. Prediction 4. Hill-form dose-response with measurable nH. Al3+-induced oxidative damage in cell-free or cellular systems with controlled SOD activity should follow a Hill curve when plotted against log[Al3+], with effective coefficient nH structurally diagnostic of the competition mechanism. Falsified by: dose-response inconsistent with any bounded sigmoidal form across at least two orders of magnitude. Prediction 5. Within-system intervention test. When pH and Al speciation are held constant and baseline SOD activity is independently manipulated, the damage threshold [Al3+]threshold should scale linearly with baseline [SOD]local, with slope 1 on log-log axes (0.8-1.2 across at least one order of magnitude). Falsified by: systematic deviation from unit slope.

Murray — Aczél-Family Structure of Pharmacology (v14) Page 30 Prediction 6 (elliptic signature — exploratory). The Bliss and Loewe compositions are both parabolic isometries of the underlying compositional structure — flows that translate the rapidity without rotation. A third class of composition, corresponding to elliptic isometries (mutual bidirectional modulation between agents), would produce a qualitatively distinct observable signature: for a fixed dose ratio of agents A and B, the combined effect ecombo as a function of total dose would exhibit a local interior maximum — a dose level beyond which increasing either agent's dose reduces the combined effect. This non-monotonic response surface is not predicted by any standard synergy model and constitutes a sharp, falsifiable signature of elliptic composition. Experimental detection requires full dose-response surfaces and monotonicity testing across at least two orders of magnitude in total dose. No confirmed example in the Al-toxicity literature is known to the author; the prediction is exploratory but falsifiable in any system where mutual modulation is suspected. 9. Implications 9.1 For combination pharmacology The framework provides a geometric foundation for combination effect analysis that complements the practical guidance long operationalised in mixture toxicology (e.g. EPA

The two-stage architecture (Section 7) implies that intervention efficacy depends on which stage limits damage. At neutral pH with low free Al3+, Stage 1 (membrane gate) is rate- limiting; chelators with neutral-pH affinity and pH-modulating interventions are predicted to be more effective than SOD modulation. In acidic compartments, plaque-core microenvironments, or pathological Al overload, Stage 2 (SOD competition) becomes rate- limiting; SOD-supporting interventions and antioxidants targeting downstream Fenton chemistry are predicted to dominate. The framework supplies a conditional intervention roadmap. 10. Limitations The framework's load-bearing empirical claim is Prediction 1. Confirmation in the saturated regime (Al-Otaibi 2018, Section 5.3) is consistent with both predictions but does not

discriminate them. The discriminating mid-range test has not been performed in Al toxicology, and its outcome is the principal open question. The paper is best characterised as a hypothesis paper in the sense of Elsevier’s editorial taxonomy: it advances new structural and methodological claims based primarily on previously published data, findings, and a strong line of mathematical reasoning, with explicit testable predictions but without new direct experimental confirmation of its load-bearing prediction. The framework’s empirical status awaits the discriminating mid-range matched-dose test, which is implementable in standard Al-toxicity infrastructure (Section 6.3). The kAl bound from Section 2.4 is order-of-magnitude only and rests on a single null result whose sensitivity to substrate-level competition has not been formally characterised. Specific The Aczél-theorem framing rests on axioms that hold approximately rather than exactly in real biological systems. Section 3.3 catalogues the principal failure modes; their occurrence is diagnostic and predicted by the framework, but they limit the precision with which the structural derivation applies to specific experimental data. In particular, feedback regulation on the perturbation timescale is a known violation of associativity in plant systems and restricts the framework's clean predictions to short exposures. The Hill-family unification (Section 4.1) recognises that four equations are mathematically identical; their independent derivations are mechanistically distinct. The structural identity does not eliminate the mechanistic distinctions — cooperative binding, surface adsorption, enzyme catalysis, and acid-base equilibrium are different biological processes — only the