Mechanistic hormesis

### PDF page 13 before damage dominates. In that scoped case, the dose-response cannot remain linear and harmful down to arbitrarily low dose, because the repair term initially contributes a positive rapidity increment before the damage term overtakes it. This conclusion applies specifically to single adaptive endpoints such as SOD activity, cell viability, DNA repair capacity, or stress-protein-mediated survival under the independence conditions stated in Section 3.2. It does not by itself invalidate every regulatory use of LNT, especially for multistage carcinogenesis, population-level precautionary policy, or endpoints in which adaptive repair is absent, delayed, or inseparable from damage. The claim is narrower and stronger: for bounded adaptive endpoints satisfying the stated premises, LNT is excluded by the composition law. 7.6 Implications for pharmaceutical dosing The model predicts that the beneficial-effect peak for agents acting through adaptive pathways occurs at approximately 2 × Da, not the maximum tolerated dose. This is consistent with evidence for metronomic chemotherapy outperforming maximum-dose regimens 28 and with the biphasic dose-response of many chemotherapeutic agents 8,29. 8. Conclusion Hormesis is a geometric necessity of bounded adaptive endpoints under the condition that repair and damage compose independently on the bounded viability interval. The derivation proceeds from two biological premises (finite repair capacity and adaptive upregulation), one scope condition (independent composition of repair and damage over the low-to-moderate dose range), one theorem (Aczél’s uniqueness of bounded composition), and one established result (the summation theorem of metabolic control analysis). No substance-specific curve shape is assumed. No parameters are fitted to hormetic data. Each parameter — Da, Dt, η — is either measured independently or derived from pathway architecture. The five quantitative predictions — peak dose location (≈ 2 × Da), zone width (scaling with Dt/Da), peak amplitude (determined by Amax,mol / η), repair-capacity dependence of zone width, and pathway-specific η — are each determined by independently measurable biological quantities. The pathway-specific derivation of η makes a novel prediction: agents acting through multi-target effectors should show systematically larger hormetic peaks than agents acting through linear pathways. This is consistent with the observed distribution in the Calabrese database. Validation comprised two parts: aggregate agreement with the Calabrese database of more than 10,000 responses — peak-amplitude range, mean amplitude, and zone width — without parameter fitting, and confirmation across three mechanistically distinct agents, using independently published data, that adaptive activation precedes toxicity, the structural condition for hormesis. Pathway-specific attenuation distinguished linear NRF2-mediated responses (130–160%) from larger multi-target heat-shock responses (200–300%); the peak-location law (peak at approximately twice the repair-activation dose) is presented as a falsifiable prediction, and the multi-target amplitude case as a testable architectural extension rather than part of the core proof. The result is a precise conditional claim: bounded adaptive composition forces hormesis when repair activation precedes toxicity and high-dose damage eventually exceeds bounded repair. ### PDF page 14 Appendix: Numerical Verification A.1 Exact model vs. product approximation The exact hyperbolic composition model (Equation 2) and the product approximation (Equation 3) were evaluated numerically across the parameter space Da = 2–20, Dt/Da = 5–20, Amax = 0.3–0.8. Both models produce biphasic dose-response curves under all parameter combinations tested. The product form approximates the exact model with less than 10% error in peak location and zone width throughout the biologically plausible parameter range. Where the two differ, the product form is conservative: it yields slightly lower amplitudes because it omits higher-order hyperbolic composition terms. The applied calculations therefore do not contradict the fundamental theory; they give closed-form, lower-bound estimates of the exact model’s predictions. A.2 Peak location derivation The peak dose ratio d/Da was computed numerically by setting dR/dd = 0 for the product form across 36 parameter combinations (4 values of Da × 3 values of Dt/Da × 3 values of Amax). Results: d/Da ranges from 1.69 (Dt/Da = 5, Amax = 0.3) to 2.43 (Dt/Da = 20, Amax = 0.8). The dependence on Amax is weak; the primary determinant is Dt/Da. A.3 η from the summation theorem The flux control coefficient Cᵢ = 1/n for each of n equivalent rate-limiting enzymes (summation theorem). Doubling one enzyme (Amax,mol = 1.0) increases flux by Cᵢ = 1/n, giving Amax,func = 1/n and η = n. For n = 2: η = 2. For n = 3: η = 3. Verified numerically for molecular fold-changes of 1.5–3.0× and pathway sizes n = 2–5. For multi-target effectors acting on m survival nodes through pathways averaging navg steps each: the effective control coefficient is predicted (by heuristic extension of the summation theorem) to be higher than for a single linear pathway, yielding a lower η. For HSP70 acting on 3–4 nodes through ~2-step pathways: the predicted η ≈ 1.3–1.5. This predicts Amax,func ≈ 0.67–0.77 for a 2-fold molecular upregulation, yielding peak amplitudes of 167–177% — substantially above the linear-pathway prediction of 133–150% and closer to the observed 200–300% for heat shock. The remaining gap may reflect the product-form approximation and non-equilibrium effects not captured by the steady-state summation theorem.

Hormesis as a Geometric Necessity of Bounded Adaptive Systems: Quantitative Predictions from First Principles Daniel John Murray Abstract Objectives: To determine whether hormesis can be derived from the geometry of bounded adaptive biological endpoints, and whether the derivation yields quantitative predictions independent of curve fitting. Methods: Bounded endpoints were represented in rapidity coordinates using Aczél’s arctanh linearisation for associative composition. Repair activation and damage accumulation were modelled as opposing bounded rapidity increments with independently measurable thresholds. The molecular-to-functional attenuation factor was derived for linear pathways from the metabolic-control summation theorem and separated from a testable multi-target extension. Predictions were compared with published H₂O₂, CdCl₂, and heat-shock dose-response data. Results: The model predicts a biphasic response whenever repair activation precedes toxicity (Da < Dt) and high-dose damage ultimately exceeds bounded repair capacity. The exact hyperbolic model is the primary model; the product form is used only as a conservative analytical approximation. Aggregate predictions — peak-amplitude range, mean amplitude, and hormetic-zone width — matched the Calabrese hormesis database of more than 10,000 responses without parameter fitting; across three mechanistically distinct agents, independently published data confirmed the structural prediction that adaptive activation precedes toxicity, and pathway-specific attenuation distinguished linear NRF2- mediated responses (130–160%) from larger multi-target heat-shock responses (200–300%). The peak- location law is presented as a falsifiable prediction. Conclusion: Within the stated scope of independently composing bounded adaptive endpoints, hormesis follows from finite repair capacity plus adaptive upregulation. The framework does not claim that all biological history is axiomatic; it identifies the conditional geometry that constrains dose-response shape once a bounded adaptive endpoint is specified.

induction for thermal stress 6, DNA repair pathway activation for genotoxic agents 7 — but the question of why mechanistically distinct responses produce quantitatively similar features remains open. The Linear No-Threshold (LNT) model, which dominates regulatory toxicology, predicts proportional harm at all doses and is, for bounded adaptive endpoints, inconsistent with the low-dose stimulation seen in hormetic data 8,9. This paper derives hormesis from first principles. The derivation is structured as follows: Section 2 establishes the mathematical framework (Aczél’s theorem applied to bounded biological observables). Section 3 derives the hormetic response as a necessary consequence of this framework. Section 4 derives the molecular-to-functional attenuation factor from metabolic control analysis. Section 5 presents five quantitative predictions with analytical expressions. Section 6 validates against published data. Section 7 discusses implications. Scope of the claim: The derivation is conditional, not universal in the unrestricted biological sense. It applies to bounded adaptive endpoints in which repair and damage contribute independently to the same measured state over the low-to-moderate dose range. Biological variability is therefore not ignored; it

then-saturating functions whose midpoints are separated — with the activating function rising at the lower dose and contributing sufficient low-dose amplitude — is biphasic and rises above the control level over a low-dose interval. The logistic is used for quantitative predictions because it is the canonical form under Aczél’s theorem with linear dose-rapidity mapping, but the qualitative result (biphasic shape, hormetic zone) is robust to relaxation of the linearity assumption. Even if the dose-rapidity relationship is mildly nonlinear, producing a sigmoid that deviates from the logistic, the biphasic structure persists. 2.4 The steepness parameter is constrained

transcription vs. direct oxidative damage to lipids) and this scope condition is satisfied. At very high doses, where molecular systems are simultaneously overwhelmed and degraded, the independence breaks down; but this regime is beyond the hormetic zone and does not affect the predictions of interest. Scope condition (stated explicitly): The rapidity-addition model applies when the repair and damage pathways compose independently on the bounded viability interval — i.e., when the rapidity increment contributed by repair does not depend on the current damage load, and vice versa. This condition is

where Amax is the fractional increase in protective capacity and σ is the logistic function (Section 2.3). Equation 3 is not used as the proof of hormesis; the proof is the rapidity-space result in Equations 1–2. The approximation is used for closed-form predictions, and the exact model is used as the reference model for numerical verification. Thus any discrepancy between Equation 2 and Equation 3 affects numerical precision, not the existence of the biphasic response. 4. Derivation of the Molecular-to-Functional Attenuation Factor 4.1 The measurement problem

nodes, the effective attenuation should be smaller than in a single linear pathway because one molecular increase contributes to multiple functional outputs. This pathway-dependent η makes a specific prediction that is not required for the core proof of biphasic shape: hormetic peak amplitude should be larger for agents whose protection is mediated by multi-target effectors than for agents acting through single linear pathways. This prediction can be tested directly by measuring molecular upregulation and functional protection in the same experimental system before fitting any hormetic curve. 4.3 Numerical verification For a 2-enzyme linear pathway (n = 2, C = 0.5): doubling one enzyme (molecular fold = 2×, Amax,mol = 1.0) 0.33). η = 3.0. For a multi-target effector acting on three survival nodes through approximately two-step pathways, the predicted effective control is higher than for one two- or three-step linear pathway, giving η ≈ 1.3–1.5. This numerical estimate is an explicit hypothesis about architecture, not an additional axiom. It is included because it explains why heat-shock responses can exceed the usual 130–160% chemical-stressor range and because it provides a falsifiable test of the model. 4.4 Pathway-specific η values for validation agents Agent Protective pathway Architecture Predicted η H₂O₂ NRF2 → NQO1/SOD/GPx Linear, 2–3 steps 2.0–3.0 CdCl₂ NRF2 → NQO1/HO-1 Linear, 2–3 steps 2.0–3.0 membranes + anti- apoptosis Multi-target, 3–4 nodes 1.3–1.5 (predicted) For linear pathways, η values are derived from the summation theorem. For multi-target effectors, η values are predicted by architecture and must be independently tested by measuring both molecular upregulation and functional protection in the same system. The core geometric predictions of peak location and zone width do not depend on this multi-target extension. 5. Quantitative Predictions 5.1 Analytical derivation of peak dose For analytical transparency, the peak of the product approximation (Equation 3) occurs where dR/dd = 0: A · σ’(d, Da) · [1 − σ(d*, Dt)] = [1 + A · σ(d*, Da)] · σ’(d, Dt) … (6) where σ’(d, D) = s · σ(d, D) · [1 − σ(d, D)] is the logistic derivative.

Numerical evaluation of the exact hyperbolic model and the product approximation across the biologically plausible parameter space (Dt/Da = 5–20, Amax = 0.3–0.8) yields the same peak-order prediction: d* / Da ≈ 1.7 to 2.4 … (7) The ratio increases with Dt/Da (wider threshold separation pushes the peak rightward because damage engages later) and with Amax (stronger repair extends the beneficial zone). For the typical range Dt/Da = 10–20, the peak sits at approximately 2.0–2.4 × Da. Prediction 1: For any substance showing hormesis, the peak dose should correlate with the independently measured repair activation EC50, with a proportionality constant of approximately 2. 5.2 Analytical derivation of zone width The hormetic zone boundaries occur where Response = 1, i.e., where (1 + A · σ(d, Da)) · (1 − σ(d, Dt)) = 1. The lower boundary dlo occurs at low dose where repair is just beginning to exceed unity; the upper Numerical evaluation yields zone widths (dhi/dlo) of 8–80 fold across the biologically plausible parameter space, encompassing and extending the Calabrese database range of 5–50 fold. Prediction 2: The zone width should scale approximately with the ratio Dt/Da. 5.3 Peak amplitude From Section 4: Peak amplitude ≈ (1 + Amax,mol / η) × 100% of control … (8) where η ≈ n (number of rate-limiting enzymes in the protective pathway). For Amax,mol = 0.8–1.5 and η = 2–3: peak amplitude = 127–175% of control. Mean ≈ 145%. Prediction 3: The peak amplitude is determined by the functional repair capacity Amax,func = Amax,mol / η, not by the molecular fold-change alone. 5.4 Repair capacity determines zone width Prediction 4: Cell types with higher Amax,func should show wider hormetic zones. Testable by comparing NRF2-competent versus NRF2-knockout cells, or young versus aged cells. 5.5 Molecular-to-functional attenuation is pathway-determined Prediction 5: The ratio η = Amax,mol / Amax,func is determined by the architecture of the protective pathway, not by the stressor. Specifically, η ≈ n for linear pathways with n rate-limiting steps, and η < n for multi- target effectors acting on m survival nodes (where the effective η ≈ n/m). This is testable by independently measuring both molecular upregulation and functional protection in the same system. The prediction distinguishes between linear-pathway agents (η ≈ 2–3, peak amplitude 130–160%) and multi- target agents (η ≈ 1.3–1.5, peak amplitude 170–200%+).

6. Validation Against Published Data 6.1 Aggregate agreement with the hormesis database Feature Model prediction Calabrese database (>10,000 responses) Peak amplitude range 127–175% 130–160% typical, up to 200% across a low-micromolar range, yet the associated functional stimulation peaks only modestly before toxicity engages at higher concentrations16. The molecular and functional scales are typically measured in separate studies and treated as unrelated; the present framework predicts that they are not, and specifies how they are connected. The summation theorem of metabolic control analysis (Section 4) accounts for this constraint. Because the flux-control coefficients of the enzymes in a pathway sum to unity (ΣCi = 1), a multi-fold increase in any single effector is compressed into a fractional change in pathway flux by a factor η ≈ n, where n is the parameters from primary studies that measured molecular activation and toxicity independently of any hormetic-curve fitting. Because matched, fine-grained dose-response data within the hormetic window are not uniformly available for these systems, the agent-level analysis tests the structural prediction — the

ordering Da < Dt and the resulting biphasic separation — rather than agent-specific numerical peak amplitudes; the sharper quantitative claim (peak ≈ 2 × Da) is treated as a falsifiable prediction (Section 5.1). Test 1: CdCl₂ in HepG2 hepatocytes (NRF2 pathway). Zhu et al.18 measured NRF2 activation and cytotoxicity in the same system. NRF2 protein rose from a control level of 0.60 to a plateau near 2.24 (relative units), with half-maximal activation between 2 and 5 μM (Da ≈ 2–3 μM); the downstream NRF2 metabolic-viability endpoint the independent-composition scope condition (Section 3.2) is not satisfied and no resolved viability peak is expected — consistent with the MTT data. Hormesis nonetheless re- emerges in the proliferation endpoint, where that condition holds. The framework therefore predicts not merely that hormesis occurs, but in which endpoint it appears and in which it does not — a discriminating prediction the data bear out. Test 2: H₂O₂ in human keratinocytes (NRF2 pathway). Yokoo et al.20 reported a hormetic functional response in NHEK-F keratinocytes: 20 μM H₂O₂, which was non-cytotoxic, extended replicative lifespan to ~160% of control, and 60 μM, which was marginally cytotoxic, to ~120%; in the same keratinocyte lineage, Han et al. applied 0.3 mM H₂O₂ as an oxidative-stress challenge in HaCaT cells21. The location of the stimulatory peak (≤ 20 μM), the graded decline toward the cytotoxic range, and the wide separation between stimulatory and toxic doses are consistent with the predicted biphasic geometry. This system is treated as a structural and falsification test rather than a quantitative anchor for the peak-location law, because same-system NRF2-activation EC50 data are not available for the keratinocyte H₂O₂ model: H₂O₂ is conventionally applied as an acute bolus challenge rather than titrated as an adaptive-pathway inducer. A direct test of the peak ≈ 2 × Da prediction requires the H₂O₂ NRF2-activation EC50 in the same keratinocyte model, which is identified here as a specific experimental target (Section 5.1). Test 3: Heat shock in human fibroblasts (HSP70 pathway). Repeated mild heat stress (41°C) produces well-documented hormetic, anti-aging effects in human skin fibroblasts, whereas severe heat stress is damaging6,22. Demirovic et al.23 showed that the dominant adaptive controller HSF1 translocates to the nucleus substantially more strongly under mild than under severe stress in young cells (~5-fold), confirming that adaptive activation is concentrated in the mild, sub-toxic regime, consistent with Da < Dt. As predicted for a multi-target effector (η ≈ 1.3–1.5; Section 4.4), heat-shock hormesis exhibits the largest peak amplitudes of the three systems (200–300%). 6.4 Summary Agent / system Real data source Structural prediction (Da < Dt; biphasic) H₂O₂ / keratinocytes Yokoo (2004); Han (2023) Supported CdCl₂ / HepG2 (NRF2) Zhu (2017); Niture (2023) Supported Heat shock / fibroblasts Demirovic (2014); Rattan (2009)

Across all three agents, the structural prediction — adaptive activation below the toxicity threshold, producing a stimulatory response separated from overt toxicity — is supported by real, independently published data (Table 4). The aggregate quantitative predictions (peak-amplitude range, mean amplitude, and zone width) match the database of more than 10,000 responses without parameter fitting (Section 6.1). The predicted class separation in amplitude is consistent with the compiled evidence: linear NRF2- mediated pathways cluster at 130–160% (η ≈ 2–3), whereas the multi-target HSP70-mediated response reaches 200–300% (η ≈ 1.3–1.5). This class separation between single-pathway and multi-target effectors — rather than an agent-by-agent numerical match — is the substantive amplitude result. The framework’s sharper quantitative claim, that the stimulatory peak occurs at approximately 2 × Da, is presented as a falsifiable prediction (Section 5.1) rather than a completed agent-by-agent validation. Testing it requires dose-resolved measurement of Da (the adaptive-marker EC50) and of the functional peak in the same system — a dedicated experiment that the datasets compiled here were not designed to provide. The validation claimed here is therefore threefold: aggregate agreement with the hormesis database (Section 6.1), the framework’s account of the molecular-to-functional attenuation that constrains

enzyme upregulation to 1/n of the molecular fold-change, with n ≈ 2–3 for typical linear pathways. Together, these yield functional Amax values of 0.3–0.6 and peak amplitudes of 130–160%. The model also predicts that agents acting through multi-target effectors (e.g., HSP70, certain growth factors) should show HIGHER peak amplitudes (170–300%), because the pathway-specific η is smaller for multi-target action. The Calabrese database is consistent with this: while 80% of hormetic responses show peaks below 200% (consistent with linear-pathway dominance), a significant minority show peaks of 200–400%, which the present model attributes to multi-target effector mechanisms with low η. This bimodal distribution of amplitudes — most clustered at 130–160%, a tail extending to 200%+ — is a novel prediction that can be tested by classifying agents according to the architecture of their protective pathways. 7.4 Morphological adaptation and the adaptation-pathology threshold The same logic applies to morphologically expressed adaptation. Hyperplasia, hypertrophy, and atrophy are not merely descriptive pathology terms; they are tissue-level examples of bounded adaptive to it: dysregulated growth represents a case in which adaptive proliferative machinery, genomic instability, apoptosis evasion27, and tissue constraints no longer compose as a stable bounded repair response. This addition clarifies why the model predicts ordinary adaptive morphology before breakdown while not claiming that multistage cancer risk is reducible to a single hormetic endpoint. These morphological responses are instructive precisely because they are multi-target in the sense of Section 7.3, and the framework therefore makes a specific qualitative prediction about them. Compensatory hyperplasia (for example, the proliferative response of hepatocytes to partial hepatectomy or of epithelium to chronic irritation), physiological hypertrophy (such as load-induced enlargement of cardiac or skeletal muscle), and adaptive atrophy (such as disuse- or nutrient-limitation–driven reduction of tissue mass) are each governed not by a single linear enzyme chain but by the joint action of cytoskeletal organisation, mechanotransduction, mitotic control, and apoptosis. Because control over the measured endpoint is distributed across many contributing nodes, the pathway-specific attenuation factor η is small, and the amplitude argument of Section 7.3 predicts that such morphologically expressed adaptations should occupy the higher-amplitude regime — a larger adaptive reserve between baseline and the adaptive ceiling — than adaptations mediated by a single linear pathway. The adaptation–pathology threshold is then reached when the distributed adaptive capacity can no longer raise the protective rapidity faster than the damage or dysregulation it offsets: at that boundary, ordinarily reversible hyperplasia,

satisfying the stated premises, LNT is excluded by the composition law. 7.6 Implications for pharmaceutical dosing The model predicts that the beneficial-effect peak for agents acting through adaptive pathways occurs at approximately 2 × Da, not the maximum tolerated dose. This is consistent with evidence for metronomic chemotherapy outperforming maximum-dose regimens 28 and with the biphasic dose-response of many chemotherapeutic agents 8,29. 8. Conclusion data. Each parameter — Da, Dt, η — is either measured independently or derived from pathway architecture. The five quantitative predictions — peak dose location (≈ 2 × Da), zone width (scaling with Dt/Da), peak amplitude (determined by Amax,mol / η), repair-capacity dependence of zone width, and pathway-specific η — are each determined by independently measurable biological quantities. The pathway-specific derivation of η makes a novel prediction: agents acting through multi-target effectors should show systematically larger hormetic peaks than agents acting through linear pathways. This is consistent with the observed distribution in the Calabrese database. Validation comprised two parts: aggregate agreement with the Calabrese database of more than 10,000 responses — peak-amplitude range, mean amplitude, and zone width — without parameter fitting, and distinguished linear NRF2-mediated responses (130–160%) from larger multi-target heat-shock responses (200–300%); the peak-location law (peak at approximately twice the repair-activation dose) is presented as a falsifiable prediction, and the multi-target amplitude case as a testable architectural extension rather than part of the core proof. The result is a precise conditional claim: bounded adaptive composition forces hormesis when repair activation precedes toxicity and high-dose damage eventually exceeds bounded repair.