Unified architecture perspective

### PDF page 11 Murray — Architectural Classification of Bounded Adaptive Systems | page 11 of 14 dissociation). The larger body of structural reframings recasts established biology in the architectural vocabulary; these four represent the programme's primary empirical commitments going forward. Mode 4c (setpoint miscalibration) remains the most operationally provisional element: its calibration tolerance δ depends on cohort regressions that have not yet been performed for any biological system [19, §6.2.3]. 7. Relation to the Historical Pattern The structural analogue is Klein's 1872 Erlangen Programme [25]. Klein observed that the apparent multiplicity of mid-nineteenth-century geometries — Euclidean, projective, hyperbolic, elliptic, affine — was a single classification problem: each geometry corresponds to the invariants of a specific transformation group acting on the underlying space. The Erlangen Programme did not derive new geometric theorems; it made explicit a structural fact that the existing theorems had collectively established but none had individually stated. This perspective makes an analogous claim. The apparent multiplicity of biphasic and U-shaped phenomena in bounded adaptive biology is a single classification problem: each is a surface projection of one of the architectural mode-types onto a measurement axis. The classification is exhaustive within stated scope; the empirical anchors are independent across the six companion papers; the falsification criteria are sharp. 8. Conclusion Bounded adaptive biological systems admit a finite architectural classification: the Möbius classification of admissible composition laws on bounded observables, and the BAS-S classification of admissible failure topologies in systems with feedback regulation and parallel cell-fate surveillance. The two classifications are independently established in six companion papers, combine structurally through the natural composition of bounded observable and architectural role, and together exhaust the structural content of the recurring biphasic and U-shaped phenomena across toxicology, oncology, cardiology, endocrinology, nutrition, and ageing. The scope conditions are precise: single bounded primary observable, identifiable slow controller, identifiable parallel surveillance branch, equilibration faster than dosing. Within this scope, the classification is exhaustive and falsifiable. Outside this scope, the mathematical extensions required are identified but not undertaken here. The synthesis introduces no new mathematics, no new calibration, and no new prediction. Its function is to state explicitly the architectural claim that the six companion papers collectively establish but none individually asserts. The claim is that biology builds robust adaptive systems by coupling a fast bounded plant to a slow internal-model controller, then bolting on an independent surveillance layer that shares the same input signal — and that this architectural template, once fixed, admits a finite, classifiable, ### PDF page 12 Murray — Architectural Classification of Bounded Adaptive Systems | page 12 of 14 falsifiable set of composition laws and failure modes. The six companion papers provide the derivations; this perspective provides the architecture. Whether the framework persists at the level of structural impact suggested by the Erlangen Programme analogy depends on empirical validation across the candidate systems identified — diabetes, hypertension, immune tolerance, bone remodelling, ER proteostasis, neural homeostasis — and on the outcome of the pre-registered Phase II NSCLC prediction. The framework is offered with scope conditions, falsification criteria, and empirical commitments stated openly. Its empirical status is measurable rather than rhetorical, and its development from here is a matter of empirical work rather than further theoretical elaboration. Appendix A. Citation Map: Tracing Synthesis Claims to Companion Papers This appendix provides the explicit map from each structural claim in the main text to its source, supporting the commitment stated in §1: the synthesis introduces no new results. Synthesis claim Source Aczél's representation on bounded observables P1 [14, §2]; Aczél [17] Exhaustive five-flow Möbius classification P1 [14, §2.4, Appendix A] σ-duality acting on the five flows P1 [14, §3.6] Bliss–Loewe gap closed form and golden-ratio location P1 [14, Appendix C]; P2 [15, §4.2] Unification of eight pharmacological equations P2 [15, §4, Table 1] Hyperbolic Fisher-information metric on (0,1) P2 [15, §4.7–4.8] Half-dose-mix methodological correction P2 [15, §6] Derivation of hormesis as geometric necessity P3 [16, §3] Kacser–Burns summation theorem for attenuation factor η P3 [16, §4] 6/6 hormesis validation P3 [16, §6] Calibrated three-variable redox model P4 [18, §2] Dual failure modes (oxidative collapse + reductive fade) P4 [18, §2.2–2.3] WHO G6PD recovery without fitting P4 [18, §3] Pre-registered Phase II NSCLC prediction with 8-oxo-dG biomarker P4 [18, §5] BAS-S four-role decomposition (R1–R4) P5 [19, §2.1] 2+2 asymmetry of component failures vs regime conditions P5 [19, §2.9] Mode 1 closed-form threshold β_c P5 [19, Theorem 1]

Aczél-family structure; (3) the derivation of hormesis as a geometric necessity of bounded adaptive repair; (4) the calibrated dual-failure-mode redox model that recovers G6PD contraindications without parameter fitting and pre-registers a Phase II NSCLC trial prediction; (5) the four-role plant–sensor– controller–surveillance architecture with block-diagonal threshold Jacobian; and (6) the cross-domain portability of this architecture to calcium handling in cardiomyocytes and to candidate systems in immunology, bone biology, and neural homeostasis. The synthesis introduces no new mathematics, no new calibration, and no new prediction. Its function is to state explicitly the architectural claim that the six companion papers collectively establish but none individually asserts, and to make legible the framing under which the recurring biphasic and U-shaped phenomena across toxicology, oncology, cardiology, endocrinology, nutrition, and ageing are recognised as surface projections of a single underlying geometric object.

transformation. The physical content is that bounded associative composition is mathematically rigid: given the axioms, the rapidity is forced. The Möbius hypothesis. Aczél's theorem alone does not specify which rapidity applies. The additional empirical hypothesis, motivated by the rational form of mass-action equilibrium dose-response curves and verified case-by-case for four canonical two-agent mechanisms via explicit mass-action derivation [14, Appendix F], is that the composition extends to a smooth one-parameter subgroup of PSL(2,ℝ) acting on ℝP¹, preserving [0,1] under semigroup action. The classification theorem. Under these conditions, the boundary-preserving one-parameter Möbius

theorem. Figure 1. The five Möbius composition laws on the bounded interval. Panel A: the five matched-diagonal predictions across e ∈ (0,1), converging at the boundaries and maximally discriminating in the mid-range; the vertical bar at e* = (√5−1)/2 ≈ 0.618 marks where the Bliss–Loewe gap is maximised. Panel B: the σ- duality involution, with Loewe ↔ Inverse-odds and Bliss ↔ Multiplicative as two dual pairs and Logit as the unique self-dual flow. Panel C: the fixed-point configurations on ℝP¹ for each flow; parabolic flows have a single fixed point on [0,1], hyperbolic flows have two, one of which may lie at infinity.

2 Regime condition Surveillance (R4) Operating region: Φ* < Φ_surv KEAP1–NSCLC prediction [18, §5] 3 Regime condition Sensor (R2) Rank/fidelity degradation SASP decoupling; pseudohypoxia [19, §5.3] 4 Component failure Controller (R3) Bandwidth, pinning, Cross-domain portability. The BAS-S architecture is not redox-specific. A second worked instantiation in cardiomyocyte calcium handling [19, §15.1] maps R1–R4 onto the Ca²⁺/calcineurin/NFAT system and demonstrates that the framework's most distinctive structural prediction — Mode 4b controller pinning → downstream chronic phenotype, rescued by controller unpinning — has a direct empirical parallel in calcineurin-transgenic dilated cardiomyopathy rescued by calcineurin/NFAT inhibition [23,24]. Candidate systems in immune tolerance, bone remodelling, ER proteostasis, and neural firing-rate homeostasis are identified as forward research [19, §15.2].

loop DDR via p53/ATM Mode 1 WHO G6PD contraindications [18, §3] Redox / KEAP1– healthy → Mode 1 6/6 predictions confirmed [16, §6] Antioxidant prevention trials

Table 4. The six companion papers and their architectural roles. Paper Manuscript ID Journal (under review) Architectural contribution P1 JTB-D-26-00695 Journal of Theoretical Biology Exhaustive five-flow Möbius classification; σ-duality; Bliss– Loewe golden-ratio gap; cross-class order-dependence prediction [14] P2 ARRES-D-26-00060 Advances in Redox Research Aczél-family unification of eight pharmacological equations; hyperbolic metric on (0,1); half-dose-mix methodological correction [15] P3 26-074-DR Dose-Response Derivation of hormesis as geometric necessity; Kacser– Burns summation theorem for attenuation factor η; 6/6 validation [16] P4 RBC-D-26-00017 Redox Biochemistry and Chemistry Calibrated dual-failure-mode redox model; G6PD recovery without fitting; pre-registered Phase II NSCLC prediction with 8-oxo-dG biomarker [18]

Murray — Architectural Classification of Bounded Adaptive Systems | page 10 of 14 Paper Manuscript ID Journal (under review) Architectural contribution P5 26-100-DR Dose-Response BAS-S architectural grammar; four-mode partition with 2+2 asymmetry; block-diagonal threshold Jacobian; six biological corollaries; calcium/cardiomyocyte portability [19] P6 (this perspective) Physics of Life Reviews (suggested) Capstone synthesis; architectural claim stated explicitly; Erlangen Programme framing; scope conditions and falsification axes [this paper] The papers are scientifically independent — each supplies its own empirical anchors and stands on its own derivations — but architecturally interdependent: P1 and P2 establish the Möbius classification; P3 derives its dose-axis projection; P4 and P5 establish the BAS-S classification with its dual failure modes and cross-domain reach; P6 states the combined architectural claim. 6. Scope Conditions and Falsification The framework's claims are precisely scoped. The Möbius classification applies to single bounded observables under boundary-preserving semigroup action with strict monotonicity; multivariate compositional data, function-valued projections, and bounded ratios that are not primitive observables require mathematical extension. The BAS-S classification applies to systems instantiating R1–R4 with prescribed information flow; systems lacking one or more roles are outside the class for the corresponding mode-types. The framework is falsifiable on five independent axes: 1.Möbius classification failure. A biological system within scope realising a composition law not among the five flows of Table 1. 2.Architectural grammar failure. An in-class BAS-S exhibiting fewer than four threshold-typed failure modes, or a phenotype that cannot be located in any of R1–R4. 3.Bliss–Loewe discriminating test failure. Matched-dose combination data at e* ≈ 0.618 falling systematically far from both Bliss and Loewe in a system within scope. 4.Forward prediction failure. Any of the novel predictions P11, P13, P14, P17 [19, §11] failing on properly designed empirical test. 5.Block-diagonality failure. Orthogonal interventions on one role block consistently moving thresholds of other role blocks beyond what shared biological parameters explain. The genuinely novel forward predictions — distinct from structural reframings of established biology — are four [19, §11]: P11 (timescale-stratified hormesis), P13 (controller-bandwidth pharmacology), P14 (V- distribution as Mode 4b diagnostic), and P17 (intervention-orthogonality stratification of V–Φ

phenomena in bounded adaptive biology is a single classification problem: each is a surface projection of one of the architectural mode-types onto a measurement axis. The classification is exhaustive within stated scope; the empirical anchors are independent across the six companion papers; the falsification criteria are sharp. 8. Conclusion Bounded adaptive biological systems admit a finite architectural classification: the Möbius classification of admissible composition laws on bounded observables, and the BAS-S classification of admissible failure The scope conditions are precise: single bounded primary observable, identifiable slow controller, identifiable parallel surveillance branch, equilibration faster than dosing. Within this scope, the classification is exhaustive and falsifiable. Outside this scope, the mathematical extensions required are identified but not undertaken here. The synthesis introduces no new mathematics, no new calibration, and no new prediction. Its function is to state explicitly the architectural claim that the six companion papers collectively establish but none individually asserts. The claim is that biology builds robust adaptive systems by coupling a fast bounded plant to a slow internal-model controller, then bolting on an independent surveillance layer that shares the same input signal — and that this architectural template, once fixed, admits a finite, classifiable,

Murray — Architectural Classification of Bounded Adaptive Systems | page 12 of 14 falsifiable set of composition laws and failure modes. The six companion papers provide the derivations; this perspective provides the architecture. Whether the framework persists at the level of structural impact suggested by the Erlangen Programme analogy depends on empirical validation across the candidate systems identified — diabetes, hypertension, immune tolerance, bone remodelling, ER proteostasis, neural homeostasis — and on the outcome of the pre-registered Phase II NSCLC prediction. The framework is offered with scope conditions, falsification criteria, and empirical commitments stated openly. Its empirical status is measurable rather than rhetorical, and its development from here is a matter of empirical work rather than further theoretical elaboration. Appendix A. Citation Map: Tracing Synthesis Claims to Companion Papers This appendix provides the explicit map from each structural claim in the main text to its source, supporting the commitment stated in §1: the synthesis introduces no new results. Synthesis claim Source Aczél's representation on bounded observables P1 [14, §2]; Aczél [17] Exhaustive five-flow Möbius classification P1 [14, §2.4, Appendix A] σ-duality acting on the five flows P1 [14, §3.6] Bliss–Loewe gap closed form and golden-ratio location P1 [14, Appendix C]; P2 [15, §4.2] Unification of eight pharmacological equations P2 [15, §4, Table 1] Hyperbolic Fisher-information metric on (0,1) P2 [15, §4.7–4.8] Half-dose-mix methodological correction P2 [15, §6] Derivation of hormesis as geometric necessity P3 [16, §3] Kacser–Burns summation theorem for attenuation factor η P3 [16, §4] 6/6 hormesis validation P3 [16, §6] Calibrated three-variable redox model P4 [18, §2] Dual failure modes (oxidative collapse + reductive fade) P4 [18, §2.2–2.3] WHO G6PD recovery without fitting P4 [18, §3] Pre-registered Phase II NSCLC prediction with 8-oxo-dG biomarker P4 [18, §5] BAS-S four-role decomposition (R1–R4) P5 [19, §2.1] 2+2 asymmetry of component failures vs regime conditions P5 [19, §2.9] Mode 1 closed-form threshold β_c P5 [19, Theorem 1]