Five canonical Mobius flows

PDF page 22 • Bistability and hysteresis. The framework’s flows are smooth one-parameter subgroups; bistable systems with discontinuous response transitions are outside scope. • Non-Möbius dose-response. P3 is an empirical hypothesis. Systems whose dose- response is genuinely non-rational (e.g., compressed exponential, complex sigmoidal forms not reducible to Hill kinetics) lie outside the Möbius scope. Most successful pharmacological assays at single mechanisms have rational dose- response (this is the motivating evidence for P3), but exceptions exist and require non-Möbius treatment. • Multi-agent combinations beyond pairs. The classification theorem applies to pairwise composition. For three or more agents in a single flow class, composition is associative addition in the rapidity (no new structure). For agents across multiple classes, the matrix-product computation generalises but produces order-dependent compositions whose effective parameter count grows with the number of cross- class transitions. • Simultaneous vs sequential dosing for cross-class combinations. Order- dependence in §4.3 is a prediction for sequential dosing with equilibration between doses. Simultaneous administration of two cross-class agents produces a combined flow generated by the sum of their generators (commutative addition in sl(2,ℝ)), and shows no order-dependence regardless of class assignment. The framework makes no claim about systems outside these scope conditions. Its predictions are for the regime where biology operates as a single-mechanism Möbius dose- response, which is the regime in which the classical pharmacological and statistical combination rules were developed. The framework’s contribution is to organise that regime, not to extend beyond it. 9. Conclusion The bounded interval (0,1), considered as a chart of the projective line ℝP¹ with marked boundary {0, 1}, supports exactly five canonical Möbius composition laws under the boundary-preserving semigroup action with strictly monotonic dose-response: Loewe additivity (parabolic at e=1), inverse-odds additivity (parabolic at e=0), logit translation (hyperbolic at {0, 1}), Bliss independence (hyperbolic at {1, ∞}), and multiplicative composition (hyperbolic at {0, ∞}). The classification is exhaustive: the case-by-case enumeration of all boundary-preserving Möbius vector fields with strict monotonicity yields exactly two parabolic configurations (root at e=0 or e=1; root at infinity gives the trivial flow) and exactly three hyperbolic configurations (fixed-point pairs from {0, 1, ∞}, with sign of the generator determined by boundary preservation), and the elliptic class is excluded by orbit topology on ℝP¹. The five flows organise into two σ-dual pairs (Loewe ↔ Inverse-odds, Bliss ↔ Multiplicative) plus one self-dual flow (Logit) under the involution exchanging the boundary points. The Bliss-Loewe gap is derivably maximised at the golden-ratio reciprocal e* = (√5−1)/2 with magnitude (5√5−11)/2; the σ-dual Inverse- PDF page 23 odds-Multiplicative gap has identical magnitude at the dual location 1 − e*, by direct application of the σ-symmetry to the gap function. The framework does not derive itself from postulates of boundedness and homomorphism alone. The Möbius hypothesis (P3) is a substantive empirical claim that the composition law is rational — algebraic in e — motivated by the mass-action kinetic structure of pharmacological dose-response and the operational success of the classical combination rules. Within this scope, the classification is closed, the predictions are deployable, and the failure modes are diagnostic. The framework solves several specific problems in the existing literature. It resolves the eighty-year Bliss-vs-Loewe debate as a fixed-point-structure question rather than a model- selection question. It places multiplicative two-hit survival and Bliss independence in their proper structural relationship as σ-dual hyperbolic flows. It gives Inverse-odds additivity — the structural content of Gaddum’s two-antagonist equation for shared-site competitive antagonism — a formal place in the classification alongside Loewe and Bliss. It identifies the molecular mechanism of inhibition (competitive at shared site versus non-competitive at independent sites with multiplicative K-shifts) as the operational discriminator between the Inverse-odds and Logit flows for two-antagonist scenarios, with explicit mass-action derivations (Appendix F). It situates logistic regression as the abelian flow with both physical boundaries fixed, with the Chentsov-privileged rapidity coincident with that flow’s natural coordinate. It identifies linear pharmacology as the affine chart of the projective structure with predictable boundary-failure modes. The framework adds three falsifiable predictions: the canonical assignment of two-antagonist scenarios to flow class by molecular mechanism, order-dependence in cross-class agent combinations under the non- commuting Möbius generator structure of sl(2,ℝ), and the ten-fold pairwise gap structure with the Bliss-Loewe gap derivably at the golden-ratio reciprocal. The framework is testable now. The matched-perturbation protocol with five pre- registered predictions discriminates the five flows at standard assay precision in any system within scope. The Inverse-odds prediction is testable in receptor antagonism (two competitive antagonists at shared orthosteric site, fixed agonist), ion channel block, transporter inhibition, and competitive enzyme inhibition under the simple operational test “do (1−e_A)/e_A and (1−e_B)/e_B add to (1−e_combo)/e_combo above the no-inhibitor baseline?” The Logit prediction is testable for two non-competitive antagonists at independent sites under the operational test “do log[(1−e_A)/e_A] and log[(1−e_B)/e_B] add to log[(1−e_combo)/e_combo]?” The order-dependence prediction is testable by cross- over administration of agents from different flow classes with sufficient inter-dose equilibration. Performance of these tests will either identify the operative flow for a given biological system or falsify a load-bearing structural claim cleanly. The five Möbius composition laws are the orbits of one Lie group acting on one bounded interval with one σ-symmetry. The familiar pharmacological and statistical combination rules are recognised as those orbits, not unified by external argument. The structural fact is the σ-pairing organising the table. The mechanistic fact is the molecular discriminator between Inverse-odds and Logit two-antagonist scenarios. The operational tool is the gap- referenced classification of single-mechanism combination data into one of five canonical PDF page 24 flows, with explicit Chou-Talalay CI predictions for each flow. The classification is exhaustive and closed; the mass-action derivations are explicit; the predictions are deployable. Appendix A. Vector fields, discriminants, and matrices Flow X(e) (a, b, c) Δ Fixed points sl(2,ℝ) matrix M Loewe (1−e)² (1, −2, 1) 0 {1} (double) [[−1, 1], [−1, 1]] Inverse-odds −e² (0, 0, −1) 0 {0} (double) [[0, 0], [1, 0]] Logit e(1−e) (0, 1, −1) 1 {0, 1} [[1/2, 0], [1, −1/2]] Bliss 1−e (1, −1, 0) 1 {1, ∞} [[−1/2, 1], [0, 1/2]] Multiplicative −e (0, −1, 0) 1 {0, ∞} [[−1/2, 0], [0, 1/2]] The sl(2,ℝ) matrix corresponding to vector field X(e) = a + be + ce² is M = [[b/2, a], [−c, −b/2]], such that the Möbius action of exp(sM) on e ∈ ℝP¹ gives the flow at parameter s. The five matrices span sl(2,ℝ) as a real vector space (verified by determinant of the coefficient matrix in any three-element subset). Each satisfies X(0) ≥ 0 and X(1) ≤ 0, the boundary-preservation condition for semigroup invariance of [0,1]. Exhaustiveness by case analysis. The classification theorem of §3.1 follows from a direct case-by-case enumeration of boundary-preserving Möbius vector fields on (0,1) with no interior fixed points. We list all cases explicitly to confirm that the table above is both complete and minimal: Parabolic cases (Δ = 0, double root). The double root must lie in {0, 1, ∞} by Lemma 1 of §2.4 (no interior fixed points under strict monotonicity). • Double root at e=0: X(e) = c·e² with c ≠ 0. Boundary preservation: X(0) = 0 ≥ 0 ✓; X(1) = c ≤ 0 forces c < 0. After rescaling, X = −e². This is Inverse-odds. • Double root at e=1: X(e) = c·(1−e)² with c ≠ 0. Boundary preservation: X(0) = c ≥ 0 forces c > 0; X(1) = 0 ≤ 0 ✓. After rescaling, X = (1−e)². This is Loewe. • Double root at e=∞: The “double root at infinity” requires both c = 0 and b = 0 (no quadratic or linear term), so X = a constant. Boundary preservation forces a = 0 (trivial flow). Excluded. Hyperbolic cases (Δ > 0, two distinct real fixed points from {0, 1, ∞}). • Fixed points at {0, 1}: X(e) = c·e(1−e) with c ≠ 0. Both signs of c preserve (0,1) under semigroup action (with c > 0 the flow points toward e=1, with c < 0 toward e=0); these are the same one-parameter group up to time reversal, giving one flow class. After rescaling, X = e(1−e). This is Logit. • Fixed points at {1, ∞}: Fixed point at infinity means c = 0; fixed point at 1 means a + b = 0, so b = −a. Then X = a(1−e). Boundary preservation: X(0) = a ≥ 0 forces a > 0; X(1) = 0 ≤ 0 ✓. After rescaling, X = (1−e). This is Bliss. (The opposite sign a < 0 would violate X(0) ≥ 0 and is excluded.)

composition for shared-site competitive antagonists), Bliss with multiplicative composition, and fixes Logit. Within the framework’s empirical scope (single-mechanism mass-action kinetics on a single bounded observable), three falsifiable predictions follow. First, ten pairwise discriminating gaps at matched mid-range effect, with the Bliss–Loewe gap derivably maximised at e* = (√5−1)/2 ≈ 0.618 with magnitude (5√5−11)/2 ≈ 0.090, and the Bliss–Multiplicative gap reaching 0.50 at e=0.5. Second, the canonical assignment of two-antagonist scenarios to flow class by molecular mechanism: competitive at shared site magnitude 0.05–0.15 in effect units at standard dosing, computable exactly by matrix multiplication. Linear pharmacology is recovered as the affine chart of the projective structure, exact in the boundary-far limit and predictably divergent near the boundaries. The framework reorganises the Bliss-vs-Loewe debate as a fixed-point-structure question rather than a model-selection question, places multiplicative two-hit survival in its proper structural relationship to Bliss as σ-dual hyperbolic flows, and gives Inverse-odds additivity (Gaddum) a formal place in the classification alongside Loewe, Bliss, Logit, and

Drug combinations have been studied quantitatively since Bliss’s 1939 analysis of joint poison toxicity and Loewe’s 1953 isobologram. The two formalisms — Bliss multiplying unaffected fractions, Loewe summing dose ratios — give different predictions and have been treated as competing models for eight decades, generating an extensive literature on combination indices (Chou and Talalay 1984), response surface methods (Greco et al. 1995), and most recently the MuSyC parameterisation (Meyer et al. 2019). The persistent disagreement has not converged on a winner; biological systems are reported as composition, and fixes the logit. The framework’s domain of applicability is the empirical regime where biological dose-response is Möbius — characterised by mass-action kinetics on a single bounded observable. Within this domain, the framework’s predictions are exact and discriminating; outside it, the deviations are themselves diagnostic. The framework clarifies what the existing combination index literature has been measuring (mid-range divergence between projective flows), why the Bliss-vs-Loewe debate has been irresolvable as posed (they are different orbits selected by mechanism, not competing models), and where linear pharmacology must fail (near the boundaries, where the projective structure asserts itself). It provides three falsifiable predictions: the canonical assignment of two-antagonist scenarios to flow class by molecular mechanism — competitive antagonists at a shared site realise Inverse-odds (Gaddum’s two-antagonist equation), non-competitive antagonists at independent sites with multiplicative K-shifts realise Logit, with explicit mass-action derivations distinguishing the two cases; the ten-

2. The framework 2.1 Postulates and scope P1 (Boundedness). A class of biological observables of interest is confined to the open interval (0,1). Receptor occupancy, enzyme saturation, viable fraction, normalised damage, fraction of channels in an open state, and steady-state ratios of aggregates to physical capacities are bounded by physical structure. P2 (Strictly monotonic compositional homomorphism). The dose-effect map θ : D → e from non-negative additive doses to bounded effects is a strictly monotonic semigroup homomorphism: for d_1, d_2 ≥ 0, θ(d_1 + d_2) = θ(d_1) ⊕ θ(d_2), with θ strictly monotonic in d. P2 is testable by sequential-dosing protocols with cross-over and time resolution. It holds where the system equilibrates faster than dosing changes, where damage from prior doses does not modify the substrate of subsequent doses, and where feedback regulation operates on a slower timescale than the experiment. The strict monotonicity requirement excludes systems where dose-response saturates at a sub- maximal effect (partial agonism); such systems are outside the framework’s scope and discussed in §8. P3 (Möbius hypothesis). The composition operation ⊕ extends to a smooth one- parameter subgroup of the Möbius group of the projective line ℝP¹, acting on the bounded interval (0,1) and preserving [0,1] as a set under its semigroup action. P3 is the framework’s substantive empirical hypothesis. It is not derivable from P1 and P2 alone. Aczél’s representation theorem (1966) guarantees that any continuous, strictly monotonic, associative composition admits a linearising rapidity, but the rapidity can in principle be any continuous monotonic bijection from (0,1) to ℝ; for example, φ(e) = tan(π(e − 1/2)) defines a valid Aczél composition that is not Möbius. P3 is motivated empirically as follows: (a) the dominant pharmacological dose-response curves (Hill, Langmuir, Michaelis-Menten, Henderson-Hasselbalch) arise from mass-action equilibria with rational form e(D) = Dn/(Kn + D^n), so single-agent rapidities (odds, logit) are Möbius functions of e; (b) the compositional structure of two-agent combinations is not a logical independent-site allosteric antagonism, independent failure events), substantiating the inheritance of Möbius form from underlying kinetics in canonical single-mechanism scenarios. P3 is the assertion that this inheritance is generic across single-mechanism regimes; the classification theorem then yields exactly five canonical flows for systems where P3 holds. The framework is testable: if a biological system within scope realises a composition law that is not one of the five flows, either P3 fails for that system or the system operates outside the single-mechanism scope. The framework’s clean predictions apply where P1, P2, and P3 hold; deviations from the five flows are diagnostic.

2.2 The Lie algebra of the bounded interval Under P3, the generating vector field of the composition flow is X(e) ∂/∂e where X is a Möbius vector field — a polynomial of degree at most two in e, by the standard correspondence between Möbius transformations of ℝP¹ and quadratic vector fields. The space of such vector fields is three-dimensional, spanned by {1·∂/∂e, e·∂/∂e, e²·∂/∂e}, and closes as the Lie algebra sl(2,ℝ) (Lang 1975; Helgason 1978). The matrix realisation maps a is PSL(2,ℝ), acting on ℝP¹ by Möbius transformations. 2.3 Boundary preservation excludes elliptic flows Under P1, the composition flow must preserve the closed interval [0,1] under its semigroup action: T_s([0,1]) ⊆ [0,1] for all s ≥ 0. For a Möbius vector field X(e) = a + be + ce² this requires X(0) = a ≥ 0 and X(1) = a + b + c ≤ 0 (the flow at e=0 cannot point left out of the interval, and at e=1 cannot point right out). The Iwasawa decomposition of PSL(2,ℝ) factors elements as products in the compact boundary-preservation constraints X(0) ≥ 0 and X(1) ≤ 0. 2.4 The fixed-point classification and exhaustiveness Lemma 1 (boundary-point lemma). Under P1, P2, and P3, the fixed points of the generating vector field on ℝP¹ lie in {0, 1, ∞}. Proof. Two parts. Part 1: no interior fixed points in (0,1). Suppose for contradiction that X has a fixed point e* ∈ (0,1), so X(e) = 0. Boundary preservation requires X(0) ≥ 0 and X(1) ≤ 0; combined with X(e)

converge to it as d → ∞; if unstable, the symmetric statement holds for d → −∞.) Take any e_0 in such a neighbourhood and consider the flow θ(d) starting from e_0: as d → ∞, θ(d) → e. Since φ : (0,1) → ℝ is by P3 a continuous monotonic bijection, φ is finite at the interior point e, and φ(θ(d)) → φ(e*), a finite limit. But by P2 with strict monotonicity, φ(θ(d)) = αd + β is linear in d with α ≠ 0, hence unbounded as d → ∞. Contradiction. Therefore no interior fixed point exists. Part 2: real fixed points lie in {0, 1} ∪ {∞}. By Part 1, X has no zero in (0,1). The boundary inequalities X(0) ≥ 0 and X(1) ≤ 0 admit equality (a fixed point at 0 or 1) but no strict zero root of the quadratic X(e) = a + be + ce² (or its limit when c = 0) lies at the projective fixed point e = ∞. □ Theorem (exhaustive classification). Under P1, P2, and P3, the boundary- preserving one-parameter Möbius flows on (0,1) are exhausted by exactly five canonical classes, indexed by their fixed-point set on ℝP¹: 1. Parabolic at {1} (Loewe additivity) 2. Parabolic at {0} (inverse-odds additivity)

3. The five canonical flows Each flow is specified by its vector field, rapidity, identity element on (0,1), two-agent matched-effect prediction, physical realisation, and σ-image. The vector field X(e) is the generator; the rapidity φ(e) satisfies φ’(e) · X(e) = 1 up to additive constant fixed by the choice of identity. 3.1 Loewe additivity (parabolic at e=1) X_L = (1−e)²; φ_L(e) = e/(1−e); identity e=0 Composition: e_combo/(1−e_combo)

framework must specify which competitive structure their inhibitors share, and the flow assignment follows. Earlier versions of this paper conflated these two cases; the framework’s predictions for the two-antagonist scenario depend on which mechanism is operative. (The Schild and Cheng-Prusoff equations describe the single-antagonist log-shift of an agonist dose-response curve, and are foundational for assigning each antagonist’s individual K_I from competition data. The two-antagonist composition law that follows

These are exact values from matrix multiplication, not BCH approximations; the BCH formula is a series expansion that organises the matrix-exp computation but the matrix-exp itself is exact and convergent for any finite (s, t). The leading-order BCH prediction, st · X_L(e_0) = 1 · 1 · (1−0.3)² = 0.49, overshoots the exact answer at large doses; at small doses it tracks the exact answer with relative error rising from ~3% at s = t = 0.02 to ~16% at s = t = 0.10 (Appendix D table). The leading-order BCH approximation is therefore reliable to within ~10% only at s, t ≤ 0.05; for standard mid-range dosing one should use the exact matrix product rather than the BCH leading order. Prediction. Cross-class agent combinations show order-dependence in cross- over experiments, with magnitude predicted exactly from the Möbius matrix product. Within-class combinations show no order-dependence. This is the framework’s strongest single-experiment prediction. Most pharmacological combination protocols administer agents simultaneously, which obscures order- dependence. A cross-over design with matched doses producing mid-range effects, varying

only the order of administration with sufficient inter-dose equilibration time, would test the framework’s cross-class prediction directly. The distinction between simultaneous and sequential dosing is mathematically fundamental and must be stated unambiguously. Simultaneous administration of two agents corresponds to the sum of their generators: the combined flow is generated by M_A + M_B, which lies in sl(2,ℝ) and produces a one-parameter Möbius semigroup whose with equilibration corresponds to the product of group elements: exp(t M_A) exp(s M_B), in which the order of factors matters whenever [M_A, M_B] ≠ 0. The order-dependence prediction is a prediction about sequential dosing; experiments using simultaneous administration cannot test it. Conversely, observing order-dependence in a cross-over experiment with adequate equilibration is direct evidence of cross-class generator non- commutation, with magnitude calibrated by the matrix product computation. 5. Discriminating tests 5.1 The five matched-diagonal predictions For two agents at matched individual effect e_A = e_B = e, the five flows predict: Flow e_combo Loewe 2e/(1+e) Inverse-odds e/(2−e) Logit e²/[e²+(1−e)²] 0.382. 5.2 The Bliss-Loewe gap derivation and the golden-ratio location The diagonal gap between Bliss and Loewe predictions is g(e) = (2e − e²) − 2e/(1+e) = e²(1−e)/(1+e), obtained by combining the matched-diagonal predictions and simplifying. To find the maximum, differentiate: g’(e) = d/de [e²(1−e)/(1+e)] = 2e(1 − e − e²)/(1+e)². Setting g’(e) = 0 in (0,1) gives the critical equation e² + e − 1 = 0, with positive root e* = (√5 − 1)/2 ≈ 0.618034 (the golden-ratio reciprocal).

Bliss-Loewe and Inverse-odds-Multiplicative gaps (0.090) and the Loewe-Logit gap near e=0.62 (0.040) are the technically demanding cases requiring matched mid-range design. Chou-Talalay CI predictions for the five flows. The Chou-Talalay combination index (CI) is a single-number summary widely used in the synergy literature. For the standard matched- effect protocol with Hill-1 single-agent kinetics — each agent dosed at its single-agent effect-e dose, both delivered together, observe combined effect e_combo — the predicted CI for each flow is CI = D_A_combo/D_A^isobole(e_combo) + D_B_combo/D_B^isobole(e_combo) = 2 · e(1−e_combo) / [(1−e)·e_combo],

effects); Inverse-odds gives CI ≈ 4 (strong “antagonism” by Loewe convention but actually describing distinct inhibitor pharmacology); Multiplicative gives CI ≈ 5–6 (well above any conventional threshold). The framework predicts that re-classification of existing CI literature against the five-flow taxonomy, restricted to single-mechanism contexts, will reveal clustering at these specific CI values rather than continuous variation. 5.4 Statistical resolution Naive resolution analysis: combination-arm CV = 10%, n replicates, standard error independent-error analysis. Realistic dose-finding analysis: matched-effect doses are estimated from single-agent dose- response curves with non-zero variance, propagating into the combination prediction. Tallarida (2000) gives the appropriate isobolographic confidence-region treatment. We provide a simulation-based power analysis below. Simulation 1 (power, true flow = Bliss): Bliss-generated data at matched e=0.618 with CV=10% and varying n. Classification by closest-flow rule (identify the operative flow as the closest of five predictions to the observed e_combo) and by strict gap-referenced rule (identify only if closest flow is significantly closer than next-closest by 2σ). n Closest-flow correct Strict (2σ) correct Ambiguous 8 94% 55% 45% 12 96% 66% 34%

dose pairs producing matched individual effects in the mid-range, with explicit confidence regions on the matched doses. 2. Pre-register the predictions of all five flows at the chosen matched effect. 3. Run six arms with replication n ≥ 8 per arm (n ≥ 24 for the Bliss-Loewe demanding case): vehicle control, single A, single B, combination at matched additive co- administration, plus two further matched dose pairs at different mid-range effects to map the diagonal. 4. For cross-class candidates (kinetically classified as different flow classes), include an additional arm with reversed dosing order and time gap appropriate for equilibration to test for predicted order-dependence. 5. Compare observed e_combo against all five predictions using gap-referenced falsification. 5.6 Falsification criteria The framework is falsified if any of the following holds in a system satisfying P1–P3 in scope: 1. The observed combination effect lies significantly far from all five flow predictions, with separation from the closest flow exceeding the gap to the next-closest flow by more than 2σ. 2. The number of independent parameters required to fit a single-mechanism k-agent combination dataset exceeds the dimension of the relevant flow orbits (one location

4. Cross-class agent combinations (verified by independent kinetics) consistently fail to show predicted order-dependence in cross-over experiments at the magnitude computed from the Möbius matrix product. 6. Linear pharmacology is the affine chart The bounded interval (0,1) is a chart of ℝP¹ with two distinguished boundary points. −log(e): slope −1/e₀ at e₀ Each composition law reduces to addition in (e − e₀) to leading order; the five laws agree to first order on combination predictions in the boundary-far regime, with quadratic corrections producing the divergence between the laws as |δ| grows. The linearisation breaks at different rates for different rapidities. For logit and the symmetric self-dual structure, the breakdown scale is δ ~ √[e₀(1−e₀)]. For Bliss and Loewe, the breakdown scale is asymmetric, faster near the relevant boundary. For Inverse-odds

Möbius family. The choice between additive-in-D^n (Loewe) and multiplicative-in-K (Logit) realisations is fixed by the kinetic class of the perturbation. What new structural prediction does the framework add? Three: (i) the canonical assignment of two-inhibitor scenarios to flow class by molecular mechanism — competitive antagonists at a shared site realise Inverse-odds additivity (Gaddum’s two- antagonist composition, here recognised as a structurally distinct flow with its proper place in the classification beside Loewe and Bliss); non-competitive antagonists at independent sites with multiplicative K-shifts realise Logit composition; both predictions with explicit mass-action derivations (Appendix F); (ii) order-dependence in cross-class agent combinations under the non-commuting Möbius generator structure of sl(2,ℝ), with magnitude calculable exactly from the matrix product — testable by cross-over administration of agents with verified differing kinetic classes; (iii) the matched-mid- range gap structure with ten pairwise discriminations and explicit Chou-Talalay CI predictions, including the Bliss-Loewe gap maximised at the golden-ratio reciprocal and its σ-dual Inverse-odds-Multiplicative gap at 1 − e*. Relation to MuSyC and combination index frameworks The MuSyC parameterisation (Meyer et al. 2019) introduces four parameters (β, α₁, α₂, γ) interpolating between Bliss and Loewe predictions and capturing efficacy and potency synergy separately. The framework’s relationship to MuSyC: MuSyC’s parameters describe

deviation from Loewe additivity. CI < 1 indicates synergy beyond Loewe, CI > 1 antagonism. With the standard matched-effect protocol (each agent at its single-agent effect-e dose, factor-of-2 sum of contributions), the framework predicts the following CI ranges in single- mechanism contexts: Bliss CI ≈ 0.55–0.67 in mid-range (mild Loewe-convention synergy); Loewe CI = 1 by definition; Logit CI ≈ 1.2–2.0 (Loewe-convention antagonism, varying with effect level); Inverse-odds CI ≈ 4 (strong “antagonism” by Loewe convention but actually describing inhibitor pharmacology at a shared site); Multiplicative CI ≈ 5–6 (well above any 8. Limitations and scope The framework’s domain of applicability is the regime where P1, P2, and P3 hold. Systems outside this scope include: • Multi-mechanism observables. A bounded observable affected by multiple distinct mechanisms simultaneously (e.g., an agent acting at one receptor by competitive antagonism and at another by allosteric modulation) does not satisfy the framework’s single-mechanism requirement. The observed combination effect is a weighted average of two or more flow predictions, and identification requires mechanism-resolved sub-experiments. The framework’s failure in such systems is itself diagnostic: deviations from any single flow with structurally specific signatures indicate multi-mechanism action. • Partial agonism and sub-maximal saturation. Strict monotonicity in P2 excludes systems where dose-response saturates at a sub-maximal effect e_max < 1. Partial agonists fall outside the framework as stated; they require either rescaling the observable to (0, e_max) or extending the framework to admit interior fixed points. The operational model of pharmacological agonism (Black and Leff 1983) provides the standard parametric framework for partial-agonist responses and is a natural complement to the present classification at the boundary of its scope. • Non-equilibrium and feedback. P2 requires the system to equilibrate faster than dosing changes. Systems with significant feedback regulation, induced enzyme expression, or receptor desensitisation on the experimental timescale violate P2 and produce time-dependent composition laws.

subgroups; bistable systems with discontinuous response transitions are outside scope. • Non-Möbius dose-response. P3 is an empirical hypothesis. Systems whose dose- response is genuinely non-rational (e.g., compressed exponential, complex sigmoidal forms not reducible to Hill kinetics) lie outside the Möbius scope. Most successful pharmacological assays at single mechanisms have rational dose- response (this is the motivating evidence for P3), but exceptions exist and require non-Möbius treatment. • Multi-agent combinations beyond pairs. The classification theorem applies to pairwise composition. For three or more agents in a single flow class, composition is associative addition in the rapidity (no new structure). For agents across multiple class transitions. • Simultaneous vs sequential dosing for cross-class combinations. Order- dependence in §4.3 is a prediction for sequential dosing with equilibration between doses. Simultaneous administration of two cross-class agents produces a combined flow generated by the sum of their generators (commutative addition in sl(2,ℝ)), and shows no order-dependence regardless of class assignment. The framework makes no claim about systems outside these scope conditions. Its predictions are for the regime where biology operates as a single-mechanism Möbius dose- response, which is the regime in which the classical pharmacological and statistical combination rules were developed. The framework’s contribution is to organise that regime, not to extend beyond it.

application of the σ-symmetry to the gap function. The framework does not derive itself from postulates of boundedness and homomorphism alone. The Möbius hypothesis (P3) is a substantive empirical claim that the composition law is rational — algebraic in e — motivated by the mass-action kinetic structure of pharmacological dose-response and the operational success of the classical combination rules. Within this scope, the classification is closed, the predictions are deployable, and the failure modes are diagnostic. The framework solves several specific problems in the existing literature. It resolves the eighty-year Bliss-vs-Loewe debate as a fixed-point-structure question rather than a model- selection question. It places multiplicative two-hit survival and Bliss independence in their physical boundaries fixed, with the Chentsov-privileged rapidity coincident with that flow’s natural coordinate. It identifies linear pharmacology as the affine chart of the projective structure with predictable boundary-failure modes. The framework adds three falsifiable predictions: the canonical assignment of two-antagonist scenarios to flow class by molecular mechanism, order-dependence in cross-class agent combinations under the non- commuting Möbius generator structure of sl(2,ℝ), and the ten-fold pairwise gap structure with the Bliss-Loewe gap derivably at the golden-ratio reciprocal. The framework is testable now. The matched-perturbation protocol with five pre- registered predictions discriminates the five flows at standard assay precision in any system within scope. The Inverse-odds prediction is testable in receptor antagonism (two competitive antagonists at shared orthosteric site, fixed agonist), ion channel block, transporter inhibition, and competitive enzyme inhibition under the simple operational test “do (1−e_A)/e_A and (1−e_B)/e_B add to (1−e_combo)/e_combo above the no-inhibitor baseline?” The Logit prediction is testable for two non-competitive antagonists at independent sites under the operational test “do log[(1−e_A)/e_A] and log[(1−e_B)/e_B] add to log[(1−e_combo)/e_combo]?” The order-dependence prediction is testable by cross- over administration of agents from different flow classes with sufficient inter-dose equilibration. Performance of these tests will either identify the operative flow for a given biological system or falsify a load-bearing structural claim cleanly. The five Möbius composition laws are the orbits of one Lie group acting on one bounded interval with one σ-symmetry. The familiar pharmacological and statistical combination rules are recognised as those orbits, not unified by external argument. The structural fact is the σ-pairing organising the table. The mechanistic fact is the molecular discriminator

flows, with explicit Chou-Talalay CI predictions for each flow. The classification is exhaustive and closed; the mass-action derivations are explicit; the predictions are deployable. Appendix A. Vector fields, discriminants, and matrices Flow X(e) (a, b, c) Δ Fixed points sl(2,ℝ) matrix M