Order and graph integrability

### PDF page 17 Daniel J. Murray Revised September 2026 11. CONCLUSION BED is retained as an empirical compression, not promoted to universal equivalence. A reversed pair built from the same component fractions has identical complete-repair LQ BED, and the usual symmetric two-fraction incomplete-repair correction remains reversal-blind at fixed exchange- symmetric timing. A resolved endpoint-distribution difference therefore means equal nominal BED is not equal predictive state for that endpoint. Reversal supplies that test without graph machinery. Proposition 1 formalizes the distributional null; the Aczél remark marks the boundary of associative one-scalar replacements. K3/K4 should be added only when integrability or potential shape changes the next decision, because graph verdicts identify neither mechanism nor state dimension. The replacement problem is biological: measure the retained state into which the next exposure arrives. A candidate state earns support only by restoring held-out prediction under a common future with state and outcome linked at the inferential-unit level. Destructive parallel mouse cohorts can motivate such a representation but cannot prove unit-level recovery. The practical sequence is therefore: qualify one reversal pair; retain BED where equivalence is demonstrated; after a resolved failure, test state recovery; use complete graphs only when their extra structural information earns the extra experimental cost.

create cycle defect from exact latent units. The resulting protocol uses independent gap qualifica- tion, end-to-start timing, equivalence-based ADEQUATE/REJECTED/UNRESOL VED decisions, and unit-linked held-out prediction for candidate state variables. BED is therefore treated as an empirically testable compression whose predictive domain must be declared rather than assumed. Key words: biologically effective dose; radiation fractionation; order dependence; history depen- dence; tumor microenvironment; integrability; stateful radiation response 1. INTRODUCTION BED compresses a fractionation history to one scalar for schedule comparison. Its utility does not imply biological sufficiency for every endpoint: sequence, spacing, waveform, radiation quality, spatial context, and pre-exposure state can matter. The question here is when BED-like compression retains enough information to predict a declared endpoint. Reversal is the minimal test. Exchange two component exposures while holding timing and readout fixed. Standard complete-repair LQ BED, BED(a,b)=a[1+a/( 𝛼/𝛽)]+b[1+b/(𝛼/𝛽)], is unchanged. 1Corresponding author: Daniel J. Murray, Independent Researcher, Melbourne, Victoria, Australia. Email: dan-

under any common one-to-one transformation. Mean reversal contrasts and graph classifications are different; they depend on the prespecified metric scale, pooling, normalization, and eligibility rules. A positive distributional reversal is therefore the robust first-stage falsifier, whereas upper structural classifications are scale-conditional. Begin with one qualified reversal pair and biological equivalence margins. Add K3/K4 only if their structural verdict changes the next experiment. State recovery is stronger: a candidate state must restore held-out prediction under a common future, with state and outcome linked within the same biological unit or matched split material. No graph verdict identifies a unique mechanism or minimal state dimension. 2. CLAIM JURISDICTION AND THE SCALAR-COMPRESSION CRITERION

margin, a residual-variance test, and adequate precision for the features being claimed equivalent. These finite-feature conditions do not establish equality of entire endpoint distributions. Proposition 1 is intentionally definitional: it turns scalar sufficiency into a falsifiable distributional null. Remark 1 (Aczél boundary). Let L map histories into a real interval I with nonempty interior

Daniel J. Murray Revised September 2026 and at least one endpoint excluded, closed under ⊕, with ⊕ continuous, strictly increasing in each argument, and associative. If L(H1H2)=L(H1) ⊕L(H2), Aczél’s representation theorem gives a monotone coordinate 𝜓 with 𝜓(x⊕y)=𝜓(x)+𝜓(y) (16); hence ⊕ is commutative and cannot encode reversal. This marks a boundary on associative one-scalar replacements, not a newly identified radiobiological victim model. 2.4 Scale and population averaging

0 ⊂ 1 ⊂ 2 ⊂ 3 ⊂ 6. (14) Here M1 denotes the observable linear-exact restriction Cij = gamma(di-dj). The one-state incomplete-repair linear-quadratic temporal model described in Section 4 predicts this restriction when its entry state is nonzero, but M1 is not a mechanistic label. 3.4 Why four doses are the mathematical minimum - and not always the practical optimum With three dose levels, K3 has three independent edges. The linear-exact class has dimension one,

Thus 6+12 and 12+6 Gy have identical nominal BED for any fixed 𝛼/𝛽. Any BED-like symmetric function of the unordered schedule components is likewise reversal-blind. A reproducible reversal does not make BED globally wrong; it shows that the declared endpoint is not predicted by that scalar alone on the tested domain. Result 1 (two-fraction symmetric-memory commutation). For acute fractions a and b at fixed gap Δ, conventional incomplete-repair LQ contains the symmetric interaction 2 𝛽r(Δ)ab (11,12). More generally, with one common amplitude-scaled waveform family, a symmetric two-time memory or other dynamical models can carry information from the first exposure into the state encountered by the second, and therefore need not be order blind. The complete-graph assay does not reject such models merely because order matters; it classifies the edge field they predict and then asks which additional state measurements recover prediction. 4.2 One-state incomplete-repair LQ as an occupant of the linear-exact class Consider the classical one-state incomplete-repair system ̇𝑋 = 𝑢(𝑡) − 𝜅𝑋, ̇𝐻 = [𝛼 + 2𝛽𝑋]𝑢(𝑡), (18) with two acute blocks a and b, fixed gap Δ, retained fraction r = e−𝜅Δ, and common entry state At fixed gap and preparation, Equation (19) lies in the one-dimensional linear-exact class M1. The boundary state x0 controls whether reversal is visible: in a genuinely naive preparation with x0 = 0, this model predicts zero two-block reversal despite carrying state between exposures. More generally, any naive stationary quadratic response with a symmetric two-time memory kernel also commutes for two blocks drawn from one common amplitude-scaled waveform family under an exchange-symmetric timing convention. The self terms and bilinear ab cross term are unchanged by exchange. Thus a reproducibly nonzero reversal in such a naive preparation rejects a substantially

S1. Equation (19) itself assumes acute blocks and first-order exponential repair. If the linear-exact retained-state class is a target of inference, a separately preregistered non-naive stratum is required: all material receives the same priming exposure before randomization to the reversal graph. The prime is not assumed to isolate one mechanism; it removes the naive boundary degeneracy within a separately declared response law. Finite-width or differently shaped blocks require model-specific re-derivation, while the complete-graph integrability result does not assume exponential repair or ments in immunocompetent mice, while clonogenic survival was not order-resolved and the in-vivo separation was not reproduced in immunodeficient mice (8). This supports a host-state contribution without claiming that BED was designed to predict every immune endpoint. Sia et al. likewise found fraction-size/BED-dependent antitumor immune effects (9). Palmqvist et al. reported no resolved neutron-gamma order difference on selected endpoints (7); without equivalence margins that result is UNRESOL VED here, not demonstrated zero reversal. History sensitivity remains endpoint-, preparation-, and timescale-dependent.

SCALE RULE. The structural class belongs to the declared observation rule. Nonlinear trans- formation can change metric classification and can even reverse a difference-of-means edge sign; preregister scale, normalization, denominator construction, and any sensitivity scales before order labels are released. 6.4 What qualifies as a primary endpoint A primary metric endpoint should be fixed before order labels are released and satisfy four condi- tions:

metric integrability; K4 adds quadratic discrimination; and a shape-specific node design is needed to overidentify a broader potential family (Section 3.4). Any graph expansion after seeing confirmatory results is a new preregistered experiment, not retrospective promotion. Graph extension must earn its cost. Do not multiply schedules merely to obtain a geometric label: K3 or K4 is justified only when exact versus non-exact, or quadratic versus broader-exact, changes the biological follow-up, model falsifier, or optimization strategy. If the one-state retained-entry-state class is a target, use a separately preregistered primed stratum because a naive quadratic-memory system can commute despite memory. Prime all material before schedule randomization and analyze naive and primed strata separately. 7.3 In vivo and non-splittable variants For indivisible in-vivo units, use blocked cohort randomization and model the hierarchy explicitly;

8.0 Distributional null and the actual tested features Proposition 1 is a distributional implication. A difference in any well-defined response feature falsifies equality of the two endpoint laws. The reverse inference is unavailable: matching means and the between-replicate variation of mean reversal contrasts does not establish equality of those laws. For example, every preparation can have arm laws 𝑁 (0, 1)and 𝑁 (0, 4): all true mean reversal contrasts and their between-preparation variation vanish, while the endpoint distributions differ. The metric ladder below therefore reports practical equivalence of its prespecified residual means and residual biological variation only. For a distributional claim, separately preregister a scien- tifically meaningful distance or measure-determining feature model, its tolerance and confidence procedure, and test that target directly. A Bernoulli endpoint is a special case in which its prob- ability determines its law. Finite testing supports only the declared range, response features and margins, even when a distributional structural null is the motivation. . (28) At the primary gap the metric spaces are nested by representation, not by mechanism. On K3: M0 is zero reversal (dimension 0); M1 is the linear-exact predictor di-dj (dimension 1); M2 is the full exact-gradient space (dimension 2), which is also the quadratic-gradient space on three nodes; and the unrestricted edge space has dimension 3. On K4 the same first two classes are followed by M2 quadratic exact (dimension 2), M3 exact/cubic-on-K4 (dimension 3), and unrestricted dimension 6. • K3 and K4 therefore share the decisive first two questions: whether scalar compositional

variation margin is the largest acceptable heterogeneous violation across biological replicates. Sup- plementary Information S1 gives a worked colony-count conversion from a prespecified percentage- point tolerance to counts at fixed seeded-cell number. Prospective sensitivity analysis should span defensible margins; instability across modest choices is a planning warning. 8.4 Randomization, residual variance, and sign cancellation An exact assignment randomization test requires a sharp no-effect null and the actual blocked/split- material assignment scheme. Zero mean reversal alone does not make order labels exchangeable.

9. DISCUSSION 9.1 BED as a conditional measurement, not a universal equivalence BED remains useful where it predicts the declared endpoint adequately (6). Vetrugno's 6+12 versus 12+6 Gy pair shows why sufficiency must be endpoint-specific: standard BED is identical, yet tumor-growth delay and immune state differed by order in immunocompetent mice while clonogenic survival did not (8). The argument is not that BED should predict every immune readout, but that its predictive domain can be tested. Proposition 1 supplies the distributional falsifier; the Aczél remark supplies only an associative- scalar boundary. If a complete graph is run, Proposition 2 asks whether the metric reversal field is a node-potential gradient. Neither zero reversal nor finite-node exactness proves a global dose law or sufficient scalar state. 9.2 Exact order fields are boundary signatures, not recovered BED 9.3 When BED fails: recover state instead of inventing another scalar After a positive reversal, the biological follow-up tests whether a measured state X improves held- out prediction under a declared future intervention. Reversal alone does not prove that a chosen pre-second-fraction measurement must differ: the reversed schedules apply different second inputs, and graph algebra does not locate the missing information in time. X and Y must be linked within the same biological unit or prospectively matched split material. Destructive state sampling in separate mouse cohorts supports schedule-level mapping, not unit-level conditional sufficiency. A failed well-powered test can reflect an incomplete or mistimed state, an incorrect outcome model, poor overlap, or measurement error. For a common future input 𝑈 , the target is ℒ(𝑌 ∣ 𝑋, 𝑈 , ℎ) = ℒ(𝑌 ∣ 𝑋, 𝑈 ) for supported histories and declared interventions . (30) Include the future dose/waveform in 𝑈 ; it must not be silently omitted when reversed schedules use different second fractions. Create distinct histories with overlapping measured-state support, then randomize the same future probe or matched probe panel in prospectively linked material. Hold entire histories/schedules and biological replicates out of state construction and outcome-model

Daniel J. Murray Revised September 2026 fitting. An X that only labels schedules with disjoint support cannot demonstrate within-state history invariance. Predictive improvement is useful but weaker than eliminating the held-out history residual within a preregistered tolerance. Such predictive evidence does not by itself establish that X mediates the causal effect; mediation requires additional intervention or causal-identification assumptions. Candidate coordinates are concrete. Vanpouille-Box et al. showed TREX1 induction above ap- proximately 12-18 Gy in the studied systems can attenuate cytosolic-DNA/cGAS-STING/type-I- tive state, and cell-cycle distribution; outcome material receives fraction 2 and a linearly pooled endpoint such as raw colony count at fixed seeding. Train on prespecified schedules and test a held- out reversal or replication. Success means X restores prediction within the declared equivalence region, not merely that a biomarker differs. Secondary in-vivo immune extension. In a sequence-sensitive syngeneic tumor model (8), prefer minimally perturbative serial measurements so X and Y are observed in the same animal. If the desired tumor-state assay is destructive, use separate mapping and outcome cohorts but label the domain; under the Aczél assumptions it also excludes the specified associative one-scalar class. That result stands alone. K3/K4 are optional structural extensions. A stronger state-recovery claim requires a measured X that restores held-out predictive equivalence with X and Y linked at the inferential-unit level. 10. LIMITATIONS Four limits are decisive. Reversal sees only the antisymmetric schedule component. Metric graph classes depend on the prespecified endpoint scale and observation rule, and nonlinear pooling or

usual symmetric two-fraction incomplete-repair correction remains reversal-blind at fixed exchange- symmetric timing. A resolved endpoint-distribution difference therefore means equal nominal BED is not equal predictive state for that endpoint. Reversal supplies that test without graph machinery. Proposition 1 formalizes the distributional null; the Aczél remark marks the boundary of associative one-scalar replacements. K3/K4 should be added only when integrability or potential shape changes the next decision, because graph verdicts identify neither mechanism nor state dimension. The replacement problem is biological: measure the retained state into which the next exposure arrives. A candidate state earns support only by restoring held-out prediction under a common future with state and outcome linked at the inferential-unit level. Destructive parallel mouse cohorts can motivate such a representation but cannot prove unit-level recovery. The practical sequence is therefore: qualify one reversal pair; retain BED where equivalence is demonstrated; after a resolved failure, test state recovery; use complete graphs only when their