Reversal predecessor

### PDF page 27 MURRAY | TESTING BED SUFFICIENCY recovery claim requires a measured X that restores held-out predictive equivalence with X and Y 552 linked at the inferential-unit level. 553 10. LIMITATIONS 554 Four limits are decisive. Reversal sees only the antisymmetric schedule component. Metric graph 555 classes depend on the prespecified endpoint scale and observation rule, and nonlinear pooling or 556 eligibility can change them. Exactness is finite-node and domain-local, not a global dose law, 557 mechanism, or recovered state. Finally, destructive parallel in-vivo state/outcome cohorts cannot 558 establish unit-level conditional sufficiency. Additional timing, covariance, model, state-recovery, 559 and translational limits are detailed in Supplementary Information S1. 560 11. CONCLUSION 561 Conventional two-fraction incomplete-repair LQ corrects timing with the symmetric interaction 562 2βr(Δ)ab. At a fixed exchange-symmetric gap it cannot distinguish a→b from b→a. Reversal is 563 therefore a direct experimental falsifier of this temporal compression: when the endpoint 564 distributions separate, the scalar has discarded predictive information carried by order. 565 The clean-start result is broader. A naive stationary quadratic response with symmetric two-time 566 memory also predicts zero reversal. A resolved reversal from a clean start rejects the stationary-567 quadratic memory class, not merely complete-repair BED. Proposition 2 and the cubic 568 Vandermonde result then classify optional graph extensions when the structure of the residual 569 order field matters. 570 The observation layer is itself part of the model. Nonlinear population pooling can create cycle 571 defect even when every latent unit is exact, so structural inference must be attached to a declared 572 measurement and pooling rule. This self-audit prevents observation-induced geometry from 573 being mistaken for latent radiation dynamics. 574 ### PDF page 28 MURRAY | TESTING BED SUFFICIENCY After a resolved reversal, the decisive biological problem is state recovery: measure the state into 575 which the next exposure arrives and require that state to restore held-out prediction in unit-linked 576 data. The resulting program is minimal and testable—qualify one reversal, reject a lossy 577 compression when order separates outcomes, and replace it only with measured state that earns 578 predictive sufficiency. 579

nominal BED was accompanied by order-dependent tumor-growth delay and immune state in 25 immunocompetent murine tumors. Reversal therefore tests whether temporal compression 26 preserves prediction, and the observation-layer analysis separates latent dynamics from structure 27 created by population processing. 28 Key words: biologically effective dose; radiation fractionation; order dependence; history 31

not imply biological sufficiency for every endpoint: sequence, spacing, waveform, radiation 35 quality, spatial context, and pre-exposure state can matter. The question here is when BED-like 36 compression retains enough information to predict a declared endpoint. 37 Reversal is the minimal test. Exchange two component exposures while holding timing and 38 readout fixed. Standard complete-repair LQ BED, BED(a,b)=a[1+a/(α/β)]+b[1+b/(α/β)], is 39 unchanged. For two fractions at a fixed exchange-symmetric gap, the conventional incomplete-40 repair correction is also unchanged because its cross term is proportional to ab (11,12). A 41

identical under any common one-to-one transformation. Mean reversal contrasts and graph 64 classifications are different; they depend on the prespecified metric scale, pooling, normalization, 65 and eligibility rules. A positive distributional reversal is therefore the robust first-stage falsifier, 66 whereas upper structural classifications are scale-conditional. 67 Begin with one qualified reversal pair and biological equivalence margins. Add K3/K4 only if 68 their structural verdict changes the next experiment. State recovery is stronger: a candidate state 69 must restore held-out prediction under a common future, with state and outcome linked within 70 the same biological unit or matched split material. No graph verdict identifies a unique 71 mechanism or minimal state dimension. 72 2. CLAIM JURISDICTION AND THE SCALAR-COMPRESSION CRITERION 73 2.1 The criterion conditions are experimental conditions 74

opposite replicate-level effects that average toward zero. Practical equivalence therefore requires 121 a declared biological margin, a residual-variance test, and adequate precision. 122 Proposition 1 is intentionally definitional: it turns scalar sufficiency into a falsifiable 123 distributional null. Remark 1 (Aczél boundary). Let L map histories into a real interval I closed 124

under ⊕, with ⊕ continuous, strictly increasing in each argument, and associative. If 125 L(H1H2)=L(H1)⊕L(H2), Aczél's representation theorem gives a monotone coordinate ψ with 126 ψ(x⊕y)=ψ(x)+ψ(y) (16); hence ⊕ is commutative and cannot encode reversal. This marks a 127 boundary on associative one-scalar replacements, not a newly identified radiobiological victim 128 model. 129 2.4 Scale and population averaging 130

encountered by the second and therefore need not be order-blind. Reversal tests whether that 223 retained information matters for the declared endpoint. 224 4.2 Naive stationary quadratic memory predicts zero reversal 225 Result 1 (clean-start quadratic-memory null). Consider a stationary quadratic response driven by 226 one common amplitude-scaled waveform family with a symmetric two-time memory kernel and 227 exchange-symmetric timing. If both orderings begin from the same naive state, the self terms and 228 the bilinear cross term are unchanged by exchange, so the reversal contrast is identically zero: 229

The acute-block derivation is given in Supplementary Information S1. 245 Equation (19) itself assumes acute blocks and first-order exponential repair. If the linear-exact 246 retained-state class is a target of inference, a separately preregistered non-naive stratum is 247 required: all material receives the same priming exposure before randomization to the reversal 248 graph. The prime is not assumed to isolate one mechanism; it removes the naive boundary 249 degeneracy within a separately declared response law. Finite-width or differently shaped blocks 250 require model-specific re-derivation, while the complete-graph integrability result does not 251

microenvironments in immunocompetent mice, while clonogenic survival was not order-resolved 272 and the in-vivo separation was not reproduced in immunodeficient mice (8). This supports a 273 host-state contribution without claiming that BED was designed to predict every immune 274 endpoint. 275 Sia et al. likewise found fraction-size/BED-dependent antitumor immune effects (9). Palmqvist 276 et al. reported no resolved neutron-gamma order difference on selected endpoints (7); without 277 equivalence margins that result is UNRESOLVED here, not demonstrated zero reversal. History 278

SCALE RULE. The structural class belongs to the declared observation rule. Nonlinear 333 transformation can change metric classification and can even reverse a difference-of-means edge 334 sign; preregister scale, normalization, denominator construction, and any sensitivity scales before 335 order labels are released. 336 6.4 What qualifies as a primary endpoint 337 A primary metric endpoint should be fixed before order labels are released and satisfy four 338 conditions: 339

itself a prespecified decision question. The K4 cycle identity, rank and dimension count, basis 378 matrices, and covariance-weighted projections are given in Supplementary Information S1. Any 379 graph expansion after outcome unblinding is a new preregistered experiment. 380 Graph extension must earn its cost: add K3 or K4 only when exact versus non-exact, or quadratic 381 versus broader-exact, changes the biological follow-up, model falsifier, or optimization strategy. 382 If the one-state retained-entry-state class is a target, use a separately preregistered primed stratum 383 because a naive quadratic-memory system can commute despite memory. Prime all material 384 before schedule randomization and analyze naive and primed strata separately. 385 7.3 In vivo and non-splittable variants 386 For indivisible in-vivo units, use blocked cohort randomization and model the hierarchy 387

scale; the variation margin is the largest acceptable heterogeneous violation across biological 444 replicates. Supplementary Information S1 gives a worked colony-count conversion from a 445 prespecified percentage-point tolerance to counts at fixed seeded-cell number. Prospective 446 sensitivity analysis should span defensible margins; instability across modest choices is a 447 planning warning. 448 8.4 Randomization, residual variance, and sign cancellation 449 For the sharp zero-flow null, order labels are exchangeable within randomized biological 450

REJECTED, test M1; if M1 is ADEQUATE, report the linear-exact edge form admissible 488 without assigning a mechanism. 489 1. If a complete graph was preregistered and M1 is REJECTED, test exactness under 490 Proposition 2; on K4, insert the quadratic-exact layer first only when that distinction was 491 prespecified. ADEQUATE exactness means a node potential exists on the tested graph; 492 REJECTED exactness means non-exact metric history dependence. The full K3/K4 493 projection sequence is specified in Supplementary Information S1. 494 9. DISCUSSION 500 9.1 BED as a conditional measurement, not a universal equivalence 501 BED remains useful where it predicts the declared endpoint adequately (6). Vetrugno's 6+12 502 versus 12+6 Gy pair shows why sufficiency must be endpoint-specific: standard BED is 503 identical, yet tumor-growth delay and immune state differed by order in immunocompetent mice 504 while clonogenic survival did not (8). The argument is not that BED should predict every 505 immune readout, but that its predictive domain can be tested. 506

MURRAY | TESTING BED SUFFICIENCY Proposition 1 supplies the distributional falsifier; the Aczél remark supplies only an associative-507 scalar boundary. If a complete graph is run, Proposition 2 asks whether the metric reversal field 508 is a node-potential gradient. Neither zero reversal nor finite-node exactness proves a global dose 509 law or sufficient scalar state. 510 9.2 Exact order fields are boundary signatures, not recovered BED 511 class alone identifies neither mechanism nor state dimension (10-14,22-24). 514 9.3 When BED fails: recover state instead of inventing another scalar 515 After a positive reversal, the central biological prediction is that the histories occupy measurably 516 different pre-second-fraction states and that an informative state X improves prediction of a 517 common future outcome Y. X and Y must be linked within the same biological unit or 518 prospectively matched split material. Destructive state sampling in separate mouse cohorts 519 supports schedule-level mapping, not unit-level conditional sufficiency. Failure means the 520 measured state is incomplete, mistimed, or both. 521 Candidate coordinates are concrete. Vanpouille-Box et al. showed TREX1 induction above 522 approximately 12-18 Gy in the studied systems can attenuate cytosolic-DNA/cGAS-523

engagement, oxidative state, and cell-cycle distribution; outcome material receives fraction 2 and 533 a linearly pooled endpoint such as raw colony count at fixed seeding. Train on prespecified 534 schedules and test a held-out reversal or replication. Success means X restores prediction within 535 the declared equivalence region, not merely that a biomarker differs. 536 Secondary in-vivo immune extension. In a sequence-sensitive syngeneic tumor model (8), prefer 537 minimally perturbative serial measurements so X and Y are observed in the same animal. If the 538 desired tumor-state assay is destructive, use separate mapping and outcome cohorts but label the 539

MURRAY | TESTING BED SUFFICIENCY recovery claim requires a measured X that restores held-out predictive equivalence with X and Y 552 linked at the inferential-unit level. 553 10. LIMITATIONS 554 Four limits are decisive. Reversal sees only the antisymmetric schedule component. Metric graph 555 classes depend on the prespecified endpoint scale and observation rule, and nonlinear pooling or 556 Conventional two-fraction incomplete-repair LQ corrects timing with the symmetric interaction 562 2βr(Δ)ab. At a fixed exchange-symmetric gap it cannot distinguish a→b from b→a. Reversal is 563 therefore a direct experimental falsifier of this temporal compression: when the endpoint 564 distributions separate, the scalar has discarded predictive information carried by order. 565 The clean-start result is broader. A naive stationary quadratic response with symmetric two-time 566 memory also predicts zero reversal. A resolved reversal from a clean start rejects the stationary-567 quadratic memory class, not merely complete-repair BED. Proposition 2 and the cubic 568 Vandermonde result then classify optional graph extensions when the structure of the residual 569 order field matters. 570 The observation layer is itself part of the model. Nonlinear population pooling can create cycle 571

After a resolved reversal, the decisive biological problem is state recovery: measure the state into 575 which the next exposure arrives and require that state to restore held-out prediction in unit-linked 576 data. The resulting program is minimal and testable—qualify one reversal, reject a lossy 577 compression when order separates outcomes, and replace it only with measured state that earns 578 predictive sufficiency. 579 ACKNOWLEDGMENTS 580 The author thanks Jack Devanney for correspondence on temporal radiation-response modelling. 581 The author is solely responsible for the derivations, interpretation, literature selection, and 582 manuscript. 583