Dose-to-time transport

### PDF page 27 Daniel J. Murray Revised September 2026 • biological validation is common-future equivalence, not retrospective curve agreement; • unknown population mixture is measured by calibrated paired probes or retained as a nuisance class; • the equivalence margin is fixed by endpoint consequences before unblinding; • infeasible power rejects the design before resources are spent; • quadratic degeneracy redirects effort from redundant challenge amplitudes to gap composition and prime-amplitude scaling; • reversible outcomes are separated by an assay-time ladder, and duration effects by isochronous schedules; • failure at any rung identifies the next state coordinate rather than licensing a profile-specific clock. This is the sense in which the framework is self-correcting: it cannot guarantee that scalar closure is true, but it can prevent a failed scalar model from being silently rescued. 12.3 Relation to DDREF and radiological protection An affine potential makes the endpoint independent of temporal profile and therefore passes the temporal audits trivially. A nonlinear model claims profile sensitivity and must declare the state, phase, recovery object, and record resolution that create it. DDREF is consequently not one universal constant inside this framework; it is a model-, profile-, endpoint-, and record-conditional quotient with a certifiable range. 12.4 Scope and limitations The falsification programme is principally a laboratory programme. Section 8 shows that a human- mortality implementation of the mixed-curvature example is infeasible. The paper does not establish any specific acute curve, transport closure across tissues or qualities, or equality between molecular repair time and epidemiological interaction time. The complete-monotonicity kernel result assumes positive independent linear modes. General non- normal linear recovery can produce a broader matrix-exponential kernel. Nonquadratic multi-state degeneracy, nonlinear recovery kernels, and non-associative endpoints remain open classes. Re- versible endpoints require an explicit score-recovery state before the cumulative-potential framework applies. 13. Conclusion An acute curve extends to a temporal increment model with one interaction coordinate only after two separate closure conditions have been earned. Every reachable challenge curve must be a translated copy of the acute curve, and waiting must evolve the resulting coordinate by one separately identified autonomous semigroup. The first is kinematic closure; the second is dynamic closure. Absolute outcomes may additionally require the accumulated ledger and a validated observation law. The quadratic endpoint potential separates them exactly. It makes every hidden state look scalar to challenge amplitudes, while leaving multirate recovery visible to gap composition. That degeneracy is the paper’s sharpest practical result: in the standard LQ class, the experiment most often treated as over-identifying state has no power, whereas an amplitude-resolved gap ladder can falsify dynamic scalar closure when signal and precision are adequate. The broader framework preserves that discipline across reset conventions, incomplete records, mixed curvature, reversible observations, and population heterogeneity. It outputs either a frozen predic- tion on a declared range or a specific measurement required to enlarge state. Nothing is repaired by convention. ---

The Acute Dose-Response Curve as a Transition Potential Predictive-State Closure, Quadratic Degeneracy, and Temporal Identifiability Daniel John Murray 7 September 2026 | audited revision Independent Researcher, Melbourne, Victoria, Australia amplitudes. Highlights • Predictive equivalence is defined by conditional future laws, not by a fitted biomarker, accu- mulated dose, or presumed mechanism. • The loading cocycle is leafwise: it gives one potential per dose trajectory. The acute curve determines off-trajectory increments only when those potentials are translates of one measured function. the transition from states that the acute experiment never visited? Histories are identified by equality of their conditional future laws within a declared intervention class, probe class, endpoint, and operating range. Fresh dose acts on the resulting predictive-state space through a semigroup Λ𝑑, and additive endpoint increments obey

mixed-curvature visibility and power criteria, the population cumulative hazard under frailty, a composition-law audit for reversible endpoints, and an isochronous design separating recovery from duration-dependent repopulation. The framework outputs either a frozen, testable scalar prediction or the specific experimental axis required to enlarge state. Keywords: predictive state; transition potential; quadratic degeneracy; incomplete recovery; split- dose design; semigroup test; archive width; population frailty 1. The measurement question before the risk model A temporal exposure is a history 𝑢(𝑡), not merely its cumulative loading 𝐷 = ∫ Radiobiology already contains explicit time dependence through Lea-Catcheside kernels, incomplete-repair models, dual-radiation-action theory, lethal-potentially-lethal lesion models and repair-misrepair kinetics [7-14,38-40]. Predictive-state, causal-state and bisimulation theories provide the complementary prospective criterion: two pasts are the same state only when no declared future experiment separates them [19-23]. The present paper connects these traditions. It asks what temporal transition is forced by an acute curve after a scalar predictive state has been earned, and what observations prove that one scalar is too small. 1.1 Four physical temporal structures and one numerical grid Five quantities are frequently merged under the word time: 1. Delivery chronology 𝑢(𝑡): when inputs occur.

an endpoint observed decades later; conversely, a population mortality curve does not identify the interaction-memory law that generated it [3-6,41]. 1.2 Three sources of an off-orbit prediction An acute experiment observes loading from a designated origin and then an endpoint. Every pre- diction from a recovered, preconditioned or chronically exposed state must come from one of three sources: 1. what the acute measurement fixes; after failure [50-52]. 1.4 Reader’s guide and principal notation Sections 2-3 construct predictive state and distinguish leafwise potential structure from genuine cross-trajectory scalar closure. Section 4 separates kinematic closure from autonomous recovery and the stochastic observation level. Sections 5-7 treat archive extrema, profile bounds, and reset phase. Section 6.5 proves the quadratic degeneracy that changes the experimental priority. Sections 8-10 address mixed curvature, population heterogeneity, established incomplete-repair theory, and panel, and operating conditions Λ𝑑 Fresh-dose action of magnitude 𝑑 on predictive state 𝑞(𝑠, 𝑑) Additive endpoint increment from state 𝑠 under fresh dose 𝑑 𝐺𝑧(𝑥) Potential on loading trajectory 𝑧 at dose coordinate 𝑥

total available interaction magnitude ℐ(𝑃 ) Prediction set compatible with record 𝑃 Figure 1. Histories are reduced by predictive equivalence. The construction returns either a complete scalar law or a specific state-growth experiment. T erminology . Recovery is decrease of the future-interacting state. Retention is the coordinate remaining after a stated gap. Kinematic closure means that challenge increments depend on one dose-equivariant coordinate. Dynamic closure means that waiting evolves that coordinate by one

Daniel J. Murray Revised September 2026 2. Predictive state before scalar state 2.1 Declared experiment Let a declared experiment be ℰ = (𝒜, 𝒱, 𝒴, 𝒲) , where 𝒜 is the set of admissible interventions, including waiting or washout when these change the system; 𝒱 is the future-probe class; 𝒴 is a separating response panel; and 𝒲 fixes scale, timing, target and baseline conditions. A history 𝑤 is a finite or continuous sequence of interventions in the declared class. For response feature 𝑔 ∈ 𝒴 , write 𝜇𝑔(𝑤𝑣) for the conditional response after history 𝑤 followed by future 𝑣. 2.2 Exact predictive equivalence Let P𝑣 𝑤 denote the conditional law of the declared response after history 𝑤 followed by future 𝑣. Definition 1 (predictive equivalence). Histories 𝑤 and 𝑤′ are equivalent, written 𝑤 ∼ ℰ 𝑤′, when P𝑣 𝑤 = P𝑣 𝑤′ for every 𝑣 ∈ 𝒱. (2) multivariate, recurrent, or general counting outcomes, equality of one mean is insufficient unless the response panel contains a measure-determining family of features. The equivalence class [𝑤] is the predictive state relative to the declared experiment. Exact equality is the mathematical target. In noisy data, a candidate partition is assessed with preregistered distances and scientific equivalence margins; a pairwise tolerance relation is not presumed transitive. Proposition 1 (right-congruence closure). If the future class is closed under prefixing by admissible action 𝑎, then 𝑤 ∼ℰ 𝑤′ ⇒ 𝑤𝑎 ∼ℰ 𝑤′𝑎. (3) Proof. Every declared continuation after 𝑎 is itself a declared future. Equality of the corresponding conditional laws is therefore inherited after prefixing. □ Proposition 2 (Nerode/causal-state minimality; standard restatement). The predictive quotient is the coarsest deterministic state representation sufficient for the declared futures [19-23]. It is unique up to relabelling and relative to ℰ; it is not an assertion that all biologically relevant state has been recovered. 2.3 Candidate scalar predictive coordinate Assumption A1 (candidate scalar coordinate). For the strong scalar predictive-state hypoth- esis, consider a calibrated component 𝐿 of the predictive quotient with the following properties: • 𝐿 is order-isomorphic to an interval; • fresh loading and recovery map 𝐿 into itself; • every state used for prediction is assigned a dose-equivariant coordinate only after its challenge family has been tested against the acute curve; • equal coordinates imply equal conditional future laws for the declared probes. A recovered history earns a scalar predictive-state interpretation only after common-future equiv- alence against the freshly loaded state labelled 𝑥 has been assessed. Equality of a fitted number

Daniel J. Murray Revised September 2026 is not enough. Sections 3 and 6.5 analyze the weaker hypothesis of a scalar challenge coordinate on a possibly multidimensional accessible set; they do not assume A1 or infer A1 from immediate challenge agreement. 2.4 State dimension depends on the declared probe class Proposition 3 (probe-relative state dimension). Let a homogeneous individual have survival

Daniel J. Murray Revised September 2026 3.2 Loading action and accessible set Let fresh dose act on predictive state through Λ𝑎+𝑏 = Λ 𝑏 ∘ Λ𝑎, Λ 0 = 𝑖𝑑. (8) Let 𝒳 be the accessible predictive-state set generated from the fresh state by dose and waiting. For the differential formulation assume a smooth manifold, a 𝐶 1 dose flow, and a nonzero fresh- dose vector field on the calibrated interior. Flow-box coordinates then exist locally. Statements about an entire loading trajectory require a nonrecurrent trajectory admitting a single-valued dose coordinate; nonzero vector field alone does not exclude periodic orbits or give a global coordinate.

recovery flow. Exponential recovery, 𝑟Δ(𝑎) = 𝑎𝑒−Δ/𝜏 , (26) is an additional substantive hypothesis and is not invariant under arbitrary reparameterisation of the state coordinate. 4.4 Stochastic endpoint, frailty, and reversible observations The accumulated log-effect 𝐻 in Equation (20) is not automatically a hazard on chronological time. An acute survival score at a later assay fixes one endpoint probability; it does not identify when ℎ′(𝑥)𝑢(𝑡) ≥ 0. Hormetic or other signed relative effects require a separate nonnegative total-hazard or observation model. Let 𝑇𝑒 be the event time, 𝑁 (𝑡) =1{𝑇𝑒 ≤ 𝑡}, and 𝑌0(𝑡) = 1{𝑇𝑒 ≥ 𝑡} the predictable at-risk indicator. Conditional on fixed frailty 𝜃 ≥ 0, specify the hazard while at risk as 𝛼𝑒(𝑡 ∣ 𝜃) = 𝜃 ̇𝐻(𝑡), 𝜆 𝑁 (𝑡 ∣ 𝜃) = 𝑌 0(𝑡)𝛼𝑒(𝑡 ∣ 𝜃). (27) For deterministic external exposure, no informative censoring, and no other observed frailty infor- mation, let 𝐿(𝐻) = 𝐸(𝑒−𝜃𝐻) and Φ(𝐻) = − log 𝐿(𝐻). The marginal hazard while at risk is

physical profile. The second concerns what can be inferred from an incomplete record. 5.1 Exact-profile representation invariance Let ℱ[𝑢] denote the prediction for a fully specified physical profile. Replacing one numerical mesh by a finer mesh without changing 𝑢 must not change the limiting value. Any exact segment flow forms a semigroup; a convergent numerical scheme approximates the same trajectory. A model that resets a biological state at every solver step has changed the model, not refined its calculation. This is a necessary lawfulness audit with deliberately limited discriminatory power. Properly speci- shift-level summaries plus dosimetric side information [29-32]. A record 𝑃 defines 𝒰(𝑃 ) = {𝑢 ∶ 𝑢 is compatible with record 𝑃 }. (30) For a declared model ℱ, define the compatible prediction set ℐ(𝑃 ) = {ℱ[𝑢] ∶ 𝑢 ∈ 𝒰(𝑃 )}. (31) Lemma 2 (record-set nesting). If record 𝑄 refines record 𝑃 , so that 𝒰(𝑄) ⊆ 𝒰(𝑃 ), then

Proof. Every profile compatible with 𝑄 is compatible with 𝑃 . Applying the same map to a subset cannot create values outside the image of the superset. □ Lemma 2 is deliberately unfalsifiable as a biological statement: it follows for every map ℱ. Its function is to audit the archive and optimisation implementation. A violation means that the feasible-set encoding, numerical extrema or uncertainty propagation are inconsistent. It does not select among biological models. 5.3 Archive-width certificate and analytic extrema For model class Θ, define ℋ(𝑃 , Θ) = {𝐻[𝑢; 𝜃] ∶ 𝑢 ∈ 𝒰(𝑃 ), 𝜃 ∈ Θ}, 𝐻 𝐿 = inf ℋ, 𝐻 𝑈 = sup ℋ. (34) A record is adequate for a preregistered scientific tolerance 𝜀 only if 𝐻𝑈 (𝑃 ) − 𝐻𝐿(𝑃 ) ≤ 𝜀. (35) Monte Carlo samples only an inner range unless extremality is certified. In the LQ-exponential class the principal extrema are analytic or convex. Proposition 5 (maximal rate-capped loading). Let ℎ be convex, 𝑅 ≥ 0 locally Lipschitz with

Goodness of fit tests kinematic closure directly in increment space. Separate inversion of each challenge is inferior near derivative plateaus because it divides by ℎ′(𝑟 + 𝑏) − ℎ ′(𝑟) before testing the prediction. For one challenge, the delta-method variance including acute-curve uncertainty is 𝑉 𝑎𝑟( ̂ 𝑟) ≈𝑉 𝑎𝑟( ̂𝐼) + 𝑉 𝑎𝑟[ ̂ℎ(𝑟 + 𝑏) − ̂ℎ(𝑟) − ̂ℎ(𝑏)] − 2𝐶𝑜𝑣[ ̂𝐼, ̂ℎ(𝑟 + 𝑏) − ̂ℎ(𝑟) − ̂ℎ(𝑏)] [ℎ′(𝑟 + 𝑏) − ℎ′(𝑟)] 2 . (49) 3. reserves futures not used to create the match; 4. compares conditional response, such as 𝑆(history+probe)/𝑆(history); 5. declares practical closure only when the confidence region lies inside a preregistered scientific margin [43]. Failure to reject a difference is not equivalence. Conversely, mathematically nonzero discrepancies may be practically negligible; the margin defines which claim is being tested. 6.5 Quadratic degeneracy and the true location of experimental power The challenge-family test is exact, but it has an exact degeneracy at the standard LQ potential. Theorem 2 (quadratic kinematic degeneracy). Let the physical predictive state be 𝑠 ∈ ℝ 𝑛, let fresh dose act as 𝑠 ↦ 𝑠 + 𝑐𝑑, and let 𝐺(𝑠) = ℓ 𝑇 𝑠 + 𝑠𝑇 𝐴𝑠, 𝐴 = 𝐴 𝑇 , 𝛽 = 𝑐 𝑇 𝐴𝑐 > 0. (50) Put 𝛼 = ℓ 𝑇 𝑐 and ℎ(𝑑) = 𝛼𝑑 + 𝛽𝑑 2. Then for every state 𝑠 and challenge 𝑏, 𝑞(𝑠, 𝑏) = ℎ(𝑟 + 𝑏) − ℎ(𝑟), 𝑟(𝑠) = 𝑐𝑇 𝐴𝑠

Daniel J. Murray Revised September 2026 Continuity, 𝑚(0) = 1 , and the nonincreasing positive-kernel hypothesis then force 𝑚(𝑡) = 𝑒 −𝜅𝑡 for 𝜅 ≥ 0 , including the no-recovery case 𝜅 = 0 . Conversely, this identity is sufficient for scalar output autonomy on the linear span reachable from zero by the same input direction. Indeed, with 𝑝 = 𝑐 𝑇 𝐴/𝛽 and 𝑠 = ∑𝑖 𝑎𝑖𝐸𝑡𝑖 𝑐, one has 𝑝𝐸𝑡𝑠 = ∑𝑖 𝑎𝑖𝑚(𝑡 + 𝑡𝑖) = 𝑒 −𝜅𝑡𝑝𝑠. It does not certify other

mortality implementation is not powered by these Gaussian log-effect examples. Geometric visibility does not imply experimental utility. The prospective score is 𝒱(𝑎, 𝑏) = max 𝑟 |𝐹𝑏(𝑟)| 𝑆𝐸[ ̂𝐹𝑏(𝑟)]

9.1 Population transition as a probe-class corollary Let individual susceptibility 𝜃 multiply cumulative potential 𝐻, with Laplace transform 𝐿(𝐻) = 𝐸(𝑒−𝜃𝐻) and Φ = − log 𝐿. Proposition 3 in Section 2 already predicts the state enlargement: survival conditioning changes the susceptibility distribution according to past 𝐻. If the individual challenge increment is 𝑞𝑖𝑛𝑑(𝑥, 𝑑), the exact conditional population increment is

𝑞𝑝𝑜𝑝(𝐻, 𝑥; 𝑑) = Φ[𝐻 + 𝑞𝑖𝑛𝑑(𝑥, 𝑑)] − Φ(𝐻). (75) Because Φ″(𝐻) = −𝑉 𝑎𝑟𝐻(𝜃) ≤ 0, this increment decreases with prior 𝐻 for fixed positive individual increment. The population prediction generally requires both the interaction coordinate 𝑥 and the exposure-dependent mixing information encoded by 𝐻. This is a sufficient coordinate description under the stated frailty model, not a universal two-dimensional lower bound: 𝐻 may already be a function of 𝑥 on a restricted reachable set. 9.2 Measuring the mixing transform when the design permits it

This is established radiobiology [9,10,12,38,44]. The new result is not the formula but Theorem 2: quadratic potentials make every hidden finite-dimensional state kinematically scalar under one dose direction. That algebraic degeneracy explains why additional challenge amplitudes cannot falsify multi-mode LQ recovery. For independent positive linear modes, replace the one exponential by the measured kernel 𝑚(Δ) = ∑ 𝑗

9. T est common-future equivalence. Match distinct routes on the candidate present and use untouched futures with nested bootstrap confidence regions. 10. F reeze and predict. Predict complete profiles not used in construction. Grow state only at the rung that fails.

covariance kinematic closure rejected; add a transverse predictive coordinate Assay-time ladder vary assay time at fixed gap and gap at fixed assay time outcome recovery and repopulation Common-future equivalence route-matched states, untouched probes, preregistered margin candidate present omits predictive history Archive certificate analytic formula, convex programme, or proved outer bound over uncertainty class chronology insufficient for one declared profile has a converged prediction solver rule changes the model R1 archive nesting and width

quotient with a certifiable range. 12.4 Scope and limitations The falsification programme is principally a laboratory programme. Section 8 shows that a human- mortality implementation of the mixed-curvature example is infeasible. The paper does not establish any specific acute curve, transport closure across tissues or qualities, or equality between molecular repair time and epidemiological interaction time. The complete-monotonicity kernel result assumes positive independent linear modes. General non- challenge amplitudes, while leaving multirate recovery visible to gap composition. That degeneracy is the paper’s sharpest practical result: in the standard LQ class, the experiment most often treated as over-identifying state has no power, whereas an amplitude-resolved gap ladder can falsify dynamic scalar closure when signal and precision are adequate. The broader framework preserves that discipline across reset conventions, incomplete records, mixed curvature, reversible observations, and population heterogeneity. It outputs either a frozen predic- tion on a declared range or a specific measurement required to enlarge state. Nothing is repaired