Finite rescue and fold regime

### PDF page 17 Daniel J. Murray Revised September 2026 Claim Required observation Failure or limitation Tumour-normal separation Tumour and relevant normal response surfaces separate under matched definitions Equal/earlier normal failure or incompatible assay normalization Spatial extension A stated coupling model predicts propagation outputs Propagation contradicts that coupling/local model combination 10. Conclusion Published experiments distinguish redox injury from the loss of a particular rescue opportunity, demonstrate dependence on exposure duration, state, intervention and environment, and show fer- roptotic propagation in suitably coupled populations. The acute rescue data and spatial descriptors are strong motivation for a dynamical theory, while durable clonogenic commitment requires its own measurement. For the proposed deterministic fold, fixed straddling finite sections give an exact arctangent passage law. Its effective exponent lies between 1/2 and 1 with the stated limiting behaviour. Symmetric sections give a monotone crossover; asymmetric sections need not. Time-varying forcing requires a derivative correction and explicit recovery dynamics. Population 𝑇50 follows the single-unit law only under an additional tested observation or heterogeneity model. These restrictions preserve a discriminating hypothesis. A calibrated depth-duration-rescue exper- iment, durable fate assay and held-out model comparison can establish whether the constrained family adds predictive value in NRF2-active cancer. Success would provide a quantified, operation- specific boundary of recoverability; failure would reject the tested model without erasing the inde- pendently established biological rescue phenomena. Glossary Term Definition Operational commitment Failure of a named rescue operation to restore a named fate endpoint. Rescue window Interval in which that operation retains efficacy under the specified assay. Survival margin 𝜇 Normalized local control parameter for the proposed fold. Fold depth 𝑎 Positive calibrated distance below the fold, 𝑎 = −𝜇. Mobility 𝑀 Positive kinetic coefficient in a fixed coordinate normalization. Finite sections Prespecified entry and rescue-boundary locations used to define passage. 𝑇𝑐 Deterministic passage time for an individual model trajectory. 𝑇50 Population duration yielding the prespecified 50% normalized survival endpoint. Trigger wave Self-regenerating propagation requiring local response dynamics and spatial coupling. ---

Daniel John Murray 7 September 2026 Independent researcher. Hypothesis and theory paper with secondary analysis of published data. Abstract NRF2-active cancers can maintain high antioxidant abundance while losing the capacity to survive redox injury after a specified rescue operation. Published studies demonstrate timed rescue, effects of cystine withdrawal, intervention-dependent survival, and spatial ferroptotic propagation. These findings motivate a local saddle-node hypothesis; they do not identify its parameters or establish durable clonogenic rescue. For the deterministic normal form ̇ 𝑥 = −𝑀 (𝑎 + 𝑥2) at constant cali- brated depth 𝑎 > 0, passage between fixed sections 𝑥𝑖 = 𝐴 > 0 and 𝑥𝑓 = −𝐵 < 0 is an arctangent interval divided by 𝑀 √𝑎. Its effective depth-duration exponent lies strictly between 1/2 and 1, approaching those limits at shallow and deep forcing within the model. A monotone crossover follows for symmetric sections; asymmetric sections can produce a nonmonotone exponent profile. This constrained finite-section family is the principal prediction. Time-varying depth requires a coordinate-derivative correction, and interrupted treatment must retain recovery dynamics. A five- depth by seven-duration starting design, adjusted by pilot calibration, combines target engagement, a fixed rescue operation and 7–14-day clonogenic follow-up. A stated observation model links deter- ministic passage to population survival. Fold, cumulative-exposure, fixed-power and smooth-hazard No existing dataset considered here jointly identifies calibrated depth, finite sections, rescue and durable fate. Failure of the prespecified model within a demonstrably tested regime would reject that operational fold hypothesis. Keywords: NRF2; KEAP1; glutathione; GPX4; ferroptosis; rescue window; saddle-node bifurca- tion; clonogenic survival. Claim discipline. Published experiments establish timed and state-dependent rescue in their stated systems and endpoints. The finite-section fold law remains a quantitative hypothesis for a specified rescue operation and biological regime. It is not a universal law of ferroptosis or a clinical treatment recommendation. 1. Introduction: abundance is not recoverability A calibrated model of glutathione homeostasis separates oxidative collapse from reductive fade under

These systems share one local fold Compatible hypothesis The normal form has not been empirically identified across them. The finite-section exponent family predicts durable rescue Open Requires calibrated depth-duration-rescue data and rival-model comparison. Figure 1: Figure 1. Evidence architecture: assay-specific rescue, supply perturbation, route depen-

4. Local fold model of recoverable survival 4.1 Normal-form assumptions and dimensions A saddle-node is not implied by every codimension-one loss of stability. The fold hypothesis assumes a smooth deterministic drift with one simple zero eigenvalue at the critical equilibrium, stable remaining modes, nonzero parameter derivative in the critical direction and nonzero quadratic coefficient there. After centre-manifold reduction and a fixed local normalization, the leading drift has the form 𝑀 (𝜇 − 𝑥 2) [23–30]. Hopf, transcritical, pitchfork, non-smooth and multiple-slow- mode transitions require different hypotheses. Molecular feedback and observed bistability motivate testing a fold; they do not prove these nondegeneracy conditions. The stochastic working model is 𝑑𝑥 = 𝑀 (𝐸)[𝜇 − 𝑥2]𝑑𝑡 + 𝜎(𝐸)𝑑𝑊𝑡. (1) Here 𝑊𝑡 is standard Brownian motion and the stochastic equation is interpreted in the Itô sense.

Grx1-roGFP2 and hydrogen-peroxide indicators can report components of state [31–33]. They do not directly measure ATP or NADPH supply flux, nor does a single reporter automatically determine 𝐸, 𝜇 or 𝑀 . A covariate earns such an interpretation through independent calibration and predictive performance. 4.3 Exact finite passage at constant depth Set 𝜇 = −𝑎 < 0 and hold 𝑎 and 𝑀 > 0 constant during a pulse. For 𝑥𝑓 < 𝑥 𝑖, direct separation of ̇ 𝑥 = −𝑀 (𝑎 + 𝑥2) gives

2(1 + 𝑞2) arctan 𝑞 , 𝑞 = 𝑠/ √𝑎. The derivative of 𝑞/[(1 + 𝑞2) arctan 𝑞]has numerator (1 − 𝑞2) arctan 𝑞 − 𝑞 < 0, and 𝑞 decreases with 𝑎. This proves the restricted monotonicity result. The experimentally testable prediction is therefore the full constrained finite-section family, not a universal monotone path between its limits.

dynamics and passes the resulting state into the next pulse. Summing 𝑀 √−𝜇 only over negative- margin intervals discards recovery and is not a general commitment criterion. For interrupted, rapidly changing or noisy treatment, the primary prediction is direct integration of a prespecified state model with the specified rescue operation. The exact constant-pulse test remains the first experimental stage.

Daniel J. Murray Revised September 2026 5. A discriminating experimental specification 5.1 Minimal falsification stage and endpoint Begin with one established ferroptosis model whose effective perturbation and rescue can be mea- sured reproducibly. An inducible GPX4-loss line provides a genetically defined perturbation, but induction strength, knockdown kinetics and rescue timing still require calibration. A pulse-washout system is acceptable only if drug removal, persistent target engagement and the resulting stress tra- cue maintained during the entire assay measures survival under continued support; it cannot be described as treatment-free restoration. T able 3. Starting design, subject to prospective pilot adjustment. Component Specification Purpose Biological system One characterized model; matched starting state and culture conditions Test the local hypothesis before generalization Crossing threshold Estimate 𝐹𝑐 independently or propagate a narrowly justified calibration uncertainty

Daniel J. Murray Revised September 2026 Component Specification Purpose Durable fate Preregistered clonogenic or equivalent reproductive endpoint Separate lasting loss from transient assay suppression

Use held-out forcing depths as the main extrapolation test, with independent repeat/batch holdouts for reproducibility. Randomly splitting technical wells while sharing all depths is a weaker test of functional shape. Preregister prediction scores, acceptable calibration error and a model-selection rule. AICc may be a secondary criterion only for comparable likelihoods, correctly counted fit- ted parameters and a defensible independent sampling unit; it must not use repeated timepoints as independent replicates. The direction of any reported difference must be stated, for example ΔAICc = AICcrival − AICcfold > 4 . Where small-sample likelihood assumptions do not support AICc, use the prespecified predictive comparison instead. The model has an exact normalization symmetry: 𝑥 ↦ 𝑐𝑥 , 𝑎 ↦ 𝑐 2𝑎, 𝐴 ↦ 𝑐𝐴 , 𝐵 ↦ 𝑐𝐵 and 𝑀 ↦ 𝑀 /𝑐 preserve Eq. (4). Consequently passage times alone do not identify the absolute coordinate scale or mobility. The sections are also exchangeable in this formula: 𝐴 and 𝐵 cannot be separately labelled as entry and exit from passage-time data alone. Fix the normalization using 5.4 Decision rules Support requires a reproducible rescue-response transition, acceptable exposure and state calibra- tion, and improved held-out prediction by the prespecified finite-section model relative to the rele- vant rivals. The effective exponent should match the fitted section geometry within the measured regime. Monotone increase is required only for independently justified symmetric sections, not for every finite-section fold. A well-powered failure of the common-parameter arctangent family within its calibrated determin- istic regime rejects that operational fold model. A rival consistently predicting better weakens the claimed distinctive value of the fold. If no transition is found despite a design established to span relevant rescue and failure states, the proposed boundary model fails for that tested scope. Re- sults remain inconclusive when threshold calibration, observation-model adequacy, accessible depth range, duration bracketing or precision prevents discrimination. These categories must be set before seeing confirmation data. Figure 5: Figure 5. Prospective test sequence and the observation bridge between unit-level passage and a population rescue surface. Pilot calibration, a frozen confirmation design and held-out-depth prediction are separate stages.

For constant 𝜇 > 0 and 𝑀 , linearizing at 𝑥𝑠 = √𝜇 gives 𝜆return = 2𝑀 √𝜇. (10) This secondary prediction requires the same local coordinate and a controlled or independently measured mobility. Near a fold, additive noise, drift and finite observation windows can obscure the deterministic return rate. In the Ornstein–Uhlenbeck approximation, stationary variance satisfies Var(𝑥)𝜆return = 𝜎 2/2. That identity holds for any stable linear scalar relaxation with additive white noise; it does not diagnose a fold. Controlled small perturbations and recovery measurements are non-destructive perturbations can assist calibration before destructive pulses, but perturbation- induced adaptation and phototoxicity require controls. Retain an independently estimated mobility covariate only if it improves prespecified held-out predictions. 6.3 Calibrating effective forcing Nominal drug concentration is not fold depth. Near a calibrated threshold, use 𝑎 = 𝜅(𝐹 phys − 𝐹𝑐,phys) + 𝑜(|𝐹phys − 𝐹𝑐,phys|) with 𝜅 > 0 in stated units, or normalize effective forcing so 𝑎 = 𝐹 − 𝐹 𝑐. Target engagement may constrain this mapping, but a merely monotone reporter does not determine nant GPX4 under their tested biochemical assay conditions [39]. That result should be read along- side cellular GPX4-associated evidence and assay-specific target-engagement information [11,15]. The hypothesis does not settle the compounds’ full target pharmacology. Washout may remove free compound while leaving persistent covalent target inhibition; a nominal pulse therefore need not be a rectangular pulse of 𝜇. Genetic perturbations or validated cellular engagement assays are preferable to equating concentration directly with 𝑎, and time-dependent forcing should be modelled when measurements require it.

geometry and relevant boundary conditions must be specified and independently tested. Outputs become initiation probability, front velocity and transmission range. The Co and Roeck results mo- tivate distinct coupling hypotheses. Failure of a proposed coupling mechanism rejects that extension without automatically disproving every uncoupled local fold. Figure 6: Figure 6. Heterogeneous and spatial extension: sample-specific trajectories and a stated rescue operation determine local fate probabilities; an additional coupling model is required for tissue-scale propagation. NRF2-active cancers provide a stringent test because high antioxidant abundance can coexist with substrate and pathway dependencies. SLC7A11/xCT activity can generate cystine and glucose de- pendencies [34]. Lipid composition, p53 state and antioxidant interventions can alter ferroptosis susceptibility or tumour progression [35–38]. The model asks how far a state is from loss of recov- erable survival and how rapidly its support can be redeployed; antioxidant abundance alone does not answer those questions. Prospective calibration should distinguish basal abundance, inducible reserve, cystine/GSH support, NADPH-linked regeneration, target engagement and at least one dynamic reporter. These are candidate explanatory variables, not automatic measurements of 𝐹 , 𝜇 or 𝑀 . Their contribution requires predictive evaluation with appropriate controls. The delayed normal-tissue protection in SCLC [18] provides an example of intervention ordering, not evidence that every constitutively NRF2-active malignancy has the same response.

No clinical regimen follows from this paper. Antioxidants, pro-oxidants, dietary interventions, fer- roptosis inducers and treatment schedules should not be changed in patients on the basis of these calculations. The proposed experiments are laboratory tests of a dynamical hypothesis. No sex- or gender-stratified inference is made. The cited studies use diverse lines and systems with incompletely reported sex information. Prospective experiments should report cell-line or donor sex where known, authentication, culture conditions and other plausible effect modifiers, and should state the scope of any comparison. T able 4. Prespecified claims and failure conditions. Claim Required observation Failure or limitation otherwise unbracketed/inconclusive Finite-section fold family Improved held-out predictions under fixed calibrated sections, mobility and observation model Rival consistently predicts better, or response shape contradicts the constrained family Symmetric monotone crossover Independently justified Mobility covariate matters Independently calibrated dynamic measurement improves prediction No reproducible incremental predictive value

normalization Spatial extension A stated coupling model predicts propagation outputs Propagation contradicts that coupling/local model combination 10. Conclusion a derivative correction and explicit recovery dynamics. Population 𝑇50 follows the single-unit law only under an additional tested observation or heterogeneity model. These restrictions preserve a discriminating hypothesis. A calibrated depth-duration-rescue exper- iment, durable fate assay and held-out model comparison can establish whether the constrained family adds predictive value in NRF2-active cancer. Success would provide a quantified, operation- specific boundary of recoverability; failure would reject the tested model without erasing the inde- pendently established biological rescue phenomena. Glossary Term Definition