Rescue predecessor

### PDF page 12 Finite rescue windows and redox commitment 10. Conclusion Published experiments now establish that redox injury and irreversible commitment are temporally distinct, that rescue efficacy depends on dwell time, cell state, intervention and microenvironment, and that cystine-restricted populations can become bistable and propagate ferroptotic trigger waves. These findings make a dynamical theory of recoverability scientifically warranted. The corrected fold model makes a sharper prediction than the earlier inverse-square-root statement. Finite biological sections generate a specific crossover in the effective depth-duration exponent from 1/2 near threshold toward 1 under deep forcing. This both narrows the claim and identifies the regime in which competing models can be distinguished. The paper supplies an executable, pre-specified falsification test. If the rescue boundary is absent or the finite- section crossover does not outperform cumulative-dose and hazard alternatives on held-out conditions, the model should be rejected or revised. If it survives, redox oncology gains a quantitative object that abundance measurements cannot provide: the time-dependent boundary between an injured cell and a cell that can no longer be brought back. Glossary Term Definition Commitment Failure of a prespecified rescue operation to restore a prespecified durable fate endpoint. Rescue window Interval after injury during which the chosen intervention can still restore durable survival. Survival margin, µ Local control parameter measuring whether a recoverable stable state exists. Fold depth, a Calibrated distance below the local crossing, a = F − F_c when µ = −a. State-transition mobility, M Kinetic scale multiplying both recovery above the fold and passage below it. Finite biological sections Operational entry and commitment locations used to define passage time. Trigger wave Self-regenerating spatial propagation through coupled bistable or excitable cells. Data and code availability No new wet-lab, animal or patient data were generated. Quantitative figures use public source-data workbooks released with Wiernicki et al. [12] and Co et al. [16]. Analysis scripts, derived data, bootstrap settings, synthetic benchmark datasets and deterministic figure-generation code are supplied as supplementary reproducibility material. Raw third-party source files should be obtained from the original publications. Ethics statement This hypothesis paper and secondary analysis used only published, de-identified aggregate or cell-line data. No new studies involving humans or animals were conducted. Funding This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors. Declaration of competing interests The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

commitment in NRF2-active cancer: fold geometry and a decisive experimental test Hypothesis paper Abstract NRF2-active cancers can maintain high antioxidant abundance yet still pass from recoverable redox injury to irreversible death. The unresolved problem is whether commitment follows a measurable dynamical law. Published experiments establish finite rescue windows after GPX4 loss, RSL3 pulse-washout and cystine passage time is an arctangent phase interval divided by M√a. The effective depth-duration exponent is therefore not fixed: it runs from 1/2 near threshold toward 1 under deep forcing. This finite-section crossover is the model's strongest discriminating prediction. Existing datasets do not jointly provide calibrated forcing depth, multiple pulse durations, a defined rescue operation and durable fate. A recommended starting experiment uses an independently estimated crossing threshold, five near-threshold depths, seven durations, washout/rescue and 7-14 d clonogenic survival, with final replication set from pilot variance. The fold model must outperform cumulative- dose, fixed-power and hazard alternatives on held-out conditions. Failure to produce a separable rescue boundary or the predicted crossover would falsify the model. Keywords: NRF2; KEAP1; glutathione; GPX4; ferroptosis; rescue window; saddle-node bifurcation Claim discipline Published experiments establish the biological phenomenon: finite and state-dependent redox rescue windows. They do not yet validate the exact fold law. The law is retained only as a quantitative hypothesis with explicit failure conditions. 1. Introduction: abundance is not recoverability A calibrated model of glutathione homeostasis has already shown how finite NADPH-linked supply can separate oxidative collapse from reductive fade in G6PD deficiency and NRF2-active lung cancer [1]. A complementary bounded-adaptive analysis derived the conditions under which separated repair and toxicity thresholds generate commitment. Second, it corrects the finite-passage mathematics and identifies a constrained exponent crossover that is more specific than a single inverse-square-root law. Third, it specifies a minimal, executable experiment that can falsify the model without requiring a new molecular construct.

A local fold unifies these observations Compatible The normal form is a parsimonious hypothesis, not an empirical identification. The finite-section crossover is universal Open It requires calibrated depth-duration-rescue data and rival-model comparison. 3. Secondary analysis and audit of available datasets

an exact arctangent phase interval. (C) Depth, duration and state-transition mobility jointly set the boundary. (D) In time-dependent treatment, phase accumulates only while the local margin is negative. 4.4 Finite sections predict an exponent crossover Let A = x_i > 0, B = −x_f > 0 and u = √a. The local effective exponent is p_eff(a) = −d ln T_c/d ln a. Differentiation gives p_eff(a) = 1/2 + {uA/(u² + A²) + uB/(u² + B²)}/[2C(a)] When a is small relative to the squared section distances, both arctangents saturate and p_eff approaches 1/2. Under deep forcing, arctan(q) ≈ q, C(a) ≈ (A + B)/√a and T_c ≈ (A + B)/(Ma), so p_eff approaches 1. The model therefore predicts a constrained crossover rather than one freely fitted power law. The crossover location is set by the entry and commitment sections. This correction changes the experimental design. Deep-forcing conditions cannot sharply distinguish the fold from cumulative dose because both approach inverse-first-power behaviour. Most forcing depths must be concentrated just above an independently estimated crossing threshold F_c.

I ≥ C_commit(R,z,Y) (8) R denotes the prespecified rescue operation, z the cellular state and Y the durable fate endpoint. The phase formulation is a hypothesis for slowly varying forcing. Rapid, non-adiabatic transitions may require direct integration of the full state model rather than an accumulated-phase approximation. 5. A decisive experimental specification 5.1 Minimal falsification stage The minimal experiment is designed to reject the theory before a larger tumour-normal programme is attempted. It can use an established inducible GPX4-loss line or a validated pulse-washout system. No new construct is required if target engagement and rescue timing are already measurable. The five-depth by seven-duration matrix below is a recommended starting grid, not a power guarantee; final replication and range should be adjusted from on log-duration, with uncertainty from independent biological repeats. This operational definition prevents metabolic suppression during exposure from being mistaken for commitment. Table 3. Minimal experiment that can falsify the fold model. Component Specification Reason System One established ferroptosis model with reliable pulse-washout or inducible GPX4 control. Minimizes construction cost and tests the law before Durable fate 7-14 d clonogenic survival or equivalent reproductive fate. Separates irreversible loss from temporary metabolic inhibition. Secondary state Lipid peroxidation plus one redox/energetic reporter. Tests whether state calibration improves prediction. 5.2 Model comparison and identifiability The exact finite-section model, cumulative-dose model, a single free power law and a smooth hazard or damage- accumulation model must be fitted to identical training conditions and compared on held-out depths or repeats. Model selection should use held-out prediction error and AICc, not visual resemblance. F_c must not be allowed to drift without constraint because threshold and exponent are jointly weakly identified near a critical point.

Finite rescue windows and redox commitment Before data collection, the laboratory should preregister a superiority rule, for example lower held-out error together with ΔAICc > 4; outcomes failing that rule remain inconclusive rather than supportive. • Finite-section fold: exact Eq. (4), with common x_i and x_f across forcing depths. • Cumulative dose: T50 = K/F or a prespecified exposure-area variant. • Single free power: T50 = K(F − F_c)^(−p), with one fixed p. 5.3 Decision rules Support requires all of the following: a separable rescued/non-rescued boundary, a finite-section model that meets the preregistered superiority rule over rivals, and an exponent profile that increases from near 1/2 toward 1 across calibrated depth. Falsification occurs if no separable boundary exists under controlled induction or if a rival model consistently predicts held-out conditions better. Results are inconclusive when F_c is poorly constrained, the boundary lies outside the sampled duration range, pilot-adjusted sampling is inadequate, or uncertainty intervals are too wide to discriminate. Figure 5. Executable falsification design. (A) Estimate the local crossing threshold, concentrate forcing depths just above it, vary pulse length, apply a fixed rescue operation and follow durable clonogenic fate. (B) The primary endpoint T50 maps the rescue boundary. (C) The full finite-section crossover must outperform cumulative-dose, single-power and smooth-hazard alternatives on held-out conditions. 6. Measurement and calibration 6.1 Return-rate scaling For µ > 0, linearization around the stable point x_s = √µ gives a return rate λ_return = 2M(E)√µ (9) The square-root relation is a secondary prediction, but passive variance alone is not diagnostic. For an Ornstein- Uhlenbeck approximation, variance multiplied by return rate is σ²/2, which holds for any linear relaxation with additive noise. A driven perturbation or controlled recovery assay is therefore preferable.

phase lag determined by λ_return. Repeated, non-destructive perturbations can estimate M and local control distance before the destructive pulse matrix. The mobility estimate earns a place in the model only if it improves held-out prediction of T50. 6.3 Calibrating effective forcing Nominal drug concentration is not fold depth. Effective forcing should be calibrated through target engagement or a monotone state variable that tracks the relevant stress axis. The model assumes that the calibrated forcing-to- depth map is locally linear near F_c. If calibration is demonstrably nonlinear, that map must be estimated and (11) Oxygenation, nutrient access, drug penetration, KEAP1/NRF2 state, FSP1, DHODH, membrane lipids and cell- cell contact alter the parameter distribution. Heterogeneity therefore predicts a distribution of rescue boundaries. It does not license fitting one threshold to a whole tumour. At larger scales, spatial coupling can generate fronts. The correct outputs become initiation probability, front velocity and transmission range. The local fold and the tissue front are related levels of description, not competing explanations.

8. NRF2-active cancer as the principal application NRF2-active cancers are a stringent test because they combine high antioxidant abundance with substrate and pathway dependencies. SLC7A11/xCT can create cystine and glucose dependencies [34]. Lipid composition, p53 state and antioxidant supplementation can further alter ferroptosis susceptibility and tumour progression [35-38]. The model does not classify cells as simply antioxidant-rich or antioxidant-poor. It asks how far each state is from loss of recoverable survival and how quickly support can be redeployed. The relevant calibration cannot be reduced to NRF2 abundance alone. A prospective test should distinguish basal antioxidant abundance, inducible reserve, cystine/GSH support, NADPH-linked regeneration, GPX4 target engagement and at least one dynamic redox or energetic reporter. These quantities are candidates for calibrating F, µ and M; none should be identified with a model variable unless it improves held-out prediction. A tumour-selective boundary is meaningful only relative to the dose-limiting normal compartment. Normal comparators should therefore be matched as closely as feasible in tissue origin and proliferative state, and tumour and normal boundaries should be estimated with the same rescue operation and durable endpoint. A controlled single-cell system is the first gate. Extension to organoids, co-cultures or tumours is justified only inducers and treatment schedules should not be altered in patients on the basis of this model. No sex- or gender-stratified inference is made. The cited studies use diverse cell lines and experimental systems for which sex information is not consistently reported. Prospective tests should report donor or cell-line sex where known and test whether it changes the calibrated boundary. Table 4. Pre-specified claims and failure conditions. Claim Required observation Failure condition Finite operational rescue boundary Timed rescue separates recovered and non-recovered better than rivals. Cumulative-dose, fixed-power or hazard model predicts consistently better. State-transition mobility matters Independent M estimate improves held-out T50 prediction. M adds no reproducible predictive information. Tumour-normal separation Tumour crosses before relevant normal compartments under a schedule. Normal compartments cross at equal or lower forcing.

populations can become bistable and propagate ferroptotic trigger waves. These findings make a dynamical theory of recoverability scientifically warranted. The corrected fold model makes a sharper prediction than the earlier inverse-square-root statement. Finite biological sections generate a specific crossover in the effective depth-duration exponent from 1/2 near threshold toward 1 under deep forcing. This both narrows the claim and identifies the regime in which competing models can be distinguished. The paper supplies an executable, pre-specified falsification test. If the rescue boundary is absent or the finite- section crossover does not outperform cumulative-dose and hazard alternatives on held-out conditions, the model should be rejected or revised. If it survives, redox oncology gains a quantitative object that abundance measurements cannot provide: the time-dependent boundary between an injured cell and a cell that can no longer be brought back. material. Raw third-party source files should be obtained from the original publications. Ethics statement This hypothesis paper and secondary analysis used only published, de-identified aggregate or cell-line data. No new studies involving humans or animals were conducted. Funding This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors.