Action and diagnosis

PDF page 10 Daniel J. Murray Revised September 2026 Gate Claim Failure or limiting evidence R2 A required selection distinction is diagnosable. A proved upper bound on best admissible distinguishability cannot meet the declared selection-error target. A failed tested diagnostic alone is insufficient. R3 Diagnosis remains timely and safe. The tested diagnostic-control policy consumes the rescue opportunity or violates safety before effective treatment. R4 Extra information earns its cost. A cheaper/coarser code achieves equivalent held-out outcomes and risk within tolerance. A practical finite assay must estimate success probabilities with simultaneous uncertainty control. If lower confidence bounds 𝐿𝑠,𝑎 jointly cover all declared state-action probabilities, use the conservative library ̂𝐴− 𝛼(𝑠) = {𝑎 ∶ 𝐿 𝑠,𝑎 ≥ 1 − 𝛼}. A common action selected from these sets inherits the stated confidence guarantee for that finite table. Empty conservative intersections establish failure of certification, not impossibility of rescue; uncertainty can exclude genuinely acceptable actions. Report inconclusive results, freeze state reconstruction and code construction before final evaluation, and assess diagnostic delay and harm as part of the complete policy. 11. Discussion: a different meaning of biological information The framework suggests a disciplined hierarchy for biological information. A distinction can be real yet irrelevant to a particular prediction. It can be predictive yet irrelevant to action because several futures admit the same safe response. It can be action-relevant yet operationally inaccessible because the required measurement is too slow or destructive. And it can be measurable but useless because control authority has already vanished. This hierarchy helps explain why living control can be low dimensional without implying that living systems are simple. The molecular world can be enormous while the current decision boundary is small. Conversely, a visually simple present can demand a richer controller if hidden histories map to incompatible actions. The correct complexity measure therefore depends on the task: predictive-state dimension for future-law closure; finite hypergraph code size for static action selection; viability geometry for physical authority; and diagnostic distinguishability under time and safety constraints for operational recoverability. The result also sharpens the notion of a “state of health. ” Health cannot generally be one scalar reserve. A state is healthy relative to a declared persistence problem when viable controls remain available and the information-control loop can still identify and execute them under expected disturbances. This is compatible with specialization: a differentiated cell can lose many possible futures while remaining viable because the relevant action set has changed. The framework does not imply maximization of future-option volume or thermodynamic negentropy. 12. Conclusion Predictive Closure answers what the present must retain so that the future law is well defined. That is not yet the end of the adaptive problem. A controller must also know which distinctions change what it should do. The minimum action problem has a simple exact form. A compressed code is valid exactly when every code cell has a common acceptable action. On finite individually feasible PDF page 11 Daniel J. Murray Revised September 2026 domains, the minimum deterministic code is the chromatic number of the minimal-conflict hyper- graph, equivalently the minimum action cover. Recursive implementation requires an additional update condition. This makes low-dimensional control a falsifiable claim: proposed code classes must survive common-action tests, including higher-order conflicts. Recovery adds the final con- straint. The existence of a successful action in the true state does not guarantee that a real bounded observer can use it. If incompatible rescues sit behind an unresolved state distinction, the distinc- tion must be acquired safely before the rescue boundary closes. Statistical distinguishability places an error floor on unacceptable-action selection under the stated one-shot assumptions. Outcome risk is separately quantified; viability and time restrict admissible diagnostic policies. The resulting chain is History → predictive state → action-sufficient code → viable control → operational recovery . This is not a claim that biology optimizes one universal objective. It is a measurement architecture for asking, at any declared scale, whether the system knows enough, can still act, and can acquire the missing distinction before action ceases to matter.

From Predictive State to Viable Action Action Sufficiency, Safe Diagnosis, and the Operational Recoverability Bound Daniel John Murray September 2026 — reviewed revision Independent Researcher, Melbourne, Victoria, Australia Central result. Predictive state may retain more information than a declared action problem requires. A deterministic code is action-sufficient exactly when every occupied cell has a common acceptable action. For finite, individually feasible states, minimum code size equals a minimal- conflict hypergraph’s chromatic number and, equivalently, a set-cover minimum over actions. Phys- ical rescue can remain unavailable operationally when safe, timely information cannot support an ferent quantities. Abstract A predictive state retains every distinction from history that can alter a declared future law. A living controller, clinician, robot, or certified AI system often needs a coarser object: enough information to choose an acceptable action. Let 𝐴𝛼(𝑠) be the actions with success probability at least 1 − 𝛼 from state 𝑠. A deterministic code is action-sufficient if and only if every code cell has a nonempty common-action intersection. For finite, individually feasible state and action sets, minimum code outcome risks require separate accounting. We supply that distinction, a robust policy definition of operational recoverability, and finite reproducible checks. The contribution is a biological and experimental synthesis of predictive representations, state abstraction, set cover, viability, and safe diagnosis. Its empirical claim is that measured hidden-state distinctions can improve held-out action selection before a rescue window closes; no biological dataset or clinical efficacy result is established here. Keywords: predictive state; action sufficiency; state abstraction; viability; recoverability; safe diagnosis; information; control; biological state; decision theory. 1. The missing arrow: prediction is not action A complete predictive representation answers a demanding question: which histories must remain distinguishable so that every admitted future experiment has the correct conditional law? In

Daniel J. Murray Revised September 2026 controlled predictive-state language, histories ℎ, ℎ′ are equivalent when all declared future action- observation tests have the same law. The quotient is the coarsest exact representation for those tests. This definition can be stronger than what a particular controller requires [1–5]. A bacterium need not encode every future molecular trajectory. A tissue need not identify every microscopic configuration. A clinician need not possess a lossless model of a patient when different question: what is the least information that must be retained, sensed, or reconstructed to choose a viable action before the opportunity disappears? 2. Predictive state and acceptable-action sets Fix a history set ℋ, a declared future-test family 𝒯, outcome records, a finite horizon 𝑇, and a response resolution. For histories and interventions in this jurisdiction, let 𝐾(ℎ, 𝜏 ) = Law(𝑌 + ∣ ℎ, do(𝜏 )). (1) The kernels are assumptions or empirical targets; a conditional observational distribution alone does not identify an intervention law. Exact predictive equivalence is ℎ ∼ ℎ ′ ⟺ 𝐾(ℎ, 𝜏 ) = 𝐾(ℎ ′, 𝜏 ) for every 𝜏 ∈ 𝒯. (2) Write 𝑠 = [ℎ] ∈ 𝑆 = ℋ/ ∼ . Every proposed decision policy and every recorded quantity used in its success criterion must belong to the predictive jurisdiction. Otherwise success need not be a function of 𝑠. Let 𝐺 be a prospectively specified success event, 𝛼 ∈ [0, 1) an outcome-failure tolerance, and 𝒜 the admissible action library. A utility requirement must first be expressed as an explicit success event or as a separate acceptable-action inequality; probability of an unspecified “utility criterion” is not defined. Set 𝑝𝑠(𝑎) = Pr(𝐺 ∣ 𝑠, do(𝑎)), 𝐴 𝛼(𝑠) = {𝑎 ∈ 𝒜 ∶ 𝑝𝑠(𝑎) ≥ 1 − 𝛼}. (3)

Because the code has finitely many occupied cells, no measurability or infinite-selection issue arises. The resulting rule satisfies the inequality for every state. □ The criterion is elementary and exact. Predictively distinct states may share one action code when their common-action intersection is nonempty. For infinite measurable spaces, a measurable selector must additionally exist; setwise nonemptiness alone is not a theorem about implementable measurable control. 3.2 Predictive state bounds deterministic action-code size If two histories are predictively equivalent in the decision jurisdiction, all declared actions have the same success probabilities and hence the same acceptable-action sets. On an individually feasible finite state domain, the identity code always suffices. Consequently, 𝑚∗ ≤ |𝑆 +|. (6) This compares finite cardinalities; it does not compare a continuous manifold dimension with a number of codewords. The converse fails: different future laws can share an acceptable action and one codeword. Thus full predictive information can exceed decision-relevant information. The result is consistent with the established model-preserving versus policy-preserving abstraction distinction [3,4]. 3.3 No unique action state without a decision declaration Predictive equivalence is canonical relative to a future-test repertoire. Minimum action compression depends additionally on the success criterion, action library, failure tolerance, costs, and the use of deterministic, randomized, robust, or adaptive policies. Distinct partitions can attain the same minimum size. Each declared problem induces a family of sufficient compressions; there is no universal action partition established here.

The architecture can be written as four related questions; these are not four nested subsets of one common space. 1. Predictive state: what distinctions can still alter future laws? 2. Action sufficiency: which of those distinctions alter the existence of a common acceptable action? 3. Control/viability: are the required actions physically available while constraints are respected? 4. Operational recoverability: can the decision-relevant distinction be acquired safely and early being action-sufficient. A treatment can work in a fully observed model while being operationally unavailable under real diagnostic limits. A normal current readout can coexist with low control authority. A highly detailed molecular state can contain more predictive information than any viable controller needs. 7. Biological interpretations and tests 7.1 Redox and finite rescue windows The glutathione model in [12] motivates a redox test of whether abundance and regenerative capacity can dissociate in experimentally relevant regimes. Two preparations can have similar glutathione concentration while differing in NADPH-linked regeneration, oxidative load, or future rescue re- sponse. The predictive-state question asks whether the matched present predicts a common future. The action question is narrower: do the states share a rescue action? The operational question

measure a fast diagnostic response or compensating effort, and then freeze a rescue policy before the final outcome is revealed. The proposed redox application fails its added-value test if the extra state distinction neither improves held-out action selection nor predicts the closing rescue boundary. Such a result would not falsify the set-theoretic theorems. 7.2 Cancer and adaptive therapy Adaptive-therapy models explicitly distinguish sensitive and resistant populations and their chang- ing composition [16]. Equal total burden therefore need not specify the treatment-relevant state. Full prediction may require a rich state, while acceptable treatment may depend only on a coarser partition of that state. A minimum action code could therefore be substantially smaller than a complete molecular taxonomy. The hypergraph theorem gives a direct assay: estimate which inter- ventions are acceptable for each reconstructed state and test whether proposed code classes possess a nonempty common-action intersection. states whenever their safe repair policies become incompatible. This yields a stronger criterion than pattern correlation. Generate heterogeneous hidden conditions, estimate 𝐴𝛼(𝑠) over a prede- clared intervention library, construct the conflict hypergraph, and prospectively test whether the compressed code chooses actions that remain safe and effective on unseen tissues. 7.4 Disease and cure A disease state can be physically reversible but operationally unrecoverable if the state-specific rescue exists only behind a diagnostic distinction that cannot be acquired safely before commitment. bounded information is adequate when it preserves the distinctions required for safe action. 9. Relation to prior work and novelty boundary The components of this paper have substantial ancestry. Predictive-state representations encode systems using action-conditional predictions. MDP abstraction distinguishes full-model preservation

not asserted to have priority: 1. placing common-action compression explicitly downstream of experimentally reconstructed predictive state; 2. giving an exact finite minimal-conflict hypergraph characterization, and its classical set-cover dual, for deterministic action-code size; 3. making the distinction between physical and operational recoverability central to biological restoration and cure; 4. treating the rescue deadline as a joint information-control boundary that can be measured prospectively; 5. providing a cross-domain experimental grammar in which hidden state earns biological rele- vance by improving held-out action selection, not merely by improving retrospective fit. The application and added-value claims can fail independently; the conditional mathematical equiv- alences are assessed by their proofs and assumptions. If predictive distinctions rarely alter viable action, action codes will remain coarse. If diagnostics are always fast relative to rescue windows, operational recoverability adds little. If proposed common-action classes fail prospectively, the code must be refined. 10. Falsification ladder The gates test proposed models, codes, or empirical usefulness. Finite data cannot certify exact state equivalence or a universal action threshold across untested conditions. Gate Claim Failure or limiting evidence A1 Proposed state predicts declared action-conditioned futures. Prior history retains held-out predictive information after conditioning on state. A2 Proposed action code is sufficient. A code cell has a verified empty

11. Discussion: a different meaning of biological information The framework suggests a disciplined hierarchy for biological information. A distinction can be real yet irrelevant to a particular prediction. It can be predictive yet irrelevant to action because several futures admit the same safe response. It can be action-relevant yet operationally inaccessible because the required measurement is too slow or destructive. And it can be measurable but useless because control authority has already vanished. This hierarchy helps explain why living control can be low dimensional without implying that living systems are simple. The molecular world can be enormous while the current decision boundary is small. Conversely, a visually simple present can demand a richer controller if hidden histories map to incompatible actions. The correct complexity measure therefore depends on the task: predictive-state dimension for future-law closure; finite hypergraph code size for static action selection; viability geometry for physical authority; and diagnostic distinguishability under time and safety constraints for operational recoverability. The result also sharpens the notion of a “state of health. ” Health cannot generally be one scalar reserve. A state is healthy relative to a declared persistence problem when viable controls remain available maximization of future-option volume or thermodynamic negentropy. 12. Conclusion Predictive Closure answers what the present must retain so that the future law is well defined. That is not yet the end of the adaptive problem. A controller must also know which distinctions change what it should do. The minimum action problem has a simple exact form. A compressed code is valid exactly when every code cell has a common acceptable action. On finite individually feasible

domains, the minimum deterministic code is the chromatic number of the minimal-conflict hyper- graph, equivalently the minimum action cover. Recursive implementation requires an additional update condition. This makes low-dimensional control a falsifiable claim: proposed code classes must survive common-action tests, including higher-order conflicts. Recovery adds the final con- straint. The existence of a successful action in the true state does not guarantee that a real bounded observer can use it. If incompatible rescues sit behind an unresolved state distinction, the distinc- tion must be acquired safely before the rescue boundary closes. Statistical distinguishability places risk is separately quantified; viability and time restrict admissible diagnostic policies. The resulting chain is History → predictive state → action-sufficient code → viable control → operational recovery . This is not a claim that biology optimizes one universal objective. It is a measurement architecture for asking, at any declared scale, whether the system knows enough, can still act, and can acquire the missing distinction before action ceases to matter. Funding, competing interests, and AI disclosure claims, proofs, references, and interpretation. References 1. Littman ML, Sutton RS, Singh S. Predictive representations of state. Advances in Neural Information Processing Systems. 2001;14:1555–1561. 2. Shalizi CR, Crutchfield JP. Computational mechanics: pattern and prediction, structure and simplicity. J Stat Phys. 2001;104:817–879. doi:10.1023/A:1010388907793. 3. Givan R, Dean T, Greig M. Equivalence notions and model minimization in Markov decision processes. Artificial Intelligence. 2003;147:163–223. doi:10.1016/S0004-3702(02)00376-4. 4. Li L, Walsh TJ, Littman ML. Towards a unified theory of state abstraction for MDPs. Pro-