Temporal/environment/adaptation backbone

### PDF page 22 Daniel J. Murray Revised September 2026 The quantum bridge is useful because it shows, at a lower physical level, that record formation and state update cannot always be separated conceptually: an instrument creates an outcome record and a conditioned successor state, while record distinguishability trades against locally accessible coherence. The biological bridge is deliberately weaker. Organisms are record-forming physical systems, but their memory mechanisms are biochemical, cellular and network processes with their own update laws. The shared architecture is event → retained distinction → altered future. The two published biological anchors also delimit the claim. Hormesis and glutathione homeostasis are mechanistically different. Their relevance is not that they prove one universal curve; it is that both become intelligible when capacities, competing channels, thresholds and future authority are declared explicitly. Mechanism chooses the parameters and substrate. Architecture constrains which observables and closure tests can be valid. The independent examples added in this revision matter because they converge on different parts of the architecture without being mutually dependent. The MWC work shows that predictive structure can change while coarse functional summaries remain smooth [29]. The anchovy-hake ap- plication operationalises robust viability under biological and economic constraints [30]. Human fear experiments separate retrieval from prediction-error-dependent updating [31,32]. Active-inference niche-construction simulations independently treat agent and environment as a coupled attract- ing system [33]. Large-scale hormesis literature establishes recurrence of the biphasic phenotype [34], while recent GPx4 work supplies an independent redox-flux constraint and threshold picture [35]. None proves the whole framework; together they make it increasingly difficult to dismiss the architecture as a metaphor assembled only from the author’s own examples. The paper therefore rejects several tempting but unsupported upgrades. The existence of a forward- invariant regime does not imply that evolution searches for fixed points. Predictive rank is not automatically neuron count, energy or organismal complexity. Robust viability is not numerically identical to allostatic load. Memory addresses are not quantum branches. And no category-theoretic reformulation is needed to obtain the central result: the concrete quotient-congruence theorem already supplies the composition closure required for temporal state. 20. Conclusion A living system is not fully described by what can be measured at one instant. It is a continuing unit whose present state is made from the distinctions of prior interaction that remain relevant to its future. The histories that reach the present have already been filtered by constraints. The state that remains must be sufficient for future prediction. Viability determines whether admissible continuation exists; robust viability asks whether it survives a disturbance class. Action changes the organism and can change the environment that supplies later constraints. The architecture can be written compactly as declared unit → constraint-filtered history → predictive state → physically certified viable futures → action → altered environment → new constraint . (25) Every arrow has an empirical failure mode. That is what makes the framework more than a metaphor. If matched histories with the same declared state diverge under a common future, the state is incomplete. If rank exceeds the proposed dimension, the linear closure is too small. If nominal viability disappears under realistic disturbance, reserve was overstated. If action does not alter later constraints, the eco-evolutionary return edge is absent. If a recovered snapshot fails rechallenge, restoration has not occurred. FINAL STATEMENT: A living present is the future-sufficient residue of a constraint-filtered past. ---

The Temporal Architecture of Living Nature Predictive state, robust viability, and the recursive construction of biological futures Daniel J. Murray September 2026 CENTRAL RESULT: For a prospectively declared biological unit, the present state is complete only when histories identified as the same state are indistinguishable under every declared future test. The histories that can reach the present have already been filtered by prior physical and viability constraints. A stated physical realization and control information pattern determine which continuations are robustly viable, and biological action can alter the environment that generates Living systems are historical objects: two systems can share an instantaneous measurement yet respond differently to the same subsequent challenge. This paper develops a formal architecture for that fact. A biological unit is declared prospectively by an operational boundary, an iden- tity criterion and a time horizon; state is then discovered rather than assumed. Histories are predictively equivalent relative to a declared class of future action-observation tests when every permitted future has the same response law. The resulting predictive quotient is the coarsest exact state description for that future class, and—when the future family is closed under continuation— admissible operations descend to well-defined transitions on the quotient when their domains also respect the quotient; continuous records require an almost-sure kernel formulation. The living present is the quotient of histories that survived prior constraints. Finite-horizon nominal contin- generate a closed-loop map whose self-maintaining object is generally a forward-invariant regime, with fixed points only as a special case. If reproductive output is included among the future tests, predictive equivalence also implies equality of the corresponding conditional fitness law without identifying fitness with viability. Finite observers are treated as compressors of interaction history; exact compression is predictive sufficiency and approximate compression is naturally related to the information bottleneck. Quantum measurement supplies a lower-level physical instance in which record distinguishability, local coherence and record-conditioned state update can be stated exactly; memory reconsolidation supplies a distinct biological instance of record-dependent state updating. Independent literature examples now provide cross-scale convergence without becoming premises: a biologically motivated Monod-Wyman-Changeux sensor model shows causal-state complexity can change qualitatively under small kinetic changes; robust viability has been computed for a Peruvian anchovy-hake management model; human fear-memory studies identify prediction error as a bound- ary condition for reconsolidation; and joint agent-environment niche-construction models produce attracting sets in the coupled state space. Two published quantitative anchors—bounded-adaptive hormesis and glutathione/NRF2 redox homeostasis—illustrate how specific mechanisms instantiate

Daniel J. Murray Revised September 2026 the broader architecture. Every claim is labelled as established theorem, proved consequence, pub- lished empirical result, literature-supported synthesis, model application or prospective hypothesis. The central thesis is testable: a present biological description is a state only if no admissible future can recover a distinction that description has erased. Keywords: predictive state; viability; robust control; biological identity; temporal composition; memory; niche construction; observer; quantum measurement; hormesis; recovery Highlights • Biological units are declared prospectively; predictive state is then discovered from future- response equivalence rather than used circularly to define the unit. • The predictive quotient is the coarsest exact state partition and, under continuation closure and quotient-compatible action domains, supports well-defined successor dynamics. • Nominal viability and robust viability are different: the latter requires preserving admissible futures against a declared disturbance class. • Order, waiting and profile effects all test temporal closure, but they are controlled by different commutators and neither generally implies the other. • If reproductive output is included in the future-test class, predictive equivalence automatically preserves the conditional fitness law. • Finite observers compress history; the exact compression criterion is predictive sufficiency, while approximate compression is an information-bottleneck problem. • Every programme theorem used here is reproduced self-contained; the two quantitative anchors are independently citable published studies. Declared unit Surviving histories Prior constraints Predictive state Declared future laws Physical viability Control + disturbance Action in environment Subsequent histories Figure 1. The temporal architecture. Prior constraints filter which histories can reach the present. Future-response equivalence compresses the surviving histories into predictive state. State determines accessible futures and action. Organismal action can modify environmental state and therefore later constraints. The loop is causal and does not imply teleology.

mitted future can distinguish histories identified as the same state. This has established precedents. Computational mechanics defines causal states by equivalence of pasts that induce the same condi- tional distribution over futures and proves minimality properties [1]. Predictive-state representations use action-conditional predictions of future observations as state coordinates [2]. Viability theory describes the states from which constraints can continue to be satisfied [3,4]. Organisational biology characterises living organisation through networks of mutually maintaining constraints [5,6], and work on organismal identity emphasises both historical and relational criteria [7]. Niche-construction theory establishes reciprocal organism-environment causation and ecological inheritance [9,10]. whose branches can fail independently. CLAIM DISCIPLINE: Established mathematics and physics remain explicitly attributed. Proposi- tions proved here are logical consequences of declared definitions or hypotheses. Published biological anchors are kept dataset- and model-specific. Structural analogies are labelled as analogies, not as identities of mechanism. 2. The biological unit: declaration before inference A predictive state is meaningless until the entity whose future is being predicted is specified. A potential circularity arises if “biological unit” is allowed to be defined retrospectively by whichever state representation happens to work. The repair is to separate unit declaration from state inference. Definition 1. Prospectively declared biological unit A unit declaration is a triple 𝒰 = (𝐵, ℐ, 𝐻). The boundary or interface 𝐵 distinguishes variables treated as internal, environmental, or coupled; ℐ is a trajectory-level identity or persistence criterion; and 𝐻 is the horizon over which that criterion is to be maintained. This declaration is made independently of the response pattern later used to test the theory.

Response laws Admissible domains Infer predictive classes Merge equal future laws T est held-out futures Report scope Unit identity is declared independently of the prediction being assessed. Figure 2. Unit declaration precedes state discovery. A: boundary, identity criterion and horizon are declared prospectively; organisational closure may justify an organism-level boundary but does not make predictive state circular. B: after the unit is declared, histories are merged only when all declared common futures have the same response law [1,2,5–7]. 3. State from futures: predictive quotient and successor closure Let H𝒰 be the admissible finite histories of the declared unit and let 𝒯 be a declared family of finite future action-observation tests. Tests may be open-loop or adaptive, with later actions chosen from earlier test observations. Write 𝑍𝑇 for the terminal response of test 𝑇. At the set-theoretic level, the following definition assumes specified response-law kernels on the chosen history domain. In statistical applications, histories, records, and responses are standard Borel; regular conditional laws and inferential conclusions are defined almost surely on declared common support. A probability- zero history is not assigned an empirical pointwise prediction by an arbitrary choice of conditional version. Definition 2. Predictive equivalence ℎ ∼ 𝒯 ℎ′ ⟺ Law(𝑍𝑇 ∣ ℎ) =Law(𝑍𝑇 ∣ ℎ ′) ∀𝑇 ∈ 𝒯. (2) The predictive state is the equivalence class 𝑆𝒯(ℎ) = [ℎ]𝒯. (3) This definition is relational and resolution-dependent. It does not claim that an organism stores a symbolic copy of its history, nor that one privileged molecular coordinate must represent state. It retains exactly the distinctions from history that remain separable by the declared future class.

Daniel J. Murray Revised September 2026 requires further measurable-factorization or realization hypotheses; it is not implied by set-theoretic minimality. Proof. Equality of laws is reflexive, symmetric, and transitive. If 𝜎 is sufficient, histories with equal 𝜎 have equal future laws and belong to one class. Conversely every future law is constant on each class by definition. The resulting factorization is set-theoretic unless a measurable quotient realization is supplied. Appendix A.1. Theorem 1. Predictive right-congruence and well-defined successor state Suppose the future family is closed under common admissible continuation: applying a next op- eration 𝑎, retaining its record 𝑟, and running any later test is represented by a joint test in the family. For an action defined on an entire quotient class, its admissibility domain must be a union of predictive classes. Then equal predictive histories give the same next-record law and the same continuation predictions. With discrete records, this holds on every positive-probability branch. With continuous records, it holds almost surely in the common record law; a countable determin- ing continuation family or another common-null-set condition is required to assert all continuation predictions simultaneously. A measurable quotient transition additionally requires a measurable realization of these predictions. [ℎ]𝒯 𝑎,𝑟 − − − → [ℎ𝑎𝑟]𝒯. (4) Proof. For discrete records, equal joint laws of the next record and later response give equal sets. Quotient-compatible action domains ensure that action availability is not changed by choosing a representative. The resulting transition is well defined on this scope; Appendix A.2. 3.1 Operational predictive rank Finite data require a resolvable rather than metaphysical dimension claim. Choose past histories h_i and future tests T_j with declared response features g_j. Form the finite past-future matrix H𝑖𝑗 = 𝐸[𝑔𝑗(𝑍𝑇𝑗 ) ∣ ℎ𝑖]. (5) If an exact d-dimensional linear predictive representation exists on this block, H factors as SR T and therefore rank(H) ≤d. Hence a statistically resolvable rank greater than d rejects that proposed linear closure. In linear predictive-state theory, rank of the full system-dynamics/Hankel object determines linear dimension under the model assumptions [2]. For biological data the empirical matrix is noisy and finite: singular values should be whitened or otherwise uncertainty-scaled, a threshold calibrated by preregistered bootstrap/simulation, and conclusions restricted to the tested histories, futures and tolerance. Rank growth can reject a proposed finite d; it cannot by itself prove infinite biological memory.

Daniel J. Murray Revised September 2026 Exact predictive classes All declared future laws are equal within a class Finite linear realization H = SRT Statistical complexity can change qualitatively Linear predictive rank and causal-state coding cost are different quantities. Figure 3. Predictive state, operational rank and an independent model precedent. A: histories merge only when all declared future laws agree. B: a finite past-future matrix can reject an undersized linear predictive closure when resolved rank exceeds its proposed dimension. C: in Marzen’s biologically motivated Monod-Wyman-Changeux sensor model, coarse observation of active/inactive output can conceal a causal-state architecture whose statistical complexity changes from finite to infinite under small kinetic changes [29]. Statistical complexity and linear predictive rank are different objects; the example warns that smooth input-output summaries need not imply smooth predictive architecture. 3.2 Independent model precedent: predictive architecture in an MWC sensor model Computational mechanics has already been applied to a biologically motivated Monod-Wyman- Changeux (MWC) sensor model. Marzen analysed the stochastic active/inactive output of a dy- namical single-site MWC model and constructed its causal-state representation. In special kinetic limits the predictive structure is finite, whereas arbitrarily small changes to certain transition rates can make the statistical complexity infinite even though excess entropy and the studied transfer function vary much more smoothly [29]. This is not experimental evidence that a measured molecule literally carries the quotient state used here. Its importance is structural: a coarse biological output can remain deceptively similar while the history required for exact prediction changes qualitatively. It therefore strengthens the paper’s insistence that predictive architecture must be tested from past-future response structure rather than inferred from a smooth transfer curve or a small set of instantaneous observables. RANK IS NOT STATISTICAL COMPLEXITY: Marzen’s C 𝜇 and the finite-matrix rank test in Section 3.1 quantify different aspects of predictive structure. An infinite causal-state coding cost does not automatically imply that every finite past-future matrix has growing algebraic rank. The correct inference is narrower: simple response summaries can fail to reveal a qualitative change in the predictive-state architecture. 4. The living present: constraint-filtered predictive state Predictive equivalence treats possible histories symmetrically. Extant biological systems add a prior filter: histories that irreversibly violate the declared unit identity or constraints do not reach the present as histories of that continuing unit. Define H𝑡 surv = {ℎ 0∶𝑡 ∶ 𝑥𝜏 (ℎ) ∈ 𝐾𝒰(𝜏 )for all relevant 0 ≤ 𝜏 ≤ 𝑡}. (6)

S𝑡 = H𝑡 surv/ ∼ 𝒯, 𝑠 𝑡 = [ℎ 0∶𝑡]𝒯 ∈ S𝑡. (7) Theorem 2. Viability-filtered predictive state Relative to the declared unit 𝒰, constraint family, and future-test class, the present predictive state is the equivalence class of a surviving history under equality of future response laws. Distinctions erased by every permitted future need not be retained; future-detectable distinctions cannot be quotiented away. Proof. Restrict the equivalence relation of Definition 2 to H𝑡 surv. Restriction preserves equivalence. Merging histories with different future laws violates Definition 2; merging histories whose future laws all agree preserves the declared predictions. This is a statement about the surviving history domain, not evidence that survival causes memory. Appendix A.3. MEANING OF “MEMORY” IN THIS THEOREM: The theorem does not reduce development, immunity, learning, injury, evolution and ecological succession to one molecular memory mechanism. It states only the operational commonality: an earlier event belongs to present state exactly when its distinction remains detectable by an admissible future. 5. Nominal and robust viability: possibility is not resilience The predictive quotient retains distinctions needed for declared predictions. Viability is a separate question in a stated physical realization 𝑋. A quotient formed from selected response tests need not preserve physical geometry or feasible action sets. Such control claims require tests that preserve the relevant state/action information or an additional refinement. For attainable velocities 𝐹 (𝑡, 𝑥) and persistence constraints 𝐾𝒰(𝑡), define the finite-horizon viability tube by 𝒱𝐻(𝑡) = {𝑥 ∈ 𝐾𝒰(𝑡) ∶

6. Time is composition: order, waiting and profile are related but not identical The scalar variable t labels sequence; it does not guarantee that sequence is predictive. Time becomes load-bearing when exchanging operations, inserting a gap or redistributing a continuous profile changes a later response. Order and waiting admit a common Lie-algebraic description, but the mathematically correct unification is commutator structure, not the claim that every order effect implies a waiting effect or vice versa.

Introduce an action policy 𝐴𝑡 = 𝜋(𝑠𝑡, 𝐸𝑡)and the combined actual state 𝑋𝑡 = (𝑠𝑡, 𝐸𝑡), where 𝑠𝑡 ∈ S𝑡. This representation assumes sufficiency for the policy, environmental evolution, and constraint variables; prediction of a smaller response family alone does not imply that closure. With time or developmental stage included when required, write the closed loop as 𝑋𝑡+1 = Φ(𝑋𝑡, 𝜉𝑡). (15) Here 𝜉𝑡 collects stochasticity or disturbances. A forward-invariant regime Ω retains trajectories beginning in Ω under the declared policy and disturbance convention. In a deterministic autonomous

environment, the natural dynamical object is the coupled organism-environment process and its invariant or attracting regimes. 8. Predictive state, viability and fitness Viability and evolutionary fitness are related but not interchangeable. Viability is a feasibility property: does at least one admissible continuation remain? Fitness is a reproductive weighting or outcome, usually defined through descendant contribution over a specified context and horizon. The Price equation formalizes change in population means through covariance with fitness and within-lineage change [8]; it does not identify fitness with volume of a viability kernel. Corollary 1. Fitness preservation by predictive equivalence Let 𝑊𝐻 be a declared reproductive-output variable over horizon 𝐻, such as descendant count or an explicit fitness proxy. If a test containing 𝑊𝐻 belongs to 𝒯, predictive equivalence preserves its law. Equality of expectations additionally assumes integrability: ℎ ∼ 𝒯 ℎ′ ⟹ Law(𝑊𝐻 ∣ ℎ) =Law(𝑊𝐻 ∣ ℎ ′) ⟹ 𝐸[𝑊 𝐻 ∣ ℎ] = 𝐸[𝑊𝐻 ∣ ℎ ′]. (16) Thus a predictive state constructed from a future class that includes reproduction is automatically sufficient for the corresponding conditional fitness law. The converse does not hold: equal expected fitness is far too coarse to imply equal predictive state, because two states can share the same reproductive expectation while differing in many other future responses.

occurs over the horizon? The definition of state Predictive state Which distinctions alter declared future laws? Necessarily finite-dimensional 9. Information and finite observer compression An observer with finite information capacity cannot retain every distinguishable detail of an arbitrar- 𝑀2 ⪰Blackwell 𝑀1 ⟹ 𝑉 ∗(𝑀2) ≥ 𝑉∗(𝑀1). (17) Biology adds costs of sensing, storage, computation, delay and tissue maintenance. Therefore the biological prediction is not “maximize information. ” It is to preserve useful future-relevant distinctions subject to energetic, material and temporal constraints. 9.2 Exact and approximate predictive compression Let 𝐻 be the fine history, 𝐹 a declared future variable or joint future-response object, and 𝑀 a representation encoded from 𝐻. Assume the encoder does not access the future beyond 𝐻: 𝑀 ⟂ 𝐹 ∣ 𝐻 . A deterministic encoder 𝑀 = 𝐶(𝐻)satisfies this condition. With standard Borel variables and well-defined information quantities, exact statistical predictive sufficiency is 𝐼(𝐻; 𝐹 ∣ 𝑀 ) = 0. (18) The information-bottleneck formulation provides a principled approximation when exact sufficiency is impossible [12]. One may seek a representation minimizing retained past information while allowing at most 𝜀 predictive loss: min 𝐼(𝐻; 𝑀 )subject to 𝐼(𝐻; 𝐹 ) − 𝐼(𝑀 ; 𝐹 ) = 𝐼(𝐻; 𝐹 ∣ 𝑀 ) ≤ 𝜀. (19) The equality in Eq. (19) follows from the chain rule and 𝑀 ⟂ 𝐹 ∣ 𝐻 , provided the difference does not subtract two infinities. Equivalently, define the constraint directly by conditional mutual information. At 𝜀 = 0 , sufficiency holds almost surely under the specified population law. At

Daniel J. Murray Revised September 2026 𝜀 > 0 , this bounds average predictive information loss; it does not bound the separation of every pair of histories merged by 𝑀 . For example, with 𝐻 ∼ Bernoulli(𝑝), 𝐹 = 𝐻 , and constant 𝑀 , the loss is the binary entropy, tending to zero as 𝑝 → 0 , yet the two conditional future laws have total-variation distance one. Uniform biological margins require a separate predictive-distance or worst-case constraint. Thermodynamics-of-prediction results distinguish predictive information from nonpredictive memory [13], without establishing that brains literally solve this optimization. NO NEURON-COUNT THEOREM: Larger nervous systems may support richer representations, but channel capacity, predictive rank, learning horizon and organismal performance do not follow monotonically from neuron count alone. Cross-species claims require explicit task, cost, phylogeny and measurement controls. 10. Observation as physical history-writing: the quantum bridge Quantum measurement is used here for a narrow reason: it supplies an exact physical formalism in a countable determining family or another justified common-null-set condition. This is statistical sufficiency, not automatic equality at every probability-zero history. Conversely, positive conditional mutual information identifies predictive information lost on the population support. Appendix A.7.

Figure 7. Observation and history. A: an ideal two-alternative quantum interaction links record distinguishability to locally accessible coherence [21]. B: any finite observer compresses a fine record into an internal state; exact compression is the 𝜀=0 predictive-sufficiency condition. C: biological record updating can share the abstract event →record→successor-state grammar without sharing quantum substrate, ontology or update law. BOUNDARY: The paper does not claim that consciousness causes collapse, that biology is “quan- tum” in the explanatory sense, or that memory traces are quantum branches. The bridge is struc- tural: physical interaction can create records, and records can condition the state from which later predictions are made. 11. Biological history-writing: memory reconsolidation as a distinct instance Memory reconsolidation supplies a biological example of history being rewritten at the level of future response. Nader, Schafe and LeDoux showed that a reactivated fear memory can return to a labile state requiring protein synthesis to persist [15]. Reconsolidation is boundary-condition dependent [16]. Crucially for the present architecture, human fear-conditioning experiments show that retrieval alone is insufficient and that prediction error can be required for destabilisation/reconsolidation: Sev- enster, Beckers and Kindt first separated retrieval from updating conditions and then demonstrated prediction error as a necessary condition in their associative fear paradigm [31,32]. A broader review places prediction error centrally in memory updating while retaining important boundary conditions and task dependence [17]. 11.1 Shared formal motif, distinct mechanism The safe correspondence with Section 10 is not that a memory “address” is mathematically the same object as a quantum branch. It is weaker and more useful: in both cases a present physical record changes which successor state is relevant for future prediction. Abstractly, one may write 𝑆+ = 𝐾 𝑟(𝑆−). (22) Here 𝑟 is an outcome, cue, or access record and 𝐾𝑟 is the domain-specific update. In quantum theory it denotes the normalized positive-probability instrument update, generally nonlinear in the normalized state; in memory biology it represents biochemical and network plasticity following

Daniel J. Murray Revised September 2026 11.2 Prospective addressed-update model A testable extension distinguishes retrieval, access, mismatch, credit assignment, write-back, and extinction. Let 𝐴 label a latent context, 𝑍𝐴 a probability law over organismic states associated with it, 𝑞𝑡(𝐴)a context responsibility, 𝑏𝑡 an access/writeability gate, and 𝑔(𝑚𝑡)a mismatch-dependent plasticity factor. Let 𝑄𝐶 be the corrective target law and 𝐸𝐴 an extinction-related target law on 𝐴 + 𝜆𝐴𝑄𝐶 + 𝜉𝐴𝐸𝐴, 𝜆 𝐴, 𝜉𝐴 ≥ 0, 𝜆 𝐴 + 𝜉𝐴 ≤ 1. (23) 𝜆𝐴 = 𝜂 𝑞 𝑡(𝐴)𝑏𝑡𝑔(𝑚𝑡). (24) Equations (23–24) remain a prospective addressed-update model. The mismatch factor g(m_t) is no longer biologically unmotivated: prediction-error dependence has direct experimental prece- dent in human fear reconsolidation [31,32]. What remains unestablished is the particular factori- sation 𝜆𝐴=𝜂q_t(A)b_tg(m_t), the latent-address responsibility term q_t(A), and the proposed access/writeability gate b_t. Those components therefore retain explicit no-fit controls: same- context correction must be separated from exposure-only, correction attributed to a different latent context, and low-access conditions. Failure of the access gate would not by itself falsify context- specific updating. All coefficients must satisfy Eq. (23); for example require the factors defining 𝜆𝐴 to lie in [0, 1] and choose 0 ≤ 𝜉 𝐴 ≤ 1 − 𝜆 𝐴. Without these constraints the update can assign negative probability. The targets and gates require prospective operational definitions before fitting; the present paper does not validate their neural implementation. 12. Hormesis: an explicit sufficient condition for a biphasic branch The phrase “opposing responses” is too vague to support a theorem. The strongest safe statement is a sufficient calculus condition, not an unjustified if-and-only-if threshold rule. published bounded-adaptive hormesis paper gives a stronger mechanistic instantiation in which adaptive activation precedes toxicity and high- dose damage ultimately exceeds bounded repair capacity, with quantitative predictions tested against published dose- response data [18]. The wider hormesis literature supplies strong evidence for recurrence of the biphasic phenotype but not, by itself, for this theorem’s mechanism. Calabrese’s quantitative reviews document thousands

of hormetic responses across diverse biological models and characteristic quantitative regularities, including a typically modest maximum stimulation [34]. The published bounded-adaptive model reports that its aggregate amplitude and hormetic-zone predictions match the >10,000-response Calabrese database without fitting and that mechanistically distinct worked agents satisfy its struc- tural ordering [18]. These observations support the prevalence and quantitative regularity of the branch; the mechanistic premises remain independently testable and should not be inferred from curve shape alone.

Daniel J. Murray Revised September 2026 14. Recovery, residue and restoration Return to a present readout is weaker than return to predictive state. Let 𝜋 be a measured output and suppose 𝜋(h_after)=𝜋(h_before). Theorem 4 says nothing about restoration until a common future is applied. If the later response law differs, the system carries residue even though the snapshot has normalized. Proposition 3. Operational restoration criterion Relative to a declared restoration test family, an operational restoration claim requires post- withdrawal future-response differences from an appropriate reference to lie within prospectively chosen biological margins. For a finite battery with response-law distances 𝑑𝑗 and margins 𝜀𝑗 > 0, use simultaneous upper bounds to certify max 𝑗 𝑑𝑗/𝜀𝑗 ≤ 1 . Failure to reject a difference is not an equivalence result. A single challenge supports only its own response and horizon; larger-family claims require explicit coverage assumptions. Finite-tolerance agreement is pairwise and need 15. Architecture-phenomenon correspondence The framework does not propose one universal response curve. It proposes that once an architecture is declared prospectively, the class of laws and falsifiers becomes constrained. The important dis- tinctions are terminal versus recursive update, nominal versus robust viability, commuting versus noncommuting transformation, full state versus projection, record-preserving versus information- losing observation, and clean versus residual return. Architecture AB versus BA with fixed ingredients Predictive quotient Merged histories agree on declared futures Common-future challenge Robust viability A strategy preserves constraints under the intervention direction Gap scan with fixed inputs Finite observer compression Exact lossless-for-prediction compression screens history Conditional information and held-out prediction Residual return A normalized snapshot can retain a response defect Withdrawal and equivalence-tested rechallenge

Daniel J. Murray Revised September 2026 Claim Status Primary support Predictive equivalence/minimality Established precedent and proved partition formulation [1,2]; Proposition 1 MWC predictive complexity Published sensor-model application [29] Quotient successor dynamics Proved with continuation, domain, and measurable-scope integrability for means Corollary 1 Predictive compression Established information theory and synthesis [12,13] Reconsolidation and prediction error Published neuroscience within paradigms [15–17,31,32] mechanisms are identical Not claimed Explicitly excluded 17. Falsification programme The paper becomes useful only if proposed closures can die. The following tests are deliberately designed to produce rejection rather than post hoc reinterpretation. 17.1 Common-future state test Construct two histories that match on the proposed present readout. Freeze the future challenge. A

Daniel J. Murray Revised September 2026 17.2 Unit-scale test Repeat the state test under a prospectively changed biological boundary or horizon. If the predictive state changes, report scale dependence rather than treating one level as universally privileged. 17.3 Robust viability test Start from a candidate nominally viable state and introduce a preregistered disturbance family. Test whether the specified controller preserves the required constraints and local inward conditions. A failure refutes that controller’s certificate in the tested conditions; it does not exclude another robust strategy. Excluding membership in a robust viability tube requires a valid impossibility argument over the entire admissible strategy class, not merely failure of one tested policy. 17.4 Order and waiting matrix Run AB versus BA and an independent gap scan. Estimate order defect and waiting transport separately. The theory predicts no universal proportionality because [X_A,X_B] and [L,X_A] are different objects. 17.5 Predictive-rank growth Construct H_ij from preregistered histories, future tests and features. Whiten by uncertainty; calibrate a singular-value threshold under a null rank-d model or bootstrap; increase history depth and future probes; and reserve untouched continuations for validation. A reproducible rank greater than d rejects that d-dimensional linear closure. Absence of further resolved rank at finite depth does not prove global finiteness. 17.7 Observer-compression test Vary record availability, sensory bandwidth or memory access while holding the external stream fixed. Test whether a coarser representation loses held-out predictive information and estimate I(H;F|M) or a task-specific divergence. 17.8 Restoration test After apparent normalization and treatment withdrawal, apply a standardized challenge. Restora- tion predicts convergence of future-response rows, not only convergence of the treatment-on snap- shot. Evidence for restoration requires within-margin equivalence bounds with adequate sensitivity, not simply a nonsignificant contrast.

Within margin Unresolved Prediction, causal attribution, and physical persistence require distinct checks. Figure 9. The common-future experiment. Match the proposed present state after two different histories, then apply a frozen challenge. If the future laws differ, the present readout is not a sufficient state. The same logic extends to order, waiting, robust viability, predictive rank, observer compression and restoration. 18. Implications for adaptive intervention The temporal architecture naturally supports closed-loop intervention, but it cannot validate a modality by mathematics alone. A safe adaptive controller must declare the unit, state estimate, admissible region, disturbance model, action set, temporal accounting, recovery criterion and stop conditions. An intervention that is safe for a nominal state can be unsafe for a different hidden history or under a narrower robust margin. A minimal architecture therefore contains: (i) prospectively declared unit/boundary and identity criterion; (ii) a predictive state estimate validated by common futures; (iii) nominal and, where rel- evant, robust safety constraints; (iv) explicit intervention and disturbance alphabets; (v) order/gap accounting; (vi) held-out prediction; (vii) recovery/residue testing after withdrawal; and (viii) fail- safe stopping rules. Specific sensory, optical, electrical, magnetic, pharmacological or behavioural modalities require independent efficacy, safety, ethics and regulatory evidence. TRANSLATIONAL BOUNDARY: The mathematics can specify what a closed-loop biological con- troller must know and how to falsify its state estimate. It does not establish clinical benefit for infrared stimulation, TMS, sensory stimulation or any other modality without domain-specific tri- als. 19. Discussion: the present as a causal interface The strongest version of the temporal architecture is more modest than “life is a fixed point” and discarded are genuinely irrelevant at the declared resolution. This formulation makes several previously separate distinctions precise. Unit identity is declared independently from state inference, avoiding circularity. Predictive right-congruence supports suc- cessor dynamics when action domains and the measurable or almost-sure realization conditions are also met. Robust viability separates a barely possible path from a disturbance-tolerant regime. Order and waiting are unified at the level of noncommutative temporal operators while remaining experimentally distinct. Reproductive fitness becomes one future functional rather than the defi- nition of state. Finite observer capacity becomes a predictive-compression problem rather than a metaphorical “frame rate. ”

observables and closure tests can be valid. The independent examples added in this revision matter because they converge on different parts of the architecture without being mutually dependent. The MWC work shows that predictive structure can change while coarse functional summaries remain smooth [29]. The anchovy-hake ap- plication operationalises robust viability under biological and economic constraints [30]. Human fear experiments separate retrieval from prediction-error-dependent updating [31,32]. Active-inference niche-construction simulations independently treat agent and environment as a coupled attract- ing system [33]. Large-scale hormesis literature establishes recurrence of the biphasic phenotype [34], while recent GPx4 work supplies an independent redox-flux constraint and threshold picture [35]. None proves the whole framework; together they make it increasingly difficult to dismiss the architecture as a metaphor assembled only from the author’s own examples. The paper therefore rejects several tempting but unsupported upgrades. The existence of a forward- invariant regime does not imply that evolution searches for fixed points. Predictive rank is not automatically neuron count, energy or organismal complexity. Robust viability is not numerically identical to allostatic load. Memory addresses are not quantum branches. And no category-theoretic reformulation is needed to obtain the central result: the concrete quotient-congruence theorem already supplies the composition closure required for temporal state. unit whose present state is made from the distinctions of prior interaction that remain relevant to its future. The histories that reach the present have already been filtered by constraints. The state that remains must be sufficient for future prediction. Viability determines whether admissible continuation exists; robust viability asks whether it survives a disturbance class. Action changes the organism and can change the environment that supplies later constraints. The architecture can be written compactly as declared unit → constraint-filtered history → predictive state → physically certified viable futures → action → altered environment → new constraint . (25) Every arrow has an empirical failure mode. That is what makes the framework more than a