Descent/composition/temporal grammar

### PDF page 31 PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 The programme’s strongest philosophical claim is therefore modest in form but broad in reach: P/H - FIBRE DISCIPLINE A representation is lawful only to the extent that its discarded distinctions remain irrelevant to every future it claims to predict. That statement is not a new physical law. It is a discipline on what counts as state, exposure, composition, and coordinate in empirical science. 15. Conclusion Predictive Closure can now be stated without separate foundational papers competing for the same middle ground. Histories define state by future equivalence. Continuation-closed futures make deterministic or outcome- labelled continuations descend to partial predictive actions, with successor kernels for unconditioned stochastic actions. Those actions admit an exact global fork. If the generator actions commute, the intervention language factors through its abelianization and order can be forgotten at the standardized- count level; with one-agent semigroup calibration, a continuous cumulative-vector representation can be tested. If the actions do not commute, order is predictive information and the appropriate branch is Temporal Grammar: weak-word compression, reciprocal commutators, higher-order context, schedule circulation, gap transport, and local curvature in the smooth regime. Neither branch ends the state problem. Past–future Hankel reconstruction determines whether the resulting representation predicts untouched continuations and how many linear predictive directions the declared experiment resolves. Only after the cumulative branch passes may a scalar summary be tested. Only after scalar compositional descent and the standard regularity axioms may an Aczél generator be used. Boundedness alone supplies none of those steps. The result is one theorem-and-no-go architecture rather than a sequence of loosely related claims: predictive state → predictive action → ( abelian cumulative branch, nonabelian temporal branch, → predictive realization → only then: scalar coordinate or richer geometry. (60) Cumulative exposure is therefore not the starting noun. It is an earned quotient of action. Temporal Grammar is not a rival theory. It is the structure left behind when that quotient fails. Lawful Coordinates are not an upstream geometry. They are a downstream scalar branch. The paper’s central scientific question is the one shared by all three: How much of the past may the present lawfully forget without changing the futures the model claims to predict? Theorem-and-no-go paper | claims are jurisdiction-relative 31 ---

VERSION 4 - UNIFIED STATE-ACTION ARCHITECTURE Predictive Closure State, action, and the experimental compression of history A theorem-and-no-go architecture for cumulative exposure, temporal grammar, predictive realization, and lawful coordinates Daniel John Murray Independent Researcher, Melbourne, Australia ORCID: 0009-0005-1794-5945 7 September 2026 | Version 0.4.1 The conclusion written first Histories define predictive state by the futures they still change. Interventions act on that state only after descent is earned. The action then has one decisive fork: commuting generators admit an unordered cumulative representation; non- commuting generators retain temporal grammar. Past–future response blocks reconstruct the predictive realization in either branch. Scalar dose is a fur- ther quotient, and an additive coordinate appears only after scalar composition passes closure, identity, continuity, strict order, and associativity. Curvature is the local obstruction to predictive abelianization in the smooth branch, not a universal geometry. Predictive Closure is therefore a calculus for how much history may lawfully be forgotten. Claim-status key P proved here or elementary from definitions I imported theorem with named hypotheses C conditional corollary once named premises are granted E empirical illustration; not validation of the general theorem G open gap with a stated closure test N countermodel, negative result, or no-go that blocks an overclaim Scope warning. This is an admissibility and reconstruction calculus for empirical state and intervention history. It does not select the laws of nature, prove that all systems have finite predictive state, derive one universal geometry, or turn every bounded observable into a hyperbolic coordinate.

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 Contents Abstract 4 1 Conclusion first: compression before equations 5 2 Declared futures and the predictive state quotient 6 2.1 Empirical jurisdiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Continuation-closed futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Partial intervention descent becomes automatic . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Nested jurisdictions and coarse dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Predictive action and the exact abelianization fork 9 3.1 The predictive action monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Predictive Abelianization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Counts are not yet cumulative amounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4 Scalar descent is a second quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Lawful scalar coordinates are a downstream branch 12 4.1 The lawful-coordinate admission theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.4 Reversal-even interaction and reversal-odd memory . . . . . . . . . . . . . . . . . . . . . . . 15 5.5 Schedule fields and Hodge obstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6 Smooth predictive transport: curvature is the local abelianization obstruction 16 6.1 Earning a challenge-visible local manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.2 The predictive connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.3 Time, waiting, washout, and recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.4 Challenge-visible curvature tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.5 Bianchi and holonomy are conditional extensions . . . . . . . . . . . . . . . . . . . . . . . . 19 7 Global predictive reconstruction from futures 20 Theorem-and-no-go paper | claims are jurisdiction-relative 2

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 7.1 The past–future Hankel object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 7.2 Cheap pair-block witnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7.3 Reachability and observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7.4 Predictive emergence spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7.5 Feature refinement and distributional state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7.6 Active state discovery and held-out closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8 The completed object: predictive action form 23 9 Experimental decision architecture 23 9.1 Stage 1: state before algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 9.2 Stage 2: descent and one-agent calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 9.3 Stage 3: abelianization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 11 No-go architecture: what the unified theory does not permit 26 12 Relation to prior frameworks and the actual novelty claim 28 13 Frozen predictions and falsifiers 28 14 Discussion: the paper the programme was trying to become 30 15 Conclusion 31 Data, code, and manuscript status 32 A Theorem and dependency ledger 32

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 Abstract Scientific models routinely compress intervention histories into cumulative dose, cumulative expo- sure, burden, biologically effective dose, or another endpoint-only summary before asking whether the discarded ordering information remains predictive. This paper builds the missing state–action architec- ture in the opposite order. A declared experimental world supplies admissible histories, finite future intervention–observation tests, outcomes, operating range, and error tolerance. Predictive equivalence identifies histories exactly when all admitted future laws agree. With a continuation-closed future family, the quotient is the coarsest exact predictive state and deterministic or outcome-labelled admitted continu- ations induce well-defined partial actions on that state. An unconditioned stochastic intervention instead induces a successor kernel; its observed branches must be retained when they affect later predictions. Inadmissibility and terminal outcomes are part of the future law rather than hidden exceptions. The action itself then becomes an empirical object. We define its predictive action monoid and prove a Predictive Abelianization Theorem: for standardized intervention labels, the ordered word action factors through the unordered multiplicity vector exactly when the generator partial maps commute pairwise, including agreement of domains. Thus cumulative exposure is an earned quotient of intervention history, not a primitive numerical convention. For continuously parameterized interventions, cumulative amount requires an additional one-agent semigroup calibration; pairwise commutation alone is insufficient. A further scalar reduction is a distinct fibre-factorization problem, and an induced scalar composition exists only when concatenation is constant on scalar fibres. Under the standard Aczél hypotheses, that scalar operation has an additive generator unique up to positive scale after identity normalization. Boundedness alone selects neither the operation nor the coordinate. When predictive abelianization fails, the framework does not repair history with another transform. It enters the noncommutative branch. Near an experimentally validated identity limit, fixed-multiset words differ through second order only by inversion counts multiplying pairwise commutators; repeated-word ladders isolate higher-order nested brackets. Reciprocal-pair matrices separate reversal-even interaction from reversal-odd memory, while schedule Hodge decomposition distinguishes additive node potentials from circulation residuals without conflating numerical exactness with ordinal sortability. In a smooth protocol-dependent branch, the same obstruction is the curvature of the predictive transport connection. Curvature is therefore the differential obstruction to abelianization, not an independently postulated biological geometry. Waiting is another protocol direction, giving mixed time–intervention curvature and adjoint gap transport. The global reconstruction is supplied by the complete past–future Hankel object. Finite Hankel rank is equivalent to a finite linear predictive realization; pair blocks provide cheap rank witnesses; reachability and observability separate what the intervention alphabet can write from what the future panel can read. For analytic finite truncations, a Smith-form emergence spectrum links weak perturbative order to the scale at which predictive directions become experimentally visible. Frozen challenge panels can therefore perform challenge-visible curvature tomography without claiming molecular completeness. The resulting theory has one exact fork. A commuting predictive action can descend toward cumulative- vector, scalar, and lawful-coordinate representations. A noncommuting action retains temporal grammar and must be reconstructed as such. Every arrow is an admission test with a named death condition. The paper’s contribution is not priority for predictive states, automata, Lie brackets, Hankel realization, functional equations, or connection curvature separately. It is the typed composition of those objects into one experimental calculus for deciding how much of history may lawfully be forgotten. Keywords: predictive state; intervention action; cumulative exposure; abelianization; temporal grammar; noncommutativity; Hankel rank; curvature; dose response; lawful coordinates; system identification; history dependence. Theorem-and-no-go paper | claims are jurisdiction-relative 4

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 1. Conclusion first: compression before equations A scientific model usually begins by naming a state variable and writing its evolution. That order is unsafe whenever the proposed present is itself a compression of history. If two histories are assigned the same present but a common future separates them, the proposed state has already failed before any differential arithmetic lawful. The correct order is therefore history → predictive state → predictive action → ( abelian cumulative branch, nonabelian temporal branch, → predictive realization (1) with scalar composition, additive generators, metric geometry, observation instruments, thermodynamic laws, and continuum limits entering only after their own premises are supplied. Figure 1: The unified Predictive Closure architecture. The predictive quotient and descended action are upstream. The central fork asks whether order can be forgotten. The abelian branch admits cumulative-vector and then scalar tests; the nonabelian branch enters Temporal Grammar. Both feed global predictive realization. Geometry is downstream, not a substitute for failed state or action compression. P/H - CONSTRUCTIONAL CLOSURE CRITERION Within a declared empirical jurisdiction, a representation is structurally closed only relative to the objects it actually claims. It must specify: (i) admissible histories and futures; (ii) when histories are the same predictive state; (iii) which interventions act on that state; (iv) whether intervention order can be quotiented out; (v) what further vector or scalar compression is claimed; (vi) which Theorem-and-no-go paper | claims are jurisdiction-relative 5

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 composition, metric, instrument, stochastic, or continuum premises are independently added; and (vii) which held-out observations would reject each compression. Passing the calculus does not show that the resulting dynamics describe nature; it shows only that the representation has earned the operations performed on it. This version makes one change to the earlier architecture that is mathematically small and conceptually decisive. It places predictive abelianization between action descent and cumulative exposure. Action descent asks whether an intervention is well defined on state. Abelianization asks whether its order can be forgotten. Scalar descent asks whether the surviving cumulative vector can be compressed further. These are different factorization problems and can fail independently. 2. Declared futures and the predictive state quotient 2.1 Empirical jurisdiction Fix a declared empirical jurisdiction J = ( H, Π, Y, R, ε), (2) where H is the set of admissible realized histories, Π is the admitted family of future intervention– observation tests, Y is the outcome language, R records operating range and timing, and ε is the tolerated discrepancy. Exact statements set ε = 0; experiments replace equality with preregistered equivalence margins and confidence regions. The state claim is necessarily jurisdiction-relative. Enlarging the intervention alphabet, future depth, feature family, or resolution can refine state. No finite experiment proves that no richer future could distinguish two histories.

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 including agreement of which futures are admissible. The predictive state is the class SΠ(h) = [ h]Π, SΠ = H/ ∼Π . (4) Figure 2: Predictive state as a future-response row. Histories h1 and h2 are the same predictive state because every admitted future response agrees; h3 and h4 remain distinct. A finite panel is an experimental projection of this ideal object, not the definition of metaphysical state. P - PROVED MODULE Predictive quotient theorem. The relation ∼Π is an equivalence relation. Its quotient SΠ is the coarsest exact state sufficient for the declared future family in the information order: every exact sufficient representation z : H → Z factors through distinctions at least as fine as ∼Π. Proof. Equality of all future laws is reflexive, symmetric, and transitive. If z(h) = z(h′) and z is exact sufficient, every admitted future law agrees, hence h ∼Π h′. Therefore z cannot identify histories that the predictive quotient separates. □ This construction is a controlled-experiment version of causal-state, predictive-state, observable-operator, sufficient-statistic, bisimulation, and automata-style future-equivalence ideas [1, 2, 3, 4, 5]. Priority is not claimed for the quotient itself. 2.3 Partial intervention descent becomes automatic First consider a deterministic continuation label a ∈ A that maps an admissible history h to the extended

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 is a well-defined partial map on SΠ. P - PROVED MODULE Continuation-descent theorem (deterministic continuations). For a continuation-closed future world with deterministic history extension and admissibility included in the future law, every

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 3. Predictive action and the exact abelianization fork 3.1 The predictive action monoid The descended intervention maps form an ordered action. For a word w = a1a2 · · · ak ∈ A ∗, define the partial composition ρ(w) = Tak ◦ · · · ◦ Ta1, (8) with the convention fixed once and used throughout. Function composition is associative wherever defined, even when the interventions do not commute. Two words are predictively action-equivalent when they have the same partial action on SΠ: u ≡ρ v ⇐ ⇒ ρ(u) = ρ(v) as partial maps, including equal domains. (9) The quotient A∗/ ≡ρ is the predictive action monoid. This is the action-side analogue of the predictive state quotient and is closely related to syntactic-monoid constructions in automata theory [6]. 3.2 Predictive Abelianization Theorem Let n : A∗ → Nm count the multiplicity of each of m standardized intervention labels. The map n forgets order and is the abelianization of the free monoid. P - PREDICTIVE ABELIANIZATION THEOREM Let T1, . . ., Tm be the descended partial generator actions on predictive state. There exists a unique partial action ¯ρ of the free commutative monoid Nm such that ρ(w) = ¯ρ(n(w)) for every word w ∈ A ∗ (10) if and only if TiTj = TjTi as partial maps for every i, j, (11) Equation (10) is the exact admission condition for forgetting order at the level of standardized intervention counts. It is stronger than saying that a particular endpoint happened not to resolve an order effect, because it is stated on predictive state and all declared futures. Theorem-and-no-go paper | claims are jurisdiction-relative 9

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 Figure 3: The abelian/nonabelian fork. When predictive actions commute, words with the same multiplicities can be quotiented to one cumulative count vector. When they do not, the same cumulative ingredients can write different predictive state, and order is part of the state-relevant input. N - NO-GO / FAILURE BOUNDARY A scalar or monotone transform cannot repair failed abelianization. Ifρ(AB)s ̸= ρ(BA)s in predictive state, every injective coordinate preserves that inequality. The repair is a richer state or action representation, not a different plotting scale. 3.3 Counts are not yet cumulative amounts Theorem (10) concerns standardized labels. To replace repeated pulses by a continuously valued cumula-

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 C - CONDITIONAL BRANCH Continuous cumulative-vector branch. If each intervention family has an additive semigroup parameter θa and the cross-family flows commute on predictive state, then the joint action factors through the additive parameter monoid Rm + (or the empirically admitted additive interval product). This does not identify physical units; calibration is a separate map. 3.4 Scalar descent is a second quotient Even after cumulative-vector admission, a scalar summary q : Θ → I is not automatically sufficient. Let G(θ) denote the complete declared future law after cumulative vector θ. The scalar is predictively sufficient exactly when q(θ) = q(θ′) = ⇒ G(θ) = G(θ′). (14) Equivalently, the future law factors asG = ¯G ◦ q. This is the ordinary fibre criterion, now placed at the correct point in the compression chain. The comparison holds at a common declared initial predictive state or preparation distribution. In general the future law is G(s0, θ); commutation does not remove dependence on s0. A still stronger claim is needed to compose scalar exposures. The vector addition law descends to a binary scalar operation ⊕ only when q(θ) = q(θ′), q(η) = q(η′) =⇒ q(θ + η) = q(θ′ + η′). (15) Then and only then is q(θ) ⊕ q(η) := q(θ + η) (16) well defined. Predictive scalar sufficiency, scalar compositional descent, and additive coordinate choice are therefore three different questions. Equations (15)–(16) assume that Θ is closed under the vector addition being used. If the operating range instead gives a partial composition domain D ⊆ Θ2, one must additionally require q(θ) = q(θ′), q(η) = q(η′) = ⇒ has constant output wherever addition is allowed, yet (0, 0) is admissible and (1, 1) is not. The scalar fibre has erased admissibility. A bounded operating range must not silently be treated as an additive monoid. P - PREDICTIVE COMPRESSION LADDER For a declared intervention world, exact replacement of ordered history by a scalar additive coordi- nate requires the following distinct gates in order: 1. future-equivalence state construction; 2. intervention descent to predictive state; 3. one-agent amount calibration if continuous exposure is claimed; 4. predictive abelianization of cross-intervention order; 5. cumulative-vector predictive sufficiency; Theorem-and-no-go paper | claims are jurisdiction-relative 11

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 6. scalar fibre sufficiency and scalar compositional descent; 7. regularity premises for an additive generator. Failure at an earlier gate cannot be repaired by satisfying a later one. 4. Lawful scalar coordinates are a downstream branch identity [15, 16, 17]. I - IMPORTED THEOREM / STANDARD RESULT The Aczél theorem is imported. Predictive Closure contributes the upstream certificate telling us when there is a scalar operation to which the theorem may legitimately be applied. The additive image need not always be all ofR. One-sided monoids can map to additive subintervals such as [0, ∞). Two-sided unbounded repeated composition forces an additive image without finite endpoints, but that is an additional closure premise.

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 N - NO-GO / FAILURE BOUNDARY Boundedness ̸⇒ artanh. Scalar nonlinearity ̸⇒ intrinsic curvature. A coordinate transform ̸⇒ a new state. The first two are branch-identification errors; the third is a state-sufficiency error. 4.3 Higher-dimensional commuting flows If d complete smooth vector fields commute, are linearly independent, and generate a free transitive action on a simply connected predictive-state sheet, their joint flow gives a vector additive chart. This is the legitimate higher-dimensional analogue of scalar addition. If the fields do not commute, no coordinate- wise scalar transform can turn the action into ordinary vector addition; the system remains in a Lie-group, semigroup, quotient-with-cocycle, gyrogroup, or more general nonabelian branch depending on the earned structure [12, 14, 18]. additive vector coordinates are themselves a theorem-dependent compression of the action. 5. The nonabelian branch is Temporal Grammar When predictive abelianization fails, order is not a nuisance parameter. It is experimentally accessible information about how interventions act on state. The goal is then not to force a cumulative scalar, but to determine the smallest ordered action structure that predicts held-out futures. 5.1 Near-identity interventions and the visibility window Let O be a normed space of declared observables or an operator representation sufficient for the chosen response functional. A standardized weak intervention i is admissible for local analysis when its baseline- relative map has an identity limit

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 If no such interval can be supported after calibration, the smooth weak grammar is empirically inadmissi- ble. This sufficient envelope assumes cp ̸= 0, 0 < η < 1, and Mp+1 > 0. Its lower endpoint uses |Cp(s)| ≥ (1 − η)|cp|sp, so an adversarially signed remainder cannot cancel the claimed detectable signal. If Mp+1 = 0, 5.3 The A2B2 graded ladder Two copies each of A and B produce six exposure-matched words distributed over five inversion lev- els. Through second order, one intercept and one directional coefficient predict the entire ladder. The duplicated central level provides a protected higher-order channel. Let X = αALA and Y = αBLB. Then UAUBUBUA − UBUAUAUB = −s3[X + Y, [X, Y]] + o(s3), (25) so

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 Figure 4: The A2B2 ladder. The second-order inversion-count model is strongly overidentified. ABBA and BAAB share the same inversion count, so their leading allowed separation is cubic and isolates a nested-commutator direction. The point is methodological. A single significant order contrast can always be absorbed into an uncon- strained history term. A graded ladder predicts which residuals must remain silent to each perturbative order. It can therefore reject the local grammar before a mechanistic state is fitted. 5.4 Reversal-even interaction and reversal-odd memory For one-letter pasts and futures define the exact pair block Nij = µ (ij). (27)

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 5.5 Schedule fields and Hodge obstruction For a scalar endpoint, the calibrated pair coefficients cij = −cji form an antisymmetric edge field on the intervention graph. A stronger compression is global sortability: cij = ui − uj. (30) potential, a scalar coordinate on biological state, or the connection curvature introduced below. These are different typed objects even when all use the language of exactness or circulation. 6. Smooth predictive transport: curvature is the local abelian- ization obstruction The noncommutative word calculus is valid without a manifold. A smoother geometric refinement becomes legitimate only after the experiment has earned a local differentiable representation of predictive state. 6.1 Earning a challenge-visible local manifold Let a frozen challenge-feature map be G(s) =
∈ Rq, (32) where each gc is a future-response feature defined on predictive state. A finite panel is not automatically the state. It earns a local challenge-visible chart only when, on a reachable sheet, the preparation-to-response map is smooth, has locally constant rank, separates the predictive states needed by the experiment, and remains dynamically closed under the tested interventions to the declared tolerance. The imageMvis ⊂ Rq can then serve as a local response manifold. Failure of any of these gates routes the analysis back to the exact discrete/Hankel branch. Theorem-and-no-go paper | claims are jurisdiction-relative 16

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 6.2 The predictive connection Let θ = ( θ1, . . ., θm) be calibrated protocol coordinates and let Va(θ, x) be the induced visible-state vector field. Define horizontal fields on protocol space times the visible state sheet by Ha = ∂a + Va. (33) Their vertical commutator is . (36) Thus an AB/BA experiment is, in the smooth local branch, a measurement of a covector projection of predictive curvature. P/I - LOCAL PREDICTIVE FLATNESS THEOREM On a regular neighbourhood where the horizontal fields (33) are smooth and transverse to the state fibres, the following are locally equivalent: 1. predictive transport between nearby protocol endpoints is independent of the path; 2. every sufficiently small protocol loop has trivial predictive transport; 3. Fab = 0 for all a, b. On a simply connected protocol domain, with regular continuation of the required lifted paths and homotopies maintained globally, flatness supports endpoint-only transport. Without the topology and continuation clauses, local flatness does not prove global path independence.

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 Figure 5: Curvature and waiting on protocol space. Reciprocal loops estimate FAB in the smooth branch. Treating time as another protocol coordinate makes recovery and spacing mixed curvature components rather than a separate mathematical story. 6.3 Time, waiting, washout, and recovery

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 B = E21, and α = β = e1. Then α⊤(BeτLA − AeτLB)β = −e−3τ, e−τLAeτL = e−2τ A. The raw rate −3 lies outside the difference set {0, ±2} without violating the generator model. A spectral falsifier must therefore name and independently calibrate the exact measured coefficient and frame. A finite-dimensional nonlinear state manifold alone does not supply this finite matrix-spectrum conclusion. 6.4 Challenge-visible curvature tomography Let the direct response matrix be Aca = dgc(Va), (40) visible closure, explicitly distinguishing them from certificates of hidden confounding; recent lattice work develops discrete diamond-curvature tests for path dependence under prerequisite constraints [23, 22]. The present contribution is the placement of such local objects on an empirically defined predictive-state quotient and their connection to the global compression ladder. 6.5 Bianchi and holonomy are conditional extensions For a genuine smooth connection, curvature obeys the Bianchi identity. It becomes an experimental consistency test only if its constituent vector and curvature fields are estimated independently over nearby baselines; algebraically recycling the same schedule contrasts would make the check tautological. Likewise, classical holonomy theorems require an additional finite transformation-group or principal- connection structure. These are useful future branches, not prerequisites for the core predictive-closure result [20, 21]. Theorem-and-no-go paper | claims are jurisdiction-relative 19

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 7. Global predictive reconstruction from futures The action algebra tells us how histories can differ. Predictive realization asks how many coordinates are required to reproduce all declared past–future responses. 7.1 The past–future Hankel object Let µ (uv) be a scalar response feature after past word u followed by future word v. Define the Hankel matrix labelled joint probabilities and their normalization. These are distinct observation conventions. Figure 6: The same response series has a weak-scale filtration and a past–future realization filtration. Hankel rows construct predictive states. Singular-value slopes across calibrated pulse scales reveal the perturbative order at which predictive directions become visible. I - IMPORTED THEOREM / STANDARD RESULT Finite Hankel realization. A scalar formal response series has finite Hankel rank d if and only if it admits a d-dimensional linear representation

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 Rank is not the number of molecules, the dimension of a chosen nonlinear physical manifold, the number of stored bits, or the number of mechanistic pathways. It is the minimal dimension of an exact linear predictive realization of the declared series. A molecularly enormous system can have low predictive rank, and a low-dimensional physical system can produce a high or infinite observable rank under a rich nonlinear observation scheme. For a probability-valued series, a finite real linear realization need not be a positive hidden-state realization. Nonnegativity, normalization, and admissible stochastic updates remain additional requirements; the

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 (47). Appending a updates it by xua = M⊤ a xu, and future v is read by β⊤M⊤ v xu. A new intervention can write a coordinate that old interventions could not reach; a new future probe can reveal a coordinate that was already written but previously invisible. 7.4 Predictive emergence spectrum A finite Hankel block may be analytic in a common pulse scale s near the identity. Over the ring of convergent real power-series germs, a finite matrix has a Smith normal form. Thus there are locally invertible analytic matrix germs P(s) and Q(s) and uniquely determined integers 0 ≤ ν1 ≤ · · · ≤ νr (53) Consequently, for sufficiently small nonzeros, σj(H(s)) = Θ(|s|νj ). (55) The exponents νj are predictive-emergence orders of the frozen finite block. The comparison uses a fixed block and analytic changes of basis invertible at s = 0. A scale-dependent normalization singular at zero can change observed slopes and is not covered by this invariance. They replace the false intuition that exact rank must jump as scale increases. Exact rank can already be full for all s ̸= 0; what changes is the visibility of directions above noise. These interpretations are design assignments, not universal labels attached by the algebra. 7.5 Feature refinement and distributional state Mean response can hide predictive distinctions carried by variance, survival, lineage, composition, or other channels. For a nested feature family G ⊆ G ′, the block Hankel rank cannot decrease. The rank increment measures additional linear predictive directions exposed by the new feature block relative to the frozen history/future truncation. A zero increment means only that the new feature adds no independent direction at that resolution; it does not prove causal irrelevance. Death is an outcome when observed, not missing data. Treatment-induced selection can be part of the intervention map if total population, survival denominators, or absorbing states are retained. Conditioning

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 3. fit the smallest representation that predicts training and validation blocks; 4. choose untouched futures that best separate unresolved rows or reduce model uncertainty; 5. test fully held-out histories and continuations; 6. stop at stable local closure, grow the representation after reproducible failure, or report inadmissibility when precision cannot decide. fresh responses independent of that choice, or a justified sequential inference procedure; reusing selected outcomes as an untouched confirmation invalidates the nominal error guarantee. 8. The completed object: predictive action form The framework does not terminate in a response curve. Once state and action have been earned, the experimentally accessible object is a predictive action form PJ = SΠ, Mρ, ρ, O, V, Jcost
, (56) where SΠ is predictive state, Mρ is the predictive action monoid, ρ is its partial action, O is the declared observation/future map, V is an optional viability set, and Jcost is an optional calibrated intervention/re- source cost. Geometry is not built into (56). A topology, metric, connection, Lie-group structure, stochastic kernel, or thermodynamic potential is added only after the relevant comparison or dynamics has been inde- the action component retains the successor kernel or outcome instrument specified in Section 2.3; it is not a single unconditioned posterior state. P/C - THE CENTRAL FORK OF THE PREDICTIVE ACTION FORM Within the declared jurisdiction: Mρ − → ( commutative image: cumulative-vector branch, (57) The two branches are not competing theories. They are alternative structural outcomes of the same descended predictive action. Global Hankel reconstruction remains available in either branch. 9. Experimental decision architecture The calculus becomes scientific only when every branch has an observable gate. Figure 7 summarizes the minimal order of attack. Theorem-and-no-go paper | claims are jurisdiction-relative 23

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 Figure 7: Experimental decision architecture. State is constructed before action algebra; abelianization is tested before cumulative exposure; the cumulative and temporal branches are both validated by held-out predictive reconstruction. 9.1 Stage 1: state before algebra Create at least two different histories and ask whether a proposed present summary is sufficient. Match the candidate state within a preregistered equivalence margin, apply a common held-out future, and compare complete outcome laws. A reproducible separation refutes the state summary. Practical equivalence requires a valid confidence bound to lie within the prespecified discrepancy margin, with simultaneous control over the claims being certified. Failure to reject equality, even in a nominally powered study, does not establish equivalence. Finite-caliper matching, measurement error, common support, popula- insufficiency finding alone does not identify a within-unit causal memory mechanism. 9.2 Stage 2: descent and one-agent calibration Once a predictive state representation is proposed, verify that each admitted intervention has the same admissibility and successor future law across histories assigned to one state. For continuously scaled actions, estimate single-agent trajectories over a range of physical pulse sizes and search for a parameter in which the baseline-relative action approaches identity and repeated pulses satisfy the claimed semigroup law within tolerance. Failure here means the cumulative amount coordinate is not yet earned.

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 For m ≥ 3 interventions, pairwise commutation claims should be treated as a family. If measurement units rescale response rows and intervention calibration rescales columns, sign incompatibilities and cycle products can provide scale-robust obstructions to a symmetric susceptibility submodel, but such reciprocity tests are distinct from the more basic commutation test. 9.4 Stage 4A: cumulative-vector and scalar branch After one-agent calibration and cross-action commutation pass, test whether the cumulative vector itself predicts held-out futures. If it does, compare candidate scalar summaries q(θ) by the fibre criterion (14) using cross-validated future prediction or formal equivalence tests. A scalar that predicts but does not support a well-defined composition remains a predictive score, not an extensive quantity. Only after compositional descent should Aczél-type coordinate tests be attempted. 9.5 Stage 4B: temporal branch If order survives, use the smallest overidentified schedule family appropriate to the signal-to-noise ratio. The AB/BA pair is the cheapest falsifier. The A2B2 ladder distinguishes second-order compression from cubic context. Three or more interventions permit schedule-field circulation and Hodge tests. Varying gaps estimates transport under baseline evolution. A frozen challenge panel can test whether direct and curvature columns expose additional state directions. 9.6 Stage 5: global reconstruction Whichever branch is followed, assemble past–future blocks and demand held-out predictive closure. The abelian branch is not exempt: commutation of action maps does not identify different starting states. A cumulative vector alone can be incomplete when preparations with the same vector begin from different predictive states, or when the claimed state/action descent was supported only on a restricted finite panel. The temporal branch is not exempt either: a beautiful commutator fit does not prove that the complete history process has finite predictive dimension. 10. Three empirical illustrations, not validations The theorem architecture is general; its scientific value depends on domains in which the required interventions and held-out futures can actually be measured. The following examples illustrate different failure gates without being pooled as evidence for universal biological geometry. dt = (V1 − V2) Gmax − G KM + Gmax − G , (58) away from saturation. Thus (e, G) is not a deterministic predictive state over that model jurisdiction unless V is equal, the prefactor vanishes, or the experiment cannot resolve the difference. The important lesson is not that V is universally “the” hidden state. It is that matched-present/common-future separation has Theorem-and-no-go paper | claims are jurisdiction-relative 25

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 an exact mechanistic instance. 10.2 Radiation: equal cumulative summaries can fail before curve shape matters Standard complete-repair linear-quadratic BED is invariant under reversal of the same two fraction sizes. A recent two-fraction radiation study reported different immune states for reversed low/high schedules The broader radiation programme therefore has a clean hierarchy: first test reversal; then classify the residual order field if desired; then measure candidate pre-second-fraction state and demand that it restore held-out prediction. A successful state-recovery experiment would not merely add an empirical correction term to BED; it would identify which information the scalar discarded. 10.3 Antibiotics: finite hysteresis exposes an order-bearing action Sequential antibiotic experiments demonstrate that pretreatment can change response to the next drug and that the direction can depend on order [ 27, 28, 29, 30]. These data are not infinitesimal curvature measurements unless a near-identity scale law is separately established. They are nevertheless exact finite examples of the action-side question: the same ingredients can write different successor states when sequenced differently. In the Predictive Closure architecture, such a result routes the system away from a cumulative order-blind representation and toward a schedule-aware or state-aware model. 11. No-go architecture: what the unified theory does not per- mit A foundational calculus is strongest when the nearest false theorem is written down and killed. Table 1 that measurement and separate under a common future. Construct predictive equivalence; enlarge the state or narrow the jurisdiction. Passing state descent makes intervention order irrelevant. Descended maps may be noncommuting. Test predictive abelianization. Pairwise order equality at one endpoint proves commutation. The endpoint may be blind to a state difference. Use a separating future panel or global predictive rows. Repeated pulse count is continuous cumulative dose. One-agent repetition can fail the semigroup law.

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 Blocked claim Why it fails Required repair A scalar output automatically has a scalar composition law. Fine composition can vary equals biological mechanism dimension. It is a minimal linear predictive dimension, not a molecular count. Add mechanistic perturbations, positive-realization constraints, absorbing outcomes; model nonrandom dropout. Predictive closure alone supplies a thermodynamic arrow. Reversible exact dynamics can preserve information or fibre models, Newtonian systems, and relativistic fields can share the predictive kernel. Construct the bridge with independent spacetime premises. The last three rows preserve the broader type discipline of the earlier Predictive Closure architecture without allowing those external branches to dominate the state–action theorem. Observation theory, Theorem-and-no-go paper | claims are jurisdiction-relative 27

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 thermodynamics, spacetime reconstruction, and continuum limits remain downstream modules. They are not derived by the core merely because the same word “state” or “curvature” appears in them. 12. Relation to prior frameworks and the actual novelty claim The mathematical ingredients of this paper have deep precedents. Predictive equivalence and causal-state ideas appear in computational mechanics and predictive-state representations [ 1, 2, 3]. Finite Hankel realization belongs to automata, formal-series, and system-identification theory [ 6, 7, 8]. Lie brackets, chronological calculus, and nonholonomic accessibility are standard in geometric control [9, 10, 11, 12]. Hodge decomposition of pairwise comparison fields is established mathematics [19]. Additive generators for regular associative interval operations are classical functional-equation results [15, 16]. Connection curvature and holonomy are classical geometry [20, 21]. The novelty claim is therefore intentionally narrower and more structural: 1. predictive state, action descent, action abelianization, cumulative amount calibration, scalar descent, and lawful coordinate are placed in one strict dependency order; 2. cumulative exposure is identified as the abelian image of the predictive action, rather than as an upstream physical state variable; 3. Temporal Grammar is identified as the nonabelian branch of the same Predictive Closure architecture, rather than as a separate foundational theory; 4. weak word contrasts and smooth curvature are linked as local diagnostics of failed predictive abelian- ization, while Hankel realization supplies the global state reconstruction; 5. every branch carries an empirical death condition that says which richer object is required when compression fails. Recent preprints sharpen the boundary. Pasechnyuk-Vilensky develops exact diamond-curvature condi- tions for path-independent edge-additive sequential interventions on ideal lattices [22]. Mahadevan’s revised preprint uses Lie-bracket geometry in an interventional screening pipeline and treats nonclosure Those works demonstrate that order-sensitive intervention geometry is an active area rather than a blank mathematical field. The present paper does not claim priority for order curvature. Its proposed contribution is the predictive-state-first compression ladder and the exact abelian/nonabelian fork linking cumulative exposure, temporal grammar, scalar descent, and predictive realization. 13. Frozen predictions and falsifiers A framework of admissibility earns scientific force only through predictions that can fail. Table 2 freezes the main tests at Version 4. Theorem-and-no-go paper | claims are jurisdiction-relative 28

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 Table 2: Frozen predictions and branch falsifiers. Programme Frozen prediction / admission test Falsifier or failure route Predictive state A specified pair/future hypothesis predicts separation at matched measured state; a proposed sufficient augmentation predicts held-out equivalence within its scope. The specified separation fails, or the augmentation fails equivalence; an untested future class remains open. Action descent Histories in one predictive class have identical intervention admissibility and successor future laws. Domain or successor-law mismatch. Persistent semigroup defect beyond tolerance. Predictive abelianization All same-multiplicity words act identically iff generator partial maps commute. Any replicated reciprocal future separation. curves require their own transfer law. Predictive realization A finite-rank representation predicts untouched past–future blocks and no smaller model does. Rank growth or held-out prediction failure. Scalar descent Candidate scalar equalizes all future laws and supports a representative-independent composition. Lawful coordinate An admitted total scalar law satisfies the representation theorem’s hypotheses and its generator reproduces the same held-out law in new coordinates. A structural premise or the claimed transformed equality fails; predictive improvement is not required by reparameterization. Redox carrier At matched fast state, future redox separation scales with the measured

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 Programme Frozen prediction / admission test Falsifier or failure route Radiation cumulative scalar Reversal of same components is null if the scalar is sufficient; a positive The earlier branches of the programme appeared to compete because they were written at different layers of one compression problem. Bounded coordinates asked how a scalar should be represented. Temporal Grammar asked how ordered interventions reveal hidden state. Predictive-state papers asked what state means. Radiation reversal asked when a cumulative scalar loses history. Predictive Flatness asked when path endpoints can replace paths. The unified architecture shows that these were not rival foundations. They are successive gates. The first object is not geometry. It is a future-equivalence quotient. The second object is not a differential equation. It is the partial action of interventions on that quotient. Only then does composition become through the Hankel realization. This ordering resolves several recurrent confusions. First, physical additivity and predictive sufficiency are different. Absorbed energy, delivered mass, elapsed time, and particle counts may be physically additive while remaining insufficient as biological state variables. Predictive Closure does not deny those physical quantities; it tests whether the future law factors through them. Second, order dependence is not nonassociativity. Intervention maps compose associatively as maps. A nonzero commutator means only that exchanging order changes the result. The correct algebraic response is a noncommutative monoid or its local Lie structure, not a claim that function composition has failed. declared jurisdiction, in which case a scalar curve may be adequate. The framework is designed to classify that jurisdiction rather than legislate one answer. Fourth, finite-dimensional reconstruction is an empirical outcome. Predictive Closure does not assume that biology is low-dimensional. It supplies a stopping rule: stable finite rank and held-out closure support compact prediction; reproducible rank growth reports unresolved or effectively long-memory state; insufficient precision yields inadmissibility rather than a forced model. Fifth, geometry is earned last. The connection curvature in Section 6 is a local property of an already reconstructed action chart. It should not be confused with intrinsic curvature inferred from a sigmoid, with spacetime curvature, or with a universal hyperbolic container. A state space may ultimately be Euclidean, spherical, hyperbolic, stratified, Finsler, directed, discrete, or nonmanifold. The predictive calculus licenses the comparison; it does not predetermine the winner. Theorem-and-no-go paper | claims are jurisdiction-relative 30

PREDICTIVE CLOSURE - VERSION 4 MURRA Y - REVISED SEPTEMBER 2026 The programme’s strongest philosophical claim is therefore modest in form but broad in reach: P/H - FIBRE DISCIPLINE A representation is lawful only to the extent that its discarded distinctions remain irrelevant to every future it claims to predict. That statement is not a new physical law. It is a discipline on what counts as state, exposure, composition, and coordinate in empirical science. 15. Conclusion Predictive Closure can now be stated without separate foundational papers competing for the same middle ground. Histories define state by future equivalence. Continuation-closed futures make deterministic or outcome- labelled continuations descend to partial predictive actions, with successor kernels for unconditioned stochastic actions. Those actions admit an exact global fork. If the generator actions commute, the intervention language factors through its abelianization and order can be forgotten at the standardized- count level; with one-agent semigroup calibration, a continuous cumulative-vector representation can be tested. If the actions do not commute, order is predictive information and the appropriate branch is Temporal Grammar: weak-word compression, reciprocal commutators, higher-order context, schedule circulation, gap transport, and local curvature in the smooth regime. Neither branch ends the state problem. Past–future Hankel reconstruction determines whether the resulting representation predicts untouched continuations and how many linear predictive directions the declared experiment resolves. Only after the cumulative branch passes may a scalar summary be tested. Only after scalar compositional descent and the standard regularity axioms may an Aczél generator be used. Boundedness alone supplies none of those steps. The result is one theorem-and-no-go architecture rather than a sequence of loosely related claims: predictive state → predictive action → ( abelian cumulative branch, nonabelian temporal branch, → predictive realization → only then: scalar coordinate or richer geometry. (60) Cumulative exposure is therefore not the starting noun. It is an earned quotient of action. Temporal Grammar is not a rival theory. It is the structure left behind when that quotient fails. Lawful Coordinates are not an upstream geometry. They are a downstream scalar branch. The paper’s central scientific question is the one shared by all three: How much of the past may the present lawfully forget without changing the futures the model claims to predict? Theorem-and-no-go paper | claims are jurisdiction-relative 31