Closure predecessor

### PDF page 17 PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM Blocked claim Why it fails Exact repair Finite / approximate re- pair Same measured value means same state A common future can split one measurement fibre Enlarge state until every fu- ture law factors Simultaneousfibrediscrep- ancy ≤ 𝜀 A precise instrument guaran- teesanadequatestatevariable Value uncertainty and represen- tation sufficiency are distinct Test future-law fibre con- stancy Report both value uncer- taintyandpredictivedefect Changing Π destroys the old state Richer futures refine equiva- lence Use the canonical surjection Φ21 Retest only newly exposed fibres/futures Anactionequationmeansthe state is dynamically closed Update can depend on the hid- den representative history Proveactiondescent/congru- ence Frozen continuation bat- terywithdomainoutcomes No observed pair-order effect proves commutation One endpoint may be blind to noncommutation Equality over a separating fu- ture family Equivalence certificate over frozen future panel Cumulative count automat- ically means cumulative amount Repeated action may violate a semigroup law Calibrate one-agent semi- group Bound semigroup defect and propagate error Commuting cumulative vec- tor implies one scalar dose Future law may vary inside scalar fibres Test scalar descent Scalar fibre discrepancy≤ 𝜀 Anonlineartransformrepairs history loss Injective transforms preserve fi- bres Enlarge state or narrow claim Noapproximatecoordinate fix for an already failed fi- bre A scalar output deserves ad- ditive arithmetic Composition may not descend or be associative Test scalar composition ax- ioms Equivalence margins on bracketed operations FinitenoisySVDprovesexact state dimension Small singular values are uncertainty-sensitive Exact rank only in exact real- ization theorem Certify lower dimensions / effective rank with pertur- bation bounds FinitePASSprovesexactlaw- fulness A finite panel and nonzero mar- gin do not quantify over all fu- tures Verify exact fibre equality overthedeclaredexactfamily Report a(Π, 𝜀, 𝛼)-indexed certificate only Low precision means out of jurisdiction Precision failure does not alter the estimand Keep the declared claim and obtain more information Classify UNRESOLVED; reserve out-of-jurisdiction for violated design premises Order effect proves curvatureLarge/thresholded protocols need not have a smooth local limit Establish a near-identity area- scaling window Report discrete order de- pendence if local model is unresolved A failed measurement ends the analysis The separating future itself car- ries missing-state information Refine representation with re- sponse rows Discovery/freeze/hold-out reconstruction cycle 12. Discussion The central theorem is elementary because the scientific mistake it forbids is elementary. If a representation claims to stand in for state, then histories identified by that representation must be interchangeable for the future claims made from it. What is usually left implicit is that the same obligation recurs at every later compression. This produces a strict order of inference. First ask whether the measured present is a sufficient predictive state. Then ask whether the interventions descend. Then ask whether temporal order may be erased. Only then ask whether the resulting cumulative object can be scalarized. Only after that ask whether the scalar operation deserves additive arithmetic. A later transform cannot repair a failed earlier quotient. Thefinite-datatheoryisdeliberatelymoreconservativethantheexacttheorem. Approximateequalityisnotpromoted to an equivalence relation. A pass is a certificate indexed by a frozen future family, discrepancy, margin, range, timing, and error budget. An unresolved result is not evidence for equivalence. An expanded future family refines the Exact theorem, finite-data certificate, and reconstructive failure protocol 17 ### PDF page 18 PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM state rather than making previous measurements meaningless. Theframeworkalsoseparatesrejectionfromreconstruction. Afuturethatsplitsafibreidentifiesadistinctionthatwas active but unmeasured. Response rows provide a direct empirical language for retaining those distinctions. In finite linear-realization settings, Hankel rank supplies the exact classical dimension theorem; outside that branch, finite blocks remain proposals and lower-resolution certificates rather than universal nonlinear state-dimension claims. The present paper therefore separates three scientific statuses that should not be conflated: theorem validity ≠ protocol executability ≠ empirical utility. (22) The theorem establishes that measurement adequacy for a declared predictive use is a factorization property, and that the same obligation governs successive compressions. The worked simulation establishes executability under a frozen finite design. The remaining empirical question is whether this staged process materially improves inference or experimental design in real systems beyond existing specialist methods. That question is intentionally left open rather than being inferred from the theorem. The next scientific step is therefore prospective rather than rhetorical: apply the frozen protocol to independent data, compare its selected representation with existing state-selection methods, and test whether its additional distinctions improve held-out intervention prediction. The present paper supplies the theorem, the finite certificate, and the reconstruction rule; nature must decide the utility claim. Final statement A lawful measurement process for an evolving system begins before calibration ends: it asks whether the measured representation preserves the future laws for which it will be used. Predictive Closure proves that admission criterion, shows how it propagates through action and compression, and turns a failed fibre into a reconstruction experiment. The theorem determines what must be true. The test determines whether a particular measurement earns the claim. A. Partial-map abelianization: domain propagation For partial maps, equality means equal domains and equal values on that domain. Suppose𝑇𝑖𝑇𝑗 = 𝑇𝑗𝑇𝑖 for every pair. Any two words with the same multiplicities are connected by a finite sequence of adjacent transpositions. Replacing one adjacent pair by the swapped pair preserves the partial composite as a map. Composition on either side with fixed partial maps preserves equality of the resulting composites. Induction over the transposition sequence therefore gives equality of the complete word maps, including domains. Conversely, factorization through multiplicity makes each 𝑎𝑖𝑎 𝑗 and 𝑎 𝑗 𝑎𝑖 equal immediately. If scientifically meaningful terminal failure is represented by an absorbing state⊥, a partial system may be totalized and the same theorem applied in an ordinary transformation monoid. This is legitimate only when⊥ is part of the phenomenon, not when it is merely missing data. B. Finite-data preregistration template Before confirmatory data are opened, record: 1. the claimed measurement𝑥 and exact scientific statement it is supposed to support; 2. the frozen confirmatory future familyΠ and any separately labelled discovery family; 3. operating range, timing, preparation, inclusion/exclusion, and terminal outcomes; 4. the rule for matching histories on𝑥 without using confirmatory future outcomes; Exact theorem, finite-data certificate, and reconstructive failure protocol 18

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM Abstract Measurement accuracy and measurement adequacy are different questions. An instrument may assign the chosen quantity with negligible uncertainty while that quantity still omits history needed to predict the next intervention. This paper formalizes that distinction for evolving systems. For a declared family of future intervention–observation tests, a candidate present measurement𝑥 is calledpredictively lawfulexactly when equal measured presents imply equal declared future laws. We prove the equivalent fibre-factorization criterion and show that the quotient of histories by complete future equivalence is the coarsest exact predictive state. Thus an exactly measured variable can fail as a measurement of the claimed state without any calibration error. The same rule generates a hierarchy of stronger measurement claims. Interventions act on predictive state only when their continuation laws descend. Ordered intervention histories may be replaced by cumulative counts exactly when the descended generator actions commute; for stochastic total actions the same statement holds for Markov kernels under composition. A continuous cumulative amount additionally requires an approximately or exactly semigroup-like one-agent calibration. A scalar burden or dose is a separate quotient and is admissible only when future laws are constant on its fibres. Only after an empirical scalar composition is well defined and satisfies the regularity, order, identity, and associativity hypotheses does the Aczél theorem license an additive coordinate. Exactstateisanidealobject; finitesciencereceivesjurisdiction-andresolution-indexedcertificates,notmetaphysical completion. We define a preregistered predictive discrepancy, require equivalence margins to be fixed independently of confirmatory outcomes, and separate PASS, FAIL, UNRESOLVED, and OUT-OF-JURISDICTION. Nested future families induce exact coarse maps, so an expanded jurisdiction refines rather than destroys an earlier state representation. Failed fibres are constructive: a reproducible forward-refinement rule selects separating futures, freezes the enlarged representation, and stops only after held-out predictive closure or a declared resource ceiling. Finite block-Hankel matrices may propose additional predictive directions, while exact Hankel-rank claims are restricted to the classical finite linear-realization branch. A fully reproducible simulation runs the complete gated protocol, and a mechanistic redox example shows the same fibre logic symbolically. The result is a measurement- admission process for deciding, with explicit scope and uncertainty, what information an evolving present is entitled to forget. Keywords:measurementtheory;statesufficiency;predictivestate;intervention;cumulativeexposure;abelianization; temporal order; Hankel realization; equivalence testing; system identification. 1. The measurement problem comes before the model Metrology asks what quantity is intended, how values are obtained, and with what uncertainty [1]. Representational measurement asks what empirical structure licenses a numerical representation [2]. Statistics asks when a statistic preserves information relevant to an inferential target. Control and predictive-state theories ask how present information supports future prediction and action. These traditions are complementary, but they leave a practical question that is routinely decided before any model is fitted: THE PRIOR QUESTION When may a measured present be treated as the state of an evolving system for the future interventions and outcomes that the investigator intends to predict? The distinction is simple. Suppose two organisms, machines, or populations have exactly the same measured value𝑥. If one common future intervention produces different outcome laws, then the instrument need not be inaccurate. The variable 𝑥 was simply an incomplete representation of the state claim built on top of it. Exact theorem, finite-data certificate, and reconstructive failure protocol 4

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM history ℎ1 history ℎ2 same measured present 𝑥 future law 𝑃2 If 𝑃1 ≠ 𝑃2, value accuracy does not rescue the state claim. Figure 1:The value question and the representation question are distinct. Predictive Closure tests the second. The entire paper is one recursive rule: a distinction may be erased only if every claimed future law is constant on the fibre created by erasing it.(1) State measurement, coarse dynamics, cumulative exposure, scalarization, and coordinate choice are different applications of that rule. The mathematical ingredients are often classical; the claim here is the measurement architecture that forces them into this dependency order and makes failure constructive. 2. Predictive measurement admission 2.1 Declared use and exact lawfulness Fix a declared exact jurisdiction J = (H , Π, Y, R) , (2) where H is the admissible history set,Π is the future intervention–observation family,Y is the outcome language, The codomain may be scalar, vector, categorical, or functional. P - PROVED HERE Definition/Theorem 1 - Predictive Measurement Admission.The measurement𝑥 is exact and predictively lawful forΠ if and only if every future law factors through𝑥: for each𝑢 ∈ Π there exists a kernel𝐾𝑢 on 𝑋 such that Law(𝑌 | ℎ, 𝑢) = 𝐾𝑢 ( · | 𝑥(ℎ)) ∀ ℎ ∈ H . Equivalently, Reparameterization may still be scientifically useful: it can improve numerical conditioning, expose a composition law, simplify comparison across studies, or provide a lawful additive coordinate after the upstream fibre tests pass. What it cannot do is restore predictive distinctions that the original measurement already erased. Exact theorem, finite-data certificate, and reconstructive failure protocol 5

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM 2.2 The coarsest exact state Define predictive equivalence by ℎ ∼Π ℎ′ ⇐ ⇒ Law(𝑌 | ℎ, 𝑢) = Law(𝑌 | ℎ′, 𝑢) ∀ 𝑢 ∈ Π, (4) including agreement on scientifically meaningful admissibility and terminal outcomes. Let SΠ = H / ∼Π, 𝑞 (ℎ) = [ℎ]Π. (5) P - PROVED HERE Theorem 2 - Minimal Predictive State.The quotient map𝑞 : H → S Π is the coarsest exact measurement sufficient for the declared future family. A candidate exact measurement𝑥 is lawful exactly when𝑞 factors through 𝑥 on its image. Proof. If 𝑥 is lawful, equal𝑥 implies predictive equivalence, so𝑞 is constant on each𝑥-fibre. Conversely,𝑞 = 𝜑 ◦ 𝑥 makes equal𝑥 imply equal predictive class.□ Predictive and causal states, sufficient statistics, belief states, and predictive state representations provide important precedents for future-relevant state [3, 6, 5, 7, 8]. Priority is not claimed for the quotient. Predictive Closure instead uses that quotient as a measurement-admission standard for arbitrary claimed present measurements and then audits every stronger compression built on top of them. 2.3 Jurisdiction refinement: state need not be rebuilt from scratch The dependence onΠ is not a defect to hide. It is what makes the claim testable. If a future family is expanded, the exploratory additions become a new refinement experiment rather than a post-hoc redefinition of the same claim. 3. From admitted state to admitted action A variable can predict passive continuation while failing under intervention. A state used for action must therefore remain sufficient after the actions it is supposed to support. For an intervention𝑎, it is enough that the declared future family be𝑎-stable: whenever a later future𝑢 is used to compare the successors of histories under𝑎, the composite continuation𝑎𝑢 is included in the state-defining family whenever scientifically admissible. Global closure under every imaginable continuation is not required.

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM P - PROVED HERE Theorem 4 - Action Descent.If Π is 𝑎-stable, then intervention𝑎 induces a well-defined partial map 𝑇𝑎 ( [ℎ]Π) = [ℎ𝑎]Π exactly when predictive equivalence is a congruence for𝑎: equivalent histories have the same𝑎-domain and, wherever 𝑎 is admissible, equivalent successor histories. If the declared future family is not𝑎-stable, the action is not thereby false; it isuncertified. The investigator may enlarge Π to the required continuation hull or narrow the action claim. Partiality is itself scientifically visible. If𝐴𝐵 is viable but𝐵 𝐴is terminal or inadmissible for the same predictive state, then the two word actions already have different domains. Death, dropout, impossibility, and absorbing failure must therefore be recorded as outcomes when they are consequences of the process rather than analysis-induced missingness. C - CONDITIONAL BRANCH Stochastic-action branch.Total Markov kernels on predictive state form a monoid under kernel composition. Therefore the same abelianization statement below holds verbatim for stochastic interventions: ordered kernel products factor through multiplicities exactly when the generator kernels commute. Partial stochastic admissibility may be totalized only when the added terminal state represents a genuine scientific outcome. 4. The compression ladder claims. history predictive state descended intervention action order can be erased: cumulative representation order remains predictive: retain temporal structure scalar future-law descent lawful additive coordinate separating futures

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM 4.1 Order removal is predictive abelianization Let A = {𝑎1, . . . , 𝑎𝑚} and let ordered words form the free monoidA∗. The descended action is 𝜌 : A∗ → PMap(SΠ). (6) Let 𝑛(𝑤) ∈ N𝑚 record only multiplicities. P - PROVED HERE Theorem 5 - Predictive Abelianization.The ordered predictive action factors through multiplicity, 𝜌(𝑤) = ¯𝜌(𝑛(𝑤)) ∀ 𝑤, if and only if all generator actions commute pairwise as partial maps, 𝑇𝑖𝑇𝑗 = 𝑇𝑗𝑇𝑖 ∀𝑖, 𝑗, with equality of domains and equality of values on that domain. identifies all same-multiplicity protocols, and is therefore exact only after this action factorization is licensed. P - PROVED HERE Corollary - Observable falsification is one-sided.If a declared separating future𝑔 satisfies 𝑔 (𝑇𝑗𝑇𝑖 𝑠) ≠ 𝑔 (𝑇𝑖𝑇𝑗 𝑠), then the state actions do not commute and cumulative counts are false for that jurisdiction. Equality of one endpoint under 𝑖 𝑗 and 𝑗𝑖 does not prove commutation; support for commutation requires equality over a separating future family. 𝑔 ∈ G 𝐷Law(𝑔 | 𝑇𝑗𝑇𝑖 𝑠), Law(𝑔 | 𝑇𝑖𝑇𝑗 𝑠)
. (7) A simultaneous lower confidence bound above a preregistered𝜀ord rejects predictive commutation. An upper bound below 𝜀ord supports only the panel-relative statement that no order-relevant distinction was resolved byG at that resolution. It supports commutation on a claimed state class only whenG has independently earned a separating certificate for that class. There is therefore no universal required number of futures: the relevant requirement is separation of the state distinctions the claim purports to erase. A repeated intervention may change its own later effect. Define the one-agent semigroup defect Δ𝑎 (𝑠, 𝑡) = 𝑑Π (𝑇𝑎 (𝑠 + 𝑡), 𝑇 𝑎 (𝑠)𝑇𝑎 (𝑡)) , (8) for a declared predictive discrepancy𝑑Π. Exact additive amount requires zero defect on the claimed domain. Finite science tests whether the defect is below a preregistered margin throughout a calibrated window. Exact theorem, finite-data certificate, and reconstructive failure protocol 8

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM C - CONDITIONAL BRANCH Continuous cumulative branch.If 𝑇𝑎 (0) = id, each one-agent family obeys the semigroup law on a common invariant domain, and distinct families commute, then the predictive action factors through an additive exposure vector 𝜃 ∈ R𝑚 +. Ifthesemigroupdefectisonlybounded,theresultingrepresentationisanapproximatecertificate whose accumulated error must be propagated rather than silently set to zero. For a locally𝐿-Lipschitz update with per-step representation error at most𝜂, an𝑛-step propagated discrepancy is severity index. P - PROVED HERE Theorem 6 - Scalar Measurement Descent.The scalar𝑞(𝑧) is an exact predictive measurement exactly when every declared future law is constant on the fibres of𝑞. Commutativity of the cumulative vector does not imply one-dimensional sufficiency. Theconditionisintentionallystrongbecausetheclaim“thisscalaristhestateforthesefutures”isstrong. Approximate scalarization is handled by the finite-data certificate in Section 6, not by weakening the exact theorem. An empirically sufficient scalar is not automatically an extensive coordinate. An operation⊕ on its interval is well defined only when the fine composition is constant on the scalar fibres. After that descent, its algebraic properties are empirical hypotheses. I - IMPORTED RESULT Aczélbranch. Foraclosedintervaloperationthatiscontinuous,strictlyincreasingineachargument,associative, and has an identity, classical functional-equation theorems give a continuous monotone generator𝜓 with 𝜓(𝑥 ⊕ 𝑦) = 𝜓(𝑥) + 𝜓(𝑦).

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM 5.1 Local order witnesses In a validated near-identity smooth regime, fix the convention[ 𝐴, 𝐵] := 𝐴𝐵 − 𝐵 𝐴. Weak interventions generated by 𝐿 𝐴, 𝐿 𝐵 then obey the standard expansion 𝑈𝐴(𝑠)𝑈𝐵 (𝑠) − 𝑈𝐵 (𝑠)𝑈𝐴(𝑠) = 𝑠2 [𝐿 𝐴, 𝐿 𝐵] + 𝑂 (𝑠3). (9) A reciprocal difference with the predicted area/scale onset is therefore a local observable witness of failed order removal. This is imported chronological-calculus structure, not a new Lie theorem [14,15,16]. If no near-identity scaling window is demonstrated, the proper conclusion is discrete order dependence, not curvature or a local bracket estimate. 5.2 Global reconstruction by future-response rows Let 𝑝 index past histories,𝑞 future words, and𝜙 𝑗 (𝑌 ) a frozen response feature. Define the block response matrix 𝐻 𝑝, (𝑞, 𝑗 ) = E[𝜙 𝑗 (𝑌 ) | 𝑝𝑞]. (10) Rows that differ are predictively distinguishable for the selected future-feature family. I - IMPORTED RESULT Finite linear realization branch.In the classical weighted-automaton/observable-operator setting, finite Hankel rank 𝑟 is equivalent to an exact𝑟-dimensional linear predictive realization, and𝑟 is the minimal linear realization dimension [11, 12, 13, 8]. This theorem is not extended here to arbitrary nonlinear or infinite-dimensional state. P - PROVED HERE Theorem 7 - Failure-to-Reconstruction.If a candidate measurement fails, then some pair of histories in 1. Freeze a discovery future libraryF𝐷, an independent validation libraryF𝑉, discrepancy 𝐷, margin𝜀, and a resource ceiling before confirmatory refinement begins. 2. Start from the candidate representation𝑧0 = 𝑥. On validation data, estimate the largest within-fibre predictive defect of𝑧𝑘 with simultaneous uncertainty. 3. If its upper bound is at most𝜀, stop and certify𝑧𝑘 for the frozen validation jurisdiction. If its interval straddles𝜀, report UNRESOLVED rather than adding dimensions merely because the estimate is noisy. 4. If failure is certified, use discovery data to choose a future–feature pair with the largest reproducible lower- bound separation among the failed fibres, freeze that response feature, and define the refined representation 𝑧𝑘+1 = (𝑧𝑘, 𝜙𝑘+1). 5. Refit the predictive representation without opening the held-out validation outcomes, then return to Step 2. If no pass occurs before the registered resource ceiling, report that finite closure was not demonstrated within the declared library and depth. Exact theorem, finite-data certificate, and reconstructive failure protocol 10

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM This greedy procedure isnot claimed to find the globally minimal exact state. It gives a reproducible empirical path toward it. Along a nested registered refinement path, the first representation that obtains held-out closure is the smallest certified representation on that path. Complexity therefore remains subordinate to predictive adequacy: dimensions are added only after a certified fibre failure, and retained only if they transfer to held-out futures. 6. Finite-data measurement admission The exact quotient is an ideal mathematical target. Finite science should not pretend that noisy approximate equality is itself an equivalence relation. Instead it issues a certificate for a frozen experimental claim. timing, discrepancy, equivalence margin, and error budget. FINITE CERTIFICATE SEMANTICS If a valid simultaneous upper confidence bound for the largest tested within-fibre predictive defect is at most𝜀 witherrorbudget 𝛼,thenthecandidatemeasurementis certifiedatresolution 𝜀 forthatfrozenfinitejurisdiction at confidence level1 − 𝛼. The wordlawful without qualification is reserved for the exact theorem. Finite certification is evidence about a declared experimental claim, not proof of equality for untested futures. 6.2 Predictive discrepancy and equivalence margin Choose before confirmatory analysis a discrepancy𝐷 between outcome laws and define 𝑑Π (ℎ, ℎ′) = sup 𝑢∈Π 𝐷 (𝑃(· | ℎ, 𝑢), 𝑃(· | ℎ′, 𝑢)) . (11) For a candidate measurement fibre, letΔΠ be the largest relevant within-fibre predictive discrepancy over the frozen design. Let 𝜀 > 0 be the largest discrepancy the measurement claim is willing to treat as scientifically negligible. HOW 𝜀 MAY BE CHOSEN Theequivalencemarginmustbejustifiedindependentlyoftheconfirmatoryfibreoutcomes. Defensiblesources are: (i) a decision threshold at which the downstream scientific/control decision would change; (ii) an external assay or intervention resolution; or (iii) a previously established domain margin. If none exists, report the estimated predictive defect and uncertainty rather than tuning𝜀 until the measurement passes. 6.3 Four outcomes, not a forced binary decision Let [𝐿, 𝑈] be a simultaneous confidence interval forΔΠ under the frozen probe battery and matching design. 𝑈 ≤ 𝜀 ⇒ PASS, 𝐿 > 𝜀 ⇒ FAIL,

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM declared range; lack of overlap/support for the matched comparison; a required intervention that cannot be executed as declared; a scientifically meaningful terminal outcome that was instead lost as missing data; or a protocol change that alters the estimand. Low precision by itself isnot out of jurisdiction. 6.4 Multiplicity, matching, and discovery/confirmation candidate futures are discovered adaptively, split the workflow: discovery − →freeze representation and probes− →independent confirmation. (13) Coordinate-wise scanning requires simultaneous error control; alternatively, a single preregistered multivariate discrepancy may be calibrated by an appropriate permutation or bootstrap null. ILLUSTRATIVE BOUNDED-OUTCOME CALIBRATION Suppose 𝑀 planned two-group mean differences use independent responses in[0, 1], with𝑛 replicates in each group for each contrast. Applying Hoeffding directly to each difference of independent sample means gives the same dimensions and ordered singular values, |𝜎𝑘 ( b𝐻) − 𝜎𝑘 (𝐻)| ≤ 𝛿. (15) Thus 𝜎𝑘 ( b𝐻) − 𝛿 > 0 certifies at least𝑘 nonzero predictive directions in that finite block. Finite noisy data do not generally certify that all smaller singular values are exactly zero. Upper bounds can instead support a declared effective-ranktolerance. This prevents numerical rank thresholding from masquerading as an exact state-dimension theorem. Exact theorem, finite-data certificate, and reconstructive failure protocol 12

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM 7. A gated protocol that an experimenter can run 0. Declare claim and jurisdiction 1. Match presents; source, matching rule, confirmatory error budget, and any hold-out futures. Stage 1 - test the state measurement.Create or identify histories equivalent on𝑥 within a predeclared matching rule; apply common futures; compute the simultaneous predictive defect; classify PASS/FAIL/UNRESOLVED/OUT- OF-JURISDICTION. Stage 2 - test action and order removal.For an admitted state representation, test whether claimed interventions descend. If cumulative exposure is proposed, compare reciprocal or same-multiplicity protocols using separating futures. A single robust separating future rejects exact abelianization. 8.1 Stage 0: a perfectly measured but incomplete present Two preparation historiesℎ0, ℎ1 both have the perfectly measured candidate present𝑥 = 0. Their unobserved predictive distinction is𝑚 ∈ { 0, 1}. Three discovery futures produce Bernoulli outcomes with true probabilities 𝐶1 𝐶2 𝐶3 ℎ0 0.25 0 .45 0 .50 ℎ1 0.75 0 .55 0 .50 (16)

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM 8.2 Stage 1: the fibre fails With random seed 20260817, the discovery data are: Future ℎ0 successes ℎ1 successes b𝑝0 b𝑝1 − b𝑝0 𝐶1 68/300 225/300 0.227 0.523  , (18) whose singular values are1.899 and 0.331. In this simulation the true model has two predictive classes, but the finite matrix is used only as a coordinate proposal, not as proof of exact rank. The independent hold-out future gives 94/300 = 0.313 for ℎ0 and 230/300 = 0.767 for ℎ1, preserving the reconstructed separation. Operationally, the reconstructed coordinate is the preparation/history class whose future-response law was separated; it is not necessary to apply the destructive𝐶1 challenge to the same individual and then call that outcome its pre-challenge state. The replacement representation therefore retains the predictive distinction that the original𝑥 erased. 8.4 Re-enter Stage 2: cumulative counts can fail after state is fixed For a purely algebraic downstream illustration, interpret𝑟 as a rapidly written response coordinate and𝑚 as retained memory on an admitted two-coordinate predictive state(𝑟, 𝑚 ). Define deterministic interventions 𝑇𝐴(𝑟, 𝑚 ) = (𝑟 + 1, 𝑚), 𝑇 𝐵 (𝑟, 𝑚 ) = (𝑟, 𝑚 + 𝑟). (19) From (0, 0), executing𝐴𝐵 (apply 𝐴 then 𝐵) gives(1, 1), whereas𝐵 𝐴gives (1, 0). Both words have count vector (1, 1), but a future challenge that reads𝑚 separates them. Hence the count representation fails by Theorem 5. This illustrates the gating logic: state was repaired first; only then was the stronger cumulative-exposure claim tested.

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM 9. Biological and empirical illustrations: exact status 9.1 A mechanistic fibre failure in glutathione dynamics In a published redox model [21], a fast damage state𝑒, glutathione pool𝐺, and slower regeneration capacity𝑉 obey a term of the form Thisdoesnotprovethat (𝑒, 𝐺, 𝑉 ) isacompletebiologicalstateforeveryfuture. Itshowsexactlyhowademonstrated failure identifies the missing coordinate within the stated model. The example is therefore an exact mechanistic instantiation, not empirical validation of Predictive Closure and not evidence for any particular geometry. 9.2 External order effects as candidate cumulative-measurement falsifiers Sequential antibiotic studies report history-dependent susceptibility and asymmetric hysteresis, including reciprocal sequence effects among beta-lactam treatments [22, 23]. Such results are relevant because a reproducible same- components/different-order future contrast is exactly the kind of observation that can reject an order-blind cumulative representation. The cited studies were not designed as full Predictive Closure tests, so they are treated here as external empirical illustrations rather than framework validation. An independent raw-data reanalysis or prospective preregistered sequence study remains an open empirical obligation. G - OPEN EMPIRICAL GATE Flagshipempiricalobligation. ExecutethefrozenStage0–4protocolononeindependentpublicorprospective dataset,includinganexternallyjustifiedequivalencemargin,confirmatorysimultaneouserrorcontrol,aheld-out future battery, and reconstruction after any demonstrated fibre failure. Until then the measurement theorem is proved, the protocol is operationalized, and empirical utility remains to be demonstrated. 10. Relation to existing theories

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM Tradition Primary question Relationship to Predictive Closure Statistical sufficiency Does a statistic retain information about a specified inferential target? Supplies the central logic of information-preserving com- pression. Here the target is a frozen family of future ning? Beliefstatessolvepartialobservabilityrelativetoamodel. Predictive Closure asks whether an arbitrary measured variabledeservesstatestatusbeforedownstreamdynamics are trusted [5]. Causal states / PSRs Can state be represented by predic- tions of future observables? Closest state precedent. Predictive Closure adopts future equivalence as the admission standard and adds recur- sive tests for intervention descent, cumulative exposure, scalarization, and constructive repair [6, 7, 8]. Causal intervention cal- be recovered from input-output be- haviour? Predictive Closure asks the prior question of what em- pirical distinctions should count as state; after failure, realization theory becomes a reconstruction tool [10,13]. Representational mea- surement a valid causal graph and adjustment assumptions, to identify the mean effect of a treatment𝐴 on an outcome𝑌 by a back-door adjustment. That result answers an intervention-identification question. It does not imply that𝑍 is a predictive state for a later family containing another intervention𝐵. If two histories with the same𝑍 have different future laws under𝐵, Predictive Closure rejects𝑍 as a state measurement for that sequential jurisdiction even though the effect of𝐴 is identifiable. Conversely, a variable may pass the predictive fibre test on observed or randomized preparations while an unrandomized causal effect remains unidentified because the intervention- assignment assumptions fail. Predictive sufficiency and causal identification therefore constrain different arrows of the scientific argument; neither substitutes for the other. 11. No-go and repair ledger Exact theorem, finite-data certificate, and reconstructive failure protocol 16

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM Blocked claim Why it fails Exact repair Finite / approximate re- pair Same measured value means same state stancy Report both value uncer- taintyandpredictivedefect Changing Π destroys the old state Richer futures refine equiva- lence stand in for state, then histories identified by that representation must be interchangeable for the future claims made from it. What is usually left implicit is that the same obligation recurs at every later compression. This produces a strict order of inference. First ask whether the measured present is a sufficient predictive state. Then ask whether the interventions descend. Then ask whether temporal order may be erased. Only then ask whether the resulting cumulative object can be scalarized. Only after that ask whether the scalar operation deserves additive arithmetic. A later transform cannot repair a failed earlier quotient. Thefinite-datatheoryisdeliberatelymoreconservativethantheexacttheorem. Approximateequalityisnotpromoted

PREDICTIVE CLOSURE MURRAY | MEASUREMENT -ADMISSION THEOREM state rather than making previous measurements meaningless. Theframeworkalsoseparatesrejectionfromreconstruction. Afuturethatsplitsafibreidentifiesadistinctionthatwas active but unmeasured. Response rows provide a direct empirical language for retaining those distinctions. In finite linear-realization settings, Hankel rank supplies the exact classical dimension theorem; outside that branch, finite The present paper therefore separates three scientific statuses that should not be conflated: theorem validity ≠ protocol executability ≠ empirical utility. (22) The theorem establishes that measurement adequacy for a declared predictive use is a factorization property, and that the same obligation governs successive compressions. The worked simulation establishes executability under a frozen finite design. The remaining empirical question is whether this staged process materially improves inference or experimental design in real systems beyond existing specialist methods. That question is intentionally left open rather than being inferred from the theorem. The next scientific step is therefore prospective rather than rhetorical: apply the frozen protocol to independent data, compare its selected representation with existing state-selection methods, and test whether its additional distinctions improve held-out intervention prediction. The present paper supplies the theorem, the finite certificate, and the reconstruction rule; nature must decide the utility claim. Final statement A lawful measurement process for an evolving system begins before calibration ends: it asks whether the measured representation preserves the future laws for which it will be used. Predictive Closure proves that admission criterion, shows how it propagates through action and compression, and turns a failed fibre into a reconstruction experiment. The theorem determines what must be true. The test determines whether a particular measurement earns the claim. A. Partial-map abelianization: domain propagation