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Trajectory Abstraction for the Science of Language Agent Behavior
Tianqiang Yan
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
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Summary
The paper proposes a recursive procedure for trajectory abstraction to study language agent behavior. It involves measuring role- and phase-indexed events, proposing temporally constrained relations, testing their stability across conditions, and constructing episode-level motif variables. This hierarchy allows for the discovery of generalizable behavioral hypotheses through explicit measurement functions and randomized protocol experiments, distinguishing the approach from semantic taxonomies and qualitative theory induction.
Entities (8)
Relation Signals (5)
Trajectory Abstraction → appliesto → Language Agents
confidence 99% · Scientific studies of language agents need behavioral variables... We formulate this research problem as learning and testing a hierarchy of trajectory abstractions.
Trajectory Abstraction → uses → Recursive Procedure
confidence 95% · We formulate this research problem as learning and testing a hierarchy of trajectory abstractions. A concrete recursive procedure...
Recursive Procedure → produces → Motif Variables
confidence 92% · It then constructs episode-level motif variables from selected relations and repeats the analysis on those variables.
Intervention Validation → tests → Trajectory Abstraction
confidence 90% · Observations and randomized protocol experiments assess the resulting hypotheses... A finite-sample test makes projected intervention consistency operational
Causal Abstraction → underpins → Intervention Validation
confidence 88% · Following causal abstraction... compare... over a specified finite family of environments and implementations.
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Abstract
Abstract:Scientific studies of language agents need behavioral variables that support hypotheses across tasks and models. We formulate this research problem as learning and testing a hierarchy of trajectory abstractions. A concrete recursive procedure first measures role- and phase-indexed events, proposes temporally constrained relations, and tests their stability across conditions. It then constructs episode-level motif variables from selected relations and repeats the analysis on those variables. Explicit measurement functions connect every abstraction level to the original trajectories. Observations and randomized protocol experiments assess the resulting hypotheses, while comparisons between intervention realizations determine whether an abstraction should be retained, refined, or restricted. We derive a finite-depth bound for accepted reductions, identify protocol effects on fixed abstractions, and characterize realization disagreement and composition of abstraction error. A finite-sample test makes projected intervention consistency operational, and constructed examples illustrate motif construction and abstraction refinement. The formulation distinguishes this experimental approach from semantic taxonomies, qualitative theory induction, and behavior-model recovery. It specifies a proposed research procedure for discovering generalizable behavioral hypotheses, with literature-relative novelty assessed separately from model-relative surprise.
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- Source: https://arxiv.org/abs/2610.09237v1
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Trajectory Abstraction for the Science of Language Agent Behavior Tianqiang Yan * Faculty of Information Technology, Monash University Tianqiang.Yan@monash.edu Abstract Scientific studies of language agents need behavioral variables that support hypotheses across tasks and models. We formulate this research problem as learning and testing a hi- erarchy of trajectory abstractions. A concrete recursive procedure first measures role- and phase-indexed events, proposes temporally con- strained relations, and tests their stability across conditions. It then constructs episode-level motif variables from selected relations and re- peats the analysis on those variables. Explicit measurement functions connect every abstrac- tion level to the original trajectories. Obser- vations and randomized protocol experiments assess the resulting hypotheses, while compar- isons between intervention realizations deter- mine whether an abstraction should be retained, refined, or restricted. We derive a finite-depth bound for accepted reductions, identify proto- col effects on fixed abstractions, and charac- terize realization disagreement and composi- tion of abstraction error. A finite-sample test makes projected intervention consistency op- erational, and constructed examples illustrate motif construction and abstraction refinement. The formulation distinguishes this experimen- tal approach from semantic taxonomies, qual- itative theory induction, and behavior-model recovery. It specifies a proposed research pro- cedure for discovering generalizable behavioral hypotheses, with literature-relative novelty as- sessed separately from model-relative surprise. 1 Introduction Language agents communicate, invoke tools, re- flect on feedback, and coordinate actions. Scientific studies of these behaviors seek explanations that remain useful across tasks and models (Rahwan et al., 2019; Chen et al., 2026). Their observations are trajectories, while their conclusions concern patterns such as checking evidence before action * Corresponding author. or completing a repair after detecting an error. The scientific problem is to construct the variables in which these patterns can be stated and tested. A useful abstraction must preserve the distinc- tions relevant to the question. Consider two mes- sages: one identifies a retrieved result and the in- tended action; another says “Okay.” Both can be counted as acknowledgments, yet the shared label can conceal differences relevant to the next action. Testing those differences determines whether to retain the count, distinguish message types, or re- strict the hypothesis to one protocol. This is an abstraction-construction problem within the scien- tific study of behavior. Recent work already provides behavioral vocab- ularies, inductive analysis, and formal trace ab- straction. Actonomy develops an action taxon- omy; AutoTraceGT revises grounded codes and relations; ATLAS learns automata from abstract traces (Gao et al., 2026; Lu et al., 2026b; Lopez- Miguel et al., 2026). ReBeCA and CAMO connect measured factors to structural analysis and exper- iments (Yan et al., 2026; Yu et al., 2026). These developments motivate a concrete formulation of how trajectory variables, relational hypotheses, and executable tests interact during discovery. Our position is that higher-level behavioral ex- planations should be constructed through explicit trajectory functions and tested against the contrasts they claim to preserve. We operationalize this po- sition with a recursive procedure: measure events, learn relation candidates, test cross-condition sta- bility, construct motif variables, and evaluate their intervention responses. An accepted reduction be- comes the input to the next level. The output is a hierarchy of measurements and tested hypotheses, including unresolved and rejected proposals. The paper contributes a critical review, a com- putational formulation of this procedure, and ana- lytical results supporting its design. Randomized assignments identify protocol effects on fixed ab- 1 arXiv:2610.09237v1 [cs.AI] 6 Oct 2026 Study questionq Scenes and backbonesE ProtocolsA; outcomeY Episode measurementsX 0 Roles, phases, referents TableT ℓ (D) Proposed relations b G ℓ Temporal constraints Cross-condition screening Motif functionsf B Disjoint blocks; carry-forward Next tableT ℓ+1 Independent confirmation Protocol response; transfer Retain, refine, or restrict Measured hierarchyH Tested hypotheses Scope; novelty new episodes accepted reduction: analyze the next level Figure 1: Recursive trajectory abstraction as a scientific research procedure. Episode measurements produce a table at each level. Relations nominate temporal motifs, whose explicit measurement functions define the next table. Independent experiments test these abstractions and their hypotheses across scenes and language-model backbones. stractions even when the proposed graph is incor- rect. Disagreement between realizations gives a criterion for rejecting a common-response claim. A finite-sample comparison implements that crite- rion, while compression and consistency bounds constrain recursion. The procedure is specified here; empirical implementation and evaluation re- main future work. 2 Related Work We review primary work available through 6 Oc- tober 2026, using topic searches, reference snow- balling, and targeted verification of relevant papers. Appendix A summarizes the selection and compar- isons. 2.1 Behavioral Measurement and Induction ReAct, Reflexion, and LATS expose reasoning, ac- tions, observations, and feedback (Yao et al., 2023; Shinn et al., 2023; Zhou et al., 2024a). ALFWorld, WebArena, SWE-bench, and PARTNR provide re- peatable tasks and verifiable outcomes (Shridhar et al., 2021; Zhou et al., 2024b; Jimenez et al., 2024; Chang et al., 2025). AgentBoard analyzes intermediate progress; MAST derives failure cat- egories; AgentDiagnose evaluates trajectory prop- erties using rubrics and visualizations (Ma et al., 2024; Cemri et al., 2025; Ou et al., 2025). These resources supply measurements and experimental settings. Actonomy organizes agent actions into a reusable three-level taxonomy. AutoTraceGT per- forms open and axial coding, revises categories and relations, and stops according to a saturation rule (Gao et al., 2026; Lu et al., 2026b). AutoTraceGT therefore already models an inductive study pro- cess. Our distinction concerns recursive construc- tion of measured motifs from relational hypothe- ses and confirmation through executable interven- tions. Category stability and saturation answer dif- ferent questions from consistency of intervention responses. Language-based measurements must connect ut- terances to context and action. Grounding theory, MindCraft, and embodied-dialogue analysis mo- tivate this connection (Clark and Brennan, 1991; Bara et al., 2021; Dongre and Hakkani-Tür, 2026). Interface-changing experiments also distinguish se- mantic use from interface reliance (Gu et al., 2026). These examples guide the definition of discrimi- nating tests rather than treating semantic labels as explanations by themselves. 2.2 Trajectory and Causal Abstraction ATLAS categorizes action–observation pairs, nor- malizes labels, and uses automata learning to re- cover Markov models; it retains concrete realiza- tions and studies symbolic transfer (Lopez-Miguel et al., 2026). Its recovered transition structure differs from the intervention-preserving response tested here. COAT and IRIS propose variables and evaluate relations from unstructured evidence (Liu et al., 2024; Feng et al., 2025). ReBeCA combines semantic reflection patterns, structure learning, sta- bility checks, and prompt interventions; CAMO constructs factors and performs paired simulator interventions (Yan et al., 2026; Yu et al., 2026). LLawCo tests cooperation laws as policy guidance (Zhou et al., 2026). We build on these concrete operations while making the per-episode map at every recursive level explicit. Causal abstraction defines consistency of in- 2 terventions across descriptions (Rubenstein et al., 2017; Beckers and Halpern, 2019; Beckers et al., 2020); causal representation learning concerns the variables themselves (Schölkopf et al., 2021; Geiger et al., 2021). Options, predictive-state repre- sentations, and bisimulation supply temporal or pre- dictive criteria (Sutton et al., 1999; Littman et al., 2001; Givan et al., 2003). The present construc- tion selects its criterion according to the behavioral hypothesis and the experiments available to test it. 2.3 Automated Scientific Research The AI Scientist, Co-Scientist, and AI-Researcher generate and execute research operations with sci- entific evaluations (Lu et al., 2026a; Gottweis et al., 2026; Tang et al., 2025). AutoDiscovery uses Bayesian surprise to direct hypothesis search (Agar- wal et al., 2025). These systems can support the proposed procedure. Its scientific outputs depend on measurements, identification, and confirmation, while surprise ranks belief changes within an ex- plicit model (Itti and Baldi, 2009). Literature- relative novelty requires an independent compari- son. 3 Problem Formulation 3.1 Study Setting A study specifies a questionq, environmentsE, joint protocol assignmentAwith lawP A , an out- come functiong, knowledgeK, and budgetB. In the categorical instantiation, every scored node has prespecified finite states, includingAand the nu- merical endpointY. Counts use fixed bins or trun- cation. Continuous endpoints require an appropri- ate likelihood and a fixed finite projection for cate- gorical response tests; their original mean contrasts use integrable numerical endpoints. Environments can be indexed by scene and language-model back- bone,e = (s,m). Task, agent composition, inter- faces, and persistent memory are fixed or included among the study factors. An episode has trajectoryτcontaining times- tamped observations, declared plans, messages, tool calls, actions, and execution feedback. Its outcome is Y = g(τ ). Data are D =(e u ,A u ,τ u ,Y u ) N u=1 .(1) Assign complete protocol policies randomly be- fore each experimental unit starts. A unit is an episode or a cluster with a prespecified endpoint and weighting; there is no unmodeled interference or carry-over between units. Background strata are preassignment. Confirmation units are con- ditionally iid within each tested environment and arm, given completed construction. Successful and failed units are retained. 3.2 Discovery Target The target is a hierarchy H = (X ℓ ) L ℓ=0 , (a ℓ ) L−1 ℓ=0 ,(2) whereX ℓ contains measured behavioral variables anda ℓ constructs the next level. Proposed graphs b G ℓ , selected relationsR ℓ , and hypothesis tests are associated with the levels where computed; a termi- nal graph is optional. All episode, coordinate and response spaces have specified sigma algebras, and all maps are measurable. Every coordinate is an explicit function of an episode. Graph edges them- selves are estimates, so they become new variables only through these functions. A hypothesis specifies a measured relation, out- come contrast, and scope. Descriptive hypotheses concern occurrence; predictive hypotheses concern later outcomes; interventional hypotheses concern feasible manipulations. Results are supported, re- futed, or unresolved relative to the tests and as- sumptions. The search seeks abstractions that re- duce unnecessary detail while preserving the con- trasts required byq. Generalization is evaluated in held-out scenes, backbones, or their combinations; novelty is assessed against K. 4 Recursive Trajectory Abstraction 4.1 Event Measurements A measurement mapφ 0 yieldsX 0 = φ 0 (τ ). Each coordinate has an event family, agent role, tempo- ral phase, support interval, and availability time. Families include communication, declared plan- ning, tool/action execution, and observed context. Phases follow observable protocol boundaries. For predictive hypotheses, fixt pred < t target and use coordinates available byt pred ; feedback revealing an already realized target is excluded. Later coor- dinates support retrospective analysis. Final out- comes are not measurement inputs. Executable predicates measure tool arguments and environmental events. Semantic predicates specify referents, coding rules, and independently assessed error. Each coordinate retains its categor- ical state and the support, referent, and eligibility metadata needed by later maps. Graph learning 3 uses the categorical states;x B in a construction map includes their associated metadata. Every macro propagates the metadata required for fur- ther construction. Missing or ineligible events are recorded explicitly. For example,D k denotes de- tection of a discrepancy in phasek,C k+1 a com- mitment referring to that discrepancy, andR k+2 an action implementing its correction. The referent links distinguish one repair sequence from three unrelated events. 4.2 Relation Proposal and Stability At levelℓ, writeW = s(X ℓ )for categorical states and setV ℓ = (A,W,Y ).Ais one joint assignment node; its factors may be dependent. Graphs have no incoming edges toA, no outgoing edges from Y, and no backward temporal edges. Equal-phase relations require an explicit order or retain their orientation ambiguity. Acyclicity is enforced by the search. One concrete choice is tier-constrained greedy equivalence search (GES) (Chickering, 2002). For categorical measurements, use a prespecified de- composable score such as s(G) = X e∈E train −2ℓ e (G) + d e (G) logN e , (3) whereℓ e is the maximized multinomial log likeli- hood,d e the parameter count, andN e the episode count. GES approximates score minimization; re- sampling or alternative scores assess sensitivity. This is a relation-proposal step. Causal interpreta- tion requires the corresponding Markov, faithful- ness, and confounding assumptions. On independent screening episodes, compare each target conditional distributionP e (W v | W S = w S ) across environments. Retain a relation when simultaneous confidence bounds on the largest between-environment TV distance are below a pre- specified marginδ S , over conditioning cells with sufficient support. Fix targets,S, margins, and the tested cell family using build data. Use cell-specific sample counts in simultaneous bounds; required cells with insufficient support remain unresolved. Adaptive screening beyond that fixed family is ex- ploratory. Invariant causal prediction supplies iden- tification results under its own assumptions (Peters et al., 2016); a generic stability pass instead nomi- nates relations. Context-aware discovery is another option (Mooij et al., 2020). Retain uncertainty in edge directions. Candidate blocks come from rela- tions connected toYor a question-specific target; temporal precedence alone does not establish an effect (Runge et al., 2019). 4.3 Motif Construction LetP ℓ contain proposed connected blocks of at least two behavioral coordinates, excludingA,Y. For each blockB, select a functionf B from a finite template family specified for the question. Templates include ordered conjunctions, counts, and categorical progress states. A binary repair motif is M repair = D k C k+1 R k+2 ,(4) where the product uses binary state components on eligible episodes with the specified order and shared referent. Its value indicates completion of the defined sequence. An alternative categorical template distinguishes detection, commitment, and completed repair. Ineligible or missing cases retain their own category. The map’s support covers all required events; it becomes available at the last member’s availabil- ity time. Assigned protocols and final outcomes are excluded fromf B . A name such as “evidence- linked repair” is attached after fixing the function. An LLM can propose predicates or names, but the resulting variable must remain computable and val- idated. For a concrete selection rule, fit outcome predic- tors on build data before and after each proposed replacement. On screening data, retain the map when its increase in clipped log loss is at most a prespecified marginδ Y in each environment and its target relations pass the stability criterion. This is a risk comparison between the fitted predictors. Select a disjoint collectionB ℓ ⊆P ℓ in increasing description length, breaking ties by fixed coordi- nate order. Description length can rank maps that meet the same criteria (Grünwald, 2007); it is a compression preference rather than a causal test. Unmatched variables are carried forward. Writing U ℓ =1,...,p ℓ \ S B∈B ℓ B for their indices, a ℓ (A,x,Y ) = (A,x ′ ,Y ), x ′ = (f B (x B )) B∈B ℓ ,x U ℓ . (5) WithT 0 (A,τ,Y ) = (A,φ 0 (τ ),Y ), the recurrence is T ℓ+1 = a ℓ ◦ T ℓ .(6) HereT ℓ (D)means rowwise application to the archive. Thus each next-level row contains mea- sured motifs. In a constructed two-level example, 4 X 0 = (D,C,R,H)describes discrepancy detec- tion, referent-linked commitment, corrective action, and a subsequent handoff. The first map forms M = DCRand retainsH, givingX 1 = (M,H). A second map formsJ = MH, measured after the handoff and meaning “completed repair followed by handoff.” It is retained only if its specified pre- dictive and intervention contrasts pass the tests. These products define motif occurrence for subse- quent relational hypotheses. 4.4 Intervention Validation A chosen implementation is a complete measurable policyκ j : h 7→ κ(j,h)on available histories, in- cluding a specified action when no eligible event occurs: for example, insert a specified message, require verification before action, or change a role protocol. Its identity is assigned before the unit starts; later realized actions can depend on history. Compliance and collateral changes enter the result- ing trajectory. The effect θ e (j,j 0 ) =E e [Y κ j ]−E e [Y κ j 0 ](7) concerns assignment to those protocols. Identi- fying a downstream behavior effect requires ad- ditional assumptions (Hernán and Robins, 2020; Robins et al., 2000). LetP i e denote the joint assignment–trajectory– outcome law under implementationi, withω(i) = j, holding non-target factors fixed or using pre- assignment strata. The validation map isΦ + ℓ = c ℓ ◦ T ℓ , wherec ℓ selects the scientific coordinates and outcome from a learning-table row. It excludes arm identifiers and construction-only timestamps or referent IDs. These remain metadata in the learning tableT ℓ . An abstract response modelQ j ℓ,e is fitted independently of confirmation data on the same selected measurable response space. Following causal abstraction (Rubenstein et al., 2017; Beck- ers et al., 2020), compare η ℓ = max e,i d TV (Φ + ℓ ) # P i e ,Q ω(i) ℓ,e (8) over a specified finite family of environments and implementations. Push-forward denotes ap- plying the map to draws from the distribution. Implementation-specific hypotheses compare their documented protocol; implementation-independent hypotheses compare multiple realizations of the same j (Figure 2). For a fixed categorical projection, such as (M B ,Y ) , letr ≥ 1be its number of states and Realization i 1 P i 1 e Realization i 2 P i 2 e R 1 states andY R 2 states andY Candidate response Q j e Φ + ℓ Φ + ℓ ω(i 1 ) = ω(i 2 ) = j Maximum response error≥ 1 2 d TV (R 1 , R 2 ) Figure 2: Different realizations of one abstract manipu- lation must support a common response. Disagreement rejects the common-response claim at the stated toler- ance and motivates refinement or scope revision. n≥ 1iid confirmation units from that law, condi- tional on construction. The confidence allocation satisfies0 < β < 1. Coordinate-wise Hoeffding bounds give b(n,r,β) = min ( 1, r 2 r log(2r/β) 2n ) .(9) With probability at least1 − β, its empirical distribution b Psatisfiesd TV (P, b P )≤ b. Hence d TV ( b P,Q)+b≤ εcertifies the projected tolerance for independently fittedQ. All tests, including re- jected proposals and revisions, allocate confidence budgetsβ t with P t β t ≤ β . The projection, map, response model, predictors, target laws, tolerances, batch size and allocation are fixed before confir- mation; eligibility determined after assignment is retained as a category. Appendix D derives the bounds. Confirm the jointly selected replacement, since separate block checks do not establish joint preser- vation. Predictors are fitted before this test; the upper confidence bound on the paired clipped-loss increase must not exceedδ Y in every specified environment/arm or fixed target mixture. Failed or unresolved criteria trigger additional sampling, refinement, or restriction. Revised functions use fresh fixed-size confirmation batches and new con- fidence allocations. A layer is accepted only after all required criteria pass. Projected checks do not establish full-trajectory consistency. 4.5 Recursion and Scientific Outputs Algorithm 1 specifies the construction. Each level uses independent build, screening, and confirma- tion data. An accepted map becomes the next-level table; lower levels remain available for compari- son. The search stops when no reduction meets its 5 criteria or resources are exhausted. Outputs retain accepted maps and hypotheses, including rejected and unresolved tests. Render each hypothesis as a measured relation with its contrast and scope, then compare it with Kto assess novelty. Condition the prior and like- lihood on completed explorationF, includingK. Forπ h = P (h|F ), independent confirmation de- fines a posteriorπ ′ h under the fixed hypothesis mod- els. The surprise scoreKL(Ber(π ′ h )∥ Ber(π h )) ranks model-relative belief change. Literature nov- elty is assessed separately; LLM belief elicitation is a heuristic unless the update is justified. 5 Analytical Results 5.1 Randomized Design and Finite Reduction Proposition 1. IfT ℓ is fixed independently of con- firmation data, valid randomized assignment iden- tifies its protocol effects. For a fixed bounded real statisticV = h(X ℓ (τ )), such as the indicator of motif completion, E[V a − V a ′ | e] =E[V | A = a,e]−E[V | A = a ′ ,e]. (10) This follows from assignment exchangeability, con- sistency, and positivity. It identifies the assigned protocol’s effect on the measured abstraction re- gardless of graph correctness. Proposition 2. Letp ℓ =|X ℓ |withp 0 ≥ 1. Dis- joint block replacement with carry-forward gives p ℓ+1 = p ℓ − X B∈B ℓ (|B|− 1).(11) Every nonempty accepted reduction decreasesp ℓ by at least one. A fixed-inventory reduction chain therefore has at mostp 0 −1transitions. Refinement, splitting, or feature additions outside that chain consume the research budget and are not covered by this bound. 5.2 Realization Disagreement Proposition 3. Ifi 1 ,i 2 realize the samejandR k is their abstract response distribution, then max k∈1,2 d TV (R k ,Q j ) ≥ 1 2 d TV (R 1 ,R 2 ). (12) The triangle inequality proves the bound. It rejects a common-response claim at insufficient tolerance; possible repairs include refining the block, imple- mentations, response model, or scope. In a constructed example,Xindicates a grounded acknowledgment,Wa generic acknowl- edgment,Z = X +Wtheir count, andY = Xthe stipulated outcome. The two protocols i 1 = do(X = 1,W = 0), i 2 = do(X = 0,W = 1) (13) both realizej =deliver one acknowledgment, with ω(i 1 ) = ω(i 2 ) = j, but produce point masses at (Z,Y ) = (1, 1)and(1, 0). Their TV distance is one. A common response at tolerance below1/2is impossible. Distinguishing types requires refining the intervention labels; alternatively the hypothesis can be restricted to one protocol. The example specifies the outcome law; it is not an empirical claim about acknowledgments. 5.3 Composition of Consistency Error Proposition 4. Fix an environment and preassign- ment stratum. Letj 0 = iandj ℓ+1 = ω ℓ (j ℓ ). On specified measurable response spaces, letR i 0 = (Φ + 0 ) # P i e be the actual initial law andR j ℓ ℓ = Q j ℓ ℓ,e forℓ ≥ 1be specified comparison laws. Sup- pose measurable outcome-preserving maps satisfy c ℓ+1 ◦ a ℓ = ̄a ℓ ◦ c ℓ and d TV (( ̄a ℓ ) # R j ℓ ℓ ,R j ℓ+1 ℓ+1 )≤ ε ℓ .(14) Then, with Ψ L = ̄a L−1 ◦·◦ ̄a 0 , d TV (Ψ L ) # R i 0 ,R j L L ≤ min n 1, L−1 X ℓ=0 ε ℓ o . (15) TV contraction and induction with the triangle in- equality prove the bound. The factorization re- quires projections retaining sufficient linkage and eligibility summaries. The first tolerance compares actual data to the first model; later tolerances com- pare specified model laws. The bound concerns the specified adjacent model comparisons. Appendix C gives the proof. 6 Discussion The formulation specifies how measured mo- tifs become objects of further scientific study. Its practical value should be evaluated against 6 Algorithm 1 Recursive trajectory abstraction Require: Study settings; φ 0 ; templates and tolerances 1: T ← T 0 ; initializeH at X 0 ; ℓ← 0 2: while resources permit do 3:Acquire independent build and screening data 4:Fit b G ℓ ; screen relations; store test results 5:Select disjoint blocks and explicit f B 6:if no qualifying reduction then break 7:Form a ℓ with carry-forward; T ′ ← a ℓ ◦ T 8:Acquire independent confirmation episodes 9:Test and record joint predictive loss, responses, and transfer 10:if required criteria fail or remain unresolved then 11:Revise a ℓ ,T ′ or intervention scope 12:if resources insufficient then break 13:Acquire fresh fixed-size batch; allocate β t ; retest 14:if criteria still unmet then break 15:end if 16:Append accepted maps and hypotheses toH 17: T ← T ′ ; ℓ← ℓ + 1 18: end while 19: Assess literature novelty and modeled belief change 20: returnH and tested hypotheses semantic-category induction, automata abstrac- tion, and prediction-driven compression. Matched- budget comparisons can assess independently re- jected hypotheses, appropriate unresolved conclu- sions, calibration, and transfer across scenes and backbones. High-dimensional conditioning and projected-response tests can require many episodes, motivating sparse templates, better experimental design (Myung et al., 2013), and partial identifi- cation. Flat semantic coding is sufficient when a question concerns event prevalence or when no val- idated reduction is available. A hierarchy is useful when a measured motif supports a new relational hypothesis with fewer variables. Descriptive stud- ies need observations, predictive studies need inde- pendent future-outcome tests, and causal hypothe- ses need identified interventions. Compression can remove precisely the distinction that matters to a hypothesis, so retaining a lower-level representa- tion is a scientific outcome. The number of levels is therefore not a quality measure. 7 Conclusion We formulate scientific discovery of language- agent behavior through a concrete recursive ab- straction procedure. Explicit episode functions construct motifs from proposed relations; inde- pendent tests determine their retention, refinement, and scope. Randomized protocols anchor scien- tific contrasts, while realization disagreement and consistency bounds constrain the hierarchy. The formulation provides inputs, outputs, construction rules, validation tests, and stopping conditions for subsequent empirical research. Limitations The method is proposed and analyzed, with a con- structed example; it has not been implemented or empirically evaluated here. The narrative re- view is bounded by its publication cutoff and selec- tion criteria. Template selection, semantic coding, phase alignment, and intervention feasibility re- main study-specific. GES and invariance procedures require model assumptions that may fail for language-agent trajec- tories or after coarsening. Hidden state, persistent memory, selection, and incomplete logs can affect identification. Stability and disjoint source coor- dinates do not establish causality or faithfulness. Protocol effects do not automatically identify the effects of downstream behavioral variables. The finite-sample bound assumes fixed categori- cal projections and independent confirmation units; it can be conservative. Projected tests support 7 only their measured contrasts. The finite-depth result concerns accepted fixed-inventory reduc- tions, and the composition result requires matched layer-wise distributions. Neither establishes scien- tific completeness or inevitable improvement with depth. Generalization beyond tested conditions and literature-relative novelty require additional evidence. Ethical Considerations Trajectory studies can expose private messages and tool outputs. 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A Review Scope and Comparisons The review covers trajectory measurement, induc- tive behavioral analysis, causal abstraction, and automated scientific research. Searches of pri- mary publication and preprint sources were com- plemented by backward and forward reference searches and verification of specific related works. Selection favored studies specifying measurements, construction procedures, experimental contrasts, or formal assumptions relevant to language-agent trajectories. The publication cutoff is 6 October 2026; topic-based searches on 7 October checked coverage against that cutoff. We distinguish peer- reviewed publications from author versions and preprints. The synthesis is a narrative review rather than an exhaustive systematic review. B Measurement Specification A coordinate is specified by its event predicate, role, phase, referent, support interval, availability time, codomain, and missingness rule. For the repair example, detection requires an observable discrepancy statement; commitment must identify that discrepancy and an intended correction; repair requires execution feedback matching the correc- tion. A label inferred from an utterance is a mea- surement of expressed content. It does not identify latent intention. An ordered conjunction requires the declared temporal order and shared referent. Letχ B (x B ) indicate eligibility. Eligibility, temporal order, and shared referents are determined from the coordi- nate metadata retained at the current level. Write w v = s v (x v )for each state component. The macro state is m B (x B ) = ( Q v∈B w v , χ B (x B ) = 1, ⊥,χ B (x B ) = 0. (16) This equation defines the macro state. The com- plete record isf B (x B ) = (m B (x B ),k B (x B )), wherek B retains the support, referent and eligi- bility summaries needed by subsequent maps. The symbol⊥records ineligibility; missing observation can be another category. Successor maps access only the current coordinate records. Products and graph scores use state components; the projection used for a finite-state test excludes identifiers and timestamps. Prospective coordinates access only their available histories. Training may use outcome supervision to select a function, but the frozen func- tion takes neither final outcome nor assigned factor as an input. The next learning table preserves those fields separately. Validation compares behavior and outcome coordinates, with non-target factors fixed or stratified; realization identifiers remain outside this response distribution. Categorical alternatives can distinguish no detec- tion, detection without commitment, commitment without completed correction, completed repair, and ineligibility. Choice of a binary or categorical 10 WorkRelevant constructionRelation to this formulation AutoTraceGTGrounded coding, category/relation revision, saturation, theory generation. Already specifies a study process; our target adds recursive measured motifs and protocol-response tests. ActonomyHierarchical action vocabulary and trajectory coding. Supplies event categories; hypothesis-specific experimental distinctions determine reductions. ATLASTrace abstraction, concretization, automata learning, symbolic transfer. Recovers transitions; our target is motif construction for tested behavioral hypotheses. ReBeCASemantic patterns, graph learning, stability checks, prompt interventions. Closely related operations; our formulation makes recursive maps and realization dependence explicit. CAMOFactor construction and paired simulator interventions. Provides experimental factor analysis; our maps define next-level episode measurements. COAT / IRISVariable proposal and relation discovery from unstructured evidence. Motivates construction; our recursion preserves availability and tests feasible realizations. Table 1: Selected trajectory-analysis methods and their relation to the proposed scientific target. map is a testable modeling decision: it changes both compression and the distinctions that later hy- potheses can express. If role or phase boundaries differ across scenes, alignment is defined before comparison or the hypothesis scope is restricted. On observed eligible phase windows, the con- structed two-level example gives the state compo- nents X 0 = (D,C,R,H), X 1 = (DCR,H) = (M,H), X 2 = MH = J. (17) For instance,(D,C,R,H) = (1, 1, 0, 1)yields (M,H) = (0, 1)andJ = 0;(1, 1, 1, 1)yields (1, 1)andJ = 1. These values demonstrate the maps. C Proofs and Identification C.1 Randomized Assignment Condition on completed construction, including the fixed measurement function and the unit-level design assumptions in Section 3. Suppose assign- mentAis randomized within environmente, with positive probability for each compared arm. For a fixed bounded statisticV = h(X ℓ (τ )), consis- tency givesV = V a whenA = a, and assignment exchangeability gives V a ⊥ A| e. Therefore E[V | A = a,e] =E[V a | A = a,e] =E[V a | e]. (18) Subtracting the corresponding identity fora ′ proves Proposition 1. Randomization identifies assign- ment effects, including noncompliance. A prompt requiring an action is an assigned protocol; replac- ing it with a perfect intervention on the resulting action requires additional identification assump- tions. Randomization of design factors does not ori- ent every relation between behavioral variables. LetUbe a fair binary variable independent of an assigned factor. ModelsX = U,Y = Uand X = U,Y = Xhave identical observational distri- butions. SettingXto zero leaves the first outcome random and fixes the second at zero. Thus behav- ioral causal effects remain unidentified from these observations, despite randomized assignment of an unrelated factor. C.2 Reduction Depth For disjoint blocks, replacing|B|coordinates by one removes|B|− 1coordinates. All unmatched coordinates remain, proving the exact recurrence in Proposition 2. Each block has size at least two, so each accepted transition decreases the positive in- tegerp ℓ . Hence there are at mostp 0 − 1transitions 11 in a fixed-inventory chain. Retaining lower-level measurements permits later scientific comparison without counting them as new coordinates within that chain. If blocks partition all coordinates and every block has size at least two, thenp ℓ+1 ≤ ⌊p ℓ /2⌋, yielding at most⌊log 2 p 0 ⌋transitions.This stronger bound does not apply to carry-forward re- ductions. Introducing features, splitting variables, or rolling back restarts the relevant chain; the re- search budget bounds those operations separately. C.3 Realization Dependence Withd k = d TV (R k ,Q)for a common responseQ, the triangle inequality gives d TV (R 1 ,R 2 )≤ d 1 + d 2 ≤ 2 max k d k . (19) This proves Proposition 3. The constructed ac- knowledgment example has disjoint outcome sup- port under the two protocols, so its TV distance equals one. Its repair options are to retain the dis- tinction between grounded and generic messages or to make an implementation-specific claim. C.4 Composition Use the fixed environment, response spaces, com- parison laws and factorization in Proposition 4. Let Ψ 0 be the identity andΨ ℓ = ̄a ℓ−1 ◦·◦ ̄a 0 . The assumed layer comparison is d TV (( ̄a ℓ ) # R j ℓ ℓ ,R j ℓ+1 ℓ+1 )≤ ε ℓ .(20) For a measurable map, TV contracts because each output event has a measurable inverse image. Insert ( ̄a ℓ ) # R j ℓ ℓ as an intermediate law: d TV ((Ψ ℓ+1 ) # R i 0 ,R j ℓ+1 ℓ+1 ) ≤ d TV ((Ψ ℓ ) # R i 0 ,R j ℓ ℓ ) + ε ℓ . (21) The initial discrepancy is zero. Induction proves the sum bound, capped at one. If two records have identical selected states but differ in linkage, the next motif may be one versus⊥. A projection discarding that distinction has no induced map; its comparisons remain level-specific. A common abstract label also does not define a unique actual law when realizations disagree. The higher-levelR laws here are explicitly chosen comparison models. An invariance test’s retention event does not au- tomatically establish causal correctness. If a dis- covery procedure has a justified conditional error guaranteeα ℓ at each adaptively chosen level, a union bound controls the probability of any corre- sponding error by P ℓ α ℓ . Such a guarantee must be proved for that procedure and its post-abstraction assumptions. Independent data alone do not supply it. D Finite-Sample Response Tests Condition on all pretest information. LetPbe the fixed law of the declaredr-state projection, with r ≥ 1,n ≥ 1, and0 < β < 1. For statev, empirical frequencybp v fromniid experimental units drawn fromPobeys Hoeffding’s inequality (Hoeffding, 1963): P (|bp v − p v | > t)≤ 2e −2nt 2 .(22) Taking a union bound over states witht = p log(2r/β)/(2n) yields d TV (P, b P ) = 1 2 X v |p v − bp v |≤ b(n,r,β) (23) with probability at least1− β. Clipping at one is valid because TV never exceeds one. For predictive confirmation, use two fixed pre- dictors and clipped losses in[0,C], withC > 0 fixed before testing. On the same iid units let ∆ u = ℓ new,u − ℓ old,u and b ∆ = n −1 P u ∆ u . Since ∆ u ∈ [−C,C], Hoeffding bounds the expected increase by E∆ u ≤ b ∆ + C r 2 log(1/β t ) n (24) with probability at least1− β t . Accept only if this upper bound is at mostδ Y , for the joint replace- ment in every required target law. The criterion concerns the expected clipped-risk contrast of the fixed predictors. For a response modelQfitted independently of the confirmation batch, the triangle inequality gives d TV (P,Q)≤ d TV ( b P,Q) + b(n,r,β).(25) For two independently sampled realizations with confidence allocationsβ 1 + β 2 ≤ β, putb k = b(n k ,r,β k ). Let b d = d TV ( b P 1 , b P 2 ). With probabil- ity at least 1− β, d TV (P 1 ,P 2 )≥ max0, b d− b 1 − b 2 , d TV (P 1 ,P 2 )≤ min1, b d + b 1 + b 2 . (26) If half the lower bound exceeds a proposed common-response tolerance, Proposition 3 rejects 12 that response scope. If the upper bounds to inde- pendently fittedQfall below the tolerance for all required arms, the projected criterion is met. Oth- erwise the conclusion is unresolved. Before attemptt, the completed historyF t−1 fixes maps, projections, models, predictors, tar- get laws, tolerances, sample sizes and allocations. Each fresh batch is conditionally iid from its de- clared law. Splitβ t among every tested arm, en- vironment, cell and quantity, or supply a simul- taneous guarantee. If failure eventE c t has condi- tional probability at mostβ t and P t β t ≤ βalmost surely, then Pr(∃t : E c t )≤E X t β t ≤ β.(27) Every confirmation attempt uses a fresh fixed-size batch. Completed batches enter the construction history for subsequent independently confirmed proposals. For clusters, this simple proof uses one prespecified finite-valued observation per iid clus- ter. Episode-weighted cluster analyses specify their corresponding sampling bound separately. A post- assignment missing or ineligible state remains in the distribution; removing it can change the proto- col comparison. Tests on(M B ,Y )establish that projected comparison, not equality of unrestricted text-trajectory laws. E Novelty and Future Evaluation A hypothesis is represented by its measured vari- ables, claimed relation, contrast, and scope. Com- parison with existing work distinguishes replica- tion, scope extension, and previously untested re- lations. The literature search and its date delimit an underexploration claim. Supported known hy- potheses remain legitimate scientific findings. For a selected hypothesis, condition on com- pleted explorationF. Specify exhaustive alterna- tive modelsh,¬h, a prior0 < π h = P (h | F ) < 1, and predictive likelihoodsL h = P (D c | h,F ) andL 0 = P (D c | ¬h,F )for the independent batch under the fixed design. Composite models integrate nuisance parameters under their specified priors. Ifπ h L h + (1− π h )L 0 is positive and finite, then π ′ h = π h L h π h L h + (1− π h )L 0 .(28) Use0 log 0 = 0for posterior endpoints. The KL score measures change within that model. It can prioritize investigation but does not replace experi- mental support or literature comparison. Future evaluation should compare the pro- posed construction with fixed action taxonomies, grounded-theory induction, automata learning, and prediction-only compression at matched data and experimental budgets. Controlled tasks with stipu- lated mechanisms can assess abstraction recovery and appropriate rejection. Held-out scenes and backbones test transfer of measured relations and intervention responses. Required analyses include coding error, eligibility rules, role/phase alignment, template choice, graph-score assumptions, indepen- dent data partitions, and calibration of unresolved conclusions. 13