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Toward Machine Learning with the Unit as a Primitive: Learning from Unit-Linked Events
Heyang Gong
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 89%
Last extracted: 8/27/2026, 4:43:44 AM
Summary
The paper proposes the 'unit' as an explicit primitive in machine learning task semantics, distinguishing persistent individuals from individual event samples. It formalizes supervised learning as learning a family of unit-conditioned response laws, where a tokenizer produces contextual unit tokens and a shared response-law form (e.g., linear predictor) reads them. This approach addresses heterogeneity across units and distinguishes between homogeneous worlds and pooled marginals.
Entities (6)
Relation Signals (5)
Unit → isprimitiveof → Machine Learning
confidence 95% · We propose the unit as an explicit primitive at the level of task semantics.
Tokenizer → produces → Contextual Unit Token
confidence 90% · a tokenizer that produces a contextual unit token
Shared Response-Law Form → reads → Contextual Unit Token
confidence 90% · one shared response-law form that reads it
Homogeneity → isspecialcaseof → Unit-Conditioned Response Laws
confidence 88% · Homogeneity is the special case in which those laws coincide
Unit Abduction → forms → Token
confidence 85% · unit abduction forms a token of the same type from factual evidence
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Abstract
Abstract:Machine learning is usually formalized through samples, while the persistent individual to which multiple observed or possible events refer often remains implicit. We propose the \emph{unit} as an explicit primitive at the level of task semantics. A learning task first declares a population of persistent referents and a sameness criterion; the realized value $u$ denotes the selected referent. Supervised learning is the main formal specialization. Its semantic object is a family of unit-conditioned response laws. Homogeneity is the special case in which those laws coincide; a sample-only conditional is silent as to whether the world is homogeneous or the observed law is only the marginal of a heterogeneous family. What is learned from data is a pair $(T_\phi,R_\theta)$: a tokenizer that produces a contextual unit token and one shared response-law form that reads it. The structured class takes that form to be a simple relation in the token; a linear predictor is the running instance. The token is the learner-side representation through which the task-side unit affects prediction, while a learner specification that omits unit information is unit-insensitive; homogeneity remains a property of the world-side response family. When identity is unresolved, the world-side law mixes unit-conditioned targets, while the learner composes its shared form with a token. A trusted resolver may fix the unit and supply a lookup token; otherwise \emph{unit abduction} forms a token of the same type from factual evidence. Unlinked single-row observations can fail to distinguish a heterogeneous unit world from a homogeneous pooled world; trusted same-unit pairs separate a restricted witness. The formal results concern this supervised specialization.
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- Source: https://arxiv.org/abs/2608.25118v1
- Canonical: https://arxiv.org/abs/2608.25118v1
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Toward Machine Learning with the Unit as a Primitive: Learning from Unit-Linked Events Heyang Gong August 26, 2026 Abstract Machine learning is usually formalized through samples, while the persistent individual to which multiple observed or possible events refer often remains implicit. We propose the unit as an explicit primitive at the level of task semantics. A learning task first declares a population of persistent referents and a sameness criterion; the realized value u denotes the selected referent. Supervised learning is the main formal specialization. Its semantic object is a family of unit-conditioned response laws. Homogeneity is the special case in which those laws coincide; a sample-only conditional is silent as to whether the world is homogeneous or the observed law is only the marginal of a heterogeneous family. What is learned from data is a pair (Tϕ,Rθ)(T_φ,R_θ): a tokenizer that produces a contextual unit token and one shared response-law form that reads it. The structured class takes that form to be a simple relation in the token; a linear predictor is the running instance. The token is the learner-side representation through which the task-side unit affects prediction, while a learner specification that omits unit information is unit-insensitive; homogeneity remains a property of the world-side response family. When identity is unresolved, the world-side law mixes unit-conditioned targets, while the learner composes its shared form with a token. A trusted resolver may fix the unit and supply a lookup token; otherwise unit abduction forms a token of the same type from factual evidence. Unlinked single-row observations can fail to distinguish a heterogeneous unit world from a homogeneous pooled world; trusted same-unit pairs separate a restricted witness. The formal results concern this supervised specialization. 1 Introduction The standard supervised-learning formulation starts from records =(xi,yi)i=1ND=\(x_i,y_i)\_i=1^N and a map f:→f:X [4, 17, 38]. The index i identifies a record, not the individual to which that record refers. If two records concern the same patient, user, device, organism, or other persistent individual, ordinary notation represents this relation only through additional metadata or an implicit modeling convention. This omission is harmless for some row-level questions, but consequential when events share an individual, queries change while that individual is held fixed, or event-to-individual attribution is uncertain. It also leaves a more basic question unstated: whether different individuals share one response law or whether the observed row-level law pools distinct unit-conditioned relationships. Persistent individuals are not new to statistics or machine learning. Repeated-measures and random-effects models link observations through a supplied subject or group and model subject-specific variation [25, 13]. Potential-outcome frameworks compare alternative treatment responses for the same experimental unit [35]. Recommender systems use supplied user IDs to address user-specific representations learned across interactions [28, 24], while record linkage and entity resolution model uncertainty about whether records concern the same underlying entity [41]. Each tradition develops a rich local theory with its own assumptions and targets. Together, these traditions suggest a common machine-learning formulation with a semantic order that is usually left implicit. A task first declares a population of persistent referents and a sameness criterion. It then declares the learning object attached to those referents and the across-unit structure that supports joint learning. Only after these choices does learner access become relevant: attribution may be supplied directly or inferred from evidence. We call the persistent referent a unit and elevate it to a primitive of the learning problem. Let U:Ω⟶,U∼Π,U=u,u∈,U: , U , U=u, u , (1) represent population-to-individual selection and its realization. Unit declaration. A learning task declares a population of possible units, a criterion under which observed or possible events concern the same persistent referent, and the span over which that relation is retained. The realization U=uU=u denotes that referent. The law Π describes population selection; the data-collection protocol separately determines the joint law of dataset attributions. The unit declaration precedes the task-specific choice of learning object and does not by itself impose a response law, loss, model class, or causal semantics. The organizing claim of the paper is that machine learning learns shared structure across task-declared units from noisy, selectively observed, unit-linked events. The primitive states what persists and which events belong together. Supervised learning is the main formal specialization: a fixed unit selects an entire unit-conditioned response law, u u ⟼[x↦Pu⋆(dy∣x)], [\,x P_u (dy x)\, ], (2) Pu⋆(dy∣x) P_u (dy x) :=P⋆(Y∈dy∣X=x,U=u). :=P (Y∈ dy X=x,U=u). Changing x queries another location on the same response surface; changing u selects another member of the family. A family of unrelated maps Ru\R_u\ is too flexible to learn from finite data, including for a previously unseen unit. Joint learning therefore requires a shared restriction across units, developed in Section 3.2: a tokenizer produces a contextual unit token, and one shared response-law form reads it. Task declaration: what counts as one unit Population statement: U∼Π⟶U=uU \ \ U=u (a) Events share persistent unit identityevent 1(x1,y1)(x_1,y_1)event 3(x3,y3)(x_3,y_3)unituAu_Aevent 2(x2,y2)(x_2,y_2)unituBu_B(b) The same unit selects a response lawevent i(xi,yi)(x_i,y_i)unituAu_A response law PuA⋆(dy∣x)P _u_A(dy x) uAu_A selects the law, not the event Figure 1: From the unit declaration to linked events and response laws. The task specifies the unit boundary and sameness criterion; U∼ΠU is the population statement and U=uU=u its realized unit. Several event records may be attributed to the same u, and that same referent indexes a world-side response law. Holding u fixed preserves the referent while allowing context and event variation. The paper makes three contributions. 1. It formalizes the unit as a task-declared population primitive: a sample records an event, and the task first declares the persistent referent to which events belong. 2. It states a shared-form interface (assumption 1) whose structured class is a simple relation in the token, with a linear predictor as the running instance (Definition 2); the learned object is the pair (Tϕ,Rθ)(T_φ,R_θ). 3. It separates oracle predictive value, evidence access, learner approximation, and a single-row impossibility boundary, with evaluation corollaries for row versus unit weighting and for record-wise versus unit-disjoint splits. 2 The Unit as a Machine-Learning Primitive 2.1 Samples, Events, and Task-Declared Units For the supervised specialization developed below, a conceptual complete-data representation with ground-truth attribution is Uworld=(xi,yi,ui)i=1N,ui∈.D_U world=\(x_i,y_i;u_i)\_i=1^N, u_i . (3) The input–response pair (xi,yi)(x_i,y_i) is the event content used in this specialization. The semicolon marks the more general role distinction between that event content and uiu_i, which identifies the individual to which the event is attributed. The sample index remains only bookkeeping. In particular, ui=uj⟺events i and j concern the same individual,u_i=u_j $i$ and $j$ concern the same individual, (4) whenever the protocol supplies reliable attribution. Equality records the same persistent referent. Predictive equivalence is weaker: two distinct units may induce the same response law. The space U is task-declared and need not be numerical. Direct access exposes three objects: k⏟trusted key,u(k)⏟task-side unit,Zu(k)c⏟learner-side lookup token. k_trusted key, u(k)_task-side unit, Z_u(k)^c_learner-side lookup token. (5) The map k↦u(k)k u(k) resolves the referent, and lookup supplies the token through which RθR_θ reads it in context c. In an ID-indexed model Zu(k)c=zθ(k),Z_u(k)^c\;=\;z_θ(k), (6) an embedding, preference factor, random effect, or other parameter block, with any remaining answer-time variation absorbed into c. Its coordinates are model-internal and need not be unique; the semantic and inferential distinctions are summarized in Appendix B [3]. What is carried consistently from attribution into task-specific learning is the resolved referent. All records attributed to u(k)u(k) read or update the token associated with that same key. Accordingly, Pθ(dy∣x,u)P_θ(dy x,u) semantically denotes a learned response law indexed by a fixed unit; computationally it is the shared form RθR_θ evaluated at that unit’s token ZucZ_u^c. In an ID-indexed model it may be computed from zθ(k)z_θ(k). A dynamic history-dependent response state, when used, belongs to the tokenizer’s context argument or to the response model, and may change while the referent stays fixed. When neither unit attribution nor a trusted referential resolver is learner-visible, the observed data may contain only obs=(xi,yi)i=1N.D obs=\(x_i,y_i)\_i=1^N. (7) The complete-data representation in Eq. 3 is therefore distinct from observations with learner-visible attribution, Uobs=(xi,yi,ui)i=1N,D_U obs=\(x_i,y_i;u_i)\_i=1^N, (8) in which persistent-unit labels, or a trusted resolver that determines them, are available to the learner. This attribution guarantee reveals which referent and parameter address an event concerns; it does not make learned embeddings, random effects, or response states observed data. 2.2 Fixing the unit does not fix the event In supervised learning, the fixed-unit response law Pu⋆(dy∣x)P_u (dy x) from Eq. 2 may remain stochastic. Holding u fixed conditions on the same individual while allowing repeated measurements, choices, or outcomes to differ. Event noise, time-varying state, and other exogenous variation can remain after unit selection. The unit primitive therefore separates “which individual?” from “what happens for that individual here?” as shown in Fig. 1. Independence, causal semantics, and deterministic responses are separate modeling commitments. The concrete three-event inset has three samples but only two units: events 1 and 3 share uAu_A while their inputs and realized responses remain event-specific. The unit declaration precedes the choice between observational and causal semantics. Causal estimands require additional commitments; one such specialization is recorded in Appendix D. Direct unit access and unit abduction provide two learner-side modes for obtaining the same task-declared unit (Section 4). 3 Learning a Family of Unit-Conditioned Response Laws This paper develops supervised learning as the main formal specialization of the unit primitive because the ordinary object P(dy∣x)P(dy x) makes the consequences of a persistent unit especially transparent. Attribution is treated separately in Section 4. Once the unit is declared, the semantic object is the family Pu⋆(dy∣⋅):u∈\P_u (dy ·):u \ in Eq. 2. The next subsection states the learner-side factorization: every path from the unit to prediction passes through a contextual token, all units use one response-law form RθR_θ, and every unit-specific difference represented by the learner must pass through the token. The supervised question is therefore not how a particular architecture encodes u, but whether a tokenizer can organize unit-specific regularities so that one shared RθR_θ becomes simple enough to generalize. Conceptually, a fixed u selects an entire response surface over the complete task-declared response input x. Changing x moves to another location on that same surface while u remains fixed. Across units, the dependence may enter through a baseline shift α(u)α(u), a changed input effect β(u)β(u), a response function fuf_u, an outcome-noise law, or the support, threshold, ranking preference, or query sensitivity of the response. Under assumption 1, all of these differences are carried by the token. Concatenation, attention, and modulation may implement the tokenizer or a richer readout; the structured class below takes the readout RθR_θ to be a simple relation in the token, with a linear predictor as the running instance. Three layers must be distinguished. If units are declared and Pu⋆(dy∣x)P_u (dy x) is the same for almost every u, the world is a homogeneous unit extension of ordinary supervised learning. Separately, a heterogeneous family can be marginalized over U to produce the same row-level conditional law P⋆(dy∣x)P (dy x). At the computational layer, a unit-omitting learner specification is unit-insensitive; a constant token or a response form that ignores its token is a canonical realization. This learner-side restriction and the world-side homogeneity layer are distinct, so row-level notation and fit alone do not choose between them. Proposition 4 makes the ambiguity formal and gives one observation protocol that can separate a restricted pair of worlds; the boundary cases are collected in Appendix B. 3.1 Response Heterogeneity Across Units The first distinction is whether changing the unit changes that conditional relation at all. Definition 1 (Unit-response homogeneity and heterogeneity). Fix the complete response inputs declared by a supervised-learning task. The declared family of unit-conditioned response laws is unit-response homogeneous if there exist a Π -null set N⊆N and a common conditional response kernel such that, for every u∉Nu∉ N and every declared input x, Pu⋆(dy∣x)=P⋆(dy∣x).P_u (dy x)=P (dy x). (9) Here the right-hand side denotes the common response kernel shared across units; it is not defined by marginalizing over U. For compatible conditional versions, however, homogeneity makes this kernel equal to the unit-marginal response law P⋆(Y∈dy∣X=x)P (Y∈ dy X=x) for PXP_X-almost every x. Thus the two uses of P⋆(dy∣x)P (dy x) agree under the homogeneous restriction. The family is unit-response heterogeneous otherwise. This definition concerns variation in the response law and is independent of a particular parameterization, architecture, or clustering device. Identity and predictive behavior remain distinct: two individuals may induce the same response law while remaining different units. The appendix gives the jointly measurable-kernel formulation, its deployment-visible qualification, and the usual almost-everywhere and off-support boundaries in Definition 3. The homogeneity distinction is defined only after the task has specified what counts as a unit, and it is a world-side property independent of a particular learner. The computational counterparts—a unit-insensitive learner, a structured pair (Tϕ,Rθ)(T_φ,R_θ), and a saturated private-law class—are developed in Section 3.2. 3.2 Shared response-law form and contextual unit tokens Heterogeneity characterizes world-side variation among response laws. Joint learning requires a separate computational restriction: we do not fit an unrelated mechanism RuR_u for each unit. That class is too flexible. When the response specification distinguishes a focal query x from an answer-time context c, write Pu⋆(dy∣x,c)P_u (dy x,c) for the complete conditional; c may otherwise be absorbed into a larger x. Given factual evidence attached to a unit and the current context, a tokenizer produces a contextual unit token Zuc=Tϕ(u,c)∈.Z_u^c=T_φ(O_u,c) . (10) Assumption 1 (Shared form, contextual tokens). For the learner in a given context, every path by which the selected unit can affect prediction is mediated by a contextual unit token, and all units use one shared response-law form. Any unit-specific difference represented by the learner must be expressed by the token. The unit u is the task-side referent and ZucZ_u^c is a learner-side representation. Assumption 1 is an interface. The structured class of this paper requires a simple shared relation in the token: unit-specific differences enter RθR_θ only through a declared low-complexity map of z, while the query maps may remain nonlinear. A finite-dimensional linear predictor (Definition 2) is the running instance; other declared simple maps of the token are the same restriction, not a second theory. Its boundary cases are collected in Appendix B. The object ZucZ_u^c need not be a deterministic vector. A deterministic tokenizer may return a vector, a distribution, or another typed computational object; a fixed embedding zθ(k)z_θ(k) is the simplest lookup special case. Alternatively, the tokenizer itself may be stochastic, in which case it returns a law over token values. These are different typed constructions. A distribution-valued token, say Zuc=μucZ_u^c= _u^c, is passed as one typed object to Rθ(dy∣x,c,μuc)R_θ(dy x,c, _u^c); it is not the sampling randomness marginalized in Eq. 14. In every case the token is learner-side information about the unit under the present evidence and context, not a second copy of the unit. It may be read loosely as a belief about the unit, but it is not required to be a posterior over identity, and it is not required to recover a unique complete latent unit. The unit–token relation is a representation map, not a bijection. The same unit may receive different tokens in different contexts, Zuc≠Zuc′.Z_u^c≠ Z_u^c . (11) Distinct units may receive the same token, or the same token law, in a given context, u≠v,Zuc=Zvc.u≠ v, Z_u^c d=Z_v^c. (12) Token coincidence is computational indistinguishability under the current query scope. It does not identify the units: referential identity remains u≠vu≠ v. For a deterministic token, the learner uses one shared response mechanism P^θ,ϕ(dy∣u;x,c):=Rθ(dy∣x,c,Zuc). P_θ,φ(dy _u;x,c):=R_θ(dy x,c,Z_u^c). (13) If the tokenizer is a stochastic kernel over token values, the learner instead uses the integral P^θ,ϕ(y∣u;x,c):=∫Rθ(y∣x,c,z)Tϕ(z∣u,c). P_θ,φ(dy _u;x,c):= R_θ(dy x,c,z)\,T_φ(dz _u,c). (14) The essential point is that RθR_θ is the same learned map for every unit; neither display defines the world target Pu⋆P_u . Definition 2 (Linear shared form). Fix a finite dimension d and let tokens take values in ℝdR^d. A shared form RθR_θ is linear in the token when there exist maps αθ(x,c)∈ℝ _θ(x,c) and ψθ(x,c)∈ℝd _θ(x,c) ^d such that ηθ(x,c,z)=αθ(x,c)+⟨ψθ(x,c),z⟩ _θ(x,c,z)= _θ(x,c)+ _θ(x,c),z (15) and Rθ(dy∣x,c,z)R_θ(dy x,c,z) depends on z only through this linear predictor, via a declared exponential-family, GLM, or scoring link. The maps αθ _θ and ψθ _θ may be nonlinear in (x,c)(x,c). Finite-type models arise when tokens are further restricted to a finite set or simplex in ℝdR^d; a low-rank model is the same class at small d. Assumption 1 remains the interface. The structured class is a restriction on how RθR_θ reads the token, not a private mechanism per unit: RθR_θ depends on z only through a declared simple relation, while TϕT_φ may remain complex. Definition 2 is the running instance. A finite-type token or another declared low-complexity map of z instantiates the same class; they are not developed here. Misspecification is residual unit-specific variation that the declared simple relation cannot absorb. Without a bound on d, or with an unrestricted nonlinear readout of z, the factorization is again a reparameterization. Random-intercept and random-slope models, matrix factorization, and a linear last-layer over a query encoder instantiate the linear running instance; they differ in how αθ _θ and ψθ _θ are parameterized, not in the role of the unit. A distribution-valued token lies outside the running instance unless it is reduced to a vector in ℝdR^d, for example a mean or a finite coefficient summary. When identity is unresolved, under response sufficiency Y⟂|(U,X,C)Y \!\!\! (U,X,C) and the external-query condition U⟂(X,C)|U \!\!\! (X,C) , the world-side target and the learner-side token composition are P⋆(dy∣x,c,)=∫Pu⋆(dy∣x,c)P(du∣),world target,P^θ,ϕ(dy∣;x,c):=∫Rθ(dy∣x,c,z)Tϕ(dz∣,c),learner composition. aligned P (dy x,c,O)&= _UP_u (dy x,c)\,P(du ),&&world target,\\ P_θ,φ(dy ;x,c)&:= _ZR_θ(dy x,c,z)\,T_φ(dz ,c),&&learner composition. aligned (16) The first line is the exact world mixture under these conditions; it contains no learned parameter. The second line is a learner defined on token space. They coincide only if the tokenizer realizes an appropriate token law for the relevant world-side unit mixture and the shared response form correctly realizes the corresponding unit-conditioned targets. A point token is a further learner-side collapse. If a trusted key fixes u(k)u(k), the world identity conditional is the Dirac law P(du∣k)=δu(k)P(du k)= _u(k), while the learner separately reads the lookup token Zu(k)cZ_u(k)^c. The theorems below use an identity-mixture learner because it is information-theoretically convenient; deployed learners may instead use the token composition. These are alternative learner coordinates, not identities that place learned parameters inside world truth. What is learned is the pair (Tϕ,Rθ)(T_φ,R_θ), not a collection of unrelated unit-specific mechanisms. Heterogeneity is thereby organized in a common token space: u→TϕZuc→Rθ(⋅,x,c)P^θ,ϕ(⋅∣u;x,c).O_u T_φZ_u^c R_θ(\,·\,,x,c) P_θ,φ(\,· _u;x,c). (17) The tokenizer is responsible for discovering structure that units can share; the response mechanism specifies how any token, faced with a query and context, produces a response. A good tokenizer converts complex unit-specific regularities into a simple, stable, shared law on token space. The shared-form class also contains a degeneration in which the response never receives the unit. In the linear running instance this is ψθ≡0 _θ≡ 0, or any token that does not enter ηθ _θ. That unit-omitting specification is the computational restriction P^θ,ϕ(dy∣u;x,c)=P^θ(dy∣x,c)for all u, P_θ,φ(dy _u;x,c)= P_θ(dy x,c) all u, (18) which has two canonical sufficient realizations under assumption 1: Zuc=z0,Z_u^c=z_0, (19) or Rθ(dy∣x,c,z)=Rθ(dy∣x,c).R_θ(dy x,c,z)=R_θ(dy x,c). (20) These equations define the unit-insensitive learner specification. The same specification can be used with a homogeneous world or with the marginal of a heterogeneous family; world homogeneity is a property of Pu⋆\P_u \, not of token equality. The tokenizer and the response learner are trained jointly: each shapes whether the other remains simple enough to generalize. When a trusted identity is available, a token may be obtained by direct lookup at the resolved address. When no resolver supplies the realized unit, a token of the same type is formed from factual evidence by unit abduction (Section 4). Both access modes still train the same pair (Tϕ,Rθ)(T_φ,R_θ). The token space also determines how observations from other units may constrain the response law for a target unit. Referential identity answers which individual is present; sharing observations across units requires an explicit similarity or borrowing rule, such as a metric, kernel, graph, or hierarchy. A learnable specification names what is shared, what varies with the unit, what remains invariant, and how evaluation distinguishes a unit-dependent relationship from identifier memorization (Appendix G). Assumption 1 answers the first three by requiring a shared RθR_θ and a contextual token; the structured class further requires that RθR_θ depend on the token only through a declared simple relation, with Definition 2 as the running instance. Structure as the condition of learnability. The pair (Tϕ,Rθ)(T_φ,R_θ) is the computational expression of a dual restriction. The response map x↦Pu⋆x P_u needs structure even for a fixed unit, and the unit axis u↦Pu⋆u P_u needs a second restriction so that observations can be shared and an unseen unit can be served. Homogeneity is one such structure. A simple shared relation in the token supplies another: units may differ, but only through a low-complexity map of ZucZ_u^c. The linear running instance uses the inner product in Eq. 15; other declared simple maps of the same token remain open. Which other units’ observations may constrain the token is a separate borrowing question. The roles of a which-unit belief and a trusted lookup are clarified in Appendix B. (a) Unit-insensitive learneruAu_AuBu_BuCu_C common token z0z_0 shared RθR_θ (b) Shared form RθR_θtokenizer TϕT_φuAu_A ZAcZ_A^c uBu_B ZBcZ_B^c uCu_C ZCcZ_C^c shared RθR_θ (c) SaturateduAu_A RuAR_u_A uBu_B RuBR_u_B uCu_C RuCR_u_C Figure 2: The computational assumption for the supervised specialization. (a) A unit-insensitive learner: every unit receives a common token z0z_0, or the shared form ignores the token. (b) Structured heterogeneity: a tokenizer forms contextual unit tokens, and all units share one response-law form RθR_θ, a simple relation in the token (linear running instance). (c) A saturated class assigns an unrelated law to each unit. 4 Learner Access to the Unit With the unit boundary, response family, and shared form (Tϕ,Rθ)(T_φ,R_θ) fixed (Section 3.2), world-side attribution determines which realized unit the event concerns. Learner access determines how that unit enters downstream learning as a contextual token. It does not replace the structure that made the family learnable. We distinguish direct unit access from unit abduction; both cases retain the same persistent referent and return tokens of the same type. They separate two questions: which unit does the current evidence concern, and, given a token for that unit, how does the shared form respond to the supplied query? Direct access resolves the first question without eliminating the second. The first question is referential. A further structural-localization question asks where the resolved or inferred referent lies in the token space used for sharing information. A stable ID may answer the referential question without supplying that position, just as a learned token geometry need not uniquely identify the individual. Direct unit access. A trusted resolver uses a key k to determine which persistent unit the event concerns and which stable address should be read. A separate lookup at that address supplies the token Zu(k)cZ_u(k)^c. With an ID-indexed model this is the learned row zθ(k)z_θ(k)—an embedding, preference factor, or random effect—updated from the records associated with that key. Personalized recommendation with a trusted user key is a common example. Direct access therefore removes which-unit uncertainty while leaving token-and-form approximation error. A history-dependent response state may also be updated for the known unit, but such state belongs to the tokenizer’s context or to RθR_θ rather than to unit attribution. A Dirac measure can record the resolved identity in common measure-valued notation without changing the token’s origin or turning lookup into abduction; details are recorded in the appendix. Unit abduction. Unit abduction applies when no resolver identifies the realized persistent unit. Let O denote the factual evidence available before the current answer. The access step forms a contextual unit token Zc=Tϕ(,c)Z^c=T_φ(O,c) (21) of the same type as a lookup token. Alternative queries then use that token through RθR_θ. This yields the learner’s token-space composition in Eq. 16. The theoretically exact predictive law remains the mixture over units. Under the data-generating distribution, P(U∈du∣)P(U∈ du ) is the world-side identity posterior: conditioning changes uncertainty about which unit was realized, while the referent itself remains fixed. The theorems below use an identity-mixture specialization and write the learner’s approximation as Qϕ(du∣).Q_φ(du ). (22) When the learning protocol explicitly targets the identity posterior, QϕQ_φ may be interpreted and evaluated as an approximate posterior; otherwise its calibration and recovery properties require separate targets and diagnostics. A tokenizer may emit a token, or a law over tokens, without first forming an identity posterior; Eq. 24 gives the corresponding prediction. Unit information rather than a mandated belief. The formed token ZcZ^c is not the unit and is not required to be a probability over U. Direct access and abduction differ by how the token is obtained, not by whether that information is a belief. Conditioning on an unobserved quantity and averaging predictions over its uncertainty are standard probabilistic-learning operations [4, 29]. The contribution claimed here is the persistent-referent formulation and the separation of learner-access modes, not those operations themselves. Established causal and logical uses of abduction are compared in Section 7. The unit boundary is declared before learning and evaluation. Within a declared same-unit query family, the same realized value u is retained across response queries. A new factual observation may update the formed token; alternative queries in the same family reuse the current token. (a) Direct unit access — lookup token trusted key k trusted resolver resolved referent u(k)u(k) stable address lookup token Zu(k)cZ_u(k)^c zθ(k)z_θ(k) lookup(b) Unit abduction — formed token factual evidence O tokenizer TϕT_φ formed token ZcZ^c Shared response form queries x1Q,x2Q,x3Qx^Q_1,\,x^Q_2,\,x^Q_3 shared form RθR_θ Rθ(dy∣xQ,c,Zc)R_θ(dy x^Q,c,Z^c) direct: Zu(k)c=zθ(k)Z_u(k)^c=z_θ(k) abduction: formed token ZcZ^c Figure 3: Direct unit access and unit abduction return tokens of the same type. A trusted resolver uses a key k to fix the persistent referent and its stable address; lookup then supplies Zu(k)cZ_u(k)^c, typically the learned row zθ(k)z_θ(k), without forming a which-unit belief. When no resolver identifies the unit, a tokenizer forms a contextual token ZcZ^c from factual evidence. A which-unit belief QϕQ_φ may approximate the world-side identity conditional (Appendix B). Both access modes use the same learned response model, with the token mode using the shared form RθR_θ. In the supervised specialization, an identity-mixture learner mirrors the world-side unit mixture in Eq. 16 by reusing one learned identity mixing measure through an identity-indexed response model: P^θ,ϕ(y∣;xQ):=∫Pθ(y∣xQ,u)Qϕ(u∣). P_θ,φ(dy ;\,x^Q):= _UP_θ(dy x^Q,u)Q_φ(du ). (23) This identity-mixture form is used in the theorems below. A different learner realization is the token-space form P^θ,ϕ(y∣;xQ,c):=∫Rθ(y∣xQ,c,z)Tϕ(z∣,c). P_θ,φ(dy ;\,x^Q,c):= R_θ(dy x^Q,c,z)\,T_φ(dz ,c). (24) The two integrals coincide with each other and with the exact world mixture only under the conditions of Eq. 16. The semicolon distinguishes the supplied response query xQx^Q from abductive evidence. The displayed learner prediction is always well-defined when its components are. To identify the corresponding exact fixed-belief mixture with P(YQ∈dy∣,xQ)P(Y^Q∈ dy ,x^Q), we use two separate conditions: response sufficiency, YQ⟂|(U,xQ)Y^Q \!\!\! (U,x^Q), and the external-query condition, U⟂XQ|U \!\!\! X^Q (or the same statement for the complete input (XQ,CQ)(X^Q,C^Q) when context is exposed). The latter holds for a fixed query and, more generally, when a query policy uses the declared factual evidence but has no hidden access to U. If a recorded variable is informative about the unit, the protocol places that factual information in O before forming the token; an alternative response query does not silently become a new abduction call. When answer-time conditions must be distinguished from the focal query, we write the complete response input as (xQ,cQ)(x^Q,c^Q) and call cQc^Q response context. It may otherwise be absorbed into a larger xQx^Q. Attribution evidence remains in O, which excludes the current prediction target. The uninformative limit. If U⟂U \!\!\! , then P(U∈du∣)=Π(du)P(U∈ du )= (du) almost surely. Evidence-based individualization is then unavailable: a learner calibrated to this target remains at population-level uncertainty, or an operational system may abstain and seek new evidence. This limitation concerns attribution information and is compatible with either homogeneous or heterogeneous responses across units. Evidence informativeness and response heterogeneity. Evidence informativeness asks whether the available evidence changes what can be known about U. Response heterogeneity asks whether changing U changes the current response law. They define separate axes of the problem. Table 1: For a unit boundary fixed by the task, learner access to the unit and unit-response dependence are independent axes. “Current evidence” means the evidence declared by the protocol. Access to U unit-homogeneous response unit-heterogeneous response resolved or informative The referent may be resolved directly, or the formed token may be informative; this response target is invariant across units. Personalization is substantive; a resolved identity or an informative token changes the prediction. unresolved; uninformative Unit uncertainty remains at the population level; the response is unit-invariant. Oracle predictive value of knowing U may exist, but the current evidence cannot access it; mix, abstain, or collect new evidence. Recommendation with a trusted resolver occupies the resolved-identity, heterogeneous cell; the lookup token zθ(k)z_θ(k) may nevertheless remain learned. The lower-right cell separates the predictive value of unit information from the accessibility of attribution evidence; the upper-left cell separates attribution from relevance to the response target. The regular-conditional and support qualifications are recorded in Section E.4. 5 Unit Information: Predictive Value, Access, and Deployment In this section, “unit information” refers to the predictive value of knowing the realized unit, decomposed into oracle value, accessibility, learned realization, and component identifiability. The proofs use the identity-mixture learner of Eq. 23. The central theoretical question is not merely whether a unit-conditioned model can be written. Under assumption 1 the learned object is the pair (Tϕ,Rθ)(T_φ,R_θ). The question is which part of the oracle predictive value of unit information is available through declared evidence, how much of that available value the pair attains, and what the resulting marginal prediction can certify about its internal token-and-form decomposition. The corresponding layers are distinct: oracle value≠predictive accessibility≠learned-pair realization≠component identifiability. oracle value accessibility -pair realization identifiability. (25) Here learnability is reserved for whether some learning rule drives the approximation penalty to zero as training information grows. The identities below are classical specializations of conditional entropy, mutual information, the relative-entropy chain rule, and data processing [7, 14]. They organize the four layers; learnability and component identification require the additional conditions listed in the appendix. 5.1 Oracle Predictive Value, Accessibility, and Residual Information Question. If oracle unit information is useful, how much of that value is available from the factual evidence that the protocol actually exposes? Write W for the complete declared response input, with W=XQW=X^Q in the base response specification and W=(XQ,CQ)W=(X^Q,C^Q) when optional response context is exposed. The letter W is not a unit token ZucZ_u^c. Consider a declared deployment law for (U,,W,YQ)(U,O,W,Y^Q) and assume response sufficiency, YQ⟂|(U,W).Y^Q \!\!\! (U,W). (26) When the relevant conditional densities exist with respect to a common outcome reference measure and the log-loss risks are finite, write RW⋆ R_W :=E[−logp(YQ∣W)], :=E[- p(Y^Q W)], R,W⋆ R_O,W :=E[−logp(YQ∣,W)], :=E[- p(Y^Q ,W)], RU,W⋆ R_U,W :=E[−logp(YQ∣U,W)]. :=E[- p(Y^Q U,W)]. (27) Proposition 1 (Value–access decomposition under log loss). Under Eq. 26, RU,W⋆≤R,W⋆≤RW⋆,R_U,W ≤ R_O,W ≤ R_W , (28) and the oracle value decomposes exactly as RW⋆−RU,W⋆⏟Voracle R_W -R_U,W _V_ oracle =RW⋆−R,W⋆⏟Vaccessible+R,W⋆−RU,W⋆⏟Vresidual = R_W -R_O,W _V_ accessible+ R_O,W -R_U,W _V_ residual =I(U;YQ∣W) =I(U;Y^Q W) (29) =I(;YQ∣W)+I(U;YQ∣,W). =I(O;Y^Q W)+I(U;Y^Q ,W). (30) Moreover, I(;YQ∣W)≤minI(U;∣W),I(U;YQ∣W).I(O;Y^Q W)≤ \! \I(U;O W),I(U;Y^Q W) \. (31) Intuition. Perfect knowledge of U defines an oracle ceiling, but evidence can expose only the response-relevant part of that information. The remainder is still predictively valuable in principle and inaccessible under the declared evidence cutoff. The three gaps vanish exactly under YQ⟂U|WY^Q \!\!\! U W, YQ⟂|WY^Q \!\!\! W, and YQ⟂U|(,W)Y^Q \!\!\! U (O,W). The last says that the evidence retains all response-relevant unit information, not that it recovers the realized unit. A trusted key that uniquely resolves U=u(k)U=u(k) zeros the residual term; estimating zθ(k)z_θ(k) and RθR_θ remains the approximation problem below. The proper-score comparison is Proposition 7; proofs are in Section E.1. Binary example. Let W be fixed, U∼Bernoulli(1/2)U (1/2), YQ=UY^Q=U, and =U⊕NO=U N with N∼Bernoulli(δ)N (δ) independent and 0≤δ≤1/20≤δ≤ 1/2. In nats, the oracle value is log2 2, the accessible value is log2−h(δ) 2-h(δ), and the residual value is h(δ)h(δ), where h(δ)=−δlogδ−(1−δ)log(1−δ)h(δ)=-δ δ-(1-δ) (1-δ). The unit matters for every δ, but the declared evidence ranges from perfectly informative to useless. 5.2 Realization by the Learned Pair and End-to-End Error Question. How do unit-belief and fixed-unit response errors limit the value realized by the deployed mixture? For the fixed-belief formulation, additionally assume the external-query condition U⟂W|.U \!\!\! W . (32) This holds for a fixed query and for a query policy that uses O without hidden access to U. At admissible (o,w)(o,w), write Po(du) P_o(du) :=P(U∈du∣=o), :=P(U∈ du =o), Qo(du) Q_o(du) :=Qϕ(du∣o), :=Q_φ(du o), Ku,w(dy) K_u,w(dy) :=P(YQ∈dy∣U=u,W=w), :=P(Y^Q∈ dy U=u,W=w), K^u,w(dy) K_u,w(dy) :=Pθ(dy∣w,u), :=P_θ(dy w,u), (33) Mo,w(dy) M_o,w(dy) :=∫Ku,w(y)Po(u), := K_u,w(dy)P_o(du), M^o,w(dy) M_o,w(dy) :=∫K^u,w(y)Qo(u). := K_u,w(dy)Q_o(du). (34) Under Eqs. 26 and 32, Mo,w=P(YQ∈⋅∣=o,W=w)M_o,w=P(Y^Q∈· =o,W=w) on the chosen versions. Proposition 2 (End-to-end predictive excess and component-error bound). Let Rdep(θ,ϕ)R_ dep(θ,φ) be the expected log loss of M^,W M_O,W. Whenever the displayed quantities are finite, Rdep(θ,ϕ)−R,W⋆ R_ dep(θ,φ)-R_O,W =E[DKL(M,W∥M^,W)]=:ℰpred(θ,ϕ), =E\! [D_ KL\! (M_O,W\, \|\, M_O,W ) ]=:E_ pred(θ,φ), (35) RW⋆−Rdep(θ,ϕ) R_W -R_ dep(θ,φ) =I(;YQ∣W)−ℰpred(θ,ϕ), =I(O;Y^Q W)-E_ pred(θ,φ), (36) Rdep(θ,ϕ)−RU,W⋆ R_ dep(θ,φ)-R_U,W =I(U;YQ∣,W)+ℰpred(θ,ϕ). =I(U;Y^Q ,W)+E_ pred(θ,φ). (37) If, for almost every (o,w)(o,w) under the deployment law of (,W)(O,W), Po≪QoP_o Q_o and Ku,w≪K^u,wK_u,w K_u,w for PoP_o-almost every u, then ℰpred(θ,ϕ) _ pred(θ,φ) ≤E[DKL(P(U∈⋅∣)∥Qϕ(⋅∣))] ≤ E_O\! [D_ KL\! (P(U∈· )\, \|\,Q_φ(· ) ) ] +E,W[∫DKL(Ku,W∥K^u,W)P(u∣)]. +E_O,W\! [ D_ KL\! (K_u,W\, \|\, K_u,W )P(du ) ]. (38) The bound remains valid in the extended sense when its right-hand side is infinite. The exact identities say that the learned pair realizes accessible response value minus its predictive excess. The upper bound follows by applying the relative-entropy chain rule to the true and learned latent-joint laws and then projecting away U; the proof is in Section E.2. For a complementary query-local statement, let dTV(μ,ν)=supA|μ(A)−ν(A)|d_ TV(μ,ν)= _A|μ(A)-ν(A)| and let oU_o be a common full-mass region under PoP_o and QoQ_o. Lemma 3 (Query-local unit-mixture error propagation). For measurable response kernels for which the displayed quantities are well-defined, dTV(M^o,w,Mo,w) d_ TV\! ( M_o,w,M_o,w ) ≤∫dTV(K^u,w,Ku,w)Qo(u) ≤ d_ TV\! ( K_u,w,K_u,w )Q_o(du) +[supu,v∈odTV(Ku,w,Kv,w)]dTV(Qo,Po). + [ _u,v _od_ TV\! (K_u,w,K_v,w ) ]d_ TV(Q_o,P_o). (39) Intuition. The KL result tracks the log-loss penalty of the deployed prediction; the total-variation result shows when unit-belief error can matter at a particular query. If all true fixed-unit response laws agree there, the heterogeneity diameter is zero and belief error cannot change the response mixture. The KL bound splits deployed excess into unit-belief error and true-posterior-averaged response-law error (Eq. 38); the total-variation lemma makes belief sensitivity heterogeneity-dependent (Section E.5). Under trusted direct access, Po=Qo=δu(k)P_o=Q_o= _u(k) with no QϕQ_φ module, so remaining error is mismatch between K^u(k),w K_u(k),w and Ku(k),wK_u(k),w—token and form, not attribution. Both bounds are one-sided: a perfect marginal identifies neither component, support mismatch can make the KL bound infinite, and total variation does not control unbounded log loss. Binary example and TV sensitivity. In the binary example above, the exact evidence-conditional predictor has ℰpred=0E_ pred=0 and realizes all accessible value. A learner that ignores O and always reports the pooled Bernoulli(1/2)Bernoulli(1/2) law has ℰpred=I(,YQ)E_ pred=I(O;Y^Q) and realizes none of it. Separately, if K0,w=K1,wK_0,w=K_1,w at a query, even a belief that swaps all mass between the two units leaves the response prediction unchanged; when the laws separate, the heterogeneity multiplier in Eq. 39 records the possible sensitivity. 5.3 Single-Row Impossibility and Repeated-Linkage Separation Question. Can perfect marginal prediction validate the internal unit structure? The law of total probability gives ∫Pu⋆(y∣x)P(U∈u∣X=x)=P(Y∈y∣X=x)for PX-almost every x. _UP_u (dy x)P(U∈ du X=x)=P(Y∈ dy X=x) $P_X$-almost every $x$. (40) The next result upgrades this identity to an observational-equivalence and testing statement, then exhibits a protocol that separates the same worlds. Proposition 4 (Single-row indistinguishability and linked-pair separation). (a) The identity in (40) holds for compatible conditional versions. Moreover, in an unrestricted response-kernel class, every sample-only law R(dy∣x)R(dy x) has, under every learner belief, the unit-constant representation KuR(dy∣x):=R(dy∣x),K_u^R(dy x):=R(dy x), (41) so marginal fit does not identify the two mixture components. (b) For a statistical witness, fix one response input. In world H, let U∼Bernoulli(1/2)U (1/2) and P(Y=1∣U=0)=14,P(Y=1∣U=1)=34.P_ H(Y=1 U=0)= 14, P_ H(Y=1 U=1)= 34. (42) In world P, let P(Y=1∣U=u)=1/2P_ P(Y=1 U=u)=1/2 for both units. If each observation comes from a fresh independently drawn unit and neither its identity nor linkage is observed, then for every n, ℒ(Y1,…,Yn)=ℒ(Y1,…,Yn)=Bernoulli(1/2)⊗n.L_ H(Y_1,…,Y_n)=L_ P(Y_1,…,Y_n)=Bernoulli(1/2) n. (43) Consequently every possibly randomized test has αn+βn=1 _n+ _n=1 and infφnmaxαn,βn=12. _ _n \ _n, _n\= 12. (44) (c) If instead two responses per unit are observed with trusted same-unit linkage and are conditionally independent given U, then Cov(Y1,Y2)=116,Cov(Y1,Y2)=0,Cov_ H(Y_1,Y_2)= 116, _ P(Y_1,Y_2)=0, (45) or equivalently their agreement probabilities are 5/85/8 and 1/21/2. If M linked pairs are sampled independently from independently drawn units, their empirical agreement rate therefore gives a consistent test as M→∞M→∞. (d) More generally, under the declared linked-pair model (Y1,Y2)|U=u∼Bernoulli(pu)⊗2(Y_1,Y_2) U=u (p_u) 2, the observable pair law identifies Var(pU)=P(Y1=1,Y2=1)−P(Y1=1)2=Cov(Y1,Y2),Var(p_U)=P(Y_1=1,Y_2=1)-P(Y_1=1)^2=Cov(Y_1,Y_2), (46) but does not by itself identify the full mixing law or realized unit. Intuition. One response from each fresh, unlinked unit reveals only the pooled Bernoulli mean. Trusted repeated linkage exposes a joint observation, whose within-unit dependence carries a heterogeneity signal. Unrestricted single-row observations cannot distinguish a homogeneous world from every heterogeneous alternative. Trusted same-unit pairs under a fixed-u product law separate the displayed worlds and identify Var(pU)Var(p_U). Repeated records without trusted linkage do not suffice; recommendation logs with user IDs are outside the fresh-unlinked regime because the IDs already supply linkage and update addresses. Restricted mixture identifiability can require further structure [43]. Proof and the explicit test are in Section E.3. Binary example. The two fixed-unit success probabilities are 1/2±1/41/2± 1/4, so their population variance is (1/4)2=1/16(1/4)^2=1/16, exactly the linked-pair covariance. Both probabilities remain strictly between zero and one: the witness retains event variation after U is fixed rather than replacing it by a deterministic label. (a) Heterogeneous fixed-unit lawsxxP(Y=1∣x,u)P(Y=1 x,u)uAu_AuBu_B puA=34p_u_A= 34, puB=14p_u_B= 14, equal weights 12 12, 12 12 (b) unit-independent lawxxP(Y=1∣x,u)P(Y=1 x,u) one law for every u, P(Y=1∣x,u)=12P(Y=1 x,u)= 12 mix over UUObservable marginalxxP(Y=1∣x)P(Y=1 x)12 12 same marginal law Figure 4: Single-row marginals do not certify unit structure. Distinct fixed-unit Bernoulli laws and a unit-independent Bernoulli law can both yield P(Y=1)=1/2P(Y=1)=1/2 after marginalizing U. In the restricted witness of Proposition 4, one response from each fresh unit remains indistinguishable for every sample size, whereas a conditionally independent pair with trusted same-unit linkage reveals within-unit covariance. 5.4 Practical Corollaries for Repeated-Unit Evaluation When attribution is observed, retaining units also changes elementary bookkeeping in ways that alter the estimand and evaluation protocol [19, 36, 34]. For M observed units, let m(u)m(u) be the number of records from u and let ℓ¯(u) (u) be its mean loss. Corollary 5 (Row and unit weighting). The empirical risks R^row=∑um(u)Nℓ¯(u),R^unit=1M∑uℓ¯(u) R_ row= _u m(u)N (u), R_ unit= 1M _u (u) (47) agree for every possible collection of unit mean losses if and only if all observed units have equal record multiplicity. Corollary 6 (Known units survive a record-wise split). If each record independently enters training with probability p∈(0,1)p∈(0,1), a unit with m(u)m(u) records appears in both train and test with probability 1−pm(u)−(1−p)m(u).1-p^m(u)-(1-p)^m(u). (48) This probability is positive exactly when m(u)≥2m(u)≥ 2. Neither weighting is universally correct; the deployment estimand determines the target. Likewise, a record-wise split generally mixes new-event prediction for known units with prediction for unseen units, so an unseen-unit claim requires an explicitly unit-disjoint protocol. Proofs are in Section E.7. Prediction-level marginalization is distinct from deleting attribution metadata: removing unit labels from a table is a data map, not the same operation as mixing a unit-conditioned family. (a) Repeated units change the estimandRow averageweight m(u)/Nm(u)/NA1A_1A2A_2A3A_3B1B_1C1C_1C2C_23/N3/N1/N1/N2/N2/NPer-unit averageweight 1/M1/MuAu_AuBu_BuCu_Cℓ¯(uA) (u_A)ℓ¯(uB) (u_B)ℓ¯(uC) (u_C)(b) Repeated units change the split questionRecord-wise splittraintestA1A_1A2A_2B1B_1A3A_3B2B_2known unit, new eventWhole-unit splittraintestA1A_1A2A_2B1B_1C1C_1C2C_2new unit Figure 5: Repeated units change both the estimand and the split question. Row averaging and unit averaging assign different weights when units contribute unequal numbers of records. A record-wise split can measure a new event for a known unit, whereas a whole-unit split targets generalization to a new unit. The declared deployment question determines the appropriate choice. An empirical study that uses the identity-mixture specialization should keep separate unit-belief quality, fixed-unit response quality, deployed marginalized prediction, and a matched unit-omitting baseline under the same information and target. A compact checklist is in Appendix G. 6 Recommendation as a Worked Setting Recommendation provides a worked instance of the unit primitive: many interaction records can share one user while candidate items change. Direct unit access uses a trusted key to resolve which persistent user is active and which stable address to read; lookup supplies Zu(k)c=zθ(k)Z_u(k)^c=z_θ(k). Unit abduction forms a token of the same type from factual evidence (Section 4). The two access modes differ in how the token is obtained, not in whether the user is the unit. The mapping is summarized in Table 2. Factorization and neural collaborative-filtering models are established instances of the ID-indexed formulation [28, 24, 18]. In the linear running instance they instantiate Eq. 15 by a user token zθ(k)z_θ(k) and an item map ψθ(xQ) _θ(x^Q); a deeper query encoder may be nonlinear, but the readout in the user token remains an inner product. Table 2: For this worked task, users instantiate the declared unit boundary. Object in the unit formulation Recommendation interpretation Population population unit U possible user individuals Individual referent realized unit u one persistent user; not a uniquely identified embedding coordinate Direct-access model trusted key k stable referential address that associates interaction records and parameter updates with the same user slot lookup token Zu(k)cZ_u(k)^c learner-side object obtained by indexing, typically represented by a learned row zθ(k)z_θ(k) (embedding, factor, random effect); not the user u(k)u(k) Learner under unit abduction factual evidence O admissible observed history or side information used to form a user token tokenizer TϕT_φ maps evidence and context to a user token ZcZ^c; when explicitly targeted, QϕQ_φ approximates the world-side identity conditional for the identity-mixture learner Query–response specification query xQx^Q candidate item or slate supplied to the response model response context cQc^Q explicit answer-time context or history-dependent response state, when used; it may change the token while user identity stays fixed shared form Rθ(dy∣xQ,cQ,Z)R_θ(dy x^Q,c^Q,Z) feedback law instantiated by the user token through a simple relation in Z; the linear running instance is η=αθ(xQ,cQ)+⟨ψθ(xQ,cQ),Z⟩η= _θ(x^Q,c^Q)+ _θ(x^Q,c^Q),Z ; a direct-ID model computes it at Zu(k)c=zθ(k)Z_u(k)^c=z_θ(k) 7 Relations to Established Learning Traditions Classical statistical learning studies how a predictor class and learning rule generalize from finite samples [17, 38]. The unit formulation does not replace that theory. Under assumption 1 it adds an explicit question when persistent units matter: how a tokenizer organizes units so that one shared response-law form generalizes across them. The components of the pair (Tϕ,Rθ)(T_φ,R_θ) have long appeared across statistics and machine learning under different objects and assumptions. Longitudinal random-effects models provide a particularly close comparison because they combine supplied same-subject linkage, a shared response form, and a subject-specific token (the random effect). Other traditions infer attribution, organize prediction around episodes or environments rather than individuals, or begin from a supplied group boundary. Table 3 compares six traditions by their persistent referent, attribution assumptions, identity uncertainty, and how they instantiate a tokenizer and a shared form. The table records a recurring structural role rather than mathematical equivalence: each tradition retains its own estimands, identification conditions, and learnability assumptions. Table 3: Relations to six established model families. Family Established object or assumption Relation to the unit formulation random effects A supplied subject or group index and a population model for subject-specific parameters [25, 13]. Direct lookup of a subject token with a shared RθR_θ; the random effect is a distributional token, not a private response mechanism. latent-variable models and mixtures of experts Incomplete-data estimation, input-dependent gating over experts, and amortized inference over latent codes [9, 20, 23]. The encoder or gating map is a tokenizer; the decoder or expert family is a shared form, provided the task assigns the latent the role of a persistent unit. multitask, grouped-latent, Neural-Process, and meta-learning methods Shared parameters or representations across related tasks, and dataset-, group-, or task-level latents or deterministic context representations derived from an already associated context set [6, 1, 10, 5, 12, 11, 16]. An episode or context encoder is a tokenizer and the decoder is a shared form, provided the outer task declares a persistent unit. In the cited methods the context-to-group association is supplied before prediction, which differs from forming a token from otherwise unattributed evidence. record linkage, entity resolution, and probabilistic identity representations Posterior linkage structures, learned entity matching, and distribution-valued instance representations [41, 42, 22, 26, 30, 39]. These methods infer which records share a task-side unit. The present formulation additionally asks for a learner-side token that instantiates one shared RθR_θ. recommenders and personalization Known user indices, latent user factors, history encoders, and user-conditioned response scores [28, 24, 37, 18, 21, 27]. A known user index provides a lookup token zθ(k)z_θ(k) while the scoring model is a shared RθR_θ. Unresolved user attribution forms a token of the same type rather than looking it up. potential outcomes, structural causal models, and Abductive Learning units under treatment comparisons, counterfactual abduction–intervention–prediction, and logical abduction coupled to learning [35, 32, 31, 8]. Here u remains a task-side referent across treatments or worlds. Their abduction objects (noise, logical facts, identity posteriors) are not the unit token of assumption 1, and each tradition keeps its own estimands and identification conditions. The history also separates persistent-unit variation from environment variation. Random effects and personalization primarily ask how responses vary across persistent individuals. Covariate shift, domain adaptation, and invariant prediction primarily organize variation across environments or selection regimes [40, 2, 33]. These axes can be crossed: a response may vary across units while remaining invariant across environments, vary across environments while remaining homogeneous across units, or vary along both. Collapsing these axes would conflate distinct sources of variation. The term abduction likewise has established neighboring meanings. In structural-causal counterfactual reasoning it infers exogenous noise before intervention and prediction, while Abductive Learning couples logical abduction to machine learning [32, 31, 8]. Unit abduction instead names the construction of a contextual unit token from factual evidence (Eq. 21). A which-unit belief may approximate the world-side identity conditional and feed the identity-mixture learner; the access step remains the formed token (Appendix B). These learner formulations may coexist in a larger model while remaining distinct inference operations. 8 Discussion and Limitations The unit primitive is useful when a learning question depends on persistence, attribution, or unit-level evaluation. For row-level deployments with no same-unit question, the ordinary sample formulation is the appropriate special case. That sufficiency does not make the rows homogeneous units: without a declared unit axis the row-level law is silent about unit-response homogeneity. Two readings of the preceding sections are methodological rather than theorems. For many applied tasks, a natural starting class admits structured response heterogeneity and lets data decide whether the unit-omitting restriction Eqs. 19 and 20 is adequate; this does not claim that every task is empirically heterogeneous. Separately, the comparisons in Section 7 can be read as an organizing program in which different fields recover partial aspects of a unit-conditioned family. That reading is not a reduction. Assumption 1 is a computational interface, not a learnability theorem. The structured class makes it a restriction: RθR_θ must read the token through a declared simple relation. The running instance Definition 2 is a finite-dimensional linear predictor; other declared simple maps of z instantiate the same class. Without a bound on d, or with an unrestricted nonlinear readout of z, the factorization is again a reparameterization. The class is misspecified when residual unit-specific variation cannot be absorbed into the declared simple relation. The paper does not prove that some learning rule drives the approximation penalty to zero as training information grows, and joint training of (Tϕ,Rθ)(T_φ,R_θ) does not by itself identify either module. The theorems in Section 5 analyze an identity-mixture learner; a deployed system may instead use the token composition. These are alternative coordinates, not a proof that a tokenizer recovers the identity posterior. The unit boundary is task-relative. Within a fixed declaration, a trusted key resolves the referent but does not identify the token: zθ(k)z_θ(k) may be reparameterized with RθR_θ, and learning it does not recover the individual. Referential identity also supplies no intrinsic similarity; borrowing requires a metric, kernel, graph, hierarchy, or other coupling. In the linear running instance the inner product in Eq. 15 is the default readout; other simple relations remain open. When identity is unresolved, identifying QϕQ_φ and the fixed-unit kernels still needs shared-unit observations or other restrictions. Response-informative observations belong in O; a merely proposed query does not trigger a second abduction step. The formal results already indicate what would test a unit-centered claim. Unlinked single-row observations cannot separate a restricted heterogeneous witness from a pooled homogeneous world; trusted same-unit pairs can (Proposition 4). A record-wise split mixes known-unit new-event prediction with unseen-unit generalization (Corollary 6), and row weighting disagrees with unit weighting whenever multiplicities differ (Corollary 5). An empirical study should therefore keep a matched unit-omitting baseline, report unit-belief quality separately from fixed-unit response quality when the identity-mixture specialization is used, and declare whether the split is unit-disjoint (Appendix G). No such study is reported here. Causal estimands require an additional layer. The primitive preserves the referent across a query family; intervention, counterfactual coupling, and identification assumptions are specified separately (Appendix D). Token reparameterization, the identity-belief specialization, and the fixed evidence cutoff are collected in Appendices B and F. 9 Conclusion The paper’s organizing claim is that machine learning learns shared structure across task-declared units from noisy, selectively observed, unit-linked events. A sample records an event; the task first declares the persistent referent, and only then chooses an ID, embedding, or other representation. Supervised learning takes a family of unit-conditioned response laws as its semantic object. Assumption 1 states a computational factorization for learning that family: every learner-side path from the unit to prediction passes through a contextual unit token, all units use one form RθR_θ, and represented unit-specific differences must pass through the token. The learned object is the pair (Tϕ,Rθ)(T_φ,R_θ); the structured class is a simple relation in the token, with a linear predictor as the running instance (Definition 2). A unit-omitting learner specification is unit-insensitive, while a constant or ignored token is a canonical realization of that specification. The conceptual boundaries are collected in Appendix B. Oracle predictive value of unit information is then separated from evidence accessibility; the learned pair incurs an additional approximation penalty; and marginal prediction alone does not reveal the internal token-and-form decomposition. Trusted same-unit linkage provides a restricted positive boundary. 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The exact mixture layer is the first line of Eq. 16: unresolved identity mixes the world targets Pu⋆P_u over units, without introducing learned parameters. The computational layer contains learner compositions under assumption 1: the default token-space composition uses Tϕ(⋅∣,c)T_φ(\,· ,c) and RθR_θ, while the theorem specialization uses QϕQ_φ and an identity-indexed response model. A learner composition coincides with the exact world mixture only under the stated access, response-realization, response-sufficiency, and external-query conditions. World-side identity posteriors P(U∈du∣)P(U∈ du ) remain well-defined probability objects whether or not the learner represents them. The theorems below use the identity-mixture learner specialization; they are not a definition of unit abduction and do not require the computational token to be an identity posterior. In Section 5 the complete response input is written W; the same object is the lowercase index w of the kernel KuwK_u^w in Appendix C. Table 4: Notation by semantic layer. Learner quantities are listed with the computational pair first; QϕQ_φ belongs to the identity-mixture learner specialization. Symbol Intended type Boundary World and population U:Ω→U: population unit selection variable which individual in U Π(du) (du) population unit law which-individual randomness under P Fixed individual and event u∈u realized persistent unit fixed value of U; indexes a unit-conditioned law but is not synonymous with a learned numerical representation Ku(dx,dy)K_u(dx,dy) fixed-individual law retains exogenous/event variation after U=uU=u X,YX,Y population variables observed values may supply pre-answer evidence; an observed evidence response is not the current target Xu,YuX_u,Y_u fixed-individual variables conditional laws under U=uU=u Yu(a)Y_u(a) potential response conventional causal notation for individual u under treatment a Observed record and attribution i sample index counts observations, not units (xi,yi)(x_i,y_i) realized record supplies values that specify population events UworldD_U world conceptual complete-data table includes the definite uiu_i whether or not attribution is observed UobsD_U obs learner-visible-attribution table special regime in which persistent-unit labels, or a trusted resolver that determines them, are learner-visible K=kK=k and u(k)u(k) trusted key and resolved unit stable referential and update address; determines which persistent unit a record concerns Learner quantities Tϕ(u,c)T_φ(O_u,c) tokenizer forms a contextual unit token from unit-attached factual evidence and context; may be deterministic or a kernel on token space; jointly learned with RθR_θ Zuc∈Z_u^c contextual unit token learner-side representation supplied to RθR_θ in context c; not the unit Rθ(dy∣x,c,z)R_θ(dy x,c,z) shared response-law form the same learned map for every unit; learner-side unit differences enter only through the token z Qϕ(du∣)Q_φ(du ) which-unit belief a law on U that may approximate P(U∈du∣)P(U∈ du ) and feeds the identity-mixture learner; not the unit, not the token, and not the definition of unit abduction zθ(k)z_θ(k) ID-indexed lookup token learned table row implementing Zu(k)cZ_u(k)^c under direct access (embedding, factor, random effect, …); not the unit u(k)u(k) f(x,u)f(x;u) unit-conditioned predictor fixed u conditions a member of the predictor family f0(x)f_0(x) ordinary row-level predictor collapse notation, not an average over units Pθ(dy∣xQ,cQ,u)P_θ(dy x^Q,c^Q,u) identity-indexed response model aimed at Pu⋆P_u and used by the theorem’s identity-mixture learner; under direct access this is Rθ(dy∣xQ,cQ,zθ(k))R_θ(dy x^Q,c^Q,z_θ(k)) Evidence, query, and target O factual evidence observed-event information available before the answer and declared usable for unit inference P(U∈du∣)P(U∈ du ) evidence-conditioned unit posterior conditional law of the already-realized unit under the data-generating distribution xQx^Q response query supplied response argument; not new abductive evidence merely by being queried cQc^Q explicit response context answer-time information declared to change YQY^Q directly beyond (U,xQ)(U,x^Q); omitted when empty W complete response input in Section 5 W=XQW=X^Q or W=(XQ,CQ)W=(X^Q,C^Q); not the unit token ZucZ_u^c KuwK_u^w unit-conditioned response kernel at query w w=(xQ,cQ)w=(x^Q,c^Q) is the realized complete response input; not a token YQY^Q current response target is excluded from the pre-answer evidence Pu⋆(dy∣xQ,cQ)P_u (dy x^Q,c^Q) population response conditional shorthand for P⋆(YQ∈dy∣XQ=xQ,CQ=cQ,U=u)P (Y^Q∈ dy X^Q=x^Q,C^Q=c^Q,U=u) under response sufficiency As summarized in Table 4, the domain U belongs to the declared population learning problem. Dataset indices label records rather than creating new unit domains. A value xix_i instead specifies the event X=xi\X=x_i\; a pair (xi,yi)(x_i,y_i) may specify X=xi,Y=yi\X=x_i,Y=y_i\. If a study makes attribution learner-visible, lowercase uiu_i may annotate its records, and ui=uju_i=u_j then records a supplied same-individual relation. This metadata provides direct unit access; it does not make a model-side representation observed. Unit abduction applies when no resolver identifies the unit. Under direct access, lowercase xi,yix_i,y_i are event-level realized values associated with the fixed-unit event law Kui(dx,dy)K_u_i(dx,dy). The notation keeps a record value distinct from the fixed-unit stochastic object, so we avoid writing Xi:=XuiX_i:=X_u_i or Yi:=YuiY_i:=Y_u_i. Repeated observations require a separately specified joint law; the population unit variable remains U rather than a foundational family (Ui)(U_i). Let Π(du) (du) denote the population law of U, and let KX,u(dx)K_X,u(dx) be the fixed-individual observation kernel. Then PX(x)=∫KX,u(x)Π(u),P(XU∈A∣U=u)=KX,u(A).P_X(dx)= _UK_X,u(dx)\, (du), P(X_U∈ A U=u)=K_X,u(A). (49) The first variation comes from drawing U; the second remains inside KX,uK_X,u after an individual is fixed. A realized record contributes an event from the selected unit rather than a new unit variable UiU_i. Repeated-record sampling and uncertain linkage require an explicitly declared joint observation model. Evidence is event-level information. Uppercase symbols denote random variables; lowercase symbols denote realized values. A positive-probability event E permits ordinary conditioning P(U∈du∣E)P(U∈ du E). When the protocol records evidence at a realized value o, P(U∈du∣=o)P(U∈ du =o) denotes evaluation of a regular conditional distribution. We use O as protocol-level shorthand for either form of pre-answer information. Thus X=xi,Y=yi\X=x_i,Y=y_i\ is shorthand for observed evidence values, whereas (xi,yi)(x_i,y_i) is their realized-value encoding. Any yiy_i admitted here is a historical or other already observed response, not the current target response. A designed candidate xQx^Q supplied to Pθ(dy∣xQ,cQ,u)P_θ(dy x^Q,c^Q,u) is a response query. It contributes evidence about U only when it was factually observed before the answer and the protocol incorporates it into O. When no resolver identifies the realized unit, unit abduction forms a contextual unit token as in Eq. 21; the default prediction is the token composition Eq. 24. The identity-mixture specialization used by the theorems is Eq. 23. These displays are not repeated here. Direct lookup and abduction differ by provenance, not by the type of the object they return. In the basic supervised protocol, i=X=xiO_i=\X=x_i\ determines the world-side conditional P(U∈du∣X=xi)P(U∈ du X=x_i). The target yiy_i scores the learner mixture but is excluded from iO_i. A history encoder, probabilistic identity rule, or other map may implement TϕT_φ; the primitive is architecture-independent. The response-sufficiency condition is YQ⟂|(U,xQ,cQ),Y^Q \!\!\! (U,x^Q,c^Q), (50) with cQc^Q omitted when (U,xQ)(U,x^Q) suffices. This assumption applies when information in the factual record that directly changes the response beyond its role in forming unit information is represented explicitly by cQc^Q, rather than by an undeclared dependence of the response kernel on the full evidence bundle. The same raw observation may support both roles when the protocol declares them separately. On the learner side, the role of O in the response is mediated by the formed token after that context has been represented. To identify the exact response conditional with a mixture using the fixed identity law P(U∈du∣)P(U∈ du ), the protocol additionally satisfies the external-query condition in Eq. 32. Otherwise the exact mixing law is generally P(U∈du∣,xQ,cQ)P(U∈ du ,x^Q,c^Q). This is not permission to update formed unit information from a merely proposed query: any unit-informative observed variable must be declared as factual evidence before the token, or, in the identity specialization, the belief, is formed. Observed identifiers and direct unit access. If an ID event uniquely determines which persistent unit is present, a trusted resolver implements k↦u(k)k u(k) and fixes a stable address. A separate lookup at that address supplies the token, typically the learned row Zu(k)c=zθ(k)Z_u(k)^c=z_θ(k). The resolver fixes the referent; the lookup selects the token slot to read or update. Neither operation makes zθ(k)z_θ(k) into the unit. Learning zθ(k)z_θ(k) is fitting the token, not a which-unit inference. The corresponding which-unit conditional P(U∈du∣ID=k)P(U∈ du =k) is a point mass (a Dirac measure). The identity-mixture specialization then uses δu(k) _u(k); no learned QϕQ_φ module need be run, and the Dirac mass is not an executed abduction. Crucially, the Dirac mass is on the stable referential unit u(k)u(k), not on the trainable and potentially reparameterized token zθ(k)z_θ(k). A history-dependent state for the resolved unit belongs to the tokenizer’s context, or to the response layer, and may change without changing this identity law. Appendix B Conceptual Clarifications and Inferential Distinctions This section is the boundary register for the main text. Four layers should be kept separate: the task-side referent u, the learner-side token ZucZ_u^c, the world-side response family Pu⋆\P_u \, and the deployed prediction produced by a particular learner formulation. A trusted key resolves a referent and a lookup selects a token; unit abduction forms a token from factual evidence. Neither access procedure fixes the token parameterization or supplies a similarity rule between units. Likewise, a unit-omitting learner can fit a row-level marginal in either a homogeneous or heterogeneous world, while the identity-mixture specialization QϕQ_φ is a theorem-specific formulation rather than the definition of the tokenizer. The subsections below collect the associated semantic, evidence, and identification qualifications so that the main text can state the positive model without repeating every boundary case. B.1 Persistent units and model representations The declared unit domain U carries the task’s persistent-referent semantics. A learned embedding, preference factor, or random effect may parameterize its lookup token, but is not thereby the realized value of U. In a trusted-ID model, u(k)u(k) is the resolved unit and zθ(k)z_θ(k) is the token Zu(k)cZ_u(k)^c. The token may be transformed jointly with the downstream model without changing predictions, so its coordinates need not be unique or scientifically identifiable [3]. A posterior over such response-side coordinates is likewise not automatically the which-unit belief Qϕ(du∣)Q_φ(du ), and neither object is a sample-inclusion probability or an importance weight. B.2 Tokens, which-unit beliefs, and point summaries The default learner-side object is the contextual unit token ZcZ^c. In the identity-belief specialization, Qϕ(du∣)Q_φ(du ) instead targets the world conditional P(U∈du∣)P(U∈ du ) when the protocol declares that objective; calibration and recovery require additional conditions. Means and selected modes, when defined, are deterministic functionals of this belief, whereas a computational draw satisfies U~ϕ∣∼Qϕ(⋅∣) U_φ Q_φ(· ). The response-level prediction propagates the full belief through Eq. 23; a downstream decision may then extract a point prediction, interval, action, or abstention. B.3 Record, evidence, and query roles The notation table separates record values, factual evidence, response queries, and current targets. In particular, i indexes an event rather than defining a world variable UiU_i, and a supplied candidate xQx^Q is evaluated at the token formed from the fixed evidence cutoff. A newly observed factual event may update that token, and may update QϕQ_φ in the identity-belief specialization; the current target remains excluded from pre-answer evidence. If a response-relevant context is neither fixed by the protocol nor absorbed into x, omitting it from the response specification generally yields, for compatible regular-conditional versions and PX,UP_X,U-almost every (x,u)(x,u), Pu⋆(y∣x)=∫Pu⋆(y∣x,c)P⋆(c∣x,U=u).P_u (dy x)= P_u (dy x,c)\,P (dc x,U=u). (51) Consequently, unit-dependent context distributions can create heterogeneity at the marginal response level even when the context-conditional response surface is common across units. Conversely, context-specific differences between units can cancel after integration and disappear from the marginal response specification. Declaring cQc^Q is therefore optional at the level of mathematical typing but substantive whenever the task must distinguish which individual differs from which condition that individual occupies. An evidence bundle may contain several observed values without asserting that they concern the same individual. Repeated-individual attribution and entity linkage require an additional joint observation model. B.4 Marginalization, erasure, and component identification Three operations associated with unit erasure have different targets. At the prediction level, marginalizing the exact world law over the conditional unit distribution yields the row-wise law in Proposition 4. At the response-structure level, f(x,u)=f0(x)f(x;u)=f_0(x) for every relevant u is a sufficient invariance condition for a sample-only rule f0f_0; it does not define f0f_0 as an average of f(x,u)f(x;u). At the data level, deleting attribution columns from a table is a deterministic map, not a mixture. The first is probabilistic marginalization, the second is a response restriction, and the third is a data edit. For the identity-belief specialization of the composition, the exact world mixture equals P(Y∈dy∣X=xi)P(Y∈ dy X=x_i) by Proposition 4. When a density or probability mass exists and i=X=xiO_i=\X=x_i\, the learner score in that specialization is logp^θ,ϕ(yi∣i;xi)=log∫pθ(yi∣xi,u)Qϕ(du∣i). p_θ,φ(y_i _i;\,x_i)= \! _Up_θ(y_i x_i,u)Q_φ(du _i). (52) The default token-space prediction instead scores ∫Rθ(dyi∣xi,c,z)Tϕ(z∣i,c) R_θ(dy_i x_i,c,z)\,T_φ(dz _i,c). Here the observed xix_i serves both as pre-target evidence and as the response covariate, while an alternative xQx^Q reuses the same formed token. The log of the mixture is generally different from the mixture of conditional log scores; for continuous Y, the likelihood is a density rather than a point probability. Matching the marginal law alone does not identify TϕT_φ or QϕQ_φ, recover a persistent unit, or establish response dependence on u. Appendix C Unit-Conditioned Response Heterogeneity Throughout this section, w=(xQ,cQ)w=(x^Q,c^Q) denotes a declared complete response input—not a unit token ZucZ_u^c. This is the same object written W in Section 5. The unit primitive makes it possible to define response heterogeneity without committing to a particular random-effect parameter, neural architecture, or clustering device. This section isolates that definition, separates it from the structural assumptions needed for learning, and then locates several established literatures in the resulting coordinates. Its scope is the precise object of heterogeneity, the restrictions that make it learnable, and its relation to neighboring forms of variation. Assumption 1 is one such restriction: differences among KuwK_u^w must be expressed by contextual unit tokens under a shared form RθR_θ. C.1 The response-law profile Let w=(xQ,cQ)w=(x^Q,c^Q) denote a declared response query, with cQc^Q omitted when it is empty, and let Q be a common admissible query family for the task. The task-level response object is a declared jointly measurable Markov kernel (u,w)↦Kuw(u,w) K_u^w on ×U× Q. Write Kuw(dy):=Pu⋆(dy∣xQ,cQ).K_u^w(dy):=P_u (dy x^Q,c^Q). (53) For fixed u, the family w↦Kuww K_u^w gives the response law over the declared query family. Equality of two such families means equality of the declared kernels on Q. If the kernel is learned or interpreted only through a regular conditional law under a design Λ(du,dw) (du,dw), its off-support values are modeling choices rather than identified features. The Λ -relative notion below states the corresponding observable condition. Definition 3 (Kernel formulation of unit-response homogeneity). For U∼ΠU , a declared query family Q, and the chosen jointly measurable kernel in Eq. 53, the task is unit-response homogeneous when there are a single Π -null set N⊆N and a common response kernel w↦Kww K^w such that Kuw=KwK_u^w=K^w for every u∉Nu∉ N and every w∈w∈ Q. It is unit-response heterogeneous otherwise. Equivalently, define query-relative response equivalence by u≡v⟺Kuw=Kvwas probability laws for every declared w∈.u _ Qv K_u^w=K_v^w\ as probability laws for every declared w∈ Q. (54) Homogeneity says that Π is concentrated on one such response-equivalence class; heterogeneity says that it is not. This quotient is predictive, not referential. Distinct individuals may satisfy u≡vu _ Qv while remaining distinct units. Conversely, changing P(X∣U=u)P(X U=u) or the frequency with which a unit is sampled does not by itself establish heterogeneity in KuwK_u^w. For a declared design Λ on ×U× Q, call the family Λ -visible homogeneous when there is a kernel KwK^w such that Kuw=KwK_u^w=K^w for Λ -almost every (u,w)(u,w), and Λ -visible heterogeneous otherwise. Structural homogeneity in Definition 3 implies visible homogeneity for every compatible design, where compatibility means that the unit marginal ΛU _U is absolutely continuous with respect to Π . The converse can fail when a design omits queries or units on which response laws differ. This qualification also makes the notion invariant to changes of a regular conditional law on a Λ -null set. The definition is query-relative for a substantive reason. Two units may be equivalent on the queries in one study and distinguishable on a larger query family. A treatment, time point, item, or context can be part of w without becoming a new unit. The response kernels in Eq. 53 are observational unless a separate causal assignment and identification assumptions license an interventional reading. C.2 Response-law model classes The family KuwK_u^w supports several qualitatively different model classes. The restrictions in Table 5 distinguish their response objects and transfer assumptions. Table 5: Response-law cases and model classes inside a unit-explicit task. Standard sample-only learning is listed separately because its basic object does not require a declared unit population. Case or class Restriction Meaning sample-only no persistent unit is declared Learns a row-level law such as P(dy∣x)P(dy x); it makes no direct statement about persistent individuals. homogeneous unit extension Kuw=KwK_u^w=K^w for Π -almost every u After persistent units are declared, unit identity adds no response information at the queries in Q. pure-unit, input-invariant Kuw=KuK_u^w=K_u while KuK_u varies with u Outcomes vary across units while remaining invariant to the supplied query. structured unit-responsive class Kuw=Kw(η,ζu)K_u^w=K^w(η, _u); simple relation in a token, linear running instance (Definition 2) Both query and unit may matter. The running instance is a linear predictor αθ+⟨ψθ,Z⟩ _θ+ _θ,Z in the unit token Z; other declared simple maps of Z instantiate the same class. ψθ _θ may be nonlinear in the query w. saturated unit-responsive class (u,w)↦Kuw(u,w) K_u^w is otherwise unrestricted across units The class permits one arbitrary response family per unit, including homogeneous families; unseen-unit learning requires additional structure or information. The final two rows describe model classes that permit heterogeneity, not a claim that every member is heterogeneous. Whether the realized response family is homogeneous or heterogeneous is still decided by Definition 3, or by its Λ -visible counterpart for a deployment-limited claim. Standard supervised learning does not, by its basic sample-indexed notation [4, 17, 38], specify a population of persistent units over which homogeneity could be assessed. A homogeneous specialization results after those units are declared and the second row of Table 5 is imposed. The exact marginal collapse in Proposition 4 is a different statement: heterogeneous fixed-unit laws may integrate to the same row-level predictive law. C.3 Shared Structure and Learnability Definition 3 determines whether the response family varies with u but does not specify how the families across units are related. Statistical learnability therefore depends on a declared structural restriction of the form (u,w)⟼Kuwbelongs to a declared structured class of kernels,(u,w) K_u^w to a declared structured class of kernels, (55) which may encode shared parameters, a hierarchy, low-rank or smooth variation, a shared representation with unit-specific heads, a finite set of response types, or another coupling across units. These are alternative structural assumptions rather than additions to the definition of heterogeneity. A saturated class of measurable unit-to-law maps permits heterogeneity but does not by itself support transfer from observed units to an unseen unit. A learnability analysis specifies which response structure is shared, which part varies, what evidence constrains the varying part, and the deployment population and query family under which that structure is testable. Under assumption 1 the shared restriction is the pair (Tϕ,Rθ)(T_φ,R_θ), with a simple relation in the token as the structured class and Definition 2 as the running instance: RθR_θ depends on z only through a declared low-complexity map, linearly in finite dimension or through another declared simple map of z. Random effects and multitask representations provide two established forms of such structure [13, 6, 1]; a full learnability analysis must additionally specify the loss, sampling regime, model complexity, and learning rule [38]. C.4 Oracle Value Under Proper Scoring Rules The definition also has a decision-theoretic reading. Let Λ(du,dw) (du,dw) be a declared deployment design over units and queries, let Y|(U=u,W=w)∼KuwY (U=u,W=w) K_u^w, and suppose a regular conditional Λ(du∣w) (du w) has been fixed. Define the pooled Bayes law K¯w(y):=∫Kuw(y)Λ(u∣w). K^w(dy):= _UK_u^w(dy)\, (du w). (56) Let the admissible reports range over a class P of probability laws that contains KuwK_u^w for Λ -almost every (u,w)(u,w) and is closed under the displayed mixtures, so that K¯w∈ K^w . Let S(P,y)S(P,y) be a strictly proper predictive loss on P, with smaller values better, and let DS(P,Q):=EY∼P[S(Q,Y)−S(P,Y)]D_S(P,Q):=E_Y P[S(Q,Y)-S(P,Y)] be its associated nonnegative regret divergence [14]. Proposition 7 (Oracle heterogeneity gap under a proper score). Assume the displayed expectations are finite. The Bayes-risk difference between the best predictor that observes (U,W)(U,W) and the best predictor that observes only W is Rpool−RUnit R_ pool-R_ Unit :=E[S(K¯W,Y)−S(KUW,Y)] :=E\! [S( K^W,Y)-S(K_U^W,Y) ] (57) =E(U,W)∼Λ[DS(KUW,K¯W)]≥0. =E_(U,W) [D_S\! (K_U^W, K^W ) ]≥ 0. (58) Under strict propriety, equality holds if and only if KUW=K¯WK_U^W= K^W for Λ -almost every (U,W)(U,W). For logarithmic loss, the gap is IΛ(U;Y∣W)I_ (U;Y W) whenever the conditional mutual information is well defined. Proof. Condition on (U,W)=(u,w)(U,W)=(u,w). By the definition of the scoring-rule regret, the conditional expected excess loss from reporting K¯w K^w instead of the true KuwK_u^w is DS(Kuw,K¯w)D_S(K_u^w, K^w). Integrating with respect to Λ proves the identity and nonnegativity. Strict propriety makes the regret zero exactly when the two conditional laws agree almost surely. Under logarithmic loss the regret is the Kullback–Leibler divergence, whose conditional expectation is IΛ(U;Y∣W)I_ (U;Y W). ∎ This proposition applies the classical proper-scoring-rule identity to give an exact, query- and deployment-relative magnitude to Λ -visible response heterogeneity. It is an oracle comparison and does not establish that U is observed, identifiable, causally useful, or learnable from finite selectively observed events. Appendix D DiscoSCM as a Causal Specialization of the Unit Primitive DiscoSCM, introduced by Gong et al. [15], provides a substantive causal specialization of the unit primitive. This relation is stronger than a shared use of the letter U: DiscoSCM separates the selected individual from event-level exogenous variation, conditions structural causal mechanisms on the selected unit, and derives intervention and counterfactual quantities within that unit-conditioned model. At the same time, the causal and cross-world assumptions of DiscoSCM are additional structure; they do not follow from the unit primitive alone. D.1 From a Response-Law Family to Unit-Conditioned Structural Mechanisms The generic supervised specialization in this paper begins with a population law U∼ΠU and a unit-conditioned response family u↦Kuw(dy)u K_u^w(dy). DiscoSCM refines the response family into a structural causal model ℳD=⟨U,,,ℱ⟩,Vj←fj(paj,Ej,U),M D= U,E,V,F , V_j← f_j(pa_j,E_j;U), (59) where U=uU=u selects the individual, E contains exogenous variables, V contains endogenous variables, and ℱF is a family of unit-conditioned structural assignments. The DiscoSCM formulation assumes U⟂U \!\!\! in its basic construction. This assumption is specific to that construction; the general unit primitive does not require independence between unit selection and every source of event variation. For a treatment or supplied cause X=xX=x and outcome Y, suppressing other parents for clarity, the structural assignment induces the fixed-unit response kernel Ku,strx(B):=P(fY(x,EY,u)∈B).K_u,str^x(B):=P\! (f_Y(x,E_Y;u)∈ B ). (60) Thus u↦Ku,strxu K_u,str^x is a structurally induced unit-conditioned response family, while the mechanisms u↦fY(⋅,⋅,u)u f_Y(·,·;u) provide its richer causal representation. It is not identified with the observational conditional Pu⋆(dy∣x)P_u (dy x) without additional conditions linking observational and interventional laws. A unit can alter the baseline response, the effect of treatment, the outcome noise law, or a larger causal mechanism. The shared causal graph and the form of ℱF then provide one possible answer to the shared-structure question posed in Section 3.2. The separation between U and E is the central correspondence. A realized event is generated by both an individual value u and an exogenous realization e. Holding u fixed preserves the referential individual; it does not freeze the event realization. In the notation of this paper, DiscoSCM therefore instantiates the distinction between which-individual variation and the stochastic variation retained by a fixed-unit kernel. Table 6: Object-level correspondence between the unit formalism and DiscoSCM. The final column records structure contributed by the causal specialization. Unit formalism DiscoSCM object Additional causal content population law U∼ΠU unit selection variable U with law P(u)P(u) Population over individuals participating in a structural causal system. realized unit u individual retained across factual and counterfactual worlds The same referent appears in every structural assignment fj(⋅,⋅,u)f_j(·,·;u). fixed-unit event variation exogenous variables E A realized event depends on both (u,)(u,e); fixing u need not make the outcome deterministic. response law KuwK_u^w law induced by Vj←fj(paj,Ej,u)V_j← f_j(pa_j,E_j;u) The response is generated by a directed mechanism rather than specified only as a conditional kernel. response query xQx^Q intervention value x in do(X=x)do(X=x) Reading a query as an intervention requires the causal graph, structural assignments, and an intervention semantics. factual evidence O an observed factual trace Evidence updates unit information about the already-realized unit; DiscoSCM represents that update as an identity posterior, which is one specialization of a formed token. The trace may also contain variables with separately declared direct response roles. conditional unit law P(du∣)P(du ) DiscoSCM unit posterior P(du∣)P(du ) DiscoSCM’s population-level Layer-3 reduction mixes against the exact identity posterior. That is DiscoSCM’s abduction object, not the definition of unit abduction in this paper. A learned QϕQ_φ approximates that identity law when the protocol uses the identity-belief specialization. D.2 The additional causal and cross-world structure The unit primitive preserves the referent across a query family but is observational by default. DiscoSCM adds an intervention operation and a family of counterfactual exogenous variables (x)E(x). For an outcome mechanism, the resulting fixed-unit counterfactual outcome has the form Yud(x)=fY(x,EY(x),u).Y_u^d(x)=f_Y(x,E_Y(x);u). (61) The superscript d distinguishes this distribution-consistent counterfactual from a traditional same-noise SCM counterfactual. The defining marginal restriction is (x)=,E(x)\ d=\ E, (62) which yields, under the model’s structural conditions, the fixed-unit distribution-consistency relation ℒ(Yud(x)∣X=x,U=u)=ℒ(Y∣X=x,U=u).L\! (Y_u^d(x) X=x,U=u )=L\! (Y X=x,U=u ). (63) This equality in distribution replaces the pointwise equality imposed by the usual consistency relation. Equation (62) fixes the marginal law in each world but does not by itself determine the coupling among E, (x)E(x), and (x′)E(x ). That coupling is a Layer-3 modeling choice because it determines joint cross-world quantities. Traditional SCM semantics occupy the same-noise endpoint (x)=E(x)=E almost surely. The population-valuation result of Gong et al. [15] instead uses the additional assumption (x)⟂E(x) \!\!\! for the factual and queried worlds. Other couplings can share the same Layer-1 and Layer-2 marginals while inducing different Layer-3 joint distributions. The unit primitive identifies the referent that is held fixed across these worlds; DiscoSCM specifies how the remaining cross-world randomness is related. D.3 Unit Abduction, causal valuation, and reduction Let O denote an observed factual trace. The law of total probability gives the exact decomposition P(Yd(x)∈dy∣)=∫ P\! (Y^d(x)∈ dy )= _U P(Yd(x)∈dy∣,U=u)P(du∣). P\! (Y^d(x)∈ dy ,U=u )P(du ). (64) Under the DiscoSCM condition that the queried counterfactual noise is independent of the factual trace after fixing the unit, the first integrand no longer depends on O. Writing its fixed-unit counterfactual kernel as Ku,cfxK_u,cf^x, the decomposition becomes P(Yd(x)∈y∣)=∫Ku,cfx(y)P(u∣).P\! (Y^d(x)∈ dy )= _UK_u,cf^x(dy)P(du ). (65) This is the causal specialization of the identity-belief composition in Eq. 23. DiscoSCM abducts a posterior over identity: factual evidence updates P(du∣)P(du ); the causal model evaluates the counterfactual response law for each fixed u; and integration reduces the unit-specific valuations to a population-level answer. Gong et al. [15] call these stages abduction, valuation, and reduction. That identity posterior is one admissible token; it is not the definition of unit abduction in this paper. The default computational prediction remains Eq. 24. There are also two important differences from the generic learner formulation. First, Eq. 65 uses the exact world identity conditional P(du∣)P(du ), whereas this paper’s default abduction object is a formed token ZcZ^c, and Qϕ(du∣)Q_φ(du ) is reserved for a learner identity belief that need not be calibrated. Second, the kernel in DiscoSCM is interventional or counterfactual because the structural model supplies that semantics; the kernel in the general supervised formulation is observational unless additional causal assumptions are stated. The stability result in Lemma 3 applies to this causal specialization after replacing KuK_u by Ku,cfxK_u,cf^x, provided the conditional independence used in Eq. 65 is stated directly and the learned counterfactual kernel is measurable. It then separates counterfactual- kernel approximation error from unit-belief error, with the latter scaled by the diameter of the true fixed-unit counterfactual laws. This is a stability statement for a specified causal target; it neither supplies the cross-world coupling nor identifies the counterfactual kernel from observational data. D.4 Scope and Provenance of the Causal Specialization DiscoSCM is a causal theoretical specialization of the unit primitive. It declares the persistent unit, replaces a merely predictive response family with unit-conditioned structural causal mechanisms, and states how factual evidence and interventional or counterfactual queries interact with that object. The hierarchy is unit primitive primitive ⟶unit-conditioned structural mechanisms -conditioned structural mechanisms (66) ⟶DiscoSCM cross-world assumptions cross-world assumptions ⟶causal valuations and identification results. valuations and identification results. The first arrow specializes the learned object; the second adds distribution-consistency and a cross-world noise law; the third derives DiscoSCM-specific causal consequences. This ordering also fixes the attribution of results: the unit primitive supplies the persistent referent and the population-to-individual decomposition, whereas DiscoSCM supplies the causal mechanisms, intervention semantics, cross-world coupling assumptions, and the theorems that depend on them. Finally, unit abduction should not be conflated with the classical SCM abduction of a full exogenous-noise realization. In this paper, unit abduction forms a contextual unit token from factual evidence. That token may be an identity posterior, but it need not be. DiscoSCM’s abduction–valuation– reduction step uses the identity posterior as its mixing measure, and deliberately keeps that question separate from the coupling or resampling of event-level noise across worlds. The two “abduction” operations are therefore distinct even when DiscoSCM is read as a causal specialization of the unit primitive. Appendix E Additional Formal Results The following results give technical statements and proofs supporting the claims in the main text. They are consequences of retaining unit identity rather than additional primitives. Unless a display is written in token variables, the proofs use the identity-belief specialization Qϕ(du∣)Q_φ(du ) of Eq. 23; that is the formulation in which value–access, mixture stability, and single-row collapse are stated. It is not a claim that every tokenizer emits a law on U. E.1 Proof of the Value–Access Decomposition Proof of Proposition 1. The conditional log-score identity gives RW⋆−RU,W⋆ R_W -R_U,W =I(U;YQ∣W), =I(U;Y^Q W), (67) RW⋆−R,W⋆ R_W -R_O,W =I(;YQ∣W). =I(O;Y^Q W). (68) The chain rule expands the same conditional mutual information in two ways: I(U,;YQ∣W) I(U,O;Y^Q W) =I(U;YQ∣W)+I(;YQ∣U,W) =I(U;Y^Q W)+I(O;Y^Q U,W) (69) =I(;YQ∣W)+I(U;YQ∣,W). =I(O;Y^Q W)+I(U;Y^Q ,W). (70) Response sufficiency makes I(;YQ∣U,W)=0I(O;Y^Q U,W)=0, which proves Eq. 30. Nonnegativity of conditional mutual information gives the risk ladder. Response sufficiency also makes →U→YQO→ U→ Y^Q a conditional Markov chain given W. Conditional data processing gives I(;YQ∣W)≤I(U;∣W)I(O;Y^Q W)≤ I(U;O W), while Eq. 30 gives the other upper bound in Eq. 31. ∎ The mutual-information decomposition itself requires response sufficiency but not the external-query condition. The latter is needed when the exact response conditional is represented using the fixed belief P(U∈du∣)P(U∈ du ) rather than P(U∈du∣,W)P(U∈ du ,W). E.2 Proof of the End-to-End Approximation Result Assume the unit, evidence, query, and outcome spaces are standard Borel and use compatible regular-conditional versions. The finite-risk statement below uses conditional densities with respect to a common sigma-finite reference measure; the component bound may instead be read in the extended relative-entropy sense. Proof of Proposition 2. Conditioning on (,W)(O,W), the difference between the learner’s expected log loss and the Bayes log loss is E[logp(YQ∣,W)p^θ,ϕ(YQ∣;W)|,W],E\! [ p(Y^Q ,W) p_θ,φ(Y^Q ;W)\, |\,O,W ], (71) which is DKL(M,W∥M^,W)D_ KL(M_O,W\| M_O,W) under Eqs. 26 and 32. Averaging proves Eq. 35. Adding and subtracting R,W⋆R_O,W and applying Proposition 1 proves Eqs. 36 and 37. For the component bound, at fixed (o,w)(o,w) define latent-joint laws Jo,w(du,dy):=Po(du)Ku,w(dy),J^o,w(du,dy):=Qo(du)K^u,w(dy).J_o,w(du,dy):=P_o(du)K_u,w(dy), J_o,w(du,dy):=Q_o(du) K_u,w(dy). (72) The relative-entropy chain rule gives DKL(Jo,w∥J^o,w) D_ KL(J_o,w\| J_o,w) =DKL(Po∥Qo)+∫DKL(Ku,w∥K^u,w)Po(du). =D_ KL(P_o\|Q_o)+ D_ KL(K_u,w\| K_u,w)P_o(du). (73) The measurable projection (u,y)↦y(u,y) y sends Jo,wJ_o,w to Mo,wM_o,w and J^o,w J_o,w to M^o,w M_o,w. Data processing therefore gives DKL(Mo,w∥M^o,w)≤DKL(Jo,w∥J^o,w).D_ KL(M_o,w\| M_o,w)≤ D_ KL(J_o,w\| J_o,w). (74) Averaging Eqs. 73 and 74 over the deployment law proves Eq. 38. ∎ This upper bound is a marginal contraction of a latent-joint discrepancy. It need not be tight: distinct latent decompositions can have identical response marginals, as the next subsection makes explicit. E.3 Single-Row Marginal Collapse and Linked-Pair Separation For compatible versions of the regular conditional kernels, the law of total probability after conditioning on X=xX=x proves Eq. 40, with its PXP_X-almost-everywhere qualification. For any sample-only conditional law R(dy∣x)R(dy x), choosing KuR(dy∣x):=R(dy∣x)K_u^R(dy x):=R(dy x) for every u in an unrestricted response-kernel class returns R after integration against every learner belief. Thus marginal response fit alone does not select a formed token, a which-unit belief, or a fixed-unit response decomposition. Proof of Proposition 4. In world H, P(Y=1)=12(14+34)=12,P_ H(Y=1)= 12( 14+ 34)= 12, (75) which is also the response probability in world P. Independence of the fresh unit draws and their event realizations gives Eq. 43 for every n. Let a possibly randomized test φn _n declare H with probability φn(Y1,…,Yn) _n(Y_1,…,Y_n) and define αn=E[φn],βn=E[1−φn]. _n=E_ P[ _n], _n=E_ H[1- _n]. (76) Equality of the observable laws implies Eφn=EφnE_ H _n=E_ P _n, so αn+βn=1 _n+ _n=1 and maxαn,βn≥1/2 \ _n, _n\≥ 1/2. Equality is attained by randomizing equally between the two worlds. For a linked pair, conditional independence gives Cov(Y1,Y2)=Var(pU).Cov(Y_1,Y_2)=Var(p_U). (77) This variance is 1/161/16 in world H and zero in world P. More explicitly, their pair probabilities are 000110115/163/163/165/161/41/41/41/4. array[]c|c&00&01&10&11\\ H&5/16&3/16&3/16&5/16\\ P&1/4&1/4&1/4&1/4. array (78) If M linked pairs are sampled independently from independently drawn units, define a^M:=1M∑j=1MYj1=Yj2. a_M:= 1M _j=1^M1\Y_j1=Y_j2\. (79) The test that declares H when a^M>9/16 a_M>9/16 is consistent; Hoeffding’s inequality bounds each error by exp(−M/128) (-M/128). Finally, P(Y1=1)=E[pU]P(Y_1=1)=E[p_U] and P(Y1=1,Y2=1)=E[pU2]P(Y_1=1,Y_2=1)=E[p_U^2], proving Eq. 46. ∎ The full mixing law is not identified by these two moments without further restrictions. For example, 12δ1/4+12δ3/4and18δ0+34δ1/2+18δ1 12 _1/4+ 12 _3/4 18 _0+ 34 _1/2+ 18 _1 (80) are distinct laws for pUp_U with the same first moment 1/21/2 and second moment 5/165/16. Trusted same-unit linkage and the conditional product law are essential to this positive statement. Arbitrary within-unit dependence, linkage error, response-relevant temporal state, or informative observation can invalidate the displayed separation. The result identifies neither the full mixing law nor membership of a realized unit in either response class. It therefore does not contradict the need for structured mixture assumptions in classical positive identifiability results [43]. E.4 Attribution information and response dependence are independent For chosen regular-conditional versions, evidence is uninformative about the unit exactly when U⟂⟺P(U∈⋅∣)=Π(⋅)almost surely.U \!\!\! P(U∈· )= (·) surely. (81) This statement concerns the conditional law under the data-generating distribution. It constrains a formed token only insofar as that token is required to carry identity information, and it constrains a learner belief QϕQ_φ only when that belief is calibrated to the declared identity target. Independently, suppose the response family is unit-homogeneous at a declared query w: Kuw=KwK_u^w=K^w for Π -almost every u. Then every learner belief with Qϕ(⋅∣)≪ΠQ_φ(· ) satisfies ∫KuwQϕ(u∣)=Kw. _UK_u^w\,Q_φ(du )=K^w. (82) Neither condition implies the other. Informative attribution can coexist with a unit-homogeneous response, and unit-heterogeneous responses can coexist with uninformative evidence. Moreover, cancellation in one particular mixture does not by itself establish response homogeneity. These are two collapse conditions for two different learner formulations, distinct again from the single-row marginal collapse in Proposition 4. E.5 Proof and qualifications for mixture stability Fix o, w, and oU_o as in Lemma 3, and suppress w by writing Ku:=Ku,wK_u:=K_u,w and K^u:=K^u,w K_u:= K_u,w. Assume the unit and outcome spaces are standard Borel, K and K K are measurable Markov kernels on the chosen versions, and the displayed total-variation integrand is measurable. These conditions ensure that all mixtures and integrals below are well-defined. The common full-mass restriction ensures that replacing U by oU_o in the diameter does not discard mixture mass. Proof of Lemma 3. The triangle inequality gives dTV(QoK^,PoK) d_ TV(Q_o K,P_oK) ≤dTV(QoK^,QoK)+dTV(QoK,PoK), ≤ d_ TV(Q_o K,Q_oK)+d_ TV(Q_oK,P_oK), (83) where, for example, QoK=∫KuQo(u)Q_oK= _UK_uQ_o(du). For the first term, the definition of total variation and the triangle inequality for integrals yield dTV(QoK^,QoK) d_ TV(Q_o K,Q_oK) ≤∫dTV(K^u,Ku)Qo(u). ≤ _Ud_ TV( K_u,K_u)Q_o(du). (84) For the second term, let Δo=Qo−Po _o=Q_o-P_o and t=dTV(Qo,Po)t=d_ TV(Q_o,P_o). If t=0t=0 there is nothing to prove. Otherwise the Jordan decomposition of the zero-mass signed measure Δo _o can be written as Δo=t(α−β) _o=t(α-β) for probability measures α and β supported on oU_o. For any measurable outcome event A, |(αK)(A)−(βK)(A)| |(α K)(A)-(β K)(A) | =|∫[Ku(A)−Kv(A)]α(u)β(v)| = | \!\! [K_u(A)-K_v(A) ]α(du)β(dv) | (85) ≤supu,v∈odTV(Ku,Kv). ≤ _u,v _od_ TV(K_u,K_v). (86) Taking the supremum over A and multiplying by t gives dTV(QoK,PoK)≤[supu,v∈odTV(Ku,Kv)]dTV(Qo,Po).d_ TV(Q_oK,P_oK)≤ [ _u,v _od_ TV(K_u,K_v) ]d_ TV(Q_o,P_o). (87) Combining Eqs. 83, 84 and 87 proves Eq. 39. ∎ The diameter term is response-input-specific. It is zero exactly when the selected versions of the true response law agree across the units included in the supremum. If QoQ_o assigns mass outside the region on which the true kernel is scientifically specified, that is a support failure rather than a small attribution error; the proposition does not repair it. For unbounded losses, including unrestricted log loss, total-variation stability alone does not provide a finite excess-risk bound without additional boundedness or density conditions. E.6 Oracle, deployed, row-weighted, and unit-weighted risks If the true u is supplied to the response predictor, the population-first oracle risk separates which-individual and fixed-individual variation: RΠoracle(f)=∫[∫ℓ(y,f(x,u))Ku(x,y)]Π(u).R_ ^oracle(f)= _U [ \! (y,f(x;u) )K_u(dx,dy) ] (du). (88) The outer integral varies U; the inner integral retains exogenous/event variation at fixed u. This is a diagnostic for the response layer, not the deployed risk of a learner that knows only a formed token, or only QϕQ_φ in the identity-belief specialization. Let a declared deployment experiment generate factual evidence O, a response query XQX^Q, any explicit response context CQC^Q, and the target response YQY^Q. Under log loss, the default deployed risk of the pair (Tϕ,Rθ)(T_φ,R_θ) is Rdep(θ,ϕ)=EPdep[−log∫rθ(YQ∣XQ,CQ,z)Tϕ(z∣,CQ)],R_dep(θ,φ)=E_P_dep\! [- \! \ _Zr_θ(Y^Q X^Q,C^Q,z)\,T_φ(dz ,C^Q) \ ], (89) when a density rθr_θ of RθR_θ exists. In the identity-belief specialization used by the theorems, the same experiment scores Rdep(θ,ϕ)=EPdep[−log∫pθ(YQ∣XQ,CQ,u)Qϕ(u∣)].R_dep(θ,φ)=E_P_dep\! [- \! \ _Up_θ(Y^Q X^Q,C^Q,u)Q_φ(du ) \ ]. (90) Other proper scores or decision losses may replace log loss. The deployment experiment and evidence cutoff are part of the risk definition. Subpopulation-law quality, oracle fixed-u response quality, and deployed marginalized quality are different quantities; the last alone need not identify the first two. A random-row design may also induce a size-biased unit law, so its empirical risk need not estimate either declared population target. E.7 Repeated-unit weighting and splitting In an observed-attribution dataset, let M=||M=|U_D|. For each observed unit define I(u)=i:ui=u,m(u)=|I(u)|.I(u)=\i:u_i=u\, m(u)=|I(u)|. (91) For sample losses ℓi _i define the mean loss of unit u by ℓ¯(u)=m(u)−1∑i∈I(u)ℓi (u)=m(u)^-1 _i∈ I(u) _i. The row-average and unit-average risks are R^row=1N∑i=1Nℓi=∑u∈m(u)Nℓ¯(u),R^unit=1M∑u∈ℓ¯(u). R_row= 1N _i=1^N _i= _u _D m(u)N (u), R_unit= 1M _u _D (u). (92) Proof of Corollary 5. The two expressions are linear in (ℓ¯(u))u∈( (u))_u _D, so they agree for every such vector exactly when their coefficients agree for every u, that is, when m(u)/N=1/Mm(u)/N=1/M. ∎ Neither objective is universally correct; the deployment estimand determines which weighting is appropriate. Proof of Corollary 6. The two excluded events are that all m(u)m(u) records enter training and that all enter test; they are disjoint and have probabilities pm(u)p^m(u) and (1−p)m(u)(1-p)^m(u). ∎ A record-wise split can therefore mix new observations of known units with any new-unit cases. An unseen-unit generalization estimand uses a test construction that is explicitly unit-disjoint from training, whether by a whole-unit split, an external new-unit cohort, or another declared protocol. Appendix F Observability Conditions Component targets and training regimes. The world target Pu⋆P_u and the learner pair (Tϕ,Rθ)(T_φ,R_θ) remain distinct. Separate token supervision, oracle-attribution response training, and marginalized end-to-end training are different identification regimes. A which-unit belief has its own target only when the protocol asks it to approximate P(U∈du∣)P(U∈ du ). Under trusted direct access there is no identity model to estimate: learning the lookup row Zu(k)c=zθ(k)Z_u(k)^c=z_θ(k) fits a token inside RθR_θ, not QϕQ_φ. Any history-derived state used at answer time must appear in the tokenizer context, in cQc^Q, or inside the shared form; otherwise response sufficiency is not justified. In Section 5 the complete response input is written W, not Z, so that it is not confused with the unit token ZucZ_u^c. Support and observational equivalence. The population law may support an individual value u⋆u even when no training record is attributed to it. Support alone does not concentrate P(U∈du∣)P(U∈ du ) at u⋆u ; the available event information may instead imply a diffuse token, a population fallback, or abstention. Likewise, two distinct individual values may induce the same response law for every query and context admitted by a study. They are observationally indistinguishable under that study while remaining distinct units. Fixed evidence and shared-unit bundles. The task declaration specifies the unit boundary, the events linked to one fixed individual, the answer-time evidence cutoff, and the query family. Within a declared same-unit query family Q, the same u and the same factual evidence O are retained while xQ∈x^Q∈ Q varies. A new factual event may update the formed token, whereas comparing alternative response queries uses the token formed at the fixed evidence cutoff. Shared attribution of several rows to one unit does not by itself imply that those rows are conditionally independent given U. The fixed-individual kernel retains exogenous variation. The kernel Pu⋆(dy∣xQ,cQ)P_u (dy x^Q,c^Q) may remain stochastic after U=uU=u is fixed. Residual within-unit variation is not which-unit variation. Appendix G Evaluation Checklist A reproducible evaluation should pre-specify: 1. the unit population and the span over which unit identity persists; 2. the factual-evidence cutoff, response query and context, and current target; 3. the access regime—direct access or unit abduction—and what attribution truth is available for evaluation; 4. the evaluated target: fixed-unit response, deployed marginalized prediction, or a unit-level estimand; 5. whether the split targets known-unit/new-event or new-unit generalization; 6. a matched unit-omitting baseline and any negative control required by the claim. A learnable unit-conditioned specification additionally names: 1. what is shared across units, here the form RθR_θ, a simple relation in the token, with a linear predictor as the running instance (Definition 2); 2. which properties of the query–response relation vary with the unit, and are therefore expressed by ZucZ_u^c; 3. which properties remain invariant across units; 4. how finitely many same-unit observations constrain that unit’s token and hence its response law; 5. how the shared form and token space support prediction for a unit not previously observed; and 6. how evaluation distinguishes a learned unit-dependent relationship from memorization of an identifier. A product likelihood adds a conditional-factorization assumption. The response law is observational by default; an interventional reading requires assignment and identification assumptions. Pre-answer evidence excludes current targets and unavailable post-query measurements.