Toward Machine Learning with the Unit as a Primitive: Learning from Unit-Linked Events

# Toward Machine Learning with the Unit as a Primitive:Learning from Unit-Linked Events
Source: [https://arxiv.org/html/2608.25118](https://arxiv.org/html/2608.25118)
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 valueuudenotes 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\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\): 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*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\.

## 1Introduction

The standard supervised\-learning formulation starts from records𝒟=\{\(xi,yi\)\}i=1N\\mathcal\{D\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{N\}and a mapf:𝒳→𝒴f:\\mathcal\{X\}\\to\\mathcal\{Y\}\[[4](https://arxiv.org/html/2608.25118#bib.bib5),[17](https://arxiv.org/html/2608.25118#bib.bib6),[38](https://arxiv.org/html/2608.25118#bib.bib7)\]\. The indexiiidentifies 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\[[25](https://arxiv.org/html/2608.25118#bib.bib2),[13](https://arxiv.org/html/2608.25118#bib.bib16)\]\. Potential\-outcome frameworks compare alternative treatment responses for the same experimental unit\[[35](https://arxiv.org/html/2608.25118#bib.bib38)\]\. Recommender systems use supplied user IDs to address user\-specific representations learned across interactions\[[28](https://arxiv.org/html/2608.25118#bib.bib26),[24](https://arxiv.org/html/2608.25118#bib.bib28)\], while record linkage and entity resolution model uncertainty about whether records concern the same underlying entity\[[41](https://arxiv.org/html/2608.25118#bib.bib34)\]\. 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,U∼Π,U=u,u∈𝒰,U:\\Omega\\longrightarrow\\mathcal\{U\},\\qquad U\\sim\\Pi,\\qquad U=u,\\quad u\\in\\mathcal\{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 realizationU=uU=udenotes that referent\. The lawΠ\\Pidescribes 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\\displaystyle u⟼\[x↦Pu⋆\(dy∣x\)\],\\displaystyle\\longmapsto\\bigl\[\\,x\\mapsto P\_\{u\}^\{\\star\}\(dy\\mid x\)\\,\\bigr\],\(2\)Pu⋆​\(d​y∣x\)\\displaystyle P\_\{u\}^\{\\star\}\(dy\\mid x\):=P⋆​\(Y∈d​y∣X=x,U=u\)\.\\displaystyle:=P^\{\\star\}\(Y\\in dy\\mid X=x,U=u\)\.Changingxxqueries another location on the same response surface; changinguuselects 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[Section3\.2](https://arxiv.org/html/2608.25118#S3.SS2): a tokenizer produces a contextual unit token, and one shared response\-law form reads it\.

Task declaration:what counts as one unitPopulation statement:U∼Π⟶U=uU\\sim\\Pi\\ \\longrightarrow\\ U=u\(a\) Events share persistent unit identityevent 1\(x1,y1\)\(x\_\{1\},y\_\{1\}\)event 3\(x3,y3\)\(x\_\{3\},y\_\{3\}\)unituAu\_\{A\}event 2\(x2,y2\)\(x\_\{2\},y\_\{2\}\)unituBu\_\{B\}\(b\) The same unit selects a response laweventii\(xi,yi\)\(x\_\{i\},y\_\{i\}\)unituAu\_\{A\}response lawPuA⋆​\(d​y∣x\)P^\{\\star\}\_\{u\_\{A\}\}\(dy\\mid x\)uAu\_\{A\}selects the law, not the eventFigure 1:From the unit declaration to linked events and response laws\.The task specifies the unit boundary and sameness criterion;U∼ΠU\\sim\\Piis the population statement andU=uU=uits realized unit\. Several event records may be attributed to the sameuu, and that same referent indexes a world\-side response law\. Holdinguufixed preserves the referent while allowing context and event variation\.The paper makes three contributions\.

1. 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. 2\.It states a shared\-form interface \([assumption1](https://arxiv.org/html/2608.25118#Thmassumption1)\) whose structured class is a simple relation in the token, with a linear predictor as the running instance \([Definition2](https://arxiv.org/html/2608.25118#Thmdefinition2)\); the learned object is the pair\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\)\.
3. 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\.

## 2The Unit as a Machine\-Learning Primitive

### 2\.1Samples, Events, and Task\-Declared Units

For the supervised specialization developed below, a conceptual complete\-data representation with ground\-truth attribution is

𝒟Uworld=\{\(xi,yi,ui\)\}i=1N,ui∈𝒰\.\\mathcal\{D\}\_\{U\}^\{\\rm world\}=\\\{\(x\_\{i\},y\_\{i\};u\_\{i\}\)\\\}\_\{i=1\}^\{N\},\\qquad u\_\{i\}\\in\\mathcal\{U\}\.\(3\)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 anduiu\_\{i\}, which identifies the individual to which the event is attributed\. The sample index remains only bookkeeping\. In particular,

ui=uj⟺eventsiandjconcern the same individual,u\_\{i\}=u\_\{j\}\\quad\\Longleftrightarrow\\quad\\text\{events $i$ 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 space𝒰\\mathcal\{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\.\\underbrace\{k\}\_\{\\text\{trusted key\}\},\\qquad\\underbrace\{u\(k\)\}\_\{\\text\{task\-side unit\}\},\\qquad\\underbrace\{Z\_\{u\(k\)\}^\{c\}\}\_\{\\text\{learner\-side lookup token\}\}\.\(5\)The mapk↦u⁡\(k\)k\\mapsto u\(k\)resolves the referent, and lookup supplies the token through whichRθR\_\{\\theta\}reads it in contextcc\. In an ID\-indexed model

Zu⁡\(k\)c=zθ​\(k\),Z\_\{u\(k\)\}^\{c\}\\;=\\;z\_\{\\theta\}\(k\),\(6\)an embedding, preference factor, random effect, or other parameter block, with any remaining answer\-time variation absorbed intocc\. Its coordinates are model\-internal and need not be unique; the semantic and inferential distinctions are summarized in[AppendixB](https://arxiv.org/html/2608.25118#A2)\[[3](https://arxiv.org/html/2608.25118#bib.bib9)\]\.

What is carried consistently from attribution into task\-specific learning is the resolved referent\. All records attributed tou⁡\(k\)u\(k\)read or update the token associated with that same key\. Accordingly,Pθ​\(d​y∣x,u\)P\_\{\\theta\}\(dy\\mid x,u\)semantically denotes a learned response law indexed by a fixed unit; computationally it is the shared formRθR\_\{\\theta\}evaluated at that unit’s tokenZucZ\_\{u\}^\{c\}\. In an ID\-indexed model it may be computed fromzθ​\(k\)z\_\{\\theta\}\(k\)\. 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

𝒟obs=\{\(xi,yi\)\}i=1N\.\\mathcal\{D\}^\{\\rm obs\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{N\}\.\(7\)The complete\-data representation in[Eq\.3](https://arxiv.org/html/2608.25118#S2.E3)is therefore distinct from observations with learner\-visible attribution,

𝒟Uobs=\{\(xi,yi,ui\)\}i=1N,\\mathcal\{D\}\_\{U\}^\{\\rm 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\.2Fixing the unit does not fix the event

In supervised learning, the fixed\-unit response lawPu⋆​\(d​y∣x\)P\_\{u\}^\{\\star\}\(dy\\mid x\)from[Eq\.2](https://arxiv.org/html/2608.25118#S1.E2)may remain stochastic\. Holdinguufixed 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](https://arxiv.org/html/2608.25118#S1.F1)\. 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 shareuAu\_\{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[AppendixD](https://arxiv.org/html/2608.25118#A4)\. Direct unit access and unit abduction provide two learner\-side modes for obtaining the same task\-declared unit \([Section4](https://arxiv.org/html/2608.25118#S4)\)\.

## 3Learning a Family of Unit\-Conditioned Response Laws

This paper develops supervised learning as the main formal specialization of the unit primitive because the ordinary objectP⁡\(d​y∣x\)P\(dy\\mid x\)makes the consequences of a persistent unit especially transparent\. Attribution is treated separately in[Section4](https://arxiv.org/html/2608.25118#S4)\. Once the unit is declared, the semantic object is the family\{Pu⋆\(dy∣⋅\):u∈𝒰\}\\\{P\_\{u\}^\{\\star\}\(dy\\mid\\cdot\):u\\in\\mathcal\{U\}\\\}in[Eq\.2](https://arxiv.org/html/2608.25118#S1.E2)\. 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 formRθR\_\{\\theta\}, and every unit\-specific difference represented by the learner must pass through the token\. The supervised question is therefore not how a particular architecture encodesuu, but whether a tokenizer can organize unit\-specific regularities so that one sharedRθR\_\{\\theta\}becomes simple enough to generalize\. Conceptually, a fixeduuselects an entire response surface over the complete task\-declared response inputxx\. Changingxxmoves to another location on that same surface whileuuremains fixed\. Across units, the dependence may enter through a baseline shiftα⁡\(u\)\\alpha\(u\), a changed input effectβ⁡\(u\)\\beta\(u\), a response functionfuf\_\{u\}, an outcome\-noise law, or the support, threshold, ranking preference, or query sensitivity of the response\. Under[assumption1](https://arxiv.org/html/2608.25118#Thmassumption1), 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 readoutRθR\_\{\\theta\}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 andPu⋆​\(d​y∣x\)P\_\{u\}^\{\\star\}\(dy\\mid x\)is the same for almost everyuu, the world is a*homogeneous unit extension*of ordinary supervised learning\. Separately, a heterogeneous family can be marginalized overUUto produce the same row\-level conditional lawP⋆​\(d​y∣x\)P^\{\\star\}\(dy\\mid x\)\. 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\.[Proposition4](https://arxiv.org/html/2608.25118#Thmproposition4)makes the ambiguity formal and gives one observation protocol that can separate a restricted pair of worlds; the boundary cases are collected in[AppendixB](https://arxiv.org/html/2608.25118#A2)\.

### 3\.1Response 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 aΠ\\Pi\-null setN⊆𝒰N\\subseteq\\mathcal\{U\}and a common conditional response kernel such that, for everyu∉Nu\\notin Nand every declared inputxx,

Pu⋆​\(d​y∣x\)=P⋆​\(d​y∣x\)\.P\_\{u\}^\{\\star\}\(dy\\mid x\)=P^\{\\star\}\(dy\\mid x\)\.\(9\)Here the right\-hand side denotes the common response kernel shared across units; it is not defined by marginalizing overUU\. For compatible conditional versions, however, homogeneity makes this kernel equal to the unit\-marginal response lawP⋆​\(Y∈d​y∣X=x\)P^\{\\star\}\(Y\\in dy\\mid X=x\)forPXP\_\{X\}\-almost everyxx\. Thus the two uses ofP⋆​\(d​y∣x\)P^\{\\star\}\(dy\\mid x\)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[Definition3](https://arxiv.org/html/2608.25118#Thmdefinition3)\.

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\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\), and a saturated private\-law class—are developed in[Section3\.2](https://arxiv.org/html/2608.25118#S3.SS2)\.

### 3\.2Shared 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 mechanismRuR\_\{u\}for each unit\. That class is too flexible\.

When the response specification distinguishes a focal queryxxfrom an answer\-time contextcc, writePu⋆​\(d​y∣x,c\)P\_\{u\}^\{\\star\}\(dy\\mid x,c\)for the complete conditional;ccmay otherwise be absorbed into a largerxx\. Given factual evidence attached to a unit and the current context, a tokenizer produces a contextual unit token

Zuc=Tϕ​\(𝒪u,c\)∈𝒵\.Z\_\{u\}^\{c\}=T\_\{\\phi\}\(\\mathcal\{O\}\_\{u\},c\)\\in\\mathcal\{Z\}\.\(10\)
###### 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 unituuis the task\-side referent andZucZ\_\{u\}^\{c\}is a learner\-side representation\.[Assumption1](https://arxiv.org/html/2608.25118#Thmassumption1)is an interface\. The structured class of this paper requires a simple shared relation in the token: unit\-specific differences enterRθR\_\{\\theta\}only through a declared low\-complexity map ofzz, while the query maps may remain nonlinear\. A finite\-dimensional linear predictor \([Definition2](https://arxiv.org/html/2608.25118#Thmdefinition2)\) 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[AppendixB](https://arxiv.org/html/2608.25118#A2)\. The objectZucZ\_\{u\}^\{c\}need not be a deterministic vector\. A deterministic tokenizer may return a vector, a distribution, or another typed computational object; a fixed embeddingzθ​\(k\)z\_\{\\theta\}\(k\)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, sayZuc=μucZ\_\{u\}^\{c\}=\\mu\_\{u\}^\{c\}, is passed as one typed object toRθ​\(d​y∣x,c,μuc\)R\_\{\\theta\}\(dy\\mid x,c,\\mu\_\{u\}^\{c\}\); it is not the sampling randomness marginalized in[Eq\.14](https://arxiv.org/html/2608.25118#S3.E14)\. 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\}\\neq Z\_\{u\}^\{c^\{\\prime\}\}\.\(11\)Distinct units may receive the same token, or the same token law, in a given context,

u≠v,Zuc​=𝑑​Zvc\.u\\neq v,\\qquad Z\_\{u\}^\{c\}\\overset\{d\}\{=\}Z\_\{v\}^\{c\}\.\(12\)Token coincidence is computational indistinguishability under the current query scope\. It does not identify the units: referential identity remainsu≠vu\\neq v\.

For a deterministic token, the learner uses one shared response mechanism

P^θ,ϕ​\(d​y∣𝒪u;x,c\):=Rθ​\(d​y∣x,c,Zuc\)\.\\boxed\{\\widehat\{P\}\_\{\\theta,\\phi\}\(dy\\mid\\mathcal\{O\}\_\{u\};x,c\):=R\_\{\\theta\}\(dy\\mid x,c,Z\_\{u\}^\{c\}\)\.\}\(13\)If the tokenizer is a stochastic kernel over token values, the learner instead uses the integral

P^θ,ϕ​\(𝑑y∣𝒪u;x,c\):=∫Rθ​\(𝑑y∣x,c,z\)​Tϕ​\(𝑑z∣𝒪u,c\)\.\\widehat\{P\}\_\{\\theta,\\phi\}\(dy\\mid\\mathcal\{O\}\_\{u\};x,c\):=\\int R\_\{\\theta\}\(dy\\mid x,c,z\)\\,T\_\{\\phi\}\(dz\\mid\\mathcal\{O\}\_\{u\},c\)\.\(14\)The essential point is thatRθR\_\{\\theta\}is the same learned map for every unit; neither display defines the world targetPu⋆P\_\{u\}^\{\\star\}\.

###### Definition 2\(Linear shared form\)\.

Fix a finite dimensionddand let tokens take values inℝd\\mathbb\{R\}^\{d\}\. A shared formRθR\_\{\\theta\}is*linear in the token*when there exist mapsαθ​\(x,c\)∈ℝ\\alpha\_\{\\theta\}\(x,c\)\\in\\mathbb\{R\}andψθ​\(x,c\)∈ℝd\\psi\_\{\\theta\}\(x,c\)\\in\\mathbb\{R\}^\{d\}such that

ηθ​\(x,c,z\)=αθ​\(x,c\)\+⟨ψθ​\(x,c\),z⟩\\eta\_\{\\theta\}\(x,c,z\)=\\alpha\_\{\\theta\}\(x,c\)\+\\langle\\psi\_\{\\theta\}\(x,c\),z\\rangle\(15\)andRθ​\(d​y∣x,c,z\)R\_\{\\theta\}\(dy\\mid x,c,z\)depends onzzonly through this linear predictor, via a declared exponential\-family, GLM, or scoring link\. The mapsαθ\\alpha\_\{\\theta\}andψθ\\psi\_\{\\theta\}may be nonlinear in\(x,c\)\(x,c\)\. Finite\-type models arise when tokens are further restricted to a finite set or simplex inℝd\\mathbb\{R\}^\{d\}; a low\-rank model is the same class at smalldd\.

[Assumption1](https://arxiv.org/html/2608.25118#Thmassumption1)remains the interface\. The structured class is a restriction on howRθR\_\{\\theta\}reads the token, not a private mechanism per unit:RθR\_\{\\theta\}depends onzzonly through a declared simple relation, whileTϕT\_\{\\phi\}may remain complex\.[Definition2](https://arxiv.org/html/2608.25118#Thmdefinition2)is the running instance\. A finite\-type token or another declared low\-complexity map ofzzinstantiates the same class; they are not developed here\. Misspecification is residual unit\-specific variation that the declared simple relation cannot absorb\. Without a bound ondd, or with an unrestricted nonlinear readout ofzz, 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 howαθ\\alpha\_\{\\theta\}andψθ\\psi\_\{\\theta\}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 inℝd\\mathbb\{R\}^\{d\}, for example a mean or a finite coefficient summary\.

When identity is unresolved, under response sufficiencyY⟂⟂𝒪\|\(U,X,C\)Y\\perp\\\!\\\!\\\!\\perp\\mathcal\{O\}\\mid\(U,X,C\)and the external\-query conditionU⟂⟂\(X,C\)\|𝒪U\\perp\\\!\\\!\\\!\\perp\(X,C\)\\mid\\mathcal\{O\}, the world\-side target and the learner\-side token composition are

P⋆​\(d​y∣x,c,𝒪\)=∫𝒰Pu⋆​\(dy∣x,c\)​P​\(du∣𝒪\),world target,P^θ,ϕ​\(d​y∣𝒪;x,c\):=∫𝒵Rθ​\(dy∣x,c,z\)​Tϕ​\(dz∣𝒪,c\),learner composition\.\\boxed\{\\begin\{aligned\} P^\{\\star\}\(dy\\mid x,c,\\mathcal\{O\}\)&=\\int\_\{\\mathcal\{U\}\}P\_\{u\}^\{\\star\}\(dy\\mid x,c\)\\,P\(du\\mid\\mathcal\{O\}\),&&\\text\{world target\},\\\\ \\widehat\{P\}\_\{\\theta,\\phi\}\(dy\\mid\\mathcal\{O\};x,c\)&:=\\int\_\{\\mathcal\{Z\}\}R\_\{\\theta\}\(dy\\mid x,c,z\)\\,T\_\{\\phi\}\(dz\\mid\\mathcal\{O\},c\),&&\\text\{learner composition\}\.\\end\{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 fixesu⁡\(k\)u\(k\), the world identity conditional is the Dirac lawP⁡\(d​u∣k\)=δu⁡\(k\)P\(du\\mid k\)=\\delta\_\{u\(k\)\}, while the learner separately reads the lookup tokenZu⁡\(k\)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\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\), not a collection of unrelated unit\-specific mechanisms\. Heterogeneity is thereby organized in a common token space:

𝒪u→TϕZuc→Rθ​\(⋅,x,c\)P^θ,ϕ\(⋅∣𝒪u;x,c\)\.\\mathcal\{O\}\_\{u\}\\xrightarrow\{T\_\{\\phi\}\}Z\_\{u\}^\{c\}\\xrightarrow\{R\_\{\\theta\}\(\\,\\cdot\\,,x,c\)\}\\widehat\{P\}\_\{\\theta,\\phi\}\(\\,\\cdot\\mid\\mathcal\{O\}\_\{u\};x,c\)\.\(17\)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 isψθ≡0\\psi\_\{\\theta\}\\equiv 0, or any token that does not enterηθ\\eta\_\{\\theta\}\. That unit\-omitting specification is the computational restriction

P^θ,ϕ​\(d​y∣𝒪u;x,c\)=P^θ​\(d​y∣x,c\)for all​u,\\widehat\{P\}\_\{\\theta,\\phi\}\(dy\\mid\\mathcal\{O\}\_\{u\};x,c\)=\\widehat\{P\}\_\{\\theta\}\(dy\\mid x,c\)\\qquad\\text\{for all \}u,\(18\)which has two canonical sufficient realizations under[assumption1](https://arxiv.org/html/2608.25118#Thmassumption1):

Zuc=z0,Z\_\{u\}^\{c\}=z\_\{0\},\(19\)or

Rθ​\(d​y∣x,c,z\)=Rθ​\(d​y∣x,c\)\.R\_\{\\theta\}\(dy\\mid x,c,z\)=R\_\{\\theta\}\(dy\\mid x,c\)\.\(20\)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\}^\{\\star\}\\\}, 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 \([Section4](https://arxiv.org/html/2608.25118#S4)\)\. Both access modes still train the same pair\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\)\.

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 \([AppendixG](https://arxiv.org/html/2608.25118#A7)\)\.[Assumption1](https://arxiv.org/html/2608.25118#Thmassumption1)answers the first three by requiring a sharedRθR\_\{\\theta\}and a contextual token; the structured class further requires thatRθR\_\{\\theta\}depend on the token only through a declared simple relation, with[Definition2](https://arxiv.org/html/2608.25118#Thmdefinition2)as the running instance\.

#### Structure as the condition of learnability\.

The pair\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\)is the computational expression of a dual restriction\. The response mapx↦Pu⋆x\\mapsto P\_\{u\}^\{\\star\}needs structure even for a fixed unit, and the unit axisu↦Pu⋆u\\mapsto P\_\{u\}^\{\\star\}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 ofZucZ\_\{u\}^\{c\}\. The linear running instance uses the inner product in[Eq\.15](https://arxiv.org/html/2608.25118#S3.E15); 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[AppendixB](https://arxiv.org/html/2608.25118#A2)\.

\(a\) Unit\-insensitive learneruAu\_\{A\}uBu\_\{B\}uCu\_\{C\}common tokenz0z\_\{0\}sharedRθR\_\{\\theta\}\(b\) Shared formRθR\_\{\\theta\}tokenizerTϕT\_\{\\phi\}uAu\_\{A\}ZAcZ\_\{A\}^\{c\}uBu\_\{B\}ZBcZ\_\{B\}^\{c\}uCu\_\{C\}ZCcZ\_\{C\}^\{c\}sharedRθR\_\{\\theta\}\(c\) 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 tokenz0z\_\{0\}, or the shared form ignores the token\. \(b\) Structured heterogeneity: a tokenizer forms contextual unit tokens, and all units share one response\-law formRθR\_\{\\theta\}, a simple relation in the token \(linear running instance\)\. \(c\) A saturated class assigns an unrelated law to each unit\.

## 4Learner Access to the Unit

With the unit boundary, response family, and shared form\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\)fixed \([Section3\.2](https://arxiv.org/html/2608.25118#S3.SS2.SSS0.Px1)\), 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 keykkto determine which persistent unit the event concerns and which stable address should be read\. A separate lookup at that address supplies the tokenZu⁡\(k\)cZ\_\{u\(k\)\}^\{c\}\. With an ID\-indexed model this is the learned rowzθ​\(k\)z\_\{\\theta\}\(k\)—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 toRθR\_\{\\theta\}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\. Let𝒪\\mathcal\{O\}denote the factual evidence available before the current answer\. The access step forms a contextual unit token

Zc=Tϕ​\(𝒪,c\)Z^\{c\}=T\_\{\\phi\}\(\\mathcal\{O\},c\)\(21\)of the same type as a lookup token\. Alternative queries then use that token throughRθR\_\{\\theta\}\. This yields the learner’s token\-space composition in[Eq\.16](https://arxiv.org/html/2608.25118#S3.E16)\.

The theoretically exact predictive law remains the mixture over units\. Under the data\-generating distribution,P⁡\(U∈d​u∣𝒪\)P\(U\\in du\\mid\\mathcal\{O\}\)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​\(d​u∣𝒪\)\.Q\_\{\\phi\}\(du\\mid\\mathcal\{O\}\)\.\(22\)When the learning protocol explicitly targets the identity posterior,QϕQ\_\{\\phi\}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](https://arxiv.org/html/2608.25118#S4.E24)gives the corresponding prediction\.

#### Unit information rather than a mandated belief\.

The formed tokenZcZ^\{c\}is not the unit and is not required to be a probability over𝒰\\mathcal\{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\[[4](https://arxiv.org/html/2608.25118#bib.bib5),[29](https://arxiv.org/html/2608.25118#bib.bib8)\]\. 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[Section7](https://arxiv.org/html/2608.25118#S7)\.

The unit boundary is declared before learning and evaluation\. Within a declared same\-unit query family, the same realized valueuuis 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 tokentrusted keykktrustedresolverresolved referentu⁡\(k\)u\(k\)stable addresslookup tokenZu⁡\(k\)cZ\_\{u\(k\)\}^\{c\}zθ​\(k\)z\_\{\\theta\}\(k\)lookup\(b\) Unit abduction — formed tokenfactual evidence𝒪\\mathcal\{O\}tokenizerTϕT\_\{\\phi\}formed tokenZcZ^\{c\}Shared response formqueriesx1Q,x2Q,x3Qx^\{Q\}\_\{1\},\\,x^\{Q\}\_\{2\},\\,x^\{Q\}\_\{3\}shared formRθR\_\{\\theta\}Rθ​\(d​y∣xQ,c,Zc\)R\_\{\\theta\}\(dy\\mid x^\{Q\},c,Z^\{c\}\)direct:Zu⁡\(k\)c=zθ​\(k\)Z\_\{u\(k\)\}^\{c\}=z\_\{\\theta\}\(k\)abduction: formed tokenZcZ^\{c\}Figure 3:Direct unit access and unit abduction return tokens of the same type\.A trusted resolver uses a keykkto fix the persistent referent and its stable address; lookup then suppliesZu⁡\(k\)cZ\_\{u\(k\)\}^\{c\}, typically the learned rowzθ​\(k\)z\_\{\\theta\}\(k\), without forming a which\-unit belief\. When no resolver identifies the unit, a tokenizer forms a contextual tokenZcZ^\{c\}from factual evidence\. A which\-unit beliefQϕQ\_\{\\phi\}may approximate the world\-side identity conditional \([AppendixB](https://arxiv.org/html/2608.25118#A2)\)\. Both access modes use the same learned response model, with the token mode using the shared formRθR\_\{\\theta\}\.In the supervised specialization, an identity\-mixture learner mirrors the world\-side unit mixture in[Eq\.16](https://arxiv.org/html/2608.25118#S3.E16)by reusing one learned identity mixing measure through an identity\-indexed response model:

P^θ,ϕ​\(𝑑y∣𝒪;xQ\):=∫𝒰Pθ​\(𝑑y∣xQ,u\)​Qϕ​\(𝑑u∣𝒪\)\.\\boxed\{\\widehat\{P\}\_\{\\theta,\\phi\}\(dy\\mid\\mathcal\{O\};\\,x^\{Q\}\):=\\int\_\{\\mathcal\{U\}\}P\_\{\\theta\}\(dy\\mid x^\{Q\},u\)Q\_\{\\phi\}\(du\\mid\\mathcal\{O\}\)\.\}\(23\)This identity\-mixture form is used in the theorems below\. A different learner realization is the token\-space form

P^θ,ϕ​\(𝑑y∣𝒪;xQ,c\):=∫Rθ​\(𝑑y∣xQ,c,z\)​Tϕ​\(𝑑z∣𝒪,c\)\.\\widehat\{P\}\_\{\\theta,\\phi\}\(dy\\mid\\mathcal\{O\};\\,x^\{Q\},c\):=\\int R\_\{\\theta\}\(dy\\mid x^\{Q\},c,z\)\\,T\_\{\\phi\}\(dz\\mid\\mathcal\{O\},c\)\.\(24\)The two integrals coincide with each other and with the exact world mixture only under the conditions of[Eq\.16](https://arxiv.org/html/2608.25118#S3.E16)\. The semicolon distinguishes the supplied response queryxQx^\{Q\}from abductive evidence\. The displayed learner prediction is always well\-defined when its components are\. To identify the corresponding exact fixed\-belief mixture withP⁡\(YQ∈d​y∣𝒪,xQ\)P\(Y^\{Q\}\\in dy\\mid\\mathcal\{O\},x^\{Q\}\), we use two separate conditions: response sufficiency,YQ⟂⟂𝒪\|\(U,xQ\)Y^\{Q\}\\perp\\\!\\\!\\\!\\perp\\mathcal\{O\}\\mid\(U,x^\{Q\}\), and the external\-query condition,U⟂⟂XQ\|𝒪U\\perp\\\!\\\!\\\!\\perp X^\{Q\}\\mid\\mathcal\{O\}\(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 toUU\. If a recorded variable is informative about the unit, the protocol places that factual information in𝒪\\mathcal\{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 callcQc^\{Q\}*response context*\. It may otherwise be absorbed into a largerxQx^\{Q\}\. Attribution evidence remains in𝒪\\mathcal\{O\}, which excludes the current prediction target\.

#### The uninformative limit\.

IfU⟂⟂𝒪U\\perp\\\!\\\!\\\!\\perp\\mathcal\{O\}, thenP⁡\(U∈d​u∣𝒪\)=Π⁡\(d​u\)P\(U\\in du\\mid\\mathcal\{O\}\)=\\Pi\(du\)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 aboutUU\. Response heterogeneity asks whether changingUUchanges 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\.Recommendation with a trusted resolver occupies the resolved\-identity, heterogeneous cell; the lookup tokenzθ​\(k\)z\_\{\\theta\}\(k\)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[SectionE\.4](https://arxiv.org/html/2608.25118#A5.SS4)\.

## 5Unit 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](https://arxiv.org/html/2608.25118#S4.E23)\.

The central theoretical question is not merely whether a unit\-conditioned model can be written\. Under[assumption1](https://arxiv.org/html/2608.25118#Thmassumption1)the learned object is the pair\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\)\. 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\.\\boxed\{\\text\{oracle value\}\\neq\\text\{predictive accessibility\}\\neq\\text\{learned\-pair realization\}\\neq\\text\{component 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\[[7](https://arxiv.org/html/2608.25118#bib.bib43),[14](https://arxiv.org/html/2608.25118#bib.bib15)\]\. They organize the four layers; learnability and component identification require the additional conditions listed in the appendix\.

### 5\.1Oracle 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? WriteWWfor the complete declared response input, withW=XQW=X^\{Q\}in the base response specification andW=\(XQ,CQ\)W=\(X^\{Q\},C^\{Q\}\)when optional response context is exposed\. The letterWWis not a unit tokenZucZ\_\{u\}^\{c\}\. Consider a declared deployment law for\(U,𝒪,W,YQ\)\(U,\\mathcal\{O\},W,Y^\{Q\}\)and assume response sufficiency,

YQ⟂⟂𝒪\|\(U,W\)\.Y^\{Q\}\\perp\\\!\\\!\\\!\\perp\\mathcal\{O\}\\mid\(U,W\)\.\(26\)When the relevant conditional densities exist with respect to a common outcome reference measure and the log\-loss risks are finite, write

RW⋆\\displaystyle R\_\{W\}^\{\\star\}:=E⁡\[−log⁡p⁡\(YQ∣W\)\],\\displaystyle:=E\[\-\\log p\(Y^\{Q\}\\mid W\)\],R𝒪,W⋆\\displaystyle R\_\{\\mathcal\{O\},W\}^\{\\star\}:=E⁡\[−log⁡p⁡\(YQ∣𝒪,W\)\],\\displaystyle:=E\[\-\\log p\(Y^\{Q\}\\mid\\mathcal\{O\},W\)\],RU,W⋆\\displaystyle R\_\{U,W\}^\{\\star\}:=E⁡\[−log⁡p⁡\(YQ∣U,W\)\]\.\\displaystyle:=E\[\-\\log p\(Y^\{Q\}\\mid U,W\)\]\.\(27\)
###### Proposition 1\(Value–access decomposition under log loss\)\.

Under[Eq\.26](https://arxiv.org/html/2608.25118#S5.E26),

RU,W⋆≤R𝒪,W⋆≤RW⋆,R\_\{U,W\}^\{\\star\}\\leq R\_\{\\mathcal\{O\},W\}^\{\\star\}\\leq R\_\{W\}^\{\\star\},\(28\)and the oracle value decomposes exactly as

RW⋆−RU,W⋆⏟Voracle\\displaystyle\\underbrace\{R\_\{W\}^\{\\star\}\-R\_\{U,W\}^\{\\star\}\}\_\{V\_\{\\rm oracle\}\}=RW⋆−R𝒪,W⋆⏟Vaccessible\+R𝒪,W⋆−RU,W⋆⏟Vresidual\\displaystyle=\\underbrace\{R\_\{W\}^\{\\star\}\-R\_\{\\mathcal\{O\},W\}^\{\\star\}\}\_\{V\_\{\\rm accessible\}\}\+\\underbrace\{R\_\{\\mathcal\{O\},W\}^\{\\star\}\-R\_\{U,W\}^\{\\star\}\}\_\{V\_\{\\rm residual\}\}=I⁡\(U;YQ∣W\)\\displaystyle=I\(U;Y^\{Q\}\\mid W\)\(29\)=I\(𝒪;YQ∣W\)\+I\(U;YQ∣𝒪,W\)\.\\displaystyle=I\(\\mathcal\{O\};Y^\{Q\}\\mid W\)\+I\(U;Y^\{Q\}\\mid\\mathcal\{O\},W\)\.\(30\)Moreover,

I⁡\(𝒪;YQ∣W\)≤min⁡\{I⁡\(U;𝒪∣W\),I⁡\(U;YQ∣W\)\}\.I\(\\mathcal\{O\};Y^\{Q\}\\mid W\)\\leq\\min\\\!\\left\\\{I\(U;\\mathcal\{O\}\\mid W\),I\(U;Y^\{Q\}\\mid W\)\\right\\\}\.\(31\)

#### Intuition\.

Perfect knowledge ofUUdefines 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 underYQ⟂⟂U\|WY^\{Q\}\\perp\\\!\\\!\\\!\\perp U\\mid W,YQ⟂⟂𝒪\|WY^\{Q\}\\perp\\\!\\\!\\\!\\perp\\mathcal\{O\}\\mid W, andYQ⟂⟂U\|\(𝒪,W\)Y^\{Q\}\\perp\\\!\\\!\\\!\\perp U\\mid\(\\mathcal\{O\},W\)\. The last says that the evidence retains all response\-relevant unit information, not that it recovers the realized unit\. A trusted key that uniquely resolvesU=u⁡\(k\)U=u\(k\)zeros the residual term; estimatingzθ​\(k\)z\_\{\\theta\}\(k\)andRθR\_\{\\theta\}remains the approximation problem below\. The proper\-score comparison is[Proposition7](https://arxiv.org/html/2608.25118#Thmproposition7); proofs are in[SectionE\.1](https://arxiv.org/html/2608.25118#A5.SS1)\.

#### Binary example\.

LetWWbe fixed,U∼Bernoulli⁡\(1/2\)U\\sim\\operatorname\{Bernoulli\}\(1/2\),YQ=UY^\{Q\}=U, and𝒪=U⊕N\\mathcal\{O\}=U\\oplus NwithN∼Bernoulli⁡\(δ\)N\\sim\\operatorname\{Bernoulli\}\(\\delta\)independent and0≤δ≤1/20\\leq\\delta\\leq 1/2\. In nats, the oracle value islog⁡2\\log 2, the accessible value islog⁡2−h⁡\(δ\)\\log 2\-h\(\\delta\), and the residual value ish⁡\(δ\)h\(\\delta\), whereh⁡\(δ\)=−δ​log⁡δ−\(1−δ\)​log⁡\(1−δ\)h\(\\delta\)=\-\\delta\\log\\delta\-\(1\-\\delta\)\\log\(1\-\\delta\)\. The unit matters for everyδ\\delta, but the declared evidence ranges from perfectly informative to useless\.

### 5\.2Realization 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\|𝒪\.U\\perp\\\!\\\!\\\!\\perp W\\mid\\mathcal\{O\}\.\(32\)This holds for a fixed query and for a query policy that uses𝒪\\mathcal\{O\}without hidden access toUU\. At admissible\(o,w\)\(o,w\), write

Po​\(d​u\)\\displaystyle P\_\{o\}\(du\):=P⁡\(U∈d​u∣𝒪=o\),\\displaystyle:=P\(U\\in du\\mid\\mathcal\{O\}=o\),Qo​\(d​u\)\\displaystyle Q\_\{o\}\(du\):=Qϕ​\(d​u∣o\),\\displaystyle:=Q\_\{\\phi\}\(du\\mid o\),Ku,w​\(d​y\)\\displaystyle K\_\{u,w\}\(dy\):=P⁡\(YQ∈d​y∣U=u,W=w\),\\displaystyle:=P\(Y^\{Q\}\\in dy\\mid U=u,W=w\),K^u,w​\(d​y\)\\displaystyle\\widehat\{K\}\_\{u,w\}\(dy\):=Pθ​\(d​y∣w,u\),\\displaystyle:=P\_\{\\theta\}\(dy\\mid w,u\),\(33\)Mo,w​\(d​y\)\\displaystyle M\_\{o,w\}\(dy\):=∫Ku,w​\(𝑑y\)​Po​\(𝑑u\),\\displaystyle:=\\int K\_\{u,w\}\(dy\)P\_\{o\}\(du\),M^o,w​\(d​y\)\\displaystyle\\widehat\{M\}\_\{o,w\}\(dy\):=∫K^u,w​\(𝑑y\)​Qo​\(𝑑u\)\.\\displaystyle:=\\int\\widehat\{K\}\_\{u,w\}\(dy\)Q\_\{o\}\(du\)\.\(34\)Under[Eqs\.26](https://arxiv.org/html/2608.25118#S5.E26)and[32](https://arxiv.org/html/2608.25118#S5.E32),Mo,w=P\(YQ∈⋅∣𝒪=o,W=w\)M\_\{o,w\}=P\(Y^\{Q\}\\in\\cdot\\mid\\mathcal\{O\}=o,W=w\)on the chosen versions\.

###### Proposition 2\(End\-to\-end predictive excess and component\-error bound\)\.

LetRdep​\(θ,ϕ\)R\_\{\\rm dep\}\(\\theta,\\phi\)be the expected log loss ofM^𝒪,W\\widehat\{M\}\_\{\\mathcal\{O\},W\}\. Whenever the displayed quantities are finite,

Rdep​\(θ,ϕ\)−R𝒪,W⋆\\displaystyle R\_\{\\rm dep\}\(\\theta,\\phi\)\-R\_\{\\mathcal\{O\},W\}^\{\\star\}=E⁡\[DKL​\(M𝒪,W∥M^𝒪,W\)\]=:ℰpred​\(θ,ϕ\),\\displaystyle=E\\\!\\left\[D\_\{\\rm KL\}\\\!\\left\(M\_\{\\mathcal\{O\},W\}\\,\\middle\\\|\\,\\widehat\{M\}\_\{\\mathcal\{O\},W\}\\right\)\\right\]=:\\mathcal\{E\}\_\{\\rm pred\}\(\\theta,\\phi\),\(35\)RW⋆−Rdep​\(θ,ϕ\)\\displaystyle R\_\{W\}^\{\\star\}\-R\_\{\\rm dep\}\(\\theta,\\phi\)=I⁡\(𝒪;YQ∣W\)−ℰpred​\(θ,ϕ\),\\displaystyle=I\(\\mathcal\{O\};Y^\{Q\}\\mid W\)\-\\mathcal\{E\}\_\{\\rm pred\}\(\\theta,\\phi\),\(36\)Rdep​\(θ,ϕ\)−RU,W⋆\\displaystyle R\_\{\\rm dep\}\(\\theta,\\phi\)\-R\_\{U,W\}^\{\\star\}=I\(U;YQ∣𝒪,W\)\+ℰpred\(θ,ϕ\)\.\\displaystyle=I\(U;Y^\{Q\}\\mid\\mathcal\{O\},W\)\+\\mathcal\{E\}\_\{\\rm pred\}\(\\theta,\\phi\)\.\(37\)If, for almost every\(o,w\)\(o,w\)under the deployment law of\(𝒪,W\)\(\\mathcal\{O\},W\),Po≪QoP\_\{o\}\\ll Q\_\{o\}andKu,w≪K^u,wK\_\{u,w\}\\ll\\widehat\{K\}\_\{u,w\}forPoP\_\{o\}\-almost everyuu, then

ℰpred​\(θ,ϕ\)\\displaystyle\\mathcal\{E\}\_\{\\rm pred\}\(\\theta,\\phi\)≤E𝒪\[DKL\(P\(U∈⋅∣𝒪\)∥Qϕ\(⋅∣𝒪\)\)\]\\displaystyle\\leq E\_\{\\mathcal\{O\}\}\\\!\\left\[D\_\{\\rm KL\}\\\!\\left\(P\(U\\in\\cdot\\mid\\mathcal\{O\}\)\\,\\middle\\\|\\,Q\_\{\\phi\}\(\\cdot\\mid\\mathcal\{O\}\)\\right\)\\right\]\+E𝒪,W​\[∫DKL​\(Ku,W∥K^u,W\)​P​\(𝑑u∣𝒪\)\]\.\\displaystyle\\quad\+E\_\{\\mathcal\{O\},W\}\\\!\\left\[\\int D\_\{\\rm KL\}\\\!\\left\(K\_\{u,W\}\\,\\middle\\\|\\,\\widehat\{K\}\_\{u,W\}\\right\)P\(du\\mid\\mathcal\{O\}\)\\right\]\.\(38\)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 awayUU; the proof is in[SectionE\.2](https://arxiv.org/html/2608.25118#A5.SS2)\.

For a complementary query\-local statement, letdTV​\(μ,ν\)=supA\|μ⁡\(A\)−ν⁡\(A\)\|d\_\{\\rm TV\}\(\\mu,\\nu\)=\\sup\_\{A\}\|\\mu\(A\)\-\\nu\(A\)\|and let𝒰o\\mathcal\{U\}\_\{o\}be a common full\-mass region underPoP\_\{o\}andQoQ\_\{o\}\.

###### Lemma 3\(Query\-local unit\-mixture error propagation\)\.

For measurable response kernels for which the displayed quantities are well\-defined,

dTV​\(M^o,w,Mo,w\)\\displaystyle d\_\{\\rm TV\}\\\!\\left\(\\widehat\{M\}\_\{o,w\},M\_\{o,w\}\\right\)≤∫dTV​\(K^u,w,Ku,w\)​Qo​\(𝑑u\)\\displaystyle\\leq\\int d\_\{\\rm TV\}\\\!\\left\(\\widehat\{K\}\_\{u,w\},K\_\{u,w\}\\right\)Q\_\{o\}\(du\)\+\[supu,v∈𝒰odTV​\(Ku,w,Kv,w\)\]​dTV​\(Qo,Po\)\.\\displaystyle\\quad\+\\left\[\\sup\_\{u,v\\in\\mathcal\{U\}\_\{o\}\}d\_\{\\rm TV\}\\\!\\left\(K\_\{u,w\},K\_\{v,w\}\\right\)\\right\]d\_\{\\rm 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 \([Eq\.38](https://arxiv.org/html/2608.25118#S5.E38)\); the total\-variation lemma makes belief sensitivity heterogeneity\-dependent \([SectionE\.5](https://arxiv.org/html/2608.25118#A5.SS5)\)\. Under trusted direct access,Po=Qo=δu⁡\(k\)P\_\{o\}=Q\_\{o\}=\\delta\_\{u\(k\)\}with noQϕQ\_\{\\phi\}module, so remaining error is mismatch betweenK^u⁡\(k\),w\\widehat\{K\}\_\{u\(k\),w\}andKu⁡\(k\),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 hasℰpred=0\\mathcal\{E\}\_\{\\rm pred\}=0and realizes all accessible value\. A learner that ignores𝒪\\mathcal\{O\}and always reports the pooledBernoulli⁡\(1/2\)\\operatorname\{Bernoulli\}\(1/2\)law hasℰpred=I⁡\(𝒪,YQ\)\\mathcal\{E\}\_\{\\rm pred\}=I\(\\mathcal\{O\};Y^\{Q\}\)and realizes none of it\. Separately, ifK0,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](https://arxiv.org/html/2608.25118#S5.E39)records the possible sensitivity\.

### 5\.3Single\-Row Impossibility and Repeated\-Linkage Separation

#### Question\.

Can perfect marginal prediction validate the internal unit structure? The law of total probability gives

∫𝒰Pu⋆​\(𝑑y∣x\)​P​\(U∈𝑑u∣X=x\)=P⁡\(Y∈𝑑y∣X=x\)forPX\-almost everyx\.\\int\_\{\\mathcal\{U\}\}P\_\{u\}^\{\\star\}\(dy\\mid x\)P\(U\\in du\\mid X=x\)=P\(Y\\in dy\\mid X=x\)\\quad\\text\{for $P\_\{X\}$\-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\(Single\-row indistinguishability and linked\-pair separation\)\.

1. \(a\)The identity in \([40](https://arxiv.org/html/2608.25118#S5.E40)\) holds for compatible conditional versions\. Moreover, in an unrestricted response\-kernel class, every sample\-only lawR⁡\(d​y∣x\)R\(dy\\mid x\)has, under every learner belief, the unit\-constant representation KuR​\(d​y∣x\):=R⁡\(d​y∣x\),K\_\{u\}^\{R\}\(dy\\mid x\):=R\(dy\\mid x\),\(41\)so marginal fit does not identify the two mixture components\.
2. \(b\)For a statistical witness, fix one response input\. In world𝖧\\mathsf\{H\}, letU∼Bernoulli⁡\(1/2\)U\\sim\\operatorname\{Bernoulli\}\(1/2\)and P𝖧​\(Y=1∣U=0\)=14,P𝖧​\(Y=1∣U=1\)=34\.P\_\{\\mathsf\{H\}\}\(Y=1\\mid U=0\)=\\tfrac\{1\}\{4\},\\qquad P\_\{\\mathsf\{H\}\}\(Y=1\\mid U=1\)=\\tfrac\{3\}\{4\}\.\(42\)In world𝖯\\mathsf\{P\}, letP𝖯​\(Y=1∣U=u\)=1/2P\_\{\\mathsf\{P\}\}\(Y=1\\mid U=u\)=1/2for both units\. If each observation comes from a fresh independently drawn unit and neither its identity nor linkage is observed, then for everynn, ℒ𝖧​\(Y1,…,Yn\)=ℒ𝖯​\(Y1,…,Yn\)=Bernoulli⁡\(1/2\)⊗n\.\\mathcal\{L\}\_\{\\mathsf\{H\}\}\(Y\_\{1\},\\ldots,Y\_\{n\}\)=\\mathcal\{L\}\_\{\\mathsf\{P\}\}\(Y\_\{1\},\\ldots,Y\_\{n\}\)=\\operatorname\{Bernoulli\}\(1/2\)^\{\\otimes n\}\.\(43\)Consequently every possibly randomized test hasαn\+βn=1\\alpha\_\{n\}\+\\beta\_\{n\}=1and infφnmax⁡\{αn,βn\}=12\.\\inf\_\{\\varphi\_\{n\}\}\\max\\\{\\alpha\_\{n\},\\beta\_\{n\}\\\}=\\tfrac\{1\}\{2\}\.\(44\)
3. \(c\)If instead two responses per unit are observed with trusted same\-unit linkage and are conditionally independent givenUU, then Cov𝖧⁡\(Y1,Y2\)=116,Cov𝖯⁡\(Y1,Y2\)=0,\\operatorname\{Cov\}\_\{\\mathsf\{H\}\}\(Y\_\{1\},Y\_\{2\}\)=\\tfrac\{1\}\{16\},\\qquad\\operatorname\{Cov\}\_\{\\mathsf\{P\}\}\(Y\_\{1\},Y\_\{2\}\)=0,\(45\)or equivalently their agreement probabilities are5/85/8and1/21/2\. IfMMlinked pairs are sampled independently from independently drawn units, their empirical agreement rate therefore gives a consistent test asM→∞M\\to\\infty\.
4. \(d\)More generally, under the declared linked\-pair model\(Y1,Y2\)\|U=u∼Bernoulli⁡\(pu\)⊗2\(Y\_\{1\},Y\_\{2\}\)\\mid U=u\\sim\\operatorname\{Bernoulli\}\(p\_\{u\}\)^\{\\otimes 2\}, the observable pair law identifies Var⁡\(pU\)=P⁡\(Y1=1,Y2=1\)−P​\(Y1=1\)2=Cov⁡\(Y1,Y2\),\\operatorname\{Var\}\(p\_\{U\}\)=P\(Y\_\{1\}=1,Y\_\{2\}=1\)\-P\(Y\_\{1\}=1\)^\{2\}=\\operatorname\{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\-uuproduct law separate the displayed worlds and identifyVar⁡\(pU\)\\operatorname\{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\[[43](https://arxiv.org/html/2608.25118#bib.bib11)\]\. Proof and the explicit test are in[SectionE\.3](https://arxiv.org/html/2608.25118#A5.SS3)\.

#### Binary example\.

The two fixed\-unit success probabilities are1/2±1/41/2\\pm 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 afterUUis fixed rather than replacing it by a deterministic label\.

\(a\) Heterogeneous fixed\-unit lawsxxP⁡\(Y=1∣x,u\)P\(Y=1\\mid x,u\)uAu\_\{A\}uBu\_\{B\}puA=34p\_\{u\_\{A\}\}=\\tfrac\{3\}\{4\},puB=14p\_\{u\_\{B\}\}=\\tfrac\{1\}\{4\},
equal weights12\\tfrac\{1\}\{2\},12\\tfrac\{1\}\{2\}\(b\) unit\-independent lawxxP⁡\(Y=1∣x,u\)P\(Y=1\\mid x,u\)one law for everyuu,
P⁡\(Y=1∣x,u\)=12P\(Y=1\\mid x,u\)=\\tfrac\{1\}\{2\}mix overUUObservable marginalxxP⁡\(Y=1∣x\)P\(Y=1\\mid x\)12\\tfrac\{1\}\{2\}same marginal lawFigure 4:Single\-row marginals do not certify unit structure\.Distinct fixed\-unit Bernoulli laws and a unit\-independent Bernoulli law can both yieldP⁡\(Y=1\)=1/2P\(Y=1\)=1/2after marginalizingUU\. In the restricted witness of[Proposition4](https://arxiv.org/html/2608.25118#Thmproposition4), 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\.4Practical Corollaries for Repeated\-Unit Evaluation

When attribution is observed, retaining units also changes elementary bookkeeping in ways that alter the estimand and evaluation protocol\[[19](https://arxiv.org/html/2608.25118#bib.bib1),[36](https://arxiv.org/html/2608.25118#bib.bib17),[34](https://arxiv.org/html/2608.25118#bib.bib18)\]\. ForMMobserved units, letm⁡\(u\)m\(u\)be the number of records fromuuand letℓ¯​\(u\)\\bar\{\\ell\}\(u\)be its mean loss\.

###### Corollary 5\(Row and unit weighting\)\.

The empirical risks

R^row=∑um⁡\(u\)N​ℓ¯​\(u\),R^unit=1M​∑uℓ¯​\(u\)\\widehat\{R\}\_\{\\rm row\}=\\sum\_\{u\}\\frac\{m\(u\)\}\{N\}\\bar\{\\ell\}\(u\),\\qquad\\widehat\{R\}\_\{\\rm unit\}=\\frac\{1\}\{M\}\\sum\_\{u\}\\bar\{\\ell\}\(u\)\(47\)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 probabilityp∈\(0,1\)p\\in\(0,1\), a unit withm⁡\(u\)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 whenm⁡\(u\)≥2m\(u\)\\geq 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[SectionE\.7](https://arxiv.org/html/2608.25118#A5.SS7)\. 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 averageweightm⁡\(u\)/Nm\(u\)/NA1A\_\{1\}A2A\_\{2\}A3A\_\{3\}B1B\_\{1\}C1C\_\{1\}C2C\_\{2\}3/N3/N1/N1/N2/N2/NPer\-unit averageweight1/M1/MuAu\_\{A\}uBu\_\{B\}uCu\_\{C\}ℓ¯​\(uA\)\\bar\{\\ell\}\(u\_\{A\}\)ℓ¯​\(uB\)\\bar\{\\ell\}\(u\_\{B\}\)ℓ¯​\(uC\)\\bar\{\\ell\}\(u\_\{C\}\)\(b\) Repeated units change the split questionRecord\-wise splittraintestA1A\_\{1\}A2A\_\{2\}B1B\_\{1\}A3A\_\{3\}B2B\_\{2\}known unit, new eventWhole\-unit splittraintestA1A\_\{1\}A2A\_\{2\}B1B\_\{1\}C1C\_\{1\}C2C\_\{2\}new unitFigure 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[AppendixG](https://arxiv.org/html/2608.25118#A7)\.

## 6Recommendation 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 suppliesZu⁡\(k\)c=zθ​\(k\)Z\_\{u\(k\)\}^\{c\}=z\_\{\\theta\}\(k\)\. Unit abduction forms a token of the same type from factual evidence \([Section4](https://arxiv.org/html/2608.25118#S4)\)\. The two access modes differ in how the token is obtained, not in whether the user is the unit\. The mapping is summarized in[Table2](https://arxiv.org/html/2608.25118#S6.T2)\. Factorization and neural collaborative\-filtering models are established instances of the ID\-indexed formulation\[[28](https://arxiv.org/html/2608.25118#bib.bib26),[24](https://arxiv.org/html/2608.25118#bib.bib28),[18](https://arxiv.org/html/2608.25118#bib.bib31)\]\. In the linear running instance they instantiate[Eq\.15](https://arxiv.org/html/2608.25118#S3.E15)by a user tokenzθ​\(k\)z\_\{\\theta\}\(k\)and an item mapψθ​\(xQ\)\\psi\_\{\\theta\}\(x^\{Q\}\); 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\.
## 7Relations to Established Learning Traditions

Classical statistical learning studies how a predictor class and learning rule generalize from finite samples\[[17](https://arxiv.org/html/2608.25118#bib.bib6),[38](https://arxiv.org/html/2608.25118#bib.bib7)\]\. The unit formulation does not replace that theory\. Under[assumption1](https://arxiv.org/html/2608.25118#Thmassumption1)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\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\)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\.[Table3](https://arxiv.org/html/2608.25118#S7.T3)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\.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\[[40](https://arxiv.org/html/2608.25118#bib.bib12),[2](https://arxiv.org/html/2608.25118#bib.bib13),[33](https://arxiv.org/html/2608.25118#bib.bib14)\]\. 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\[[32](https://arxiv.org/html/2608.25118#bib.bib40),[31](https://arxiv.org/html/2608.25118#bib.bib41),[8](https://arxiv.org/html/2608.25118#bib.bib42)\]\. Unit abduction instead names the construction of a contextual unit token from factual evidence \([Eq\.21](https://arxiv.org/html/2608.25118#S4.E21)\)\. A which\-unit belief may approximate the world\-side identity conditional and feed the identity\-mixture learner; the access step remains the formed token \([AppendixB](https://arxiv.org/html/2608.25118#A2)\)\. These learner formulations may coexist in a larger model while remaining distinct inference operations\.

## 8Discussion 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](https://arxiv.org/html/2608.25118#S3.E19)and[20](https://arxiv.org/html/2608.25118#S3.E20)is adequate; this does not claim that every task is empirically heterogeneous\. Separately, the comparisons in[Section7](https://arxiv.org/html/2608.25118#S7)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\.

[Assumption1](https://arxiv.org/html/2608.25118#Thmassumption1)is a computational interface, not a learnability theorem\. The structured class makes it a restriction:RθR\_\{\\theta\}must read the token through a declared simple relation\. The running instance[Definition2](https://arxiv.org/html/2608.25118#Thmdefinition2)is a finite\-dimensional linear predictor; other declared simple maps ofzzinstantiate the same class\. Without a bound ondd, or with an unrestricted nonlinear readout ofzz, 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\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\)does not by itself identify either module\. The theorems in[Section5](https://arxiv.org/html/2608.25118#S5)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\_\{\\theta\}\(k\)may be reparameterized withRθR\_\{\\theta\}, 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](https://arxiv.org/html/2608.25118#S3.E15)is the default readout; other simple relations remain open\. When identity is unresolved, identifyingQϕQ\_\{\\phi\}and the fixed\-unit kernels still needs shared\-unit observations or other restrictions\. Response\-informative observations belong in𝒪\\mathcal\{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 \([Proposition4](https://arxiv.org/html/2608.25118#Thmproposition4)\)\. A record\-wise split mixes known\-unit new\-event prediction with unseen\-unit generalization \([Corollary6](https://arxiv.org/html/2608.25118#Thmproposition6)\), and row weighting disagrees with unit weighting whenever multiplicities differ \([Corollary5](https://arxiv.org/html/2608.25118#Thmproposition5)\)\. 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 \([AppendixG](https://arxiv.org/html/2608.25118#A7)\)\. 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 \([AppendixD](https://arxiv.org/html/2608.25118#A4)\)\. Token reparameterization, the identity\-belief specialization, and the fixed evidence cutoff are collected in[AppendicesB](https://arxiv.org/html/2608.25118#A2)and[F](https://arxiv.org/html/2608.25118#A6)\.

## 9Conclusion

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\.[Assumption1](https://arxiv.org/html/2608.25118#Thmassumption1)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 formRθR\_\{\\theta\}, and represented unit\-specific differences must pass through the token\. The learned object is the pair\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\); the structured class is a simple relation in the token, with a linear predictor as the running instance \([Definition2](https://arxiv.org/html/2608.25118#Thmdefinition2)\)\. 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[AppendixB](https://arxiv.org/html/2608.25118#A2)\.

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\. When preserving the unit changes none of a declared target, estimand, admissible answer, or evaluation protocol, the unit\-marginal formulation is sufficient\.

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## Appendix ANotation and Semantic Roles

The main text separates three layers that this appendix must not collapse\. The*semantic*layer is the task\-declared unitu∈𝒰u\\in\\mathcal\{U\}and its response lawPu⋆P\_\{u\}^\{\\star\}\. The*exact mixture*layer is the first line of[Eq\.16](https://arxiv.org/html/2608.25118#S3.E16): unresolved identity mixes the world targetsPu⋆P\_\{u\}^\{\\star\}over units, without introducing learned parameters\. The*computational*layer contains learner compositions under[assumption1](https://arxiv.org/html/2608.25118#Thmassumption1): the default token\-space composition usesTϕ\(⋅∣𝒪,c\)T\_\{\\phi\}\(\\,\\cdot\\mid\\mathcal\{O\},c\)andRθR\_\{\\theta\}, while the theorem specialization usesQϕQ\_\{\\phi\}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 posteriorsP⁡\(U∈d​u∣𝒪\)P\(U\\in du\\mid\\mathcal\{O\}\)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[Section5](https://arxiv.org/html/2608.25118#S5)the complete response input is writtenWW; the same object is the lowercase indexwwof the kernelKuwK\_\{u\}^\{w\}in[AppendixC](https://arxiv.org/html/2608.25118#A3)\.

Table 4:Notation by semantic layer\. Learner quantities are listed with the computational pair first;QϕQ\_\{\\phi\}belongs to the identity\-mixture learner specialization\.As summarized in[Table4](https://arxiv.org/html/2608.25118#A1.T4), the domain𝒰\\mathcal\{U\}belongs to the declared population learning problem\. Dataset indices label records rather than creating new unit domains\. A valuexix\_\{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, lowercaseuiu\_\{i\}may annotate its records, andui=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, lowercasexi,yix\_\{i\},y\_\{i\}are event\-level realized values associated with the fixed\-unit event lawKui​\(d​x,d​y\)K\_\{u\_\{i\}\}\(dx,dy\)\. The notation keeps a record value distinct from the fixed\-unit stochastic object, so we avoid writingXi:=XuiX\_\{i\}:=X\_\{u\_\{i\}\}orYi:=YuiY\_\{i\}:=Y\_\{u\_\{i\}\}\. Repeated observations require a separately specified joint law; the population unit variable remainsUUrather than a foundational family\(Ui\)\(U\_\{i\}\)\.

LetΠ⁡\(d​u\)\\Pi\(du\)denote the population law ofUU, and letKX,u​\(d​x\)K\_\{X,u\}\(dx\)be the fixed\-individual observation kernel\. Then

PX​\(𝑑x\)=∫𝒰KX,u​\(𝑑x\)​Π​\(𝑑u\),P⁡\(XU∈A∣U=u\)=KX,u​\(A\)\.P\_\{X\}\(dx\)=\\int\_\{\\mathcal\{U\}\}K\_\{X,u\}\(dx\)\\,\\Pi\(du\),\\qquad P\(X\_\{U\}\\in A\\mid U=u\)=K\_\{X,u\}\(A\)\.\(49\)The first variation comes from drawingUU; the second remains insideKX,uK\_\{X,u\}after an individual is fixed\. A realized record contributes an event from the selected unit rather than a new unit variableUiU\_\{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 eventEEpermits ordinary conditioningP⁡\(U∈d​u∣E\)P\(U\\in du\\mid E\)\. When the protocol records evidence at a realized valueoo,P⁡\(U∈d​u∣𝒪=o\)P\(U\\in du\\mid\\mathcal\{O\}=o\)denotes evaluation of a regular conditional distribution\. We use𝒪\\mathcal\{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\. Anyyiy\_\{i\}admitted here is a historical or other already observed response, not the current target response\. A designed candidatexQx^\{Q\}supplied toPθ​\(d​y∣xQ,cQ,u\)P\_\{\\theta\}\(dy\\mid x^\{Q\},c^\{Q\},u\)is a response query\. It contributes evidence aboutUUonly when it was factually observed before the answer and the protocol incorporates it into𝒪\\mathcal\{O\}\.

When no resolver identifies the realized unit, unit abduction forms a contextual unit token as in[Eq\.21](https://arxiv.org/html/2608.25118#S4.E21); the default prediction is the token composition[Eq\.24](https://arxiv.org/html/2608.25118#S4.E24)\. The identity\-mixture specialization used by the theorems is[Eq\.23](https://arxiv.org/html/2608.25118#S4.E23)\. 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,𝒪i=\{X=xi\}\\mathcal\{O\}\_\{i\}=\\\{X=x\_\{i\}\\\}determines the world\-side conditionalP⁡\(U∈d​u∣X=xi\)P\(U\\in du\\mid X=x\_\{i\}\)\. The targetyiy\_\{i\}scores the learner mixture but is excluded from𝒪i\\mathcal\{O\}\_\{i\}\. A history encoder, probabilistic identity rule, or other map may implementTϕT\_\{\\phi\}; the primitive is architecture\-independent\.

The response\-sufficiency condition is

YQ⟂⟂𝒪\|\(U,xQ,cQ\),Y^\{Q\}\\perp\\\!\\\!\\\!\\perp\\mathcal\{O\}\\mid\(U,x^\{Q\},c^\{Q\}\),\(50\)withcQc^\{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 bycQc^\{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 of𝒪\\mathcal\{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 lawP⁡\(U∈d​u∣𝒪\)P\(U\\in du\\mid\\mathcal\{O\}\), the protocol additionally satisfies the external\-query condition in[Eq\.32](https://arxiv.org/html/2608.25118#S5.E32)\. Otherwise the exact mixing law is generallyP⁡\(U∈d​u∣𝒪,xQ,cQ\)P\(U\\in du\\mid\\mathcal\{O\},x^\{Q\},c^\{Q\}\)\. 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 implementsk↦u⁡\(k\)k\\mapsto u\(k\)and fixes a stable address\. A separate lookup at that address supplies the token, typically the learned rowZu⁡\(k\)c=zθ​\(k\)Z\_\{u\(k\)\}^\{c\}=z\_\{\\theta\}\(k\)\. The resolver fixes the referent; the lookup selects the token slot to read or update\. Neither operation makeszθ​\(k\)z\_\{\\theta\}\(k\)into the unit\. Learningzθ​\(k\)z\_\{\\theta\}\(k\)is fitting the token, not a which\-unit inference\. The corresponding which\-unit conditionalP⁡\(U∈d​u∣ID=k\)P\(U\\in du\\mid\\mathrm\{ID\}=k\)is a point mass \(a Dirac measure\)\. The identity\-mixture specialization then usesδu⁡\(k\)\\delta\_\{u\(k\)\}; no learnedQϕQ\_\{\\phi\}module need be run, and the Dirac mass is not an executed abduction\. Crucially, the Dirac mass is on the stable referential unitu⁡\(k\)u\(k\), not on the trainable and potentially reparameterized tokenzθ​\(k\)z\_\{\\theta\}\(k\)\. 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 BConceptual Clarifications and Inferential Distinctions

This section is the boundary register for the main text\. Four layers should be kept separate: the task\-side referentuu, the learner\-side tokenZucZ\_\{u\}^\{c\}, the world\-side response family\{Pu⋆\}\\\{P\_\{u\}^\{\\star\}\\\}, 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 specializationQϕQ\_\{\\phi\}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\.1Persistent units and model representations

The declared unit domain𝒰\\mathcal\{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 ofUU\. In a trusted\-ID model,u⁡\(k\)u\(k\)is the resolved unit andzθ​\(k\)z\_\{\\theta\}\(k\)is the tokenZu⁡\(k\)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\[[3](https://arxiv.org/html/2608.25118#bib.bib9)\]\. A posterior over such response\-side coordinates is likewise not automatically the which\-unit beliefQϕ​\(d​u∣𝒪\)Q\_\{\\phi\}\(du\\mid\\mathcal\{O\}\), and neither object is a sample\-inclusion probability or an importance weight\.

### B\.2Tokens, which\-unit beliefs, and point summaries

The default learner\-side object is the contextual unit tokenZcZ^\{c\}\. In the identity\-belief specialization,Qϕ​\(d​u∣𝒪\)Q\_\{\\phi\}\(du\\mid\\mathcal\{O\}\)instead targets the world conditionalP⁡\(U∈d​u∣𝒪\)P\(U\\in du\\mid\\mathcal\{O\}\)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 satisfiesU~ϕ∣𝒪∼Qϕ\(⋅∣𝒪\)\\widetilde\{U\}\_\{\\phi\}\\mid\\mathcal\{O\}\\sim Q\_\{\\phi\}\(\\cdot\\mid\\mathcal\{O\}\)\. The response\-level prediction propagates the full belief through[Eq\.23](https://arxiv.org/html/2608.25118#S4.E23); a downstream decision may then extract a point prediction, interval, action, or abstention\.

### B\.3Record, evidence, and query roles

The notation table separates record values, factual evidence, response queries, and current targets\. In particular,iiindexes an event rather than defining a world variableUiU\_\{i\}, and a supplied candidatexQx^\{Q\}is evaluated at the token formed from the fixed evidence cutoff\. A newly observed factual event may update that token, and may updateQϕQ\_\{\\phi\}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 intoxx, omitting it from the response specification generally yields, for compatible regular\-conditional versions andPX,UP\_\{X,U\}\-almost every\(x,u\)\(x,u\),

Pu⋆​\(𝑑y∣x\)=∫Pu⋆​\(𝑑y∣x,c\)​P⋆​\(𝑑c∣x,U=u\)\.P\_\{u\}^\{\\star\}\(dy\\mid x\)=\\int P\_\{u\}^\{\\star\}\(dy\\mid x,c\)\\,P^\{\\star\}\(dc\\mid x,U=u\)\.\(51\)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\. DeclaringcQc^\{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\.4Marginalization, 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[Proposition4](https://arxiv.org/html/2608.25118#Thmproposition4)\. At the response\-structure level,f​\(x,u\)=f0​\(x\)f\(x;u\)=f\_\{0\}\(x\)for every relevantuuis a sufficient invariance condition for a sample\-only rulef0f\_\{0\}; it does not definef0f\_\{0\}as an average off⁡\(x,u\)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 equalsP⁡\(Y∈d​y∣X=xi\)P\(Y\\in dy\\mid X=x\_\{i\}\)by[Proposition4](https://arxiv.org/html/2608.25118#Thmproposition4)\. When a density or probability mass exists and𝒪i=\{X=xi\}\\mathcal\{O\}\_\{i\}=\\\{X=x\_\{i\}\\\}, the learner score in that specialization is

logp^θ,ϕ\(yi∣𝒪i;xi\)=log∫𝒰pθ\(yi∣xi,u\)Qϕ\(du∣𝒪i\)\.\\log\\widehat\{p\}\_\{\\theta,\\phi\}\(y\_\{i\}\\mid\\mathcal\{O\}\_\{i\};\\,x\_\{i\}\)=\\log\\\!\\int\_\{\\mathcal\{U\}\}p\_\{\\theta\}\(y\_\{i\}\\mid x\_\{i\},u\)Q\_\{\\phi\}\(du\\mid\\mathcal\{O\}\_\{i\}\)\.\(52\)The default token\-space prediction instead scores∫Rθ​\(d​yi∣xi,c,z\)​Tϕ​\(𝑑z∣𝒪i,c\)\\int R\_\{\\theta\}\(dy\_\{i\}\\mid x\_\{i\},c,z\)\\,T\_\{\\phi\}\(dz\\mid\\mathcal\{O\}\_\{i\},c\)\. Here the observedxix\_\{i\}serves both as pre\-target evidence and as the response covariate, while an alternativexQx^\{Q\}reuses the same formed token\. The log of the mixture is generally different from the mixture of conditional log scores; for continuousYY, the likelihood is a density rather than a point probability\. Matching the marginal law alone does not identifyTϕT\_\{\\phi\}orQϕQ\_\{\\phi\}, recover a persistent unit, or establish response dependence onuu\.

## Appendix CUnit\-Conditioned Response Heterogeneity

Throughout this section,w=\(xQ,cQ\)w=\(x^\{Q\},c^\{Q\}\)denotes a declared complete response input—not a unit tokenZucZ\_\{u\}^\{c\}\. This is the same object writtenWWin[Section5](https://arxiv.org/html/2608.25118#S5)\. 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\.[Assumption1](https://arxiv.org/html/2608.25118#Thmassumption1)is one such restriction: differences amongKuwK\_\{u\}^\{w\}must be expressed by contextual unit tokens under a shared formRθR\_\{\\theta\}\.

### C\.1The response\-law profile

Letw=\(xQ,cQ\)w=\(x^\{Q\},c^\{Q\}\)denote a declared response query, withcQc^\{Q\}omitted when it is empty, and let𝔔\\mathfrak\{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\(u,w\)\\mapsto K\_\{u\}^\{w\}on𝒰×𝔔\\mathcal\{U\}\\times\\mathfrak\{Q\}\. Write

Kuw​\(d​y\):=Pu⋆​\(d​y∣xQ,cQ\)\.K\_\{u\}^\{w\}\(dy\):=P\_\{u\}^\{\\star\}\(dy\\mid x^\{Q\},c^\{Q\}\)\.\(53\)For fixeduu, the familyw↦Kuww\\mapsto K\_\{u\}^\{w\}gives the response law over the declared query family\. Equality of two such families means equality of the declared kernels on𝔔\\mathfrak\{Q\}\. If the kernel is learned or interpreted only through a regular conditional law under a designΛ⁡\(d​u,d​w\)\\Lambda\(du,dw\), its off\-support values are modeling choices rather than identified features\. TheΛ\\Lambda\-relative notion below states the corresponding observable condition\.

###### Definition 3\(Kernel formulation of unit\-response homogeneity\)\.

ForU∼ΠU\\sim\\Pi, a declared query family𝔔\\mathfrak\{Q\}, and the chosen jointly measurable kernel in[Eq\.53](https://arxiv.org/html/2608.25118#A3.E53), the task is*unit\-response homogeneous*when there are a singleΠ\\Pi\-null setN⊆𝒰N\\subseteq\\mathcal\{U\}and a common response kernelw↦Kww\\mapsto K^\{w\}such thatKuw=KwK\_\{u\}^\{w\}=K^\{w\}for everyu∉Nu\\notin Nand everyw∈𝔔w\\in\\mathfrak\{Q\}\. It is*unit\-response heterogeneous*otherwise\.

Equivalently, define query\-relative response equivalence by

u≡𝔔v⟺Kuw=Kvwas probability laws for every declaredw∈𝔔\.u\\equiv\_\{\\mathfrak\{Q\}\}v\\quad\\Longleftrightarrow\\quad K\_\{u\}^\{w\}=K\_\{v\}^\{w\}\\ \\text\{as probability laws for every declared \}w\\in\\mathfrak\{Q\}\.\(54\)Homogeneity says thatΠ\\Piis concentrated on one such response\-equivalence class; heterogeneity says that it is not\. This quotient is predictive, not referential\. Distinct individuals may satisfyu≡𝔔vu\\equiv\_\{\\mathfrak\{Q\}\}vwhile remaining distinct units\. Conversely, changingP⁡\(X∣U=u\)P\(X\\mid U=u\)or the frequency with which a unit is sampled does not by itself establish heterogeneity inKuwK\_\{u\}^\{w\}\.

For a declared designΛ\\Lambdaon𝒰×𝔔\\mathcal\{U\}\\times\\mathfrak\{Q\}, call the family*Λ\\Lambda\-visible homogeneous*when there is a kernelKwK^\{w\}such thatKuw=KwK\_\{u\}^\{w\}=K^\{w\}forΛ\\Lambda\-almost every\(u,w\)\(u,w\), and*Λ\\Lambda\-visible heterogeneous*otherwise\. Structural homogeneity in Definition[3](https://arxiv.org/html/2608.25118#Thmdefinition3)implies visible homogeneity for every compatible design, where compatibility means that the unit marginalΛU\\Lambda\_\{U\}is absolutely continuous with respect toΠ\\Pi\. 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 aΛ\\Lambda\-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 ofwwwithout becoming a new unit\. The response kernels in[Eq\.53](https://arxiv.org/html/2608.25118#A3.E53)are observational unless a separate causal assignment and identification assumptions license an interventional reading\.

### C\.2Response\-law model classes

The familyKuwK\_\{u\}^\{w\}supports several qualitatively different model classes\. The restrictions in[Table5](https://arxiv.org/html/2608.25118#A3.T5)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\.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](https://arxiv.org/html/2608.25118#Thmdefinition3), or by itsΛ\\Lambda\-visible counterpart for a deployment\-limited claim\.

Standard supervised learning does not, by its basic sample\-indexed notation\[[4](https://arxiv.org/html/2608.25118#bib.bib5),[17](https://arxiv.org/html/2608.25118#bib.bib6),[38](https://arxiv.org/html/2608.25118#bib.bib7)\], 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[Table5](https://arxiv.org/html/2608.25118#A3.T5)is imposed\. The exact marginal collapse in[Proposition4](https://arxiv.org/html/2608.25118#Thmproposition4)is a different statement: heterogeneous fixed\-unit laws may integrate to the same row\-level predictive law\.

### C\.3Shared Structure and Learnability

Definition[3](https://arxiv.org/html/2608.25118#Thmdefinition3)determines whether the response family varies withuubut 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,\(u,w\)\\longmapsto K\_\{u\}^\{w\}\\quad\\text\{belongs 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[assumption1](https://arxiv.org/html/2608.25118#Thmassumption1)the shared restriction is the pair\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\), with a simple relation in the token as the structured class and[Definition2](https://arxiv.org/html/2608.25118#Thmdefinition2)as the running instance:RθR\_\{\\theta\}depends onzzonly through a declared low\-complexity map, linearly in finite dimension or through another declared simple map ofzz\. Random effects and multitask representations provide two established forms of such structure\[[13](https://arxiv.org/html/2608.25118#bib.bib16),[6](https://arxiv.org/html/2608.25118#bib.bib3),[1](https://arxiv.org/html/2608.25118#bib.bib4)\]; a full learnability analysis must additionally specify the loss, sampling regime, model complexity, and learning rule\[[38](https://arxiv.org/html/2608.25118#bib.bib7)\]\.

### C\.4Oracle Value Under Proper Scoring Rules

The definition also has a decision\-theoretic reading\. LetΛ⁡\(d​u,d​w\)\\Lambda\(du,dw\)be a declared deployment design over units and queries, letY\|\(U=u,W=w\)∼KuwY\\mid\(U=u,W=w\)\\sim K\_\{u\}^\{w\}, and suppose a regular conditionalΛ⁡\(d​u∣w\)\\Lambda\(du\\mid w\)has been fixed\. Define the pooled Bayes law

K¯w​\(𝑑y\):=∫𝒰Kuw​\(𝑑y\)​Λ​\(𝑑u∣w\)\.\\overline\{K\}^\{w\}\(dy\):=\\int\_\{\\mathcal\{U\}\}K\_\{u\}^\{w\}\(dy\)\\,\\Lambda\(du\\mid w\)\.\(56\)Let the admissible reports range over a class𝒫\\mathcal\{P\}of probability laws that containsKuwK\_\{u\}^\{w\}forΛ\\Lambda\-almost every\(u,w\)\(u,w\)and is closed under the displayed mixtures, so thatK¯w∈𝒫\\overline\{K\}^\{w\}\\in\\mathcal\{P\}\. LetS⁡\(P,y\)S\(P,y\)be a strictly proper predictive loss on𝒫\\mathcal\{P\}, with smaller values better, and letDS​\(P,Q\):=EY∼P​\[S⁡\(Q,Y\)−S⁡\(P,Y\)\]D\_\{S\}\(P,Q\):=E\_\{Y\\sim P\}\[S\(Q,Y\)\-S\(P,Y\)\]be its associated nonnegative regret divergence\[[14](https://arxiv.org/html/2608.25118#bib.bib15)\]\.

###### 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 onlyWWis

Rpool−RUnit\\displaystyle R\_\{\\rm pool\}\-R\_\{\\rm Unit\}:=E⁡\[S⁡\(K¯W,Y\)−S⁡\(KUW,Y\)\]\\displaystyle:=E\\\!\\left\[S\(\\overline\{K\}^\{W\},Y\)\-S\(K\_\{U\}^\{W\},Y\)\\right\]\(57\)=E\(U,W\)∼Λ​\[DS​\(KUW,K¯W\)\]≥0\.\\displaystyle=E\_\{\(U,W\)\\sim\\Lambda\}\\left\[D\_\{S\}\\\!\\left\(K\_\{U\}^\{W\},\\overline\{K\}^\{W\}\\right\)\\right\]\\geq 0\.\(58\)Under strict propriety, equality holds if and only ifKUW=K¯WK\_\{U\}^\{W\}=\\overline\{K\}^\{W\}forΛ\\Lambda\-almost every\(U,W\)\(U,W\)\. For logarithmic loss, the gap isIΛ​\(U;Y∣W\)I\_\{\\Lambda\}\(U;Y\\mid W\)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 reportingK¯w\\overline\{K\}^\{w\}instead of the trueKuwK\_\{u\}^\{w\}isDS​\(Kuw,K¯w\)D\_\{S\}\(K\_\{u\}^\{w\},\\overline\{K\}^\{w\}\)\. Integrating with respect toΛ\\Lambdaproves 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 isIΛ​\(U;Y∣W\)I\_\{\\Lambda\}\(U;Y\\mid W\)\. ∎

This proposition applies the classical proper\-scoring\-rule identity to give an exact, query\- and deployment\-relative magnitude toΛ\\Lambda\-visible response heterogeneity\. It is an oracle comparison and does not establish thatUUis observed, identifiable, causally useful, or learnable from finite selectively observed events\.

## Appendix DDiscoSCM as a Causal Specialization of the Unit Primitive

DiscoSCM, introduced by[Gong et al\. \[15\]](https://arxiv.org/html/2608.25118#bib.bib39), provides a substantive causal specialization of the unit primitive\. This relation is stronger than a shared use of the letterUU: 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\.1From a Response\-Law Family to Unit\-Conditioned Structural Mechanisms

The generic supervised specialization in this paper begins with a population lawU∼ΠU\\sim\\Piand a unit\-conditioned response familyu↦Kuw​\(d​y\)u\\mapsto K\_\{u\}^\{w\}\(dy\)\. DiscoSCM refines the response family into a structural causal model

ℳD=⟨U,𝐄,𝐕,ℱ⟩,Vj←fj​\(paj,Ej,U\),\\mathcal\{M\}^\{\\rm D\}=\\langle U,\\mathbf\{E\},\\mathbf\{V\},\\mathcal\{F\}\\rangle,\\qquad V\_\{j\}\\leftarrow f\_\{j\}\(\\operatorname\{pa\}\_\{j\},E\_\{j\};U\),\(59\)whereU=uU=uselects the individual,𝐄\\mathbf\{E\}contains exogenous variables,𝐕\\mathbf\{V\}contains endogenous variables, andℱ\\mathcal\{F\}is a family of unit\-conditioned structural assignments\. The DiscoSCM formulation assumesU⟂⟂𝐄U\\perp\\\!\\\!\\\!\\perp\\mathbf\{E\}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 causeX=xX=xand outcomeYY, 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,\\mathrm\{str\}\}^\{x\}\(B\):=P\\\!\\left\(f\_\{Y\}\(x,E\_\{Y\};u\)\\in B\\right\)\.\(60\)Thusu↦Ku,strxu\\mapsto K\_\{u,\\mathrm\{str\}\}^\{x\}is a structurally induced unit\-conditioned response family, while the mechanismsu↦fY​\(⋅,⋅,u\)u\\mapsto f\_\{Y\}\(\\cdot,\\cdot;u\)provide its richer causal representation\. It is not identified with the observational conditionalPu⋆​\(d​y∣x\)P\_\{u\}^\{\\star\}\(dy\\mid x\)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 ofℱ\\mathcal\{F\}then provide one possible answer to the shared\-structure question posed in[Section3\.2](https://arxiv.org/html/2608.25118#S3.SS2)\.

The separation betweenUUand𝐄\\mathbf\{E\}is the central correspondence\. A realized event is generated by both an individual valueuuand an exogenous realization𝐞\\mathbf\{e\}\. Holdinguufixed 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\.
### D\.2The 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 variables𝐄⁡\(x\)\\mathbf\{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 superscriptdddistinguishes this distribution\-consistent counterfactual from a traditional same\-noise SCM counterfactual\. The defining marginal restriction is

𝐄⁡\(x\)​=𝑑​𝐄,\\mathbf\{E\}\(x\)\\ \\overset\{d\}\{=\}\\ \\mathbf\{E\},\(62\)which yields, under the model’s structural conditions, the fixed\-unit distribution\-consistency relation

ℒ⁡\(Yud​\(x\)∣X=x,U=u\)=ℒ⁡\(Y∣X=x,U=u\)\.\\mathcal\{L\}\\\!\\left\(Y\_\{u\}^\{d\}\(x\)\\mid X=x,U=u\\right\)=\\mathcal\{L\}\\\!\\left\(Y\\mid X=x,U=u\\right\)\.\(63\)This equality in distribution replaces the pointwise equality imposed by the usual consistency relation\.

Equation \([62](https://arxiv.org/html/2608.25118#A4.E62)\) fixes the marginal law in each world but does not by itself determine the coupling among𝐄\\mathbf\{E\},𝐄⁡\(x\)\\mathbf\{E\}\(x\), and𝐄⁡\(x′\)\\mathbf\{E\}\(x^\{\\prime\}\)\. That coupling is a Layer\-3 modeling choice because it determines joint cross\-world quantities\. Traditional SCM semantics occupy the same\-noise endpoint𝐄⁡\(x\)=𝐄\\mathbf\{E\}\(x\)=\\mathbf\{E\}almost surely\. The population\-valuation result of[Gong et al\. \[15\]](https://arxiv.org/html/2608.25118#bib.bib39)instead uses the additional assumption𝐄⁡\(x\)⟂⟂𝐄\\mathbf\{E\}\(x\)\\perp\\\!\\\!\\\!\\perp\\mathbf\{E\}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\.3Unit Abduction, causal valuation, and reduction

Let𝒪\\mathcal\{O\}denote an observed factual trace\. The law of total probability gives the exact decomposition

P\(Yd\(x\)∈dy∣𝒪\)=∫𝒰\\displaystyle P\\\!\\left\(Y^\{d\}\(x\)\\in dy\\mid\\mathcal\{O\}\\right\)=\\int\_\{\\mathcal\{U\}\}P⁡\(Yd​\(x\)∈d​y∣𝒪,U=u\)​P​\(d​u∣𝒪\)\.\\displaystyle P\\\!\\left\(Y^\{d\}\(x\)\\in dy\\mid\\mathcal\{O\},U=u\\right\)P\(du\\mid\\mathcal\{O\}\)\.\(64\)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 on𝒪\\mathcal\{O\}\. Writing its fixed\-unit counterfactual kernel asKu,cfxK\_\{u,\\mathrm\{cf\}\}^\{x\}, the decomposition becomes

P⁡\(Yd​\(x\)∈𝑑y∣𝒪\)=∫𝒰Ku,cfx​\(𝑑y\)​P​\(𝑑u∣𝒪\)\.P\\\!\\left\(Y^\{d\}\(x\)\\in dy\\mid\\mathcal\{O\}\\right\)=\\int\_\{\\mathcal\{U\}\}K\_\{u,\\mathrm\{cf\}\}^\{x\}\(dy\)P\(du\\mid\\mathcal\{O\}\)\.\(65\)This is the causal specialization of the identity\-belief composition in[Eq\.23](https://arxiv.org/html/2608.25118#S4.E23)\. DiscoSCM abducts a posterior over identity: factual evidence updatesP⁡\(d​u∣𝒪\)P\(du\\mid\\mathcal\{O\}\); the causal model evaluates the counterfactual response law for each fixeduu; and integration reduces the unit\-specific valuations to a population\-level answer\.[Gong et al\. \[15\]](https://arxiv.org/html/2608.25118#bib.bib39)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](https://arxiv.org/html/2608.25118#S4.E24)\.

There are also two important differences from the generic learner formulation\. First,[Eq\.65](https://arxiv.org/html/2608.25118#A4.E65)uses the exact world identity conditionalP⁡\(d​u∣𝒪\)P\(du\\mid\\mathcal\{O\}\), whereas this paper’s default abduction object is a formed tokenZcZ^\{c\}, andQϕ​\(d​u∣𝒪\)Q\_\{\\phi\}\(du\\mid\\mathcal\{O\}\)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[Lemma3](https://arxiv.org/html/2608.25118#Thmproposition3)applies to this causal specialization after replacingKuK\_\{u\}byKu,cfxK\_\{u,\\mathrm\{cf\}\}^\{x\}, provided the conditional independence used in[Eq\.65](https://arxiv.org/html/2608.25118#A4.E65)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\.4Scope 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\\displaystyle\\text\{unit primitive\}⟶unit\-conditioned structural mechanisms\\displaystyle\\longrightarrow\\text\{unit\-conditioned structural mechanisms\}\(66\)⟶DiscoSCM cross\-world assumptions\\displaystyle\\longrightarrow\\text\{DiscoSCM cross\-world assumptions\}⟶causal valuations and identification results\.\\displaystyle\\longrightarrow\\text\{causal 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 EAdditional 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 specializationQϕ​\(d​u∣𝒪\)Q\_\{\\phi\}\(du\\mid\\mathcal\{O\}\)of[Eq\.23](https://arxiv.org/html/2608.25118#S4.E23); 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 on𝒰\\mathcal\{U\}\.

### E\.1Proof of the Value–Access Decomposition

###### Proof of[Proposition1](https://arxiv.org/html/2608.25118#Thmproposition1)\.

The conditional log\-score identity gives

RW⋆−RU,W⋆\\displaystyle R\_\{W\}^\{\\star\}\-R\_\{U,W\}^\{\\star\}=I⁡\(U;YQ∣W\),\\displaystyle=I\(U;Y^\{Q\}\\mid W\),\(67\)RW⋆−R𝒪,W⋆\\displaystyle R\_\{W\}^\{\\star\}\-R\_\{\\mathcal\{O\},W\}^\{\\star\}=I⁡\(𝒪;YQ∣W\)\.\\displaystyle=I\(\\mathcal\{O\};Y^\{Q\}\\mid W\)\.\(68\)The chain rule expands the same conditional mutual information in two ways:

I⁡\(U,𝒪;YQ∣W\)\\displaystyle I\(U,\\mathcal\{O\};Y^\{Q\}\\mid W\)=I\(U;YQ∣W\)\+I\(𝒪;YQ∣U,W\)\\displaystyle=I\(U;Y^\{Q\}\\mid W\)\+I\(\\mathcal\{O\};Y^\{Q\}\\mid U,W\)\(69\)=I\(𝒪;YQ∣W\)\+I\(U;YQ∣𝒪,W\)\.\\displaystyle=I\(\\mathcal\{O\};Y^\{Q\}\\mid W\)\+I\(U;Y^\{Q\}\\mid\\mathcal\{O\},W\)\.\(70\)Response sufficiency makesI\(𝒪;YQ∣U,W\)=0I\(\\mathcal\{O\};Y^\{Q\}\\mid U,W\)=0, which proves[Eq\.30](https://arxiv.org/html/2608.25118#S5.E30)\. Nonnegativity of conditional mutual information gives the risk ladder\. Response sufficiency also makes𝒪→U→YQ\\mathcal\{O\}\\to U\\to Y^\{Q\}a conditional Markov chain givenWW\. Conditional data processing givesI⁡\(𝒪;YQ∣W\)≤I⁡\(U;𝒪∣W\)I\(\\mathcal\{O\};Y^\{Q\}\\mid W\)\\leq I\(U;\\mathcal\{O\}\\mid W\), while[Eq\.30](https://arxiv.org/html/2608.25118#S5.E30)gives the other upper bound in[Eq\.31](https://arxiv.org/html/2608.25118#S5.E31)\. ∎

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 beliefP⁡\(U∈d​u∣𝒪\)P\(U\\in du\\mid\\mathcal\{O\}\)rather thanP⁡\(U∈d​u∣𝒪,W\)P\(U\\in du\\mid\\mathcal\{O\},W\)\.

### E\.2Proof 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[Proposition2](https://arxiv.org/html/2608.25118#Thmproposition2)\.

Conditioning on\(𝒪,W\)\(\\mathcal\{O\},W\), the difference between the learner’s expected log loss and the Bayes log loss is

E\[logp⁡\(YQ∣𝒪,W\)p^θ,ϕ​\(YQ∣𝒪;W\)\|𝒪,W\],E\\\!\\left\[\\log\\frac\{p\(Y^\{Q\}\\mid\\mathcal\{O\},W\)\}\{\\widehat\{p\}\_\{\\theta,\\phi\}\(Y^\{Q\}\\mid\\mathcal\{O\};W\)\}\\,\\middle\|\\,\\mathcal\{O\},W\\right\],\(71\)which isDKL\(M𝒪,W∥M^𝒪,W\)D\_\{\\rm KL\}\(M\_\{\\mathcal\{O\},W\}\\\|\\widehat\{M\}\_\{\\mathcal\{O\},W\}\)under[Eqs\.26](https://arxiv.org/html/2608.25118#S5.E26)and[32](https://arxiv.org/html/2608.25118#S5.E32)\. Averaging proves[Eq\.35](https://arxiv.org/html/2608.25118#S5.E35)\. Adding and subtractingR𝒪,W⋆R\_\{\\mathcal\{O\},W\}^\{\\star\}and applying[Proposition1](https://arxiv.org/html/2608.25118#Thmproposition1)proves[Eqs\.36](https://arxiv.org/html/2608.25118#S5.E36)and[37](https://arxiv.org/html/2608.25118#S5.E37)\.

For the component bound, at fixed\(o,w\)\(o,w\)define latent\-joint laws

Jo,w​\(d​u,d​y\):=Po​\(d​u\)​Ku,w​\(d​y\),J^o,w​\(d​u,d​y\):=Qo​\(d​u\)​K^u,w​\(d​y\)\.J\_\{o,w\}\(du,dy\):=P\_\{o\}\(du\)K\_\{u,w\}\(dy\),\\qquad\\widehat\{J\}\_\{o,w\}\(du,dy\):=Q\_\{o\}\(du\)\\widehat\{K\}\_\{u,w\}\(dy\)\.\(72\)The relative\-entropy chain rule gives

DKL\(Jo,w∥J^o,w\)\\displaystyle D\_\{\\rm KL\}\(J\_\{o,w\}\\\|\\widehat\{J\}\_\{o,w\}\)=DKL\(Po∥Qo\)\+∫DKL\(Ku,w∥K^u,w\)Po\(du\)\.\\displaystyle=D\_\{\\rm KL\}\(P\_\{o\}\\\|Q\_\{o\}\)\+\\int D\_\{\\rm KL\}\(K\_\{u,w\}\\\|\\widehat\{K\}\_\{u,w\}\)P\_\{o\}\(du\)\.\(73\)The measurable projection\(u,y\)↦y\(u,y\)\\mapsto ysendsJo,wJ\_\{o,w\}toMo,wM\_\{o,w\}andJ^o,w\\widehat\{J\}\_\{o,w\}toM^o,w\\widehat\{M\}\_\{o,w\}\. Data processing therefore gives

DKL\(Mo,w∥M^o,w\)≤DKL\(Jo,w∥J^o,w\)\.D\_\{\\rm KL\}\(M\_\{o,w\}\\\|\\widehat\{M\}\_\{o,w\}\)\\leq D\_\{\\rm KL\}\(J\_\{o,w\}\\\|\\widehat\{J\}\_\{o,w\}\)\.\(74\)Averaging[Eqs\.73](https://arxiv.org/html/2608.25118#A5.E73)and[74](https://arxiv.org/html/2608.25118#A5.E74)over the deployment law proves[Eq\.38](https://arxiv.org/html/2608.25118#S5.E38)\. ∎

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\.3Single\-Row Marginal Collapse and Linked\-Pair Separation

For compatible versions of the regular conditional kernels, the law of total probability after conditioning onX=xX=xproves[Eq\.40](https://arxiv.org/html/2608.25118#S5.E40), with itsPXP\_\{X\}\-almost\-everywhere qualification\. For any sample\-only conditional lawR⁡\(d​y∣x\)R\(dy\\mid x\), choosingKuR​\(d​y∣x\):=R⁡\(d​y∣x\)K\_\{u\}^\{R\}\(dy\\mid x\):=R\(dy\\mid x\)for everyuuin an unrestricted response\-kernel class returnsRRafter 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[Proposition4](https://arxiv.org/html/2608.25118#Thmproposition4)\.

In world𝖧\\mathsf\{H\},

P𝖧​\(Y=1\)=12​\(14\+34\)=12,P\_\{\\mathsf\{H\}\}\(Y=1\)=\\tfrac\{1\}\{2\}\(\\tfrac\{1\}\{4\}\+\\tfrac\{3\}\{4\}\)=\\tfrac\{1\}\{2\},\(75\)which is also the response probability in world𝖯\\mathsf\{P\}\. Independence of the fresh unit draws and their event realizations gives[Eq\.43](https://arxiv.org/html/2608.25118#S5.E43)for everynn\.

Let a possibly randomized testφn\\varphi\_\{n\}declare𝖧\\mathsf\{H\}with probabilityφn​\(Y1,…,Yn\)\\varphi\_\{n\}\(Y\_\{1\},\\ldots,Y\_\{n\}\)and define

αn=E𝖯​\[φn\],βn=E𝖧​\[1−φn\]\.\\alpha\_\{n\}=E\_\{\\mathsf\{P\}\}\[\\varphi\_\{n\}\],\\qquad\\beta\_\{n\}=E\_\{\\mathsf\{H\}\}\[1\-\\varphi\_\{n\}\]\.\(76\)Equality of the observable laws impliesE𝖧​φn=E𝖯​φnE\_\{\\mathsf\{H\}\}\\varphi\_\{n\}=E\_\{\\mathsf\{P\}\}\\varphi\_\{n\}, soαn\+βn=1\\alpha\_\{n\}\+\\beta\_\{n\}=1andmax⁡\{αn,βn\}≥1/2\\max\\\{\\alpha\_\{n\},\\beta\_\{n\}\\\}\\geq 1/2\. Equality is attained by randomizing equally between the two worlds\.

For a linked pair, conditional independence gives

Cov⁡\(Y1,Y2\)=Var⁡\(pU\)\.\\operatorname\{Cov\}\(Y\_\{1\},Y\_\{2\}\)=\\operatorname\{Var\}\(p\_\{U\}\)\.\(77\)This variance is1/161/16in world𝖧\\mathsf\{H\}and zero in world𝖯\\mathsf\{P\}\. More explicitly, their pair probabilities are

00011011𝖧5/163/163/165/16𝖯1/41/41/41/4\.\\begin\{array\}\[\]\{c\|cccc\}&00&01&10&11\\\\ \\hline\\cr\\mathsf\{H\}&5/16&3/16&3/16&5/16\\\\ \\mathsf\{P\}&1/4&1/4&1/4&1/4\.\\end\{array\}\(78\)IfMMlinked pairs are sampled independently from independently drawn units, define

a^M:=1M∑j=1M𝟏\{Yj​1=Yj​2\}\.\\widehat\{a\}\_\{M\}:=\\frac\{1\}\{M\}\\sum\_\{j=1\}^\{M\}\\mathbf\{1\}\\\{Y\_\{j1\}=Y\_\{j2\}\\\}\.\(79\)The test that declares𝖧\\mathsf\{H\}whena^M\>9/16\\widehat\{a\}\_\{M\}\>9/16is consistent; Hoeffding’s inequality bounds each error byexp\(−M/128\)\\exp\(\-M/128\)\. Finally,P⁡\(Y1=1\)=E⁡\[pU\]P\(Y\_\{1\}=1\)=E\[p\_\{U\}\]andP⁡\(Y1=1,Y2=1\)=E⁡\[pU2\]P\(Y\_\{1\}=1,Y\_\{2\}=1\)=E\[p\_\{U\}^\{2\}\], proving[Eq\.46](https://arxiv.org/html/2608.25118#S5.E46)\. ∎

The full mixing law is not identified by these two moments without further restrictions\. For example,

12​δ1/4\+12​δ3/4and18​δ0\+34​δ1/2\+18​δ1\\tfrac\{1\}\{2\}\\delta\_\{1/4\}\+\\tfrac\{1\}\{2\}\\delta\_\{3/4\}\\quad\\text\{and\}\\quad\\tfrac\{1\}\{8\}\\delta\_\{0\}\+\\tfrac\{3\}\{4\}\\delta\_\{1/2\}\+\\tfrac\{1\}\{8\}\\delta\_\{1\}\(80\)are distinct laws forpUp\_\{U\}with the same first moment1/21/2and second moment5/165/16\.

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\[[43](https://arxiv.org/html/2608.25118#bib.bib11)\]\.

### E\.4Attribution information and response dependence are independent

For chosen regular\-conditional versions, evidence is uninformative about the unit exactly when

U⟂⟂𝒪⟺P\(U∈⋅∣𝒪\)=Π\(⋅\)almost surely\.U\\perp\\\!\\\!\\\!\\perp\\mathcal\{O\}\\quad\\Longleftrightarrow\\quad P\(U\\in\\cdot\\mid\\mathcal\{O\}\)=\\Pi\(\\cdot\)\\quad\\text\{almost 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 beliefQϕQ\_\{\\phi\}only when that belief is calibrated to the declared identity target\.

Independently, suppose the response family is unit\-homogeneous at a declared queryww:Kuw=KwK\_\{u\}^\{w\}=K^\{w\}forΠ\\Pi\-almost everyuu\. Then every learner belief withQϕ\(⋅∣𝒪\)≪ΠQ\_\{\\phi\}\(\\cdot\\mid\\mathcal\{O\}\)\\ll\\Pisatisfies

∫𝒰Kuw​Qϕ​\(𝑑u∣𝒪\)=Kw\.\\int\_\{\\mathcal\{U\}\}K\_\{u\}^\{w\}\\,Q\_\{\\phi\}\(du\\mid\\mathcal\{O\}\)=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[Proposition4](https://arxiv.org/html/2608.25118#Thmproposition4)\.

### E\.5Proof and qualifications for mixture stability

Fixoo,ww, and𝒰o\\mathcal\{U\}\_\{o\}as in[Lemma3](https://arxiv.org/html/2608.25118#Thmproposition3), and suppresswwby writingKu:=Ku,wK\_\{u\}:=K\_\{u,w\}andK^u:=K^u,w\\widehat\{K\}\_\{u\}:=\\widehat\{K\}\_\{u,w\}\. Assume the unit and outcome spaces are standard Borel,KKandK^\\widehat\{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 replacing𝒰\\mathcal\{U\}by𝒰o\\mathcal\{U\}\_\{o\}in the diameter does not discard mixture mass\.

###### Proof of[Lemma3](https://arxiv.org/html/2608.25118#Thmproposition3)\.

The triangle inequality gives

dTV​\(Qo​K^,Po​K\)\\displaystyle d\_\{\\rm TV\}\(Q\_\{o\}\\widehat\{K\},P\_\{o\}K\)≤dTV​\(Qo​K^,Qo​K\)\+dTV​\(Qo​K,Po​K\),\\displaystyle\\leq d\_\{\\rm TV\}\(Q\_\{o\}\\widehat\{K\},Q\_\{o\}K\)\+d\_\{\\rm TV\}\(Q\_\{o\}K,P\_\{o\}K\),\(83\)where, for example,Qo​K=∫𝒰Ku​Qo​\(𝑑u\)Q\_\{o\}K=\\int\_\{\\mathcal\{U\}\}K\_\{u\}Q\_\{o\}\(du\)\. For the first term, the definition of total variation and the triangle inequality for integrals yield

dTV​\(Qo​K^,Qo​K\)\\displaystyle d\_\{\\rm TV\}\(Q\_\{o\}\\widehat\{K\},Q\_\{o\}K\)≤∫𝒰dTV​\(K^u,Ku\)​Qo​\(𝑑u\)\.\\displaystyle\\leq\\int\_\{\\mathcal\{U\}\}d\_\{\\rm TV\}\(\\widehat\{K\}\_\{u\},K\_\{u\}\)Q\_\{o\}\(du\)\.\(84\)
For the second term, letΔo=Qo−Po\\Delta\_\{o\}=Q\_\{o\}\-P\_\{o\}andt=dTV​\(Qo,Po\)t=d\_\{\\rm TV\}\(Q\_\{o\},P\_\{o\}\)\. Ift=0t=0there is nothing to prove\. Otherwise the Jordan decomposition of the zero\-mass signed measureΔo\\Delta\_\{o\}can be written asΔo=t⁡\(α−β\)\\Delta\_\{o\}=t\(\\alpha\-\\beta\)for probability measuresα\\alphaandβ\\betasupported on𝒰o\\mathcal\{U\}\_\{o\}\. For any measurable outcome eventAA,

\|\(α​K\)​\(A\)−\(β​K\)​\(A\)\|\\displaystyle\\left\|\(\\alpha K\)\(A\)\-\(\\beta K\)\(A\)\\right\|=\|∫∫⁡\[Ku​\(A\)−Kv​\(A\)\]​α​\(𝑑u\)​β​\(𝑑v\)\|\\displaystyle=\\left\|\\int\\\!\\\!\\int\\bigl\[K\_\{u\}\(A\)\-K\_\{v\}\(A\)\\bigr\]\\alpha\(du\)\\beta\(dv\)\\right\|\(85\)≤supu,v∈𝒰odTV​\(Ku,Kv\)\.\\displaystyle\\leq\\sup\_\{u,v\\in\\mathcal\{U\}\_\{o\}\}d\_\{\\rm TV\}\(K\_\{u\},K\_\{v\}\)\.\(86\)Taking the supremum overAAand multiplying byttgives

dTV​\(Qo​K,Po​K\)≤\[supu,v∈𝒰odTV​\(Ku,Kv\)\]​dTV​\(Qo,Po\)\.d\_\{\\rm TV\}\(Q\_\{o\}K,P\_\{o\}K\)\\leq\\left\[\\sup\_\{u,v\\in\\mathcal\{U\}\_\{o\}\}d\_\{\\rm TV\}\(K\_\{u\},K\_\{v\}\)\\right\]d\_\{\\rm TV\}\(Q\_\{o\},P\_\{o\}\)\.\(87\)Combining[Eqs\.83](https://arxiv.org/html/2608.25118#A5.E83),[84](https://arxiv.org/html/2608.25118#A5.E84)and[87](https://arxiv.org/html/2608.25118#A5.E87)proves[Eq\.39](https://arxiv.org/html/2608.25118#S5.E39)\. ∎

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\. IfQoQ\_\{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\.6Oracle, deployed, row\-weighted, and unit\-weighted risks

If the trueuuis supplied to the response predictor, the population\-first oracle risk separates which\-individual and fixed\-individual variation:

RΠoracle​\(f\)=∫𝒰\[∫ℓ⁡\(y,f⁡\(x,u\)\)​Ku​\(𝑑x,𝑑y\)\]​Π​\(𝑑u\)\.R\_\{\\Pi\}^\{\\mathrm\{oracle\}\}\(f\)=\\int\_\{\\mathcal\{U\}\}\\left\[\\int\\ell\\\!\\left\(y,f\(x;u\)\\right\)K\_\{u\}\(dx,dy\)\\right\]\\Pi\(du\)\.\(88\)The outer integral variesUU; the inner integral retains exogenous/event variation at fixeduu\. This is a diagnostic for the response layer, not the deployed risk of a learner that knows only a formed token, or onlyQϕQ\_\{\\phi\}in the identity\-belief specialization\.

Let a declared deployment experiment generate factual evidence𝒪\\mathcal\{O\}, a response queryXQX^\{Q\}, any explicit response contextCQC^\{Q\}, and the target responseYQY^\{Q\}\. Under log loss, the default deployed risk of the pair\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\)is

Rdep​\(θ,ϕ\)=EPdep​\[−log⁡\{∫𝒵rθ​\(YQ∣XQ,CQ,z\)​Tϕ​\(𝑑z∣𝒪,CQ\)\}\],R\_\{\\mathrm\{dep\}\}\(\\theta,\\phi\)=E\_\{P\_\{\\mathrm\{dep\}\}\}\\\!\\left\[\-\\log\\\!\\left\\\{\\int\_\{\\mathcal\{Z\}\}r\_\{\\theta\}\(Y^\{Q\}\\mid X^\{Q\},C^\{Q\},z\)\\,T\_\{\\phi\}\(dz\\mid\\mathcal\{O\},C^\{Q\}\)\\right\\\}\\right\],\(89\)when a densityrθr\_\{\\theta\}ofRθR\_\{\\theta\}exists\. In the identity\-belief specialization used by the theorems, the same experiment scores

Rdep​\(θ,ϕ\)=EPdep​\[−log⁡\{∫𝒰pθ​\(YQ∣XQ,CQ,u\)​Qϕ​\(𝑑u∣𝒪\)\}\]\.R\_\{\\mathrm\{dep\}\}\(\\theta,\\phi\)=E\_\{P\_\{\\mathrm\{dep\}\}\}\\\!\\left\[\-\\log\\\!\\left\\\{\\int\_\{\\mathcal\{U\}\}p\_\{\\theta\}\(Y^\{Q\}\\mid X^\{Q\},C^\{Q\},u\)Q\_\{\\phi\}\(du\\mid\\mathcal\{O\}\)\\right\\\}\\right\]\.\(90\)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\-uuresponse 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\.7Repeated\-unit weighting and splitting

In an observed\-attribution dataset, letM=\|𝒰𝒟\|M=\|\\mathcal\{U\}\_\{\\mathcal\{D\}\}\|\. For each observed unit define

I⁡\(u\)=\{i:ui=u\},m⁡\(u\)=\|I⁡\(u\)\|\.I\(u\)=\\\{i:u\_\{i\}=u\\\},\\qquad m\(u\)=\|I\(u\)\|\.\(91\)For sample lossesℓi\\ell\_\{i\}define the mean loss of unituubyℓ¯​\(u\)=m​\(u\)−1​∑i∈I⁡\(u\)ℓi\\bar\{\\ell\}\(u\)=m\(u\)^\{\-1\}\\sum\_\{i\\in I\(u\)\}\\ell\_\{i\}\. The row\-average and unit\-average risks are

R^row=1N​∑i=1Nℓi=∑u∈𝒰𝒟m⁡\(u\)N​ℓ¯​\(u\),R^unit=1M​∑u∈𝒰𝒟ℓ¯​\(u\)\.\\widehat\{R\}\_\{\\mathrm\{row\}\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\ell\_\{i\}=\\sum\_\{u\\in\\mathcal\{U\}\_\{\\mathcal\{D\}\}\}\\frac\{m\(u\)\}\{N\}\\bar\{\\ell\}\(u\),\\qquad\\widehat\{R\}\_\{\\mathrm\{unit\}\}=\\frac\{1\}\{M\}\\sum\_\{u\\in\\mathcal\{U\}\_\{\\mathcal\{D\}\}\}\\bar\{\\ell\}\(u\)\.\(92\)
###### Proof of[Corollary5](https://arxiv.org/html/2608.25118#Thmproposition5)\.

The two expressions are linear in\(ℓ¯​\(u\)\)u∈𝒰𝒟\(\\bar\{\\ell\}\(u\)\)\_\{u\\in\\mathcal\{U\}\_\{\\mathcal\{D\}\}\}, so they agree for every such vector exactly when their coefficients agree for everyuu, that is, whenm⁡\(u\)/N=1/Mm\(u\)/N=1/M\. ∎

Neither objective is universally correct; the deployment estimand determines which weighting is appropriate\.

###### Proof of[Corollary6](https://arxiv.org/html/2608.25118#Thmproposition6)\.

The two excluded events are that allm⁡\(u\)m\(u\)records enter training and that all enter test; they are disjoint and have probabilitiespm⁡\(u\)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 FObservability Conditions

#### Component targets and training regimes\.

The world targetPu⋆P\_\{u\}^\{\\star\}and the learner pair\(Tϕ,Rθ\)\(T\_\{\\phi\},R\_\{\\theta\}\)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 approximateP⁡\(U∈d​u∣𝒪\)P\(U\\in du\\mid\\mathcal\{O\}\)\. Under trusted direct access there is no identity model to estimate: learning the lookup rowZu⁡\(k\)c=zθ​\(k\)Z\_\{u\(k\)\}^\{c\}=z\_\{\\theta\}\(k\)fits a token insideRθR\_\{\\theta\}, notQϕQ\_\{\\phi\}\. Any history\-derived state used at answer time must appear in the tokenizer context, incQc^\{Q\}, or inside the shared form; otherwise response sufficiency is not justified\. In[Section5](https://arxiv.org/html/2608.25118#S5)the complete response input is writtenWW, notZZ, so that it is not confused with the unit tokenZucZ\_\{u\}^\{c\}\.

#### Support and observational equivalence\.

The population law may support an individual valueu⋆u^\{\\star\}even when no training record is attributed to it\. Support alone does not concentrateP⁡\(U∈d​u∣𝒪\)P\(U\\in du\\mid\\mathcal\{O\}\)atu⋆u^\{\\star\}; 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 family𝔔\\mathfrak\{Q\}, the sameuuand the same factual evidence𝒪\\mathcal\{O\}are retained whilexQ∈𝔔x^\{Q\}\\in\\mathfrak\{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 givenUU\.

#### The fixed\-individual kernel retains exogenous variation\.

The kernelPu⋆​\(d​y∣xQ,cQ\)P\_\{u\}^\{\\star\}\(dy\\mid x^\{Q\},c^\{Q\}\)may remain stochastic afterU=uU=uis fixed\. Residual within\-unit variation is not which\-unit variation\.

## Appendix GEvaluation Checklist

A reproducible evaluation should pre\-specify:

1. 1\.the unit population and the span over which unit identity persists;
2. 2\.the factual\-evidence cutoff, response query and context, and current target;
3. 3\.the access regime—direct access or unit abduction—and what attribution truth is available for evaluation;
4. 4\.the evaluated target: fixed\-unit response, deployed marginalized prediction, or a unit\-level estimand;
5. 5\.whether the split targets known\-unit/new\-event or new\-unit generalization;
6. 6\.a matched unit\-omitting baseline and any negative control required by the claim\.

A learnable unit\-conditioned specification additionally names:

1. 1\.what is shared across units, here the formRθR\_\{\\theta\}, a simple relation in the token, with a linear predictor as the running instance \([Definition2](https://arxiv.org/html/2608.25118#Thmdefinition2)\);
2. 2\.which properties of the query–response relation vary with the unit, and are therefore expressed byZucZ\_\{u\}^\{c\};
3. 3\.which properties remain invariant across units;
4. 4\.how finitely many same\-unit observations constrain that unit’s token and hence its response law;
5. 5\.how the shared form and token space support prediction for a unit not previously observed; and
6. 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\.