Ontic state recovery through prediction

# Ontic State Recovery through Prediction
Source: [https://arxiv.org/html/2610.27139](https://arxiv.org/html/2610.27139)
###### Abstract

We encounter a system through its responses, yet its state concerns how the system is\. Predicting further responses promises knowledge beyond what is immediately observed\. What that knowledge tells us about the system’s state depends on what we mean by a state in the first place\. We give this notion a model\-relative meaning: a state is the complete response profile over a specified query universe, and configurations agreeing on every query are regarded as the same state\. From this perspective, a learned state is understood through the distinctions between these profiles that it retains\. Prediction concerns selected observations; recovery concerns the profile as a whole\. We examine their relationship through query ambiguity, state\-space geometry, and the choice of prediction targets\. Synthetic systems with known response laws and neural recordings provide complementary settings for this investigation\. The aim is to clarify how prediction can become a way of knowing a system, and what it would mean for a learned state to preserve the state being studied\.

## 1Introduction

Figure 1:Fromobservationto prediction\. Theencodermapsinput observationsto alearned state; thepredictorestimatesobservationsat thetarget queries\.Learning a system’s state requires deciding which differences between its possible conditions matter\. Knowing that state would let us reason beyond the aspects we happen to observe\. For an animal, a neural recording offers one such view: a pattern of activity from which we seek to understand the condition that gives rise to it\([Cunningham and Yu, 2014](https://arxiv.org/html/2610.27139#bib.bib7);[Churchland et al\., 2012](https://arxiv.org/html/2610.27139#bib.bib5)\)\. Giving this underlying condition a precise meaning is part of the problem of knowing it\.

Predicting further observations offers a way to study this problem\. Neural population models learn to predict held\-out activity\([Pei et al\., 2021](https://arxiv.org/html/2610.27139#bib.bib34)\), and models trained across recordings use related objectives to support downstream decoding\([Ye et al\., 2023](https://arxiv.org/html/2610.27139#bib.bib46)\)\. A learned state is useful here because it carries information about observations beyond its input\.

*“When does learning to predict a system amount to learning its state?”*

We make this question precise relative to a query universe\. A*query*specifies what is measured, and an*observation*is the resulting value\. Within that universe, the*ontic state*is the complete response profile: configurations agreeing on every query count as the same state\. In a neural recording, a query specifies a unit and a trailing\-window duration\. Anencodermaps input observations to a learned state, from which apredictorestimates observations at target queries \([figure1](https://arxiv.org/html/2610.27139#S1.F1)\)\. Theoretical work relates auxiliary prediction and context learning to downstream performance\([Lee et al\., 2021](https://arxiv.org/html/2610.27139#bib.bib24);[Zhai et al\., 2025](https://arxiv.org/html/2610.27139#bib.bib48)\)\. We ask how much of the ontic state can be recovered from the learned state that supports these predictions\.

Consider targets that all depend on one coordinate of a state\. Exact prediction constrains that coordinate while leaving the others free\.*Query ambiguity*measures the unresolved state differences\. It connects the distinctions tested by prediction to the profile we seek to recover\.

This connection motivates three contributions:

1. 1\.Ontic\-state formulation\.We formulate ontic states as complete response profiles over a query universe\.
2. 2\.Recovery guarantees\.We establish prediction–recovery guarantees in both general settings and the low\-dimensional settings commonly studied in neuroscience\.
3. 3\.Fixed\-family sharpness\.We prove matching budget bounds on a fixed state family, distinguishing targets shared acrossencodersfrom targets tailored to eachencoder\.

## 2Ontic State

### 2\.1State, Query, and Observation

Figure 2:State,query, andobservation\. Thestatef∈ℱf\\in\\mathcal\{F\}andquery setQ=\{q1,q2\}⊆𝒰Q=\\\{q\_\{1\},q\_\{2\}\\\}\\subseteq\\mathcal\{U\}determine theobservation vectorf⁡\(Q\)f\(Q\)\. Its entries aref⁡\(q1\)f\(q\_\{1\}\)andf⁡\(q2\)f\(q\_\{2\}\); unlabeled points denote queries outsideQQ\.What does it mean for a system to be in a particular state, independently of how we describe it? We need a criterion of identity\. Leibniz’s account of individual substance locates identity in a complete concept determining the predicates of its subject\([Leibniz, 1686](https://arxiv.org/html/2610.27139#bib.bib25)\)\. For a scientific account of state, which properties should determine this identity?

We give these properties empirical meaning through the observations the system could produce\([Peirce, 1878](https://arxiv.org/html/2610.27139#bib.bib35);[James, 1907](https://arxiv.org/html/2610.27139#bib.bib19)\)\. Each query specifies one such observation\. Two configurations may agree in position and differ in velocity: whether we distinguish them depends on the queries admitted\. We therefore fix a query universe before defining its states\.

If no question in this universe can distinguish two putative states, we regard them as one ontic state: a difference in description alone does not constitute a difference in state\. Their common response profile fixes this identity independently of the coordinates used to represent it\([Poincaré, 1905](https://arxiv.org/html/2610.27139#bib.bib36)\)\. An experiment reveals part of the profile, while the state specifies responses throughout the universe\. We thus model a state as a function from queries to observations\.

###### Definition 2\.1\(Ontic State\)\.

An*ontic state*is an admissible bounded function

f:𝒰→ℝ,f:\\mathcal\{U\}\\to\\mathbb\{R\},\(1\)where𝒰\\mathcal\{U\}is a fixed nonempty set\. The collectionℱ\\mathcal\{F\}of all admissible ontic states is called the*ontic state space*\.

###### Definition 2\.2\(Query Universe\)\.

The common domain𝒰\\mathcal\{U\}of the ontic states is called the*query universe*, and its elements are called*queries*\. Forf∈ℱf\\in\\mathcal\{F\}andq∈𝒰q\\in\\mathcal\{U\}, the valuef⁡\(q\)f\(q\)is the*observation*returned by queryqqin stateff\.

An experiment records responses to a finite set of queries\. We collect these readings into an observation vector\.

###### Definition 2\.3\(Observation\)\.

A*query set*is a nonempty finite subsetQ⊆𝒰Q\\subseteq\\mathcal\{U\}\. The*observation vector*of a statef∈ℱf\\in\\mathcal\{F\}onQQis

f⁡\(Q\)≔\(f⁡\(q\)\)q∈Q∈ℝQ\.f\(Q\)\\coloneqq\\bigl\(f\(q\)\\bigr\)\_\{q\\in Q\}\\in\\mathbb\{R\}^\{Q\}\.\(2\)HereℝQ\\mathbb\{R\}^\{Q\}denotes real\-valued vectors indexed byQQ\.

To compare two profiles, we ask how far their responses can differ over the queries being considered\. Taking the largest difference gives a common distance for full states and finite observation vectors\.

###### Definition 2\.4\(Ontic Distance\)\.

For a nonempty index setIIand bounded real\-valued functionsu,vu,vwhose domains containII, define their distance onIIby

dI​\(u,v\)≔supi∈I\|u⁡\(i\)−v⁡\(i\)\|\.d\_\{I\}\(u,v\)\\coloneqq\\sup\_\{i\\in I\}\|u\(i\)\-v\(i\)\|\.\(3\)This is the supremum metric between their restrictions toII\. Vectors are functions on their index set; for finiteII, the supremum is a maximum\.

The*ontic distance*isd𝒰​\(f,g\)d\_\{\\mathcal\{U\}\}\(f,g\); the*observation distance*onQQisdQ​\(f​\(Q\),g​\(Q\)\)d\_\{Q\}\(f\(Q\),g\(Q\)\)\. The latter is a metric on observation vectors and a pseudometric on states, since distinct states can yield identical observations onQQ\.

[Figure2](https://arxiv.org/html/2610.27139#S2.F2)shows how a state and query set determine an observation vector\. The noisy extension treats the profile as a conditional mean response and measured activity as a noisy observation \([appendixE](https://arxiv.org/html/2610.27139#A5)\)\.

### 2\.2Query Ambiguity

A complete response profile defines a state, but how much of it do our observations determine? Hume’s question about inference beyond present observation brings this gap into focus\([Hume, 1902](https://arxiv.org/html/2610.27139#bib.bib18), sec\. IV\)\. Within a specified model class, optimal recovery makes the uncertainty quantitative\([Micchelli and Rivlin, 1977](https://arxiv.org/html/2610.27139#bib.bib28);[Binev et al\., 2024](https://arxiv.org/html/2610.27139#bib.bib3)\)\. We ask how different two full states can be when their recorded observations agree within a given tolerance\.

###### Definition 2\.5\(Query Ambiguity\)\.

Fix a query setQ⊆𝒰Q\\subseteq\\mathcal\{U\}\. For a tolerancer≥0r\\geq 0, define the*query ambiguity*onQQby

ωQ​\(r\)≔supf,g∈ℱdQ​\(f⁡\(Q\),g⁡\(Q\)\)≤rd𝒰​\(f,g\)\.\\omega\_\{Q\}\(r\)\\coloneqq\\sup\_\{\\begin\{subarray\}\{c\}f,g\\in\\mathcal\{F\}\\\\ d\_\{Q\}\(f\(Q\),g\(Q\)\)\\leq r\\end\{subarray\}\}d\_\{\\mathcal\{U\}\}\(f,g\)\.\(4\)

Atr=0r=0, ambiguity measures the state differences compatible with identical observations onQQ\. Positive tolerances allow for prediction error\. Anobservation decoderrecovers a state from the observation vector\.

###### Definition 2\.6\(Observation Decoder\)\.

Fix a query setQ⊆𝒰Q\\subseteq\\mathcal\{U\}\. Define theobservation decoder classby

𝒜Q≔\{aQ:ℝQ→ℱ\}\.\\mathcal\{A\}\_\{Q\}\\coloneqq\\\{a\_\{Q\}:\\mathbb\{R\}^\{Q\}\\to\\mathcal\{F\}\\\}\.\(5\)For anobservation decoderaQ∈𝒜Qa\_\{Q\}\\in\\mathcal\{A\}\_\{Q\}, its*recovery error*is

ℰQ​\(aQ\)≔supf∈ℱd𝒰​\(aQ​\(f⁡\(Q\)\),f\)\.\\mathcal\{E\}\_\{Q\}\(a\_\{Q\}\)\\coloneqq\\sup\_\{f\\in\\mathcal\{F\}\}d\_\{\\mathcal\{U\}\}\\bigl\(a\_\{Q\}\(f\(Q\)\),f\\bigr\)\.\(6\)The*optimal recovery error*from observations onQQis

ℰQ⋆≔infaQ∈𝒜QℰQ​\(aQ\)\.\\mathcal\{E\}\_\{Q\}^\{\\star\}\\coloneqq\\inf\_\{a\_\{Q\}\\in\\mathcal\{A\}\_\{Q\}\}\\mathcal\{E\}\_\{Q\}\(a\_\{Q\}\)\.\(7\)

Additional queries can resolve distinctions left open by the current observations\. We describe this accumulation of information by a nested query sequence\.

###### Definition 2\.7\(Query Sequence\)\.

A*query sequence*is a family\(Qp\)p≥1\(Q\_\{p\}\)\_\{p\\geq 1\}of query sets withQp⊆Qp\+1⊆𝒰Q\_\{p\}\\subseteq Q\_\{p\+1\}\\subseteq\\mathcal\{U\}for allp≥1p\\geq 1\. The stage\-ppobservation vector isf⁡\(Qp\)f\(Q\_\{p\}\)\.

###### Definition 2\.8\(State Separation\)\.

A query sequence\(Qp\)p≥1\(Q\_\{p\}\)\_\{p\\geq 1\}is said to*separate states inℱ\\mathcal\{F\}*if

∀f≠g∈ℱ,∃p≥1:f⁡\(Qp\)≠g⁡\(Qp\)\.\\forall f\\neq g\\in\\mathcal\{F\},\\quad\\exists p\\geq 1:\\quad f\(Q\_\{p\}\)\\neq g\(Q\_\{p\}\)\.\(8\)

Separating queries progressively resolve observation ambiguity \([figure8](https://arxiv.org/html/2610.27139#A1.F8)\)\. Compactness makes this resolution stable under vanishing tolerances\.

###### Lemma 2\.9\(Vanishing Ambiguity\)\.

Let\(ℱ,d𝒰\)\(\\mathcal\{F\},d\_\{\\mathcal\{U\}\}\)be nonempty and compact, and let\(Pm\)m≥1\(P\_\{m\}\)\_\{m\\geq 1\}be a query sequence separating states\. For nonnegative tolerancesrm→0r\_\{m\}\\to 0,

ωPm​\(rm\)→0\.\\omega\_\{P\_\{m\}\}\(r\_\{m\}\)\\to 0\.\(9\)

### 2\.3Prediction and Recovery

What must a learned representation preserve for us to recover the state it describes? Hertz’s account of representation centers on the correspondence between consequences\([Hertz, 1899](https://arxiv.org/html/2610.27139#bib.bib17), Introduction\)\. Here we test that correspondence by predicting further responses from encoded observations\. The preceding lemma explains why the targets matter: agreement on sufficiently rich readings constrains the difference between full states\. Prediction can thereby test whether a representation retains the distinctions defining the state\.

###### Definition 2\.10\(Prediction and Recovery\)\.

The learning pipeline uses two disjoint query sets:QQindexes the input observationsf⁡\(Q\)f\(Q\), andPPindexes the target observationsf⁡\(P\)f\(P\)\. Anencodere:ℝQ→𝒵e:\\mathbb\{R\}^\{Q\}\\to\\mathcal\{Z\}mapsf⁡\(Q\)f\(Q\)to the learned statee⁡\(f⁡\(Q\)\)e\(f\(Q\)\)in a nonempty space𝒵\\mathcal\{Z\}\. From this learned state, apredictorb:𝒵→ℝPb:\\mathcal\{Z\}\\to\\mathbb\{R\}^\{P\}estimatesf⁡\(P\)f\(P\), while astate decodera:𝒵→ℱa:\\mathcal\{Z\}\\to\\mathcal\{F\}reconstructs the ontic stateff\. Their errors are

ℒQ,P​\(b,e\)\\displaystyle\\mathcal\{L\}\_\{Q,P\}\(b;e\)≔supf∈ℱdP​\(\(b∘e\)​\(f⁡\(Q\)\),f⁡\(P\)\),\\displaystyle\\coloneqq\\sup\_\{f\\in\\mathcal\{F\}\}d\_\{P\}\\bigl\(\(b\\circ e\)\(f\(Q\)\),f\(P\)\\bigr\),\(10\)ℛQ​\(a,e\)\\displaystyle\\mathcal\{R\}\_\{Q\}\(a;e\)≔supf∈ℱd𝒰​\(\(a∘e\)​\(f⁡\(Q\)\),f\)\.\\displaystyle\\coloneqq\\sup\_\{f\\in\\mathcal\{F\}\}d\_\{\\mathcal\{U\}\}\\bigl\(\(a\\circ e\)\(f\(Q\)\),f\\bigr\)\.LetℬP\\mathcal\{B\}\_\{P\}and𝒜\\mathcal\{A\}be the classes of all maps of these respective types\. For a fixedencoder, define

ℒQ,P⋆​\(e\)≔infb∈ℬPℒQ,P​\(b,e\),ℛQ⋆​\(e\)≔infa∈𝒜ℛQ​\(a,e\)\.\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\\coloneqq\\inf\_\{b\\in\\mathcal\{B\}\_\{P\}\}\\mathcal\{L\}\_\{Q,P\}\(b;e\),\\qquad\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\\coloneqq\\inf\_\{a\\in\\mathcal\{A\}\}\\mathcal\{R\}\_\{Q\}\(a;e\)\.\(11\)

Both optimizations hold theencoderfixed;ℛQ⋆​\(e\)\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)measures the recovery accuracy available from its learned state\.

###### Corollary 2\.11\(Input Ambiguity\)\.

For a nonempty state family, query setQQ, and encoderee,

ℛQ⋆​\(e\)≥ℰQ⋆≥12​ωQ​\(0\)\.\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\\geq\\mathcal\{E\}\_\{Q\}^\{\\star\}\\geq\\frac\{1\}\{2\}\\omega\_\{Q\}\(0\)\.\(12\)

Input queries supply distinctions to theencoder; targets test their preservation in its learned state\.

### 2\.4Low\-Dimensional Geometry

Figure 3:From population activity to state geometry\. The neural inset motivates a low\-dimensional model for theontic state spaceℱ\\mathcal\{F\}\.A neural recording contains many coordinates, yet activity within a task can vary through a few latent variables\([Gallego et al\., 2017](https://arxiv.org/html/2610.27139#bib.bib11);[Jazayeri and Ostojic, 2021](https://arxiv.org/html/2610.27139#bib.bib20)\)\. This organization can constrain which patterns are learned\([Sadtler et al\., 2014](https://arxiv.org/html/2610.27139#bib.bib37)\)and take forms such as the toroidal geometry of grid\-cell activity\([Gardner et al\., 2022](https://arxiv.org/html/2610.27139#bib.bib12)\)\. The resulting state family may curve through a much larger linear space\([De and Chaudhuri, 2023](https://arxiv.org/html/2610.27139#bib.bib9)\)\.

Recovering a state within such a family requires responses that distinguish its members\. Classical population decoding illustrates how neural responses carry this information: movement direction can be read from motor\-cortical ensembles\([Georgopoulos et al\., 1986](https://arxiv.org/html/2610.27139#bib.bib13)\), and position reconstructed from hippocampal place\-cell activity\([Zhang et al\., 1998](https://arxiv.org/html/2610.27139#bib.bib49)\)\. Models of population codes explain how neuronal tuning and background activity affect readout accuracy\([Seung and Sompolinsky, 1993](https://arxiv.org/html/2610.27139#bib.bib42)\)\.

These observations lead to two quantitative requirements \([figure3](https://arxiv.org/html/2610.27139#S2.F3)\)\. The state family should admit a controlled number of representatives at each resolution, and queries outside the inputs should distinguish its members\. Covering numbers express the first requirement; observability expresses the second\.

###### Assumption 2\.12\(Low\-Dimensional Observability\)\.

For a query setQQwith𝒰∖Q≠∅\\mathcal\{U\}\\setminus Q\\neq\\varnothing, there are constantsC≥1C\\geq 1,d\>0d\>0, andα∈\(0,1\]\\alpha\\in\(0,1\]such that

𝒩ℱ​\(ε\)≤C​ε−d,0<ε≤1,\\mathcal\{N\}\_\{\\mathcal\{F\}\}\(\\varepsilon\)\\leq C\\varepsilon^\{\-d\},\\qquad 0<\\varepsilon\\leq 1,\(13\)where𝒩ℱ​\(ε\)\\mathcal\{N\}\_\{\\mathcal\{F\}\}\(\\varepsilon\)is the covering number: the fewest closedd𝒰d\_\{\\mathcal\{U\}\}\-balls of radiusε\\varepsilon, centered inℱ\\mathcal\{F\}, that coverℱ\\mathcal\{F\}\. The queries outsideQQdistinguish states quantitatively:

supp∈𝒰∖Q\|f⁡\(p\)−g⁡\(p\)\|≥α​d𝒰​\(f,g\),f,g∈ℱ\.\\sup\_\{p\\in\\mathcal\{U\}\\setminus Q\}\|f\(p\)\-g\(p\)\|\\geq\\alpha d\_\{\\mathcal\{U\}\}\(f,g\),\\qquad f,g\\in\\mathcal\{F\}\.\(14\)

## 3Main Results

Prediction tests the distinctions retained by a learned state\. If anencodermaps two states to the same learned state, everypredictormust return the same target observations for both\. Their target difference therefore imposes an unavoidable prediction error\.

###### Lemma 3\.1\(Prediction within Fibers\)\.

Assume\(ℱ,d𝒰\)\(\\mathcal\{F\},d\_\{\\mathcal\{U\}\}\)is nonempty and bounded\. Fix a query setQ⊆𝒰Q\\subseteq\\mathcal\{U\}, a nonempty finite target query setP⊆𝒰∖QP\\subseteq\\mathcal\{U\}\\setminus Q, and an encodere:ℝQ→𝒵e:\\mathbb\{R\}^\{Q\}\\to\\mathcal\{Z\}\. The optimal prediction error is attained and equals

ℒQ,P⋆​\(e\)=12​supf,g∈ℱe⁡\(f⁡\(Q\)\)=e⁡\(g⁡\(Q\)\)dP​\(f⁡\(P\),g⁡\(P\)\)\.\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)=\\frac\{1\}\{2\}\\sup\_\{\\begin\{subarray\}\{c\}f,g\\in\\mathcal\{F\}\\\\ e\(f\(Q\)\)=e\(g\(Q\)\)\\end\{subarray\}\}d\_\{P\}\(f\(P\),g\(P\)\)\.\(15\)

In the one\-coordinate example, two states can have identical target readings while differing elsewhere\. Apredictorcan be exact on those targets even if itsencodermerges the states\. Recovery depends on how much full\-state difference those same readings permit\.

### 3\.1General Case

Target ambiguity controls how prediction accuracy translates into state recovery\.

###### Theorem 3\.2\(General Prediction\-Recovery Bound\)\.

Let\(ℱ,d𝒰\)\(\\mathcal\{F\},d\_\{\\mathcal\{U\}\}\)be nonempty and bounded\. For disjoint query setsQ,PQ,Pand any encodere:ℝQ→𝒵e:\\mathbb\{R\}^\{Q\}\\to\\mathcal\{Z\},

ℒQ,P⋆​\(e\)≤ℛQ⋆​\(e\)≤ωP​\(2​ℒQ,P⋆​\(e\)\)\.\\mathmakebox\[0\.28\]\[r\]\{\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\}\\leq\\boxed\{\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\}\\leq\\mathmakebox\[0\.28\]\[l\]\{\\omega\_\{P\}\\bigl\(2\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\\bigr\)\.\}\(16\)The upper bound also holds withℒQ,P​\(b,e\)\\mathcal\{L\}\_\{Q,P\}\(b;e\)for any predictorbbwith finite error\.

When the state space is compact, a growing set of separating targets resolves the remaining ambiguity\. Prediction and recovery then converge together\.

###### Theorem 3\.3\(General Prediction\-Recovery Equivalence\)\.

Let\(ℱ,d𝒰\)\(\\mathcal\{F\},d\_\{\\mathcal\{U\}\}\)be nonempty and compact, and let\(Pm\)m≥1\(P\_\{m\}\)\_\{m\\geq 1\}be a nested query sequence separating states\. For eachmm, choose a query setQm⊆𝒰∖PmQ\_\{m\}\\subseteq\\mathcal\{U\}\\setminus P\_\{m\}and an encoderem:ℝQm→𝒵me\_\{m\}:\\mathbb\{R\}^\{Q\_\{m\}\}\\to\\mathcal\{Z\}\_\{m\}\. Then

ℒQm,Pm⋆\(em\)→0⟺ℛQm⋆\(em\)→0\.\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\to 0\\quad\\Longleftrightarrow\\quad\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\to 0\.\(17\)

For fixed input queries and a fixedencoder, vanishing prediction error along such a separating sequence leaves no distinct ontic states with the same learned state: an exactstate decoderexists \([corollaryC\.1](https://arxiv.org/html/2610.27139#A3.Thmtheorem1)\)\. If the map to a Euclidean learned state is also continuous, compactness makes it a homeomorphism onto its image \([corollaryC\.2](https://arxiv.org/html/2610.27139#A3.Thmtheorem2)\)\. Under these conditions, preserving the distinctions that define states also preserves the topology of the state space\.

### 3\.2Low\-Dimensional Case

[Assumption2\.12](https://arxiv.org/html/2610.27139#S2.Thmtheorem12)combines a bound on the number of representative states with the ability to distinguish them through queries outsideQQ\. Together, these conditions give a finite target budget\.

###### Theorem 3\.4\(Low\-Dimensional Prediction\-Recovery Bound\)\.

Letℱ\\mathcal\{F\}be nonempty and let\(ℱ,Q\)\(\\mathcal\{F\},Q\)satisfy[assumption2\.12](https://arxiv.org/html/2610.27139#S2.Thmtheorem12)with constantsC,d,αC,d,\\alpha\. For every integerm≥C2/2m\\geq C^\{2\}/2, there is a target setP⊆𝒰∖QP\\subseteq\\mathcal\{U\}\\setminus Q,1≤\|P\|≤m1\\leq\|P\|\\leq m, such that every encodereesatisfies

ℒQ,P⋆​\(e\)≤ℛQ⋆​\(e\)≤2α​ℒQ,P⋆​\(e\)\+2\+αα​\(C22​m\)12​d\.\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\\leq\\boxed\{\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\}\\leq\\frac\{2\}\{\\alpha\}\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\+\\frac\{2\+\\alpha\}\{\\alpha\}\\left\(\\frac\{C^\{2\}\}\{2m\}\\right\)^\{\\frac\{1\}\{2d\}\}\.\(18\)The upper bound also holds withℒQ,P​\(b,e\)\\mathcal\{L\}\_\{Q,P\}\(b;e\)for any predictorbbwith finite error\.

The recovery allowance vanishes as the target budget grows, yielding convergence and rate equivalences\.

###### Theorem 3\.5\(Low\-Dimensional Prediction\-Recovery Equivalence\)\.

Letℱ\\mathcal\{F\}be nonempty\. Assume\(ℱ,Qm\)\(\\mathcal\{F\},Q\_\{m\}\)satisfies[assumption2\.12](https://arxiv.org/html/2610.27139#S2.Thmtheorem12)with commonC,d,αC,d,\\alphafor every integerm≥C2/2m\\geq C^\{2\}/2, and choosePmP\_\{m\}by[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4)\. Then every encoder sequence\(em\)\(e\_\{m\}\)satisfies

ℒQm,Pm⋆\(em\)→0⟺ℛQm⋆\(em\)→0\.\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\to 0\\quad\\Longleftrightarrow\\quad\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\to 0\.\(19\)Moreover,

ℒQm,Pm⋆\(em\)=O\(m−1/\(2d\)\)⟺ℛQm⋆\(em\)=O\(m−1/\(2d\)\)\.\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)=O\(m^\{\-1/\(2d\)\}\)\\quad\\Longleftrightarrow\\quad\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(e\_\{m\}\)=O\(m^\{\-1/\(2d\)\}\)\.\(20\)

### 3\.3Sharpness on a Fixed State Family

We next examine whether the budget rate can be improved when one target set serves everyencoder\. Forα=1\\alpha=1, we measure the additive recovery cost beyond2​ℒQ,P⋆​\(e\)2\\mathcal\{L\}^\{\\star\}\_\{Q,P\}\(e\)on a fixed state family\. Hereddis the covering exponent; the family has topological dimension one\.

###### Definition 3\.6\(Uniform Transfer Residual\)\.

LetℰLip\\mathcal\{E\}\_\{\\mathrm\{Lip\}\}be the globally Lipschitz mapse:ℝQ→ℝ2e:\\mathbb\{R\}^\{Q\}\\to\\mathbb\{R\}^\{2\}\. For an integerm≥1m\\geq 1, the*uniform transfer residual*is

𝒯m​\(ℱ,Q\)≔infP⊆𝒰∖Q1≤\|P\|≤msupe∈ℰLip\[ℛQ⋆​\(e\)−2​ℒQ,P⋆​\(e\)\]\.\\mathcal\{T\}\_\{m\}\(\\mathcal\{F\},Q\)\\coloneqq\\inf\_\{\\begin\{subarray\}\{c\}P\\subseteq\\mathcal\{U\}\\setminus Q\\\\ 1\\leq\|P\|\\leq m\\end\{subarray\}\}\\sup\_\{e\\in\\mathcal\{E\}\_\{\\mathrm\{Lip\}\}\}\\bigl\[\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\-2\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\\bigr\]\.\(21\)

The target set is selected before theencoder\. The family below has identifying inputs and even a single query outsideQQwhose exact response identifies the state \([equation92](https://arxiv.org/html/2610.27139#A3.E92)\)\. Anencodercan nevertheless merge distinct states\. The residual measures the additive cost of controlling their recovery by twice the prediction error, uniformly acrossencoders\.

###### Theorem 3\.7\(Sharp Uniform Transfer\)\.

For everyd\>1d\>1, there exist a fixed countable query universe𝒰\\mathcal\{U\}, a singleton input setQQ, and a compact state familyℱ\\mathcal\{F\}homeomorphic to\[0,1\]\[0,1\], of diameter11\. The pair\(ℱ,Q\)\(\\mathcal\{F\},Q\)satisfies[assumption2\.12](https://arxiv.org/html/2610.27139#S2.Thmtheorem12)withC=CdC=C\_\{d\}andα=1\\alpha=1, and its input observation map is injective\. For every integerm≥Cd2/2m\\geq C\_\{d\}^\{2\}/2,

16m−1/\(2d\)≤𝒯m​\(ℱ,Q\)≤3\(Cd22​m\)1/\(2​d\)\.\\mathmakebox\[0\.28\]\[r\]\{\\frac\{1\}\{6\}m^\{\-1/\(2d\)\}\}\\leq\\boxed\{\\mathcal\{T\}\_\{m\}\(\\mathcal\{F\},Q\)\}\\leq\\mathmakebox\[0\.28\]\[l\]\{3\\left\(\\frac\{C\_\{d\}^\{2\}\}\{2m\}\\right\)^\{1/\(2d\)\}\.\}\(22\)The supremum in[equation21](https://arxiv.org/html/2610.27139#S3.E21)has the same value when restricted to11\-Lipschitz encoders\.

The order of selection matters as well\. On this same family, a target selected for one fixed11\-Lipschitzencodercan achieveℛQ⋆​\(e\)≤2​ℒQ,\{p\}⋆​\(e\)\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\\leq 2\\mathcal\{L\}\_\{Q,\\\{p\\\}\}^\{\\star\}\(e\)with a single query \([corollaryC\.10](https://arxiv.org/html/2610.27139#A3.Thmtheorem10)\)\. This selection uses the complete state family\. Requiring one target set to serve everyencodergives the sharp budget in[theorem3\.7](https://arxiv.org/html/2610.27139#S3.Thmtheorem7)\. Classical optimal recovery studies the radius left by fixed information\([Traub et al\., 1983](https://arxiv.org/html/2610.27139#bib.bib43);[Krieg and Ullrich, 2026](https://arxiv.org/html/2610.27139#bib.bib23)\); our budget measures how many prediction targets suffice to control that radius uniformly across learned representations\.

### 3\.4From Recorded Error to Recovery

The bounds above use prediction error over the entire state family\. Training supplies finitely many observation pairs\. To pass from recorded error to a recovery guarantee, we need to know both how these records cover possible observations and how predictions vary between them\.

###### Definition 3\.8\(Recorded Error and Coverage\)\.

Fix disjoint query setsQ,PQ,P, a pair\(e,b\)\(e,b\), and statesf1,…,fn∈ℱf\_\{1\},\\ldots,f\_\{n\}\\in\\mathcal\{F\}withn≥1n\\geq 1indexing the recorded pairs\(fi​\(Q\),fi​\(P\)\)\(f\_\{i\}\(Q\),f\_\{i\}\(P\)\)\. The*empirical prediction error*and*observation coverage radius*are

ℒ^Q,P,n​\(b,e\)≔max1≤i≤n⁡dP​\(\(b∘e\)​\(fi​\(Q\)\),fi​\(P\)\),\\widehat\{\\mathcal\{L\}\}\_\{Q,P,n\}\(b;e\)\\coloneqq\\max\_\{1\\leq i\\leq n\}d\_\{P\}\\bigl\(\(b\\circ e\)\(f\_\{i\}\(Q\)\),f\_\{i\}\(P\)\\bigr\),\(23\)hn≔supf∈ℱmin1≤i≤n⁡dQ∪P​\(f⁡\(Q∪P\),fi​\(Q∪P\)\)\.h\_\{n\}\\coloneqq\\sup\_\{f\\in\\mathcal\{F\}\}\\min\_\{1\\leq i\\leq n\}d\_\{Q\\cup P\}\\bigl\(f\(Q\\cup P\),f\_\{i\}\(Q\\cup P\)\\bigr\)\.\(24\)

The radius measures coverage jointly in inputs and targets: every possible observation pair lies withinhnh\_\{n\}of a record\. Prediction stability controls how the learned map responds over this distance\.

###### Assumption 3\.9\(Prediction Stability\)\.

For the pair\(e,b\)\(e,b\), there is a nondecreasing functionρ:\[0,∞\)→\[0,∞\)\\rho:\[0,\\infty\)\\to\[0,\\infty\)withρ⁡\(0\)=0\\rho\(0\)=0andρ⁡\(t\)→0\\rho\(t\)\\to 0ast↓0t\\downarrow 0such that, for allf,g∈ℱf,g\\in\\mathcal\{F\},

dP​\(\(b∘e\)​\(f⁡\(Q\)\),\(b∘e\)​\(g⁡\(Q\)\)\)≤ρ⁡\(dQ​\(f⁡\(Q\),g⁡\(Q\)\)\)\.d\_\{P\}\\bigl\(\(b\\circ e\)\(f\(Q\)\),\(b\\circ e\)\(g\(Q\)\)\\bigr\)\\leq\\rho\\bigl\(d\_\{Q\}\(f\(Q\),g\(Q\)\)\\bigr\)\.\(25\)

For a nearby record, the prediction changes by at mostρ⁡\(hn\)\\rho\(h\_\{n\}\)and the target reading by at mosthnh\_\{n\}\. Adding these changes to the recorded error controls prediction at the unrecorded state\. A Lipschitz prediction map hasρ⁡\(t\)=K​t\\rho\(t\)=Kt\.

###### Proposition 3\.10\(Recovery from Records\)\.

Let\(ℱ,d𝒰\)\(\\mathcal\{F\},d\_\{\\mathcal\{U\}\}\)be nonempty and compact\. For a pair\(e,b\)\(e,b\)satisfying[assumption3\.9](https://arxiv.org/html/2610.27139#S3.Thmtheorem9), the prediction error obeys

En≔ℒ^Q,P,n​\(b,e\)\+ρ⁡\(hn\)\+hn,\\displaystyle E\_\{n\}\\coloneqq\\widehat\{\\mathcal\{L\}\}\_\{Q,P,n\}\(b;e\)\+\\rho\(h\_\{n\}\)\+h\_\{n\},\(26\)ℒ^Q,P,n​\(b,e\)≤ℒQ,P​\(b,e\)≤En,\\displaystyle\\mathmakebox\[0\.20\]\[r\]\{\\widehat\{\\mathcal\{L\}\}\_\{Q,P,n\}\(b;e\)\}\\leq\\boxed\{\\mathcal\{L\}\_\{Q,P\}\(b;e\)\}\\leq\\mathmakebox\[0\.20\]\[l\]\{E\_\{n\},\}\(27\)and the recovery error satisfies

ℛQ⋆​\(e\)≤ωP​\(2​En\)\.\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\\leq\\omega\_\{P\}\(2E\_\{n\}\)\.\(28\)Under[assumption2\.12](https://arxiv.org/html/2610.27139#S2.Thmtheorem12), for targets selected by[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4),

ℛQ⋆​\(e\)≤2α​En\+2\+αα​\(C22​m\)1/\(2​d\)\.\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\\leq\\frac\{2\}\{\\alpha\}E\_\{n\}\+\\frac\{2\+\\alpha\}\{\\alpha\}\\left\(\\frac\{C^\{2\}\}\{2m\}\\right\)^\{1/\(2d\)\}\.\(29\)

The terms answer different questions\. The empirical error measures fit on recorded pairs\. Their coverage and the prediction modulus control what this fit implies between records\. The target budget controls distinctions left unresolved even by a uniformly accuratepredictor\. More training records and more target queries address these two sources of uncertainty separately\. The fine\-tuning results in[theoremsD\.4](https://arxiv.org/html/2610.27139#A4.Thmtheorem4)and[D\.5](https://arxiv.org/html/2610.27139#A4.Thmtheorem5)include model\-class error and optimization tolerance;[appendixE](https://arxiv.org/html/2610.27139#A5)also treats measurement perturbations\.

## 4Experiment

The experiments examine prediction and recovery with two kinds of access to the state\. OnSphere,Torus, andGeneralfamilies, the known response law makes the complete profile available for evaluation\. Random target sampling tests separation and budget dependence;[theorem3\.7](https://arxiv.org/html/2610.27139#S3.Thmtheorem7)supplies the sharpness result analytically\. Neural recordings then let us examine covering geometry, held\-out prediction, and recovery of a fixed model’s hidden state as an ontic\-state surrogate\.

### 4\.1Synthetic Experiment

Figure 4:Synthetic state families\. Representative low\-dimensional views of the sphere, product of circles and independent box\.We use three synthetic state families \([figure4](https://arxiv.org/html/2610.27139#S4.F4)\):Sphereis uniform on a unit sphere;Torusconsists of independent circle factors;Generalhas independent uniform coordinates in a high\-dimensional box\. A common observation law embeds them into the same query universe\. We jointly train anencoderandpredictoron input–target observations, then recover the state through a deterministic ground\-truth \(GT\) map, following[theorem3\.2](https://arxiv.org/html/2610.27139#S3.Thmtheorem2)\. Here the coordinate structure determines the random\-sampling budget; the experiment tests how target separation transfers prediction accuracy to recovery\.

For a latent vectorx∈\[−1,1\]rx\\in\[\-1,1\]^\{r\}in one of the three families, define

fx​\(j,k\)=tanh⁡\(aj​k​sj​xi⁡\(j\)\+cj​k\),f\_\{x\}\(j,k\)=\\tanh\\bigl\(a\_\{jk\}s\_\{j\}x\_\{i\(j\)\}\+c\_\{jk\}\\bigr\),\(30\)wherei⁡\(j\)i\(j\)selects a coordinate,sj∈\{−1,1\}s\_\{j\}\\in\\\{\-1,1\\\}, andaj​k\>0,cj​k∈ℝa\_\{jk\}\>0,c\_\{jk\}\\in\\mathbb\{R\}are fixed channel coefficients\. Each family uses the same coefficient law\. Repeating low\-dimensional coordinates across the query universe changes the geometry of the state family while retaining this observation mechanism\.

Given predicted target observationsy∈ℝPy\\in\\mathbb\{R\}^\{P\}, the ideal deterministic GT map returns

f^​\(y\)∈arg​ming∈ℱ⁡dP​\(g⁡\(P\),y\)\.\\widehat\{f\}\(y\)\\in\\operatorname\*\{arg\\,min\}\_\{g\\in\\mathcal\{F\}\}d\_\{P\}\\bigl\(g\(P\),y\\bigr\)\.\(31\)The known observation law converts target\-error tolerances into coordinate intervals\. A deterministic feasibility calculation approximates this map \([appendixF](https://arxiv.org/html/2610.27139#A6)\) and supplies the complete recovered profile, including responses outside the target set\.

Observability check\.To apply[theorem3\.2](https://arxiv.org/html/2610.27139#S3.Thmtheorem2), we need to control target ambiguity\. Complete coordinate coverage gives an explicit constant before training\.

###### Proposition 4\.1\(Coordinate Observability\)\.

For the family in[equation30](https://arxiv.org/html/2610.27139#S4.E30), supposePPobserves every latent coordinate\. Letℓi\\ell\_\{i\}be the largestaj​k​sech2⁡\(aj​k\+\|cj​k\|\)a\_\{jk\}\\operatorname\{sech\}^\{2\}\(a\_\{jk\}\+\|c\_\{jk\}\|\)among target channels observing coordinateii, and letAiA\_\{i\}be the largestaj​ka\_\{jk\}among all channels observing it\. Then, for allf,g∈ℱf,g\\in\\mathcal\{F\},

αP=mini⁡ℓiAi\>0,dP​\(f,g\)≥αP​d𝒰​\(f,g\)\.\\alpha\_\{P\}=\\min\_\{i\}\\frac\{\\ell\_\{i\}\}\{A\_\{i\}\}\>0,\\qquad d\_\{P\}\(f,g\)\\geq\\alpha\_\{P\}d\_\{\\mathcal\{U\}\}\(f,g\)\.\(32\)

This inequality givesωP​\(t\)≤t/αP\\omega\_\{P\}\(t\)\\leq t/\\alpha\_\{P\}, so[theorem3\.2](https://arxiv.org/html/2610.27139#S3.Thmtheorem2)yieldsℛQ⋆​\(e\)≤2​ℒQ,P​\(b,e\)/αP\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\\leq 2\\mathcal\{L\}\_\{Q,P\}\(b;e\)/\\alpha\_\{P\}\. The observation law and target set determine this global constant; predictive training determines the target error\. Together they control differences in the complete profile\. The budget calculation and covering constants appear in[tables7](https://arxiv.org/html/2610.27139#A6.T7)and[F](https://arxiv.org/html/2610.27139#A6)\.

Error transfer\.With target separation established,[figure6](https://arxiv.org/html/2610.27139#S4.F6)follows the prediction\-to\-recovery relation\. The left panel compares finite\-test errors with the exact\-map benchmark\. The middle panel gives the sufficient separation budget under the coordinate\-sampling law\. The right panel extends the recovery comparison to the full known family through the conditional numerical audit below\.

Table 1:Conditional whole\-domain bounds for the ideal GT map: mean±\\pmsample standard deviation across paired runs\. Arithmetic conditions:[sectionF\.5](https://arxiv.org/html/2610.27139#A6.SS5)\.Whole\-domain audit\.A test set samples prediction error; the theoretical quantity is a supremum over the state family\. Where[proposition3\.10](https://arxiv.org/html/2610.27139#S3.Thmtheorem10)controls this supremum through observation coverage, the known coordinatewise law permits a scalar\-domain audit\. For each target, its error is a scalar function of the observed latent coordinate\. A fixed grid and a sensitivity bound computed from the saved weights enclose its error between grid points\. Under the arithmetic\-error assumptions in[sectionF\.5](https://arxiv.org/html/2610.27139#A6.SS5), the grid audit gives a conditional upper bound throughout the known family\. Applying the ideal GT map yields[table1](https://arxiv.org/html/2610.27139#S4.T1)\.

Every paired run at the audited separating budgets yields an ideal\-map bound below the full\-profile diameter \([table1](https://arxiv.org/html/2610.27139#S4.T1)\)\. The bound is evaluated for each trainedpredictorusing the protocol in[sectionF\.5](https://arxiv.org/html/2610.27139#A6.SS5)\.

For joint queries,[appendixG](https://arxiv.org/html/2610.27139#A7)derives a global prediction\-to\-recovery certificate from calibration records at known latent coordinates\.

Figure 5:Prediction\-to\-recovery transfer\.*Left:*test errors and exact\-map benchmark\.*Middle:*budgets for95%95\\%separation\.*Right:*conditional whole\-domain bounds\. Dashed lines: independent\-family budget and profile diameter\. Bars: repetition standard deviations\. See[tables7](https://arxiv.org/html/2610.27139#A6.T7)and[1](https://arxiv.org/html/2610.27139#S4.T1)\.
Figure 6:HC\-3 covering geometry\.*Left:*fixed\-radius cover sizes; bars span shift controls\.*Middle:*session reductions relative to median shift controls, grouped by animal; bars mark animal medians\.*Right:*median covering curves with session interquartile bands\. See[sectionH\.2](https://arxiv.org/html/2610.27139#A8.SS2)\.

### 4\.2Real Experiment

In neural recordings, we examine covering complexity, held\-out target prediction, and recovery of a fixed reference model’s hidden coordinates\. These coordinates define the surrogate state whose preservation we evaluate\.

Table 2:Diagnostic\-block greedy covers at the fixed radius in[table12](https://arxiv.org/html/2610.27139#A8.T12)\. Variance\-retention ranks use reference\-derived directions\.Observation geometry\.The HC\-3 hippocampal–entorhinal recordings\([Mizuseki et al\., 2014](https://arxiv.org/html/2610.27139#bib.bib30)\)span animals and behavioral contexts\. Chronological blocks separate preprocessing, reference geometry, diagnostic geometry, and prediction evaluation \([sectionH\.2](https://arxiv.org/html/2610.27139#A8.SS2)\)\.

We cover the diagnostic block with observed representatives in the maximum\-coordinate metric \([figure6](https://arxiv.org/html/2610.27139#S4.F6)\)\. At a fixed resolution, every session needs a smaller greedy cover than its median time\-shift control, with a median reduction of15\.2%15\.2\\%\. These controls preserve each unit’s marginal distribution and circular autocorrelation while changing alignment between units\. The comparison holds within every selected animal and across shorter and longer counting windows \([tables2](https://arxiv.org/html/2610.27139#S4.T2),[3](https://arxiv.org/html/2610.27139#S4.T3)and[13](https://arxiv.org/html/2610.27139#A8.T13)\)\. Covering curves measure the finite\-record analog of the approximation complexity in[equation13](https://arxiv.org/html/2610.27139#S2.E13)\. Variance\-retention ranks measure concentration along reference\-derived covariance directions\.

Table 3:Cover\-size reduction within each selected animal\. Values are session medians relative to the matched time\-shift median at the fixed radius\. Every selected session contributes\.Data and controls use matched radii and sample sizes \([sectionH\.2](https://arxiv.org/html/2610.27139#A8.SS2)\)\.[Table3](https://arxiv.org/html/2610.27139#S4.T3)compares counting windows within each animal\.

Target prediction\.We next ask whether a learned representation predicts a disjoint target set\. Anencoder–predictorlearns from the same sessions’ training blocks and is evaluated on later held\-out observations\. At the larger target budgets, prediction improves on a constantpredictorderived from the training block \([figure7](https://arxiv.org/html/2610.27139#S4.F7), left\)\.

Reference state recovery\.The trainedencoderabove supplies the reference model: its full hidden\-coordinate vector defines each observation’s surrogate state\. A smaller model learns the same prediction task independently, using the same observations and chronological split\. We then recover the reference state by local interpolation in the smaller model’s learned space: reference states at nearby training inputs are averaged with distance\-based weights\. The reference states enter the recovery map after predictive training\.

Figure 7:Prediction and reference\-state recovery\.*Left:*prediction gain\.*Middle:*recovery gain with session \(S\) or animal \(A\) weighting\.*Right:*animal gains\. Repetition means and standard deviations; see[tables19](https://arxiv.org/html/2610.27139#A9.T19)and[21](https://arxiv.org/html/2610.27139#A10.T21)\.The recovery score measures reconstruction of the surrogate’s full coordinates on held\-out observations\. Predictive training improves reconstruction of these reference coordinates relative to the same smallencoderat initialization, with a session\-weighted gain of\.0239±\.0118\.0239\\pm\.0118inR2R^\{2\}\([figures7](https://arxiv.org/html/2610.27139#S4.F7)and[21](https://arxiv.org/html/2610.27139#A10.T21)\)\. The mean gain remains positive within each animal and each paired seed group, and after leaving out any animal\. The protocol and cohort summaries appear in[appendixJ](https://arxiv.org/html/2610.27139#A10)\.

## 5Related Work

Prediction and state learning\.Predictive state models describe dynamics through action\-conditioned future observations\([Littman et al\., 2001](https://arxiv.org/html/2610.27139#bib.bib26)\)\. Auxiliary prediction and context learning link predictive objectives to downstream performance\([Lee et al\., 2021](https://arxiv.org/html/2610.27139#bib.bib24);[Zhai et al\., 2025](https://arxiv.org/html/2610.27139#bib.bib48)\)\.[Zhang et al\. \(2025\)](https://arxiv.org/html/2610.27139#bib.bib50)recover generating latent variables up to coordinate negations and permutations as tasks increase, under a low\-degree task distribution and preference for low\-degree realizations\. Our target is the complete query\-response profile; the fixed\-family bound quantifies the cost of one target set serving allencoders\.

Identifiability\.Identifiability depends on the learning setup\([Locatello et al\., 2019](https://arxiv.org/html/2610.27139#bib.bib27)\)\. Auxiliary variables\([Khemakhem et al\., 2020](https://arxiv.org/html/2610.27139#bib.bib21)\), contrastive objectives with generative assumptions\([Zimmermann et al\., 2021](https://arxiv.org/html/2610.27139#bib.bib51)\), and multiple modalities\([Daunhawer et al\., 2023](https://arxiv.org/html/2610.27139#bib.bib8)\)can identify factors up to transformations\. Our recovery target is specified by query responses before choosing latent coordinates\.

Optimal recovery and sampling\.Optimal recovery studies uncertainty under partial observations\([Micchelli and Rivlin, 1977](https://arxiv.org/html/2610.27139#bib.bib28)\)\. Here, states sharing anencoderoutput form a set whose diameter limits recovery; prediction constrains that set through its targets\. Structured model classes also yield recovery and sampling guarantees\([Binev et al\., 2024](https://arxiv.org/html/2610.27139#bib.bib3);[Krieg and Ullrich, 2026](https://arxiv.org/html/2610.27139#bib.bib23)\)\.

Geometry and stable approximation\.Embeddings\([Sauer et al\., 1991](https://arxiv.org/html/2610.27139#bib.bib40)\)and random manifold projections\([Baraniuk and Wakin, 2009](https://arxiv.org/html/2610.27139#bib.bib2)\)relate geometry to measurement requirements\. Stable manifold widths link approximation to metric entropy\([Cohen et al\., 2022](https://arxiv.org/html/2610.27139#bib.bib6)\)\. Here, low\-dimensional covering controls the distinctions to resolve; observability supplies the queries\.

Neural geometry and computation\.Population geometry relates to computation\([Chung and Abbott, 2021](https://arxiv.org/html/2610.27139#bib.bib4);[Jazayeri and Ostojic, 2021](https://arxiv.org/html/2610.27139#bib.bib20)\), cross\-task generalization\([Wakhloo et al\., 2026](https://arxiv.org/html/2610.27139#bib.bib44)\), and cross\-animal decoding\([Safaie et al\., 2023](https://arxiv.org/html/2610.27139#bib.bib38)\)\. Intrinsic\-dimension estimation characterizes the degrees of freedom in neural activity\([Fadanni et al\., 2026](https://arxiv.org/html/2610.27139#bib.bib10)\)\. These studies motivate our geometric assumptions, which connect covering complexity to query budgets\.

Neural state learning\.LFADS infers population dynamics\([Pandarinath et al\., 2018](https://arxiv.org/html/2610.27139#bib.bib33)\); the Neural Latents Benchmark evaluates held\-out activity and behavioral readouts\([Pei et al\., 2021](https://arxiv.org/html/2610.27139#bib.bib34)\)\. Multi\-context, multi\-task, and foundation models learn across recordings\([Ye et al\., 2023](https://arxiv.org/html/2610.27139#bib.bib46);[Azabou et al\., 2025](https://arxiv.org/html/2610.27139#bib.bib1);[Wang et al\., 2025](https://arxiv.org/html/2610.27139#bib.bib45)\)\. CEBRA uses behavioral or temporal context\([Schneider et al\., 2023](https://arxiv.org/html/2610.27139#bib.bib41)\), DPAD prioritizes behaviorally relevant dynamics\([Sani et al\., 2024](https://arxiv.org/html/2610.27139#bib.bib39)\), and MARBLE represents local manifold dynamics\([Gosztolai et al\., 2025](https://arxiv.org/html/2610.27139#bib.bib14)\)\. These objectives reward different distinctions; we ask when preserving them suffices to recover the complete response profile\.

## 6Conclusion

We studied when learning to predict a system enables recovery of its ontic state\. Defining that state as a complete response profile lets us relate prediction to the distinctions a learned representation preserves\. Query ambiguity makes this connection quantitative, yielding recovery guarantees in general and low\-dimensional settings\. In the latter, we establish query\-budget bounds whose order is sharp on a fixed state family when one target set serves everyencoder\. These results give a precise sense in which predicting observations can become a way of knowing the state that produces them\. We hope that our work offers theoretical support for using deep learning models as computational surrogates for complex systems, including those studied in neuroscience\.

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\(Appendix\)

Contents of the Appendix

## Appendix AState, Query, and Observation

We begin with observation ambiguity, then derive the prediction–recovery bounds and their sharpness\. The training and noisy\-observation results extend this chain to finite records and perturbed measurements\. Experimental details follow the main\-text order: synthetic recovery, joint\-query calibration, and neural observation geometry and surrogate\-state recovery\.

### A\.1Distance and Restriction

Distances contract when we restrict the query set\. For query setsQ⊆Q′⊆𝒰Q\\subseteq Q^\{\\prime\}\\subseteq\\mathcal\{U\}and statesf,g∈ℱf,g\\in\\mathcal\{F\},

dQ​\(f⁡\(Q\),g⁡\(Q\)\)≤dQ′​\(f⁡\(Q′\),g⁡\(Q′\)\)≤d𝒰​\(f,g\)\.d\_\{Q\}\(f\(Q\),g\(Q\)\)\\leq d\_\{Q^\{\\prime\}\}\(f\(Q^\{\\prime\}\),g\(Q^\{\\prime\}\)\)\\leq d\_\{\\mathcal\{U\}\}\(f,g\)\.\(33\)Each inequality follows by taking the supremum over a subset of the same observation coordinates\. In particular, every observation mapf↦f⁡\(Q\)f\\mapsto f\(Q\)is11\-Lipschitz and hence continuous\.

###### Lemma A\.1\(Ontic Metric\)\.

The functiond𝒰:ℱ×ℱ→\[0,∞\)d\_\{\\mathcal\{U\}\}:\\mathcal\{F\}\\times\\mathcal\{F\}\\to\[0,\\infty\)is a metric onℱ\\mathcal\{F\}\. Hence\(ℱ,d𝒰\)\(\\mathcal\{F\},d\_\{\\mathcal\{U\}\}\)is a metric space\.

###### Proof\.

Boundedness off,g∈ℱf,g\\in\\mathcal\{F\}gives

d𝒰​\(f,g\)≤supq∈𝒰\|f⁡\(q\)\|\+supq∈𝒰\|g⁡\(q\)\|<∞\.d\_\{\\mathcal\{U\}\}\(f,g\)\\leq\\sup\_\{q\\in\\mathcal\{U\}\}\|f\(q\)\|\+\\sup\_\{q\\in\\mathcal\{U\}\}\|g\(q\)\|<\\infty\.\(34\)Nonnegativity and symmetry follow from absolute value\. Moreover,d𝒰​\(f,g\)=0d\_\{\\mathcal\{U\}\}\(f,g\)=0holds exactly whenf⁡\(q\)=g⁡\(q\)f\(q\)=g\(q\)at every query, which meansf=gf=gas functions\. Finally, for everyq∈𝒰q\\in\\mathcal\{U\},

\|f⁡\(q\)−g⁡\(q\)\|≤\|f⁡\(q\)−h⁡\(q\)\|\+\|h⁡\(q\)−g⁡\(q\)\|≤d𝒰​\(f,h\)\+d𝒰​\(h,g\)\.\|f\(q\)\-g\(q\)\|\\leq\|f\(q\)\-h\(q\)\|\+\|h\(q\)\-g\(q\)\|\\leq d\_\{\\mathcal\{U\}\}\(f,h\)\+d\_\{\\mathcal\{U\}\}\(h,g\)\.\(35\)Taking the supremum overqqproves the triangle inequality\. ∎

### A\.2A Common Readout

States sharing the same input must receive the same output\. Their target spread limits the accuracy of a common readout\.

###### Lemma A\.2\(Fiberwise Error Bound\)\.

Letℱ\\mathcal\{F\}be nonempty and\(𝒱,d\)\(\\mathcal\{V\},d\)a metric space\. The mapT:ℱ→𝒴T:\\mathcal\{F\}\\to\\mathcal\{Y\}specifies the input available to a readout, andG:ℱ→𝒱G:\\mathcal\{F\}\\to\\mathcal\{V\}specifies its target\. Define the largest target diameter among states with the same input by

Δ≔supf,g∈ℱT⁡\(f\)=T⁡\(g\)d⁡\(G⁡\(f\),G⁡\(g\)\)\.\\Delta\\coloneqq\\sup\_\{\\begin\{subarray\}\{c\}f,g\\in\\mathcal\{F\}\\\\ T\(f\)=T\(g\)\\end\{subarray\}\}d\(G\(f\),G\(g\)\)\.\(36\)Then

Δ2≤infc:𝒴→𝒱supf∈ℱd\(c\(T\(f\)\),G\(f\)\)≤Δ\.\\frac\{\\Delta\}\{2\}\\leq\\boxed\{\\inf\_\{c:\\mathcal\{Y\}\\to\\mathcal\{V\}\}\\sup\_\{f\\in\\mathcal\{F\}\}d\(c\(T\(f\)\),G\(f\)\)\}\\leq\\Delta\.\(37\)If\(𝒱,d\)=\(ℝP,dP\)\(\\mathcal\{V\},d\)=\(\\mathbb\{R\}^\{P\},d\_\{P\}\)withPPnonempty and finite andG⁡\(ℱ\)G\(\\mathcal\{F\}\)bounded, the coordinatewise midpoint attains the lower boundΔ/2\\Delta/2\.

###### Proof\.

*Choose a common output\.*For each attainable inputy∈T⁡\(ℱ\)y\\in T\(\\mathcal\{F\}\), letSy≔\{f∈ℱ:T⁡\(f\)=y\}S\_\{y\}\\coloneqq\\\{f\\in\\mathcal\{F\}:T\(f\)=y\\\}be its fiber\. Choosehy∈Syh\_\{y\}\\in S\_\{y\}and definec⁡\(y\)=G⁡\(hy\)c\(y\)=G\(h\_\{y\}\)\. Assign a fixed element of𝒱\\mathcal\{V\}at unattainable inputs\. For anyf∈Syf\\in S\_\{y\},

d⁡\(c⁡\(T⁡\(f\)\),G⁡\(f\)\)=d⁡\(G⁡\(hy\),G⁡\(f\)\)≤Δ\.d\(c\(T\(f\)\),G\(f\)\)=d\(G\(h\_\{y\}\),G\(f\)\)\\leq\\Delta\.\(38\)Taking the supremum overf∈ℱf\\in\\mathcal\{F\}gives the general upper bound\.

The state\-valued caseG⁡\(f\)=fG\(f\)=fis shown below for three fibers\.

*Compare any common output\.*IfT⁡\(f\)=T⁡\(g\)T\(f\)=T\(g\), every readoutccgives the same output to both states\. The triangle inequality gives

d⁡\(G⁡\(f\),G⁡\(g\)\)≤d⁡\(G⁡\(f\),c⁡\(T⁡\(f\)\)\)\+d⁡\(c⁡\(T⁡\(g\)\),G⁡\(g\)\)≤2​suph∈ℱd⁡\(c⁡\(T⁡\(h\)\),G⁡\(h\)\)\.d\(G\(f\),G\(g\)\)\\leq d\(G\(f\),c\(T\(f\)\)\)\+d\(c\(T\(g\)\),G\(g\)\)\\leq 2\\sup\_\{h\\in\\mathcal\{F\}\}d\(c\(T\(h\)\),G\(h\)\)\.\(39\)Taking the supremum over pairs and then the infimum over readouts gives the lower boundΔ/2\\Delta/2\.

*Use coordinate midpoints\.*When\(𝒱,d\)=\(ℝP,dP\)\(\\mathcal\{V\},d\)=\(\\mathbb\{R\}^\{P\},d\_\{P\}\)andG⁡\(ℱ\)G\(\\mathcal\{F\}\)is bounded, define the coordinatewise midpoint

\[c⋆​\(y\)\]​\(p\)≔12​\(infg∈Sy\[G⁡\(g\)\]​\(p\)\+supg∈Sy\[G⁡\(g\)\]​\(p\)\),p∈P\.\[c^\{\\star\}\(y\)\]\(p\)\\coloneqq\\frac\{1\}\{2\}\\Bigl\(\\inf\_\{g\\in S\_\{y\}\}\[G\(g\)\]\(p\)\+\\sup\_\{g\\in S\_\{y\}\}\[G\(g\)\]\(p\)\\Bigr\),\\qquad p\\in P\.\(40\)The endpoints are finite by boundedness\. Setc⋆​\(y\)=0∈ℝPc^\{\\star\}\(y\)=0\\in\\mathbb\{R\}^\{P\}at unattainable inputs\. For anyf∈Syf\\in S\_\{y\}andp∈Pp\\in P,

\|\[c⋆​\(y\)\]​\(p\)−\[G⁡\(f\)\]​\(p\)\|≤12​supg,h∈Sy\|\[G⁡\(g\)\]​\(p\)−\[G⁡\(h\)\]​\(p\)\|≤Δ2\.\\bigl\|\[c^\{\\star\}\(y\)\]\(p\)\-\[G\(f\)\]\(p\)\\bigr\|\\leq\\frac\{1\}\{2\}\\sup\_\{g,h\\in S\_\{y\}\}\\bigl\|\[G\(g\)\]\(p\)\-\[G\(h\)\]\(p\)\\bigr\|\\leq\\frac\{\\Delta\}\{2\}\.\(41\)Taking the maximum overp∈Pp\\in Pand the supremum overf∈ℱf\\in\\mathcal\{F\}shows thatc⋆c^\{\\star\}attains the lower bound\. ∎

###### Lemma A\.3\(Exact Decoding\)\.

For nonemptyℱ\\mathcal\{F\}, the following are equivalent:ℛQ⋆​\(e\)=0\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)=0; the mapf↦e⁡\(f⁡\(Q\)\)f\\mapsto e\(f\(Q\)\)is injective; and some state decoder satisfies\(a∘e\)​\(f​\(Q\)\)=f\(a\\circ e\)\(f\(Q\)\)=ffor everyf∈ℱf\\in\\mathcal\{F\}\.

###### Proof\.

IfℛQ⋆​\(e\)=0\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)=0,[lemmaA\.2](https://arxiv.org/html/2610.27139#A1.Thmtheorem2)givesd𝒰​\(f,g\)=0d\_\{\\mathcal\{U\}\}\(f,g\)=0whenevere⁡\(f⁡\(Q\)\)=e⁡\(g⁡\(Q\)\)e\(f\(Q\)\)=e\(g\(Q\)\), so the map is injective\. Its inverse on attainable learned states, extended by a fixed member ofℱ\\mathcal\{F\}elsewhere, is an exactdecoder\. Conversely, an exactdecoderhas zero error, soℛQ⋆​\(e\)=0\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)=0\. ∎

### A\.3Ambiguity and Recovery

The common\-readout construction now applies to all states compatible with an observation vector\. Compactness will ensure that their ambiguity vanishes as separating queries accumulate\.

###### Assumption A\.4\(Compactness\)\.

The ontic state space\(ℱ,d𝒰\)\(\\mathcal\{F\},d\_\{\\mathcal\{U\}\}\)is nonempty and compact\.

###### Lemma A\.5\(Ambiguity and Recovery\)\.

Assumeℱ\\mathcal\{F\}is nonempty\. For every query setQ⊆𝒰Q\\subseteq\\mathcal\{U\},

12​ωQ​\(0\)≤ℰQ⋆≤ωQ​\(0\)\.\\frac\{1\}\{2\}\\omega\_\{Q\}\(0\)\\leq\\boxed\{\\mathcal\{E\}\_\{Q\}^\{\\star\}\}\\leq\\omega\_\{Q\}\(0\)\.\(42\)

###### Proof\.

By[lemmaA\.1](https://arxiv.org/html/2610.27139#A1.Thmtheorem1),\(ℱ,d𝒰\)\(\\mathcal\{F\},d\_\{\\mathcal\{U\}\}\)is a metric space\. Apply[lemmaA\.2](https://arxiv.org/html/2610.27139#A1.Thmtheorem2)withT⁡\(f\)=f⁡\(Q\)T\(f\)=f\(Q\),G⁡\(f\)=fG\(f\)=fand\(𝒱,d\)=\(ℱ,d𝒰\)\(\\mathcal\{V\},d\)=\(\\mathcal\{F\},d\_\{\\mathcal\{U\}\}\)\. Its fiber diameter isΔ=ωQ​\(0\)\\Delta=\\omega\_\{Q\}\(0\), so[equation37](https://arxiv.org/html/2610.27139#A1.E37)gives both bounds\. ∎

###### Proof of[lemma2\.9](https://arxiv.org/html/2610.27139#S2.Thmtheorem9)\.

Suppose the conclusion fails\. There areε\>0\\varepsilon\>0, increasing indicesmkm\_\{k\}, and statesfk,gk∈ℱf\_\{k\},g\_\{k\}\\in\\mathcal\{F\}such that

dPmk​\(fk​\(Pmk\),gk​\(Pmk\)\)≤rmk,d𝒰​\(fk,gk\)≥ε\.d\_\{P\_\{m\_\{k\}\}\}\\bigl\(f\_\{k\}\(P\_\{m\_\{k\}\}\),g\_\{k\}\(P\_\{m\_\{k\}\}\)\\bigr\)\\leq r\_\{m\_\{k\}\},\\qquad d\_\{\\mathcal\{U\}\}\(f\_\{k\},g\_\{k\}\)\\geq\\varepsilon\.\(43\)Compactness gives a common subsequence withfk→ff\_\{k\}\\to fandgk→gg\_\{k\}\\to g\. Continuity of the metric givesd𝒰​\(f,g\)≥εd\_\{\\mathcal\{U\}\}\(f,g\)\\geq\\varepsilon\.

Fix any stagemm\. Eventuallymk≥mm\_\{k\}\\geq m, so nestedness and[equation33](https://arxiv.org/html/2610.27139#A1.E33)give

dPm​\(fk​\(Pm\),gk​\(Pm\)\)≤rmk→0\.d\_\{P\_\{m\}\}\\bigl\(f\_\{k\}\(P\_\{m\}\),g\_\{k\}\(P\_\{m\}\)\\bigr\)\\leq r\_\{m\_\{k\}\}\\to 0\.\(44\)Taking the limit yieldsf⁡\(Pm\)=g⁡\(Pm\)f\(P\_\{m\}\)=g\(P\_\{m\}\)\. This holds at every stage, so state separation forcesf=gf=g, contradictingd𝒰​\(f,g\)≥εd\_\{\\mathcal\{U\}\}\(f,g\)\\geq\\varepsilon\. ∎

Before introducing a learned representation, we characterize when the observations themselves suffice for recovery\.

###### Theorem A\.6\(Separation and Recovery\)\.

Assume\(ℱ,d𝒰\)\(\\mathcal\{F\},d\_\{\\mathcal\{U\}\}\)is nonempty and compact \([assumptionA\.4](https://arxiv.org/html/2610.27139#A1.Thmtheorem4)\) and let\(Qp\)p≥1\(Q\_\{p\}\)\_\{p\\geq 1\}be a query sequence\. Then

State separation⟺ℰQp⋆→p→∞0\.\\text\{State separation\}\\quad\\Longleftrightarrow\\quad\\mathcal\{E\}\_\{Q\_\{p\}\}^\{\\star\}\\xrightarrow\{p\\to\\infty\}0\.\(45\)

###### Proof\.

If the queries separate states, apply[lemma2\.9](https://arxiv.org/html/2610.27139#S2.Thmtheorem9)with zero tolerance and then[lemmaA\.5](https://arxiv.org/html/2610.27139#A1.Thmtheorem5):

0≤ℰQp⋆≤ωQp​\(0\)→0\.0\\leq\\mathcal\{E\}\_\{Q\_\{p\}\}^\{\\star\}\\leq\\omega\_\{Q\_\{p\}\}\(0\)\\to 0\.\(46\)Conversely, if two distinct statesf,gf,gagree on everyQpQ\_\{p\}, then[lemmaA\.5](https://arxiv.org/html/2610.27139#A1.Thmtheorem5)gives

ℰQp⋆≥12​ωQp​\(0\)≥12​d𝒰​\(f,g\)\>0\\mathcal\{E\}\_\{Q\_\{p\}\}^\{\\star\}\\geq\\frac\{1\}\{2\}\\omega\_\{Q\_\{p\}\}\(0\)\\geq\\frac\{1\}\{2\}d\_\{\\mathcal\{U\}\}\(f,g\)\>0\(47\)at every stage\. Thus vanishing recovery error requires state separation\. ∎

The diagram shows candidate sets under state separation\.

Figure 8:Schematic candidate sets for a fixedff: exact observations \(solid\) and tolerancermr\_\{m\}\(dashed\)\. Both have vanishing diameter under compactness and state separation, withrm→0r\_\{m\}\\to 0for the tolerance sets\.

## Appendix BPrediction Setup

We now compare prediction and recovery from the learned statee⁡\(f⁡\(Q\)\)e\(f\(Q\)\), using the error criteria in[definition2\.10](https://arxiv.org/html/2610.27139#S2.Thmtheorem10)\. In the figures,zi=e⁡\(fi​\(Q\)\)z\_\{i\}=e\(f\_\{i\}\(Q\)\)\.

###### Proof of[corollary2\.11](https://arxiv.org/html/2610.27139#S2.Thmtheorem11)\.

Anystate decoderaacan also recover from the original observations: first applyee, thenaa\. This gives anobservation decoderwith the same error, so

ℰQ⋆≤ℰQ​\(a∘e\)=ℛQ​\(a,e\)\.\\mathcal\{E\}\_\{Q\}^\{\\star\}\\leq\\mathcal\{E\}\_\{Q\}\(a\\circ e\)=\\mathcal\{R\}\_\{Q\}\(a;e\)\.\(48\)Taking the infimum overaagivesℰQ⋆≤ℛQ⋆​\(e\)\\mathcal\{E\}\_\{Q\}^\{\\star\}\\leq\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\. The remaining boundℰQ⋆≥ωQ​\(0\)/2\\mathcal\{E\}\_\{Q\}^\{\\star\}\\geq\\omega\_\{Q\}\(0\)/2follows from[lemmaA\.5](https://arxiv.org/html/2610.27139#A1.Thmtheorem5)\. ∎

The recovery criterion isℛQm⋆​\(em\)→0\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\to 0: for every desired accuracy, eachencoderat a sufficiently late stage admits adecoderachieving that accuracy uniformly over states and ontic queries\. Thedecodermay depend on the stage\.

###### Proof of[lemma3\.1](https://arxiv.org/html/2610.27139#S3.Thmtheorem1)\.

Apply[lemmaA\.2](https://arxiv.org/html/2610.27139#A1.Thmtheorem2)withT⁡\(f\)=e⁡\(f⁡\(Q\)\)T\(f\)=e\(f\(Q\)\),G⁡\(f\)=f⁡\(P\)G\(f\)=f\(P\)and\(𝒱,d\)=\(ℝP,dP\)\(\\mathcal\{V\},d\)=\(\\mathbb\{R\}^\{P\},d\_\{P\}\)\. Sinceℱ\\mathcal\{F\}is bounded and restriction contracts distances,G⁡\(ℱ\)G\(\\mathcal\{F\}\)is bounded\. The coordinatewise midpoint therefore attains the common lower bound, giving[equation15](https://arxiv.org/html/2610.27139#S3.E15)\. ∎

## Appendix CPrediction and Recovery

We prove[theorems3\.2](https://arxiv.org/html/2610.27139#S3.Thmtheorem2)and[3\.3](https://arxiv.org/html/2610.27139#S3.Thmtheorem3)first, followed by[theorems3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4)and[3\.5](https://arxiv.org/html/2610.27139#S3.Thmtheorem5)\. In each case, fixing theencodergives an exact\-recovery corollary\.

### C\.1General Case

Thepredictorassigns a common target vector to states merged by theencoder\. Their target spread controls their full\-state spread through query ambiguity\.

###### Proof of[theorem3\.2](https://arxiv.org/html/2610.27139#S3.Thmtheorem2)\.

We apply[lemmaA\.2](https://arxiv.org/html/2610.27139#A1.Thmtheorem2)to ontic states mapped to the same learned state, takingT⁡\(f\)=e⁡\(f⁡\(Q\)\)T\(f\)=e\(f\(Q\)\),G⁡\(f\)=fG\(f\)=fand\(𝒱,d\)=\(ℱ,d𝒰\)\(\\mathcal\{V\},d\)=\(\\mathcal\{F\},d\_\{\\mathcal\{U\}\}\)\.

*Lower bound\.*Restricting a decoded state to the target queries gives apredictor, as illustrated below\.

Target observations differ by at most the full state distance\. Combining this with[lemma3\.1](https://arxiv.org/html/2610.27139#S3.Thmtheorem1)and the lower bound in[lemmaA\.2](https://arxiv.org/html/2610.27139#A1.Thmtheorem2)gives

ℒQ,P⋆​\(e\)=12​supf,g∈ℱe⁡\(f⁡\(Q\)\)=e⁡\(g⁡\(Q\)\)dP​\(f⁡\(P\),g⁡\(P\)\)≤12​supf,g∈ℱe⁡\(f⁡\(Q\)\)=e⁡\(g⁡\(Q\)\)d𝒰​\(f,g\)≤ℛQ⋆​\(e\)\.\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)=\\frac\{1\}\{2\}\\sup\_\{\\begin\{subarray\}\{c\}f,g\\in\\mathcal\{F\}\\\\ e\(f\(Q\)\)=e\(g\(Q\)\)\\end\{subarray\}\}d\_\{P\}\(f\(P\),g\(P\)\)\\leq\\frac\{1\}\{2\}\\sup\_\{\\begin\{subarray\}\{c\}f,g\\in\\mathcal\{F\}\\\\ e\(f\(Q\)\)=e\(g\(Q\)\)\\end\{subarray\}\}d\_\{\\mathcal\{U\}\}\(f,g\)\\leq\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\.\(49\)
*Upper bound\.*By[lemma3\.1](https://arxiv.org/html/2610.27139#S3.Thmtheorem1), any pair withe⁡\(f⁡\(Q\)\)=e⁡\(g⁡\(Q\)\)e\(f\(Q\)\)=e\(g\(Q\)\)has target distance at most2​ℒQ,P⋆​\(e\)2\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\. The definition of query ambiguity and[lemmaA\.2](https://arxiv.org/html/2610.27139#A1.Thmtheorem2)give

ℛQ⋆​\(e\)≤supf,g∈ℱe⁡\(f⁡\(Q\)\)=e⁡\(g⁡\(Q\)\)d𝒰​\(f,g\)≤ωP​\(2​ℒQ,P⋆​\(e\)\)\.\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\\leq\\sup\_\{\\begin\{subarray\}\{c\}f,g\\in\\mathcal\{F\}\\\\ e\(f\(Q\)\)=e\(g\(Q\)\)\\end\{subarray\}\}d\_\{\\mathcal\{U\}\}\(f,g\)\\leq\\omega\_\{P\}\\bigl\(2\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\\bigr\)\.\(50\)For apredictorbbwith finite error, the same bound follows fromℒQ,P⋆​\(e\)≤ℒQ,P​\(b,e\)\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\\leq\\mathcal\{L\}\_\{Q,P\}\(b;e\)and the monotonicity ofωP\\omega\_\{P\}\. ∎

###### Proof of[theorem3\.3](https://arxiv.org/html/2610.27139#S3.Thmtheorem3)\.

IfℒQm,Pm⋆​\(em\)→0\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\to 0, apply[lemma2\.9](https://arxiv.org/html/2610.27139#S2.Thmtheorem9)with tolerance2​ℒQm,Pm⋆​\(em\)2\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\. By[theorem3\.2](https://arxiv.org/html/2610.27139#S3.Thmtheorem2),

0≤ℛQm⋆​\(em\)≤ωPm​\(2​ℒQm,Pm⋆​\(em\)\)→0\.0\\leq\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\leq\\omega\_\{P\_\{m\}\}\\bigl\(2\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\bigr\)\\to 0\.\(51\)Conversely,0≤ℒQm,Pm⋆​\(em\)≤ℛQm⋆​\(em\)0\\leq\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\leq\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(e\_\{m\}\)gives vanishing prediction error whenever recovery error vanishes\. ∎

For a fixedencoder, vanishing prediction error gives exact state recovery\.

###### Corollary C\.1\(General Exact Recovery\)\.

Under[assumptionA\.4](https://arxiv.org/html/2610.27139#A1.Thmtheorem4), fix a query setQQand an encodere:ℝQ→𝒵e:\\mathbb\{R\}^\{Q\}\\to\\mathcal\{Z\}into a nonempty learned state space\. Let\(Pm\)m≥1\(P\_\{m\}\)\_\{m\\geq 1\}be a query sequence separating states, withPm⊆𝒰∖QP\_\{m\}\\subseteq\\mathcal\{U\}\\setminus Q\. If

ℒQ,Pm⋆​\(e\)→m→∞0,\\mathcal\{L\}\_\{Q,P\_\{m\}\}^\{\\star\}\(e\)\\xrightarrow\{m\\to\\infty\}0,\(52\)then there exists a state decodera∈𝒜a\\in\\mathcal\{A\}such that

\(a∘e\)​\(f⁡\(Q\)\)=f,∀f∈ℱ\.\(a\\circ e\)\(f\(Q\)\)=f,\\qquad\\forall f\\in\\mathcal\{F\}\.\(53\)

###### Proof\.

By[theorem3\.3](https://arxiv.org/html/2610.27139#S3.Thmtheorem3), the recovery error tends to zero\. SinceQQandeeare fixed,ℛQ⋆​\(e\)\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)is independent of the stage and therefore equals zero\. Apply[lemmaA\.3](https://arxiv.org/html/2610.27139#A1.Thmtheorem3)to obtain an exactdecoder\. ∎

Embedding theory relates state distinguishability to the geometry preserved by observations\([Sauer et al\., 1991](https://arxiv.org/html/2610.27139#bib.bib40)\)\. Here, exact state recovery has a topological counterpart when the state space is compact and the map to learned states is continuous\.

###### Corollary C\.2\(Topology Preservation\)\.

Under[assumptionA\.4](https://arxiv.org/html/2610.27139#A1.Thmtheorem4), fix a query setQQand an encodere:ℝQ→𝒵=ℝℓe:\\mathbb\{R\}^\{Q\}\\to\\mathcal\{Z\}=\\mathbb\{R\}^\{\\ell\}with a continuous map to learned statesf↦e⁡\(f⁡\(Q\)\)f\\mapsto e\(f\(Q\)\)\. If there exists a state decodera∈𝒜a\\in\\mathcal\{A\}such that

\(a∘e\)​\(f⁡\(Q\)\)=f,∀f∈ℱ,\(a\\circ e\)\(f\(Q\)\)=f,\\qquad\\forall f\\in\\mathcal\{F\},\(54\)then this map is a homeomorphism onto its image\. In particular, ifℱ\\mathcal\{F\}is a topological manifold, possibly with boundary, its image under this map is a manifold of the same dimension\.

###### Proof\.

*Injectivity\.*By[lemmaA\.3](https://arxiv.org/html/2610.27139#A1.Thmtheorem3), the mapf↦e⁡\(f⁡\(Q\)\)f\\mapsto e\(f\(Q\)\)is injective\. It is therefore a continuous bijection onto its image\.

*Continuity of the inverse\.*Every closed subset of the compact spaceℱ\\mathcal\{F\}is compact\. Its image is compact by continuity, and therefore closed inℝℓ\\mathbb\{R\}^\{\\ell\}\. The bijection is thus a closed map, so its inverse is continuous\. This proves that it is a homeomorphism onto its image\.

*Manifold structure\.*Composing each local chart ofℱ\\mathcal\{F\}with this inverse gives a local chart on the image, with the same dimension and boundary structure\.

∎

### C\.2Low\-Dimensional Case

The general case controls recovery through target ambiguity\. In the low\-dimensional case, covering and observability let us bound that ambiguity using finitely many queries, yielding the recovery and convergence results in[theorems3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4)and[3\.5](https://arxiv.org/html/2610.27139#S3.Thmtheorem5)\.

###### Lemma C\.3\(Covering Queries\)\.

Letℱ\\mathcal\{F\}be nonempty and let\(ℱ,Q\)\(\\mathcal\{F\},Q\)satisfy[assumption2\.12](https://arxiv.org/html/2610.27139#S2.Thmtheorem12)with constantsC,d,αC,d,\\alpha\. Fix a covering radius0<ε≤10<\\varepsilon\\leq 1and a query allowanceτ\>0\\tau\>0\.

There is a finite representative setℱε⊆ℱ\\mathcal\{F\}\_\{\\varepsilon\}\\subseteq\\mathcal\{F\}, withN=\|ℱε\|≤C​ε−dN=\|\\mathcal\{F\}\_\{\\varepsilon\}\|\\leq C\\varepsilon^\{\-d\}, such that every state inℱ\\mathcal\{F\}is withind𝒰d\_\{\\mathcal\{U\}\}\-distanceε\\varepsilonof a representative\.

There also exists a target setP⊆𝒰∖QP\\subseteq\\mathcal\{U\}\\setminus Q, with1≤\|P\|≤max⁡\{1,N⁡\(N−1\)/2\}1\\leq\|P\|\\leq\\max\\\{1,N\(N\-1\)/2\\\}, that distinguishes these representatives: for everyf,g∈ℱεf,g\\in\\mathcal\{F\}\_\{\\varepsilon\},

dP​\(f⁡\(P\),g⁡\(P\)\)≥α​d𝒰​\(f,g\)−τ\.d\_\{P\}\\bigl\(f\(P\),g\(P\)\\bigr\)\\geq\\alpha d\_\{\\mathcal\{U\}\}\(f,g\)\-\\tau\.\(55\)Equivalently,

d𝒰​\(f,g\)≤1α​dP​\(f⁡\(P\),g⁡\(P\)\)\+τα\.d\_\{\\mathcal\{U\}\}\(f,g\)\\leq\\frac\{1\}\{\\alpha\}d\_\{P\}\\bigl\(f\(P\),g\(P\)\\bigr\)\+\\frac\{\\tau\}\{\\alpha\}\.\(56\)The representatives and targets are selected independently of the encoder\.

###### Proof\.

*Choose the representatives\.*By[equation13](https://arxiv.org/html/2610.27139#S2.E13), take anε\\varepsilon\-cover with centersℱε⊆ℱ\\mathcal\{F\}\_\{\\varepsilon\}\\subseteq\\mathcal\{F\}and1≤N=\|ℱε\|≤C​ε−d1\\leq N=\|\\mathcal\{F\}\_\{\\varepsilon\}\|\\leq C\\varepsilon^\{\-d\}\.

*Distinguish each pair\.*For each unordered pair\{f,g\}⊆ℱε\\\{f,g\\\}\\subseteq\\mathcal\{F\}\_\{\\varepsilon\}of distinct representatives, observability[equation14](https://arxiv.org/html/2610.27139#S2.E14)andτ\>0\\tau\>0let us selectp\{f,g\}∈𝒰∖Qp\_\{\\\{f,g\\\}\}\\in\\mathcal\{U\}\\setminus Qsuch that

\|f⁡\(p\{f,g\}\)−g⁡\(p\{f,g\}\)\|≥α​d𝒰​\(f,g\)−τ\.\|f\(p\_\{\\\{f,g\\\}\}\)\-g\(p\_\{\\\{f,g\\\}\}\)\|\\geq\\alpha d\_\{\\mathcal\{U\}\}\(f,g\)\-\\tau\.\(57\)The positive allowance permits this choice even if no query attains the supremum\.

*Collect the queries\.*IfN≥2N\\geq 2, letP=\{p\{f,g\}:f,g∈ℱε,f≠g\}P=\\\{p\_\{\\\{f,g\\\}\}:f,g\\in\\mathcal\{F\}\_\{\\varepsilon\},\\ f\\neq g\\\}\. Then

1≤\|P\|≤N⁡\(N−1\)2\.1\\leq\|P\|\\leq\\frac\{N\(N\-1\)\}\{2\}\.\(58\)For distinctf,g∈ℱεf,g\\in\\mathcal\{F\}\_\{\\varepsilon\}, their selected query belongs toPP, so

dP​\(f⁡\(P\),g⁡\(P\)\)≥\|f⁡\(p\{f,g\}\)−g⁡\(p\{f,g\}\)\|≥α​d𝒰​\(f,g\)−τ\.d\_\{P\}\\bigl\(f\(P\),g\(P\)\\bigr\)\\geq\|f\(p\_\{\\\{f,g\\\}\}\)\-g\(p\_\{\\\{f,g\\\}\}\)\|\\geq\\alpha d\_\{\\mathcal\{U\}\}\(f,g\)\-\\tau\.\(59\)Forf=gf=g, both distances are zero\. This proves[equation55](https://arxiv.org/html/2610.27139#A3.E55); rearranging gives[equation56](https://arxiv.org/html/2610.27139#A3.E56)\.

IfN=1N=1, take any singletonP⊆𝒰∖QP\\subseteq\\mathcal\{U\}\\setminus Q, which exists by[assumption2\.12](https://arxiv.org/html/2610.27139#S2.Thmtheorem12); the distance bounds are immediate\. ∎

We next extend the representative bound to arbitrary states\.

###### Lemma C\.4\(Decoding a Cover\)\.

Letℱε⊆ℱ\\mathcal\{F\}\_\{\\varepsilon\}\\subseteq\\mathcal\{F\}be a finiteε\\varepsilon\-cover, and let a nonempty finite target setPPsatisfy[equation56](https://arxiv.org/html/2610.27139#A3.E56)with constantsα\>0\\alpha\>0andτ≥0\\tau\\geq 0\. Fory∈ℝPy\\in\\mathbb\{R\}^\{P\}, choose a representativeg∈arg​minh∈ℱε⁡dP​\(y,h⁡\(P\)\)g\\in\\operatorname\*\{arg\\,min\}\_\{h\\in\\mathcal\{F\}\_\{\\varepsilon\}\}d\_\{P\}\(y,h\(P\)\)\. Then, for everyf∈ℱf\\in\\mathcal\{F\},

d𝒰​\(f,g\)≤2α​dP​\(y,f⁡\(P\)\)\+\(2\+α\)​ε\+τα\.d\_\{\\mathcal\{U\}\}\(f,g\)\\leq\\frac\{2\}\{\\alpha\}d\_\{P\}\(y,f\(P\)\)\+\\frac\{\(2\+\\alpha\)\\varepsilon\+\\tau\}\{\\alpha\}\.\(60\)

###### Proof\.

Takefε∈ℱεf\_\{\\varepsilon\}\\in\\mathcal\{F\}\_\{\\varepsilon\}withd𝒰​\(f,fε\)≤εd\_\{\\mathcal\{U\}\}\(f,f\_\{\\varepsilon\}\)\\leq\\varepsilon\. By[equation56](https://arxiv.org/html/2610.27139#A3.E56), the minimizing property ofgg, and restriction of the full metric,

d𝒰​\(f,g\)\\displaystyle d\_\{\\mathcal\{U\}\}\(f,g\)≤d𝒰​\(f,fε\)\+d𝒰​\(fε,g\)≤ε\+1α​dP​\(fε​\(P\),g⁡\(P\)\)\+τα\\displaystyle\\leq d\_\{\\mathcal\{U\}\}\(f,f\_\{\\varepsilon\}\)\+d\_\{\\mathcal\{U\}\}\(f\_\{\\varepsilon\},g\)\\leq\\varepsilon\+\\frac\{1\}\{\\alpha\}d\_\{P\}\(f\_\{\\varepsilon\}\(P\),g\(P\)\)\+\\frac\{\\tau\}\{\\alpha\}\(61\)≤ε\+2α​dP​\(y,fε​\(P\)\)\+τα≤2α​dP​\(y,f⁡\(P\)\)\+\(2\+α\)​ε\+τα\.\\displaystyle\\leq\\varepsilon\+\\frac\{2\}\{\\alpha\}d\_\{P\}\(y,f\_\{\\varepsilon\}\(P\)\)\+\\frac\{\\tau\}\{\\alpha\}\\leq\\frac\{2\}\{\\alpha\}d\_\{P\}\(y,f\(P\)\)\+\\frac\{\(2\+\\alpha\)\\varepsilon\+\\tau\}\{\\alpha\}\.∎

To prove[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4), we fit this construction to the prescribed query budget and decode to a representative whose target observations are closest to the prediction\.

###### Proof of[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4)\.

*Decode to a representative\.*For radiusε\\varepsilonand allowanceτ\\tau, specified below,[lemmaC\.3](https://arxiv.org/html/2610.27139#A3.Thmtheorem3)supplies a finite representative setℱε\\mathcal\{F\}\_\{\\varepsilon\}and a target query setPP\. We now use these sets to define adecoder\. Fix anyencodereeand finite\-errorpredictorbb\. Given a learned statezz, compare the predicted target vectorb⁡\(z\)b\(z\)with each representative’s target vectorg⁡\(P\)g\(P\), and return a closest representative:

a⁡\(z\)∈arg​ming∈ℱε⁡dP​\(b⁡\(z\),g⁡\(P\)\),a\(z\)\\in\\operatorname\*\{arg\\,min\}\_\{g\\in\\mathcal\{F\}\_\{\\varepsilon\}\}d\_\{P\}\\bigl\(b\(z\),g\(P\)\\bigr\),\(62\)resolving ties by a fixed ordering\.

The diagram shows the returned representativeg=a⁡\(z\)g=a\(z\), whose target readings are closest tob⁡\(z\)b\(z\)\.

*Bound the recovery error\.*Apply[lemmaC\.4](https://arxiv.org/html/2610.27139#A3.Thmtheorem4)withy=\(b∘e\)​\(f​\(Q\)\)y=\(b\\circ e\)\(f\(Q\)\)andg=\(a∘e\)​\(f​\(Q\)\)g=\(a\\circ e\)\(f\(Q\)\):

d𝒰​\(f,\(a∘e\)​\(f⁡\(Q\)\)\)≤2α​dP​\(\(b∘e\)​\(f⁡\(Q\)\),f⁡\(P\)\)\+\(2\+α\)​ε\+τα\.d\_\{\\mathcal\{U\}\}\(f,\(a\\circ e\)\(f\(Q\)\)\)\\leq\\frac\{2\}\{\\alpha\}d\_\{P\}\(\(b\\circ e\)\(f\(Q\)\),f\(P\)\)\+\\frac\{\(2\+\\alpha\)\\varepsilon\+\\tau\}\{\\alpha\}\.\(63\)
*Set the query budget\.*The radius controls the number of representatives; the allowance absorbs the approximation of separating queries\. Letδm=\(C2/\(2​m\)\)1/\(2​d\)≤1\\delta\_\{m\}=\(C^\{2\}/\(2m\)\)^\{1/\(2d\)\}\\leq 1\. Choose0<ε<δm0<\\varepsilon<\\delta\_\{m\}sufficiently close toδm\\delta\_\{m\}that

N≤C​ε−d<⌊C​δm−d⌋\+1=⌊2​m⌋\+1,N\\leq C\\varepsilon^\{\-d\}<\\lfloor C\\delta\_\{m\}^\{\-d\}\\rfloor\+1=\\lfloor\\sqrt\{2m\}\\rfloor\+1,\(64\)henceN≤⌊2​m⌋N\\leq\\lfloor\\sqrt\{2m\}\\rfloor\. Setτ=\(2\+α\)​\(δm−ε\)\>0\\tau=\(2\+\\alpha\)\(\\delta\_\{m\}\-\\varepsilon\)\>0\. By[lemmaC\.3](https://arxiv.org/html/2610.27139#A3.Thmtheorem3),

\|P\|≤max⁡\{1,N⁡\(N−1\)2\}≤max⁡\{1,N22\}≤m\.\|P\|\\leq\\max\\left\\\{1,\\frac\{N\(N\-1\)\}\{2\}\\right\\\}\\leq\\max\\left\\\{1,\\frac\{N^\{2\}\}\{2\}\\right\\\}\\leq m\.\(65\)These choices are independent ofee\. Taking the supremum of the recovery\-error bound gives

ℛQ⋆​\(e\)\\displaystyle\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)≤ℛQ​\(a,e\)≤2α​ℒQ,P​\(b,e\)\+\(2\+α\)​ε\+τα\\displaystyle\\leq\\mathcal\{R\}\_\{Q\}\(a;e\)\\leq\\frac\{2\}\{\\alpha\}\\mathcal\{L\}\_\{Q,P\}\(b;e\)\+\\frac\{\(2\+\\alpha\)\\varepsilon\+\\tau\}\{\\alpha\}\(66\)=2α​ℒQ,P​\(b,e\)\+\(2\+α\)​ε\+\(2\+α\)​\(δm−ε\)α=2α​ℒQ,P​\(b,e\)\+2\+αα​δm\.\\displaystyle=\\frac\{2\}\{\\alpha\}\\mathcal\{L\}\_\{Q,P\}\(b;e\)\+\\frac\{\(2\+\\alpha\)\\varepsilon\+\(2\+\\alpha\)\(\\delta\_\{m\}\-\\varepsilon\)\}\{\\alpha\}=\\frac\{2\}\{\\alpha\}\\mathcal\{L\}\_\{Q,P\}\(b;e\)\+\\frac\{2\+\\alpha\}\{\\alpha\}\\delta\_\{m\}\.Taking the infimum overbbproves the upper bound\. The lower bound follows from[theorem3\.2](https://arxiv.org/html/2610.27139#S3.Thmtheorem2), since the covering assumption makesℱ\\mathcal\{F\}bounded\. ∎

The bound in[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4)now gives the convergence and rate equivalences\. These target sets need not be nested: each budget has its own uniform recovery bound\.

###### Proof of[theorem3\.5](https://arxiv.org/html/2610.27139#S3.Thmtheorem5)\.

By[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4), for every admissible budget,

0≤ℒQm,Pm⋆​\(em\)≤ℛQm⋆​\(em\)≤2α​ℒQm,Pm⋆​\(em\)\+2\+αα​\(C22​m\)1/\(2​d\)\.0\\leq\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\leq\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\leq\\frac\{2\}\{\\alpha\}\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\+\\frac\{2\+\\alpha\}\{\\alpha\}\\left\(\\frac\{C^\{2\}\}\{2m\}\\right\)^\{1/\(2d\)\}\.\(67\)*Convergence\.*

*Left\-to\-right\.*IfℒQm,Pm⋆​\(em\)→0\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\to 0, both terms in the upper bound tend to zero, soℛQm⋆​\(em\)→0\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\to 0\.

*Right\-to\-left\.*IfℛQm⋆​\(em\)→0\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\to 0, the lower bound givesℒQm,Pm⋆​\(em\)→0\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\to 0\.

*Rate\.*

*Left\-to\-right\.*AssumeℒQm,Pm⋆\(em\)=O\(m−1/\(2d\)\)\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)=O\(m^\{\-1/\(2d\)\}\)\. By definition, there is a constantc≥0c\\geq 0, independent ofmm, such thatℒQm,Pm⋆\(em\)≤cm−1/\(2d\)\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\leq c\\,m^\{\-1/\(2d\)\}for all sufficiently largemm\. Substituting this into[equation67](https://arxiv.org/html/2610.27139#A3.E67)gives

ℛQm⋆​\(em\)\\displaystyle\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(e\_\{m\}\)≤2α​ℒQm,Pm⋆​\(em\)\+2\+αα​\(C22​m\)1/\(2​d\)\\displaystyle\\leq\\frac\{2\}\{\\alpha\}\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\+\\frac\{2\+\\alpha\}\{\\alpha\}\\left\(\\frac\{C^\{2\}\}\{2m\}\\right\)^\{1/\(2d\)\}\(68\)≤2​cαm−1/\(2d\)\+2\+αα\(C22\)1/\(2​d\)m−1/\(2d\)\\displaystyle\\leq\\frac\{2c\}\{\\alpha\}m^\{\-1/\(2d\)\}\+\\frac\{2\+\\alpha\}\{\\alpha\}\\left\(\\frac\{C^\{2\}\}\{2\}\\right\)^\{1/\(2d\)\}m^\{\-1/\(2d\)\}=\[2​cα\+2\+αα\(C22\)1/\(2​d\)\]⋅m−1/\(2d\)\.\\displaystyle=\\left\[\\frac\{2c\}\{\\alpha\}\+\\frac\{2\+\\alpha\}\{\\alpha\}\\left\(\\frac\{C^\{2\}\}\{2\}\\right\)^\{1/\(2d\)\}\\right\]\\cdot m^\{\-1/\(2d\)\}\.The bracket is independent ofmm, soℛQm⋆\(em\)=O\(m−1/\(2d\)\)\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(e\_\{m\}\)=O\(m^\{\-1/\(2d\)\}\)\.

*Right\-to\-left\.*AssumeℛQm⋆\(em\)=O\(m−1/\(2d\)\)\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(e\_\{m\}\)=O\(m^\{\-1/\(2d\)\}\)\. There is a constantc≥0c\\geq 0, independent ofmm, such that, for all sufficiently largemm,

0≤ℒQm,Pm⋆\(em\)≤ℛQm⋆\(em\)≤cm−1/\(2d\)\.0\\leq\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\leq\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(e\_\{m\}\)\\leq c\\,m^\{\-1/\(2d\)\}\.\(69\)ThusℒQm,Pm⋆\(em\)=O\(m−1/\(2d\)\)\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\star\}\(e\_\{m\}\)=O\(m^\{\-1/\(2d\)\}\), completing[equation20](https://arxiv.org/html/2610.27139#S3.E20)\. ∎

For a fixedencoder, the selected targets give the same recovery conclusion\.

###### Corollary C\.5\(Low\-Dimensional Exact Recovery\)\.

Under the assumptions of[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4), fix an encodere:ℝQ→𝒵e:\\mathbb\{R\}^\{Q\}\\to\\mathcal\{Z\}into a nonempty learned state space\. For each integerm≥C2/2m\\geq C^\{2\}/2, choosePmP\_\{m\}as in that theorem for the same input setQQ\. If

ℒQ,Pm⋆​\(e\)→m→∞0,\\mathcal\{L\}\_\{Q,P\_\{m\}\}^\{\\star\}\(e\)\\xrightarrow\{m\\to\\infty\}0,\(70\)then there exists a state decodera∈𝒜a\\in\\mathcal\{A\}such that

\(a∘e\)​\(f⁡\(Q\)\)=f,∀f∈ℱ\.\(a\\circ e\)\(f\(Q\)\)=f,\\qquad\\forall f\\in\\mathcal\{F\}\.\(71\)

###### Proof\.

By[theorem3\.5](https://arxiv.org/html/2610.27139#S3.Thmtheorem5), the recovery error tends to zero\. SinceQQandeeare fixed,ℛQ⋆​\(e\)\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)is independent of the stage and therefore equals zero\. Apply[lemmaA\.3](https://arxiv.org/html/2610.27139#A1.Thmtheorem3)to obtain an exactdecoder\. ∎

### C\.3Budget Sharpness

We first prove the fixed\-family result supporting[theorem3\.7](https://arxiv.org/html/2610.27139#S3.Thmtheorem7), then give a finite\-query construction for comparison\.

A fixed family at every budget\.A multiscale arc places the obstruction inside a fixed family: finer blocks contain more distinctions, while their response amplitudes decrease\. Theencodermerges a pair whose distinguishing query was omitted, and the radius of that pair gives the recovery cost\.

We construct the fixed family in[lemmaC\.6](https://arxiv.org/html/2610.27139#A3.Thmtheorem6), find a pair missed by the target set in[lemmaC\.7](https://arxiv.org/html/2610.27139#A3.Thmtheorem7), and merge that pair in[lemmaC\.8](https://arxiv.org/html/2610.27139#A3.Thmtheorem8)\. Combining their bounds proves[lemmaC\.9](https://arxiv.org/html/2610.27139#A3.Thmtheorem9)\.

###### Lemma C\.6\(Multiscale Arc\)\.

For everyd\>1d\>1, there exist a countable query universe𝒰\\mathcal\{U\}, a singleton input setQQ, and a compact state familyℱ\\mathcal\{F\}homeomorphic to\[0,1\]\[0,1\], of diameter11\. The input observation map is injective, and\(ℱ,Q\)\(\\mathcal\{F\},Q\)satisfies[assumption2\.12](https://arxiv.org/html/2610.27139#S2.Thmtheorem12)withC=CdC=C\_\{d\}andα=1\\alpha=1\. The covering constant is

Cd≔1\+121−21/d−1\.C\_\{d\}\\coloneqq 1\+\\frac\{12\}\{1\-2^\{1/d\-1\}\}\.\(72\)

###### Proof\.

*Construct the family\.*Fixd\>1d\>1\. Forj≥1j\\geq 1, put

Nj\\displaystyle N\_\{j\}=8⋅2j−1,\\displaystyle=8\\cdot 2^\{j\-1\},sj\\displaystyle s\_\{j\}=122−\(j−1\)/d,\\displaystyle=\\frac\{1\}\{2\}\\,2^\{\-\(j\-1\)/d\},\(73\)ℓj\\displaystyle\\ell\_\{j\}=2−j−1,\\displaystyle=2^\{\-j\-1\},Tj\\displaystyle T\_\{j\}=1−21−j\.\\displaystyle=1\-2^\{1\-j\}\.\(74\)Letv1,…,vNjv\_\{1\},\\ldots,v\_\{N\_\{j\}\}denote the standard basis inℝNj\\mathbb\{R\}^\{N\_\{j\}\}\. On\[Tj,Tj\+ℓj\]\[T\_\{j\},T\_\{j\}\+\\ell\_\{j\}\], the vectorλj​\(t\)\\lambda^\{j\}\(t\)follows the polygonal pathv1→⋯→vNjv\_\{1\}\\to\\cdots\\to v\_\{N\_\{j\}\}, spending equal time on each edge; the vectors in other blocks are zero\. On\[Tj\+ℓj,Tj\+1\]\[T\_\{j\}\+\\ell\_\{j\},T\_\{j\+1\}\], join this block directly to the next: setλj​\(t\)=w​vNj\\lambda^\{j\}\(t\)=wv\_\{N\_\{j\}\}andλj\+1​\(t\)=\(1−w\)​v1\\lambda^\{j\+1\}\(t\)=\(1\-w\)v\_\{1\}, wherewwdecreases linearly from11to00\. Att=1t=1, all block vectors are zero\. These intervals cover\[0,1\)\[0,1\)becauseTj\+1−Tj=2​ℓjT\_\{j\+1\}\-T\_\{j\}=2\\ell\_\{j\}\.

TakeQ=\{q0\}Q=\\\{q\_\{0\}\\\}\. The remaining queries are a duplicate parameter queryp0p\_\{0\}, a height queryphjp^\{j\}\_\{\\mathrm\{h\}\}for each block, and a private querypa​bjp^\{j\}\_\{ab\}for each1≤a<b≤Nj1\\leq a<b\\leq N\_\{j\}\. Define the stateftf\_\{t\}, for0≤t≤10\\leq t\\leq 1, by

ft​\(q0\)=ft​\(p0\)\\displaystyle f\_\{t\}\(q\_\{0\}\)=f\_\{t\}\(p\_\{0\}\)=t/2,\\displaystyle=t/2,\(75\)ft​\(pa​bj\)\\displaystyle f\_\{t\}\(p^\{j\}\_\{ab\}\)=sj​\(λaj​\(t\)−λbj​\(t\)\),\\displaystyle=s\_\{j\}\\bigl\(\\lambda\_\{a\}^\{j\}\(t\)\-\\lambda\_\{b\}^\{j\}\(t\)\\bigr\),\(76\)ft​\(phj\)\\displaystyle f\_\{t\}\(p^\{j\}\_\{\\mathrm\{h\}\}\)=2​sj​\(1−‖λj​\(t\)‖1\)\.\\displaystyle=2s\_\{j\}\\bigl\(1\-\\\|\\lambda^\{j\}\(t\)\\\|\_\{1\}\\bigr\)\.\(77\)Letℱ=\{ft:0≤t≤1\}\\mathcal\{F\}=\\\{f\_\{t\}:0\\leq t\\leq 1\\\}\.

*Geometry and observability\.*All responses are bounded\. The parameter query makest↦ftt\\mapsto f\_\{t\}injective, and its duplicate gives target observability withα=1\\alpha=1\. Each finite part of the path is continuous in the full response metric\. From blockjjonward, the distance tof1f\_\{1\}is at most2​sj2s\_\{j\}: active private coordinates have magnitude at mostsjs\_\{j\}, height differences are at most2​sj2s\_\{j\}, and the parameter difference is at most2−j≤sj2^\{\-j\}\\leq s\_\{j\}\. Thus the path is also continuous at11\. Compactness and injectivity make it a homeomorphism ontoℱ\\mathcal\{F\}\. Every coordinate has range at most11, and two vertices in the first block differ by2​s1=12s\_\{1\}=1, so the diameter is exactly11\.

The shrinking blocks accumulate at the endpoint\.

*Covering bound\.*Each edge within blockjjis a straight response\-space segment of length2​sj2s\_\{j\}, as is its outgoing connector\. Indeed, a private coordinate attains this length on a block edge, and the height queryphjp^\{j\}\_\{\\mathrm\{h\}\}attains it on the connector; the parameter increment is smaller\. TheNj−1N\_\{j\}\-1block edges and one outgoing connector giveNjN\_\{j\}closed segments per block, including their endpoints\. For0<ε<10<\\varepsilon<1, letJJbe the last index with2​sJ\>ε2s\_\{J\}\>\\varepsilon\. The tail beginning at blockJ\+1J\+1lies in the closedε\\varepsilon\-ball aboutf1f\_\{1\}\. A segment of length2​sj2s\_\{j\}is covered by⌈sj/ε⌉\\lceil s\_\{j\}/\\varepsilon\\rceilclosed balls centered on the segment\. Consequently,

𝒩ℱ​\(ε\)\\displaystyle\\mathcal\{N\}\_\{\\mathcal\{F\}\}\(\\varepsilon\)≤1\+∑j≤JNj​\(sjε\+1\)≤1\+3ε​∑j≤JNj​sj,\\displaystyle\\leq 1\+\\sum\_\{j\\leq J\}N\_\{j\}\\left\(\\frac\{s\_\{j\}\}\{\\varepsilon\}\+1\\right\)\\leq 1\+\\frac\{3\}\{\\varepsilon\}\\sum\_\{j\\leq J\}N\_\{j\}s\_\{j\},\(78\)∑j≤JNj​sj\\displaystyle\\sum\_\{j\\leq J\}N\_\{j\}s\_\{j\}≤NJ​sJ1−21/d−1≤4​ε1−d1−21/d−1\.\\displaystyle\\leq\\frac\{N\_\{J\}s\_\{J\}\}\{1\-2^\{1/d\-1\}\}\\leq\\frac\{4\\varepsilon^\{1\-d\}\}\{1\-2^\{1/d\-1\}\}\.This gives the statedCd​ε−dC\_\{d\}\\varepsilon^\{\-d\}bound\. Atε=1\\varepsilon=1, one ball suffices\. Moreover, theNjN\_\{j\}vertices of blockjjare mutually2​sj2s\_\{j\}apart\. At radiussj/2s\_\{j\}/2, each covering ball contains at most one of them\. SinceNj=8​\(2​sj\)−dN\_\{j\}=8\(2s\_\{j\}\)^\{\-d\}, the covering exponent is exactlydd\. ∎

The next two lemmas use the family and block notation constructed in the proof of[lemmaC\.6](https://arxiv.org/html/2610.27139#A3.Thmtheorem6)\.

###### Lemma C\.7\(Omitted Pair\)\.

For every integerm≥1m\\geq 1and nonemptyP⊆𝒰∖QP\\subseteq\\mathcal\{U\}\\setminus Qwith\|P\|≤m\|P\|\\leq m, some blockjjcontains two nonadjacent vertices with parametersti,tkt\_\{i\},t\_\{k\}such that

dP\(fti\(P\),ftk\(P\)\)≤sj,sj≥12m−1/\(2d\)\.d\_\{P\}\(f\_\{t\_\{i\}\}\(P\),f\_\{t\_\{k\}\}\(P\)\)\\leq s\_\{j\},\\qquad s\_\{j\}\\geq\\frac\{1\}\{2\}m^\{\-1/\(2d\)\}\.\(79\)

###### Proof\.

FixmmandPP\. Choose the smallestjjwith\(Nj−1\)​\(Nj−2\)/2\>m\(N\_\{j\}\-1\)\(N\_\{j\}\-2\)/2\>m\. This is the number of nonadjacent vertex pairs in blockjj, so some pairi<ki<k, withk−i≥2k\-i\\geq 2, haspi​kj∉Pp^\{j\}\_\{ik\}\\notin P\. Writeti,tkt\_\{i\},t\_\{k\}for their parameter values\. Their private querypi​kjp^\{j\}\_\{ik\}has response gap2​sj2s\_\{j\}\. Every other private query has gap at mostsjs\_\{j\}, every height query has gap zero, and the parameter gap is at mostℓj/2≤sj/4\\ell\_\{j\}/2\\leq s\_\{j\}/4\. Hence

dP​\(fti​\(P\),ftk​\(P\)\)≤sj\.d\_\{P\}\(f\_\{t\_\{i\}\}\(P\),f\_\{t\_\{k\}\}\(P\)\)\\leq s\_\{j\}\.\(80\)Whenj=1j=1,Nj=8≤8​mN\_\{j\}=8\\leq 8\\sqrt\{m\}\. Otherwise minimality ofjjgives\(Nj−1−1\)​\(Nj−1−2\)/2≤m\(N\_\{j\-1\}\-1\)\(N\_\{j\-1\}\-2\)/2\\leq m\. HenceNj−1≤2​m\+2N\_\{j\-1\}\\leq\\sqrt\{2m\}\+2andNj=2​Nj−1≤8​mN\_\{j\}=2N\_\{j\-1\}\\leq 8\\sqrt\{m\}\. It follows that

sj3=16\(Nj/8\)−1/d≥16m−1/\(2d\)\.\\frac\{s\_\{j\}\}\{3\}=\\frac\{1\}\{6\}\(N\_\{j\}/8\)^\{\-1/d\}\\geq\\frac\{1\}\{6\}m^\{\-1/\(2d\)\}\.\(81\)∎

###### Lemma C\.8\(Two\-State Fiber\)\.

Fix two nonadjacent verticesfti,ftkf\_\{t\_\{i\}\},f\_\{t\_\{k\}\}withi<ki<kin blockjjand a nonempty finite target setP⊆𝒰∖QP\\subseteq\\mathcal\{U\}\\setminus Q\. There is a globally Lipschitz encodere:ℝQ→ℝ2e:\\mathbb\{R\}^\{Q\}\\to\\mathbb\{R\}^\{2\}whose only non\-singleton state fiber is this pair, with

ℒQ,P⋆​\(e\)\\displaystyle\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)=12​dP​\(fti​\(P\),ftk​\(P\)\),\\displaystyle=\\frac\{1\}\{2\}d\_\{P\}\(f\_\{t\_\{i\}\}\(P\),f\_\{t\_\{k\}\}\(P\)\),\(82\)ℛQ⋆​\(e\)\\displaystyle\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)≥43​sj\.\\displaystyle\\geq\\frac\{4\}\{3\}s\_\{j\}\.\(83\)

###### Proof\.

*A two\-state fiber\.*Putui=ti/2u\_\{i\}=t\_\{i\}/2anduk=tk/2u\_\{k\}=t\_\{k\}/2\. Define

e⁡\(u\)=\{\(u−ui,0\),u≤ui,\(sin⁡2​π​\(u−ui\)uk−ui,1−cos⁡2​π​\(u−ui\)uk−ui\),ui≤u≤uk,\(u−uk,0\),u≥uk\.e\(u\)=\\\!\\\!\\begin\{cases\}\(u\-u\_\{i\},0\),&u\\leq u\_\{i\},\\\\ \(\\sin\\\!\\frac\{2\\pi\(u\-u\_\{i\}\)\}\{u\_\{k\}\-u\_\{i\}\},1\-\\cos\\\!\\frac\{2\\pi\(u\-u\_\{i\}\)\}\{u\_\{k\}\-u\_\{i\}\}\),&u\_\{i\}\\leq u\\leq u\_\{k\},\\\\ \(u\-u\_\{k\},0\),&u\\geq u\_\{k\}\.\\end\{cases\}\(84\)The circular part has positive second coordinate in its interior, while the two tails lie on opposite halves of the horizontal axis\. Thus the only merged input pair isui,uku\_\{i\},u\_\{k\}\. The pieces agree at their endpoints, and theencoderis globally Lipschitz with constant at mostmax⁡\{1,2​π/\(uk−ui\)\}\\max\\\{1,2\\pi/\(u\_\{k\}\-u\_\{i\}\)\\\}\. By[lemma3\.1](https://arxiv.org/html/2610.27139#S3.Thmtheorem1),

ℒQ,P⋆​\(e\)=12​dP​\(fti​\(P\),ftk​\(P\)\)\.\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)=\\frac\{1\}\{2\}d\_\{P\}\(f\_\{t\_\{i\}\}\(P\),f\_\{t\_\{k\}\}\(P\)\)\.\(85\)
*The admissible recovery center\.*Every singleton fiber can be decoded exactly\. It remains to bound the radius of the pair about anyg∈ℱg\\in\\mathcal\{F\}\.

Forg=ftg=f\_\{t\}in blockjj, the vectorλj​\(t\)\\lambda^\{j\}\(t\)has mass one on at most two adjacent vertices\. Exchangei,ki,kif necessary so thatλij​\(t\)=0\\lambda\_\{i\}^\{j\}\(t\)=0, and take a largest coordinateλbj​\(t\)≥1/2\\lambda\_\{b\}^\{j\}\(t\)\\geq 1/2\. Thenb≠ib\\neq i, and the private query indexed by\{i,b\}\\\{i,b\\\}gives response difference

sj​\|1−λij​\(t\)\+λbj​\(t\)\|=sj​\(1\+λbj​\(t\)\)≥32​sj\.s\_\{j\}\\bigl\|1\-\\lambda\_\{i\}^\{j\}\(t\)\+\\lambda\_\{b\}^\{j\}\(t\)\\bigr\|=s\_\{j\}\\bigl\(1\+\\lambda\_\{b\}^\{j\}\(t\)\\bigr\)\\geq\\frac\{3\}\{2\}s\_\{j\}\.\(86\)On an adjacent connector, the block vector iswwtimes an endpoint, giving private\-coordinate error at leastsj​\(1\+w\)s\_\{j\}\(1\+w\)and height error2​sj​\(1−w\)2s\_\{j\}\(1\-w\)\. Elsewhere, the block mass is zero and the height error is2​sj2s\_\{j\}\. Thus

1sj​max⁡\{d𝒰​\(g,fti\),d𝒰​\(g,ftk\)\}≥\{3/2,in block​j,max⁡\{1\+w,2​\(1−w\)\},on a connector,2,elsewhere≥43\.\\frac\{1\}\{s\_\{j\}\}\\max\\\{d\_\{\\mathcal\{U\}\}\(g,f\_\{t\_\{i\}\}\),d\_\{\\mathcal\{U\}\}\(g,f\_\{t\_\{k\}\}\)\\\}\\geq\\begin\{cases\}3/2,&\\text\{in block \}j,\\\\ \\max\\\{1\+w,2\(1\-w\)\\\},&\\text\{on a connector\},\\\\ 2,&\\text\{elsewhere\}\\end\{cases\}\\geq\\frac\{4\}\{3\}\.\(87\)The connector bound follows by minimizing over0≤w≤10\\leq w\\leq 1, with equality atw=1/3w=1/3\. Taking the infimum over common decoded states gives the recovery bound\. ∎

The three lemmas now give the fixed\-family lower bound\.

###### Lemma C\.9\(Fixed\-Family Sharpness\)\.

For everyd\>1d\>1, there exist a fixed countable query universe𝒰\\mathcal\{U\}, a singleton input setQQ, and a compact state familyℱ\\mathcal\{F\}homeomorphic to\[0,1\]\[0,1\], of diameter11\. The input observation map is injective, and\(ℱ,Q\)\(\\mathcal\{F\},Q\)satisfies[assumption2\.12](https://arxiv.org/html/2610.27139#S2.Thmtheorem12)withC=CdC=C\_\{d\}andα=1\\alpha=1\.

For every integerm≥1m\\geq 1and every nonemptyP⊆𝒰∖QP\\subseteq\\mathcal\{U\}\\setminus Qwith\|P\|≤m\|P\|\\leq m, there is a globally Lipschitz encodere:ℝQ→ℝ2e:\\mathbb\{R\}^\{Q\}\\to\\mathbb\{R\}^\{2\}such that

ℛQ⋆\(e\)−2ℒQ,P⋆\(e\)≥16m−1/\(2d\)\.\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\-2\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\\geq\\frac\{1\}\{6\}m^\{\-1/\(2d\)\}\.\(88\)

###### Proof\.

[LemmaC\.6](https://arxiv.org/html/2610.27139#A3.Thmtheorem6)supplies the fixed family and its geometric properties\. For any admissiblem,Pm,P, take the pair from[lemmaC\.7](https://arxiv.org/html/2610.27139#A3.Thmtheorem7)and theencoderfrom[lemmaC\.8](https://arxiv.org/html/2610.27139#A3.Thmtheorem8)\. Then

ℛQ⋆\(e\)−2ℒQ,P⋆\(e\)≥43sj−dP\(fti\(P\),ftk\(P\)\)≥sj3≥16m−1/\(2d\)\.\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\-2\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\\geq\\frac\{4\}\{3\}s\_\{j\}\-d\_\{P\}\(f\_\{t\_\{i\}\}\(P\),f\_\{t\_\{k\}\}\(P\)\)\\geq\\frac\{s\_\{j\}\}\{3\}\\geq\\frac\{1\}\{6\}m^\{\-1/\(2d\)\}\.\(89\)∎

The uniform transfer residual\.For the fixed family above, the uniform transfer residual in[definition3\.6](https://arxiv.org/html/2610.27139#S3.Thmtheorem6)measures the recovery cost of choosing targets before theencoder\. The lower bound is supplied by[lemmaC\.9](https://arxiv.org/html/2610.27139#A3.Thmtheorem9); the upper bound follows from[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4)\.

###### Proof of[theorem3\.7](https://arxiv.org/html/2610.27139#S3.Thmtheorem7)\.

Use the fixed family in[lemmaC\.9](https://arxiv.org/html/2610.27139#A3.Thmtheorem9), and fix an integerm≥Cd2/2m\\geq C\_\{d\}^\{2\}/2\.

*Lower bound\.*For every admissible target setPP,[lemmaC\.9](https://arxiv.org/html/2610.27139#A3.Thmtheorem9)supplies a globally Lipschitzencoderwitnessing

supe∈ℰLip\[ℛQ⋆\(e\)−2ℒQ,P⋆\(e\)\]≥16m−1/\(2d\)\.\\sup\_\{e\\in\\mathcal\{E\}\_\{\\mathrm\{Lip\}\}\}\\bigl\[\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\-2\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\\bigr\]\\geq\\frac\{1\}\{6\}m^\{\-1/\(2d\)\}\.\(90\)Taking the infimum overPPgives the lower bound for𝒯m​\(ℱ,Q\)\\mathcal\{T\}\_\{m\}\(\\mathcal\{F\},Q\)\.

*Upper bound\.*Apply[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4)withC=CdC=C\_\{d\}andα=1\\alpha=1\. It supplies one admissible target setPPfor which everyencodersatisfies

ℛQ⋆​\(e\)−2​ℒQ,P⋆​\(e\)≤3​\(Cd22​m\)1/\(2​d\)\.\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\-2\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\\leq 3\\left\(\\frac\{C\_\{d\}^\{2\}\}\{2m\}\\right\)^\{1/\(2d\)\}\.\(91\)Taking the supremum overℰLip\\mathcal\{E\}\_\{\\mathrm\{Lip\}\}for thisPP, then the infimum over all admissible target sets, gives the upper bound\.

*11\-Lipschitz encoders\.*Ifeehas a Lipschitz boundK≥1K\\geq 1, thene/Ke/Kis11\-Lipschitz\. Composing a readout withz↦K​zz\\mapsto Kztransfers it fromeetoe/Ke/K;z↦z/Kz\\mapsto z/Kreverses this correspondence\. Predictions and recovered states are preserved, so both optimal errors are unchanged\. Every residual value is thus attained in the smaller class; inclusion gives the reverse inequality between suprema\. ∎

Recovery and sampling\.Classical optimal recovery describes the best readout from fixed information by the radius of its consistency sets\([Traub et al\., 1983](https://arxiv.org/html/2610.27139#bib.bib43), Chapters 1 and 3\)\. Here the information ise⁡\(f⁡\(Q\)\)e\(f\(Q\)\), and the same consistency set determines two errors: its radius in the admissible state family and its radius in the target observation space\. The fixed arc makes their difference persist at every finite target budget\.

Optimal sampling instead chooses queries together with a reconstruction map\([Krieg and Ullrich, 2026](https://arxiv.org/html/2610.27139#bib.bib23), Sections 2 and 9\.1\)\. Write the arc asℱ=\{ft:0≤t≤1\}\\mathcal\{F\}=\\\{f\_\{t\}:0\\leq t\\leq 1\\\}, with parameter queryp0∈𝒰∖Qp\_\{0\}\\in\\mathcal\{U\}\\setminus Qsatisfyingft​\(p0\)=t/2f\_\{t\}\(p\_\{0\}\)=t/2\. This query determines the state exactly\. Witha⁡\(y\)=f2​ya\(y\)=f\_\{2y\}for0≤y≤1/20\\leq y\\leq 1/2, extended byf0f\_\{0\}elsewhere,

infa:ℝ→ℱsup0≤t≤1d𝒰\(a\(ft\(p0\)\),ft\)=0\.\\inf\_\{a:\\mathbb\{R\}\\to\\mathcal\{F\}\}\\sup\_\{0\\leq t\\leq 1\}d\_\{\\mathcal\{U\}\}\\bigl\(a\(f\_\{t\}\(p\_\{0\}\)\),f\_\{t\}\\bigr\)=0\.\(92\)The residual in[equation21](https://arxiv.org/html/2610.27139#S3.E21)measures the budget needed to transfer prediction accuracy to recovery uniformly acrossencoders\. On the constructed family, a residual toleranceε\\varepsilonrequires a budget of orderε−2​d\\varepsilon^\{\-2d\}, while direct reconstruction from the parameter query is exact\.

Uniform\-norm discretization gives a related view of how finitely many queries preserve distinctions in a linear function space\([Kosov and Temlyakov, 2024](https://arxiv.org/html/2610.27139#bib.bib22), Theorems 1\.1 and 1\.2\)\. Its criterion is multiplicative norm preservation over that space\. The criterion here couples prediction and state\-valued recovery on a fixed nonlinear compact family\. The covering bound controls the number of states needed at a given resolution; the pairwise witnesses in the construction determine the target budget\.

Encoder regularity and target order\.[Theorem3\.7](https://arxiv.org/html/2610.27139#S3.Thmtheorem7)gives a commonencoderregularity bound\. Keeping this bound fixed, we compare targets selected before and after theencoder\.

###### Corollary C\.10\(Target Selection Order\)\.

On the fixed state family of[lemmaC\.9](https://arxiv.org/html/2610.27139#A3.Thmtheorem9), letℰ1\\mathcal\{E\}\_\{1\}consist of the11\-Lipschitz mapse:ℝQ→ℝ2e:\\mathbb\{R\}^\{Q\}\\to\\mathbb\{R\}^\{2\}\. For everye∈ℰ1e\\in\\mathcal\{E\}\_\{1\}, there is a queryp∈𝒰∖Qp\\in\\mathcal\{U\}\\setminus Qsuch thatℛQ⋆​\(e\)≤2​ℒQ,\{p\}⋆​\(e\)\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\\leq 2\\mathcal\{L\}\_\{Q,\\\{p\\\}\}^\{\\star\}\(e\)\. Consequently, for every integerm≥1m\\geq 1,

supe∈ℰ1infP⊆𝒰∖Q1≤\|P\|≤m\[ℛQ⋆​\(e\)−2​ℒQ,P⋆​\(e\)\]=0\.\\sup\_\{e\\in\\mathcal\{E\}\_\{1\}\}\\inf\_\{\\begin\{subarray\}\{c\}P\\subseteq\\mathcal\{U\}\\setminus Q\\\\ 1\\leq\|P\|\\leq m\\end\{subarray\}\}\\bigl\[\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\-2\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\\bigr\]=0\.\(93\)

###### Proof\.

For a fixede∈ℰ1e\\in\\mathcal\{E\}\_\{1\}, put

Δe≔maxf,g∈ℱe⁡\(f⁡\(Q\)\)=e⁡\(g⁡\(Q\)\)⁡d𝒰​\(f,g\)\.\\Delta\_\{e\}\\coloneqq\\max\_\{\\begin\{subarray\}\{c\}f,g\\in\\mathcal\{F\}\\\\ e\(f\(Q\)\)=e\(g\(Q\)\)\\end\{subarray\}\}d\_\{\\mathcal\{U\}\}\(f,g\)\.\(94\)The maximum exists: the mapf↦e⁡\(f⁡\(Q\)\)f\\mapsto e\(f\(Q\)\)is continuous, so its equality set is a nonempty closed subset ofℱ×ℱ\\mathcal\{F\}\\times\\mathcal\{F\}\. By[lemmaA\.2](https://arxiv.org/html/2610.27139#A1.Thmtheorem2), a state\-valued representativedecodergivesℛQ⋆​\(e\)≤Δe\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\\leq\\Delta\_\{e\}\.

Each state on the arc has nonzero block mass in at most two blocks\. For any pair, differences in private and height responses are confined to the finite union of their active blocks\. The remaining differing coordinates areq0,p0q\_\{0\},p\_\{0\}, whose differences coincide\. A query outsideQQtherefore attains the full distance of the pair\. Apply this to a pair attainingΔe\\Delta\_\{e\}, and call its querypp\. The largestpp\-response gap among pairs sharingencoderoutput equalsΔe\\Delta\_\{e\}: this pair attains it, and every such gap is bounded by the full\-state gap\. By[lemma3\.1](https://arxiv.org/html/2610.27139#S3.Thmtheorem1),

ℛQ⋆​\(e\)≤Δe=2​ℒQ,\{p\}⋆​\(e\)\.\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\\leq\\Delta\_\{e\}=2\\mathcal\{L\}\_\{Q,\\\{p\\\}\}^\{\\star\}\(e\)\.\(95\)WhenΔe=0\\Delta\_\{e\}=0, the same conclusion holds with any available query\. Eachencoderthus has a singleton target giving a nonpositive residual\. Conversely,e⁡\(u\)=\(u,0\)e\(u\)=\(u,0\)belongs toℰ1\\mathcal\{E\}\_\{1\}, and the input observation identifies every state\. For thisencoderboth optimal errors are zero for every target set\. Taking the indicated infimum and supremum proves[equation93](https://arxiv.org/html/2610.27139#A3.E93)\. ∎

A finite\-query comparison\.Allowing the family to depend on the budget yields a complementary construction: theencoderis linear, the query universe is finite, and the exponent range includesd=1d=1\. The same order is attained at prediction\-error coefficient22andα=1\\alpha=1\.

###### Proposition C\.11\(Budget Sharpness\)\.

For everyd≥1d\\geq 1, there is a constantcd\>0c\_\{d\}\>0such that, for each integerm≥1m\\geq 1, there exist a finite query universe𝒰\\mathcal\{U\}, a query setQQ, and a compact connected one\-dimensional state manifoldℱ\\mathcal\{F\}of diameter11, such that\(ℱ,Q\)\(\\mathcal\{F\},Q\)satisfies[assumption2\.12](https://arxiv.org/html/2610.27139#S2.Thmtheorem12)withC=3C=3andα=1\\alpha=1\. The input observation map is injective\. For every nonemptyP⊆𝒰∖QP\\subseteq\\mathcal\{U\}\\setminus Qwith\|P\|≤m\|P\|\\leq m, there is a linear encoder into a Euclidean learned state space such that

ℛQ⋆\(e\)≥2ℒQ,P⋆\(e\)\+cdm−1/\(2d\)\.\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\\geq 2\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\+c\_\{d\}m^\{\-1/\(2d\)\}\.\(96\)

###### Proof\.

*State space\.*Fixd≥1d\\geq 1andm≥1m\\geq 1, and set

N≔⌈2​m⌉\+3,s≔12N−1/d\.N\\coloneqq\\lceil\\sqrt\{2m\}\\rceil\+3,\\qquad s\\coloneqq\\frac\{1\}\{2\}N^\{\-1/d\}\.\(97\)Letv1,…,vNv\_\{1\},\\ldots,v\_\{N\}be the standard basis ofℝN\\mathbb\{R\}^\{N\}, withvN\+1=v1v\_\{N\+1\}=v\_\{1\}\. On the polygonal cycle⋃i=1N\[vi,vi\+1\]\\bigcup\_\{i=1\}^\{N\}\[v\_\{i\},v\_\{i\+1\}\], letH⁡\(λ\)=2​min⁡\{λ1,λ2\}H\(\\lambda\)=2\\min\\\{\\lambda\_\{1\},\\lambda\_\{2\}\\\}\. This continuous function is zero off the edge\[v1,v2\]\[v\_\{1\},v\_\{2\}\]and reaches11at its midpoint\. TakeQ=\{q0,q1,…,qN\}Q=\\\{q\_\{0\},q\_\{1\},\\ldots,q\_\{N\}\\\}and𝒰∖Q=\{p0\}∪\{pi​j:1≤i<j≤N\}\\mathcal\{U\}\\setminus Q=\\\{p\_\{0\}\\\}\\cup\\\{p\_\{ij\}:1\\leq i<j\\leq N\\\}\. For eachλ\\lambdaon the cycle, define, for1≤i≤N1\\leq i\\leq Nand1≤i<j≤N1\\leq i<j\\leq N,

fλ​\(qi\)≔s​λi,fλ​\(pi​j\)≔s⁡\(λi−λj\)\.f\_\{\\lambda\}\(q\_\{i\}\)\\coloneqq s\\lambda\_\{i\},\\qquad f\_\{\\lambda\}\(p\_\{ij\}\)\\coloneqq s\(\\lambda\_\{i\}\-\\lambda\_\{j\}\)\.\(98\)Also setfλ​\(q0\)=fλ​\(p0\)=H⁡\(λ\)f\_\{\\lambda\}\(q\_\{0\}\)=f\_\{\\lambda\}\(p\_\{0\}\)=H\(\\lambda\), lifting one edge while leaving all vertices unchanged\. Letℱ\\mathcal\{F\}consist of these states, and writefi=fvif\_\{i\}=f\_\{v\_\{i\}\}\. The mapλ↦fλ\\lambda\\mapsto f\_\{\\lambda\}is continuous, with continuous inverseλi=fλ​\(qi\)/s\\lambda\_\{i\}=f\_\{\\lambda\}\(q\_\{i\}\)/s\. Thusℱ\\mathcal\{F\}is homeomorphic to the cycle, a compact connected one\-dimensional manifold, and the input observation map is injective\. Since∑iλi=∑iμi=1\\sum\_\{i\}\\lambda\_\{i\}=\\sum\_\{i\}\\mu\_\{i\}=1, we have

d𝒰​\(fλ,fμ\)=max⁡\{s⁡\[maxk⁡\(λk−μk\)−mink⁡\(λk−μk\)\],\|H⁡\(λ\)−H⁡\(μ\)\|\}\.d\_\{\\mathcal\{U\}\}\(f\_\{\\lambda\},f\_\{\\mu\}\)=\\max\\bigl\\\{s\[\\max\_\{k\}\(\\lambda\_\{k\}\-\\mu\_\{k\}\)\-\\min\_\{k\}\(\\lambda\_\{k\}\-\\mu\_\{k\}\)\],\|H\(\\lambda\)\-H\(\\mu\)\|\\bigr\\\}\.\(99\)The target queries capture both terms, so target observability holds withα=1\\alpha=1\. Since2​s≤12s\\leq 1andHHranges from00to11, the diameter is exactly11\.

*Covering bound\.*Each unlifted edge has length2​s2sind𝒰d\_\{\\mathcal\{U\}\}\. For0<ε<s0<\\varepsilon<s, the unlifted part can be covered by at mostN​⌈s/ε⌉N\\lceil s/\\varepsilon\\rceilballs centered along these edges\. Sinced≥1d\\geq 1,

N⁡⌈s/ε⌉≤2​N​sε≤2​N​sd​ε−d=21−d​ε−d≤ε−d\.N\\lceil s/\\varepsilon\\rceil\\leq\\frac\{2Ns\}\{\\varepsilon\}\\leq 2Ns^\{d\}\\varepsilon^\{\-d\}=2^\{1\-d\}\\varepsilon^\{\-d\}\\leq\\varepsilon^\{\-d\}\.\(100\)Fors≤ε<2​ss\\leq\\varepsilon<2s, theNNvertices cover the unlifted part andN=\(2​s\)−d≤ε−dN=\(2s\)^\{\-d\}\\leq\\varepsilon^\{\-d\}\. For2​s≤ε≤12s\\leq\\varepsilon\\leq 1, one ball centered at a vertex covers that part, and1≤ε−d1\\leq\\varepsilon^\{\-d\}\. The lifted edge consists of two segments of length11each, so it admits a11\-Lipschitz parametrization on an interval of length22\. It can therefore be covered by⌈1/ε⌉≤2​ε−1\\lceil 1/\\varepsilon\\rceil\\leq 2\\varepsilon^\{\-1\}additional balls centered along it\. Consequently,

𝒩ℱ​\(ε\)≤ε−d\+2​ε−1≤3​ε−d,0<ε≤1\.\\mathcal\{N\}\_\{\\mathcal\{F\}\}\(\\varepsilon\)\\leq\\varepsilon^\{\-d\}\+2\\varepsilon^\{\-1\}\\leq 3\\varepsilon^\{\-d\},\\qquad 0<\\varepsilon\\leq 1\.\(101\)
*Encoder construction\.*Fix any allowedPP\. The cycle hasN⁡\(N−3\)/2\>mN\(N\-3\)/2\>munordered nonadjacent vertex pairs, so some such pairi<ji<jhaspi​j∉Pp\_\{ij\}\\notin P\. Define a linearencodere:ℝQ→𝒵=ℝNe:\\mathbb\{R\}^\{Q\}\\to\\mathcal\{Z\}=\\mathbb\{R\}^\{N\}by

e⁡\(x\)≔\(x⁡\(q0\),x⁡\(qi\)\+x⁡\(qj\),\(x⁡\(qk\)\)k∈\{1,…,N\}∖\{i,j\}\)\.e\(x\)\\coloneqq\\bigl\(x\(q\_\{0\}\),\\,x\(q\_\{i\}\)\+x\(q\_\{j\}\),\\,\(x\(q\_\{k\}\)\)\_\{k\\in\\\{1,\\ldots,N\\\}\\setminus\\\{i,j\\\}\}\\bigr\)\.\(102\)It combines the two corresponding input coordinates and retains all others\. In particular,fif\_\{i\}andfjf\_\{j\}have the same learned state\. The diagram illustrates the nonadjacent pair and its common learned state\.

*Recovery lower bound\.*Anystate decodermust give one outputg=fλ∈ℱg=f\_\{\\lambda\}\\in\\mathcal\{F\}to the common learned state offi,fjf\_\{i\},f\_\{j\}\. A point on the cycle has at most two nonzero coordinates, at adjacent vertices\. Sincei,ji,jare nonadjacent, at least one ofλi,λj\\lambda\_\{i\},\\lambda\_\{j\}is zero\. Alsomaxk⁡λk≥1/2\\max\_\{k\}\\lambda\_\{k\}\\geq 1/2\. By[equation99](https://arxiv.org/html/2610.27139#A3.E99),

max⁡\{d𝒰​\(fi,g\),d𝒰​\(fj,g\)\}≥s⁡\(1\+maxk⁡λk\)≥3​s2\.\\max\\\{d\_\{\\mathcal\{U\}\}\(f\_\{i\},g\),d\_\{\\mathcal\{U\}\}\(f\_\{j\},g\)\\\}\\geq s\\bigl\(1\+\\max\_\{k\}\\lambda\_\{k\}\\bigr\)\\geq\\frac\{3s\}\{2\}\.\(103\)Thus everydecoderhas recovery error at least3​s/23s/2, soℛQ⋆​\(e\)≥3​s/2\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\\geq 3s/2\.

*Prediction upper bound\.*Ife⁡\(fλ​\(Q\)\)=e⁡\(fμ​\(Q\)\)e\(f\_\{\\lambda\}\(Q\)\)=e\(f\_\{\\mu\}\(Q\)\), thenλ−μ=t⁡\(vi−vj\)\\lambda\-\\mu=t\(v\_\{i\}\-v\_\{j\}\)for some\|t\|≤1\|t\|\\leq 1, andH⁡\(λ\)=H⁡\(μ\)H\(\\lambda\)=H\(\\mu\)\. The querypi​jp\_\{ij\}differs by2​s​\|t\|2s\|t\|; every other pair query differs by at mosts​\|t\|s\|t\|, andp0p\_\{0\}has identical observations\. Sincepi​j∉Pp\_\{ij\}\\notin P,[lemma3\.1](https://arxiv.org/html/2610.27139#S3.Thmtheorem1)gives

ℒQ,P⋆​\(e\)≤s2\.\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\\leq\\frac\{s\}\{2\}\.\(104\)Combining the two error bounds and usingN≤2​m\+4≤6​mN\\leq\\sqrt\{2m\}\+4\\leq 6\\sqrt\{m\}, we obtain

ℛQ⋆\(e\)−2ℒQ,P⋆\(e\)≥s2=14N−1/d≥14⋅61/dm−1/\(2d\)\.\\mathcal\{R\}\_\{Q\}^\{\\star\}\(e\)\-2\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)\\geq\\frac\{s\}\{2\}=\\frac\{1\}\{4\}N^\{\-1/d\}\\geq\\frac\{1\}\{4\\cdot 6^\{1/d\}\}m^\{\-1/\(2d\)\}\.\(105\)Takingcd=1/\(4⋅61/d\)c\_\{d\}=1/\(4\\cdot 6^\{1/d\}\)completes the proof\. ∎

## Appendix DFine\-Tuning and Recovery

The preceding results start from prediction error over the entire state family\. Fine\-tuning instead gives an error on training observations\. Coverage and prediction stability connect these quantities: a nearby training record controls the prediction at an unobserved state\. The general and low\-dimensional bounds then transfer that control to recovery\.

### D\.1Fine\-Tuning Setup

We work under[assumptionA\.4](https://arxiv.org/html/2610.27139#A1.Thmtheorem4)\. Fix disjoint query setsQ,PQ,Pand mapse:ℝQ→𝒵e:\\mathbb\{R\}^\{Q\}\\to\\mathcal\{Z\},b:𝒵→ℝPb:\\mathcal\{Z\}\\to\\mathbb\{R\}^\{P\}, where𝒵\\mathcal\{Z\}is nonempty\. We first control this pair’s prediction error from its errors on the training records\.

Use the recorded error and observation coverage radius from[definition3\.8](https://arxiv.org/html/2610.27139#S3.Thmtheorem8), together with[assumption3\.9](https://arxiv.org/html/2610.27139#S3.Thmtheorem9)\. By[equation33](https://arxiv.org/html/2610.27139#A1.E33),hnh\_\{n\}is no larger than the coverage radius in the full ontic metric\. The asymptotic results use a common stability modulus across the returned outputs\.

###### Lemma D\.1\(Record Extension\)\.

Let\(ℱ,d𝒰\)\(\\mathcal\{F\},d\_\{\\mathcal\{U\}\}\)be nonempty and compact, and letf1,…,fn∈ℱf\_\{1\},\\ldots,f\_\{n\}\\in\\mathcal\{F\}withn≥1n\\geq 1index the training records\. For disjoint query setsQ,PQ,P, every pair\(e,b\)\(e,b\)satisfying[assumption3\.9](https://arxiv.org/html/2610.27139#S3.Thmtheorem9)obeys

ℒ^Q,P,n​\(b,e\)≤ℒQ,P​\(b,e\)≤ℒ^Q,P,n​\(b,e\)\+ρ⁡\(hn\)\+hn,\\widehat\{\\mathcal\{L\}\}\_\{Q,P,n\}\(b;e\)\\leq\\boxed\{\\mathcal\{L\}\_\{Q,P\}\(b;e\)\}\\leq\\widehat\{\\mathcal\{L\}\}\_\{Q,P,n\}\(b;e\)\+\\rho\(h\_\{n\}\)\+h\_\{n\},\(106\)wherehnh\_\{n\}is the observation coverage radius in[equation24](https://arxiv.org/html/2610.27139#S3.E24)\.

###### Proof\.

The empirical error is a maximum over states inℱ\\mathcal\{F\}, so it cannot exceed the full\-family error\. For the upper bound, fixf∈ℱf\\in\\mathcal\{F\}and choose a nearest training observation pair\. Its generating statefif\_\{i\}satisfiesdQ∪P​\(f⁡\(Q∪P\),fi​\(Q∪P\)\)≤hnd\_\{Q\\cup P\}\(f\(Q\\cup P\),f\_\{i\}\(Q\\cup P\)\)\\leq h\_\{n\}\. Prediction stability and the triangle inequality give

dP​\(\(b∘e\)​\(f⁡\(Q\)\),f⁡\(P\)\)\\displaystyle d\_\{P\}\\bigl\(\(b\\circ e\)\(f\(Q\)\),f\(P\)\\bigr\)≤dP​\(\(b∘e\)​\(f⁡\(Q\)\),\(b∘e\)​\(fi​\(Q\)\)\)\+dP​\(\(b∘e\)​\(fi​\(Q\)\),fi​\(P\)\)\\displaystyle\\leq d\_\{P\}\\bigl\(\(b\\circ e\)\(f\(Q\)\),\(b\\circ e\)\(f\_\{i\}\(Q\)\)\\bigr\)\+d\_\{P\}\\bigl\(\(b\\circ e\)\(f\_\{i\}\(Q\)\),f\_\{i\}\(P\)\\bigr\)\(107\)\+dP​\(fi​\(P\),f⁡\(P\)\)≤ρ⁡\(hn\)\+ℒ^Q,P,n​\(b,e\)\+hn\.\\displaystyle\+d\_\{P\}\\bigl\(f\_\{i\}\(P\),f\(P\)\\bigr\)\\leq\\rho\(h\_\{n\}\)\+\\widehat\{\\mathcal\{L\}\}\_\{Q,P,n\}\(b;e\)\+h\_\{n\}\.Taking the supremum overffproves the upper bound\. Both the empirical maximum and the coverage radius are finite, so the full\-family error is finite as well\. ∎

###### Proof of[proposition3\.10](https://arxiv.org/html/2610.27139#S3.Thmtheorem10)\.

[LemmaD\.1](https://arxiv.org/html/2610.27139#A4.Thmtheorem1)gives the prediction bound\. Since the optimal prediction error is no larger,[theorems3\.2](https://arxiv.org/html/2610.27139#S3.Thmtheorem2)and[3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4)and monotonicity ofωP\\omega\_\{P\}give the two recovery bounds\. ∎

We now choose the pair by fine\-tuning\. Letℋ\\mathcal\{H\}be the nonempty class ofencoder–predictorpairs accessible from the initialization by changing the permitted parameters\. The empirical error is defined in[equation23](https://arxiv.org/html/2610.27139#S3.E23)\.

###### Definition D\.2\(Fine\-Tuning\)\.

The*optimal prediction error in the fine\-tuning class*is

ℒQ,Pℋ,⋆≔inf\(e,b\)∈ℋℒQ,P​\(b,e\)\.\\mathcal\{L\}\_\{Q,P\}^\{\\mathcal\{H\},\\star\}\\coloneqq\\inf\_\{\(e,b\)\\in\\mathcal\{H\}\}\\mathcal\{L\}\_\{Q,P\}\(b;e\)\.\(108\)Given an optimization toleranceη≥0\\eta\\geq 0, an*η\\eta\-optimal fine\-tuning output*is a pair\(e^,b^\)∈ℋ\(\\hat\{e\},\\hat\{b\}\)\\in\\mathcal\{H\}satisfying

ℒ^Q,P,n​\(b^,e^\)≤inf\(e,b\)∈ℋℒ^Q,P,n​\(b,e\)\+η\.\\widehat\{\\mathcal\{L\}\}\_\{Q,P,n\}\(\\hat\{b\};\\hat\{e\}\)\\leq\\inf\_\{\(e,b\)\\in\\mathcal\{H\}\}\\widehat\{\\mathcal\{L\}\}\_\{Q,P,n\}\(b;e\)\+\\eta\.\(109\)

For everyη\>0\\eta\>0, anη\\eta\-optimal output exists by the definition of the infimum\. Atη=0\\eta=0, the rule selects an empirical minimizer whenever the infimum is attained\. The quantityℒQ,Pℋ,⋆\\mathcal\{L\}\_\{Q,P\}^\{\\mathcal\{H\},\\star\}optimizes jointly over the permittedencoder–predictorpairs, whileℒQ,P⋆​\(e\)\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(e\)holds theencoderfixed\.

For a fine\-tuning output, the same lemma combines with empirical near\-optimality to relate its prediction error to the best error in the model class\.

###### Corollary D\.3\(Fine\-Tuning Consistency\)\.

Everyη\\eta\-optimal fine\-tuning output satisfying[assumption3\.9](https://arxiv.org/html/2610.27139#S3.Thmtheorem9)obeys

ℒQ,Pℋ,⋆≤ℒQ,P​\(b^,e^\)≤ℒQ,Pℋ,⋆\+η\+ρ⁡\(hn\)\+hn\.\\mathcal\{L\}\_\{Q,P\}^\{\\mathcal\{H\},\\star\}\\leq\\boxed\{\\mathcal\{L\}\_\{Q,P\}\(\\hat\{b\};\\hat\{e\}\)\}\\leq\\mathcal\{L\}\_\{Q,P\}^\{\\mathcal\{H\},\\star\}\+\\eta\+\\rho\(h\_\{n\}\)\+h\_\{n\}\.\(110\)For stagesk≥1k\\geq 1, allow the training data, query sets, model classes, and learned state spaces to vary, with a common modulusρ\\rhofor the returned outputs\. Let\(e^k,b^k\)\(\\hat\{e\}\_\{k\},\\hat\{b\}\_\{k\}\)beηk\\eta\_\{k\}\-optimal outputs and lethkh\_\{k\}be the corresponding observation coverage radius\. If

ℒQk,Pkℋk,⋆\+ηk\+hk→k→∞0,\\mathcal\{L\}\_\{Q\_\{k\},P\_\{k\}\}^\{\\mathcal\{H\}\_\{k\},\\star\}\+\\eta\_\{k\}\+h\_\{k\}\\xrightarrow\{k\\to\\infty\}0,\(111\)then

0≤ℒQk,Pk⋆​\(e^k\)≤ℒQk,Pk​\(b^k,e^k\)→k→∞0\.0\\leq\\mathcal\{L\}\_\{Q\_\{k\},P\_\{k\}\}^\{\\star\}\(\\hat\{e\}\_\{k\}\)\\leq\\mathcal\{L\}\_\{Q\_\{k\},P\_\{k\}\}\(\\hat\{b\}\_\{k\};\\hat\{e\}\_\{k\}\)\\xrightarrow\{k\\to\\infty\}0\.\(112\)

###### Proof\.

For every pair inℋ\\mathcal\{H\}, the empirical error is no larger than its full\-family error\. Hence the fine\-tuning rule gives

ℒ^Q,P,n​\(b^,e^\)≤inf\(e,b\)∈ℋℒ^Q,P,n​\(b,e\)\+η≤ℒQ,Pℋ,⋆\+η\.\\widehat\{\\mathcal\{L\}\}\_\{Q,P,n\}\(\\hat\{b\};\\hat\{e\}\)\\leq\\inf\_\{\(e,b\)\\in\\mathcal\{H\}\}\\widehat\{\\mathcal\{L\}\}\_\{Q,P,n\}\(b;e\)\+\\eta\\leq\\mathcal\{L\}\_\{Q,P\}^\{\\mathcal\{H\},\\star\}\+\\eta\.\(113\)Apply[lemmaD\.1](https://arxiv.org/html/2610.27139#A4.Thmtheorem1)to the returned pair to obtain the upper bounds\. The lower bound follows because\(e^,b^\)∈ℋ\(\\hat\{e\},\\hat\{b\}\)\\in\\mathcal\{H\}\.

Under[equation111](https://arxiv.org/html/2610.27139#A4.E111), all three nonnegative terms tend to zero\. In particular,hk→0h\_\{k\}\\to 0, soρ⁡\(hk\)→0\\rho\(h\_\{k\}\)\\to 0\. The same bound givesℒQk,Pk​\(b^k,e^k\)→0\\mathcal\{L\}\_\{Q\_\{k\},P\_\{k\}\}\(\\hat\{b\}\_\{k\};\\hat\{e\}\_\{k\}\)\\to 0, and the optimal prediction error is no larger\. ∎

### D\.2General Case

###### Theorem D\.4\(General Fine\-Tuning Bound\)\.

Everyη\\eta\-optimal fine\-tuning output satisfying[assumption3\.9](https://arxiv.org/html/2610.27139#S3.Thmtheorem9)obeys

ℒQ,P⋆​\(e^\)≤ℛQ⋆​\(e^\)≤ωP​\(2​\[ℒQ,Pℋ,⋆\+η\+ρ⁡\(hn\)\+hn\]\)\.\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(\\hat\{e\}\)\\leq\\boxed\{\\mathcal\{R\}\_\{Q\}^\{\\star\}\(\\hat\{e\}\)\}\\leq\\omega\_\{P\}\\\!\\Bigl\(2\\bigl\[\\mathcal\{L\}\_\{Q,P\}^\{\\mathcal\{H\},\\star\}\+\\eta\+\\rho\(h\_\{n\}\)\+h\_\{n\}\\bigr\]\\Bigr\)\.\(114\)For stagesk≥1k\\geq 1, allow the training states, query sets, model classes and learned state spaces to vary, with a common modulusρ\\rhofor the returned outputs\. Let\(e^k,b^k\)\(\\hat\{e\}\_\{k\},\\hat\{b\}\_\{k\}\)denote theηk\\eta\_\{k\}\-optimal outputs\. If\(Pk\)k≥1\(P\_\{k\}\)\_\{k\\geq 1\}is a query sequence separating states inℱ\\mathcal\{F\}and

ℒQk,Pkℋk,⋆\+ηk\+hk→k→∞0,\\mathcal\{L\}\_\{Q\_\{k\},P\_\{k\}\}^\{\\mathcal\{H\}\_\{k\},\\star\}\+\\eta\_\{k\}\+h\_\{k\}\\xrightarrow\{k\\to\\infty\}0,\(115\)thenℛQk⋆​\(e^k\)→0\\mathcal\{R\}\_\{Q\_\{k\}\}^\{\\star\}\(\\hat\{e\}\_\{k\}\)\\to 0\. Herehkh\_\{k\}is the coverage radius of the training observations onQk∪PkQ\_\{k\}\\cup P\_\{k\}at stagekk, defined as in[equation24](https://arxiv.org/html/2610.27139#S3.E24)\.

###### Proof\.

Substitute[equation110](https://arxiv.org/html/2610.27139#A4.E110)into[theorem3\.2](https://arxiv.org/html/2610.27139#S3.Thmtheorem2), usingℒQ,P⋆​\(e^\)≤ℒQ,P​\(b^,e^\)\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(\\hat\{e\}\)\\leq\\mathcal\{L\}\_\{Q,P\}\(\\hat\{b\};\\hat\{e\}\)and the monotonicity ofωP\\omega\_\{P\}\.

For the asymptotic claim,[corollaryD\.3](https://arxiv.org/html/2610.27139#A4.Thmtheorem3)gives vanishing optimal prediction error\. State separation and[theorem3\.3](https://arxiv.org/html/2610.27139#S3.Thmtheorem3)then give vanishing recovery error\. ∎

### D\.3Low\-Dimensional Case

Using the targets constructed in[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4)makes the query budget explicit under the same training rule and stability condition\.

###### Theorem D\.5\(Low\-Dimensional Fine\-Tuning Bound\)\.

Suppose\(ℱ,Q\)\(\\mathcal\{F\},Q\)satisfies[assumption2\.12](https://arxiv.org/html/2610.27139#S2.Thmtheorem12)with constantsC,d,αC,d,\\alpha\. For every integerm≥C2/2m\\geq C^\{2\}/2, chooseP⊆𝒰∖QP\\subseteq\\mathcal\{U\}\\setminus Qas in[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4), with\|P\|≤m\|P\|\\leq m\. Everyη\\eta\-optimal fine\-tuning output satisfying[assumption3\.9](https://arxiv.org/html/2610.27139#S3.Thmtheorem9)obeys

ℒQ,P⋆​\(e^\)≤ℛQ⋆​\(e^\)≤2α​\[ℒQ,Pℋ,⋆\+η\+ρ⁡\(hn\)\+hn\]\+2\+αα​\(C22​m\)12​d\.\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(\\hat\{e\}\)\\leq\\boxed\{\\mathcal\{R\}\_\{Q\}^\{\\star\}\(\\hat\{e\}\)\}\\leq\\frac\{2\}\{\\alpha\}\\bigl\[\\mathcal\{L\}\_\{Q,P\}^\{\\mathcal\{H\},\\star\}\+\\eta\+\\rho\(h\_\{n\}\)\+h\_\{n\}\\bigr\]\+\\frac\{2\+\\alpha\}\{\\alpha\}\\left\(\\frac\{C^\{2\}\}\{2m\}\\right\)^\{\\frac\{1\}\{2d\}\}\.\(116\)For budgetsm→∞m\\to\\infty, allow the training states, input sets, model classes, and learned state spaces to vary, selecting eachPmP\_\{m\}by the same construction\. Let\(e^m,b^m\)\(\\hat\{e\}\_\{m\},\\hat\{b\}\_\{m\}\)denote theηm\\eta\_\{m\}\-optimal outputs\. IfC,d,αC,d,\\alphaandρ\\rhoare common across budgets and

ℒQm,Pmℋm,⋆\+ηm\+hm→m→∞0,\\mathcal\{L\}\_\{Q\_\{m\},P\_\{m\}\}^\{\\mathcal\{H\}\_\{m\},\\star\}\+\\eta\_\{m\}\+h\_\{m\}\\xrightarrow\{m\\to\\infty\}0,\(117\)thenℛQm⋆​\(e^m\)→0\\mathcal\{R\}\_\{Q\_\{m\}\}^\{\\star\}\(\\hat\{e\}\_\{m\}\)\\to 0\. Herehmh\_\{m\}is the coverage radius of the training observations onQm∪PmQ\_\{m\}\\cup P\_\{m\}at budgetmm, defined as in[equation24](https://arxiv.org/html/2610.27139#S3.E24)\.

###### Proof\.

SelectPPby[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4), before the training class and output are specified\. Substitute[equation110](https://arxiv.org/html/2610.27139#A4.E110)into that theorem, usingℒQ,P⋆​\(e^\)≤ℒQ,P​\(b^,e^\)\\mathcal\{L\}\_\{Q,P\}^\{\\star\}\(\\hat\{e\}\)\\leq\\mathcal\{L\}\_\{Q,P\}\(\\hat\{b\};\\hat\{e\}\)\.

Under the stated convergence condition,[corollaryD\.3](https://arxiv.org/html/2610.27139#A4.Thmtheorem3)gives vanishing optimal prediction error\. Applying[theorem3\.5](https://arxiv.org/html/2610.27139#S3.Thmtheorem5)to the selected target sets gives vanishing recovery error\. ∎

The bounds identify the contributions of the model class, optimization tolerance, training coverage, and prediction stability\. The low\-dimensional result also makes the target\-query budget explicit:nncounts training records, whilemmcounts target queries\.

## Appendix ERecovery from Noisy Observations

The construction in[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4)also applies when a query’s response is a conditional mean and its observation is noisy\. The state specifies the mean response profile; the measurement law describes how a recording is drawn from it\. Prediction error and measurement perturbation then enter the same distance to a covering center\. Restricting the construction to a frequently encountered state family gives a probability guarantee and an integrated recovery bound\.

Letℱ\\mathcal\{F\}be compact with diameterDD, and letG⊆ℱG\\subseteq\\mathcal\{F\}be nonempty and compact\. Suppose\(G,Q\)\(G,Q\)satisfies[assumption2\.12](https://arxiv.org/html/2610.27139#S2.Thmtheorem12)with constantsCG,d,αC\_\{G\},d,\\alpha\. For an integerm≥CG2/2m\\geq C\_\{G\}^\{2\}/2, selectPPand covering centers by[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4)onGG\.

LetΦ=b∘e\\Phi=b\\circ ebe Borel measurable andKK\-Lipschitz on the full input space, with prediction error at mostℓ\\ellonGG\. Decode to the covering center closest tob⁡\(z\)b\(z\)onPP, resolving ties by index\. This defines astate decodera⁡\(z\)a\(z\); the compositiona∘ea\\circ eis measurable because it is a finite selection determined byΦ\\Phi\.

###### Lemma E\.1\(Recovery under Perturbation\)\.

Under this setup, iff∈Gf\\in GanddQ​\(x,f⁡\(Q\)\)≤ηd\_\{Q\}\(x,f\(Q\)\)\\leq\\eta, then

d𝒰​\(\(a∘e\)​\(x\),f\)≤2α​\(ℓ\+K​η\)\+2\+αα​\(CG22​m\)1/\(2​d\)\.d\_\{\\mathcal\{U\}\}\(\(a\\circ e\)\(x\),f\)\\leq\\frac\{2\}\{\\alpha\}\(\\ell\+K\\eta\)\+\\frac\{2\+\\alpha\}\{\\alpha\}\\left\(\\frac\{C\_\{G\}^\{2\}\}\{2m\}\\right\)^\{1/\(2d\)\}\.\(118\)

###### Proof\.

The triangle inequality and input sensitivity give

dP​\(Φ⁡\(x\),f⁡\(P\)\)≤dP​\(Φ⁡\(x\),Φ⁡\(f⁡\(Q\)\)\)\+dP​\(Φ⁡\(f⁡\(Q\)\),f⁡\(P\)\)≤K​η\+ℓ\.d\_\{P\}\(\\Phi\(x\),f\(P\)\)\\leq d\_\{P\}\(\\Phi\(x\),\\Phi\(f\(Q\)\)\)\+d\_\{P\}\(\\Phi\(f\(Q\)\),f\(P\)\)\\leq K\\eta\+\\ell\.\(119\)Apply[lemmaC\.4](https://arxiv.org/html/2610.27139#A3.Thmtheorem4)onGGwithy=Φ⁡\(x\)y=\\Phi\(x\)and the radius and allowance fixed in[theorem3\.4](https://arxiv.org/html/2610.27139#S3.Thmtheorem4)\. Substitution gives[equation118](https://arxiv.org/html/2610.27139#A5.E118)\. ∎

The pointwise bound becomes a probability guarantee when both the state and the measurement lie in their specified ranges\.

###### Corollary E\.2\(Recovery under Noise\)\.

Under the preceding setup, letFFhave a Borel lawμ\\muwithμ⁡\(G\)≥1−δ\\mu\(G\)\\geq 1\-\\delta, whereδ∈\[0,1\)\\delta\\in\[0,1\)\. Suppose the measured inputXXsatisfies, for everyf∈Gf\\in G,

Pr⁡\(dQ​\(X,f⁡\(Q\)\)≤η∣F=f\)≥1−ζ,\\Pr\\bigl\(d\_\{Q\}\(X,f\(Q\)\)\\leq\\eta\\mid F=f\\bigr\)\\geq 1\-\\zeta,\(120\)whereη≥0\\eta\\geq 0andζ∈\(0,1\)\\zeta\\in\(0,1\)\. With

B≔min⁡\{D,2α​\(ℓ\+K​η\)\+2\+αα​\(CG22​m\)1/\(2​d\)\},B\\coloneqq\\min\\left\\\{D,\\frac\{2\}\{\\alpha\}\(\\ell\+K\\eta\)\+\\frac\{2\+\\alpha\}\{\\alpha\}\\left\(\\frac\{C\_\{G\}^\{2\}\}\{2m\}\\right\)^\{1/\(2d\)\}\\right\\\},\(121\)the recovery error obeys

Pr⁡\(d𝒰​\(\(a∘e\)​\(X\),F\)≤B\)\\displaystyle\\Pr\\bigl\(d\_\{\\mathcal\{U\}\}\(\(a\\circ e\)\(X\),F\)\\leq B\\bigr\)≥1−δ−ζ,\\displaystyle\\geq 1\-\\delta\-\\zeta,\(122\)𝔼​d𝒰​\(\(a∘e\)​\(X\),F\)2\\displaystyle\\mathbb\{E\}\\,d\_\{\\mathcal\{U\}\}\(\(a\\circ e\)\(X\),F\)^\{2\}≤B2\+D2​\(δ\+ζ\)\.\\displaystyle\\leq B^\{2\}\+D^\{2\}\(\\delta\+\\zeta\)\.\(123\)

###### Proof\.

The eventF∈GF\\in GanddQ​\(X,F⁡\(Q\)\)≤ηd\_\{Q\}\(X,F\(Q\)\)\\leq\\etahas probability at least

μ⁡\(G\)​\(1−ζ\)≥\(1−δ\)​\(1−ζ\)≥1−δ−ζ\.\\mu\(G\)\(1\-\\zeta\)\\geq\(1\-\\delta\)\(1\-\\zeta\)\\geq 1\-\\delta\-\\zeta\.\(124\)On this event,[lemmaE\.1](https://arxiv.org/html/2610.27139#A5.Thmtheorem1)and the diameter bound give error at mostBB\. Off it, the error is at mostDDbecause both states lie inℱ\\mathcal\{F\}\. This proves the probability bound; splitting the squared\-error expectation over the two events gives the second bound\. ∎

Count observations\.For a conditional Poisson measurement law, takeXq=Nq/TX\_\{q\}=N\_\{q\}/T, whereNq\|F=fN\_\{q\}\\mid F=fhas meanT​f​\(q\)Tf\(q\)and0≤f⁡\(q\)≤Λ0\\leq f\(q\)\\leq\\LambdaonGG\. HereTTis the exposure\. A valid input\-noise radius is

η⁡\(T,ζ\)=2​Λ​log⁡\(2​\|Q\|/ζ\)T\+2​log⁡\(2​\|Q\|/ζ\)3​T\.\\eta\(T,\\zeta\)=\\sqrt\{\\frac\{2\\Lambda\\log\(2\|Q\|/\\zeta\)\}\{T\}\}\+\\frac\{2\\log\(2\|Q\|/\\zeta\)\}\{3T\}\.\(125\)Indeed, the centered Poisson moment\-generating function obeyslog⁡𝔼​eu⁡\(Nq−T​f​\(q\)\)=T​f​\(q\)​\(eu−1−u\)\\log\\mathbb\{E\}e^\{u\(N\_\{q\}\-Tf\(q\)\)\}=Tf\(q\)\(e^\{u\}\-1\-u\)\. The boundeu−1−u≤u2/\[2​\(1−u/3\)\]e^\{u\}\-1\-u\\leq u^\{2\}/\[2\(1\-u/3\)\]for0≤u<30\\leq u<3, and the corresponding lower\-tail bound, give

Pr⁡\(\|Nq−T​f​\(q\)\|\>2​T​Λ​v\+2​v3\|F=f\)≤2​e−v\.\\Pr\\left\(\\left\|N\_\{q\}\-Tf\(q\)\\right\|\>\\sqrt\{2T\\Lambda v\}\+\\frac\{2v\}\{3\}\\,\\middle\|\\,F=f\\right\)\\leq 2e^\{\-v\}\.\(126\)Choosev=log⁡\(2​\|Q\|/ζ\)v=\\log\(2\|Q\|/\\zeta\)and take a union bound\. This uses each channel’s conditional Poisson law and permits dependence between channels\. Exposure controls the measurement contribution to recovery, throughK​η​\(T,ζ\)K\\eta\(T,\\zeta\)\.

Prediction sensitivity\.For a continuously differentiablepredictorcomposition, a sufficient full\-input constant is

K≥supx∈ℝQ‖JΦ​\(x\)‖∞→∞=supx∈ℝQmax⁡∑q∈Qp∈P⁡\|∂Φp​\(x\)∂xq\|\.K\\geq\\sup\_\{x\\in\\mathbb\{R\}^\{Q\}\}\\\|J\_\{\\Phi\}\(x\)\\\|\_\{\\infty\\to\\infty\}=\\sup\_\{x\\in\\mathbb\{R\}^\{Q\}\}\\max\_\{p\\in P\}\\sum\_\{q\\in Q\}\\left\|\\frac\{\\partial\\Phi\_\{p\}\(x\)\}\{\\partial x\_\{q\}\}\\right\|\.\(127\)The derivative bound may instead be taken on a convex domain containing every input and intervening segment used in the guarantee\.

Functional observability\.Functional observers recover selected state coordinates in structured dynamical networks; algebraic and graph conditions characterize which targets are accessible\([Montanari et al\., 2022](https://arxiv.org/html/2610.27139#bib.bib31);[Montanari et al\., 2023](https://arxiv.org/html/2610.27139#bib.bib32)\)\. Observability also guides the eigenstructure and sparsity of neural state\-space architectures\([Gracyk, 2026](https://arxiv.org/html/2610.27139#bib.bib15);[Hamdan et al\., 2024](https://arxiv.org/html/2610.27139#bib.bib16)\)\. Here, the state is specified by its complete query\-response profile, and quantitative target observability compares its full metric with the remaining responses\. The query\-budget construction uses this comparison to turn target prediction into a state\-valued recovery map\. Retracing adds cycle consistency to predictive state learning in reinforcement learning\([Yu et al\., 2022](https://arxiv.org/html/2610.27139#bib.bib47)\); our bounds concern the distinctions preserved by a fixedencoderand the error of recovering its response profile\.

## Appendix FSynthetic Experiment Details

The generator specifies the full state family and every query response\. This makes recovery error, target separation, and the constants in the error bound directly computable\. We describe the generator and inverse map first, then examine the budget dependence and the assumptions responsible for it\.

The coordinatewise construction is followed by a second, joint\-query construction in[appendixG](https://arxiv.org/html/2610.27139#A7), which supplies the calibration records for a global recovery certificate\.

### F\.1State Family and Query

The GT observation law is a deterministic function of a sampled latent vectorxx\. ForSphere,xxis uniform on the unit sphere inℝd\+1\\mathbb\{R\}^\{d\+1\}; forTorus, its coordinates are cosine–sine pairs of independent uniform angles\. ForGeneral,xxranges over the box\[−1,1\]D\[\-1,1\]^\{D\}with independent uniform coordinates\. A fixed signed coordinate\-repetition map expands the latent vector to the common ambient dimension\. The observation law is given in[equation30](https://arxiv.org/html/2610.27139#S4.E30)\. The coefficients are sampled once and shared across families and repetitions\. The mapi⁡\(j\)i\(j\)cycles through the latent coordinates; signssjs\_\{j\}are fixed random signs for the low\-dimensional families and positive for the general family\. Each coordinate has input and target channels\. Inputs contain every input channel, while target sets are nested prefixes of a fixed random permutation of the target pool\. The query coefficients are public metadata supplied to theencoderandpredictor\. These are structured, separable observation laws; their geometry and full observation profiles can be evaluated exactly\.

Table 4:Synthetic design\. The circle is included once, as the sphere of intrinsic dimension one\. The general family has independent ambient coordinates\.Table 5:Random\-repetition design\. Reported dispersions are sample standard deviations across repetitions of the same condition, with the biological aggregation held fixed\. A repetition changes the sources of randomness specified by its protocol\.Throughout the paper, decimals are rounded to at most four places and omit the leading zero\. Exact zeros in decimal summaries appear as\.0000\.0000; nonzero values rounding to zero appear as\.0001\.0001in magnitude, retaining their sign\. All calculations use unrounded values\.

### F\.2Prediction and Deterministic Recovery

A shared coordinatewise MLP encodes input observations and their query coefficients\. A second shared MLP combines this learned state with each target channel’s coefficients to predict its observation\. We minimize the mean, over training states, of the maximum absolute target error\. Every configuration is evaluated at the final training epoch specified in advance\. Matching pilot configurations enter the paired\-repetition design once\.

Table 6:Synthetic training and deterministic recovery settings\. Gradient updates train theencoderandpredictor\.The ideal GT map in[equation31](https://arxiv.org/html/2610.27139#S4.E31)minimizes target discrepancy; the implementation approximates it by bisection\. Monotonicity of each observation channel converts an allowed discrepancy into a latent\-coordinate interval\. We intersect these intervals and bisect the discrepancy until the resulting box intersects the known latent domain\. For a sphere, intersection is decided by the minimum and maximum attainable squared norms; for a torus, the same criterion is applied to each circle factor\. A deterministic feasible point then defines the recovered state\. The inverse uses the predicted responses and the known latent domain; the test state’s latent vector enters the evaluation of recovery error\.

For each test stateff, we record prediction errordP​\(\(b∘e\)​\(f⁡\(Q\)\),f⁡\(P\)\)d\_\{P\}\(\(b\\circ e\)\(f\(Q\)\),f\(P\)\)and recovery errord𝒰​\(f^​\(\(b∘e\)​\(f⁡\(Q\)\)\),f\)d\_\{\\mathcal\{U\}\}\(\\widehat\{f\}\(\(b\\circ e\)\(f\(Q\)\)\),f\)\. Recovery is evaluated over every channel of the full query universe\. Summaries report mean±\\pmsample standard deviation over paired repetitions\. Random seed sets are listed in[table5](https://arxiv.org/html/2610.27139#A6.T5)\. Bound checks instead use the maximum over the test states within each run\. Seed pairing fixes initialization, state samples, and target order across budgets; seeds jointly change these sources of randomness\.

Training choices and computation\.Dimensions and target counts form geometric grids spanning the three state families\. The shared MLP widths, batch size, and sample counts are fixed design parameters; every reported configuration uses the same optimizer and two\-stage schedule\. Preliminary convergence checks informed the200200\-epoch endpoint, which is held fixed across the full suite\. AdamW uses weight decay\.01\.01, moments\(\.9,\.999\)\(\.9,\.999\), and numerical stabilizer10−810^\{\-8\}\. Training uses Float32; numerical inversion and the domain audit use Float64\. The seed table and supplied case manifest specify initialization, state generation, target order, and minibatch permutations\.

With the network widths fixed, a batch ofBBstates costsO⁡\(B⁡\(D\+m\)\)O\(B\(D\+m\)\)time and activation memory: the sharedencoderprocessesDDcoordinates and thepredictorevaluatesmmtargets\. Parameters are shared across coordinates and queries\. Each numerical inverse uses4444feasibility rounds, each linear in the target count and latent\-coordinate count; full\-profile scoring is linear inDD\. The uniform audit evaluates each target on the grid in[table9](https://arxiv.org/html/2610.27139#A6.T9); target and grid batching bound working memory without changing the evaluated domain\.

Reproduction materials and infrastructure\.The accompanying archiveontic\-reproducibility\.zipsupplies source code, case manifests, dependency versions, original numerical endpoints, and commands for preprocessing, training, evaluation, and aggregation\. Figure and table sources are included with the manuscript source\. Synthetic data are generated by the supplied response laws and seeds\. HC\-3 recordings are obtained from their provider under the terms in[appendixH](https://arxiv.org/html/2610.27139#A8)\. The archive retains all endpoints of the included suites and separates within\-condition seed variation from biological aggregation\.

Experiments ran in a Linux batch\-computing environment\. The supplied training configuration uses one NVIDIA A100 \(80 GB\); observation\-geometry calculations use CPU numerical linear algebra\. The reproduction environment is Python 3\.10\.19, PyTorch 2\.10\.0, NumPy 1\.26\.4, SciPy 1\.14\.1, Matplotlib 3\.8\.4, and mpmath 1\.3\.0\. The archive includes version\-pinned dependencies\. Checkpoints retain optimizer, sampler, and random\-generator state, with separate latest and previous copies\. Resumption uses the same device type and PyTorch version\.

The main\-text observability inequality and constantαP\\alpha\_\{P\}are given in[equation32](https://arxiv.org/html/2610.27139#S4.E32)\. We check coordinate coverage and its budget requirement before applying the transfer bound\.

### F\.3Assumption Checks

Observability\.Writerrfor the number of ambient latent coordinates:d\+1d\+1for the sphere,2​d2dfor the torus, andDDfor the box\. LetI⁡\(P\)=\{i⁡\(j\):\(j,k\)∈P\}I\(P\)=\\\{i\(j\):\(j,k\)\\in P\\\}\. For the observation law in[equation30](https://arxiv.org/html/2610.27139#S4.E30),

Pseparatesℱ⟺I\(P\)=\{1,…,r\}\.P\\text\{ separates \}\\mathcal\{F\}\\quad\\Longleftrightarrow\\quad I\(P\)=\\\{1,\\ldots,r\\\}\.\(128\)Full coordinate coverage gives separation by[proposition4\.1](https://arxiv.org/html/2610.27139#S4.Thmtheorem1)\. Conversely, each latent domain is invariant under changing the sign of one coordinate and contains a state with that coordinate nonzero\. Reflecting an unobserved coordinate therefore gives two distinct states with identical target responses\. Their full profiles differ because the query universe observes every coordinate\.

###### Proof of[proposition4\.1](https://arxiv.org/html/2610.27139#S4.Thmtheorem1)\.

*Bound each channel\.*Fixf=fxf=f\_\{x\}andg=fx′g=f\_\{x^\{\\prime\}\}inℱ\\mathcal\{F\}\. For every channel\(j,k\)\(j,k\)and\|t\|≤1\|t\|\\leq 1,

aj​k​sech2⁡\(aj​k\+\|cj​k\|\)≤\|dd​t​tanh⁡\(aj​k​sj​t\+cj​k\)\|≤aj​k\.a\_\{jk\}\\operatorname\{sech\}^\{2\}\(a\_\{jk\}\+\|c\_\{jk\}\|\)\\leq\\left\|\\frac\{d\}\{dt\}\\tanh\(a\_\{jk\}s\_\{j\}t\+c\_\{jk\}\)\\right\|\\leq a\_\{jk\}\.\(129\)The mean value theorem, followed by maximization over the selected target channels and over all channels, respectively, gives

dP​\(f,g\)≥maxi⁡ℓi​\|xi−xi′\|,d𝒰​\(f,g\)≤maxi⁡Ai​\|xi−xi′\|\.d\_\{P\}\(f,g\)\\geq\\max\_\{i\}\\ell\_\{i\}\|x\_\{i\}\-x\_\{i\}^\{\\prime\}\|,\\qquad d\_\{\\mathcal\{U\}\}\(f,g\)\\leq\\max\_\{i\}A\_\{i\}\|x\_\{i\}\-x\_\{i\}^\{\\prime\}\|\.\(130\)
A single increasing channel turns coordinate separation into response separation:

*Compare the distances\.*Every coordinate has a target channel with positive slope magnitude, soℓi\>0\\ell\_\{i\}\>0andAi\>0A\_\{i\}\>0\. There are finitely many coordinates, henceαP=mini⁡ℓi/Ai\>0\\alpha\_\{P\}=\\min\_\{i\}\\ell\_\{i\}/A\_\{i\}\>0\. Sinceℓi≥αP​Ai\\ell\_\{i\}\\geq\\alpha\_\{P\}A\_\{i\}for everyii,

dP​\(f,g\)≥maxi⁡ℓi​\|xi−xi′\|≥αP​maxi​Ai​\|xi−xi′\|≥αP​d𝒰​\(f,g\)\.d\_\{P\}\(f,g\)\\geq\\max\_\{i\}\\ell\_\{i\}\|x\_\{i\}\-x\_\{i\}^\{\\prime\}\|\\geq\\alpha\_\{P\}\\max\_\{i\}A\_\{i\}\|x\_\{i\}\-x\_\{i\}^\{\\prime\}\|\\geq\\alpha\_\{P\}d\_\{\\mathcal\{U\}\}\(f,g\)\.\(131\)This proves[equation32](https://arxiv.org/html/2610.27139#S4.E32)\. ∎

For each certified target set, we recompute this constant and the test\-error transfer\. The input and candidate\-target pools cover all latent coordinates\.

Query budget\.The actual target sampler permits a finite\-budget calculation\. LetMMbe the target\-pool size, and letnin\_\{i\}count its channels observing latent coordinateii\. A uniform subset of sizemmmisses coordinateiiwith probability\(M−nim\)/\(Mm\)\\binom\{M\-n\_\{i\}\}\{m\}/\\binom\{M\}\{m\}\. Hence

Pr⁡\{P​fails to separate​ℱ\}≤min⁡\{1,∑i=1r\(M−nim\)\(Mm\)\},\\Pr\\\{P\\text\{ fails to separate \}\\mathcal\{F\}\\\}\\leq\\min\\left\\\{1,\\sum\_\{i=1\}^\{r\}\\frac\{\\binom\{M\-n\_\{i\}\}\{m\}\}\{\\binom\{M\}\{m\}\}\\right\\\},\(132\)where a binomial coefficient is zero when its lower argument exceeds its upper argument\. On the complementary event, the global error transfer holds simultaneously for everypredictorand every state, using the correspondingαP\\alpha\_\{P\}\. In particular, this event is determined before predictive training\. Each term in the sum is at mostexp\(−mminini/M\)\\exp\(\-m\\min\_\{i\}n\_\{i\}/M\)\. Thusm≥\(M/mini⁡ni\)​log⁡\(r/η\)m\\geq\(M/\\min\_\{i\}n\_\{i\}\)\\log\(r/\\eta\)suffices for failure probability at mostη\\etawhenever this budget is available; querying the full pool always separates\. For the repeated\-coordinate low\-dimensional families,M/mini⁡niM/\\min\_\{i\}n\_\{i\}is of the order ofrr, giving a budget of orderr​log⁡\(r/η\)r\\log\(r/\\eta\)\.

We evaluate the without\-replacement bound using the actual channel counts \([table7](https://arxiv.org/html/2610.27139#A6.T7)\)\. The low\-dimensional families repeat each latent coordinate across many channels, so their sufficient budgets are much smaller\. These are family\-specific sampling guarantees; the general covering theorem accommodates arbitrary admissible response laws\.

Table 7:Smallest budget for which[equation132](https://arxiv.org/html/2610.27139#A6.E132)gives at least95%95\\%probability of separation\. The pool contains32,76832\{,\}768queries;rrcounts latent coordinates, whileddis intrinsic dimension\. For the largest sphere and torus at budget512512, failure\-probability upper bounds are4\.18×10−64\.18\\times 10^\{\-6\}and\.0189\.0189\.Covering number\.The same generator supplies explicit covering constants\. PutA=maxj​k⁡aj​kA=\\max\_\{jk\}a\_\{jk\}\. Angular coordinates for the sphere and torus give, respectively,

𝒩ℱ​\(ε\)≤\(1\+2​π​d​Aε\)d,𝒩ℱ​\(ε\)≤\(1\+2​π​Aε\)d\.\\mathcal\{N\}\_\{\\mathcal\{F\}\}\(\\varepsilon\)\\leq\\left\(1\+\\frac\{2\\pi dA\}\{\\varepsilon\}\\right\)^\{d\},\\qquad\\mathcal\{N\}\_\{\\mathcal\{F\}\}\(\\varepsilon\)\\leq\\left\(1\+\\frac\{2\\pi A\}\{\\varepsilon\}\\right\)^\{d\}\.\(133\)Indeed, changing a sphere angle changes its unit vector by at most the angular displacement in Euclidean norm; summing over angles gives the first grid radius\. Each sine–cosine factor of a torus is Lipschitz in its own angle, giving the second radius in the maximum\-coordinate metric\. The observation map isAA\-Lipschitz in that metric\. Choosing a nearest grid point in the parameter domain therefore covers every state\. Forε≤1\\varepsilon\\leq 1, these bounds have the required formC​ε−dC\\varepsilon^\{\-d\}on the low\-dimensional families\. The constants describe a conservative existence construction for these state families\. A regular grid on the box gives𝒩ℱ​\(ε\)≤\(1\+2​A/ε\)D\\mathcal\{N\}\_\{\\mathcal\{F\}\}\(\\varepsilon\)\\leq\(1\+2A/\\varepsilon\)^\{D\}\.

### F\.4Error Transfer

When every latent coordinate has a target channel,[equation32](https://arxiv.org/html/2610.27139#S4.E32)supplies the global observability constantαP\\alpha\_\{P\}\. For the ideal GT map, the true state is a feasible candidate, so the fitted target residual is at most the prediction error\. For a numerical outputg∈ℱg\\in\\mathcal\{F\}whose target residual is withinξ≥0\\xi\\geq 0of the minimum, observability and the triangle inequality give

αP​d𝒰​\(g,f\)≤dP​\(g⁡\(P\),y\)\+dP​\(y,f⁡\(P\)\)≤2​dP​\(y,f⁡\(P\)\)\+ξ\.\\alpha\_\{P\}d\_\{\\mathcal\{U\}\}\(g,f\)\\leq d\_\{P\}\(g\(P\),y\)\+d\_\{P\}\(y,f\(P\)\)\\leq 2d\_\{P\}\(y,f\(P\)\)\+\\xi\.\(134\)The exact map hasξ=0\\xi=0\. With exact feasibility tests and feasible\-point construction,4444bisections of\[0,‖y‖∞\+1\]\[0,\\\|y\\\|\_\{\\infty\}\+1\], followed by a10−1010^\{\-10\}feasibility expansion, giveξ≤2−44​\(‖y‖∞\+1\)\+10−10\\xi\\leq 2^\{\-44\}\(\\\|y\\\|\_\{\\infty\}\+1\)\+10^\{\-10\}\. For a feasible Float64 output,ξ\\xialso includes arithmetic effects on the target residual\. Finite\-test plots compare the numerical recovery errors with the exact\-map benchmark2​L^/αP2\\widehat\{L\}/\\alpha\_\{P\}, whereL^\\widehat\{L\}is the maximum test prediction error\. Coordinate coverage gives the separation test in[equation128](https://arxiv.org/html/2610.27139#A6.E128)\.

Table 8:Numerical recovery compared with the exact\-map benchmark on separating target sets\. All observed errors lie below the benchmark, computed from maximum test prediction errors and global derivative bounds\.
### F\.5Uniform Prediction and Recovery

The preceding transfer checks use test states\. The coordinatewise architecture also permits a bound over the entire known family\. For a targett=\(j,k\)t=\(j,k\), putv=sj​xi⁡\(j\)v=s\_\{j\}x\_\{i\(j\)\}and writeht​\(v\)h\_\{t\}\(v\)for the predicted observation minustanh⁡\(aj​k​v\+cj​k\)\\tanh\(a\_\{jk\}v\+c\_\{jk\}\)\. Writingeshe\_\{\\rm sh\}andbshb\_\{\\rm sh\}for the shared MLPs in[table6](https://arxiv.org/html/2610.27139#A6.T6), the implemented coordinatewise composition is

ht​\(v\)=\\displaystyle h\_\{t\}\(v\)=\{\}bsh​\(esh​\(tanh⁡\(aj​1​v\+cj​1\),tanh⁡\(aj​2​v\+cj​2\),aj​1,aj​2,cj​1,cj​2\),aj​k,cj​k\)\\displaystyle b\_\{\\rm sh\}\\\!\\Bigl\(e\_\{\\rm sh\}\\bigl\(\\tanh\(a\_\{j1\}v\+c\_\{j1\}\),\\tanh\(a\_\{j2\}v\+c\_\{j2\}\),a\_\{j1\},a\_\{j2\},c\_\{j1\},c\_\{j2\}\\bigr\),a\_\{jk\},c\_\{jk\}\\Bigr\)\(135\)−tanh⁡\(aj​k​v\+cj​k\)\.\\displaystyle\-\\tanh\(a\_\{jk\}v\+c\_\{jk\}\)\.Thus each target prediction depends only onvvand its query coefficients\. Each latent coordinate ranges over\[−1,1\]\[\-1,1\]in every family, so

ℒQ,P​\(b,e\)=max⁡supv∈\[−1,1\]t∈P⁡\|ht​\(v\)\|\.\\mathcal\{L\}\_\{Q,P\}\(b;e\)=\\max\_\{t\\in P\}\\sup\_\{v\\in\[\-1,1\]\}\|h\_\{t\}\(v\)\|\.\(136\)The supremum commutes with the finite maximum: each target’s extremum can be attained at a state of its own\. Agreement between the network forward pass and its scalar form checks this reduction before the domain audit\.

We bound prediction sensitivity in the full input\-observation norm using the trained weights\. LetW1,…,W6W\_\{1\},\\ldots,W\_\{6\}be the successiveencoderandpredictormatrices,w1,w2w\_\{1\},w\_\{2\}the varying observation columns ofW1W\_\{1\}, andW4,hW\_\{4,h\}the hidden\-state columns of the firstpredictormatrix\. Using the GELU derivative bound1\.131\.13, a valid constant is

K=1\.134​‖W6‖2​‖W5‖2​‖W4,h‖2⋅‖W3‖2​‖W2‖2​maxσ∈\{−1,1\}​‖w1\+σ​w2‖2\.K=1\.13^\{4\}\\\|W\_\{6\}\\\|\_\{2\}\\\|W\_\{5\}\\\|\_\{2\}\\\|W\_\{4,h\}\\\|\_\{2\}\\cdot\\\|W\_\{3\}\\\|\_\{2\}\\\|W\_\{2\}\\\|\_\{2\}\\max\_\{\\sigma\\in\\\{\-1,1\\\}\}\\\|w\_\{1\}\+\\sigma w\_\{2\}\\\|\_\{2\}\.\(137\)Every varying input observation enters this calculation\. Theencoder–predictorcomposition therefore satisfies[assumption3\.9](https://arxiv.org/html/2610.27139#S3.Thmtheorem9)withρ⁡\(t\)=K​t\\rho\(t\)=Kt\. The scalar error has Lipschitz constantHt=K​max⁡\(aj​1,aj​2\)\+aj​kH\_\{t\}=K\\max\(a\_\{j1\},a\_\{j2\}\)\+a\_\{jk\}, sincetanh\\tanhis Lipschitz with constant one\.

On a regular gridvℓ=−1\+2​ℓ/Nv\_\{\\ell\}=\-1\+2\\ell/N, letMt=max0≤ℓ≤N⁡\|ht​\(vℓ\)\|M\_\{t\}=\\max\_\{0\\leq\\ell\\leq N\}\|h\_\{t\}\(v\_\{\\ell\}\)\|\. Every point of the interval lies within1/N1/Nof the grid\. Thus

maxt∈P⁡Mt≤ℒQ,P​\(b,e\)≤maxt∈P⁡\(Mt\+HtN\)\.\\max\_\{t\\in P\}M\_\{t\}\\leq\\boxed\{\\mathcal\{L\}\_\{Q,P\}\(b;e\)\}\\leq\\max\_\{t\\in P\}\\left\(M\_\{t\}\+\\frac\{H\_\{t\}\}\{N\}\\right\)\.\(138\)For the reported numerical values, letM~t\\widetilde\{M\}\_\{t\}be the Float64 grid maximum and use the arithmetic allowance10−610^\{\-6\}\. We use

L¯=max⁡\{0,maxt∈P⁡M~t−10−6\},L¯=maxt∈P⁡\(M~t\+Ht/N\)\+10−6\.\\underline\{L\}=\\max\\\{0,\\max\_\{t\\in P\}\\widetilde\{M\}\_\{t\}\-10^\{\-6\}\\\},\\qquad\\overline\{L\}=\\max\_\{t\\in P\}\(\\widetilde\{M\}\_\{t\}\+H\_\{t\}/N\)\+10^\{\-6\}\.\(139\)The numerical enclosure assumes that10−610^\{\-6\}covers grid\-evaluation and enclosure\-arithmetic error, and that the evaluatedHtH\_\{t\}remain valid Lipschitz upper bounds\. Recovery additionally assumes that the evaluatedαP\>0\\alpha\_\{P\}\>0remains a valid observability lower bound\. Tables report Float64 evaluations of the resulting conditional analytic bounds at the stated display precision; rounding the final quotient and its displayed value is separate from the analytic inequality\. The implementation uses high\-precision spot checks and ordinary Float64 arithmetic\.[Table9](https://arxiv.org/html/2610.27139#A6.T9)records the allowance and checks\.

The grid is an offline domain audit of the fixedpredictor\. Its resolution is determined by the weight bound and a preset interpolation tolerance\. Under the arithmetic conditions above, the ideal GT map in[equation31](https://arxiv.org/html/2610.27139#S4.E31)obeys

supf∈ℱd𝒰​\(f^​\(\(b∘e\)​\(f⁡\(Q\)\)\),f\)≤2​L¯αP,\\sup\_\{f\\in\\mathcal\{F\}\}d\_\{\\mathcal\{U\}\}\\bigl\(\\widehat\{f\}\(\(b\\circ e\)\(f\(Q\)\)\),f\\bigr\)\\leq\\frac\{2\\overline\{L\}\}\{\\alpha\_\{P\}\},\(140\)whereL¯\\overline\{L\}denotes the numerical upper enclosure\. For feasible numerical outputs inℱ\\mathcal\{F\}, a uniform residual slackξ\\xiaddsξ/αP\\xi/\\alpha\_\{P\}by[equation134](https://arxiv.org/html/2610.27139#A6.E134)\.

The audit includes all repetitions for the largest sphere and torus at budgets above their sufficient sampling thresholds, and for the independent family at its full budget\. Each bound uses the final training checkpoint and its training target set\. All reported ideal\-map bounds improve on the full\-profile diameter \([table1](https://arxiv.org/html/2610.27139#S4.T1)\)\.

Table 9:Whole\-domain audit settings\. Bounds use the final training checkpoints and their training target sets\. Seed sets and grid sizes appear in[table5](https://arxiv.org/html/2610.27139#A6.T5)\.

## Appendix GJoint\-Query Calibration

This joint\-query family makes the mechanism in[theorem3\.2](https://arxiv.org/html/2610.27139#S3.Thmtheorem2)explicit: target separation converts prediction error into recovery of the full state\. Its linear inverse link lets records at known latent coordinates control global prediction error directly\.

For the calibration certificate below, each query depends on the whole latent configuration\. Forx∈𝕊r−1x\\in\\mathbb\{S\}^\{r\-1\}, definefx​\(q\)=tanh⁡\(aq⊤​x\)f\_\{x\}\(q\)=\\tanh\(a\_\{q\}^\{\\top\}x\)\. The input matrixAQA\_\{Q\}and base target matrixAPA\_\{P\}have independently drawn unit rows\. Target sets vary throughAλ=AP​diag⁡\(1,…,1,λ\)A\_\{\\lambda\}=A\_\{P\}\\operatorname\{diag\}\(1,\\ldots,1,\\lambda\)with0≤λ≤10\\leq\\lambda\\leq 1, so every row has Euclidean norm at most one; the query universe contains the input rows and every target variant\. This keeps the ontic state family fixed while changing the distinctions available to prediction\.

Consider anencoderapplying the inverse response link and a matrix, followed by a jointpredictor:

\(b∘e\)​\(fx​\(Q\)\)=tanh⁡\(WP​WE​arctanh⁡\(fx​\(Q\)\)\)\.\(b\\circ e\)\(f\_\{x\}\(Q\)\)=\\tanh\\bigl\(W\_\{P\}W\_\{E\}\\operatorname\{arctanh\}\(f\_\{x\}\(Q\)\)\\bigr\)\.\(141\)Inputs are clipped to the valid observation range before applying the inverse link\. The resulting full\-input prediction map is Lipschitz with constant at mostcosh2⁡\(1\)​‖WP​WE‖∞→∞\\cosh^\{2\}\(1\)\\\|W\_\{P\}W\_\{E\}\\\|\_\{\\infty\\to\\infty\}\.

We obtain prediction, observability and recovery bounds from the same calibration records below\. Throughout this section,σmin​\(A\)=inf‖v‖2=1‖A​v‖2\\sigma\_\{\\min\}\(A\)=\\inf\_\{\\\|v\\\|\_\{2\}=1\}\\\|Av\\\|\_\{2\}, so positivity means full column rank\.

### G\.1From Training Residuals to a Global Bound

Calibration records connect prediction error to recovery in three steps: they bound the global prediction error, certify that the targets distinguish states, and determine a recovery map\.

For the joint\-query family, setAP=AλA\_\{P\}=A\_\{\\lambda\}andM=WP​WEM=W\_\{P\}W\_\{E\}\. Let the calibration matrixX∈ℝn×rX\\in\\mathbb\{R\}^\{n\\times r\}have full column rank, withσ=σmin​\(X\)\>0\\sigma=\\sigma\_\{\\min\}\(X\)\>0\. The inverse\-link recordsU∈ℝn×\|Q\|U\\in\\mathbb\{R\}^\{n\\times\|Q\|\}andV∈ℝn×\|P\|V\\in\\mathbb\{R\}^\{n\\times\|P\|\}approximateX​AQ⊤XA\_\{Q\}^\{\\top\}andX​AP⊤XA\_\{P\}^\{\\top\}with entrywise errors at mostηQ\\eta\_\{Q\}andηP\\eta\_\{P\}, respectively\. The weights may be fitted to these records\.

###### Lemma G\.1\(Calibration Error\)\.

Under the calibration setup, letY∈ℝn×kY\\in\\mathbb\{R\}^\{n\\times k\}andB∈ℝr×kB\\in\\mathbb\{R\}^\{r\\times k\}satisfy\|\(Y−X​B\)i​j\|≤η\|\(Y\-XB\)\_\{ij\}\|\\leq\\eta\. Then

maxj⁡‖\(X†​Y−B\)⋅j‖2≤n​ησ,‖X†​Y−B‖2→2≤n​k​ησ\.\\max\_\{j\}\\\|\(X^\{\\dagger\}Y\-B\)\_\{\\cdot j\}\\\|\_\{2\}\\leq\\frac\{\\sqrt\{n\}\\eta\}\{\\sigma\},\\qquad\\\|X^\{\\dagger\}Y\-B\\\|\_\{2\\to 2\}\\leq\\frac\{\\sqrt\{nk\}\\eta\}\{\\sigma\}\.\(142\)

###### Proof\.

SinceX†​X=IX^\{\\dagger\}X=I,

X†​Y−B=X†​\(Y−X​B\)\.X^\{\\dagger\}Y\-B=X^\{\\dagger\}\(Y\-XB\)\.\(143\)Each column ofY−X​BY\-XBhas norm at mostn​η\\sqrt\{n\}\\eta, and its Frobenius norm is at mostn​k​η\\sqrt\{nk\}\\eta\. Multiplication byX†X^\{\\dagger\}, whose operator norm is1/σ1/\\sigma, gives both bounds\. ∎

###### Lemma G\.2\(Calibrated Prediction\)\.

Define the residual and its global error bound by

R\\displaystyle R≔U​M⊤−V,\\displaystyle\\coloneqq UM^\{\\top\}\-V,\(144\)Δ\\displaystyle\\Delta≔maxp∈P⁡‖X†​R⋅p‖2\+nσ​\(‖M‖∞→∞​ηQ\+ηP\)\.\\displaystyle\\coloneqq\\max\_\{p\\in P\}\\\|X^\{\\dagger\}R\_\{\\cdot p\}\\\|\_\{2\}\+\\frac\{\\sqrt\{n\}\}\{\\sigma\}\\bigl\(\\\|M\\\|\_\{\\infty\\to\\infty\}\\eta\_\{Q\}\+\\eta\_\{P\}\\bigr\)\.\(145\)Then

ℒQ,P​\(b,e\)≤maxp∈P⁡‖\(M​AQ−AP\)p‖2≤Δ\.\\mathcal\{L\}\_\{Q,P\}\(b;e\)\\leq\\max\_\{p\\in P\}\\\|\(MA\_\{Q\}\-A\_\{P\}\)\_\{p\}\\\|\_\{2\}\\leq\\Delta\.\(146\)For exact records,ηQ=ηP=0\\eta\_\{Q\}=\\eta\_\{P\}=0, the second inequality is an equality\.

###### Proof\.

The residual satisfies

R−X​\(M​AQ−AP\)⊤=\(U−X​AQ⊤\)​M⊤−\(V−X​AP⊤\)\.R\-X\(MA\_\{Q\}\-A\_\{P\}\)^\{\\top\}=\(U\-XA\_\{Q\}^\{\\top\}\)M^\{\\top\}\-\(V\-XA\_\{P\}^\{\\top\}\)\.\(147\)Its entries have magnitude at most‖M‖∞→∞​ηQ\+ηP\\\|M\\\|\_\{\\infty\\to\\infty\}\\eta\_\{Q\}\+\\eta\_\{P\}\. Apply[lemmaG\.1](https://arxiv.org/html/2610.27139#A7.Thmtheorem1)withY=RY=RandB=\(M​AQ−AP\)⊤B=\(MA\_\{Q\}\-A\_\{P\}\)^\{\\top\}; the column bound and triangle inequality give the boundΔ\\Deltaon every row\. The11\-Lipschitz linktanh\\tanhgives the prediction bound\. With exact records,R=X​\(M​AQ−AP\)⊤R=X\(MA\_\{Q\}\-A\_\{P\}\)^\{\\top\}, soX†​R=\(M​AQ−AP\)⊤X^\{\\dagger\}R=\(MA\_\{Q\}\-A\_\{P\}\)^\{\\top\}and the second inequality is an equality\. ∎

The same records estimate the target matrix\. Its smallest singular value measures whether the targets retain every latent direction\.

###### Lemma G\.3\(Calibrated Observability\)\.

Define

A^P\\displaystyle\\widehat\{A\}\_\{P\}≔\(X†​V\)⊤,τ≔n​\|P\|​ηPσ,\\displaystyle\\coloneqq\(X^\{\\dagger\}V\)^\{\\top\},\\qquad\\tau\\coloneqq\\frac\{\\sqrt\{n\|P\|\}\\eta\_\{P\}\}\{\\sigma\},\(148\)α^P\\displaystyle\\widehat\{\\alpha\}\_\{P\}≔sech2⁡\(1\)\|P\|​\(σmin​\(A^P\)−τ\)\+\.\\displaystyle\\coloneqq\\frac\{\\operatorname\{sech\}^\{2\}\(1\)\}\{\\sqrt\{\|P\|\}\}\\bigl\(\\sigma\_\{\\min\}\(\\widehat\{A\}\_\{P\}\)\-\\tau\\bigr\)\_\{\+\}\.\(149\)Then‖A^P−AP‖2→2≤τ\\\|\\widehat\{A\}\_\{P\}\-A\_\{P\}\\\|\_\{2\\to 2\}\\leq\\tauand

dP​\(f,g\)≥α^P​d𝒰​\(f,g\),f,g∈ℱ\.d\_\{P\}\(f,g\)\\geq\\widehat\{\\alpha\}\_\{P\}d\_\{\\mathcal\{U\}\}\(f,g\),\\qquad f,g\\in\\mathcal\{F\}\.\(150\)Thusα^P\>0\\widehat\{\\alpha\}\_\{P\}\>0certifies separation\. For exact target records,A^P=AP\\widehat\{A\}\_\{P\}=A\_\{P\}\.

###### Proof\.

Apply[lemmaG\.1](https://arxiv.org/html/2610.27139#A7.Thmtheorem1)withY=VY=V,B=AP⊤B=A\_\{P\}^\{\\top\}andη=ηP\\eta=\\eta\_\{P\}\. The operator\-norm bound gives‖A^P−AP‖2→2≤τ\\\|\\widehat\{A\}\_\{P\}\-A\_\{P\}\\\|\_\{2\\to 2\}\\leq\\tau, hence

σmin​\(AP\)≥\(σmin​\(A^P\)−τ\)\+\.\\sigma\_\{\\min\}\(A\_\{P\}\)\\geq\\bigl\(\\sigma\_\{\\min\}\(\\widehat\{A\}\_\{P\}\)\-\\tau\\bigr\)\_\{\+\}\.\(151\)All query rows have norm at most one, sod𝒰​\(fx,fx′\)≤‖x−x′‖2d\_\{\\mathcal\{U\}\}\(f\_\{x\},f\_\{x^\{\\prime\}\}\)\\leq\\\|x\-x^\{\\prime\}\\\|\_\{2\}for unitx,x′x,x^\{\\prime\}\. The derivative oftanh\\tanhon\[−1,1\]\[\-1,1\]is at leastsech2⁡\(1\)\\operatorname\{sech\}^\{2\}\(1\); hence

dP​\(fx,fx′\)\\displaystyle d\_\{P\}\(f\_\{x\},f\_\{x^\{\\prime\}\}\)≥sech2⁡\(1\)​‖AP​\(x−x′\)‖∞\\displaystyle\\geq\\operatorname\{sech\}^\{2\}\(1\)\\\|A\_\{P\}\(x\-x^\{\\prime\}\)\\\|\_\{\\infty\}\(152\)≥sech2⁡\(1\)​σmin​\(AP\)\|P\|​‖x−x′‖2≥α^P​d𝒰​\(fx,fx′\)\.\\displaystyle\\geq\\frac\{\\operatorname\{sech\}^\{2\}\(1\)\\sigma\_\{\\min\}\(A\_\{P\}\)\}\{\\sqrt\{\|P\|\}\}\\\|x\-x^\{\\prime\}\\\|\_\{2\}\\geq\\widehat\{\\alpha\}\_\{P\}d\_\{\\mathcal\{U\}\}\(f\_\{x\},f\_\{x^\{\\prime\}\}\)\.This proves[equation150](https://arxiv.org/html/2610.27139#A7.E150)\. For exact target records,V=X​AP⊤V=XA\_\{P\}^\{\\top\}givesA^P=AP\\widehat\{A\}\_\{P\}=A\_\{P\}\. ∎

When this certificate is positive, the calibrated inverse estimates the latent vector\. Normalization returns that estimate to the sphere\.

###### Proposition G\.4\(Calibrated Recovery\)\.

UseΔ\\Delta,A^P\\widehat\{A\}\_\{P\}andτ\\taufrom[lemmasG\.2](https://arxiv.org/html/2610.27139#A7.Thmtheorem2)and[G\.3](https://arxiv.org/html/2610.27139#A7.Thmtheorem3), and assumeσmin​\(A^P\)\>τ\\sigma\_\{\\min\}\(\\widehat\{A\}\_\{P\}\)\>\\tau\. Letν\\nunormalize nonzero vectors to the sphere and send zero to the first coordinate vector\. Define

c⁡\(y\)\\displaystyle c\(y\)≔arctanh⁡\(clip\[−tanh⁡\(1\),tanh⁡\(1\)\]⁡\(y\)\),\\displaystyle\\coloneqq\\operatorname\{arctanh\}\\\!\\left\(\\operatorname\{clip\}\_\{\[\-\\tanh\(1\),\\tanh\(1\)\]\}\(y\)\\right\),\(153\)x^​\(y\)\\displaystyle\\widehat\{x\}\(y\)≔ν⁡\(A^P†​c​\(y\)\)\.\\displaystyle\\coloneqq\\nu\\\!\\left\(\\widehat\{A\}\_\{P\}^\{\\dagger\}c\(y\)\\right\)\.\(154\)This readout satisfies

supx∈𝕊r−1d𝒰​\(fx^​\(\(b∘e\)​\(fx​\(Q\)\)\),fx\)≤2​\(\|P\|​Δ\+τ\)σmin​\(A^P\)\.\\sup\_\{x\\in\\mathbb\{S\}^\{r\-1\}\}d\_\{\\mathcal\{U\}\}\\bigl\(f\_\{\\widehat\{x\}\(\(b\\circ e\)\(f\_\{x\}\(Q\)\)\)\},f\_\{x\}\\bigr\)\\leq\\frac\{2\\bigl\(\\sqrt\{\|P\|\}\\Delta\+\\tau\\bigr\)\}\{\\sigma\_\{\\min\}\(\\widehat\{A\}\_\{P\}\)\}\.\(155\)

###### Proof\.

Fix a unitxxand writey=\(b∘e\)​\(fx​\(Q\)\)=tanh⁡\(M​AQ​x\)y=\(b\\circ e\)\(f\_\{x\}\(Q\)\)=\\tanh\(MA\_\{Q\}x\)\. Sincec⁡\(y\)=clip\[−1,1\]⁡\(M​AQ​x\)c\(y\)=\\operatorname\{clip\}\_\{\[\-1,1\]\}\(MA\_\{Q\}x\)andAP​x∈\[−1,1\]\|P\|A\_\{P\}x\\in\[\-1,1\]^\{\|P\|\}, clipping and[lemmasG\.2](https://arxiv.org/html/2610.27139#A7.Thmtheorem2)and[G\.3](https://arxiv.org/html/2610.27139#A7.Thmtheorem3)give

‖c⁡\(y\)−A^P​x‖2≤‖c⁡\(y\)−AP​x‖2\+‖\(AP−A^P\)​x‖2≤\|P\|​Δ\+τ\.\\\|c\(y\)\-\\widehat\{A\}\_\{P\}x\\\|\_\{2\}\\leq\\\|c\(y\)\-A\_\{P\}x\\\|\_\{2\}\+\\\|\(A\_\{P\}\-\\widehat\{A\}\_\{P\}\)x\\\|\_\{2\}\\leq\\sqrt\{\|P\|\}\\Delta\+\\tau\.\(156\)For a unitxx, normalization is a nearest\-point projection onto the sphere, including at zero\. Thus, for every vectorvv,

‖ν⁡\(v\)−x‖2≤‖ν⁡\(v\)−v‖2\+‖v−x‖2≤2​‖v−x‖2\.\\\|\\nu\(v\)\-x\\\|\_\{2\}\\leq\\\|\\nu\(v\)\-v\\\|\_\{2\}\+\\\|v\-x\\\|\_\{2\}\\leq 2\\\|v\-x\\\|\_\{2\}\.\(157\)SinceA^P\\widehat\{A\}\_\{P\}has full column rank and all query rows have norm at most one,

d𝒰​\(fx^​\(y\),fx\)≤2​‖A^P†​c​\(y\)−x‖2≤2​‖c⁡\(y\)−A^P​x‖2σmin​\(A^P\)≤2​\(\|P\|​Δ\+τ\)σmin​\(A^P\)\.d\_\{\\mathcal\{U\}\}\(f\_\{\\widehat\{x\}\(y\)\},f\_\{x\}\)\\leq 2\\\|\\widehat\{A\}\_\{P\}^\{\\dagger\}c\(y\)\-x\\\|\_\{2\}\\leq\\frac\{2\\\|c\(y\)\-\\widehat\{A\}\_\{P\}x\\\|\_\{2\}\}\{\\sigma\_\{\\min\}\(\\widehat\{A\}\_\{P\}\)\}\\leq\\frac\{2\\bigl\(\\sqrt\{\|P\|\}\\Delta\+\\tau\\bigr\)\}\{\\sigma\_\{\\min\}\(\\widehat\{A\}\_\{P\}\)\}\.\(158\)Taking the supremum overxxproves[equation155](https://arxiv.org/html/2610.27139#A7.E155)\. ∎

## Appendix HObservation Geometry

We examine how many representative observations cover a later diagnostic block, using the same geometry protocol and time\-shift controls\. Covariance statistics provide a complementary description\. Prediction and model\-state recovery use this cohort in[appendicesI](https://arxiv.org/html/2610.27139#A9)and[J](https://arxiv.org/html/2610.27139#A10)\.

Data access and reuse\.The recordings were contributed by Mizuseki, Sirota, Pastalkova, Diba, and Buzsáki and distributed as HC\-3 through CRCNS\.org\([Mizuseki et al\., 2013](https://arxiv.org/html/2610.27139#bib.bib29);[Mizuseki et al\., 2014](https://arxiv.org/html/2610.27139#bib.bib30)\)\. The provider makes the data available for scientific use under the[CRCNS data\-use terms](https://crcns.org/terms), which require attribution and citation and restrict redistribution\. The[HC\-3 dataset page](https://crcns.org/data-sets/hc/hc-3/about-hc-3)identifies the source files and access procedure\. Our analysis reuses these existing animal recordings under that published provision for scientific use\. The reproduction archive supplies the session manifest and preprocessing code; readers obtain recordings directly from the provider\. Original code and aggregate outputs are supplied for confidential peer\-review reproduction, and dependencies retain their upstream licenses, listed in the archive\.

### H\.1Session and Observation

The HC\-3 cohort\([Mizuseki et al\., 2014](https://arxiv.org/html/2610.27139#bib.bib30)\)spans open\-field, linear\-track, and wheel recordings \([table10](https://arxiv.org/html/2610.27139#A8.T10)\)\. We select sessions using metadata to balance data availability with breadth across recording groups\. Eligible sessions have aligned position, recording metadata, and spike files\. We allocate sessions across recording prefixes, cycling through the groups within each prefix\. Within a group, priority follows processing availability, matched shanks, estimated unit counts, and recording length\. This procedure supplies varied recording contexts for comparing observation geometry\.

Every included session meets the unit\-quality criteria in[table11](https://arxiv.org/html/2610.27139#A8.T11)\. Electrode metadata identifies recording prefixes belonging to the same animal\. All analyses use this fixed cohort; detailed session statistics appear in[table16](https://arxiv.org/html/2610.27139#A8.T16)\.

Table 10:HC\-3 study cohort relative to the complete local catalog, with session counts by animal\. The ec015 recording prefix belongs to ec014\.
Each query specifies a unit’s spike count over a trailing window\. At each sampled observation time, the window ends at that time; evaluating the fixed collection of queries gives the corresponding observation vector\. Each query returns the count, logarithmically transformed and scaled using the earlier training block \([table11](https://arxiv.org/html/2610.27139#A8.T11)\)\. This normalization stays fixed throughout the recording\. Each window lies fully inside its assigned block, with gaps at the boundaries\. We derive geometric coordinates from a reference block, test their transfer to a later diagnostic block, and reserve the final block for subsequent experiments\. Shorter and longer counting windows test sensitivity to temporal resolution\. The calculations operate directly on recorded observations\.

Table 11:Observation construction and time blocks\. Block percentages refer to recording duration\.
### H\.2Covering Geometry

For each recording block, letXXbe the set of sampled observation vectors, equipped with the maximum\-coordinate metric\. A farthest\-first traversal constructs a greedy cover with sizeN^X​\(ε\)\\widehat\{N\}\_\{X\}\(\\varepsilon\)\. The same traversal provides a packing lower boundN¯X​\(ε\)\\underline\{N\}\_\{X\}\(\\varepsilon\), givingN¯X​\(ε\)≤𝒩X​\(ε\)≤N^X​\(ε\)\\underline\{N\}\_\{X\}\(\\varepsilon\)\\leq\\mathcal\{N\}\_\{X\}\(\\varepsilon\)\\leq\\widehat\{N\}\_\{X\}\(\\varepsilon\)with centers inXX\. The lower bound counts selected observations separated by more than2​ε2\\varepsilon\. We evaluate the full radius grid, nested sample sizes, and time\-shift replicates specified in[table12](https://arxiv.org/html/2610.27139#A8.T12)\.

The reference block fixes the distance scale\. Real and shifted diagnostic observations use identical sample indices, sample sizes, and radii\. The main comparison uses the grid radius nearest a prespecified fraction of the reference diameter\. Curve bands show session interquartile ranges; control bars span the time\-shift controls\.

Table 12:Covering protocol, using the observations and chronological blocks in[table11](https://arxiv.org/html/2610.27139#A8.T11)\.Table 13:Fixed\-radius cover\-size comparison over the full cohort\. Each session has a smaller cover than its median time\-shift control at every window\.The fixed\-resolution comparison shows that recorded population observations admit smaller constructed covers than time\-shift controls\. The animal and session summaries use the same radius and retain the entire cohort\. These are covering statistics for the recorded observations in the specified maximum\-coordinate metric\.

Table 14:Complete primary\-window covering results at the fixed radius\. The ratio divides the real greedy\-cover size by its median time\-shift cover size\.Computational cost\.Fornnsampled observations ofuuunits, forming the maximum\-coordinate distance matrix costsO⁡\(n2​u\)O\(n^\{2\}u\)time andO⁡\(n2\)O\(n^\{2\}\)memory\. Farthest\-first traversal updates the nearest\-center distance inO⁡\(n2\)O\(n^\{2\}\)time; the same traversal supplies all reported radii\. Covariance formation and eigendecomposition costO⁡\(n​u2\+u3\)O\(nu^\{2\}\+u^\{3\}\)time andO⁡\(u2\)O\(u^\{2\}\)matrix storage\. Time\-shift replicates repeat these operations, with shifts preserving each unit’s empirical marginal\. The sample cap and number of controls are fixed in[tables12](https://arxiv.org/html/2610.27139#A8.T12)and[15](https://arxiv.org/html/2610.27139#A8.T15)\.

### H\.3Covariance Geometry

We compute the covariance spectrum and participation\-ratio effective dimension in each block\. Reference covariance eigenvectors define the projection applied to the later diagnostic block\. The number of directions needed to retain a given fraction of variance measures covariance concentration; the covering exponent in[equation13](https://arxiv.org/html/2610.27139#S2.E13)controls how many representatives are needed to approximate every state at a given accuracy\.

Table 15:Covariance geometry settings\. All use the same diagnostic block\.The controls independently circularly shift each unit’s observation sequence, preserving its marginal distribution and circular autocorrelation while changing alignment between units\. The comparison reflects shared temporal modulation and nonstationarity as well as coordination between units\. We compare all controls within each session and assess uncertainty by resampling contiguous time blocks\. Complete session results and window sensitivity appear in[tables16](https://arxiv.org/html/2610.27139#A8.T16)and[17](https://arxiv.org/html/2610.27139#A8.T17)\.

The reference subspace captures substantial variance in the later block; the covariance summaries retain the complete cohort and window comparisons\.

Table 16:Complete primary\-window results\. Effective dimension and shift reduction refer to the diagnostic block\. The 90% rank uses directions from the reference basis\.Table 17:Window sensitivity\. Entries are session medians in the diagnostic block; reduction is relative to the median shift control\. Every eligible session lies below every shift control at each window length\.

## Appendix IHC\-3 Prediction Details

Matched prediction task\.

We use the same metadata\-selected session panel and preprocessing as[appendixH](https://arxiv.org/html/2610.27139#A8)\. Units are partitioned into input queriesQQ, candidate target queries, and an independent evaluation setVV\. Nested target budgets share the input set and evaluation units within each seed\. Training uses neural observations and unit identities\. A cross\-attentionencodercompresses the input observations into learned tokens, and a query\-conditionedpredictorestimates target observations\. All parameters are trained jointly from scratch\. The full design and fixed endpoint are listed in[table18](https://arxiv.org/html/2610.27139#A9.T18)\.

Table 18:HC\-3 prediction protocol\. Percentages denote within\-session chronological blocks or unit partitions\. Normalization uses the earlier training portion specified in the geometry protocol\. All configurations use the fixed final epoch\.Training choices\.The reference architecture and200200\-epoch endpoint were fixed after preliminary training checks and then shared by every session, target budget, and repetition\. The small model reduces depth, width, and learned\-token count while retaining the same observation task\. These architecture settings are design choices, not quantities selected separately on the final test block\. AdamW uses weight decay10−410^\{\-4\}, moments\(\.9,\.999\)\(\.9,\.999\), and stabilizer10−810^\{\-8\}; gradients are clipped to norm11\. Training uses Float32, with TF32 disabled and deterministic PyTorch operations\. The loss is the batch mean of the maximum absolute target error\. The cosine schedule, chronological split, and fixed final epoch are given in[table18](https://arxiv.org/html/2610.27139#A9.T18); all remaining initialization and library settings are specified in the archived implementation\.

Attention cost\.Letq=\|Q\|q=\|Q\|,p=\|P\|p=\|P\|,ttbe the learned\-token count, andhhthe hidden width\. Per observation and layer, the dense attention and projection operations costO⁡\(\(q\+p\+t\)​h2\+\(t2\+t​q\+p2\+p​t\)​h\)O\(\(q\+p\+t\)h^\{2\}\+\(t^\{2\}\+tq\+p^\{2\}\+pt\)h\)time\. Materialized attention matrices requireO⁡\(t2\+t​q\+p2\+p​t\)O\(t^\{2\}\+tq\+p^\{2\}\+pt\)storage for a fixed number of heads, alongside the token activations\. Batch size and layer count multiply these costs; the unit\-embedding table adds storage linear in the number of recorded units andhh\. The tables specify each architecture and the supplied manifests give the session\-dependent query counts\.

Prediction score\.The prediction score averages the maximum absolute target\-observation error across test windows\. The constant baseline is the coordinatewise training median on the same target set\. Relative gain is one minus the ratio of model error to baseline error\. We average seed\-level gains within each session, then weight sessions equally\. Session\-to\-session spread is descriptive because recordings within an animal share biological variation\. Different budgets evaluate different target sets, each with its own matched baseline\.

Table 19:Prediction gain, mean±\\pmsample standard deviation over repetitions after averaging the same sessions\. Each target set uses its own training\-median baseline\.
## Appendix JLocal State Recovery

Reference state\.The ontic\-state surrogate is the largeencoder’s complete post\-normalization token state, flattened into a reference vector\. For each session and seed, the full\-target large model is held fixed\. The smallerencoder–predictorlearns the same input\-to\-target task from scratch, with identical input and target queries, preprocessing, training observations, and chronological validation and test blocks\. Its training objective uses target observations\. After predictive training, the reference states determine the recovery maps and their evaluation\. We compare the final smallencoderwith its exact saved initialization on the same held\-out inputs\.

Table 20:Matched prediction models\. The common optimization schedule and chronological split are given in[table18](https://arxiv.org/html/2610.27139#A9.T18)\. Recovery uses the full target pool, with2525sessions and33paired seeds\. Parameter counts include theencoderandpredictor\.Local recovery\.For a held\-out input, we locate the nearest neighbors of its learned state in the smallencoder’s training\-state library and average their reference states\. Distances use Euclidean coordinates, centered by the training mean and divided by a common training root\-mean\-square scale\. This scalar normalization preserves neighbor order\. A neighbor’s weight is proportional to the exponential of its negative squared distance divided by the squared distance of the furthest selected neighbor, with a numerical floor\. At test time, these training neighbors and their distance\-based weights directly determine the recovered reference state\.

Selection and cost\.The neighborhood grid spans individual neighbors through broader local averages\. The count minimizing validation mean squared error is selected separately for the trained and initial representations; ties retain the first count in ascending order\. Evaluation uses the fixed final test block\. For a library ofnntraining states withsslearned coordinates, a batch ofBBqueries requiresO⁡\(B​n​s\)O\(Bns\)distance computation\. Neighbor selection adds the cost of selecting the smallestkkdistances per query; averaging reference vectors of dimensionrrcostsO⁡\(B​k​r\)O\(Bkr\)\. Batches of128128bound the distance workspace toO⁡\(128​n\)O\(128n\)entries, in addition to stored learned and reference states\. No gradient updates are used to fit this local recovery map\.

Recovery score\.We score all reference coordinates together using squared reconstruction error divided by the squared error of the training\-mean reference state; the reportedR2R^\{2\}is one minus this ratio\. Both conditions use this same baseline\. We first average paired seeds within each session and then weight sessions equally\. Seeds change the query partition and the independently trained model pair\. Animal groups follow recording metadata; recording\-name prefixes can differ from animal identities\.

Table 21:Local\-recovery gains, mean±\\pmsample standard deviation over repetitions with the session or animal weights held fixed\.