The Lattice Representation Hypothesis of Large Language Models
Authors: Bo Xiong
Venue: ICLR 2026 Poster (2026)
Keywords: Interpretability
formal concept analysis
language models
ontology
Primary area: interpretability and explainable AI
TL;DR: We uncover the hidden lattice geometry of LLMs, showing that linear attribute directions induce concept lattices that support symbolic reasoning via meet and join operations

Abstract: We propose the Lattice Representation Hypothesis of large language models: a symbolic backbone that grounds conceptual hierarchies and logical operations in embedding geometry.  Our framework unifies the Linear Representation Hypothesis with Formal Concept Analysis (FCA), showing that linear attribute directions with separating thresholds induce a concept lattice via half-space intersections. This geometry enables symbolic reasoning through geometric meet (intersection) and join (union) operations, and admits a canonical form when attribute directions are linearly independent. Experiments on WordNet sub-hierarchies provide empirical evidence that LLM embeddings encode concept lattices and their logical structure, revealing a principled bridge between continuous geometry and symbolic abstraction.

THE LATTICE REPRESENTATION HYPOTHESIS OF LARGE LANGUAGE MODELS
Bo Xiong
Stanford University, United States x ongbo@stannoodd.
ABSTRACT
We propose the Lattice Representation Hypothesis of large language models: a symbolic backbone that grounds conceptual hierarchies and logical operations in embedding geometry. Our framework unifies the Linear Representation Hypothesis with Formal Concept Analysis (FCA), showing that linear attribute directions with separating thresholds induce a concept lattice via half-space intersections. This geometry enables symbolic reasoning through geometric meet (intersection) and join (union) operations, and admits canonical form when attribute directions are linearly independent. Experiments on WordNet provide empirical evidence that LLM embeddings encode concept lattices and their logical structure, revealing principled bridge between continuous geometry and symbolic abstraction.
INTRODUCTION
Large language models (LLMs) (Achiam et al., 2023; Grattafiori et al., 2024; Mesnard al., 2024) are surprisingly effective in capturing conceptual knowledge (Petroni et al., 2019; Wu et al., 2023; Lin & Ng. 2022; Xiong & Staab, 2025) and performing logical reasoning, capabilities traditionally associated with symbolic AI. Yet, it remains fundamentally unclear how exactly such conceptual knowledge, including concepts, hierarchies, and their logical semantics, is encoded within the continuous geometry of LLM representation spaces. Unlocking this hidden geometry is crucial not only for interpreting how LLMS representing symbolic knowledge, but also for reliably controlling and steering their inference behavior (Han et al., 2024), fundamental step for advancing Al safety,
To understand concept representations in L.MMs, a promising direction is the Linear Representation Hypothesis (Mikolov et al., 201 3b; Park et al., 2025; 2024a; Gurnee & Tegmark, 2024), which posits that semantic features and concepts are encoded as linear directions or subspaces in a model's embedding space. This idea, rooted in early work word embeddings (Pennington et al., 2014a), has since been extended to modern LLMs, where such directions can be interpreted as embedding difference, logistic probing, or steering vectors in different contexts (Gurnee & Tegmark, 2024; Nanda et al., 2023; Zhao et al., 2025). Park et al. (Park et al., 2024a) unifies them through causal inner product that respects the semantic structure of concepts in the sense that causally separable concepts are represented by orthogonal vectors. However, these works mainly focus on exploring the existence of the lincarity (binary) concepts, but offer limited insights for interpreting compositional or set-theoretic semantics such as concept inclusion, concept intersection, and union, which lies at the heart of symbolic abstraction.
Recently, Park et al. (Park et al., 2025) extended the Linear Representation Hypothesis to formalize categorical concepts as geometric regions like polytopes in the representation space, and show that semantic hierarchy corresponds to orthogonality. However, they model concepts purely in terms of their extensions, that is, as sets of tokens or objects that fall under the category, such as Y (animal)) (predator, bird, ..g....). While this extensional view is useful for evaluating membership, it overlooks their intensional nature, i.c., the attributes and relations that ground categories in logic and philosophy, making difficult to interprete how concepts related each others through set-theoretic semantics like concept subsumption, intersection, or union.
We draw inspiration from Formal Concept Analysis (FCA) (Ganter et al., 2005). principled and philosophy-insping framework that defines concepts through both their instances and their attributes. In FCA, each concept is represented pair: extent (the set of objects) and an intent (the set of

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fly A hunt
predator
predotor A bird
hunt
bird of prey
predator
bird
bird of prey
predator bird
bird
predator
fly
bird of prey
bird
(a) concepts as symbols (human view)
(b) concepts as pure directions (extensional)
(c) concepts as regions bounded by attribute directions (intensional)
How LLMs encode conceptual structure. (a) Humans represent concepts as symbols and compose them using logical operators. (b) Under the standard extensional view, LLMs encode concepts as directions, where subsumption is interpreted through the relative orientation of vectors. (c) Under our intensional view, concept is represented as the intersection of half-spaces defined by its attributes, and compositional semantics emerges through region intersection and union.
Figure 1:
shared attributes). For example, the concept bird may be defined by attributes such as can fly, has feathers, and lays eggs, while eagle (a bird of prey) refines this category by denoting a subset of birds with additional features such as can hunt. As Figure shows, unlike extensional view defining concepts as instances, such dual intent-extent view defines concepts as convex regions bounded by their attributes defining them. For example, the concept bird may be defined by attributes like can fly, has feathers, and lays eggs, while eagle (a bird of prey) refines this category by denoting subset of birds with additional features such as can hunt. This dual formulation naturally induces concept lattice because every pair of concepts can ordered by inclusion of their extents (or, dually, their intents), and any two concepts have well-defined intersection (their common subconcept) and union (their common superconcept), corresponding to the meet and join operations of the lattice.
In this work, propose the Lattice Representation Hypothesis that unifies two perspectives concept representation: the Linear Representation Hypothesis, which views semantics as directions in embedding geometry, and Formal Concept Analysis, which formalizes concepts through the incidence relation between objects and attributes. Our key insight is that these views coincide, revealing hidden lattice geometry in LLMs: attribute directions correspond to FCA intents, object embeddings to extents, and symbolic abstractions such as subsumption, intersection, and union emerge naturally from the induced closure structure. Building on this connection, we formalize half-space model of concepts and projection-based notion of concept inclusion that together recover a lattice geometry from LLM representations. Our contributions are threefold: (i) a theoretical framework linking Linear Representation Hypothesis to FCA via half-space intersections, (ii) soft inclusion measure and concept algebra (meet/join) defined directly on embeddings, and (iii) empirical evidence on WordNet sub- hi-hherrachhes showing that LLM embeddings encode concept lattices, enabling coherent generalization (join) and refinement (meet). These results demonstrate that LLMs implicitly organize conceptual knowledge into lattice geometry, providing symbolic backbone for interpretability.
PRELIMINARIES
2.1 LINEAR REPRESENTATION HYPOTHESIS IN LLMS
We consider the autoregressive family of LLMs that predict the next tokens given its context.
Definition 1 (Large Language Model). An LLM defines probability distribution over the next token
y given a context via the softmux function Pr(y a) xx exp (><x)"<<))) where X : X A Bd
maps the input context x to a context embedding vector X(x), and yy : V - ~ Ted maps each vocabulary token y to its unembedding vector 7(y).
This definition involves context embedding space A and token unembedding space I, which together define the geometry of the softmax distribution. These two spaces can be unified via the causal inner product (Park et al., 2024a). Specifically, there exists invertible matrix A E dddd and a constant vector % E Id such that defining g == A (7()) 50) and f(x) = A A=))(z), reparameterizes token and context embeddings into shared semantic space, where their interaction
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is captured by the Euclidean inner product {(x) g This transformation preserves the model's output distribution, as the softmax Pr(y z) remains invariant under any choice of A and yv-
In this unified space, Linear Representation Hypothesis states that certain semantic attributes corre- spond to specific directions. There is an explicit definition for binary attributes.
Definition 2 (Linear representation of a binary attribute/concept (Park et al., 2024a)). A vector m E Rd is said to linearly represent a binary attribute m E {0, 1 if. for all context embeddings l E TRd all scalars a > and all attributes + m that are causally separable from m, the following conditions hold:
Attribute activation: Pr(m =
alm) > Pr(m= = 1
Causal selectivity: Pr(z | + al ) = Pr(z (
In other words, moving the direction e increases the likelihood of the attribute m without affecting any causally unrelated attributes. The direction merely encodes the semantic direction of the attribute, not its strength, as any positive scalar multiple al has the same qualitative effect.
2.2 FORMAL CONCEPT ANALYSTS (FCA)
FCA (Ganter et al., 2003) is mathematical framework for modeling concepts structured relation- ships between objects and their attributes. Unlike extensional views that define concepts simply as sets of objects (as in (Cowsik et al., 2024)), FCA treats concept of objects characterized by a common set of attributes. This duality between objects and attributes allows FCA to capture both the semantic content of concept and its compositional structure. FCA begins with a binary relation between objects and attributes, formalized as a formal context:
intensional abstraction: set
Definition 3 (Formal context). A formal context
triple (G, (G,M,D I). where objects, M is finite set of attributes, and I C G x M is binary relation (called the incidence relation) such that (g, m) E I indicates that object g E G possesses attribute m E M.
finite set of
From this, FCA defines concepts maximal sets of objects and attributes that mutually consistent:
Definition 4 (Formal concept). Given formal context (G, M, consider a pair (A, B) with A and B C M. Define the Galois connections as A' {m M Vg E A, (g, m) E H. B' = {g E G Vm E B, (g, m) E T}. The pair (A, B) is called a formal concept if and only if A' and B' = A, where is the extent and B is the intent.
The set of formal concepts is partially ordered by inclusion extents (i.e., A C A2) or equivalently by reverse inclusion intents (i.e., B2 C B1).
Definition 5 (Concept lattice). Let (A₁, B,) and (A2. B-2) be formal concepts. Then (A1, B1) So (A2, B2) and only if A1 S. A2 (equivalently, B2 C B,),, where So denotes the partial order relationship. Under the partial order Sc, the set of all formal concepts forms a complete lattice: every subset concepts have a greatest lower bound (meet) and a least upper bound (join).
3 THE LATTICE REPRESENTATION GEOMETRY IN LLMS
We now establish connection between the Linear Representation Hypothesis and Formal Concept Analysis (FCA), showing that the linear geometry LLM embeddings gives rise to concept lattice. The key idea is that each binary attribute can be modeled as a direction in the unified space, with membership approximated by thresholded inner product. This naturally induces binary object-attribute relation, from which formal context and concept lattice can be constructed. We refer to this construction as the lattice geometry of LLMs. Fig. illustrates such correspondence.
3.1 FROM LINEAR TO LATTICE GEOMETRY
Geometric interpretation of attribute membership. Under the Linear Representation Hypothesis, semantic attributes are encoded as directions while objects (tokens or contexts) are encoded as vectors. Let Em E nd denote the direction for attribute m, and V, E Rd the embedding of object g. The extent to which g possesses be estimated by the projection Vg Emm In the idealized case, there
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fly
Swim
hunt
Top
eagle
fly A swim
dove
dove
fly
hunt
Swim
fly hunt
wolf snake
dove
wolf
fish
eagle
fish
fly hunt
huntt swim
swim
frog
frog
fish
eagle
snake
frog
hunt
wolf
snake fly A hunt
(a) Object-atrribute relation (formal context)
(b) Discrete lattice
(c) Lattice geometry linearly encoded in LLMs
constructed via FCA
Figure 2: How FCA connects to the linear lattice geometry of LLMs. (a) A formal context describing which objects satisfy which attributes. (b) The discrete concept lattice constructed exactly from this formal context. (c) The corresponding lattice geometry encoded in LLM embeddings, where each attribute represented as linear direction and each object as a point, and concept composition (meet and join) emerges as intersection or union of half-spaces.
exists threshold Tm such that m(g) holds in practice, so we define a soft incidence relation:
= Vg Im Tm- However, perfect separation rarely
(m(gg) = 1) == a(a ( (m))
(I)
where a > is sharpness parameter controlling the smoothness of the logistic incidence function. A larger a makes the boundary steeper and smaller makes the boundary smoother. As - oo, this recovers the mard-thresholded case. This assigns each object-atrribute pair a fuzzy degree of membership, enabling smoother definitions of concepts.
Theorem 1 (Existence of Lattice Geometry). Let G be finite set of objects and M finite set of attrhutes.. Let V (Vg E Tee 9 E G} be object embeddings and 'D (em Rdd m E M) attribute directions. Suppose for each m E M there exists threshold T. E R such that membership modeled by the soft incidence function above, For any confidence level å (0, 1), define the binary incidence relation
Ia = {(Vg. Em)
( = 2
Then the induced concept set
Fs Fs=
satisfies: (i) closure under the Galois connection, and (ii) forms a complete lattice under extent inclusion (equivalently, reverse intent inclusion). Proof is detailed in Appendix B.
Thus, when attributes are encoded thresholded linear projections, symbolic concept lattice can be recovered from embedding geometry, capturing semantic abstraction through graded boundaries.
Canonical representation under soft incidence. Theorem allows arbitrary thresholds. We now show that under mild conditions, these thresholds can be absorbed into a global shift embeddings, yielding canonical origin-passing form:
Proposition 1 (Canonical representation). Let each attribute m, M be defined by direction d, and threshold Tú. Let D E k he the matrix with rows d.., and T = (Ti,..., "h....,Tkk. If there exists c E & such that De = T, then
a(a(v, d, T:)) (a(((v c) -))) Vg.
That is, the probabilities remain invariant under the transformation v, + V. c, reducing the model to canonical half-space form while preserving the induced lattice. Proof is detailed in Appendix C.
3.2 HALF-SPACE MODEL AND CONCEPT ALGEBRA
Under the canonical representation, where all attribute thresholds have been absorbed via a global shift, each attribute defines origin-passinn half-space in the embedding space. A concept composed of multiple attributes can be interpreted geometrically as the intersection of those half-spaces, i... the region where all attribute constraints are simultaneously satisfied.
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Definition 6 (Concept half space) Let M be set of attributes, each represented by direction dn E R concept defined by a subset Y S M corresponds to the set of object embeddings that satisfy all associated directional constraints:
R(Y) = { RR v v. 2 0 all EY
Geometrically, R(Y) is convex polyhedral cone defined by intersecting sagii-passing half-spaces. However, this definition assumes perfect attribute separation, which may not hold in practice due to noise, uncertainty, and overlapping concept boundaries in LLM representations. To address this, we move from a hard region view to a soft formulation, where membership modeled as graded alignment rather than binary inclusion.
Concept representation via normalized projection profiles. Given a concept C' (e.g.. a lexical term) and its associated context embeddings (V1,-... Vn) C Rd. we define its semantic representa- tion using the average projection profile over the attribute directions {dm (me.. For each attribute m M, the projection value
TC = vi dm
The resulting projection vector we IR encodes a soft attribute profile of C, reflecting how strongly the concept aligns with each attribute. This can be seen as a continuous analogue an FCA intent. To ensure comparability across concepts, all projection vectors are ez-normalized.
Concept inclusion as region containment. With projection profiles in place, we can now define a graded notion of subsumption between two concepts. Intuitively, a concept is included in another concept B if the attributes emphasized by B are also strongly expressed in A. We capture this by soft inclusion score that evaluates how well A's profile satisfies the attribute activations of B:
Inclusion(A A E B) EmEm ((TT(m)) (TA(m))
where
((x) = log(1
(5)
Emer (mT)
Here, the sigmoid (.)) maps A's projection value soft likelihood of satisfying attribute m, while the sofiplus 6(-) weights attributes according to their salience in B. This formulation smoothly downweights weakly expressed inactive attributes in B, while strongly positive ones dominate the inclusion score. Thus, concept inclusion is modeled not as strict set-theoretic containment but as continuous, geometry-driven compatibility measure between attribute profiles.
We operationalize concept algebra in the half-space model by defining meet and join directly on concept regions R(Y).. Meet is the intersection of regions, while join is the least region subsuming both concepts, approximated by the conic hull of their defining directions.
Concept Meet and Join.
Definition 7 (Concept algebra: meet and join). Let R(Y denote the region associated with a concept defined by attribute set Y S. M (Definition 6).
Meet (intersection). For two concepts A R(YA) and B R(BB), their meet is the region satisfying all attributes from both sets AAB:= R(Y UBB) Geometrically, this corresponds to intersecting the half-spaces that define A and B.
Join (union or generalization). Their join the least upper bound in the lattice, i.e., the most specific concept region that subsumes both A and B. In the half-space model, this corresponds to the minimal region that covers R(Ya) and R(BB): AV R(YA) RYB)), which can be approximated by the conic hull spanned by the attribute directions of and B.
Soft measure of meet/join. While Definition 7 specifies meet and join geometrically, we also require a graded way to evaluate how well a concept corresponds to these symbolic operations.
Soft meet/join profiles. Given concepts A, B with projection profiles T TB. define the projection profile of their meet and join using fuzzy t-norm/co-norm combinations:
(m) min min(=a(m), (mmmmmmm))))))) [n(m)]
TAvB(m) m (m)))
(6)
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Degrees of inclusion. The degree to which C is subsumed by the meet or join is
deg(C' E\ AA Inclusion(C E AAB),
deg(C E B) Inclusion(C E AVB), (7)
where the inclusion function is as defined in Eq. (2).
Degrees equality. Soft equality is defined by symmetrizing inclusion with the harmonic mean:
2 Incl B) B
deg(C AA B)
(8)
Iucl(C B) I BCC)
2 InclCC InclA -nelAAAAE deg(C AV B)
Icc In
4 EVALUATION
In this section, we empirically evaluate the extent to which the linear structure of LLM embeddings induces a lattice geometry over concepts. We focus on testing the two core assumptions underlying framework: 1) Existence of half-space model: How well do the linear representations of binary attributes recover the ground-trume truth formal context? 2) Existence of lattice geometry in LLMs: Does the approximated formal context recover valid partial order concept lattice?
4.1 DATASET AND EXPERIMENT SETUP
Dataset construction. We construct five object-attribute datasets derived from the WordNet hi- erarchy. Three datasets correspond to physical domains (WN-Animal, WN-Plant, WN-Foood,, and correspond abstract domains (WN-Event, WN-Cognition). Each dataset represents distinct semantic domain. Statistics of datasets are shown in Table 4 in the Appendix. For each domain, we extract all concept terms that fall under the corresponding hierarchy using the WordNet is_a (hypernym) relation. We expand each concept by retrieving its synonyms and hyponyms to ensure lexical coverage and semantic granularity. Since WordNet does not provide explicit attribute annotations, we use large language model (GPT-4o) to generate the attribute schema and populate the object-attribute matrix. Specifically, for each category, we first prompt the model produce a concise set of salient binary attributes relevant to classification within the category (c.g., can fly, has fur. lays eggs for animals). We then use few-shot prompting to annotate each object with binary attribute vector. This produces complete binary incidence matrix, which treat the ground-truth formal context for evaluation. We use the WordNet is_a relation to define a symbolic subsumption hierarchy, and treat it the target concept lattice structure. Using the annotated formal context and corresponding LLM embeddings for objects and attributes, we evaluate whether the geometry of the embedding space can reconstruct the symbolic structure implied by the context.
Object embedding Each object g is represented by aggregating the embeddings of its lexical synonyms. For every WordNet object g. we retrieve all lemma names in its synset and treat them as synonymous surface forms. For each synonym string s, we compute its embedding Emb(s) Rd as the last hidden state of the model averaged across token positions. The final object embedding is defined as the mean over its synonym embeddings,
Vg
Syn(g)] SESyn(g)
Emb(s),
(10)
where Syn(g) denotes the of lemma names associated with g. For example, the synset of German shepherd includes | GGerman shepherd, German shepherd dog, German police dog, alsatian }- Averaging across these variants reduces lexical noise, stabilizes surface-form effects, and yields more faithful representation the underlying concept rather than a specific phrasing.
Attribute embedding To estimate the semantic direction associated with each attribute, we apply a linear discriminative analysis approach using object embeddings labeled as positive negative with respect to that attribute. Given a set of positive and negative object embeddings for attribute m, we first compute the class means μ4 and μm, and the class covariance matrices E., and E_, estimated

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Active dask
Aestivates periods
Fn Negative Threshord
Deraity
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Into sumsundinngs
Capable clmbing
Capable of fiiht
Deretyy
Projectoon
Projection
Distribution of projection lengths for positive and negative objects onto the directions of the first eight attributes (sorted alphabetically) in the WN-Animal dataset.
Figure 3:
using Ledoit-Wolf shrinkage to improve robustness in high-dimensional settings. We then define the attribute direction f. as the solution regularized Fisher separation criterion:
Lm = (E+ + Σ_ AT) (μ+ p-),
(11)
where À > O is small regularization constant to ensure numerical stability. This procedure yields direction vector that best separates the positive and negative object clusters embedding space under linear discriminant model. The resulting vector Lm. is used as the attribute direction in all downstream projections and geometric reasoning.
Threshold estimation Given an attribute direction f... we determine threshold Tm E IR to separate positive and negative objects along this direction. For each object embedding Vg. we compute its projection onto the normalized direction, and then calculate the threshold as the average of the mean projections of positive and negative object sets:
2 [BEG |Projmm (v,)] + Eyea_ (Proj,,, (v,)))
(12)
where G+ and G_ denote the sets of positive and negative objects for attribute m, respectively. This threshold minimizes classification error under the assumption of linear separability and equal cost.
4.2 EXISTENCE OF HALF-SPACE MODEL
We first evaluate whether semantic attributes in LLM embedding space adhere to the half-space model, i.e., whether single linear direction and threshold can reliably separate objects that possess given attribute from those that do not. For each attribute m, we estimate a direction d and threshold Tm using the training set (Section 4.1). Given object embedding Vg. predict whether the object possesses the attribute using a hard decision rule ri(g) I I[v dm 2 Tm/ where [[[]] is the indicator function. We compare this prediction the ground-truth incidence m(g) E (0, 1} from the annotated formal context. For each attribute, we compute precision, recall, and F1 score, and report averages over all attributes within each dataset.
Results Table reports results for recovering the formal context across five WordNet domains. The Linear method achieves the best precision, recall, and FI every model and domain, consistently above 78% on physical domains (WN-Animal, WN-Plant, WN-Food) and still strong on the more abstract Event and Cognition domains (above 70%). This demonstrates that LDA-estimated attribute directions align closely with ground -trut concept-attribute structure, even in semantically diffuse domains. The Mean baseline performs moderately (59-68% Fl), capturing coarse centroids lacking fine-gra fine-grained discriminative power. The Random baseline stays near chance (45-48%) confirming that attribute recovery is far from trivial and requires meaningful geometric structure. Across models, Gemma-7B and LLaMA3-8B show the strongest linear separability, with Gemma-7B reaching 83.2% Fl on Animal and Plant, and Mistral-7B following closely. Overall, these results provide clear evidence that LLM embeddings support the half-space model of concepts proposed in Section 3.2. Figure further illustrates the separation between positive and negative projection distributions, showing clear margins around the estimated thresholds.
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Table Evaluation recovering concept-attrrbuie relations (formal context). Bold denotes the best across models, shading denotes the best within model. All metrics are percentages.
WN-Animal
WN-Plant
WN-Food Rec.
WN-Event Rec.
WN-Cognition
Model
Rec.
Random
49.7
49.6
45.3
50.1
50.1
47.3
50.0
50.0
46.4
50.2
50.2
48.6
50.2
50.2
50.1
LLaMA3.1-8B
Mean
63.7
69.3
63.7
63.7
67.2
63.3
67.2
71.4
68.1
63.7
65.1
63.9
68.4
68.4
68.4
Linear
81.4
84.0
82.5
81.6
82.4
82.4
79.7
80.6
80.1
71.4
71.6
75.0
74.9
Random
49.8
49.7
45,3
50.1
50.1
47.3
50.1
50.1
46.3
49.4
49.3
47.8
50.1
50.1
50.1
Gemma- 7B
Mean
53.5
55.2
50.1
53.5
54.4
51.3
53.7
55.1
51.2
53.3
53.8
52.2
56.4
56.4
56.3
Linear
82.0
84.8
83.2
82.3
84.3
83.2
79.2
80.9
80.0
71.1
71.7
71.4
75.4
75.4
75.4
Random
49.4
49.2
45.0
50.3
50.4
47.5
49.5
49.5
45.5
50.5
50.6
49.0
49.4
49.4
49.3
Mistral-7B
Mean
62.2
67.1
62.0
61.9
64.9
61.4
62.4
66.6
62.1
57.2
58.1
56.5
63.3
63.3
63.3
Linear
80.6
83.4
81.8
80.8
82.8
81.7
77.6
78.9
78.2
69.7
69.8
69.7
74.2
74.1
74.1
Table Evaluation of partial order inference from projection- based concept representations. Bold denotes the best across models, shading denotes the best within model. All metrics are percentages.
WN-Animal
WN-Plant Rec.
WN-Food Rec.
WN-Event Rec.
WN-Cognition Rec.
Model
Pre.
Rec.
Fl
Pre.
Fl
Prc.
F1
Prc.
Fl
Prc.
F1
Random
52.3
43.2
47.3
51.1
49.5
47.6
25.0
50.0
33.3
50.2
50.2
50.2
49.8
49.9
49.8
L-aMA3.1-8B
Mean
68.4
65.0
66.7
64.2
63.3
63.8
61.5
58.5
55.7
60.3
57.9
59.1
58.2
55.4
56.8
Lincar
75.9
78.3
77.1
71.2
70.0
70.4
78.1
75.9
69.1
67.5
70.4
68.9
69.6
Random
54.0
51.2
50.6
52.7
50.3
49.5
53.4
50.8
39.1
49.8
50.1
49.9
49.4
49.7
49.5
Gemmma-7B
Mcan
66.2
63.4
62.4
50.7
56.3
55.0
54.1
52.8
53.4
Linear
74.4
76.0
75.1
72.9
70.1
71.4
76.3
75.8
75.6
66.2
65.1
65.6
67.0
65.9
66.4
Random
53.3
45.9
49.3
50.0
51.1
48.2
25.0
50.0
33.3
49.3
49.2
49.2
48.7
49.0
48.8
Mistral-7R
Mean
66.8
63.4
64.9
61.2
60.0
60.5
56.0
55.6
54.88
55.7
54.3
55.0
53.3
51.9
52.6
Linear
71.7
72.6
72.1
65.7
60.6
57.1
68.8
64.3
62.0
62.9
60.8
61.8
62.0
60.3
61.1
Random
Mean
Ours
0.6
0.6
0.4
0.4
MRR
MRR
0.2
0.2
WN-Animal
WN-Plant
WN-Food
WN- WN--vvent
WN-Coguition
WN-Animal
WN-Plant
WN--ood
WN- WN--Event
WV-CCogution
Figure 4: Quantitative evaluation (MRR) of concept algebra for Meet (left) and Join (right) operators.
4.3 EXISTENCE OF LATTICE GEOMETRY
To test the existence of lattice geometry, we use the concept inclusion score defined in Section 3.2, where projection profiles over attribute directions are compared via the soft inclusion formula. This allows us to infer subsumptions directly from embedding geometry without access to ground-truth hierarchies. As Table 2 shows, the LINEAR method consistently outperforms centroid-based (MEAN) and random baselines across all domains, achieving F1 scores up to 77.1 (LLaMMA, WN-Animal) and 75.6 (Gemma, WN-Food). These gains demonstrate that discriminatively estimated attribute directions capture intensional information sufficient to recover hierarchical relations: concepts with stronger projection activation on the attributes another concept are reliably inferred as subconcepts. The empirical alignment between projection-basee based subsumption and WordNet ground- truth provides strong evidence that LLM embeddings indeed admit a partial order structure, supporting hypothesis that lattice geometry is embedded within their representation space.
8

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Table 3: Top-10 10 terms related to the join and meet of selected WordNet concept pairs.
B
AVB
A
dog
wolf
predator. animal, canine, meat-cater,
dog. hound, puppy. terrier, mutt, bengle, retriever, spaniel, shepherd, pooch
hunter, wild, mammaal, quadruped, pel
cal
lion
predator, feline, animal, meal-eater, beast,
kilten, liger, panther, leopard, tomcat, felinc, chectah, tabby, lynx sparrow, robin, songbird, warbler, finch, canary, thrush, chickadee, pipit, titmouse horse, pony, stallion, mare. foul, filly, mustang, gelding, thoroughbred, colt carrot, parsnip, radish, beet, turnip. tuber, rooterop. veg. sprout, crop eagle, falcon, hawk, osprey, kestrel, buzzard, kite, harrier, condor, bird of prey
camivore, hunter, whiskers, mammal, wild
sparrow
robin
avian, songbird, fowl, feathered, finch, beak, chirp, nest, perching, small
horse
zebra
equid, hoofed, animal, ungulate, mammal, quadruped, stallion, beast, herhivore
carrot
parsnip
mot, edihle, produce, food, green, vegetable. crunchy, fresh, garden, plant animal, raptor, bird, predator, creature, wingspan, talon, Byer, sky, sharp-eyed
eagle
falcon
ILLLMMASB
8B
|| 70B
grasses
80
walk on the ground eat p ants
PCA Component2 2
live in sea
Macro-FI
70
-.
laaess
swim the water
live DE
svent other animas
We groups gigh
grow pitchengouund
Live
dative
dusk
averrdudd drerdaly
50
hunt insects
WN-Animal
WN-Pllatt
WN-Food
WN WN- Event
WV- WN--Cognition
fly
sky
jays eggs
PCA Component it
Figure 5: (a) PCA-based visualization of attribute directions in WN-Animal (top 20 most frequent attributes); (b) Performance of LLaMA-3 models of different sizes across WordNet domains.
Quantitative evaluation of concept algebra We quantitatively evaluate the concept algebra using the degree of equality metrics in Eq. 8-9. For each WordNet domain, we randomly sample 200 concept pairs (A.B) that have at least one shared descendant and one shared ancestor, ensuring well-defined symbolic meets and joins. Gold meets are the lowest shared descendants; gold joins are the least common hypernyms. We compare three approaches: a Random scorer, a Mean baseline that ranks candidates by similarity to the averaged embeddings of (A, B), and our concept-algebra operator. For each predicted meet or join, we rank all candidates and report the MRR of the gold labels. Figure 4 shows that method consistently outperforms both baselines, with the largest improvements on physical domains and slightly lower gains on more abstract ones.
Qualitative evaluation of concept algebra We further qualitatively evaluate the concept algebra by randomly selecting concept pairs and inspecting their top-ranked meet and join candidates. As shown in Table the join operation (A v B) reliably returns higher-level abstractions subsuming both inputs (e.g., predator for dog and wolf. or avian for sparrow and robin), mirroring WordNet hypernyms. Conversely, the meet operation (A A B) produces refined category intersections such as ponyIstallionlfoall for horse and zebra, or hawklosprey for eagle and falcon. These examples illustrate that meet and join behave as geometric analogues of set-theoretic conjunction and abstraction, providing qualitative evidence that LLM embeddings encode coherent latent lattice structure suitable for compositional reasoning.
4.4 ADDITIONAL. ANALYSIS
Physical vs. abstract domains Table Table 2, Figure 4, and Figure 5b show a consistent tendency: physical domains (Animal, Plant, Food) achieve relatively better results than abstract domains (Event, Cognition). We conjecture that this is because physical concepts are grounded in concrete, human- perceptual attributes (shape, movement, habitat, function), while abstract concepts rely on more complicated situational attributes that are less directly encoded in L.MMs.

---

Attribute correlation analysis We visualize the top 20 most frequent attributes in WN-Animal using PCA (Figure Sa) and report their pairwise correlations in the Appendix. The PCA plot reveals coherent semantic clusters, for example, eat grasses and eat plants appear close togeteer. while swim in water and live in the sea form a tight group capturing aquatic behaviors. In contrast, attributes associated with unrelated ecological or behavioral properties (e.g., lay eggs live in water) lie far apart in both the PCA and correlation map. These patterns indicate that the learned attribute directions naturally organize into meaningful semantic subspaces.
Effect of model scaling We also analyze how lattice-geometry performance scales with model size by comparing LLaMA-3 models from 3B to 70B parameters. As shown in Figure 5b, scaling leads to consistent but modest gains on physical domains (WN-Animal, WN-Plant, WN-Food), where even smaller models already capture grounded attributes that structure these categories. In contrast, scaling yields much larger improvements on abstract domains (WN-Event, WN-Cognition), where success depends on representing non- con-perceptual and relational attributes. This pattern suggests that larger models allocate more capacity to abstract conceptual structure, resulting in more coherent projection geometry in domains less tied to physical properties.
RELATED WORK AND DISCUSSION
Conceptual Knowledge in Language Models. Pretrained language models (LMs) have demonstrated
remarkable capabilities in capturing conceptual knowledge (Wu al., 2023; Lin & Ng, 2022). A variety of methods have been developed to probe such knowledge, most commonly binary probing classifiers (Aspillaga et al., 2021; Michael al., 2020) and hierarchical clustering (Sajjad et al., 2022; Hawasly et al., 2024), with validation against human-define ontologies such as WordNet (Miller, 1995). These approaches primarily provide empirical insights into what LMs capture, but they leave open the question of how LMs learn such conceptual structures. Moreover, it has been argued that LMs can develop novel concepts not strictly aligned with existing ontologies (Dalvi et al., 2022), suggesting that vntology-basee evaluations may underestimate their conceptual capacity. Xiong & Staab (2025) first show the connection of FCA to language models by define concepts in the context of FCA, but their study is limited to masked language models.
Linear Representation Geometry of Concepts. A line of research focuses on the geometric nature of concept representations. Several studies suggest that concepts in LMs correspond to distinct directions in activation space (Elhage et al., 2022a; Park et al., 2024a). This linear hypothesis builds on earlier work in distributional semantics and embeddings (Mikolov al., 201 2013a; Pennington et al., 2014b; Arora et al., 2016), and has since been connected to multiple notions of linearity, including embedding offsets, probing classifiers, and steering vectors (Park al., 2024b). Empirical studies have also shown the emergence of polytopes in toy models (Elhage et al., 2022b), pointing to more structured geometry beyond individual directions. Other theoretical work further explores why linear representations arise, linking them to properties of word embedding models (Arora et al., 2016; 2018) and the implicit bias of gradient descent in LLM training (Jiang et al., 2024).
CONCLUSION
We presented a new perspective on the geometry of large language models by linking the linear representation of attributes to Formal Concept Analysis (FCA). Our main contribution the notion of lattice geometry, which shows that when attribute directions are treated as separating half-spaces, the resulting abject-attribute relation induces a concept lattice. This framework unifies continuous embedding geometry with symbolic abstraction, and provides formal conditions under which logical structure can be recovered from L.LM representations. Empirically, we demonstrated on WordNet aub-hierarchies that (i) many attributes are linearly separable, (ii) subsumption can be predicted directly from projection profiles, and (iii) meet and join operations yield meaningful refinements and generalizations. Together, these findings indicate that LLMs encode not only rich conceptual knowledge but also the algebraic backbone of concept lattices. Our results suggest that symbolic abstraction is not an incidental byproduct but structured geometric property of LLMs. This opens up new directions for -uuro-yymboic AI: aligning geometric and symbolic reasoning, enhancing interpretability, and enabling controllable manipulation of concepts in embedding space.
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ETHICS STATEMENT
This work does not involve human subjects or sensitive personal data. Our experiments are conducted on publicly available lexical resources (WordNet) and pretrained LLM embeddings. We follow the ICLR Code of Ethics'.
REPRODUCIBILITY STATEMENT
We have made efforts to ensure the reproducibility of our results. The formal definitions, theo- rems, and proofs are fully detailed in the main text and appendix. Experimental settings, dataset construction (WordNet sub-hicrarchies and attribute annotation), attribute direction estimation, and evaluation metrics are described in Section 4. Detailed proofs are provided in the Appendix. Code for reproducing the experiments, including dataset processing and evaluation, are uploaded and will be made available in the supplementary materials.
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A USE OF LARGE LANGUAGE MODELS (LLMS)
Large language models (LLMs) were used as a writing assistant to polish sentences and improve clarity of exposition. Also, used LLMs to generate annotations of data used for evaluation. No parts of the research ideation, theoretical development, experimental design, or analysis relied on LLMs. All technical contributions, proofs, and empirical results are the work of the authors.
B PROOF OF THEOREM
We restate the theorem for convenience.
Theorem (Existence Lattice Geometry). Let G be finite set of objects and M finite set of attributes. Let V = VVv,, E IRd I E G} be the set of object embeddings and D = { E IR m E M) the set of attribute directions. Fix a > For each m E M, let Tm TR and define the soft incidence probability
Pa(m(g)) = 1) == = Im - Tm=)).
For any confidence level 8 E (0, 1), define the (crisp) incidence relation
Ig {(g, m) G M Pa(m(g) 1) 8). Let Fs be the set of pairs (X,Y) with X C G and Y C M such that
X Y' {g G: Vm E Y, (g.m) E 1s} and
X' {m E M Vg X, (g.m) 1s).
Then (i) F& is closed under the Galois connection, and (ii) Fs, ordered by extent inclusion (equiva- lently, reverse intent inclusion), forms complete lattice.
Plan of the proof. The probabilistic scoring is only used to induce the crisp relation Ig. Once Ls is fixed, the statement becomes standard FCA result. For completeness, we give self-contained proof.
B.1 GALOIS CONNECTION INDUCED BY Ts
Lemma (Antitone Galois connection). Define maps (.)' 20 + 2 and (.) 2M 20 by
:= {m M Vg E A, (g.m) Is),
B' := [g E G Vm B, (g.m) E Is).
Then for all A C G and B C M, have
AÇB' + A'. Consequently, both primes antitone: if A A2 then A²2 C Al, and if B Bz then B/ C By.
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Proof. (=) A B', then for any B and any g A we have (g, m) Is, hence A' and thus B C A'. (=) If S A', then for any g E and m E we have g) E I1, i.e, 9 E B'. hence A AÇ B'.
Lemma 2 (Closure operators). The double-prime operators ΦG(A) == A" on 2 and @M (B) == R" on M closures: for all A G. (i) A S A" (extensivity), (äi A B = A" B" (monotonicity). and (iii) (A")" A" (idempotence). Analogous properties hold on 2
Proof. Extensivity: A C A" follows from Lemma with B A': AC A", Monotonicity: A C B + B' C A' (antitone), hence A" = -AAY S (B") B". Idempotence: (A")" = A". The 2d case symmetric.
B.2 FORMAL CONCEPTS AS CLOSED PAIRS
Lemma (Characterizaaton of concepts). pair (X, (X) with X C and Y M satisfies Y X and X = Y' iff X and Y are closed, i.e., X = X" and Y = Y"
Proof. I=) XX, then X (X')= Xf Y', then = (Y''= (f X" set Y = X' so X = (X') Y'. The converse from Y = Y" is symmetric.
B.3 LATTICE STRUCTURE AND COMPLETENESS
Proposition 2 (Partial order). For formal concepts (X1, Vi) and (X2, Y2), define
(X1, X1) S (2)
: X X2 (equivalently Y2 S Yi).
Then & is partial order Fo.
Proof. Reflexivityytransiiiiity follow from C. Antisymmetry: X1 C X2 and X₂ S X1 imply X1 X2, hence Yi X X Y2.
Proposition 3 (Meets joins.. For any family {(X,, Yi)r of formal concepts,
A Y2) = (nxx x. (nx.)),
V(X,,))- =(Uxx)", nz).
and both pairs are formal concepts.
Proof. Meet: Let X- = D.X, Then (X, X!) < (X,, Y2) for all i. If W) S (X,, Y2) for all i,
then Z X., so (Z, W) S X!).
Join: Let X = (U,X,)" (closed by Lemma 2). Then (X,, Y2) < (X, X') for all If (X,, Y.
(Z, W) for all i, then U,X; C Z, hence X C z"
and (X, X') S (Z,
Corollary (Completeness). (Fs, S) is complete lattice: every subset has both meet and join as above.
B.4 DISCUSSION OF THE ROLE OF & AND
The parameter a > only rescales the logits inside o and does not affect order-theoretic conclusions, which depend solely Is. The confidence 5 E (0, 1) controls the incidence monotonically: if 5, < 52 then Is2 C Ig.- Thus increasing removes incidences and yields a different (typically coarser) concept lattice. The theorem holds for each fixed 8.
14

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Table 4: Statistics of the five WordNet-derived datasets.
Category
WN-Animal
WN-Plant
WN-Food
WN-Event
WN-Cognition
#Objects
7342
7704
2506
1009
2802
#Attributes
100
145
184
60
107
#Hyperyys
7473
8051
2628
1079
3003
C PROOF OF PROPOSITION 1
Proof of Proposition I. Let D E Rexd have i-th row d and suppose there exists Rd with
De = T. Then, for each attribute E (l,...,k),
d/c = (Dc); = T..
Hence, for any object embedding g
(Vg c). Vg d, d. = Vg d, d
Applying the sigmoid with sharpness a > 0 yields
aa((v, d, - n.))
a(a (v, c) d,).
for all and i, proving that Pa (g) = 1) is invariant under the global shift v, + V, -c Therefore, the soft-incidence model admits a canonical, higin-passing form without changing any probabilities or the induced incidence relation for fixed threshold on Pa.
Remarks. ii) The condition De = T is equivalent to T E rowspace(D). the rows {d. Vizz
are linearly independent (i.e., rank D) = k) and k S d, then rowspace( D) RR so (generally non-unique) solution exists for any (ii) When solutions exist, they are unique up to addition of any vector ()); all such choices yield the same invariance because D + z) = T for any
ker (D).
D ADDITIONAL DISCUSSION AND ANALYSIS
Connection to neuro-symbolic methods and interpretability The soft inclusion and meet/join operators introduced in our framework provide differentiable analogues of logical subsumption and concept composition. These scores make it possible to evaluate whether an LLM's embedding geometry behaves logically within given domain, offering a practical interpretability tool for assessing whether the model has learned coherent concept structure. In addition, because the operators are fully differentiable, they can be used as logic-guided regularizers that softly enforce symbolic constraints during training. Examples include subsumption, attribute satisfaction, and type consistency. This creates a natural interface with neuro-symbolic learning, where symbolic rules influence geometric representations through regularization.
Our findings also suggest a natural extension of linear steering to multi-attribute steering (Oozeer al., 2025). Whereas standard steering modifies representations along single attribute direction, lattice geometry enables logical steering. Enforcing two attributes corresponds to moving the embedding toward their meet. Promoting abstraction corresponds steering toward their join. Negation corresponds to crossing the relevant threshold hyperplane. These operators provide a simple, differentiable, and interpretable way to compose and control conceptual constraints inside LLMs.
Although our experiments focus on WordNet-style taxonomic domains, the framework is broadly applicable to any setting where concepts can be represented by an object-atrribute matrix and attribute inclusion induces a partial order. Examples include semantic fields, where words serve as objects and semantic components serve as attributes, and verb classification or speech-act semantics, where verbs are represented by propositional pragmatic features (Priss, 2005). In linguistic setting where constructions or grammatical patterns are annotated with syntactic or morphological features, the resulting feature matrix be treated as an FCA context and modeled through the proposed half-space formulation. In general, any domain that provides Aatributellabele objects admits the lattice geometry developed in this work without requiring modification.
15

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Correlation Matrix Attribute Directions
1.0
prefer live groups
crowl on the ground walk on the ground
0..8
active during the day hunt Insects
0.66
cat both plarts and animals
0.4
eggs
night
hunt other animals
swim in the water
plants
0.0
leaves
ealt meats
eat fruits
-0.2
live water
tly sky
-0.1
live sea
dawn and groups
crawe on ground
walk on the ground live an land is active during the day hunt insects
eat both plants and animals
swirn the water
eat plants
eat leaves
eac meats
eatt fruits
live weltrr
grasses
fly sky uwv son
Figure 6: Correlation analysis of attribute directions in the WordNet-Animal dataset.
Limitation Our framework assumes that attributes correspond to approximately lincar directions, assumption supported by prior work the Linear Representation Hypothesis. However, noted by Engels et al. (2025), concept types with inherently non-linear, cyclic, or continuously varying structure (for example, months of the year) may violate this linear assumption. Another limitation is that our experimental study focuses on general domains. is an interesting direction to explore how well the lattice geometry hypothesis holds in domain-specifie areas such biomedical ontologies.
16


Reviews:
[{"review_id": "UlGgQnfxet", "reviewer": "Reviewer_Lfek", "summary": "The paper proposes to unify Linear Representation Hypothesis with Formal Concept Analysis to project the discrete symbolic abstraction in a continuous embedding space. The authors formulate a half-space model of concepts and operations based on soft inclusion to introduce a lattice geometry in language models. They empirically evaluate the existence and partial order correlations using WordNet subhierarchical data.", "questions": "1. Can the framework recover non-hierarchical or cross-cutting concept structures (e.g., color vs. shape in objects)? \n\n2. Could you note the possible correlations between the attributes?\n\nMy score is mostly based on my lack of understanding of the experiment's validity. I’m willing to update my score as authors clarify this point.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. I think the authors are targeting an important and ambitious question to resolve the conflicts between continuous representation space in neural language models and symbolic abstractions. While I am not an expert in the related set theory to judge the rigor and correctness of the framework,  I appreciate the principled approach. \n\n\n2. While limited, the authors attempted to empirically validate their theoretical statements with the controlled dataset. \n\n3. The paper is generally well written, especially the motivation and its contribution are clearly developed.", "weaknesses": "1. I am unclear about the WordNet experiment; how exactly is the $v_g$? Is it simply a token-embedding output? While the theoretical framework is presented as a general account of “the hidden lattice geometry of LLMs”, I am concerned that the empirical experiments with WordNet appear to rely exclusively on static token embeddings. It is unclear if the experiment concerns contextual forward passes or layer-wise activations. Consequently, the experiments probe the lexical geometry of the embedding table rather than the representational geometry arising from the LLM’s contextual computations.\nAlso, I think the explanation for soft-inclusion parameters is lacking - what alpha is for the experiment, and how it was chosen. \nThe authors did not discuss the limitations of the framework. \n\n2. The authors did not discuss the limitations of the framework.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0xX2pbFV5A", "reviewer": "Reviewer_Ej1E", "summary": "The authors unify two perspectives: the linear representation hypothesis from LLM representation literature and formal concept analysis from information science. They describe a useful formal framework which allows one to reason about concept lattices in LLMs and extract them accordingly. The work is novel, very well organized, thorough, and well-motived by the literature. Although the experimental section is rather light (just 3 LLMs and 3 ground truth concept lattices), it is convincing, and the findings support the main theoretical claims made in the paper. Overall, this is a good paper that makes useful contributions and would be of high interest at ICLR. It should be accepted.\n\nMinor Comments:\n\nLine 135 – it would be helpful to introduce the pair (A,B) prior to defining A’ & B’\n\nLine 172 – “Discreate” should probably be “Discrete”", "questions": "Are there cases of concept structures in LLMs for which this concept hierarchy would not be appropriate?\n\nHow does LLM size impact the ability of LLM representations to match the ground truth lattice? I would think that higher dimensional embeddings would make it easier to extract a clean lattice.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "Well organized, thorough treatment of related work and preliminaries, easy to follow, clear\n\nNovel combination of the Linear Representation Hypothesis with a formal, well-defined framework\n\nThe claims introduced in the theoretical section (Section 3) are substantiated by the lattice extraction results from the experiment section (Section 4)\n\nVery good reproducibility information, particularly in section 4.1", "weaknesses": "Limited number of datasets and LLMs used in the evaluation (3 datasets, 3 LLMs)\n\nLittle to no discussion on which types of concepts (e.g., those arranged on a curved manifold such as months of the year or days of the week) could or could not be represented by this framework. Somewhat recent work has indicated that “Not All Language Model Features are One-Dimensionally Linear” (Engels et al 2025)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "d9E3dzOK9e", "reviewer": "Reviewer_gkW1", "summary": "This paper proposes a new framework for understanding conceptual knowledge in LLMs by unifying the Linear Representation Hypothesis with Formal Concept Analysis (FCA). The central claim is that LLMs implicitly encode a \"lattice geometry.\" In this framework, semantic attributes (e.g., \"can fly\") are represented as linear directions, which define separating half-spaces. Concepts (e.g., \"bird\") are then represented \"intensionally\" as the geometric intersection of these attribute half-spaces (i.e., a convex polyhedral cone).\n\nThis geometric structure naturally induces a concept lattice, allowing for symbolic operations like meet (intersection, e.g., \"bird\" $\\land$ \"predator\" $\\rightarrow$ \"bird of prey\") and join (union/generalization). The authors derive soft, projection-based measures for concept inclusion and these lattice operations. Empirical evidence using WordNet sub-hierarchies demonstrates that attribute directions induce the half-spaces, and that the projection-based inclusion model correlates strongly with the ground-truth WordNet hierarchy.", "questions": "1. Motivation for Eq. 6 (min/max): Could the authors elaborate on the theoretical motivation for using min/max as the soft measures for meet/join profiles? While this t-norm/co-norm is a standard choice in fuzzy logic, it is unclear how this operation on projection profiles relates to the geometric intersection of half-spaces defined in Definition 7. Why not estimate meet/join directly from the intersection of context embeddings? A more natural alternative might be Eq. (4), applied to the context embeddings for the concepts A $\\land$ B.\n\n2. Quantitative Evaluation of Meet/Join: The paper defines a \"degree of equality\" (Eqs. 8 and 9), but only provides a qualitative evaluation of meet/join in Table 3. Could the authors provide a quantitative evaluation using these metrics? For example, how well does the embedding for \"eagle\" (as concept C) match the deg(C = A $\\land$ B) score, where A = \"bird\" and B = \"predator\"? It is unclear whether this metric demonstrates anything non-trivial, as one might expect high alignment simply from the set-inclusion properties of the underlying hierarchy.\n\n3. Ambiguity in Embedding Definition: To clarify the experimental setup (Section 4.1): What exactly is $\\text{Emb}(s)$? Is it the token unembedding vector $\\gamma(s)$ from the original model, the re-parameterized vector $g(s)$ from the \"unified semantic space\" mentioned in Section 2.1, or some context embeddings $\\lambda(s)$? Additionally, how are synonyms defined, and how were they obtained? More details on the experimental setup would be helpful.\n\n4. Disconnect in Theoretical Machinery: Section 2.1 introduces the 'causal inner product' and 'causal selectivity' (Definition 2) as key preliminaries. Why was the causal inner product used? While it connects the geometry of the space to relationships between concepts, is there any orthogonality theory within the proposed lattice geometry? Furthermore, what was the intended role of Definition 2, and was causal selectivity applied in the experiments?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 1, "strengths": "The paper provides a valuable theoretical connection between the extensional view of concepts and a more structured intensional view based on Formal Concept Analysis (FCA). This shift from \"sets of tokens\" to \"intersections of attributes\" is both compelling and logically sound.\n\nThe experimental results, particularly those reported in Tables 1 and 2, are strong. The high F1 scores provide substantial empirical evidence that the \"half-space\" and \"projection-profile\" models are not merely theoretical constructs, but reflect genuine geometric structure in the models.", "weaknesses": "1. Missing Geometric Visualization: The paper’s title and central thesis emphasize \"lattice geometry.\" While Figure 2(b) presents an excellent illustration of this idea, the paper does not provide an equivalent visualization for real, recovered data. We observe 1D projection distributions (Figure 3), but not the actual geometric arrangement of attribute directions and concept embeddings (e.g., via PCA or a similar projection) to visually confirm the half-space intersections. This represents a significant missed opportunity to make the core claim more tangible.\n\n2. Limited Novelty over Existing Frameworks: The paper relies heavily on the foundations laid by Park et al. (2024, 2025). The core idea that a linear direction represents a feature (and thus defines a half-space) is a direct logical consequence of the Linear Representation Hypothesis, as discussed in that prior work. In particular, the vector representation for a single feature in Park et al. (2025) is arguably more precise than the half-space construction. The primary contribution of this paper appears to be the application of Formal Concept Analysis (FCA) as a descriptive label for this pre-existing geometric structure, rather than the discovery of a new structure. The paper does not clearly articulate what new insights are gained by this FCA-based \"intensional\" reframing that were not already present in the \"extensional\" view (Park et al., 2025).\n\n3. Lack of Intervention Experiments: A stronger test of the proposed framework would involve using the lattice geometry for an intervention or model editing task. For example, could the authors construct a representation for a new concept by geometrically performing a meet operation (e.g., meet(bird, live in water)) and then demonstrate that the model can reason about this novel concept? Without such experiments, the work remains descriptive, rather than demonstrating a practical mechanism.\n\n4. Arbitrary Distinction Between 'Attributes' and 'Concepts': The framework relies on a strict distinction between 'attributes' (e.g., 'can fly') and 'concepts' (e.g., 'bird'), which feels somewhat arbitrary. In the linear representation framework (Park et al.), 'bird' could itself be treated as a single feature direction, similar to 'can fly'. The paper does not justify why its 'intensional' view (concepts as intersections of attributes) is fundamentally more valid or distinct from a simpler 'extensional' view, in which 'bird' and 'bird of prey' are simply hierarchical feature directions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "g12ahHrvnb", "reviewer": "Reviewer_hstW", "summary": "The authors propose a novel framework that uncovers a hidden symbolic and logical structure within the continuous embedding spaces of Large Language Models. By bridging the Linear Representation Hypothesis with Formal Concept Analysis (FCA), they demonstrate that linear directions representing attributes can be modeled as separating half-spaces, whose intersections naturally induce a concept lattice. This \"lattice geometry\" provides a principled link between continuous geometry and symbolic abstraction. The paper's primary contributions are threefold: a theoretical framework formalizing this connection, a \"concept algebra\" that defines logical meet (intersection) and join (union) operations directly on embeddings, and strong empirical evidence from WordNet datasets showing that this structure can effectively recover conceptual hierarchies and enable meaningful compositional reasoning. Ultimately, the work suggests that symbolic abstraction is not an emergent accident but a structured geometric property of LLMs, opening new avenues for interpretability and control.", "questions": "My questions and suggestions are listed in the \"Weaknesses\" section. I just want to make clear that I am open to raising my score if all my weaknesses are addressed.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "Originality:\n- The application of formal concept analysis is an ingenious and original idea that seems to provide the missing key to the problem the authors are approaching\n\nClarity: \n- The introduction is very well written and explains what the paper does well, even for someone with very little background in the related work\n\nSignificance:\n- Although I am not familiar with the related work for this paper, it seems like the authors have found a concrete way to unify discrete/symbolic representations with the continuous representation space of LLMs, which has profound implications for both interpretability and AI safety", "weaknesses": "Typos:\n- Line 37: these them\n- Figure 2: Discreate concept lattice\n\nIn general, I think the presentation could use some work:\n- Figure 1 and 2 (especially figure 2) look almost like draft figures, and just by looking at the figures/captions it is hard to understand what they are conveying\n\nFor table 3, it would be good to know how these A and B concept pairs are selected? Are they cherrypicked? Randomly chosen? Are there any failure examples of these join and meet operations?\n\nI think the paper could benefit from more experiments on other words than animals, plants, and food. These are all very concrete and real-world concepts where these concept relations make sense, but how does the framework handle more abstract concepts? If it doesn't handle them well this is fine but should be discussed as a limitation\n\nI also think the paper should provide more discussion on how this method can be used in combination with existing neuro-symbolic/interpretability methods, as well as how the framework might apply to other kinds of concepts (grammatical relationships, etc.).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)