知识库补全忠实度标准的层次结构

# A hierarchy of faithfulness criteria for knowledge base completion
Source: [https://arxiv.org/html/2609.27863](https://arxiv.org/html/2609.27863)
Olga Mashkova and Robert HoehndorfOrcID:[0000\-0002\-4916\-1660](https://orcid.org/0000-0002-4916-1660)OrcID:[0000\-0001\-8149\-5890](https://orcid.org/0000-0001-8149-5890)Affiliation:Computer Science Program, Computer, Electrical, and Mathematical Sciences & Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955, Saudi ArabiaAffiliation:Computer Science Program, Computer, Electrical, and Mathematical Sciences & Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia

###### Abstract

Knowledge graph completion is evaluated by ranking observed triples above randomly corrupted ones, which treats every unobserved fact as false\. When the object being completed is a description logic knowledge base rather than a plain graph, the open world assumption and deductive closure make this inadequate: relative to the knowledge base, a candidate axiom is entailed, contradictory, or undetermined, and a model that cannot separate a logically impossible axiom from a plausible novel one is not merely less accurate but semantically incorrect\. We ask what it means for a knowledge base completion model to be logically faithful, and whether current embedding models are\. We define a hierarchy of four increasingly strict criteria, discrimination, logical admissibility, monotonic logical faithfulness, and probabilistic logical faithfulness, and prove that they form a strict chain of implications\. We ground the strongest criterion in the relative model countP⁡\(α∣𝒪\)=\#⁡\(𝒪∪\{α\}\)/\#⁡\(𝒪\)P\(\\alpha\\mid\\mathcal\{O\}\)=\\\#\(\\mathcal\{O\}\\cup\\\{\\alpha\\\}\)/\\\#\(\\mathcal\{O\}\), which recovers the trichotomy at its endpoints and ranks undetermined axioms in between\. Evaluating knowledge graph and logic\-geometric embedding models onℰ​ℒ\\mathcal\{EL\}ontologies, with entailed, contradictory, and undetermined test sets generated by a reasoner, we find that ranking accuracy does not imply logical faithfulness and that none of the evaluated models is faithful across the hierarchy\. The code is available at[https://github\.com/bio\-ontology\-research\-group/kbc](https://github.com/bio-ontology-research-group/kbc)\.

## 1Introduction

Knowledge graph completion \(KGC\) is formulated as the prediction of missing facts in the form of\(h​e​a​d,r​e​l​a​t​i​o​n,t​a​i​l\)\(head,relation,tail\)triples, identifying new edges between entities within a graph structure\[[12](https://arxiv.org/html/2609.27863#bib.bib12)\]\. Standard metrics such as Hits@nnand mean reciprocal rank \(MRR\) rank observed test triples against corrupted or otherwise unobserved ones\[[8](https://arxiv.org/html/2609.27863#bib.bib8)\], and the dominant training objective \(random negative sampling\) treats unobserved triples uniformly as negatives\. Both the training signal and the evaluation protocol thus implicitly identify “not in the KG” with “false”\.

The transition from knowledge graphs to knowledge bases \(KBC\) introduces phenomena that this collapse cannot accommodate\. Knowledge bases \(or ontologies\) include terminological axioms \(TBox\), such as concept subsumptionC⊑DC\\sqsubseteq D, equivalenceC≡DC\\equiv D, and disjointnessC⊓D⊑⊥C\\sqcap D\\sqsubseteq\\bot, and under the open world assumption \(OWA\) the absence of an axiom does not imply that it is false\. The*entailment status*of a candidate axiomα\\alpharelative to a knowledge base𝒪\\mathcal\{O\}is therefore not binary but a trichotomy:α\\alphamay be*entailed*\(provably true\),𝒪⊧α\\mathcal\{O\}\\models\\alpha;*contradictory*\(provably false\),𝒪⊧¬α\\mathcal\{O\}\\models\\neg\\alpha, equivalently𝒪∪\{α\}⊧⊥\\mathcal\{O\}\\cup\\\{\\alpha\\\}\\models\\bot; or*undetermined*\(logically possible\), neither proven nor disproven and hence a valid potential extension of the KB\.

A KBC model whose scores conflate*contradictory*with*undetermined*axioms is not merely suboptimal: it is unable to distinguish a logical impossibility from a plausible novel prediction\. By training to minimise the score of all unobserved triples via random negative sampling, mainstream KGC methods force statements in𝒮?\\mathcal\{S\}\_\{?\}, which are exactly the candidate*discoveries*a KBC system is meant to surface, into the same low\-scoring region as statements in𝒮⊥\\mathcal\{S\}\_\{\\bot\}\. This is not just a calibration problem in the sense of\[[41](https://arxiv.org/html/2609.27863#bib.bib41)\]: any downstream consumer that uses scores to filter candidates will accept logically impossible axioms as often as plausible ones in expectation\.

We address this gap by introducing a four\-level*hierarchy of faithfulness criteria*that any KB completion model can be evaluated against\. From weakest to strongest:*discrimination*\(L0\) requires every entailed axiom to outscore every contradiction, an averaged surrogate of which classical KGC metrics measure;*logical admissibility*\(L1\) requires that no contradiction outscore any consistent axiom;*monotonic logical faithfulness*\(MLF, L2\) requires the strict trichotomy orderingf⁡\(αΓ\)​¡​f​\(α?\)​¡​f​\(α\+\)f\(\\alpha^\{\-\}\)<f\(\\alpha^\{?\}\)<f\(\\alpha^\{\+\}\); and*probabilistic logical faithfulness*\(PLF, L3\) requiresffto be monotonic in the relative model countP⁡\(α∣𝒪\)=\#⁡\(𝒪∪\{α\}\)/\#⁡\(𝒪\)P\(\\alpha\\mid\\mathcal\{O\}\)=\\\#\(\\mathcal\{O\}\\cup\\\{\\alpha\\\}\)/\\\#\(\\mathcal\{O\}\), so that ranking within𝒮?\\mathcal\{S\}\_\{?\}matches the proportion of models in whichα\\alphaholds \(Section[5](https://arxiv.org/html/2609.27863#S5)gives formal definitions\)\.

Our contributions are as follows\. We formalise the KBC task as distinct from KGC under deductive closure and OWA \(Section[3](https://arxiv.org/html/2609.27863#S3)\), make the entailment trichotomy precise, and show that the relative model countP⁡\(α∣𝒪\)P\(\\alpha\\mid\\mathcal\{O\}\)recovers it at its endpoints \(P=0P=0on𝒮⊥\\mathcal\{S\}\_\{\\bot\},P=1P=1on𝒮⊧\\mathcal\{S\}\_\{\\models\}\) and orders𝒮?\\mathcal\{S\}\_\{?\}in between \(Section[4](https://arxiv.org/html/2609.27863#S4)\)\. We define the four\-level hierarchy and prove the strict implication chainPLF⇒MLF⇒admissibility⇒discrimination\\mathrm\{PLF\}\\Rightarrow\\mathrm\{MLF\}\\Rightarrow\\text\{admissibility\}\\Rightarrow\\text\{discrimination\}\(Section[5](https://arxiv.org/html/2609.27863#S5), Theorem[5\.1](https://arxiv.org/html/2609.27863#S5.Thmtheorem1)\), and identify training\-time interventions that target the faithfulness gap \(Section[6\.3](https://arxiv.org/html/2609.27863#S6.SS3)\)\. Finally, we evaluate seven geometric and logic\-geometric models against the hierarchy onℰ​ℒ\\mathcal\{EL\}ontologies, with reasoner\-generated test sets, and find that ranking accuracy does not imply faithfulness, that a model’s position on the hierarchy depends on the logical construct, and, via a negative\-sampling sweep, that substituting contradiction\-based for random negatives improves faithfulness at a cost in ranking accuracy \(Section[6](https://arxiv.org/html/2609.27863#S6)\)\.

## 2Related work

Embedding\-based KGC, including TransE\[[8](https://arxiv.org/html/2609.27863#bib.bib8)\]and the factorisation family \(DistMult\[[45](https://arxiv.org/html/2609.27863#bib.bib45)\], ComplEx\[[42](https://arxiv.org/html/2609.27863#bib.bib42)\], RESCAL\[[34](https://arxiv.org/html/2609.27863#bib.bib34)\]\), relies on graph topology and ranks observed triples against random corruptions\. That this misbehaves under the open world assumption has been argued semantically\[[10](https://arxiv.org/html/2609.27863#bib.bib10)\]and for the standard ranking metrics\[[46](https://arxiv.org/html/2609.27863#bib.bib46)\]; removing entailed triples that leak from train into test through deductive closure shows reported scores to be inflated\[[2](https://arxiv.org/html/2609.27863#bib.bib2)\]; surveys catalogue the gap between predicted scores and logical admissibility\[[29](https://arxiv.org/html/2609.27863#bib.bib29),[36](https://arxiv.org/html/2609.27863#bib.bib36)\]; and region\-based analyses show that common embedding geometries cannot capture even simple rule sets\[[17](https://arxiv.org/html/2609.27863#bib.bib17)\]\. Closer to our goal, consistency\-aware methods filter predictions that violate the ontology\[[22](https://arxiv.org/html/2609.27863#bib.bib22)\], and semantic\-aware metrics such as Sem@K score type\-constraint satisfaction in the top ranks\[[19](https://arxiv.org/html/2609.27863#bib.bib19)\]; both check a necessary admissibility condition, but neither isolates entailed axioms nor ranks the undetermined region\. Our hierarchy makes the gap quantitative: current models partially satisfy discrimination \(L0\) but fail MLF \(L2\) by construction, because the loss gives no incentive to distinguish𝒮⊥\\mathcal\{S\}\_\{\\bot\}from𝒮?\\mathcal\{S\}\_\{?\}\.

A separate line of work explicitly encodes the geometry of subsumption and disjointness: ELEmbeddings\[[25](https://arxiv.org/html/2609.27863#bib.bib25)\]mapsℰ​ℒ\\mathcal\{EL\}concepts tonn\-balls, EmEL\+\+\[[33](https://arxiv.org/html/2609.27863#bib.bib33)\]extends this to richer constructors, ELBE\[[37](https://arxiv.org/html/2609.27863#bib.bib37)\]and Box2EL\[[20](https://arxiv.org/html/2609.27863#bib.bib20)\]use boxes so that conjunction and existential restriction are handled geometrically, cone embeddings\[[35](https://arxiv.org/html/2609.27863#bib.bib35)\]target𝒜​ℒ​𝒞\\mathcal\{ALC\}, and BoxE\[[1](https://arxiv.org/html/2609.27863#bib.bib1)\]captures some inference patterns without ontology\-level subsumption\. These models can in principle satisfy levels above L0, but their training losses are still binary, positive versus corrupted, and disjointness is not separately regularised\. Our framework predicts, and the experiments below test, that the additional logical structure helps with MLF more than with PLF: ranking*within*𝒮?\\mathcal\{S\}\_\{?\}remains essentially unconstrained\.

The term*faithfulness*already appears in this literature, but in a different, representation\-level sense: an embedding is faithful if, at zero training loss, it*is*a model of the ontology and reproduces its entailments\. This is the sense of BoxEL\[[44](https://arxiv.org/html/2609.27863#bib.bib44)\], the strong\-faithfulness results forℰ​ℒ​ℋ\\mathcal\{ELH\}\[[26](https://arxiv.org/html/2609.27863#bib.bib26)\], and the semantic analysis of\[[9](https://arxiv.org/html/2609.27863#bib.bib9)\]; a query\-level variant asks an embedding to reproduce deductive query answers\[[40](https://arxiv.org/html/2609.27863#bib.bib40)\], while in interpretability “faithful” instead means an explanation reflects a model’s computation\[[21](https://arxiv.org/html/2609.27863#bib.bib21)\]\. Our criterion is orthogonal and ranking\-level: it constrains the*order*a scoring function induces over candidate axioms, not whether the embedding realises a model, and a representation\-faithful embedding may still rank an undetermined axiom no higher than a contradictory one\.

Probabilistic description logics\[[30](https://arxiv.org/html/2609.27863#bib.bib30)\], the distribution semantics for logic programs\[[39](https://arxiv.org/html/2609.27863#bib.bib39)\], and inconsistency\-tolerant query answering\[[28](https://arxiv.org/html/2609.27863#bib.bib28)\]attach probabilities or possibilistic weights to axioms\. Our PLF criterion uses a related object, the relative model count, in a different role: not as a semantics the KB itself carries, but as a*ground\-truth target*for evaluating embedding\-based scoring functions on a classical KB\.

Model counting \(\#SAT\) quantifies solution spaces\[[14](https://arxiv.org/html/2609.27863#bib.bib14)\]and, in weighted form, underlies probabilistic inference\[[11](https://arxiv.org/html/2609.27863#bib.bib11)\]\. The relative model count of Section[4\.2](https://arxiv.org/html/2609.27863#S4.SS2)is the random\-worlds degree of belief of\[[6](https://arxiv.org/html/2609.27863#bib.bib6)\]; our contribution is to use it as the*evaluation target*for embedding\-based KBC, which to our knowledge is new\.

Neuro\-symbolic frameworks such as DeepProbLog\[[31](https://arxiv.org/html/2609.27863#bib.bib31)\], NeurASP\[[47](https://arxiv.org/html/2609.27863#bib.bib47)\], Logic Tensor Networks\[[7](https://arxiv.org/html/2609.27863#bib.bib7)\], and Logical Neural Networks\[[38](https://arxiv.org/html/2609.27863#bib.bib38)\]couple differentiable models with declarative logic; because our hierarchy is defined on the scoring functionffalone, any such model can be placed on it \(Appendix[0\.E](https://arxiv.org/html/2609.27863#Pt0.A5)\)\.

## 3Preliminaries

### 3\.1Description logics

Description logics \(DLs\) are logic\-based knowledge representation formalisms that provide the semantics of the Web Ontology Language \(OWL\)\[[18](https://arxiv.org/html/2609.27863#bib.bib18)\]; most are decidable fragments of first\-order logic, often within the two\-variable fragment with counting𝒞2\\mathcal\{C\}^\{2\}\[[43](https://arxiv.org/html/2609.27863#bib.bib43)\]\.

A DL ontology is defined over a signatureΣ=\(NC,NR,NI\)\\Sigma=\(N\_\{C\},N\_\{R\},N\_\{I\}\)consisting of disjoint sets of concept namesNCN\_\{C\}\(unary predicates\), role namesNRN\_\{R\}\(binary predicates\), and individual namesNIN\_\{I\}\(constants\)\[[4](https://arxiv.org/html/2609.27863#bib.bib4)\]\. Complex concepts are constructed recursively from these primitives\. The logicℰ​ℒ\\mathcal\{EL\}\[[3](https://arxiv.org/html/2609.27863#bib.bib3)\], used in our experiments, is a tractable fragment that allows top \(⊤\\top\), conjunction \(⊓\\sqcap\), and existential restriction \(∃r\.C\\exists r\.C\); subsumption checking inℰ​ℒ\\mathcal\{EL\}runs in polynomial time\[[3](https://arxiv.org/html/2609.27863#bib.bib3)\]\.

An interpretationℐ=\(Δℐ,⋅ℐ\)\\mathcal\{I\}=\(\\Delta^\{\\mathcal\{I\}\},\\cdot^\{\\mathcal\{I\}\}\)maps individuals to elements of a non\-empty domain, concept names to subsets, and role names to binary relations, extending to complex concepts inductively in the standard way\[[4](https://arxiv.org/html/2609.27863#bib.bib4)\]\. A knowledge base𝒪=\(𝒯,ℛ,𝒜\)\\mathcal\{O\}=\(\\mathcal\{T\},\\mathcal\{R\},\\mathcal\{A\}\)comprises a non\-empty TBox𝒯\\mathcal\{T\}of terminological axioms \(GCIsC⊑DC\\sqsubseteq D, equivalencesC≡DC\\equiv D\), an RBoxℛ\\mathcal\{R\}governing role characteristics, and an ABox𝒜\\mathcal\{A\}of assertionsC⁡\(a\)C\(a\)andr⁡\(a,b\)r\(a,b\)\. An interpretation is a*model*of𝒪\\mathcal\{O\}if it satisfies every axiom; we write𝑀𝑜𝑑⁡\(𝒪\)\\mathit\{Mod\}\(\\mathcal\{O\}\)for the set of models of𝒪\\mathcal\{O\}, and we assume throughout that𝒪\\mathcal\{O\}is consistent \(i\.e\.𝑀𝑜𝑑⁡\(𝒪\)≠∅\\mathit\{Mod\}\(\\mathcal\{O\}\)\\neq\\emptyset\)\.

### 3\.2Knowledge base completion

Knowledge base completion \(KBC\) extends KGC link prediction to the expressive power of description logics\. While KGC predicts missing edges \(triples\(h,r,t\)\(h,r,t\), equivalent to ABox role assertions\)\[[29](https://arxiv.org/html/2609.27863#bib.bib29),[36](https://arxiv.org/html/2609.27863#bib.bib36)\], KBC predicts missing logical axioms that may include complex concepts and schema\-level constraints\[[5](https://arxiv.org/html/2609.27863#bib.bib5),[27](https://arxiv.org/html/2609.27863#bib.bib27),[25](https://arxiv.org/html/2609.27863#bib.bib25)\]\.

Formally, a*candidate axiom*α\\alphais any well\-formed formula overΣ\\Sigma, and a KBC method learns a scoring functionf:ℒ×𝒦→ℝf:\\mathcal\{L\}\\times\\mathcal\{K\}\\to\\mathbb\{R\}\(axioms×\\timesknowledge bases\) used to rank candidate extensions\. Unlike standard KGC,α\\alphamay be an ABox assertion \(C⁡\(a\)C\(a\),r⁡\(a,b\)r\(a,b\)\) or a TBox axiom \(C⊑DC\\sqsubseteq D,C≡DC\\equiv D, or a disjointnessC⊓D⊑⊥C\\sqcap D\\sqsubseteq\\bot\)\.

## 4The logical status of candidate axioms and relative model counting

This section establishes the two ingredients the faithfulness criteria are built on: the three\-way logical status of a candidate axiom under the open world assumption, and a graded, computable refinement of that status given by relative model counting\.

### 4\.1The logical status of candidate axioms

Because KBC operates under the open world assumption with full deductive closure, the entailment status of a candidate axiom is not binary\. We adopt the following standard trichotomy\[[4](https://arxiv.org/html/2609.27863#bib.bib4)\]\.

###### Definition 1\(Logical status under OWA\)

For a consistent𝒪\\mathcal\{O\}, every candidate axiomα\\alphais exactly one of:*entailed*\(provably true\),𝒪⊧α\\mathcal\{O\}\\models\\alpha, iff𝑀𝑜𝑑⁡\(𝒪\)⊆𝑀𝑜𝑑⁡\(α\)\\mathit\{Mod\}\(\\mathcal\{O\}\)\\subseteq\\mathit\{Mod\}\(\\alpha\);*contradictory*\(provably false\),𝒪⊧¬α\\mathcal\{O\}\\models\\neg\\alpha\(equivalently𝒪∪\{α\}⊧⊥\\mathcal\{O\}\\cup\\\{\\alpha\\\}\\models\\bot\), iff𝑀𝑜𝑑⁡\(𝒪\)∩𝑀𝑜𝑑⁡\(α\)=∅\\mathit\{Mod\}\(\\mathcal\{O\}\)\\cap\\mathit\{Mod\}\(\\alpha\)=\\emptyset; or*undetermined*\(contingent\), iff𝒪⊧̸α\\mathcal\{O\}\\not\\models\\alphaand𝒪⊧̸¬α\\mathcal\{O\}\\not\\models\\neg\\alpha, equivalently both𝑀𝑜𝑑⁡\(𝒪\)∩𝑀𝑜𝑑⁡\(α\)≠∅\\mathit\{Mod\}\(\\mathcal\{O\}\)\\cap\\mathit\{Mod\}\(\\alpha\)\\neq\\emptysetand𝑀𝑜𝑑⁡\(𝒪\)∖𝑀𝑜𝑑⁡\(α\)≠∅\\mathit\{Mod\}\(\\mathcal\{O\}\)\\setminus\\mathit\{Mod\}\(\\alpha\)\\neq\\emptyset\.

The set of candidate axioms then partitions into three disjoint classes𝒮⊧​\(𝒪\)\\mathcal\{S\}\_\{\\models\}\(\\mathcal\{O\}\),𝒮⊥​\(𝒪\)\\mathcal\{S\}\_\{\\bot\}\(\\mathcal\{O\}\), and𝒮?​\(𝒪\)\\mathcal\{S\}\_\{?\}\(\\mathcal\{O\}\); when𝒪\\mathcal\{O\}is fixed by context we drop the argument\. Standard KGC training generates negatives by uniform corruption from𝒮⊥∪𝒮?\\mathcal\{S\}\_\{\\bot\}\\cup\\mathcal\{S\}\_\{?\}and suppresses them without distinction; fixing this conflation is the premise of the paper\.

### 4\.2Bounded\-domain model counting

Let𝒪\\mathcal\{O\}be over signatureΣ=\(NC,NR,NI\)\\Sigma=\(N\_\{C\},N\_\{R\},N\_\{I\}\)withj​NI⊧n\|N\_\{I\}\\models n, and fix a finite domainΔ=\{a1,…,an\}\\Delta=\\\{a\_\{1\},\\ldots,a\_\{n\}\\\}identified with the named individuals\. The*Herbrand base*ofΣ\\SigmaoverΔ\\Deltais

ℋ​ℬ​\(Σ,Δ\)=⋃C∈NC\{C⁡\(ai\)\}i=1n∪⋃r∈NR\{r⁡\(ai,aj\)\}i,j=1n,\\mathcal\{HB\}\(\\Sigma,\\Delta\)=\\bigcup\_\{C\\in N\_\{C\}\}\\\{C\(a\_\{i\}\)\\\}\_\{i=1\}^\{n\}\\;\\cup\\;\\bigcup\_\{r\\in N\_\{R\}\}\\\{r\(a\_\{i\},a\_\{j\}\)\\\}\_\{i,j=1\}^\{n\},and a*Herbrand interpretation*is any subsetI⊆ℋ​ℬ​\(Σ,Δ\)I\\subseteq\\mathcal\{HB\}\(\\Sigma,\\Delta\)\. We write𝑀𝑜𝑑H​\(𝒪\)\\mathit\{Mod\}\_\{H\}\(\\mathcal\{O\}\)for the set of Herbrand interpretations satisfying every axiom of𝒪\\mathcal\{O\}, with concept\-language constructors interpreted in the standard way over the finite domainΔ\\Delta\[[6](https://arxiv.org/html/2609.27863#bib.bib6)\]and metalevel negation as set complement inℋ​ℬ\\mathcal\{HB\}:¬A⁡\(a\)≡a∉AI\\neg A\(a\)\\equiv a\\notin A^\{I\}\.

Restricting to Herbrand interpretations on the named individuals is a domain\-closure assumption that differs from the open\-domain Tarskian semantics of DLs; we adopt it because it makes\#⁡\(𝒪\)\\\#\(\\mathcal\{O\}\)finite and well\-defined while preserving satisfiability for fragments with the finite model property \(ℰ​ℒ,𝒜​ℒ​𝒞\\mathcal\{EL\},\\mathcal\{ALC\}, and others\) at sufficiently largej​Δ​j\|\\Delta\|, and because the resulting relative count is a plausibility*target*, not a redefinition of entailment, which we still take classically\.

### 4\.3Relative model count

Define\#⁡\(𝒪\)=j​𝑀𝑜𝑑H​\(𝒪\)​j\\\#\(\\mathcal\{O\}\)=\|\\mathit\{Mod\}\_\{H\}\(\\mathcal\{O\}\)\|and, for any candidate axiomα\\alpha,

P⁡\(α∣𝒪\)=\#⁡\(𝒪∪\{α\}\)\#⁡\(𝒪\)∈\[0,1\]\.P\(\\alpha\\mid\\mathcal\{O\}\)\\;=\\;\\frac\{\\\#\(\\mathcal\{O\}\\cup\\\{\\alpha\\\}\)\}\{\\\#\(\\mathcal\{O\}\)\}\\;\\in\\;\[0,1\]\.\(1\)
###### Proposition 1\(Boundary conditions\)

For any consistent𝒪\\mathcal\{O\}and any candidateα\\alpha, we haveP⁡\(α∣𝒪\)=1P\(\\alpha\\mid\\mathcal\{O\}\)=1iff𝑀𝑜𝑑H​\(𝒪\)⊆𝑀𝑜𝑑H​\(α\)\\mathit\{Mod\}\_\{H\}\(\\mathcal\{O\}\)\\subseteq\\mathit\{Mod\}\_\{H\}\(\\alpha\)\(Herbrand entailment\);P⁡\(α∣𝒪\)=0P\(\\alpha\\mid\\mathcal\{O\}\)=0iff𝑀𝑜𝑑H​\(𝒪\)∩𝑀𝑜𝑑H​\(α\)=∅\\mathit\{Mod\}\_\{H\}\(\\mathcal\{O\}\)\\cap\\mathit\{Mod\}\_\{H\}\(\\alpha\)=\\emptyset\(Herbrand contradiction\); andP⁡\(α∣𝒪\)∈\(0,1\)P\(\\alpha\\mid\\mathcal\{O\}\)\\in\(0,1\)iffα\\alphais Herbrand\-undetermined w\.r\.t\.𝒪\\mathcal\{O\}\.

The proof is immediate from \([1](https://arxiv.org/html/2609.27863#S4.E1)\)\. For DL fragments with the finite model property andj​Δ​j\|\\Delta\|taken large enough \(polynomial inj​𝒪​j\|\\mathcal\{O\}\|forℰ​ℒ\\mathcal\{EL\}\[[3](https://arxiv.org/html/2609.27863#bib.bib3)\]\), Herbrand entailment coincides with classical entailment, so the boundary conditions also hold for the trichotomy\(𝒮⊧,𝒮⊥,𝒮?\)\(\\mathcal\{S\}\_\{\\models\},\\mathcal\{S\}\_\{\\bot\},\\mathcal\{S\}\_\{?\}\)as defined in Section[4\.1](https://arxiv.org/html/2609.27863#S4.SS1)\.

Inside𝒮?\\mathcal\{S\}\_\{?\},P⁡\(α∣𝒪\)∈\(0,1\)P\(\\alpha\\mid\\mathcal\{O\}\)\\in\(0,1\)grades how constrainingα\\alphais: near11it holds in almost every model \(“almost entailed”\), near00in only a sliver \(“almost contradictory”\)\. PLF requiresffto respect this ranking\.

Exact model counting is \#P\-complete\[[14](https://arxiv.org/html/2609.27863#bib.bib14)\], so the tractability lever is the size of the bounded Herbrand domain rather than the counting algorithm\. Appendix[0\.A](https://arxiv.org/html/2609.27863#Pt0.A1)details the exact and approximate counting regimes and works a complete example:𝒪=\{A⊑B\}\\mathcal\{O\}=\\\{A\\sqsubseteq B\\\}overj​Δ⊧2\|\\Delta\\models 2gives\#⁡\(𝒪\)=576\\\#\(\\mathcal\{O\}\)=576, and adding∃r\.A⊑C\\exists r\.A\\sqsubseteq Cgives\#⁡\(𝒪\+\)=425\\\#\(\\mathcal\{O\}^\{\+\}\)=425, henceP=425/576≈0\.74P=425/576\\approx 0\.74\.

## 5A hierarchy of faithfulness criteria

Throughout this section, fix a consistent KB𝒪\\mathcal\{O\}and a scoring functionf⁡\(⋅,𝒪\):ℒ→ℝf\(\\cdot,\\mathcal\{O\}\):\\mathcal\{L\}\\to\\mathbb\{R\}provided by some completion model\. Higher scores indicate higher predicted plausibility\. We define four progressively stronger properties offf\.

### 5\.1The four criteria

We define four criteria of increasing strictness, each a property of the scoring functionffevaluated on the trichotomy\(𝒮⊧,𝒮⊥,𝒮?\)\(\\mathcal\{S\}\_\{\\models\},\\mathcal\{S\}\_\{\\bot\},\\mathcal\{S\}\_\{?\}\)\. Each criterion strictly implies the one before it, as Theorem[5\.1](https://arxiv.org/html/2609.27863#S5.Thmtheorem1)makes formal\.

###### Definition 2\(Discrimination\)

ffis*discriminative*on𝒪\\mathcal\{O\}iff for allα\+∈𝒮⊧\\alpha^\{\+\}\\in\\mathcal\{S\}\_\{\\models\}andαΓ∈𝒮⊥\\alpha^\{\-\}\\in\\mathcal\{S\}\_\{\\bot\},f⁡\(α\+,𝒪\)\>f⁡\(αΓ,𝒪\)f\(\\alpha^\{\+\},\\mathcal\{O\}\)\>f\(\\alpha^\{\-\},\\mathcal\{O\}\)\.

Classical KGC metrics \(MRR, Hits@kk\) approximate this and are silent about𝒮?\\mathcal\{S\}\_\{?\}\.

###### Definition 3\(Logical admissibility\)

ffis*logically admissible*on𝒪\\mathcal\{O\}iff no contradiction outranks any consistent axiom, i\.e\. for allαΓ∈𝒮⊥\\alpha^\{\-\}\\in\\mathcal\{S\}\_\{\\bot\}and allβ∈𝒮⊧∪𝒮?\\beta\\in\\mathcal\{S\}\_\{\\models\}\\cup\\mathcal\{S\}\_\{?\},f⁡\(αΓ,𝒪\)​¡​f​\(β,𝒪\)f\(\\alpha^\{\-\},\\mathcal\{O\}\)<f\(\\beta,\\mathcal\{O\}\); equivalently,maxαΓ∈𝒮⊥⁡f⁡\(αΓ,𝒪\)​¡​minβ∈𝒮⊧∪𝒮?⁡f⁡\(β,𝒪\)\\max\_\{\\alpha^\{\-\}\\in\\mathcal\{S\}\_\{\\bot\}\}f\(\\alpha^\{\-\},\\mathcal\{O\}\)<\\min\_\{\\beta\\in\\mathcal\{S\}\_\{\\models\}\\cup\\mathcal\{S\}\_\{?\}\}f\(\\beta,\\mathcal\{O\}\)\.

Equivalently, there is a thresholdθ\\thetaat which the accepted set\{α:f⁡\(α,𝒪\)≥θ\}\\\{\\alpha:f\(\\alpha,\\mathcal\{O\}\)\\geq\\theta\\\}contains every consistent candidate and no contradiction: a practitioner taking the top\-ranked axioms for review gets*no contradictions*among them\. Admissibility is strictly stronger than discrimination, which constrains only𝒮⊧\\mathcal\{S\}\_\{\\models\}versus𝒮⊥\\mathcal\{S\}\_\{\\bot\}and is silent on𝒮?\\mathcal\{S\}\_\{?\}\. We report the empiricalAdm​@​k\\mathrm\{Adm@\}kin Section[6](https://arxiv.org/html/2609.27863#S6)\.

###### Definition 4\(Monotonic logical faithfulness, MLF\)

ffis*monotonically logically faithful*on𝒪\\mathcal\{O\}iff for every\(αΓ,α?,α\+\)\(\\alpha^\{\-\},\\alpha^\{?\},\\alpha^\{\+\}\)in𝒮⊥×𝒮?×𝒮⊧\\mathcal\{S\}\_\{\\bot\}\\times\\mathcal\{S\}\_\{?\}\\times\\mathcal\{S\}\_\{\\models\},

f⁡\(αΓ,𝒪\)​¡​f​\(α?,𝒪\)​¡​f​\(α\+,𝒪\)\.f\(\\alpha^\{\-\},\\mathcal\{O\}\)<f\(\\alpha^\{?\},\\mathcal\{O\}\)<f\(\\alpha^\{\+\},\\mathcal\{O\}\)\.

MLF is strictly stronger than admissibility: admissibility only keeps𝒮⊥\\mathcal\{S\}\_\{\\bot\}below the consistent axioms as a block, whereas MLF inserts𝒮?\\mathcal\{S\}\_\{?\}as a distinct middle band ordered strictly below𝒮⊧\\mathcal\{S\}\_\{\\models\}\.

MLF is purely ordinal across classes and imposes no order*within*𝒮?\\mathcal\{S\}\_\{?\}, yet two undetermined axioms can differ sharply in plausibility: one holding in999,000999\{,\}000of a KB’s million models is “almost entailed”, one holding in only1,0001\{,\}000“almost contradictory”\. The relative model count captures this\.

###### Definition 5\(Probabilistic logical faithfulness, PLF\)

ffis*probabilistically logically faithful*on𝒪\\mathcal\{O\}iff for every pair of candidate axiomsα1,α2\\alpha\_\{1\},\\alpha\_\{2\},

P⁡\(α1∣𝒪\)\>P⁡\(α2∣𝒪\)⟹f⁡\(α1,𝒪\)\>f⁡\(α2,𝒪\),P\(\\alpha\_\{1\}\\mid\\mathcal\{O\}\)\>P\(\\alpha\_\{2\}\\mid\\mathcal\{O\}\)\\;\\Longrightarrow\\;f\(\\alpha\_\{1\},\\mathcal\{O\}\)\>f\(\\alpha\_\{2\},\\mathcal\{O\}\),whereP⁡\(α∣𝒪\)=\#⁡\(𝒪∪\{α\}\)/\#⁡\(𝒪\)P\(\\alpha\\mid\\mathcal\{O\}\)=\\\#\(\\mathcal\{O\}\\cup\\\{\\alpha\\\}\)/\\\#\(\\mathcal\{O\}\)is the relative model count over a fixed finite\-domain semantics \(Section[4\.2](https://arxiv.org/html/2609.27863#S4.SS2)\)\.

PLF is strictly stronger than MLF: it is silent only on pairs with identical relative model counts \(and there is noff\-constraint within𝒮⊧\\mathcal\{S\}\_\{\\models\}or within𝒮⊥\\mathcal\{S\}\_\{\\bot\}themselves, where the count is constant at 1 and 0 respectively\)\.

### 5\.2The hierarchy theorem

The four criteria differ in operational feasibility\. L0–L2 are testable on any ontology whose entailment is decidable, using reasoner\-generated𝒮⊧\\mathcal\{S\}\_\{\\models\},𝒮?\\mathcal\{S\}\_\{?\},𝒮⊥\\mathcal\{S\}\_\{\\bot\}samples; L3 additionally requires the\#​P\\\#P\-complete relative model countP⁡\(α∣𝒪\)P\(\\alpha\\mid\\mathcal\{O\}\), so we treat it as a benchmark criterion rather than a runtime property \(Section[6](https://arxiv.org/html/2609.27863#S6)\)\.

###### Theorem 5\.1\(Strict hierarchy\)

Let𝒪\\mathcal\{O\}be consistent, assume the relative model countP\(⋅∣𝒪\)P\(\\cdot\\mid\\mathcal\{O\}\)is well\-defined \(Section[4\.2](https://arxiv.org/html/2609.27863#S4.SS2)\), and assume𝒮⊧\\mathcal\{S\}\_\{\\models\},𝒮⊥\\mathcal\{S\}\_\{\\bot\}, and𝒮?\\mathcal\{S\}\_\{?\}are all non\-empty \(otherwise the criteria degenerate\)\. Then

PLF⟹MLF⟹Logical​Admissibility⟹Discrimination,\\mathrm\{PLF\}\\;\\Longrightarrow\\;\\mathrm\{MLF\}\\;\\Longrightarrow\\;\\mathrm\{Logical\\;Admissibility\}\\;\\Longrightarrow\\;\\mathrm\{Discrimination\},and none of the implications can be reversed: there exist scoring functions satisfying any one level but failing the next stronger one\.

A proof is given in Appendix[0\.B](https://arxiv.org/html/2609.27863#Pt0.A2):PLF⇒MLF\\mathrm\{PLF\}\\Rightarrow\\mathrm\{MLF\}follows from the boundary valuesP⁡\(αΓ\)=0P\(\\alpha^\{\-\}\)=0,P⁡\(α\+\)=1P\(\\alpha^\{\+\}\)=1and0​¡​P​\(α?\)​¡​10<P\(\\alpha^\{?\}\)<1; the remaining forward implications are ordinal, and the non\-reversals use small explicit constructions\. See Appendix[0\.D](https://arxiv.org/html/2609.27863#Pt0.A4)for a working example\.

## 6Experimental evaluation

### 6\.1Setup

We organise the evaluation around three questions\. RQ1: do existing KGC and KBC models satisfy MLF, and at what rate do they violate the ordering across𝒮⊧,𝒮?,𝒮⊥\\mathcal\{S\}\_\{\\models\},\\mathcal\{S\}\_\{?\},\\mathcal\{S\}\_\{\\bot\}? RQ2: conditional on L0, how large is the residual confusion of𝒮?\\mathcal\{S\}\_\{?\}with𝒮⊥\\mathcal\{S\}\_\{\\bot\}, the failure mode our critique targets? RQ3: on fragments whereP⁡\(α∣𝒪\)P\(\\alpha\\mid\\mathcal\{O\}\)is exactly computable, do logic\-geometric models satisfy PLF, i\.e\. rank𝒮?\\mathcal\{S\}\_\{?\}by relative model count?

We use two ontologies spanning the model\-counting tractability spectrum: the Pizza fragment, anℰ​ℒ\\mathcal\{EL\}version of the well\-known Pizza ontology small enough that\#⁡\(𝒪\)\\\#\(\\mathcal\{O\}\)is feasible to compute exactly via \#SAT \(RQ3\); and the GO\-plus slice, a connected slice of the Gene Ontology \(go\-plus\) ofO⁡\(103\)O\(10^\{3\}\)classes, whose trichotomy is computed via ELK\[[23](https://arxiv.org/html/2609.27863#bib.bib23)\]and whose relative model counts are computed exactly withpyganak\(Ganak\) over the same bounded Herbrand domain \(RQ1–RQ3\)\. We use bounded Herbrand domain sizej​Δ⊧2\|\\Delta\\models 2for both ontologies, matching Appendix[0\.A](https://arxiv.org/html/2609.27863#Pt0.A1)’s worked example: the smallest domain admitting non\-trivial role instantiation while keeping exact \#SAT tractable over the full test suite\. The boundary conditions of Proposition[1](https://arxiv.org/html/2609.27863#Thmproposition1)hold at this size for theℰ​ℒ\\mathcal\{EL\}axioms considered, and PLF requires only the ranking ofPP\-values on𝒮?\\mathcal\{S\}\_\{?\}, which is stable acrossj​Δ​j\|\\Delta\|choices\.

We evaluate geometric KGC models \(TransE\[[8](https://arxiv.org/html/2609.27863#bib.bib8)\], DistMult, ComplEx\), a hybrid model \(BoxE\[[1](https://arxiv.org/html/2609.27863#bib.bib1)\]\), and logic\-geometric KBC models \(ELEmbeddings\[[25](https://arxiv.org/html/2609.27863#bib.bib25)\], EmEL\+\+\[[33](https://arxiv.org/html/2609.27863#bib.bib33)\], Box2EL\[[20](https://arxiv.org/html/2609.27863#bib.bib20)\]\)\.

We use the ELEmbeddings and EmEL\+\+ losses as in their reference implementations, which deviate in two normal forms from the formulas as originally published\.111The released code omits the superclass\-radius term of the conjunctive\-subsumption loss \(Eq\. \(2\) of\[[25](https://arxiv.org/html/2609.27863#bib.bib25)\]\) and uses a corrected loss for∃r\.C⊑D\\exists r\.C\\sqsubseteq D; Appendix[0\.C](https://arxiv.org/html/2609.27863#Pt0.A3)gives the details\. We report the implementations as released and widely used\.We therefore do not attribute the conjunctive\-subsumption behaviour of ball\-based models to geometry alone, and anchor the cross\-model comparison on the unaffected box\-based and translational models\.

We sample test setsT\+⊂𝒮⊧T\_\{\+\}\\subset\\mathcal\{S\}\_\{\\models\},T?⊂𝒮?T\_\{?\}\\subset\\mathcal\{S\}\_\{?\},TΓ⊂𝒮⊥T\_\{\-\}\\subset\\mathcal\{S\}\_\{\\bot\}via the reasoner and report one metric per level\. For the GO\-plus slice, no negatives were generated for GCI1, GCI2, and GCI3 normal forms, so we report full metrics only for GCI0 and GCI1⊥settings in this case\.AUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}, the AUC offfseparatingT\+T\_\{\+\}fromTΓT\_\{\-\}, tests L0;Adm​@​k\\mathrm\{Adm\}@k, the fraction of the topkknot in𝒮⊥\\mathcal\{S\}\_\{\\bot\}, averaged over ranking queries, tests L1; the faithfulness violation rateFVR=PrTΓ×T?×T\+⁡\[¬\(f⁡\(αΓ\)​¡​f​\(α?\)​¡​f​\(α\+\)\)\]\\mathrm\{FVR\}=\\Pr\_\{T\_\{\-\}\\times T\_\{?\}\\times T\_\{\+\}\}\[\\neg\(f\(\\alpha^\{\-\}\)<f\(\\alpha^\{?\}\)<f\(\\alpha^\{\+\}\)\)\]tests L2, with the directionalPair\-FVR=PrTΓ×T?\[f\(α?\)≤f\(αΓ\)\]\\text\{Pair\-FVR\}=\\Pr\_\{T\_\{\-\}\\times T\_\{?\}\}\[f\(\\alpha^\{?\}\)\\leq f\(\\alpha^\{\-\}\)\]isolating the𝒮?\\mathcal\{S\}\_\{?\}\-versus\-𝒮⊥\\mathcal\{S\}\_\{\\bot\}confusion; and the Spearman correlationρP\\rho\_\{P\}betweenffandP\(⋅∣𝒪\)P\(\\cdot\\mid\\mathcal\{O\}\)onT?T\_\{?\}tests L3\. Standard MRR and Hits@kkonT\+T\_\{\+\}are a ranking\-accuracy sanity check\. In all reported tables, column headers are marked↑\\uparrow\(higher is better\) or↓\\downarrow\(lower is better\)\.

For each model and dataset pair we train on𝒜\\mathcal\{A\}\(and𝒯\\mathcal\{T\}where the model accepts it\), evaluate on the three classes above, and report all metrics\. Ontology sizes and the per\-normal\-form test\-set composition are given in Appendix[0\.F](https://arxiv.org/html/2609.27863#Pt0.A6)\. Each configuration is trained over multiple random seeds; the body tables report the mean and the standard deviation across seeds, full results are contained in Appendix[0\.G](https://arxiv.org/html/2609.27863#Pt0.A7)\. The effects we report are large relative to this seed\-to\-seed variation, which is small throughout \(standard deviations are mostly below0\.050\.05\)\. The training\-time intervention sweep of Section[6\.3](https://arxiv.org/html/2609.27863#S6.SS3)is the one experiment we still run at a single seed, as noted there\.

### 6\.2Results

Table 1:Faithfulness across models on atomic subsumption \(C⊑DC\\sqsubseteq D, GCI0\) over the Pizza ontology\. HigherAUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}andρP\\rho\_\{P\}are better; Pair\-FVR near0\.50\.5means𝒮?\\mathcal\{S\}\_\{?\}is not separated from𝒮⊥\\mathcal\{S\}\_\{\\bot\}\. Ranking accuracy \(MRR, Hits@10\) does not predict faithfulness\. Full metrics, the GO\-plus slice, and all normal forms are in Appendix[0\.G](https://arxiv.org/html/2609.27863#Pt0.A7)\.The one normal form on which the ball\-based models invert rather than merely underperform is conjunctive subsumption \(GCI1\), where ELEmbeddings and EmEL\+\+ fall well below chance \(AUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}near0\.020\.02–0\.050\.05\); an AUC this close to00signals near\-total inversion, with the loss ordering entailments and contradictions almost perfectly backwards rather than merely failing to separate them\. These are exactly the two models whose released loss forC⊓D⊑EC\\sqcap D\\sqsubseteq Eignores the size of the right\-hand concept \(Appendix[0\.C](https://arxiv.org/html/2609.27863#Pt0.A3)\), so a contradictory axiom with a centrally placed superclass can outscore an entailed one\. Box2EL, which represents conjunction through the exact intersection of axis\-aligned boxes, is far less affected \(AUC\+/Γ=0\.39\\mathrm\{AUC\}\_\{\+/\-\}=0\.39\), so the collapse tracks the loss as implemented rather than an intrinsic limit of region\-based embeddings\.

Two findings stand out\. First, faithfulness is a distinct axis from ranking accuracy: in Table[1](https://arxiv.org/html/2609.27863#S6.T1)the ordering of models by MRR \(TransE, Box2EL, ELEmbeddings\) does not match their ordering by Pair\-FVR orρP\\rho\_\{P\}, and TransE attains the highest MRR yet scores𝒮?\\mathcal\{S\}\_\{?\}no higher than𝒮⊥\\mathcal\{S\}\_\{\\bot\}\. Logic\-geometric models reach higherAUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}andρP\\rho\_\{P\}than the geometric baselines, but on atomic subsumption even the best leave Pair\-FVR near0\.50\.5, so they do not cleanly satisfy L2\. Second, where a model sits on the hierarchy depends on the logical construct \(Table[3](https://arxiv.org/html/2609.27863#Pt0.A7.T3), Appendix[0\.G](https://arxiv.org/html/2609.27863#Pt0.A7)\): disjointness is handled faithfully, conjunctive subsumption inverts, and atomic and existential subsumption are only weakly separated\. No setting reachesρP\\rho\_\{P\}close to11, so probabilistic logical faithfulness \(L3\) is met by none of the current models, consistent with the absence of any model\-count signal in their training objectives\.

### 6\.3Training\-time interventions

The hierarchy also indicates where to intervene during training\. We sketch three directions, specialising two existing lines, type\-constrained negative sampling\[[24](https://arxiv.org/html/2609.27863#bib.bib24)\]and rule\-injected embeddings\[[15](https://arxiv.org/html/2609.27863#bib.bib15),[16](https://arxiv.org/html/2609.27863#bib.bib16)\], to the trichotomy\.*Type\-aware negative sampling*\(L1–L2\) replaces uniform corruption with a stratified sampler that draws a fractionp⊥p\_\{\\bot\}of negatives from𝒮⊥\\mathcal\{S\}\_\{\\bot\}via cheap reasoning shortcuts \(asserted disjointness, domain/range violations\), introducing an explicit signal that separates𝒮⊥\\mathcal\{S\}\_\{\\bot\}from𝒮?\\mathcal\{S\}\_\{?\}\. Two further directions modify the loss rather than the sampler: a*contradiction margin loss*\(L2\) that penalises scores of sampled contradictions below a threshold, and an*entailed\-positives objective*\(L0–L1\) that adds entailed axioms as further positives\. None targets PLF directly, since ranking within𝒮?\\mathcal\{S\}\_\{?\}would require the loss to be informed by relative model counts, intractable for non\-toy𝒪\\mathcal\{O\}; the two loss\-based interventions are left for future work\.

We evaluate the stratified sampler by sweeping the mixture for each logic\-geometric model on the Pizza ontology fromp⊥=1p\_\{\\bot\}=1\(all negatives from𝒮⊥\\mathcal\{S\}\_\{\\bot\}\) top⊥=0p\_\{\\bot\}=0\(all random corruptions\), per normal form \(GCI0, GCI2, GCI1⊥, GCI3; GCI1 excluded, its failure driven by the released losses, Appendix[0\.C](https://arxiv.org/html/2609.27863#Pt0.A3)\)\. The endpoint effect is uniform: in all 12 model–construct settings \(Figure[1](https://arxiv.org/html/2609.27863#Pt0.A7.F1)\), training only on contradiction negatives yields lower FVR than training only on random corruptions, from noise\-level on GCI0 for the ball\-based models \(≤0\.01\\leq 0\.01\) to0\.130\.13absolute for Box2EL on GCI3\. A five\-seed replication of the full sweep \(Appendix[0\.G](https://arxiv.org/html/2609.27863#Pt0.A7), Tables[14](https://arxiv.org/html/2609.27863#Pt0.A7.T14)–[25](https://arxiv.org/html/2609.27863#Pt0.A7.T25)\) confirms this endpoint contrast on every panel\. The response is not monotone in between, so the sweep supports the endpoint contrast, not a calibrated mixing schedule\. MRR mostly moves the opposite way \(6 of 12 settings\), reproducing the rank–faithfulness dissociation of Table[1](https://arxiv.org/html/2609.27863#S6.T1)as an intervention within a single model\. Even atp⊥=1p\_\{\\bot\}=1, FVR stays above0\.430\.43for most settings: the sampler supplies the missing𝒮⊥\\mathcal\{S\}\_\{\\bot\}\-versus\-𝒮?\\mathcal\{S\}\_\{?\}signal but does not by itself make a model faithful\.

## 7Discussion and conclusion

The move from KGC to KBC is a change of evaluation target, not just of dataset: once OWA and deductive closure are in scope, ranking observed triples above random corruptions is no longer sufficient, and the hierarchy says what*is*required and at which strictness\.

The four criteria are operationalized on two tiers\. L0–L2 depend only on the reasoner\-computable trichotomy\(𝒮⊧,𝒮?,𝒮⊥\)\(\\mathcal\{S\}\_\{\\models\},\\mathcal\{S\}\_\{?\},\\mathcal\{S\}\_\{\\bot\}\)and are testable on any ontology whose entailment is decidable; they are practitioner tools, applicable at deployment scale\. L3 additionally requires the relative model countP⁡\(α∣𝒪\)P\(\\alpha\\mid\\mathcal\{O\}\), which is\#​P\\\#P\-complete: exact via\#\\\#SAT on small ontologies, approximate via ApproxMC \(Appendix[0\.A](https://arxiv.org/html/2609.27863#Pt0.A1)\) on medium ones, and infeasible at scale\. This is a design property rather than a limitation: the hierarchy orders the*strength*of ordinal constraints onff, while operational feasibility is a separate axis on which L3 sits at the benchmark end, providing a reference criterion on ontologies whereP⁡\(α∣𝒪\)P\(\\alpha\\mid\\mathcal\{O\}\)is computable, against which models trained on any ontology can be certified or falsified\. The tension between the bounded Herbrand semantics ofPPand the open\-domain semantics of DLs is by construction:PPis a plausibility target, not a redefinition of entailment \(Section[4\.2](https://arxiv.org/html/2609.27863#S4.SS2)\), aligned with the classical trichotomy on𝒮⊧\\mathcal\{S\}\_\{\\models\}and𝒮⊥\\mathcal\{S\}\_\{\\bot\}by Proposition[1](https://arxiv.org/html/2609.27863#Thmproposition1)\. Our scope isℰ​ℒ\\mathcal\{EL\}, with𝒜​ℒ​𝒞\\mathcal\{ALC\}/𝒮​ℛ​𝒪​ℐ​𝒬\\mathcal\{SROIQ\}and broaderℰ​ℒ/ℰ​ℒ\+⁣\+\\mathcal\{EL\}/\\mathcal\{EL\}^\{\+\+\}validation as future work; samplingTΓT\_\{\-\}requires disjointness axioms, absent from natural candidates such as GALEN\.

The hierarchy is defined on the scoring function alone, so any KBC method can be placed on it, including neuro\-symbolic frameworks whose logical layer differs from the geometric embeddings evaluated here; Appendix[0\.E](https://arxiv.org/html/2609.27863#Pt0.A5)places representative frameworks \(DeepProbLog, NeurASP, LTN, LNN\) on the hierarchy and identifies what each would need to reach MLF and PLF\.

We have argued that knowledge base completion requires criteria that respect the OWA trichotomy, and organised them into a hierarchy grounded at the top in model counting\. Empirically, current models cluster near the bottom, satisfying discrimination but violating MLF; where a model sits depends on the logical construct; and the random\-corruption sampler is itself implicated: contradiction negatives improve faithfulness at a cost in ranking accuracy\. The broader programme is to make KBC*about*logical correctness, not ranking in a TBox\-stripped graph\.

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## Appendix 0\.AModel counting example \(extended\)

In our experiments we compute exact \#SAT withGanak\(viapyganak\) over a bounded domain for both the Pizza fragment and the GO\-plus slice, so an exact relative count is available for every test axiom; the GO\-plus slice is kept tractable by the domain bound, not by approximation\. For ontologies beyond this scale two fallbacks are available but not used here: approximate counting \(ApproxMC, with multiplicative\(1±ε\)\(1\\pm\\varepsilon\)error at confidence1​Γ​δ1\-\\delta\), and, when even that is infeasible, only the trichotomy\(𝒮⊧,𝒮⊥,𝒮?\)\(\\mathcal\{S\}\_\{\\models\},\\mathcal\{S\}\_\{\\bot\},\\mathcal\{S\}\_\{?\}\)via a DL reasoner such as ELK or HermiT\[[13](https://arxiv.org/html/2609.27863#bib.bib13),[23](https://arxiv.org/html/2609.27863#bib.bib23)\], which suffices for MLF but not for PLF beyond bounded samples\.

We now work the example cited in Section[4\.3](https://arxiv.org/html/2609.27863#S4.SS3)\. Let𝒪=\{A⊑B\}\\mathcal\{O\}=\\\{A\\sqsubseteq B\\\}over signatureΣ=\(NC,NR,NI\)\\Sigma=\(N\_\{C\},N\_\{R\},N\_\{I\}\)withNC=\{A,B,C\}N\_\{C\}=\\\{A,B,C\\\},NR=\{r\}N\_\{R\}=\\\{r\\\},NI=\{a1,a2\}N\_\{I\}=\\\{a\_\{1\},a\_\{2\}\\\}, and Herbrand domainΔ=\{a1,a2\}\\Delta=\\\{a\_\{1\},a\_\{2\}\\\}\. The Herbrand base contains

ℋ​ℬ​\(Σ,Δ\)=\{A⁡\(ai\),B⁡\(ai\),C⁡\(ai\)\}i=1,2∪\{r⁡\(ai,aj\)\}i,j=1,2,\\mathcal\{HB\}\(\\Sigma,\\Delta\)=\\\{A\(a\_\{i\}\),B\(a\_\{i\}\),C\(a\_\{i\}\)\\\}\_\{i=1,2\}\\;\\cup\\;\\\{r\(a\_\{i\},a\_\{j\}\)\\\}\_\{i,j=1,2\},i\.e\. ten ground atoms\.

First, we compute\#⁡\(𝒪\)\\\#\(\\mathcal\{O\}\)\. ReadingA⊑BA\\sqsubseteq Bclausally as∀i\.¬A⁡\(ai\)∨B⁡\(ai\)\\forall i\.\\,\\neg A\(a\_\{i\}\)\\lor B\(a\_\{i\}\)yields two clauses, each over the two atoms\{A⁡\(ai\),B⁡\(ai\)\}\\\{A\(a\_\{i\}\),B\(a\_\{i\}\)\\\}\. Each clause has33satisfying assignments out of44, and the two clauses share no atoms, so there are99valid assignments to\{A⁡\(a1\),A⁡\(a2\),B⁡\(a1\),B⁡\(a2\)\}\\\{A\(a\_\{1\}\),A\(a\_\{2\}\),B\(a\_\{1\}\),B\(a\_\{2\}\)\\\}\. The remaining six atoms

\{C⁡\(a1\),C⁡\(a2\),r⁡\(a1,a1\),r⁡\(a1,a2\),r⁡\(a2,a1\),r⁡\(a2,a2\)\}\\\{C\(a\_\{1\}\),C\(a\_\{2\}\),r\(a\_\{1\},a\_\{1\}\),r\(a\_\{1\},a\_\{2\}\),r\(a\_\{2\},a\_\{1\}\),r\(a\_\{2\},a\_\{2\}\)\\\}are unconstrained, contributing a factor26=642^\{6\}=64\. Hence\#⁡\(𝒪\)=9⋅64=576\\\#\(\\mathcal\{O\}\)=9\\cdot 64=576\.

Next, we compute\#\(𝒪∪\{∃r\.A⊑C\}\)\\\#\(\\mathcal\{O\}\\cup\\\{\\exists r\.A\\sqsubseteq C\\\}\)\. The added axiom reads, again clausally,

∀i,j\.¬r⁡\(ai,aj\)∨¬A⁡\(aj\)∨C⁡\(ai\),\\forall i,j\.\\;\\neg r\(a\_\{i\},a\_\{j\}\)\\lor\\neg A\(a\_\{j\}\)\\lor C\(a\_\{i\}\),i\.e\. four ground clauses, one per\(i,j\)\(i,j\)\. A direct enumeration over the99valid\(A,B\)\(A,B\)\-assignments and, for each, a count of therr\- andCC\-assignments that satisfy the four added clauses, gives a total of425425models\. The intermediate counts split byAA\-extension as follows:

Multiplying matchingBB\-assignments by\(r,C\)\(r,C\)\-tuples:4⋅64\+2⋅36\+2⋅36\+1⋅25=256\+72\+72\+25=4254\\cdot 64\+2\\cdot 36\+2\\cdot 36\+1\\cdot 25=256\+72\+72\+25=425\.

The relative model count is therefore

P\(∃r\.A⊑C∣𝒪\)=425576≈0\.7378\.P\(\\exists r\.A\\sqsubseteq C\\mid\\mathcal\{O\}\)=\\frac\{425\}\{576\}\\approx 0\.7378\.Since0​¡​425/576​¡​10<425/576<1, by Proposition[1](https://arxiv.org/html/2609.27863#Thmproposition1)the candidate axiom is Herbrand\-undetermined relative to𝒪\\mathcal\{O\}, and a PLF\-faithful model would rank it above any axiomβ∈𝒮?\\beta\\in\\mathcal\{S\}\_\{?\}withP⁡\(β∣𝒪\)​¡​0\.7378P\(\\beta\\mid\\mathcal\{O\}\)<0\.7378and below anyβ′\\beta^\{\\prime\}withP⁡\(β′∣𝒪\)\>0\.7378P\(\\beta^\{\\prime\}\\mid\\mathcal\{O\}\)\>0\.7378\.

## Appendix 0\.BProof of Theorem[5\.1](https://arxiv.org/html/2609.27863#S5.Thmtheorem1)

Throughout,𝒮⊧\\mathcal\{S\}\_\{\\models\},𝒮⊥\\mathcal\{S\}\_\{\\bot\}, and𝒮?\\mathcal\{S\}\_\{?\}are non\-empty as assumed in Theorem[5\.1](https://arxiv.org/html/2609.27863#S5.Thmtheorem1)\. We prove the three forward implications in turn, then exhibit counterexamples for the converses\.

For PLF⇒\\RightarrowMLF, letffbe PLF and let\(αΓ,α?,α\+\)∈𝒮⊥×𝒮?×𝒮⊧\(\\alpha^\{\-\},\\alpha^\{?\},\\alpha^\{\+\}\)\\in\\mathcal\{S\}\_\{\\bot\}\\times\\mathcal\{S\}\_\{?\}\\times\\mathcal\{S\}\_\{\\models\}\. By Proposition[1](https://arxiv.org/html/2609.27863#Thmproposition1),P⁡\(αΓ∣𝒪\)=0P\(\\alpha^\{\-\}\\mid\\mathcal\{O\}\)=0,P⁡\(α?∣𝒪\)∈\(0,1\)P\(\\alpha^\{?\}\\mid\\mathcal\{O\}\)\\in\(0,1\),P⁡\(α\+∣𝒪\)=1P\(\\alpha^\{\+\}\\mid\\mathcal\{O\}\)=1, soP⁡\(αΓ∣𝒪\)​¡​P​\(α?∣𝒪\)​¡​P​\(α\+∣𝒪\)P\(\\alpha^\{\-\}\\mid\\mathcal\{O\}\)<P\(\\alpha^\{?\}\\mid\\mathcal\{O\}\)<P\(\\alpha^\{\+\}\\mid\\mathcal\{O\}\)\. Applying PLF to each of the two strict inequalities yieldsf⁡\(αΓ\)​¡​f​\(α?\)​¡​f​\(α\+\)f\(\\alpha^\{\-\}\)<f\(\\alpha^\{?\}\)<f\(\\alpha^\{\+\}\), which is exactly MLF\.

For MLF⇒\\Rightarrowlogical admissibility, letffbe MLF, fix anyαΓ∈𝒮⊥\\alpha^\{\-\}\\in\\mathcal\{S\}\_\{\\bot\}and any consistentβ∈𝒮⊧∪𝒮?\\beta\\in\\mathcal\{S\}\_\{\\models\}\\cup\\mathcal\{S\}\_\{?\}, and pick \(using non\-emptiness\) someα?∈𝒮?\\alpha^\{?\}\\in\\mathcal\{S\}\_\{?\}andα\+∈𝒮⊧\\alpha^\{\+\}\\in\\mathcal\{S\}\_\{\\models\}\. Ifβ∈𝒮?\\beta\\in\\mathcal\{S\}\_\{?\}, the triple\(αΓ,β,α\+\)\(\\alpha^\{\-\},\\beta,\\alpha^\{\+\}\)givesf⁡\(αΓ\)​¡​f​\(β\)f\(\\alpha^\{\-\}\)<f\(\\beta\); ifβ∈𝒮⊧\\beta\\in\\mathcal\{S\}\_\{\\models\}, the triple\(αΓ,α?,β\)\(\\alpha^\{\-\},\\alpha^\{?\},\\beta\)givesf⁡\(αΓ\)​¡​f​\(α?\)​¡​f​\(β\)f\(\\alpha^\{\-\}\)<f\(\\alpha^\{?\}\)<f\(\\beta\)\. Either wayf⁡\(αΓ\)​¡​f​\(β\)f\(\\alpha^\{\-\}\)<f\(\\beta\)for all suchαΓ,β\\alpha^\{\-\},\\beta, i\.e\.max𝒮⊥⁡f​¡​min𝒮⊧∪𝒮?​f\\max\_\{\\mathcal\{S\}\_\{\\bot\}\}f<\\min\_\{\\mathcal\{S\}\_\{\\models\}\\cup\\mathcal\{S\}\_\{?\}\}f, which is admissibility \(Definition[3](https://arxiv.org/html/2609.27863#Thmdefinition3)\)\.

For logical admissibility⇒\\Rightarrowdiscrimination, admissibility givesf⁡\(αΓ\)​¡​f​\(β\)f\(\\alpha^\{\-\}\)<f\(\\beta\)for everyαΓ∈𝒮⊥\\alpha^\{\-\}\\in\\mathcal\{S\}\_\{\\bot\}and everyβ∈𝒮⊧∪𝒮?\\beta\\in\\mathcal\{S\}\_\{\\models\}\\cup\\mathcal\{S\}\_\{?\}, in particular for everyα\+∈𝒮⊧\\alpha^\{\+\}\\in\\mathcal\{S\}\_\{\\models\}, which is discrimination\.

For non\-reversal of PLF⇒\\RightarrowMLF, take𝒪\\mathcal\{O\}admitting two undetermined axiomsα1,α2\\alpha\_\{1\},\\alpha\_\{2\}withP⁡\(α1∣𝒪\)\>P⁡\(α2∣𝒪\)P\(\\alpha\_\{1\}\\mid\\mathcal\{O\}\)\>P\(\\alpha\_\{2\}\\mid\\mathcal\{O\}\)\. Defineffto satisfy MLF and to setf⁡\(α1\)​¡​f​\(α2\)f\(\\alpha\_\{1\}\)<f\(\\alpha\_\{2\}\)within𝒮?\\mathcal\{S\}\_\{?\}; this is consistent with MLF \(which is silent within𝒮?\\mathcal\{S\}\_\{?\}\) but violates PLF\.

For non\-reversal of MLF⇒\\Rightarrowadmissibility, takeαΓ∈𝒮⊥\\alpha^\{\-\}\\in\\mathcal\{S\}\_\{\\bot\},α?∈𝒮?\\alpha^\{?\}\\in\\mathcal\{S\}\_\{?\},α\+∈𝒮⊧\\alpha^\{\+\}\\in\\mathcal\{S\}\_\{\\models\}and setf⁡\(αΓ\)​¡​f​\(α\+\)​¡​f​\(α?\)f\(\\alpha^\{\-\}\)<f\(\\alpha^\{\+\}\)<f\(\\alpha^\{?\}\)\. Then every contradiction is below every consistent axiom \(admissibility holds\), but MLF fails on\(αΓ,α?,α\+\)\(\\alpha^\{\-\},\\alpha^\{?\},\\alpha^\{\+\}\)becausef⁡\(α?\)\>f⁡\(α\+\)f\(\\alpha^\{?\}\)\>f\(\\alpha^\{\+\}\)\.

For non\-reversal of admissibility⇒\\Rightarrowdiscrimination, takeαΓ∈𝒮⊥\\alpha^\{\-\}\\in\\mathcal\{S\}\_\{\\bot\},α?∈𝒮?\\alpha^\{?\}\\in\\mathcal\{S\}\_\{?\},α\+∈𝒮⊧\\alpha^\{\+\}\\in\\mathcal\{S\}\_\{\\models\}and setf⁡\(α?\)​¡​f​\(αΓ\)​¡​f​\(α\+\)f\(\\alpha^\{?\}\)<f\(\\alpha^\{\-\}\)<f\(\\alpha^\{\+\}\)\. Thenf⁡\(α\+\)\>f⁡\(αΓ\)f\(\\alpha^\{\+\}\)\>f\(\\alpha^\{\-\}\)\(discrimination holds\), but the contradictionαΓ\\alpha^\{\-\}outranks the consistent axiomα?\\alpha^\{?\}, so admissibility fails\.

□\\square

## Appendix 0\.CImplementation deviations and corrections

The ELEmbeddings and EmEL\+\+ baselines are run with their reference implementations\. For two normal forms these implementations differ from the loss functions in the original paper\[[25](https://arxiv.org/html/2609.27863#bib.bib25)\], and the deviation for the conjunctive normal form is the proximate cause of the GCI1 inversion reported in Section[6\.2](https://arxiv.org/html/2609.27863#S6.SS2)\. We record these deviations, together with a correction to the normalisation toolchain, here for transparency\. Throughout, a conceptCCis annn\-ball with centreccand radiusrCr\_\{C\},γ\\gammais the margin, and the normalisation and regularisation terms are elided\.

#### Conjunctive subsumptionC⊓D⊑EC\\sqcap D\\sqsubseteq E\(GCI1, the NF2 normal form\)\.

Equation \(2\) of\[[25](https://arxiv.org/html/2609.27863#bib.bib25)\]defines four penalty terms:

ℓC⊓D⊑E\\displaystyle\\ell\_\{C\\sqcap D\\sqsubseteq E\}=max⁡\(0,∥c​Γ​d∥​Γ​rC​Γ​rD​Γ​γ\)\\displaystyle=\\max\(0,\\ \\lVert c\-d\\rVert\-r\_\{C\}\-r\_\{D\}\-\\gamma\)\(i\)\+max⁡\(0,∥c​Γ​e∥​Γ​rC​Γ​γ\)\\displaystyle\\quad\+\\max\(0,\\ \\lVert c\-e\\rVert\-r\_\{C\}\-\\gamma\)\(ii\)\+max⁡\(0,∥d​Γ​e∥​Γ​rD​Γ​γ\)\\displaystyle\\quad\+\\max\(0,\\ \\lVert d\-e\\rVert\-r\_\{D\}\-\\gamma\)\(iii\)\+max⁡\(0,min⁡\(rC,rD\)​Γ​rE​Γ​γ\)\.\\displaystyle\\quad\+\\max\(0,\\ \\min\(r\_\{C\},r\_\{D\}\)\-r\_\{E\}\-\\gamma\)\.\(iv\)Term \(i\) forces the balls ofCCandDDto overlap; terms \(ii\) and \(iii\) pull the centre ofEEinside both balls, that is, into the lensC⊓DC\\sqcap D; and term \(iv\), the only term involving the radiusrEr\_\{E\}of the right\-hand concept, forcesEEto be at least as large as the smaller ofCCandDD, which is what makes the intersection lie*within*EErather than merely overlap it\. The released implementation drops term \(iv\) and keeps only \(i\)–\(iii\); the radiusrEr\_\{E\}of the right\-hand concept never enters the loss at all \(the code fetches the radii ofCCandDDbut never reads the radius ofEE\)\. Consequently the score ofC⊓D⊑EC\\sqcap D\\sqsubseteq Eis independent of the size ofEE: it rewards any configuration in which the centre ofEEfalls inside the lens, regardless of whether the lens is actually contained inEE\. A contradictory axiom whose right\-hand concept happens to be large or centrally placed then receives a lower \(more plausible\) loss than a genuinely entailed axiom whose right\-hand concept is specific and off\-centre\. Because the evaluation score is the training loss itself \(the model has no separate scorer\), this surrogate is what the discrimination AUC measures, and it is anti\-correlated with entailment on the GCI1 test sets, drivingAUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}below0\.50\.5for ELEmbeddings and EmEL\+\+\. The defect is shared by the original authors’ released code, where term \(iv\) appears commented out; it is therefore a property of the implementation as used by the community, not of our re\-implementation\. Box\-based models such as Box2EL representC⊓DC\\sqcap Das the exact intersection of axis\-aligned boxes and do not exhibit the near\-total inversion of the ball\-based models, though theirAUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}on GCI1 still sits somewhat below0\.50\.5\.

BecauseAUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}is a rank statistic, a value near00is the sign\-flip of a value near11\(AUC\+/Γ\(Γf\)=1ΓAUC\+/Γ\(f\)\\mathrm\{AUC\}\_\{\+/\-\}\(\-f\)=1\-\\mathrm\{AUC\}\_\{\+/\-\}\(f\)exactly, ties included\), so one might be tempted to simply negate the GCI1 score and recover a near\-perfect classifier\. This does not yield a faithful model\. The negation would have to be applied to GCI1 alone \(the other normal forms already score above0\.50\.5\), which presupposes the symbolic normal form the score is meant to be agnostic to, and the discriminating signal is the size and placement of the right\-hand concept rather than entailment itself, a feature of how the contradiction set is sampled\. The principled fix is to restore term \(iv\) so the loss penalises an undersizedEEdirectly, recovering separation in the correct direction rather than reading the inverted surrogate backwards\.

For comparison, the NF1 loss for atomic subsumptionC⊑DC\\sqsubseteq D\(Equation \(1\) of the same paper\),ℓC⊑D=max⁡\(0,∥c​Γ​d∥\+rC​Γ​rD​Γ​γ\)\\ell\_\{C\\sqsubseteq D\}=\\max\(0,\\ \\lVert c\-d\\rVert\+r\_\{C\}\-r\_\{D\}\-\\gamma\), does use the superclass radiusrDr\_\{D\}, so the omission is specific to the conjunctive normal form\.

#### Existential subsumption∃r\.C⊑D\\exists r\.C\\sqsubseteq D\(GCI3, the NF4 normal form\)\.

Here the implemented loss*corrects*the published formula\. The reference code usesℓ∃r\.C⊑D=max⁡\(0,∥c​Γ​r​Γ​d∥\+rC​Γ​rD​Γ​γ\)\\ell\_\{\\exists r\.C\\sqsubseteq D\}=\\max\(0,\\ \\lVert c\-r\-d\\rVert\+r\_\{C\}\-r\_\{D\}\-\\gamma\), a containment\-style margin\+rC​Γ​rD\+r\_\{C\}\-r\_\{D\}matching the NF3 loss forC⊑∃r\.DC\\sqsubseteq\\exists r\.D, whereas Equation \(4\) of the original paper printsΓ​rC​Γ​rD\-r\_\{C\}\-r\_\{D\}, which encodes a disjointness\-like rather than a subsumption constraint\. We use the corrected implementation; unlike the GCI1 case this deviation makes the loss more, not less, faithful to the intended semantics\.

We report all EL embedding results with the implementations as released and widely used\. The two deviations are confined to the NF2 and NF4 normal forms; the box\-based \(Box2EL, BoxE\) and translational \(TransE, DistMult, ComplEx\) baselines are unaffected, and the cross\-model comparison in Section[6\.2](https://arxiv.org/html/2609.27863#S6.SS2)is anchored on them\.

#### Normalisation \(jcel\)\.

The GCI normal\-form splits \(GCI0 to GCI3, and the disjointness form\) are produced by normalising each ontology intoℰ​ℒ\+⁣\+\\mathcal\{EL\}^\{\+\+\}normal form withjcel\[[32](https://arxiv.org/html/2609.27863#bib.bib32)\]\. We found and fixed a bug injcel’s normaliser that produced incorrect normalised axioms; left uncorrected it would have corrupted the normal\-form train and test sets and, through them, the deductive closure and the relative model counts that define the trichotomy\. All results reported here use the corrected normaliser\.

## Appendix 0\.DWorking example

###### Example 1

Let𝒪\\mathcal\{O\}have𝒯=\{PhDStudent⊑Student,Student⊓Professor⊑⊥\}\\mathcal\{T\}=\\\{\\text\{PhDStudent\}\\sqsubseteq\\text\{Student\},\\;\\text\{Student\}\\sqcap\\text\{Professor\}\\sqsubseteq\\bot\\\}and𝒜=\{PhDStudent​\(alice\)\}\\mathcal\{A\}=\\\{\\text\{PhDStudent\}\(\\text\{alice\}\)\\\}\. For ABox candidates,Student​\(alice\)∈𝒮⊧\\text\{Student\}\(\\text\{alice\}\)\\in\\mathcal\{S\}\_\{\\models\},Professor​\(alice\)∈𝒮⊥\\text\{Professor\}\(\\text\{alice\}\)\\in\\mathcal\{S\}\_\{\\bot\}\(disjoint with the entailedStudent​\(alice\)\\text\{Student\}\(\\text\{alice\}\)\), andTeachingAssistant​\(alice\)∈𝒮?\\text\{TeachingAssistant\}\(\\text\{alice\}\)\\in\\mathcal\{S\}\_\{?\}, so a faithfulffmust orderf⁡\(Professor\)​¡​f​\(TA\)​¡​f​\(Student\)f\(\\text\{Professor\}\)<f\(\\text\{TA\}\)<f\(\\text\{Student\}\)\. The same holds for TBox candidates:PhDStudent⊑Student⊔Professor∈𝒮⊧\\text\{PhDStudent\}\\sqsubseteq\\text\{Student\}\\sqcup\\text\{Professor\}\\in\\mathcal\{S\}\_\{\\models\}\(already implied\),PhDStudent⊑Professor∈𝒮⊥\\text\{PhDStudent\}\\sqsubseteq\\text\{Professor\}\\in\\mathcal\{S\}\_\{\\bot\}\(it would force the disjointStudentandProfessorto sharealice\), andTeachingAssistant⊑Student∈𝒮?\\text\{TeachingAssistant\}\\sqsubseteq\\text\{Student\}\\in\\mathcal\{S\}\_\{?\}\.

## Appendix 0\.EPlacing neuro\-symbolic frameworks on the hierarchy

The hierarchy is defined on the scoring functionffalone \(Section[5](https://arxiv.org/html/2609.27863#S5)\), so any KBC method can be placed on it, including neuro\-symbolic frameworks whose logical layer differs substantially from the geometric embeddings evaluated in Section[6](https://arxiv.org/html/2609.27863#S6)\. Frameworks with an*exact logical layer*\(DeepProbLog\[[31](https://arxiv.org/html/2609.27863#bib.bib31)\], NeurASP\[[47](https://arxiv.org/html/2609.27863#bib.bib47)\]\) can in principle reach MLF: their inference layers enforce zero probability on contradictions and unit probability on entailed axioms, aligning with the boundary conditions of Proposition[1](https://arxiv.org/html/2609.27863#Thmproposition1)\. Reaching PLF additionally requires the inference to align with the relative model count; DeepProbLog’s weighted model counting is the closest existing mechanism and would in principle targetρP\\rho\_\{P\}directly, though translating DL axioms with existential restrictions into ProbLog programs is non\-trivial\. Frameworks with a*fuzzy or weighted logical layer*\(LTN\[[7](https://arxiv.org/html/2609.27863#bib.bib7)\], LNN\[[38](https://arxiv.org/html/2609.27863#bib.bib38)\]\) reach L0 by construction, since fuzzy satisfaction losses trainffto distinguish satisfied from violated axioms\. MLF depends on whether contradictions produce truth collapse rather than intermediate fuzzy scores, which the disjointness axioms can enforce in principle\. PLF is out of reach without an additional model\-counting mechanism, since fuzzy or interval\-based semantics do not compose into a probability measure over models\. These are ceiling placements determined by the logical layer’s expressivity; empirical placement additionally depends on the training regime and on data coverage of contradictory and undetermined axioms\. Full experimental placement of these frameworks onρP\\rho\_\{P\}is a direct extension of this work: our released evaluation pipeline accepts any scoring functionff, so the barrier is engineering rather than framework design\.

## Appendix 0\.FDataset statistics

Table[2](https://arxiv.org/html/2609.27863#Pt0.A6.T2)reports the size of each ontology and the composition of the evaluation splits\. Axiom counts are by normal form, excluding trivial axioms \(reflexiveC⊑CC\\sqsubseteq C,C⊑⊤C\\sqsubseteq\\top,⊥⊑C\\bot\\sqsubseteq C, andC⊑⊥C\\sqsubseteq\\bot\)\. For each split,T\+T\_\{\+\},TΓT\_\{\-\}, andT?T\_\{?\}are the reasoner\-generated entailed, contradictory, and undetermined candidate sets used to evaluate L0–L3; the domain sizej​Δ​j\|\\Delta\|is the bounded Herbrand domain used for model counting \(Section[6](https://arxiv.org/html/2609.27863#S6)\)\. The GO slice is evaluated with a full trichotomy on the two normal forms for which a contradictory set is available \(GCI0 and GCI1⊥\); for GCI1, GCI2, and GCI3 no contradictory set is generated, so onlyT\+T\_\{\+\}andT?T\_\{?\}are defined and the reduced metrics \(ρP\\rho\_\{P\}, MRR, Hits@10\) are reported\. Training\-axiom counts are reported for the GCI0 split\.

Table 2:Ontology sizes and evaluation\-split composition\. Axiom counts exclude trivial axioms; “n/a” marks a quantity that is not applicable \(the GO\-plus GCI1, GCI2, and GCI3 splits have no contradictory set, whereas Pizza has one for every normal form\)\.
## Appendix 0\.GFull experimental results

Tables[4](https://arxiv.org/html/2609.27863#Pt0.A7.T4)to[13](https://arxiv.org/html/2609.27863#Pt0.A7.T13)report all metrics \(AUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}, Adm@1010, FVR, Pair\-FVR,ρP\\rho\_\{P\}, MRR, Hits@10\) for every model, normal form, and dataset\. The geometric baselines \(TransE, DistMult, ComplEx, BoxE\) operate on atomic and existential axioms and are therefore evaluated on GCI0 and GCI2; for the remaining forms, and for the GO\-plus slice normal forms without a contradictory set, we reportρP\\rho\_\{P\}, MRR, and Hits@10\. The summary Tables[1](https://arxiv.org/html/2609.27863#S6.T1)and[3](https://arxiv.org/html/2609.27863#Pt0.A7.T3)of Section[6\.2](https://arxiv.org/html/2609.27863#S6.SS2)are drawn from these numbers\. Table[3](https://arxiv.org/html/2609.27863#Pt0.A7.T3)aggregates them by normal form, and Figure[1](https://arxiv.org/html/2609.27863#Pt0.A7.F1)reports the full negative\-sampling sweep of Section[6\.3](https://arxiv.org/html/2609.27863#S6.SS3): all three logic\-geometric models on all four normal forms\.

Table 3:Faithfulness is construct\-dependent: mean over the logic\-geometric models \(ELEmbeddings, EmEL\+\+, Box2EL\) by normal form, on the Pizza ontology\. Disjointness \(C⊓D⊑⊥C\\sqcap D\\sqsubseteq\\bot\) is the one construct handled faithfully \(high AUC, Pair\-FVR near00\); conjunctive subsumption \(C⊓D⊑EC\\sqcap D\\sqsubseteq E\) inverts \(AUC below0\.50\.5, negativeρP\\rho\_\{P\}\), driven by the ball models’ loss \(Section[6\.2](https://arxiv.org/html/2609.27863#S6.SS2); Appendix[0\.C](https://arxiv.org/html/2609.27863#Pt0.A3)\)\.![Refer to caption](https://arxiv.org/html/2609.27863v1/figures/intervention_1_box2el_pizza_gci0.png)\(a\)Box2EL, GCI0
![Refer to caption](https://arxiv.org/html/2609.27863v1/figures/intervention_1_box2el_pizza_gci2.png)\(b\)Box2EL, GCI2
![Refer to caption](https://arxiv.org/html/2609.27863v1/figures/intervention_1_box2el_pizza_gci3.png)\(c\)Box2EL, GCI3
![Refer to caption](https://arxiv.org/html/2609.27863v1/figures/intervention_1_elembeddings_pizza_gci0.png)\(d\)ELEmbeddings, GCI0
![Refer to caption](https://arxiv.org/html/2609.27863v1/figures/intervention_1_elembeddings_pizza_gci2.png)\(e\)ELEmbeddings, GCI2
![Refer to caption](https://arxiv.org/html/2609.27863v1/figures/intervention_1_elembeddings_pizza_gci3.png)\(f\)ELEmbeddings, GCI3
![Refer to caption](https://arxiv.org/html/2609.27863v1/figures/intervention_1_emelpp_pizza_gci0.png)\(g\)EmEL\+\+, GCI0
![Refer to caption](https://arxiv.org/html/2609.27863v1/figures/intervention_1_emelpp_pizza_gci2.png)\(h\)EmEL\+\+, GCI2
![Refer to caption](https://arxiv.org/html/2609.27863v1/figures/intervention_1_emelpp_pizza_gci3.png)\(i\)EmEL\+\+, GCI3
![Refer to caption](https://arxiv.org/html/2609.27863v1/figures/intervention_1_elembeddings_pizza_gci1_bot.png)\(j\)ELEmbeddings, GCI1⊥
![Refer to caption](https://arxiv.org/html/2609.27863v1/figures/intervention_1_emelpp_pizza_gci1_bot.png)\(k\)EmEL\+\+, GCI1⊥
![Refer to caption](https://arxiv.org/html/2609.27863v1/figures/intervention_1_box2el_pizza_gci1_bot.png)\(l\)Box2EL, GCI1⊥

Figure 1:Full negative\-sampling sweep \(Section[6\.3](https://arxiv.org/html/2609.27863#S6.SS3)\) on the Pizza ontology: FVR \(left axis, lower is better\) and MRR \(right axis\) as the fraction of random\-corruption negatives varies; the remainderp⊥=1​Γ​xp\_\{\\bot\}=1\-xis drawn from𝒮⊥\\mathcal\{S\}\_\{\\bot\}\. In all 12 panels the all\-contradiction endpoint \(x=0x=0\) has lower FVR than the all\-random endpoint \(x=1x=1\); the response is not monotone in between, and on GCI0 the ball\-based models’ effect is within single\-seed noise\.Model nameAUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}↑\\uparrowAdm@10↑\\uparrowFVR↓\\downarrowPair\-FVR↓\\downarrowρP\\rho\_\{P\}↑\\uparrowMRR↑\\uparrowHits@10↑\\uparrowELEm0\.70±\\pm0\.010\.98±\\pm0\.000\.74±\\pm0\.000\.48±\\pm0\.000\.62±\\pm0\.020\.14±\\pm0\.000\.42±\\pm0\.01Box2EL0\.74±\\pm0\.010\.99±\\pm0\.000\.76±\\pm0\.010\.60±\\pm0\.010\.53±\\pm0\.000\.17±\\pm0\.010\.51±\\pm0\.01EmEL\+\+0\.73±\\pm0\.010\.98±\\pm0\.010\.73±\\pm0\.010\.49±\\pm0\.000\.63±\\pm0\.050\.15±\\pm0\.010\.41±\\pm0\.04BoxE0\.42±\\pm0\.060\.94±\\pm0\.010\.90±\\pm0\.030\.73±\\pm0\.020\.16±\\pm0\.040\.10±\\pm0\.000\.23±\\pm0\.04ComplEx0\.53±\\pm0\.030\.94±\\pm0\.010\.82±\\pm0\.010\.45±\\pm0\.01\-0\.13±\\pm0\.030\.08±\\pm0\.010\.11±\\pm0\.01DistMult0\.52±\\pm0\.090\.94±\\pm0\.010\.82±\\pm0\.050\.47±\\pm0\.08\-0\.03±\\pm0\.080\.05±\\pm0\.020\.08±\\pm0\.06TransE0\.61±\\pm0\.040\.94±\\pm0\.010\.79±\\pm0\.030\.61±\\pm0\.050\.18±\\pm0\.070\.20±\\pm0\.000\.32±\\pm0\.03

Table 4:GCI0, PizzaModel nameAUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}↑\\uparrowAdm@10↑\\uparrowFVR↓\\downarrowPair\-FVR↓\\downarrowρP\\rho\_\{P\}↑\\uparrowMRR↑\\uparrowHits@10↑\\uparrowELEm0\.84±\\pm0\.011\.00±\\pm0\.000\.60±\\pm0\.010\.47±\\pm0\.010\.53±\\pm0\.010\.16±\\pm0\.010\.46±\\pm0\.01Box2EL0\.82±\\pm0\.011\.00±\\pm0\.000\.50±\\pm0\.010\.25±\\pm0\.020\.63±\\pm0\.010\.09±\\pm0\.000\.26±\\pm0\.01EmEL\+\+0\.83±\\pm0\.011\.00±\\pm0\.000\.62±\\pm0\.020\.49±\\pm0\.020\.57±\\pm0\.010\.15±\\pm0\.010\.45±\\pm0\.02BoxE0\.61±\\pm0\.011\.00±\\pm0\.000\.74±\\pm0\.010\.33±\\pm0\.020\.24±\\pm0\.020\.16±\\pm0\.000\.20±\\pm0\.00ComplEx0\.55±\\pm0\.051\.00±\\pm0\.000\.78±\\pm0\.040\.49±\\pm0\.080\.01±\\pm0\.050\.16±\\pm0\.000\.19±\\pm0\.00DistMult0\.27±\\pm0\.071\.00±\\pm0\.000\.95±\\pm0\.030\.76±\\pm0\.130\.01±\\pm0\.010\.04±\\pm0\.020\.09±\\pm0\.02TransE0\.61±\\pm0\.061\.00±\\pm0\.000\.73±\\pm0\.020\.48±\\pm0\.060\.34±\\pm0\.050\.24±\\pm0\.010\.36±\\pm0\.00

Table 5:GCI0, GO\-plus sliceModel nameAUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}↑\\uparrowAdm@10↑\\uparrowFVR↓\\downarrowPair\-FVR↓\\downarrowρP\\rho\_\{P\}↑\\uparrowMRR↑\\uparrowHits@10↑\\uparrowELEm0\.57±\\pm0\.011\.00±\\pm0\.000\.78±\\pm0\.000\.52±\\pm0\.010\.51±\\pm0\.010\.10±\\pm0\.000\.29±\\pm0\.01Box2EL0\.35±\\pm0\.021\.00±\\pm0\.000\.90±\\pm0\.010\.73±\\pm0\.020\.22±\\pm0\.010\.09±\\pm0\.000\.18±\\pm0\.01EmEL\+\+0\.59±\\pm0\.011\.00±\\pm0\.000\.76±\\pm0\.000\.48±\\pm0\.010\.55±\\pm0\.010\.10±\\pm0\.000\.29±\\pm0\.01BoxE0\.34±\\pm0\.011\.00±\\pm0\.000\.91±\\pm0\.000\.75±\\pm0\.010\.22±\\pm0\.030\.14±\\pm0\.010\.20±\\pm0\.01ComplEx0\.51±\\pm0\.031\.00±\\pm0\.000\.83±\\pm0\.010\.51±\\pm0\.03\-0\.05±\\pm0\.010\.11±\\pm0\.010\.19±\\pm0\.02DistMult0\.47±\\pm0\.021\.00±\\pm0\.000\.86±\\pm0\.010\.48±\\pm0\.06\-0\.04±\\pm0\.030\.09±\\pm0\.010\.18±\\pm0\.02TransE0\.45±\\pm0\.041\.00±\\pm0\.000\.88±\\pm0\.020\.69±\\pm0\.030\.25±\\pm0\.090\.12±\\pm0\.010\.22±\\pm0\.03

Table 6:GCI2, PizzaTable 7:GCI2, GO\-plus sliceModel nameAUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}↑\\uparrowAdm@10↑\\uparrowFVR↓\\downarrowPair\-FVR↓\\downarrowρP\\rho\_\{P\}↑\\uparrowMRR↑\\uparrowHits@10↑\\uparrowELEm0\.02±\\pm0\.001\.00±\\pm0\.001\.00±\\pm0\.000\.94±\\pm0\.01\-0\.34±\\pm0\.010\.06±\\pm0\.000\.11±\\pm0\.01Box2EL0\.39±\\pm0\.011\.00±\\pm0\.000\.91±\\pm0\.000\.55±\\pm0\.01\-0\.09±\\pm0\.020\.05±\\pm0\.000\.13±\\pm0\.01EmEL\+\+0\.05±\\pm0\.011\.00±\\pm0\.001\.00±\\pm0\.000\.91±\\pm0\.02\-0\.31±\\pm0\.030\.06±\\pm0\.000\.11±\\pm0\.00

Table 8:GCI1, PizzaTable 9:GCI1, GO\-plus sliceModel nameAUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}↑\\uparrowAdm@10↑\\uparrowFVR↓\\downarrowPair\-FVR↓\\downarrowρP\\rho\_\{P\}↑\\uparrowMRR↑\\uparrowHits@10↑\\uparrowELEm0\.96±\\pm0\.001\.00±\\pm0\.000\.47±\\pm0\.010\.08±\\pm0\.010\.43±\\pm0\.000\.03±\\pm0\.000\.02±\\pm0\.00Box2EL1\.00±\\pm0\.001\.00±\\pm0\.000\.17±\\pm0\.010\.01±\\pm0\.000\.41±\\pm0\.000\.05±\\pm0\.000\.01±\\pm0\.00EmEL\+\+0\.96±\\pm0\.001\.00±\\pm0\.000\.47±\\pm0\.010\.07±\\pm0\.010\.43±\\pm0\.000\.03±\\pm0\.000\.02±\\pm0\.00

Table 10:GCI1⊥, PizzaModel nameAUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}↑\\uparrowAdm@10↑\\uparrowFVR↓\\downarrowPair\-FVR↓\\downarrowρP\\rho\_\{P\}↑\\uparrowMRR↑\\uparrowHits@10↑\\uparrowELEm0\.95±\\pm0\.011\.00±\\pm0\.000\.65±\\pm0\.010\.06±\\pm0\.000\.61±\\pm0\.010\.01±\\pm0\.000\.03±\\pm0\.01Box2EL0\.99±\\pm0\.001\.00±\\pm0\.000\.50±\\pm0\.010\.03±\\pm0\.000\.73±\\pm0\.000\.01±\\pm0\.000\.04±\\pm0\.00EmEL\+\+0\.95±\\pm0\.011\.00±\\pm0\.000\.65±\\pm0\.010\.06±\\pm0\.000\.61±\\pm0\.010\.01±\\pm0\.000\.03±\\pm0\.01

Table 11:GCI1⊥, GO\-plus sliceModel nameAUC\+/Γ\\mathrm\{AUC\}\_\{\+/\-\}↑\\uparrowAdm@10↑\\uparrowFVR↓\\downarrowPair\-FVR↓\\downarrowρP\\rho\_\{P\}↑\\uparrowMRR↑\\uparrowHits@10↑\\uparrowELEm0\.75±\\pm0\.011\.00±\\pm0\.000\.71±\\pm0\.020\.56±\\pm0\.030\.26±\\pm0\.010\.23±\\pm0\.030\.65±\\pm0\.01Box2EL0\.97±\\pm0\.001\.00±\\pm0\.000\.62±\\pm0\.010\.55±\\pm0\.010\.09±\\pm0\.010\.39±\\pm0\.020\.77±\\pm0\.02EmEL\+\+0\.79±\\pm0\.021\.00±\\pm0\.000\.68±\\pm0\.010\.54±\\pm0\.010\.31±\\pm0\.010\.22±\\pm0\.020\.60±\\pm0\.03

Table 12:GCI3, PizzaTable 13:GCI3, GO\-plus slice### Multi\-seed intervention\-1 sweep

Tables[14](https://arxiv.org/html/2609.27863#Pt0.A7.T14)onwards report the mean and standard deviation across five seeds of the intervention\-1 sweep of Section[6\.3](https://arxiv.org/html/2609.27863#S6.SS3), for all three logic\-geometric models on the four normal forms of Figure[1](https://arxiv.org/html/2609.27863#Pt0.A7.F1)\(atomic subsumption GCI0, disjointness GCI1⊥, and the existential forms GCI2 and GCI3\)\. Each table variesxx, the fraction of random\-corruption negatives, with the remaining1​Γ​x1\-xdrawn from𝒮⊥\\mathcal\{S\}\_\{\\bot\}\. Across all twelve panels the all\-contradiction endpoint \(x=0x=0\) attains lower mean FVR than the all\-random endpoint \(x=1x=1\); the effect is largest on Box2EL/GCI3 \(a difference of0\.130\.13in mean FVR\) and smallest on the disjointness form, where for Box2EL FVR is already close to its floor\. Consistent with Table[1](https://arxiv.org/html/2609.27863#S6.T1), MRR does not track FVR across the sweep, reproducing the rank–faithfulness dissociation as an intervention within a single model\.

Table 14:Multi\-seed \(5 seeds\) sweep for Box2EL, Pizza ontology, GCI0\.xxis the fraction of random\-corruption negatives, with the remaining1​Γ​x1\-xdrawn from𝒮⊥\\mathcal\{S\}\_\{\\bot\}\(matching Figure[1](https://arxiv.org/html/2609.27863#Pt0.A7.F1)\)\. Values are mean±\\pmstandard deviation\.Table 15:Multi\-seed \(5 seeds\) sweep for ELEmbeddings, Pizza ontology, GCI0; conventions as in Table[14](https://arxiv.org/html/2609.27863#Pt0.A7.T14)\.Table 16:Multi\-seed \(5 seeds\) sweep for EmEL\+\+, Pizza ontology, GCI0; conventions as in Table[14](https://arxiv.org/html/2609.27863#Pt0.A7.T14)\.Table 17:Multi\-seed \(5 seeds\) sweep for Box2EL, Pizza ontology, GCI1⊥\(disjointness\); conventions as in Table[14](https://arxiv.org/html/2609.27863#Pt0.A7.T14)\.Table 18:Multi\-seed \(5 seeds\) sweep for ELEmbeddings, Pizza ontology, GCI1⊥\(disjointness\); conventions as in Table[14](https://arxiv.org/html/2609.27863#Pt0.A7.T14)\.Table 19:Multi\-seed \(5 seeds\) sweep for EmEL\+\+, Pizza ontology, GCI1⊥\(disjointness\); conventions as in Table[14](https://arxiv.org/html/2609.27863#Pt0.A7.T14)\.Table 20:Multi\-seed \(5 seeds\) sweep for Box2EL, Pizza ontology, GCI2 \(C⊑∃r\.DC\\sqsubseteq\\exists r\.D\); conventions as in Table[14](https://arxiv.org/html/2609.27863#Pt0.A7.T14)\.Table 21:Multi\-seed \(5 seeds\) sweep for ELEmbeddings, Pizza ontology, GCI2 \(C⊑∃r\.DC\\sqsubseteq\\exists r\.D\); conventions as in Table[14](https://arxiv.org/html/2609.27863#Pt0.A7.T14)\.Table 22:Multi\-seed \(5 seeds\) sweep for EmEL\+\+, Pizza ontology, GCI2 \(C⊑∃r\.DC\\sqsubseteq\\exists r\.D\); conventions as in Table[14](https://arxiv.org/html/2609.27863#Pt0.A7.T14)\.Table 23:Multi\-seed \(5 seeds\) sweep for Box2EL, Pizza ontology, GCI3 \(∃r\.C⊑D\\exists r\.C\\sqsubseteq D\); conventions as in Table[14](https://arxiv.org/html/2609.27863#Pt0.A7.T14)\.Table 24:Multi\-seed \(5 seeds\) sweep for ELEmbeddings, Pizza ontology, GCI3 \(∃r\.C⊑D\\exists r\.C\\sqsubseteq D\); conventions as in Table[14](https://arxiv.org/html/2609.27863#Pt0.A7.T14)\.Table 25:Multi\-seed \(5 seeds\) sweep for EmEL\+\+, Pizza ontology, GCI3 \(∃r\.C⊑D\\exists r\.C\\sqsubseteq D\); conventions as in Table[14](https://arxiv.org/html/2609.27863#Pt0.A7.T14)\.