Research

Analogy Discovery In Semantic Space

A Geometric Framework for Cross-Domain Reasoning

Authors: Scott Senkeresty (Chief Architect, Semantic OS), Tia (Chief Semantic Agent)
Affiliation: Semantic Infrastructure Lab
Date: 2025-12-12
Status: Research Framework
Document Type: Theoretical Research Paper
Related SIL Components: Semantic Memory (Layer 0), USIR (Layer 1), Multi-Agent Orchestration (Layer 3)

Abstract

Vector embeddings capture semantic similarity through proximity in high-dimensional space. However, the most powerful form of semantic reasoning—cross-domain analogy—requires discovering and navigating directional transformations that preserve relational structure across conceptually distant domains.

This paper presents a geometric framework for analogy discovery as manifold navigation, where analogies are formalized as structure-preserving transformations between semantic submanifolds. We show that:

  1. Analogies form a low-dimensional subspace of the embedding manifold (the analogy manifold M_A)
  2. High-quality cross-domain analogies preserve topological neighborhoods under transport
  3. Domain-to-domain translation can be systematized through centroid operations and residual analysis
  4. This framework enables computational discovery of novel analogies, not just retrieval of known ones

We connect this work to SIL's broader research on semantic manifold transport (RAG systems), semantic observability (intent-execution alignment), and hierarchical agency (multi-agent reasoning with provenance).

Keywords: semantic analogies, cross-domain reasoning, manifold transport, vector embeddings, relational structure preservation, computational creativity

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1. Introduction: From Similarity to Analogy

1.1 The Limitation of Proximity-Based Semantics

Modern embedding models excel at capturing distributional similarity:
- hot is close to warm, cold, temperature
- dog is close to canine, puppy, animal
- hot dog is close to sausage, frankfurter, food

This proximity-based semantics works well for retrieval, clustering, and similarity search. However, it fundamentally misses the kind of reasoning that drives scientific breakthroughs, literary insight, and creative problem-solving: cross-domain analogy.

1.2 Analogies as Directional Transformations

The canonical example from word embedding research:

king - man + woman ≈ queen

This is not about similarity. Instead, it reveals a directional relationship:
- The vector from man to king captures "becomes monarch"
- The vector from woman to queen captures the same relationship
- The analogy works because these vectors are parallel (or nearly so)

Key insight: Analogies are relationships between directions, not between positions.

1.3 The Schrödinger Example: Cross-Domain Analogy as Scientific Method

Erwin Schrödinger's 1926 derivation of quantum mechanics provides a canonical example of analogy-driven discovery:

He noticed a structural similarity:

Hamilton-Jacobi equation     ∂S/∂t + H(q, ∇S) = 0     [classical mechanics]
       ↕ (analogy)
Eikonal equation             ∂φ/∂t + c|∇φ| = 0        [geometric optics]

The analogy mapping:

Action S        ↔  Optical phase φ
Momentum ∇S     ↔  Wave vector k
Trajectories    ↔  Light rays
Energy surfaces ↔  Wavefronts

This wasn't just pattern-matching—it was structure-preserving cross-domain transport. Schrödinger followed this analogy to complexify the action (S → ψ = e^(iS/ℏ)) and derive the Schrödinger equation.

Research question: Can we systematize this kind of discovery?

1.4 The Goal of This Research

We propose a computational framework for:

  1. Discovering high-quality analogies between conceptually distant domains
  2. Measuring analogy quality through geometric criteria (structure preservation)
  3. Generating candidate analogies for exploration (computational creativity)
  4. Navigating the space of possible analogies (the analogy manifold M_A)

This is not pattern-matching or keyword search. It is geometric reasoning over semantic structure.

2. Theoretical Framework: Analogies as Manifold Transformations

2.1 Notation and Definitions

Let M_E be the embedding manifold—the high-dimensional space where semantic vectors live.

Let D_A and D_B be two semantic domains (submanifolds of M_E):
- D_A might be "classical mechanics" (concepts: action, momentum, trajectory, Hamiltonian...)
- D_B might be "geometric optics" (concepts: phase, wave vector, ray, eikonal...)

An analogy is a mapping φ: D_A → D_B that preserves relational structure.

Formal definition:

An analogy a:b :: c:d holds when the vector transformation v_ab = embed(b) - embed(a) is approximately parallel to v_cd = embed(d) - embed(c), and this parallelism preserves local neighborhoods.

2.2 Analogy Quality Metrics

Not all vector arithmetic produces meaningful analogies. We define quality through structural preservation:

Criterion 1: Vector Parallelism

parallel_score(a, b, c, d) = cosine_similarity(v_ab, v_cd)

where v_ab = embed(b) - embed(a) and v_cd = embed(d) - embed(c).

High parallelism (>0.8) suggests the same relational transformation operates in both contexts.

Criterion 2: Neighborhood Preservation

An analogy should preserve semantic neighborhoods under transport.

Define the transported neighborhood:

N_a = {n : n is among k-nearest neighbors of a in D_A}
N_c_transported = {n + (embed(c) - embed(a)) : n ∈ N_a}

Then:

neighborhood_preservation(a, c) = |N_c_transported ∩ N_c| / |N_c|

where N_c is the actual neighborhood of c in D_B.

If neighborhoods preserve well (>0.6), the analogy respects local semantic structure.

Criterion 3: Centroid Alignment (Domain-Level)

For cross-domain analogies, we expect domain centroids to align:

C_A = (1/|D_A|) Σ embed(x) for x in D_A
C_B = (1/|D_B|) Σ embed(y) for y in D_B

domain_translation = C_B - C_A

Individual concept analogies should roughly follow this global translation:

alignment_score(a, c) = cosine_similarity(embed(c) - embed(a), domain_translation)

High alignment (>0.7) suggests the analogy follows systematic domain mapping.

2.3 The Analogy Manifold M_A

Hypothesis: High-quality analogies do not span the entire embedding space randomly. They occupy a low-dimensional subspace of M_E.

Define the analogy manifold M_A as:

M_A = {(a, b, c, d) ∈ M_E^4 : quality(a, b, c, d) > threshold}

Research questions:
1. What is the intrinsic dimensionality of M_A?
2. Can we learn a projection π: M_E^4 → M_A that filters for high-quality analogies?
3. Do different types of analogies (taxonomic, functional, causal) cluster within M_A?

2.4 Cross-Domain Translation as Geometric Transport

Given a concept a in domain D_A, find its analogy c in domain D_B:

Algorithm: Domain-Centroid Transport

  1. Compute domain centroids C_A and C_B
  2. Compute concept's position relative to its domain:
    offset = embed(a) - C_A
  3. Transport to target domain:
    candidate = C_B + offset
  4. Find nearest neighbors in D_B to candidate
  5. Score candidates by full analogy quality metrics

Intuition: This treats domains as parallel coordinate systems. The analogy a ↔ c preserves relative position within domain.

3. Connection to Existing SIL Research

3.1 RAG as Semantic Manifold Transport

SIL's existing research on RAG as Semantic Manifold Transport identifies four misaligned manifolds:

  • M_H: Human conceptual space
  • M_E: Embedding space
  • M_L: LLM latent space
  • M_F: Fusion space

Analogy discovery operates primarily in M_E, but its outputs must be interpretable in M_H and usable in M_L:

  • M_E → M_H transport: Discovered analogies must be explainable to humans ("Action is like optical phase because...")
  • M_E → M_L transport: LLMs must be able to reason with discovered analogies

The centroid-based transport method proposed here is a structured way to minimize distortion during M_E → M_E cross-domain transport.

3.2 Semantic Observability: Analogy as Alignment Detection

SIL's work on Semantic Observability measures alignment between intent and execution.

Analogies reveal alignment across domains:

If classical_mechanics ↔ geometric_optics has high-quality mappings, it suggests deep structural alignment—not just surface similarity.

This can be measured through:
- Residual analysis: How much structure is lost during cross-domain transport?
- Bidirectional consistency: Does D_A → D_B → D_A recover the original concepts?

From the Hierarchical Agency Framework, analogy discovery can be orchestrated across multiple agents:

  • Sage: Explores high-level cross-domain patterns
  • Beth: Provides domain concept graphs for neighborhood analysis
  • Goob: Scores analogy quality through geometric validation
  • Cora: Tracks provenance of discovered analogies

Multi-agent workflow:
1. Beth identifies domain boundaries and concept clusters
2. Sage proposes candidate cross-domain mappings
3. Goob validates through geometric criteria
4. Cora records derivation lineage (how was this analogy discovered?)

3.4 USIR: Encoding Analogies as Operators

The Universal Semantic IR provides structured representation.

Discovered analogies become USIR operators:

operator:
  id: "classical_mechanics_to_geometric_optics"
  type: "cross_domain_analogy"
  from_domain: "classical_mechanics"
  to_domain: "geometric_optics"

  mappings:
    - source_concept: "action_S"
      target_concept: "optical_phase_φ"
      quality_score: 0.94
      preserves: ["first_order_PDE_structure", "variational_principle"]

    - source_concept: "momentum_p"
      target_concept: "wave_vector_k"
      quality_score: 0.91
      preserves: ["gradient_relationship", "conserved_quantity"]

  domain_translation_vector: [...]  # Centroid offset
  neighborhood_preservation: 0.87
  discovered_by: "agent:sage"
  validated_by: "agent:goob"
  provenance: "centroid_transport_method"

This makes analogies first-class semantic objects with provenance, quality metrics, and structural guarantees.

4. Experimental Design: Seven Research Directions

Experiment 1: Analogy Subspace Discovery (Dimensionality Reduction)

Goal: Find the intrinsic dimensionality of M_A.

Method:
1. Collect 10,000+ known analogies (linguistic, scientific, literary)
2. Compute 4D embedding tuples (a, b, c, d) for each analogy
3. Run PCA/UMAP to find low-dimensional structure
4. Measure: Do high-quality analogies cluster in low-dimensional subspace?

Expected outcome: M_A has intrinsic dimension ~50-200 (much lower than M_E's 768/1536).

Experiment 2: Cross-Domain Discovery (Schrödinger Test)

Goal: Rediscover Schrödinger's classical mechanics ↔ optics analogy.

Method:
1. Define domain D_A = {classical mechanics concepts from textbooks}
2. Define domain D_B = {geometric optics concepts}
3. Apply centroid-transport algorithm for each concept in D_A
4. Measure: Do we recover known mappings (action ↔ phase, momentum ↔ wave vector)?

Success criterion: Top-3 candidates include historically correct analogies with quality >0.85.

Experiment 3: Novel Analogy Generation

Goal: Discover new cross-domain analogies not documented in literature.

Method:
1. Pick two unrelated domains (e.g., "thermodynamics" and "information theory")
2. Systematically search for high-quality mappings
3. Human expert validation: Are these meaningful?

Expected discoveries:
- Entropy ↔ Shannon entropy (already known, validates method)
- Temperature ↔ ??? (research question!)
- Partition function ↔ ??? (research question!)

Experiment 4: Analogy Type Classification

Goal: Do different analogy types (functional, causal, taxonomic, proportional) cluster differently in M_A?

Method:
1. Label analogies by type
2. Train classifier to predict analogy type from embedding geometry
3. Measure: Can we distinguish "is-a" from "causes" from "analogous-to" purely geometrically?

Experiment 5: Temporal Analogy Drift

Goal: Do analogies change over time as language/science evolves?

Method:
1. Use historical corpora (1900, 1950, 2000, 2025)
2. Measure analogy quality across time periods
3. Track: How did "atom ↔ solar system" analogy decay as quantum mechanics emerged?

Experiment 6: Multi-Hop Analogy Chains

Goal: Can we chain analogies? If A ↔ B and B ↔ C, does A ↔ C?

Method:
1. Find analogy A ↔ B (e.g., classical mechanics ↔ optics)
2. Find analogy B ↔ C (e.g., optics ↔ quantum mechanics)
3. Test: Does A ↔ C hold directly? (classical ↔ quantum)

Expected result: Multi-hop analogies degrade (quality score decreases) but may reveal novel connections.

Experiment 7: Bidirectional Consistency

Goal: Are analogies symmetric? Does D_A → D_B → D_A recover original concepts?

Method:
1. Transport concept a from D_A to D_B → get c
2. Transport c back from D_B to D_A → get a'
3. Measure: distance(a, a')

Hypothesis: High-quality analogies have low round-trip error (<0.2).

5. Implementation Considerations (Architecture-Agnostic)

While this research is independent of specific tooling, any implementation requires:

5.1 Core Capabilities

  1. Vector embeddings (e.g., OpenAI, Voyage, sentence-transformers)
  2. Domain identification (clustering, topic modeling, manual curation)
  3. Centroid computation (efficient aggregation over domain subsets)
  4. Nearest-neighbor search (FAISS, Annoy, or similar)
  5. Quality scoring (vector operations, neighborhood lookups)
  6. Provenance tracking (how was this analogy discovered?)

5.2 Computational Complexity

  • Single analogy validation: O(k) for k-NN lookup
  • Cross-domain search: O(|D_A| × k) for full systematic search
  • Analogy manifold PCA: O(n × d²) where n = number of analogies, d = embedding dimension

For 10M embeddings and 1000-concept domains, this is computationally tractable on modern hardware.

5.3 Data Requirements

Minimum viable:
- 100K+ embeddings spanning multiple domains
- 1K+ known analogies for validation
- Domain labels or clustering

Ideal:
- 10M+ embeddings (comprehensive coverage)
- 100K+ known analogies (training/validation)
- Multi-modal data (text, code, scientific literature)
- Temporal versions (historical corpora)

6. Open Research Questions

6.1 Theoretical

  • Q1: What is the intrinsic dimensionality of the analogy manifold M_A?
  • Q2: Can we prove bounds on analogy quality degradation under multi-hop transport?
  • Q3: Are there universal analogy types that appear across all domains?
  • Q4: What geometric invariants distinguish high-quality from spurious analogies?

6.2 Empirical

  • Q5: What threshold (cosine similarity, neighborhood preservation) defines "good enough" analogy?
  • Q6: How stable are analogies across different embedding models?
  • Q7: Can humans reliably validate machine-discovered analogies?
  • Q8: Do LLMs already implicitly use analogical reasoning in their latent space?

6.3 Applied

  • Q9: Can analogy discovery accelerate scientific hypothesis generation?
  • Q10: Can we build "analogy search engines" for researchers?
  • Q11: Can cross-domain analogies improve transfer learning in ML?
  • Q12: Can we detect failed analogies (e.g., atom ≠ solar system in quantum regime)?

7. Relation to Existing Work

7.1 Cognitive Science: Structure-Mapping Theory

Gentner's Structure-Mapping Theory (1983) proposes that analogies preserve relational structure, not surface features.

SIL contribution: Operationalize this geometrically. "Relational structure" becomes topological neighborhood preservation in embedding space.

7.2 NLP: Word Embeddings and Analogy Tasks

Mikolov et al. (2013) demonstrated king - man + woman = queen in Word2Vec.

SIL contribution: Generalize from single-word analogies to cross-domain conceptual mappings with quality metrics and provenance.

7.3 Knowledge Graphs: Cross-Domain Reasoning

Knowledge graphs encode explicit relations (e.g., DBpedia, Wikidata).

SIL contribution: Discover implicit cross-domain analogies that graphs don't encode. "Action" and "optical phase" have no explicit link in Wikipedia, but geometric analogy reveals their relationship.

7.4 AI Creativity: GOFAI Analogical Reasoning

Systems like SME (Structure-Mapping Engine) and FARG (Fluid Analogies Research Group) explored symbolic analogies.

SIL contribution: Use continuous geometric methods instead of symbolic pattern-matching. This handles fuzzy/partial analogies and scales to millions of concepts.

8. Impact and Applications

8.1 Scientific Discovery

Use case: Systematically search for cross-domain analogies to accelerate hypothesis generation.

Example: A biologist studying protein folding could search for analogies in "thermodynamics," "information theory," "topology," discovering connections not apparent through literature search.

8.2 Education

Use case: Help students understand difficult concepts through analogy mapping.

Example: "Quantum tunneling" is hard. The system finds analogies to "wave interference," "barrier penetration," "exponential decay" with visualization of how concepts map.

8.3 Multi-Agent Reasoning (Layer 3)

Use case: Agents discover cross-domain solutions by analogy transport.

Example: Agent solving optimization problem in domain A queries for analogies in "physics," discovers simulated annealing (from thermodynamics), applies cooling schedule by analogy.

8.4 RAG Enhancement

Use case: Improve retrieval by analogy, not just keyword match.

Example: User asks about "memory management in operating systems." System retrieves documents about "garbage collection," "cache eviction," and by analogy, "resource allocation in ecology" (surprisingly relevant conceptual parallels).

9. Roadmap and Validation

Phase 1: Foundation (Months 1-3)

  • [ ] Literature review: cognitive science, NLP, knowledge graphs
  • [ ] Formalize analogy quality metrics (parallelism, neighborhood, alignment)
  • [ ] Implement centroid-transport algorithm (proof-of-concept)
  • [ ] Validate on known analogies (king/queen, Schrödinger)

Deliverable: Working prototype demonstrating cross-domain transport.

Phase 2: Empirical Validation (Months 4-6)

  • [ ] Run Experiments 1-4 (subspace discovery, Schrödinger test, novel generation, type classification)
  • [ ] Build analogy quality benchmark dataset (1K+ validated examples)
  • [ ] Measure stability across embedding models (OpenAI, Voyage, open-source)

Deliverable: Empirical characterization of analogy manifold M_A.

Phase 3: Advanced Research (Months 7-12)

  • [ ] Run Experiments 5-7 (temporal drift, multi-hop chains, bidirectional consistency)
  • [ ] Develop analogy-aware search algorithms
  • [ ] Integration with USIR (analogies as first-class operators)
  • [ ] Human expert validation studies

Deliverable: Research paper submission (MSR, ICSE, or AI venue).

Phase 4: Application (Months 12+)

  • [ ] Build analogy search API
  • [ ] Integration with multi-agent systems (Layer 3)
  • [ ] Educational applications (concept mapping tools)
  • [ ] Scientific discovery case studies

Deliverable: Production-ready analogy discovery system.

10. Success Criteria

The research succeeds if:

  1. Rediscovery: We can rediscover historically documented analogies (Schrödinger, Darwin's "tree of life," etc.)
  2. Novel discovery: We find new cross-domain analogies validated by domain experts
  3. Dimensionality: M_A is demonstrably lower-dimensional than M_E
  4. Stability: Analogies are stable across embedding models (not artifacts)
  5. Utility: Researchers find the system useful for hypothesis generation
  6. Provenance: Every discovered analogy has traceable derivation

11. Conclusion

Analogy is not keyword matching or distributional similarity. It is structure-preserving geometric transformation across semantic manifolds.

By formalizing analogies as:
- Parallel directional transformations (vector parallelism)
- Neighborhood-preserving mappings (topological structure)
- Domain-centroid transport (systematic cross-domain reasoning)

...we can build computational systems that discover, validate, and generate analogies at scale.

This research sits at the intersection of:
- Cognitive science (how humans reason)
- Geometry (manifold structure)
- Semantics (meaning representation)
- AI (computational discovery)

It extends SIL's existing work on semantic manifold transport, semantic observability, and hierarchical agency. And it provides theoretical foundations for the next generation of analogy-aware semantic infrastructure.

The potential impact:
- Scientific discovery: Accelerate cross-domain hypothesis generation
- Education: Help learners build conceptual bridges
- AI reasoning: Enable multi-agent systems to transfer knowledge across domains
- Creativity: Computational support for analogical thinking

This is rigorous, implementable, and revolutionary.

The work ahead is methodical, long-term, and necessary. As semantic systems become central to knowledge work, they must move beyond retrieval and similarity. They must reason by analogy—the hallmark of human insight.

References

SIL Internal Research:
- RAG as Semantic Manifold Transport
- Semantic Observability
- Hierarchical Agency Framework
- SIL Technical Charter
- SIL Glossary

External Work (for formal publication):
- Gentner, D. (1983). Structure-mapping: A theoretical framework for analogy. Cognitive Science.
- Mikolov, T. et al. (2013). Linguistic regularities in continuous space word representations. NAACL.
- Falkenhainer, B., Forbus, K., Gentner, D. (1989). The structure-mapping engine. Artificial Intelligence.
- Hofstadter, D., & Sander, E. (2013). Surfaces and Essences: Analogy as the Fuel and Fire of Thinking.
- Turney, P. D. (2006). Similarity of semantic relations. Computational Linguistics.

Physics Examples:
- Schrödinger, E. (1926). Quantization as eigenvalue problem.
- Maxwell, J. C. (1861). On physical lines of force (electromagnetic-mechanical analogies).
- Feynman, R. (1985). QED: The Strange Theory of Light and Matter (path integral ↔ least action).

Document Version: 1.0
Last Updated: 2025-12-12
License: CC BY 4.0 (documentation), to be determined for research publication

For questions or collaboration: See SIL repository for contact information.

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