Inside Generative Engines:
A Mathematical and System-Level Breakdown

1. The Generative Engine as a Function

At its core, a generative engine is a mapping from a user query to a response:

fGE: (qu, PU) → r

where q_u is the user's query, P_U is the personalization context (such as location or intent history), and r is the generated response (structured text with inline citations).

Unlike a classical search engine that ranks documents, a GE synthesizes an answer by reading, reasoning, and rewriting through multiple neural modules.

2. The Multi-Model Pipeline

A modern GE is a composition of specialized subsystems:

fGE = Gresp ∘ Gsum ∘ SE ∘ Gqr

2.1 Query Reformulation (G_qr)

Expands q_u into semantically diverse sub-queries:

Q1 = {q1, q2, …, qn} ∼ p(Q1 | qu; θqr)

Each q_i represents a decomposed intent of the original query.

2.2 Retrieval Engine (SE)

Fetches a ranked set of sources using information retrieval:

S = {s1, s2, …, sm} ∼ p(S | Q1; θret)

2.3 Summarization Model (G_sum)

Compresses each document into a short, citation-ready summary using automatic summarization:

Sumj = Gsum(sj), αj = |Sumj| / |sj|

2.4 Response Synthesizer (G_resp)

Constructs the final response:

r = Gresp(qu, Sum)

Each factual unit in r is grounded in the retrieved sources through inline citations.

3. Sentence-Level Structure and Citations

Let the response be a sequence of o sentences:

r = ⟨ℓ1, ℓ2, …, ℓo⟩

Each sentence ℓ_t is annotated with a citation set C_t ⊆ S.

For attribution integrity:

• Citation precision is the fraction of citations that truly support ℓ_t.
• Citation recall is the fraction of factual claims in ℓ_t that are cited.

An ideal generative engine maximizes both.

4. Quantifying Visibility Inside a Generative Response

Visibility in a generative engine is embedded within the synthesized text. It is not defined by rank but by where and how much a source contributes to the generated answer.

4.1 Word-Share Impression

Impwc(ci, r) = (Σs∈S_ci |s|) / (Σs∈S_r |s|)

where S_ci is the set of sentences citing c_i and |s| is the number of words in s.

4.2 Position-Weighted Impression

To model attention decay across sentences:

Imppwc(ci, r) = (Σs∈S_ci |s| · e-pos(s)/|S|) / (Σs∈S_r |s|)

This approximates reading probability as a function of position in the generated text.

4.3 Subjective Impression

LLM-based evaluators score each citation across six dimensions:

Subj(ci) = [Rel, Inf, Uniq, Pos, Click, Div]
Impsubj(ci) = Σk wk · Subjk(ci)

where each weight w_k is normalized such that Σ_i Imp(c_i, r) = 1.

5. Optimization Objectives

The generative engine optimizes for expected answer quality:

maxr E[f(Imp(ci, r), Rel(ci, q, r))]

Content creators, on the other hand, optimize visibility:

maxci Imp(ci, r)

This dual optimization forms the basis of Generative Engine Optimization (GEO).

6. Measuring Visibility Change

After a content update, visibility improvement is defined as:

Improvesi = (Impsi(r') - Impsi(r)) / Impsi(r) × 100

Empirically, factual enrichment and structural clarity yield the highest lifts, confirming that GEs reward grounded and information-dense content rather than keyword repetition.

7. Probabilistic Model of Answer Generation

Each GE stage is a stochastic mapping:

Q1 ∼ p(Q1 | qu; θqr) S ∼ p(S | Q1; θret) Sum ∼ p(Sum | S; θsum) r ∼ p(r | qu, Sum; θresp)

The overall likelihood of producing r given q_u is:

p(r | qu) = ΣQ1,S,Sum p(Q1 | qu) p(S | Q1) p(Sum | S) p(r | qu, Sum)

8. DAG Representation

qu → Q1 → S → Sum → r

Each node performs a transformation, and each edge defines a conditional probability distribution.

Critical hyperparameters include fan-out size n, retrieval depth k, summarization ratio α, answer length L, and citation density d_c.

9. Why GEO Works

Optimized content affects two conditional probabilities:

Imp(si, r) ∝ P(si ∈ S | qu) × P(si ∈ Ct | si ∈ S)

By improving both retrieval likelihood and synthesis attribution, GEO enables even lower-ranked sources to capture higher visibility in final LLM answers.

10. Multi-Turn Extension

For conversational engines, context history H = ⟨(q_t, r_t)⟩_t=1^T conditions the next response:

rT+1 ∼ p(rT+1 | H; θ)

This defines a temporal generative process that continuously updates latent context distributions.

11. Computational Characteristics

If k is the number of retrieved sources and L is the total token length, then inference cost is approximately:

O(k · Lsum + L2)

Summarization is linear in document size, while synthesis scales quadratically with attention, explaining why most engines restrict k ≤ 5 and compress summaries aggressively.

12. Summary

Generative engines are probabilistic pipelines that optimize for contextual answer quality under strict latency and memory constraints.

Understanding their mathematical structure is essential to improving your brand's visibility within AI-driven ecosystems.