Quantum Brand Modeling (QBM) Framework & Case Study

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Introduction:
The Quantum Brand Modeling (QBM) System is a structured framework designed to model how a brand exists, behaves, and competes within AI-driven ecosystems such as ChatGPT, Gemini, Perplexity, and enterprise copilots.

Unlike traditional branding or SEO models that rely on static rankings or keyword visibility, QBM represents a brand as a probabilistic state distributed across AI systems, endpoints, categories, and contexts.

By applying concepts inspired by quantum systems—such as state vectors, distributions, and mixed competitive states—QBM enables a multidimensional understanding of brand presence, allowing businesses to visualize how their brand appears in different AI-generated responses, query intents, and competitive environments.

Why it is needed:

Traditional digital marketing frameworks are no longer sufficient because AI systems do not operate on simple rankings—they generate answers based on context, probability, and semantic relevance.

QBM is needed because it:

1. Captures AI-native visibility (not just search rankings)

Brands are now surfaced inside AI responses across multiple platforms and contexts. QBM models this as a dynamic probability distribution, giving a more accurate picture of real visibility.

2. Enables context-level strategic insights

A brand may perform well in informational queries but poorly in direct-intent queries. QBM identifies these differences, allowing precise optimization based on user intent and context, not just traffic.

3. Reveals true AI competition

Your real competitors in AI are not always your traditional competitors. QBM identifies overlap in AI-generated spaces, showing where brands actually compete inside AI systems.

4. Bridges multiple AI ecosystems

Visibility is fragmented across:

ChatGPT
Gemini
Copilots
Perplexity

QBM unifies them into a single coherent model, enabling cross-platform strategy.

5. Transforms brand strategy into a measurable system

Instead of intuition-based branding, QBM provides:

measurable states
quantifiable dominance
actionable optimization paths

How we will do it:

(Stage 1) for Quantum Brand Baseline Simulation (Initial AI Visibility State)

Takes your inputs: HubSpot AI Grader scores for ChatGPT / Gemini / Perplexity + SEMrush AI visibility score

Sample code:

Here is the Google Colab Experiment: https://colab.research.google.com/drive/10-MG03KJEM7ozc57MWR78vf5yxLhISRl

Input:

Output:

State interpretation

Combined_Signal_0_1 blends:
- Platform-specific HubSpot visibility
- Global SEMrush AI visibility
Then converts the combined signals into a probabilistic state:
- Probability pᵢ = share of your visibility across AI systems
- Amplitude |ψᵢ| = √pᵢ (a “quantum-like” state vector form)

QBM Stage 1.2 — Brand-as-Probabilistic-State Definition Framework

Input: QBM_Stage1_Quantum_Brand_Baseline.xlsx (from Stage 1)

Sample code:

Here is the Google Colab experiment:

https://colab.research.google.com/drive/17jxZYGrV7Gszq5lr43bw3Gy0XEBINZZl

Here is the output:

Stage 1.2 Output Explanation

Brand-as-Probabilistic-State Definition Framework

This stage answers one core question:

“What is the current probabilistic state of my brand across AI systems?”

Instead of “scores,” you now have a state that can later be transformed, compared, and competed.

1️⃣ State Vector Table (Excel: State_Vector)

Columns

Column	Meaning
AI_System	Each AI surface where your brand can exist (ChatGPT, Gemini, Perplexity)
Probability_p_i	Share of your total AI visibility captured by that system
Amplitude_psi_i	Square-root of probability → quantum-style state amplitude

Interpretation

Probability answers:

“If an AI mention of my category happens, how likely is it to come from this system?”

Amplitude allows future state transformations (basis shifts, interference, context weighting).

📌 Example interpretation:

ChatGPT p = 0.52 → dominant AI surface
Gemini p = 0.31 → secondary influence
Perplexity p = 0.17 → long-tail discovery layer

Later stages modify amplitudes, not raw scores.

2️⃣ Probability Bar Chart

(Visual: QBM_stage1_2_probability_state.png)

What it shows

A distribution, not a ranking
Makes imbalance immediately visible

Why it matters

This diagram becomes your:

Baseline snapshot
Reference point for:
- before vs after optimization
- competitor overlap
- AI-model updates

📌 If one bar dominates → platform dependence risk

3️⃣ State Vector (Amplitude) Chart

(Visual: QBM_stage1_2_state_vector.png)

What this is

Same data, different representation
Amplitudes are what get “rotated” in future steps

Why amplitudes matter

In later stages:

Context changes = amplitude re-weighting
Category expansion = amplitude redistribution
Prompt bias = amplitude skew

This is the mathematical object the QBM system operates on.

4️⃣ Density Matrix ρ

(Excel: Density_Matrix + Heatmap QBM_stage1_2_density_matrix.png)

What it represents

The full brand state, including cross-system coherence

Mathematically:

ρ=∣ψ⟩⟨ψ∣\rho = |\psi\rangle\langle\psi|ρ=∣ψ⟩⟨ψ∣

How to read it

Diagonal elements → individual system strength
Off-diagonal elements → cross-system influence / coherence

📌 High off-diagonal values mean:

“Strength in one AI system reinforces visibility in another”

Low off-diagonal values mean:

“Systems behave independently”

This becomes critical when:

Comparing competitors
Detecting fragmentation
Modeling AI ecosystem shifts

5️⃣ Simplex Triangle (3-System Projection)

(Visual: QBM_stage1_2_simplex.png)

What this shows

A geometric location of your brand in AI-space

Each corner = 100% dominance in one AI
Center = perfect balance
Edges = two-system dominance

📌 Strategic meaning:

Near a corner → platform lock-in
Near center → robust multi-AI presence
On an edge → binary dependency

This visual becomes extremely powerful when:

Overlaying competitors
Tracking movement over time

6️⃣ State Metrics Table

(Excel: State_Metrics)

Key metrics explained

🔹 Entropy

Measures distribution uncertainty
High entropy = spread, low control
Low entropy = focus, but risk

🔹 Normalized Entropy (0–1)

Comparable across brands and categories

🔹 Purity (Tr(ρ²))

= 1 → pure, uncontested state
< 1 (future stages) → mixed / competitive state

Right now it’s pure because:

You haven’t added competitors yet

🔹 HHI Concentration

Economic concentration metric
0.45 = dangerous dependence
<0.33 = healthy distribution

🔹 Top System + Share

Your primary AI leverage point

QBM Stage 1.3 — (Stage 1.3): AI-System Mapping

Input: QBM_Stage1_2_Brand_State_Framework.xls (Output of the module 1.2)

Sample code:

Here is the google colab experiment link:
https://colab.research.google.com/drive/1fJjLc9dP8sEzsYdk3gIn9a2jeURWP0Lw

Output:

What this step is doing overall:
In Stage 1.2, your brand existed in a simple state like:

ChatGPT
Gemini
Perplexity

That is useful, but still too broad.

In Stage 1.3, we break those into real AI environments such as:

OpenAI ChatGPT consumer
OpenAI API
Microsoft Copilot
Gemini consumer
Gemini for Workspace
Vertex AI / Gemini API
Perplexity consumer
Perplexity Pro
Perplexity enterprise/API

So this step answers:

Where exactly does my AI visibility live inside each AI ecosystem?

Input used in this step

This program reads the Excel from the previous step:

Input file

QBM_Stage1_2_Brand_State_Framework.xlsx

Sheets it uses

Primarily:

State_Vector
State_Metrics

From State_Vector, it reads:

AI_System
Probability_p_i
Amplitude_psi_i

That means it takes the broad probability from Stage 1.2 and redistributes it into more detailed endpoints.

Main outputs you get

This step creates:

Excel file

QBM_Stage1_3_AI_System_Mapping.xlsx

Visual files

QBM_stage1_3_endpoint_probability.png
QBM_stage1_3_system_endpoint_composition.png
QBM_stage1_3_mapping_flow.png

Step-by-step explanation of each output

1) System_to_Endpoint_Mapping sheet

This is the mapping logic table.

It shows how each high-level AI system is broken into smaller endpoint surfaces.

Example idea

For ChatGPT, the program may split it into:

OpenAI ChatGPT (Consumer)
OpenAI API (Developer Apps)
Microsoft Copilot (M365)
Microsoft Copilot (Windows/Edge)

Important columns

AI_System

The original top-level AI system from Stage 1.2.

Examples:

ChatGPT
Gemini
Perplexity

Endpoint

A more specific AI touchpoint or environment inside that ecosystem.

Examples:

OpenAI ChatGPT (Consumer)
Gemini for Workspace
Perplexity Pro

Endpoint_Weight_within_System

This is the percentage split inside that AI system.

For example, if ChatGPT has 0.40 overall probability, and:

Consumer ChatGPT weight = 0.55
OpenAI API = 0.15
M365 Copilot = 0.20
Windows/Edge Copilot = 0.10

Then the 0.40 is distributed across those endpoints using those weights.

Endpoint_Type

This classifies each endpoint.

Examples:

Consumer
API
Enterprise Copilot

This helps later when you want to compare:

B2B visibility
B2C visibility
API exposure
enterprise assistant exposure

Notes

A short explanation of what that endpoint means.

Why this sheet matters

This is the bridge between broad AI visibility and practical business interpretation.

Without this sheet, “Gemini” is just one number.

With this sheet, you can say things like:

My Gemini visibility is mostly in consumer-facing Gemini
My ChatGPT presence also spills into Microsoft Copilot
My Perplexity share is concentrated in search-like discovery

2) Endpoint_State_Vector sheet

This is the main output of the step.

It tells you your new brand state across the expanded endpoint space.

You can think of this as:

Stage 1.2 gave you a 3-dimensional state
Stage 1.3 turns it into a 9- or 10-dimensional state

Key columns

AI_System

The parent system the endpoint belongs to.

Endpoint

The specific endpoint/surface.

Endpoint_Type

Consumer / API / Enterprise Copilot.

Base_System_Probability

This is the original probability from Stage 1.2.

Example:

ChatGPT = 0.50
Gemini = 0.30
Perplexity = 0.20

This value is copied to all endpoints under that system before splitting.

Endpoint_Weight_within_System

The internal split ratio.

Example:
If ChatGPT = 0.50 and consumer ChatGPT has weight 0.55, then:

endpoint probability = 0.50 × 0.55 = 0.275

Endpoint_Probability

This is the most important column in this sheet.

It tells you:

“How much of the total AI visibility state belongs to this exact endpoint?”

This is your brand’s endpoint-level probability.

Endpoint_Amplitude_psi

This is the square root of the endpoint probability.

Same logic as earlier:

Probability is used for interpretation
Amplitude is used for future state transformations

This matters later when:

you apply context weighting
compare competitive overlap
rotate the state into category-specific bases

Notes

Description of the endpoint.

Why this sheet matters

This is your expanded brand-state model.

It lets you answer questions like:

Which AI endpoint currently carries the most brand presence?
Is my visibility concentrated in consumer assistants only?
Do I have enough enterprise assistant presence?
How much of my state exists in APIs versus chat products?

3) Endpoint_State_Metrics sheet

This sheet summarizes the whole expanded endpoint state.

It compresses the endpoint distribution into interpretable metrics.

Metrics explained

Brand

Your brand/domain label.

Num_Endpoints

How many endpoint surfaces are included in the mapped state.

Example:

9 endpoints total

This matters because a higher number means a more detailed state space.

Endpoint_Entropy_bits

Measures how spread out your brand is across endpoints.

High entropy means:

your visibility is more distributed
less dependent on one endpoint

Low entropy means:

more concentrated
possibly stronger focus, but more risk

Endpoint_Normalized_Entropy_(0-1)

Same idea, but scaled to a standard range.

This is more useful when comparing different brands or future versions of the model.

Interpretation:

closer to 1 = balanced spread
closer to 0 = concentrated on a few endpoints

Endpoint_HHI_Concentration_sum(p^2)

This is a concentration score.

Higher HHI means:

a few endpoints dominate

Lower HHI means:

more evenly distributed presence

This is useful for platform dependency analysis.

Top_Endpoint

The single endpoint where your brand has the highest probability.

Example:

OpenAI ChatGPT (Consumer)

This tells you your strongest AI exposure surface.

Top_Endpoint_Share

The actual probability share of that top endpoint.

Example:

0.31 means 31% of the total expanded state sits in that endpoint

Timestamp

When the model was run.

Useful for tracking changes across runs.

Why this sheet matters

It gives you a quick executive summary of the expanded state without having to inspect every row.

This is useful for dashboards and reporting.

4) QBM_stage1_3_endpoint_probability.png

This is the endpoint probability bar chart.

What it shows

Every endpoint gets one bar.

The bar height = Endpoint_Probability

What you learn from it

You instantly see:

which endpoint dominates
which endpoints are weak
whether your brand is concentrated or balanced
whether enterprise/API surfaces are underrepresented

Example interpretation

Suppose the tallest bars are:

OpenAI ChatGPT (Consumer)
Google Gemini (Consumer)

and the smaller ones are:

Vertex AI / Gemini API
Perplexity API/Enterprise

That means:

your current AI presence is mostly consumer-facing
you have weaker enterprise/developer visibility

Why this chart matters

This is the cleanest visual answer to:

“Where does my AI visibility actually live?”

It is much more actionable than the Stage 1.2 system-level chart.

5) QBM_stage1_3_system_endpoint_composition.png

This is the stacked composition chart.

What it shows

Each top-level AI system is one bar.

That bar is divided into endpoint segments.

So instead of showing the total only, it shows the internal composition.

Example

A ChatGPT bar may be split into:

ChatGPT consumer
OpenAI API
M365 Copilot
Windows/Edge Copilot

A Gemini bar may be split into:

Gemini consumer
Gemini for Workspace
Vertex AI

What this helps you understand

It answers:

How is each system internally distributed?
Is ChatGPT mostly consumer or enterprise for this model?
Is Gemini mostly Workspace or public assistant?
Does Perplexity behave mostly like search discovery or enterprise usage?

Why this chart matters

This chart is useful when you want to inspect the internal shape of each ecosystem.

The endpoint bar chart tells you absolute endpoint size.

The stacked chart tells you composition within each parent system.

So:

endpoint bar chart = overall endpoint comparison
stacked composition = within-system breakdown

6) QBM_stage1_3_mapping_flow.png

This is the flow diagram.

What it shows

It draws connections from:

left side: AI systems
right side: endpoints

Each line represents one mapping connection.

Thickness of the line

The thicker the line, the larger the resulting endpoint probability.

So thick lines show where the most visibility mass flows.

What it means strategically

This helps you see how high-level system visibility gets distributed.

For example:

A thick line from ChatGPT to OpenAI ChatGPT consumer means most ChatGPT visibility sits there
A thinner line from ChatGPT to OpenAI API means less of that system’s visibility is represented in developer/API contexts

Why this chart matters

This is your best visual for understanding the state expansion process itself.

It is especially useful for presentations because it explains:

where the original state came from
how it was redistributed
what the endpoint structure looks like

7) Mapping weight check table

Before expanding the probabilities, the program checks whether endpoint weights for each system sum to 1.

Example:
For ChatGPT:

0.55 + 0.15 + 0.20 + 0.10 = 1.00

This ensures the system’s probability is distributed correctly.

Why it matters

If weights do not sum to 1, the model becomes mathematically inconsistent.

The program also auto-normalizes if needed.

So this step protects the quality of the output.

8) Carry-forward sheets in the output Excel

The final Excel also carries older sheets forward, such as:

Stage1_2_State_Vector
Stage1_2_Density_Matrix
Stage1_2_State_Metrics
Stage1_State_Data
Stage1_Summary

Why this matters

This makes the workbook cumulative.

So later stages can use one file as the full working artifact instead of requiring many separate files.

How to interpret the outputs together

These outputs work as a set.

Together they tell you:

First

What broad AI systems your brand occupies
from Stage 1.2

Then

How those systems expand into specific operational AI surfaces
from Stage 1.3

Then

How concentrated or diversified that expanded state is
from Endpoint_State_Metrics

Then

Where your strongest and weakest endpoint presence exists
from the charts and Endpoint_State_Vector

Practical business meaning of this step

This step helps you answer strategic questions like:

Consumer dominance

Are we mostly visible in public assistants but weak in workplace copilots?

Enterprise gap

Do we have low presence in M365 Copilot or Gemini for Workspace?

Developer gap

Are we underrepresented in API-driven environments?

Platform dependency

Is too much of our AI visibility sitting in one endpoint?

Visibility architecture

Does our AI presence exist as a broad ecosystem footprint, or is it narrow and fragile?

In one line, what each output is for

System_to_Endpoint_Mapping

Defines the model architecture.

Endpoint_State_Vector

Your actual expanded AI visibility state.

Endpoint_State_Metrics

Summary of concentration, spread, and dominance.

endpoint_probability.png

Shows which endpoints are strongest.

system_endpoint_composition.png

Shows how each AI system is internally structured.

mapping_flow.png

Shows how visibility mass flows from systems into endpoints.

Why this step is important before the next stage

You need this step before:

category boundary definition
enterprise vs consumer weighting
competitor overlap modeling
AI system prioritization

Because those later steps should act on actual endpoint surfaces, not only on top-level systems.

Top-level systems are too abstract for competitive modeling.

Endpoint-level state is much more operational.

QBM Stage 1.4 — (Stage 1.4): Core Category & Context Boundary Definition

Input: QBM_Stage1_3_AI_System_Mapping.xlsx(Output of module 3)

Sample code:

Here is the Google Colab Experiment link: https://colab.research.google.com/drive/1t73GhAT39uQkKE0nQCvD4N4zHUKKDDGY

Input sample:

Outputs:

This stage is where your model stops being only an AI visibility map and becomes a semantic positioning model.

Up to Stage 1.3, the system knew:

which AI systems matter
which endpoints matter
how your brand is distributed across them

Now Stage 1.4 adds:

what category your brand belongs to
how tightly it belongs there
in what context AI is likely to surface it

So this step answers:

“In which category space and in which query-intent contexts does this brand exist inside AI systems?”

1. What goes into this step

This notebook takes the output from the previous stage:

Input file

QBM_Stage1_3_AI_System_Mapping.xlsx

Main sheet used

Endpoint_State_Vector

That means it starts with the endpoint-level state you already built, such as:

OpenAI ChatGPT (Consumer)
Gemini for Workspace
Perplexity Pro
Microsoft Copilot

Each of those already has a probability.

Then Stage 1.4 adds a new layer:

category boundaries
context boundaries

2. What new inputs you gave

You entered:

Brand / Domain label
Core Category
Adjacent Categories
Peripheral Categories
Category strengths
Context weights

These are the conceptual inputs that shape the semantic space.

3. What the program is doing conceptually

It takes your endpoint state and asks:

First:

How strongly does this brand belong to:

the core category
adjacent categories
peripheral categories

Then:

Within the core category, how does the brand behave across:

Direct Intent
Commercial
Informational
Comparative contexts

So instead of only saying:

“The brand is strong in ChatGPT consumer”

it can now say:

“The brand is strongest in ChatGPT consumer under direct-intent and informational contexts within the core category.”

That is much more useful.

4. Main outputs you get

This stage creates:

Excel file

QBM_Stage1_4_Core_Category_Context_Boundary.xlsx

Visual files

QBM_stage1_4_category_boundary_strength.png
QBM_stage1_4_core_context_distribution.png
QBM_stage1_4_top_endpoints_in_core_category.png
QBM_stage1_4_endpoint_context_heatmap.png

5. Step-by-step explanation of each output

Output 1: Category_Layers

This sheet defines the category structure of your brand.

What it contains

It lists category layers like:

Core
Adjacent
Peripheral

Along with:

category names
category strength values

Example

For your use case, it may look like:

Core → Psychological Clinic
Adjacent → Psychological therapy
Adjacent → Individual therapy
Adjacent → Psychological therapist
Peripheral → Mental health awareness
Peripheral → Stress management
Peripheral → Self-help
Peripheral → Emotional wellbeing

Category_Strength

This is the numerical strength you entered, converted to 0–1.

For example:

Core = 90 becomes 0.90
Adjacent = 70 becomes 0.70
Peripheral = 30 becomes 0.30

What it means

This sheet defines how tightly your brand is associated with each layer of meaning.

Why it matters

This is the semantic boundary architecture of your brand.

Without it, the model only knows platform visibility.
With it, the model knows:

where the brand belongs strongly
where it belongs partially
where it is only loosely relevant

Output 2: Context_Weights

This sheet defines the query-intent context model.

What it contains

It lists contexts like:

Direct Intent
Commercial
Informational
Comparative

and their weights.

Example

If you entered:

Direct Intent = 90
Commercial = 70
Informational = 75
Comparative = 60

the sheet stores them as:

0.90
0.70
0.75
0.60

What it means

This shows the likelihood that your brand appears in each context type.

Why it matters

Brands are not surfaced equally across all query types.

For example:

A clinic may be strong in direct-intent queries
A blog might be stronger in informational queries
A SaaS brand might be stronger in commercial/comparative queries

This sheet tells the model where your brand is expected to appear.

Output 3: Core_Context_State

This is the main state table of the stage.

This is the most important output.

It combines:

the endpoint probabilities from Stage 1.3
the context weights
the core category strength

What it contains

Each row represents a combination of:

one endpoint
one context

So if you had 10 endpoints and 4 contexts, you get 40 rows.

Main columns explained

Brand

Your brand label.

Core_Category

The main category you entered.

AI_System

Parent ecosystem, such as ChatGPT, Gemini, Perplexity.

Endpoint

Specific endpoint, such as:

OpenAI ChatGPT (Consumer)
Gemini for Workspace
Perplexity Pro

Endpoint_Type

Consumer, API, Enterprise Copilot, etc.

Context

The query-intent bucket:

Direct Intent
Commercial
Informational
Comparative

Base_Endpoint_Probability

This comes from Stage 1.3.

It is the original endpoint-level probability before context is applied.

Context_Weight

This is your assigned context weight.

Category_Strength

This is the strength of the core category.

Weighted_Context_Mass

This is the first weighted result.

It is computed as:

Endpoint_Probability × Context_Weight × Core_Category_Strength

What that means

It estimates how much of your brand state exists in that exact combination of:

endpoint
context
core-category membership

Normalized_Context_Probability

This normalizes all rows so the full core-context state sums to 1.

This becomes the new probability distribution.

Context_Amplitude_psi

Square root of normalized probability.

This creates a quantum-style amplitude for later transformations.

Why Core_Context_State matters

This is the point where your brand becomes a context-conditioned state.

The model now knows not only:

where you appear

but also:

in what kind of AI answer context you appear

This is essential for:

competitive overlap
context-specific optimization
future interference modeling

Output 4: Core_Context_Summary

This summarizes the total distribution across contexts.

What it shows

It groups the core-context state by context and sums the probabilities.

So you get something like:

Direct Intent = 0.34
Informational = 0.29
Commercial = 0.22
Comparative = 0.15

What it means

This answers:

“Inside the core category, which context type dominates the brand’s AI presence?”

How to interpret it

If Direct Intent is highest:

your brand is strongest when users are actively searching for a service/provider

If Informational is highest:

your brand appears more in educational/explanatory AI answers

If Comparative is high:

your brand is often being surfaced alongside alternatives

Why it matters

This is one of the most strategic outputs of the stage.

It tells you what kind of AI demand-space your brand occupies.

Output 5: Core_Endpoint_Summary

This summarizes the total core-category probability by endpoint.

What it shows

It groups by:

AI system
endpoint
endpoint type

and sums the normalized probabilities across all contexts.

What it answers

Inside the core category, which endpoints matter most?

For example:

OpenAI ChatGPT (Consumer)
Gemini Consumer
Gemini for Workspace
M365 Copilot

Why it matters

Stage 1.3 told you the strongest endpoints overall.

This sheet tells you:

“Which endpoints matter most after category and context are applied?”

That is more meaningful because raw endpoint strength and category-conditioned endpoint strength are not always the same in future versions of the model.

Output 6: Endpoint_Context_Matrix

This is a matrix view of the core context state.

Structure

rows = endpoints
columns = contexts
values = normalized probabilities

Example

A row for OpenAI ChatGPT (Consumer) may have values like:

Direct Intent = 0.08
Commercial = 0.06
Informational = 0.07
Comparative = 0.05

What it means

It shows how each endpoint distributes across contexts.

Why it matters

This is one of the most useful technical outputs for later stages.

It will be useful for:

competitor overlap
endpoint-specific context weakness
interference modeling
optimization targeting

You can use it to answer things like:

which endpoint is best for commercial presence?
where are we weak in comparative context?
which endpoint carries most of our informational visibility?

Output 7: Boundary_State_All_Layers

This is the broader category-boundary state.

What it contains

It computes boundary mass not only for the core category, but also for:

adjacent categories
peripheral categories

combined with all contexts.

What it means

It gives a larger semantic map of the brand beyond the core category only.

Why it matters

The core category is your main identity.

But AI sometimes places brands into nearby or broader semantic areas.

This sheet helps capture that broader semantic spread.

Output 8: Boundary_Summary

This summarizes the total mass across category layers.

What it shows

It groups boundary mass by:

category layer
category name

and totals the values.

What it answers

How much of the brand’s semantic mass sits in:

core
adjacent
peripheral areas

Interpretation

If the Core category dominates:

the brand is tightly defined
strong semantic precision

If Adjacent is large:

the brand has broader relevance nearby

If Peripheral is too large:

the brand identity may be diffuse or overly broad

Why it matters

This is your main output for understanding category focus vs category diffusion.

Output 9: Boundary_Metrics

This is the executive summary sheet for the whole stage.

It contains the key metrics.

Let’s explain each one.

Brand

Your brand label.

Core_Category

The central category used in this model run.

Num_Endpoints

How many endpoints are included from Stage 1.3.

Num_Context_States

How many endpoint-context combinations exist.

Usually:
number of endpoints × number of contexts

Boundary_Tightness_(Core_vs_Total)

This is a simple measure of how dominant the core category is relative to all category layers.

Interpretation

Higher value means:

stronger semantic focus
tighter category identity

Lower value means:

more semantic spread across adjacent/peripheral zones

Why it matters

This is one of the most important stage-level KPIs.

It tells you whether the brand is sharply defined or semantically diffuse.

Core_Context_Entropy_bits

Measures how spread out the core-context state is.

High entropy:

more distributed across many endpoint-context combinations

Low entropy:

concentrated in fewer combinations

Core_Context_Normalized_Entropy_(0-1)

A normalized version of entropy so you can compare across runs.

Higher:

more spread

Lower:

more concentrated

Core_Context_HHI_Concentration

Measures concentration of the core-context state.

Higher HHI:

more concentrated
dependence on fewer endpoint-context pairs

Lower HHI:

more balanced distribution

Top_Context

The strongest context for the brand in the core category.

For example:

Direct Intent

What it means

This is the context where the brand has the strongest modeled presence.

Top_Context_Share

How much of the total core-context probability sits in that top context.

For example:

0.33 means 33%

Top_Endpoint_within_Core_Category

The strongest endpoint after category and context weighting.

This may or may not match the top endpoint from Stage 1.3.

Top_Endpoint_Share_within_Core_Category

The share of the total core-category state that belongs to that endpoint.

Context_Dispersion_(0-1)

This measures how balanced the brand is across the context buckets themselves.

High dispersion:

context presence is spread across multiple intents

Low dispersion:

one or two contexts dominate strongly

Why it matters

This tells you whether your brand appears across many AI use-cases or is concentrated in a narrower intent profile.

6. Visual outputs explained

Visual 1: QBM_stage1_4_category_boundary_strength.png

This is the category boundary strength chart.

What it shows

A bar chart of the total boundary mass for:

core category
adjacent categories
peripheral categories

What it means

It shows the brand’s semantic gravity across category layers.

How to interpret

If the Core bar is much taller:

strong semantic focus
well-defined category identity

If Adjacent is also substantial:

good expansion potential into related areas

If Peripheral is too high:

broader awareness, but weaker category clarity

Visual 2: QBM_stage1_4_core_context_distribution.png

This is the context distribution chart.

What it shows

A bar chart of the total normalized probability across:

Direct Intent
Commercial
Informational
Comparative

What it means

This is the easiest visual answer to:

“In what type of AI answer context does this brand mainly live?”

Example

If Direct Intent is tallest:

strongest in service-seeking queries

If Informational is tallest:

strongest in educational or advice-oriented answers

Visual 3: QBM_stage1_4_top_endpoints_in_core_category.png

This chart shows the top endpoints within the core category.

What it shows

The endpoints with the highest total normalized probability after context and category weighting.

What it means

This identifies your strongest operational AI surfaces inside your most important category.

Why it matters

This is more actionable than overall endpoint strength because it is already filtered through:

category fit
context relevance

Visual 4: QBM_stage1_4_endpoint_context_heatmap.png

This is the endpoint × context heatmap.

What it shows

rows = endpoints
columns = contexts
color intensity = normalized probability

What it means

It shows where each endpoint is strongest across contexts.

Why it matters

This is very useful diagnostically.

You can visually identify:

strong endpoint-context zones
weak zones
imbalances
potential optimization targets

For example:

strong Direct Intent in ChatGPT consumer
weaker Commercial presence in Perplexity Pro
low Comparative presence in enterprise copilots

7. How all outputs fit together

Here is the logic of the outputs as a system.

Category_Layers

Defines semantic category architecture.

Context_Weights

Defines query-intent architecture.

Core_Context_State

Creates the actual category+context-conditioned brand state.

Core_Context_Summary

Shows dominant contexts.

Core_Endpoint_Summary

Shows dominant endpoints inside the core category.

Endpoint_Context_Matrix

Shows endpoint-by-context structure.

Boundary_Summary

Shows category spread across core, adjacent, peripheral.

Boundary_Metrics

Gives executive KPIs for this stage.

Charts

Make category concentration and context dominance visually obvious.

8. Strategic meaning of this stage

This stage tells you:

Category clarity

Is the brand tightly associated with its main category?

Semantic spread

How much does it extend into nearby or broader topics?

Context dominance

Does the brand mainly surface in direct, commercial, informational, or comparative AI answers?

Endpoint-context interaction

Which AI surfaces matter most for which kinds of queries?

9. Why this stage matters before the competitor stage

The next stage is:

Competitive AI Landscape Scoping

You need this stage first because competition in AI is not just:

brand vs brand

It is:

brand vs brand
within a category
under specific contexts
across different AI endpoints

A competitor may be strong:

in comparative context but not direct intent
in informational context but not commercial
in Perplexity but not Gemini Workspace

Stage 1.4 creates the structure needed to model that properly.

10. Simplified example

Suppose after running this stage, your outputs imply:

Core category is dominant
Direct Intent is the strongest context
Top endpoint is OpenAI ChatGPT consumer
Informational is second strongest
Peripheral boundary mass is low

That means:

your brand is semantically well-defined, strongly tied to its primary category, and most likely to appear in AI when people are actively seeking that service, especially through major consumer AI assistants.

That is the kind of interpretation this stage is meant to enable.

QBM Stage 1.4 — (Stage 1.4): Core Category & Context Boundary Definition.

What this stage does

Until now:

You modeled your brand only

Now:

You introduce competitors
Convert your model from a pure state → mixed competitive state
Identify:
- overlap
- dominance zones
- context competition
- AI visibility gaps

Input from previous stage

Uses:

QBM_Stage1_4_Core_Category_Context_Boundary.xlsx

Main sheet used:

Core_Context_State

Sample code:

Here is the Google Colab Experiment link:
https://colab.research.google.com/drive/1ztVfuA5uXxxT5wDH-e0zy9jjKjIcYl63

Input:

Big Picture of Stage 1.5

Before:

You had a pure state (only your brand)

Now:

You have a mixed competitive state
Multiple brands coexist in the same AI space

This stage answers:

“Where do I compete, where do I win, and where do I lose in AI visibility?”

Outputs You Get:

Excel file:

QBM_Stage1_5_Competitive_Landscape.xlsx

Charts:

Overlap Heatmap
Context Competition
Market Share

Step-by-step explanation of each output

Competitive_State (MAIN DATASET)

What it is

This is the full expanded dataset containing:

Your brand
All competitors
Across:
- endpoints
- contexts
- AI systems

Key columns

Brand

Each row belongs to:

your brand
or a competitor

Context

From previous stage:

Direct Intent
Commercial
Informational
Comparative

Endpoint

AI surfaces like:

ChatGPT consumer
Gemini Workspace
Copilot
Perplexity

Competitive_Probability

This is the most important column

It represents:

“How much of the total AI visibility space this brand occupies in this exact combination of endpoint + context”

What changed from previous stage?

Before:

Only your probabilities existed

Now:

Each competitor has its own probability distribution

So this is now a multi-brand state space

How to interpret

You can now answer:

Where does competitor X dominate?
Where does my brand appear less?
Which contexts are crowded?

Overlap_Matrix

What it is

A matrix comparing every brand with every other brand.

How it is calculated

For two brands:

It calculates how similar their distributions are

Using:

minimum overlap across all state points

What values mean

Value	Meaning
High (~0.7–1.0)	Strong competition / similar presence
Medium (~0.3–0.6)	Partial overlap
Low (~0–0.2)	Different positioning

Example interpretation

If:

You vs Competitor A = 0.85
→ You are directly competing in the same spaces
You vs Competitor B = 0.25
→ Different positioning or audience

Why this matters

This tells you:

Who your real AI competitors are (not just business competitors)

Visual: Overlap Heatmap

What it shows

Colors represent overlap intensity

How to read it

Bright = strong competition
Dark = weak competition

Context_Competition

What it is

A table showing:

How each brand performs across:

Direct Intent
Commercial
Informational
Comparative

Structure

Brand

Direct

Commercial

Informational

Comparative

What it means

Each number represents:

Share of visibility in that context

How to interpret

Example:

Brand	Direct	Informational
You	0.35	0.20
Competitor	0.25	0.40

Meaning:

You dominate direct intent
Competitor dominates informational

Why this matters

This tells you:

Where you are winning vs losing in user intent space

Visual: Context Competition Chart

What it shows

Bar chart per context
Each brand compared side-by-side

What to look for

Who dominates each context
Gaps where you can expand

Endpoint_Competition

What it is

Same idea as context, but across endpoints.

Structure

Brand

ChatGPT

Gemini

Copilot

Perplexity

What it means

Shows:

“Which AI platform each brand dominates”

Interpretation

Example:

You strong in ChatGPT
Competitor strong in Gemini

That means:

Different platform strategies
Different optimization opportunities

Why this matters

AI visibility is platform-dependent.

This shows:

Where to focus your optimization efforts

Market_Share

What it is

A simple aggregation:

Total probability per brand

What it represents

“Overall AI visibility share”

Interpretation

Brand	Share
You	0.30
Competitor A	0.40
Competitor B	0.30

Competitor A dominates AI presence overall

Important note

This is simulated, not real data.

But it still gives:

relative positioning
strategic direction

Visual: Market Share Chart

What it shows

Bar chart of total visibility

Use it for:

quick comparison
stakeholder presentations

Competitive_Metrics

This is the executive summary

Each row = one brand

Metrics explained

Entropy

Measures how spread the brand is.

High → broad presence
Low → concentrated

HHI

Measures concentration.

High → dependent on few areas
Low → diversified

Top_Context

Where the brand is strongest.

Examples:

Direct Intent
Informational

Top_Endpoint

Where the brand is strongest technically.

Examples:

ChatGPT Consumer
Gemini Workspace

Why this matters

This gives quick answers:

What kind of brand is this in AI?
Where does it dominate?
Is it broad or focused?

Most important insights from this stage

1. Real competition ≠ traditional competition

Some competitors:

overlap heavily in AI
even if they are not direct business competitors

2. Context matters more than platform

You might:

win in Direct Intent
lose in Informational

That defines strategy

3. Endpoint differences matter

You may:

dominate ChatGPT
be weak in Gemini

That shows platform gaps

4. Overlap reveals threat level

High overlap = direct threat
Low overlap = different positioning

How all outputs connect

Output	Purpose
Competitive_State	Full dataset
Overlap_Matrix	Who competes with you
Context_Competition	Where you compete
Endpoint_Competition	Where you operate
Market_Share	Who dominates
Metrics	Strategic summary