Why Accessibility Checkers Can't Tell You Who Will Struggle — And What Can
How a branch of 18th-century mathematics is changing the way we think about cognitive accessibility on the web.
A municipal government website passes every automated accessibility check. Contrast ratios? Fine. Alt attributes? Present. WCAG compliance: 98%.
Then a user with ADHD lands on the page. She needs to apply for housing assistance. She sees a navigation bar with 14 categories, a three-column layout with a scrolling news sidebar, 800 words of policy text, and a small "Apply Now" button below the fold.
She leaves after 23 seconds.
The accessibility checker has nothing to say about this. It checked the rules. The rules were followed. But the user still could not complete her task. Nothing was broken. The page just demanded more cognitive effort than her brain could comfortably provide.
This is the gap we set out to close.
The wrong question
Most accessibility tools ask: does this page violate a rule?
That's a useful question. But it's the wrong one if you care about whether real people can use your site. A page can follow every WCAG guideline and still be unusable for someone with ADHD, dyslexia, motor tremor, or cognitive impairment. The barriers are not code violations. They are cognitive mismatches.
The question that matters is: how much extra mental work does this specific type of user have to do on this specific page?
No rule-based checker can answer that. You need to know something about the user: their attention span, reading speed, tolerance for complexity, motor precision. You also need to know what the page demands on each of those same dimensions.
Then you measure the gap.
Two piles of sand
Optimal Transport theory is a branch of mathematics with practical roots. In 1781, French mathematician Gaspard Monge posed a simple question: what is the cheapest way to move a pile of earth from one place to another?
The modern version of this math came from Leonid Kantorovich, who won the Nobel Prize for it. He generalized the idea. Given two distributions of "stuff" — not just dirt, but anything that can be described as a shape spread across a space — the Wasserstein distance tells you the minimum effort needed to reshape one distribution into the other.
Here is the insight that connects this to accessibility.
Every user brings a distribution of cognitive abilities. Not a single number, but a profile. How long can they sustain attention? How fast do they process text? How precisely can they control a mouse? How patient are they with confusing layouts? We track about 26 dimensions like these. And because people within any group vary — not all users with ADHD are the same — we represent each group as a range of values, not a fixed point.
Every web page imposes a distribution of cognitive demands. Tiny buttons demand high motor precision. Dense legal text demands high reading ability. Fourteen competing navigation options demand strong selective attention.
The Wasserstein distance between these two distributions — what the user brings versus what the page requires — is the cognitive transport cost (CTC). It is a single, mathematically grounded number. It represents how much extra mental work a user must do to bridge the gap.
Small gap? Smooth experience. Large gap? Users struggle, make errors, get frustrated, and leave.
This isn't a metaphor
The natural reaction: "Nice analogy, but cognition is not literally moving dirt around."
The neuroscience suggests it is closer than you might think.
In 2022, researchers showed that the visual cortex performs computations along Wasserstein geodesics — the same geometric paths that optimal transport theory defines. In 2023, another team found that when the brain reorganizes spatial memories, the effort tracks proportionally with the earth mover's distance between old and new representations.
A landmark 2020 Nature paper revealed something even more striking. Dopamine neurons do not just encode an average expected reward. They maintain full distributions of expected outcomes — exactly the kind of structure that optimal transport operates on.
In 2025, researchers found that the Wasserstein distance between letter shapes predicts neural processing time at sub-225-millisecond latency. That is before you are even consciously aware of reading.
We do not claim the brain is an optimal transport computer. But growing evidence suggests OT captures something real about how we process information. That makes it a principled foundation for modeling cognitive effort — not just a convenient formalism.
Six kinds of difficulty
A single number — "this page is hard for this user" — is useful but not actionable. A designer needs to know why it is hard and where to fix it.
The framework breaks total CTC into six layers. Each one corresponds to a different stage of how users process a web page:
1. Saliency — What do they actually see? Not everyone perceives the same visual hierarchy. A user with ADHD has their attention pulled toward animations and high-contrast novelty. A user with low vision registers only large, high-contrast elements. This layer predicts where each user type will look — and whether that matches the designer's intent.
2. Cognitive load — Is the page overwhelming? This covers visual complexity, information density, and competing elements. It measures how close the page pushes a user toward their processing capacity. For a user with ADHD, that threshold is much lower than for a neurotypical power user.
3. Decision complexity — Are there too many choices? More options means harder decisions. But the penalty varies by user type. Research suggests users with ADHD face roughly 2.5x the decision cost per additional option compared to neurotypical users.
4. Motor accessibility — Can they physically interact? Small buttons, precise sliders, closely packed links. This layer models each user's click accuracy as a probability distribution. It flags elements where the chance of hitting the target drops below a safe threshold.
5. Frustration and abandonment — When will they give up? Every user carries expectations about how an interaction should go. When reality deviates — a form rejects input without explanation, a page loads slowly, navigation leads to a dead end — frustration builds. This layer predicts when that cumulative frustration exceeds a user's tolerance.
6. Readability — Can they process the text? This goes beyond Flesch-Kincaid reading level. It includes font effects (dyslexic users need sans-serif), text size (low-vision users need 14px+), and processing speed (dyslexic users require 2.5x longer fixation time per word).
Each layer produces its own score. Together, they answer not just that a page is inaccessible, but which users will struggle, at which step, and for what reason.
Fast enough to be useful
Mathematical elegance is worthless if the computation takes an hour. This framework runs in under a millisecond.
How? A deliberate modeling choice. We represent cognitive profiles as 26-dimensional Gaussian distributions. With that assumption, every Wasserstein distance, every persona comparison, and every adversarial worst-case analysis has a closed-form mathematical solution. No machine learning model to train. No GPU required. Pure math, running in TypeScript, on any laptop.
That speed makes real integration possible. Run it in CI/CD. Check every pull request. Monitor production pages continuously. Accessibility evaluation does not have to be a quarterly audit. It can be a continuous signal.
What this doesn't replace
This framework handles early-stage screening. It catches cognitive accessibility problems before real users encounter them. It simulates 21 different cognitive profiles — users with ADHD, dyslexia, motor tremor, low vision, autism, and cognitive impairment — each grounded in published psychometric research.
But simulation is not substitution. No mathematical model replaces the insight you get from watching a real person with a real disability use your product. The framework complements participatory evaluation. It does not replace it. Think of it as a spell checker for accessibility: it catches the obvious problems so your limited testing time with real users can focus on the subtle ones.
The mismatch framing matters
We are deliberate about one thing: this framework does not model disability as deficit. It frames accessibility problems as mismatches between what interfaces demand and what user populations bring.
The user is not broken. The interface is poorly fitted.
A power user on a simple page also incurs transport cost — forced through unnecessary steps, under-stimulated, expertise going unused. The math treats this symmetrically. Difficulty is not about the user being "less than." It is about the interface failing to meet the user where they are.
This distinction matters ethically and practically. When the problem is framed as mismatch, the solution is clear: change the interface, not the user.
What's next
We are writing this up as a research paper, with empirical validation against public eye-tracking datasets and a human study testing whether the framework's predictions match real users' reported cognitive load.
The framework is implemented in CBrowser, our open-source cognitive browser automation platform. You can run a cognitive journey simulation on your own site today.
We will share more as the research progresses. In the meantime, if you have ever wondered why your page passes every accessibility check but users still struggle — now you know there is math for that.
Cognitive Optimal Transport is part of CBrowser's cognitive simulation engine, available in both the open-source package and the enterprise server. The framework covers 25 cognitive dimensions across 29 personas and is grounded in 60+ peer-reviewed sources.