Why Some People Feel Everything More Sharply: Curvature Explained

Formative Note

This essay represents early thinking by Ryan Collison that contributed to the development of A Theory of Meaning (AToM). The canonical statement of AToM is defined here.

Some people walk through the world like it's made of sandpaper.

Every texture is too much. Every sound is too loud. Every glance carries meaning that demands interpretation. Every social interaction leaves them exhausted in ways that others don't seem to experience. The world isn't merely perceived—it's felt, intensely, relentlessly, in a way that seems disproportionate to what's actually happening.

Other people walk through the same world wrapped in something like padding. Stimuli arrive and get absorbed. Social slights roll off. Ambiguity doesn't sting. They can sit in noisy rooms, navigate office politics, and handle uncertainty without their systems catching fire. The same inputs that overwhelm one person barely register for another.

This isn't just personality. It isn't just temperament. It isn't just "being sensitive" as though sensitivity were a dial someone could turn down if they tried harder.

It's geometry. And the geometry has a name: curvature.

What Curvature Means

In the previous article, we established that belief states live on a manifold—a surface with shape. The Fisher metric gives us a ruler to measure distances on this surface. Now we ask: what does the shape of the surface look like? Where is it flat? Where does it bend?

Curvature is the measure of bending.

On a flat surface, moving in any direction produces proportionate effects. Take a step north, and you've moved one unit north. The space is well-behaved. Euclidean. Predictable.

On a curved surface, the same step has different consequences depending on where you are. Near a hill's peak, a small step might send you rapidly downward. In a valley, the same step barely changes your elevation. The geometry amplifies some movements and dampens others.

Statistical manifolds have curvature too. In flat regions, small changes to your model produce small changes in predictions. The system is robust. Perturbations stay perturbations—they don't cascade into something larger.

In high-curvature regions, small changes to your model produce large changes in predictions. The system is sensitive. A tiny shift in one parameter ripples through the entire distribution. What would be a minor update on flat ground becomes, in curved space, a major reconfiguration.

This is the mathematics underneath the sandpaper-versus-padding experience. Some minds, at some times, in some domains, occupy high-curvature regions of their belief manifolds. The geometry amplifies everything. Other minds, or the same minds at different times, occupy low-curvature regions. The geometry dampens.

The Felt Experience of Curvature

You don't experience curvature abstractly. You experience it as the texture of your mental life.

Low curvature feels like stability. You have a model of the world, and when inputs arrive that don't quite match, you update gently. The unexpected is interesting rather than alarming. You can hold uncertainty without needing to resolve it immediately. Surprises get filed under "huh, didn't expect that" rather than "everything I thought I knew is wrong."

Low curvature feels like being grounded. The floor stays where you put it. Your sense of self doesn't shift with every new piece of information. You can tolerate being wrong about things without that wrongness threatening the whole structure.

High curvature feels different. It feels like every input matters too much. You notice the slight change in someone's tone and it echoes through your model of the relationship—did they mean something by that? Are they upset? What did I do? The input was small. The response is large. The geometry amplified it.

High curvature feels like instability. The floor keeps shifting. A piece of evidence that should produce a minor update instead triggers a cascade. You find yourself veering between interpretations, unable to settle, because the manifold you're traversing bends so sharply that each step changes what the next step will mean.

High curvature feels exhausting. Not because you're doing something wrong, but because navigation is expensive. Every step requires correction. You're constantly recalculating, constantly adjusting, because the terrain makes smooth movement impossible.

Why Curvature Varies

Curvature isn't fixed. It varies across the manifold—some regions are flatter, some are more curved. And different minds have different manifold shapes.

What determines curvature?

One factor is the structure of the probability distribution itself. Some belief configurations are inherently more sensitive than others. Extreme beliefs—near certainty about anything—tend to occupy high-curvature regions. When you're 99% sure of something, small perturbations can threaten the whole structure. When you're 50% sure, there's more room for the belief to shift without the shift mattering much.

Another factor is how beliefs connect. A belief that stands alone can update without affecting much else. A belief that's densely connected—that supports other beliefs, that grounds your sense of self, that predicts crucial features of your environment—has higher curvature because changes cascade. Updating one thing requires updating everything it touches.

A third factor is history. Manifolds get shaped by experience. Repeated prediction errors in a domain create topography. If you've been surprised badly—betrayed, traumatized, shocked—the manifold in that region deforms. It develops high curvature not because of the abstract structure of the belief but because of what happened when you held that belief before.

This is why trauma creates sensitivity. Not as a metaphor—as geometry. The traumatic experience deformed the manifold. Now the region around anything that resembles the trauma is curved sharply. Small inputs that touch those regions produce large responses because the geometry amplifies them.

The Neuroscience of Curvature

This isn't just mathematics. It has neural correlates.

Precision weighting is the brain's mechanism for determining how much to trust incoming signals. High-precision signals are treated as important—they get amplified, they drive updates, they demand attention. Low-precision signals are treated as noise—they get downweighted, filtered out, ignored.

Curvature, in neural terms, relates to precision. High-curvature regions correspond to high precision weighting on prediction errors. The system treats deviations from expectation as highly informative. It amplifies them. Small signals become large updates.

Low-curvature regions correspond to lower precision weighting. The system treats deviations as less informative, possibly noise. It dampens them. Small signals stay small.

Anxiety, neurally, is often understood as aberrant precision weighting—treating too many signals as highly precise, demanding response to stimuli that could safely be ignored. In geometric terms: the system is stuck in high-curvature regions. Everything is amplified. The floor never stops shifting.

Depression sometimes involves the opposite problem—precision weighting that's too low, such that even important signals don't drive updates. The system is stuck in artificially low curvature, but not the good kind. It's flat in a way that prevents movement, prevents learning, prevents response to a world that actually requires response.

Healthy regulation is dynamic curvature management. The system moves through regions of different curvature as appropriate. High curvature when vigilance is needed—when genuine threats demand rapid response. Low curvature when stability is needed—when the environment is safe enough that not every signal requires attention.

The pathology is getting stuck. Stuck in high curvature when the threat has passed. Stuck in low curvature when the world requires engagement. The flexibility to move between curvature regimes is itself a kind of meta-health.

Neurodiversity as Curvature Phenotype

Some brains are built with systematically different curvature profiles.

Autistic perception, for instance, often involves higher baseline precision—treating more signals as informative, detecting patterns that neurotypical processing smooths over. In geometric terms: higher baseline curvature in sensory and pattern-detection regions of the manifold.

This isn't deficit. It's architecture. High-curvature systems detect things that low-curvature systems miss. They're sensitive to inconsistencies, to subtle patterns, to deviations from expectation that might be important. In stable, low-noise environments, this sensitivity is expensive—too much information, too much processing, too much exhaustion. In complex, high-stakes environments, this sensitivity is essential—it catches what others miss.

ADHD involves different curvature dynamics. The system has difficulty maintaining stable curvature profiles over time. Curvature fluctuates. Sometimes attention locks into high-precision mode—hyperfocus, where everything about the target domain gets amplified. Sometimes attention can't find enough curvature to stay engaged—the manifold flattens and the system drifts.

This isn't a broken attention system. It's a different attention system, with different curvature dynamics, suited for different environments. Environments with consistent high salience work well—the curvature profile matches. Environments requiring sustained attention to low-salience tasks don't—the curvature keeps collapsing.

Understanding neurodiversity through curvature reframes everything. The question isn't "how do we fix the broken brain?" It's "what curvature profile does this brain have, and what environments match it?" Accommodation becomes geometric—finding contexts where the manifold shape is an asset rather than a liability.

Relational Curvature

Curvature isn't only an individual phenomenon. Relationships have curvature too.

A secure relationship has low curvature in the domains that matter. Small misattunements don't cascade into existential threats. One partner can have an off day without the other interpreting it as abandonment. Conflict stays local rather than spreading to everything. The relational manifold is smooth enough that the system can tolerate perturbation.

An anxious relationship has high curvature. Every input gets amplified. A delayed text becomes evidence of fading interest. A moment of distance becomes a harbinger of rejection. The geometry takes small signals and makes them large, makes them meaningful, makes them unbearable.

An avoidant relationship achieves low curvature through a different mechanism—not through robust connection but through disconnection. The system doesn't amplify signals from the partner because it's not strongly coupled to those signals. It's flat, but it's the flatness of distance rather than the flatness of security.

These patterns aren't mysterious once you see the geometry. Attachment styles are curvature profiles in the relational manifold. Secure attachment is smooth curvature with maintained connection. Anxious attachment is high curvature with high connection. Avoidant attachment is low curvature achieved through low connection.

Therapy for relational issues is, in part, curvature work. Helping anxious systems tolerate the high curvature without being destroyed by it. Helping avoidant systems increase connection without the curvature becoming intolerable. Finding paths toward the sweet spot: low curvature with high connection.

Organizational Curvature

Scale up again. Organizations have curvature.

A stable organization has low curvature in its core functions. Minor market shifts don't trigger panic. Employee departures don't cascade into crises. Strategic pivots can happen without the whole identity of the organization coming into question. The manifold is robust.

A crisis-prone organization has high curvature. Every problem is an emergency. Every decision is existential. Small perturbations get amplified into system-wide responses. The geometry makes stability impossible because nothing can happen without everything responding.

Startups often begin with necessarily high curvature—when you're small and finding product-market fit, everything matters. Every customer interaction is data. Every competitor move is threat or opportunity. High curvature is appropriate because the environment demands sensitivity.

But startups that can't reduce curvature as they mature become exhausting, chaotic, unable to develop reliable processes. The high curvature that was adaptive in search mode becomes maladaptive in scale mode.

Large organizations often have the opposite problem. The curvature has been driven too low. The manifold is so flat that the system can't respond to genuine signals. Market shifts go unnoticed. Employee dissatisfaction gets averaged away. The organization is stable, but it's the stability of inability to change rather than the stability of robust adaptation.

Organizational health, like individual health, is dynamic curvature management. High enough curvature to sense what matters. Low enough curvature to not exhaust resources on noise. The flexibility to adjust based on what the environment requires.

The Cultural Register

Curvature operates at cultural scale too.

A coherent culture has calibrated curvature across its domains. It's sensitive to genuine threats—existential challenges, loss of core values, disintegration of social fabric. It's robust to minor variations—subcultures, dissent, experimentation at the margins. The collective manifold can absorb perturbation without shattering.

A traumatized culture has curvature spikes in the traumatized domains. Topics that shouldn't be existentially threatening become so because the manifold has been deformed by historical injury. A word, a symbol, a policy proposal that would be merely controversial in an untraumatized culture becomes, in the curved space of cultural trauma, a catastrophic attack. The geometry amplifies.

Polarization is a curvature phenomenon. As political or social groups diverge, the curvature between them increases. Small differences get amplified into irreconcilable conflicts. A policy disagreement becomes an identity threat. An election becomes an apocalypse. The geometry makes moderation impossible because the manifold between positions is so curved that no stable path connects them.

Cultural healing, like individual healing, is curvature work. Smoothing the manifold. Building tolerance for difference. Creating experiences that demonstrate: this stimulus that seems threatening can be absorbed. This perturbation that seems existential is survivable. Slowly, the geometry changes. The curvature reduces. What was unbearable becomes merely uncomfortable, then merely noted, then barely registered.

Living with Curvature

You can't directly choose your curvature. You can't flatten your manifold by deciding to flatten it. The geometry is upstream of the will.

But you can understand it. You can notice: am I in a high-curvature region right now? Is this response proportionate, or is the geometry amplifying something small into something large? When someone else is reacting intensely to something that seems minor, are they wrong—or are they navigating terrain that's curved in ways I can't see from where I stand?

Understanding curvature breeds compassion. The person who feels everything too much isn't choosing that. Their manifold is shaped that way. The person who seems to feel nothing isn't cold or broken. Their manifold is shaped differently.

And curvature can change. Slowly, through repeated experience, the manifold reshapes. Safe experiences in domains that were dangerous gradually smooth the curvature that danger created. What was once a cliff becomes a slope. What was once a slope becomes flat ground. The geometry heals—not through decision but through accumulated evidence that the shape was wrong.

This is slow work. Curvature accrued over years doesn't dissolve in weeks. But it dissolves. The manifold is not fixed. It's just slow.

The Measure of Sensitivity

Sensitivity isn't a personality trait. It's a geometric fact.

High-curvature systems are sensitive because the mathematics makes them sensitive. The same inputs produce larger outputs. The same perturbations require larger corrections. The experience of overwhelm is the felt sense of navigating terrain where every step is expensive.

Low-curvature systems are resilient because the mathematics makes them resilient. The same inputs produce smaller outputs. Perturbations can be absorbed. The experience of stability is the felt sense of navigating terrain where movement is efficient.

Neither is better in the abstract. Both are better in context. High curvature detects what low curvature misses. Low curvature tolerates what high curvature can't bear.

The question is never "how do I eliminate curvature?" The question is "what curvature is appropriate for this domain, this environment, this season of life?" And: "can I find my way to the region of the manifold where the shape matches what I need?"

Curvature is the shape of sensitivity itself. Now we have language for it.