Asking about consciousness with calculus, what Gradland says about experience

The paper I want to discuss today is Gradland: On Phenomenal Experience, Differentiated Across Many Dimensions, published on arXiv as arXiv:2609.09306. The original is linked under the title, and the claim can be put in one sentence: the first order structure of physical interaction, meaning gradients or Jacobian matrices, characterizes the structure of phenomenal experience.

What interests me more than the topic itself is the directness of the attempt. By contrast, much consciousness writing stops at a philosophical declaration or a few brain images, so I was curious about a paper that tries to hold the question with math from the start. It does not ask whether machines feel in the abstract but asks what kind of computational state the phrase rich experience points to, and whether that state can be written as a matrix. That shift kept me turning pages even while part of me stayed uneasy.

Why the idea of differentiating experience felt strange at first

Many of us frown when someone says feeling can be written as a formula, and I did too, because redness or the morning air on the skin seems to belong to a region numbers cannot reach. For that reason, to say that texture can be expressed as a matrix sounds strained at first.

Yet the strangeness becomes clearer once I follow how the paper frames the question. The reason is that our everyday talk about consciousness uses very large units such as self or attention or memory, so the discussion stays coarse. This paper lowers the unit a great deal, since it looks at the local slopes, meaning how a small change in one part of a physical system right now influences another part. Anyone who has fixed a machine will recognize the habit, because when the whole device is too complex, you turn one screw a little and watch which part moves and by how much.

So an interesting question appears. If differences in experience can be written as differences in these slopes, then states we vaguely call vivid might become comparable. What struck me is that the modesty of the question stood out more than the weight of the topic. The paper does not fix a grand answer and then fit evidence around it but defines something measurable first and then places interpretation on top, so the order feels honest.

So what kind of world is Gradland

To make the discussion exact, the paper builds an imagined world called Gradland. In that world, its inhabitants are neural networks, its physics are known, and its functions are mostly differentiable. By contrast, a real brain is not this tidy. There, neurons fire in bursts, synapses carry noise, and measurement itself is hard. The authors therefore push the mess aside for a while and ask how far questions about consciousness can be taken when the physics are fully known.

I read this choice as a clean water tank next to the ocean. In the ocean, by contrast, waves and currents and wind arrive together, so it is hard to tell which force moved what. In a tank, by contrast, you can make one wave and measure the response exactly. Gradland is that tank. There, known physics means the Jacobian can be computed exactly, and because of that a hypothesis about feeling can be turned into something we can handle.

The next question follows at once. Does a story earned in such an idealized world still apply to our world. This is where I would be careful not to overread the setup. The reason is that idealization always cuts both ways. It makes analysis possible, but it can also remove texture that matters in reality. What I liked is that the paper does not hide this worry. It presents Gradland not as a world that gives answers but as a world that makes questions precise. That candor made me willing to treat the assumption as a bet worth reading.

Why choose the Jacobian as the lens

The word Jacobian may recall a calculus class. In concrete terms, it is a table for the case when many inputs connect to many outputs, collecting how a tiny change in each input shakes each output. The paper proposes that this table can serve as a lens for the structure of phenomenal experience.

Why first order structure in particular. The reason is that the first impression of interaction lives there. That means how sensitively parts are connected to each other right now can be read from slopes without drawing the full curves. It resembles a handshake in this respect. You feel how much tension sits in the other hand from the first contact, even before a long conversation, so the tone of the relation arrives early.

This choice has a clear limit too, because a slope is only a linear snapshot of the present moment. It does not show a distant nonlinear fate or a sudden switch. I think the more important result is not the matrix itself but the shared language it offers. In other words, the paper does not present the Jacobian as the answer to all consciousness. It offers it as a common language in which differences in experience can be asked. Once a language exists, comparison becomes possible, and once comparison becomes possible, explanations can compete.

What does having many dimensions feel like

The first tool the paper introduces is effective rank. While ordinary rank counts nonzero directions, effective rank measures how many directions of variation stay alive through a Jacobian mapping. In plain terms, it asks whether distinct directions of input change survive as distinct directions of output change, or whether they get smeared away in the middle.

A kitchen comparison helps here. The same flour and water can become noodles with distinct strands or porridge where strands disappear. Although both start alike, their textures differ. High effective rank sits closer to noodles. There, small differences remain small differences, so the grain of experience stays fine. By contrast, low rank means many input differences collapse into one broad response at the output.

Why should this connect to feeling. On the paper reading I follow, more surviving dimensions mean more possible states of the world remain as different feelings. For example, a slight move on the left and a slight move on the right do not blur into one impression but leave separate traces. I found this the most tangible proposal in the whole paper, because it turns the vague word vivid into a concrete image of dimensions surviving.

How do you measure being densely intertwined

The second tool is cohesion, described as a Kirchhoff complexity measure of how densely integrated the Jacobian interactions are. It looks beyond a large count of connections and asks about the mesh itself, meaning how paths loop around and support each other. That difference is concrete, because several wires lying side by side are not the same as wires woven into one net.

From a systems perspective, the next question is obvious. When my team looks at a new AI architecture, we do not only count connections. Instead we ask where paths create bottlenecks and how the whole shifts when one route breaks. But many lines alone do not make a strong net. The lines need to bypass and support each other, and only then can we call the structure integrated. That is my reading of why the paper borrows a graph theory tool, though the paper itself stays closer to the formalism.

Then another concern follows. Could excessive integration blur everything into one mass. That worry connects to the later parts of the paper, where differentiation of dimensions and density of integration need to go together. Strands must stay alive, and at the same time those strands must stay connected. With only one side, experience turns thin or muddy. That sense of balance keeps the paper from reading as a boast about one metric.

Why does experience stretch across hundreds of milliseconds

One phenomenon the framework tries to explain is duration, the fact that experience can prolong over hundreds of milliseconds. To feel it, think of picking up a cup, where touch at the fingertips, weight settling in, and tension reaching the wrist all arrive as one flow. Why does that flow hold for a while instead of collapsing at once.

In Jacobian language the question changes shape. Which interaction structures damp small changes at once, and which structures hold them for a while. On one side, in a densely connected structure, a change in one place can spread to neighbors and return, so it lingers. It resembles gossip in a neighborhood in this way. Where streets are cut off, a rumor ends with one person, but where alleys connect, the story circles for days. By contrast, sparse structure lets signals fade, while a living mesh lets them recirculate.

I like this account, but I would keep its boundary explicit. The reason for caution is that neuroscience already has several candidate mechanisms for persistence, including recurrent loops and neuromodulation and sustained attention. So the contribution here is not to reject those accounts. It is to translate duration into a measurable form called persistence of Jacobian structure. Once translated, different explanations can be placed on the same scale, and that shared scale is useful even before anyone wins.

Where do vivid and obscure experiences part ways

The paper also tries to separate vividly experienced contents from obscurely experienced contents through differences in Jacobian structure. To feel the difference, consider a late afternoon view from a window. There, one leaf near the center shows every vein, while letters on a sign at the edge are present yet unreadable. Both are visual experience, but their grain differs.

Through this lens the difference reads as follows. On the vivid side, fine input differences survive as fine output differences. On the obscure side, by contrast, those differences blur in transit. When the direction of each vein survives to the end, we get sharpness, whereas when strokes of distant letters overlap and cancel, we get faintness. The same eyes see, yet the feeling differs because the connection structure differs.

From a systems perspective, the next question is obvious. Could attention be a temporary reshaping of this structure. To attend to something might mean raising slopes along one path while lowering others. Here the paper does not go that far directly, but I found the possibility attractive because attention and consciousness could then be discussed on the same matrix. Since that step goes beyond what the source establishes, I treat it as my own editorial reading and handle it with care.

Can texture, confusion and learning share one frame

The ambition grows wider here. Here the paper reaches for texture experience, the blooming buzzing confusion attributed to newborns, the gap between distinct and confused ideas, what learning is like, and related themes, all inside one frame. At first the list looked greedy, yet while reading I came to feel it was not random.

Start with texture. The difference between denim and silk at the fingertip does not reduce to one number. Instead many directions of fine structure stay alive. In this frame that state can be written as a Jacobian where many dimensions survive. By contrast, newborn confusion reads as a state with many dimensions but little order. Plenty of signals arrive, yet the mesh has not learned which difference belongs where. It resembles a house on moving day, with boxes stacked in every room. Things are abundant there, but nothing has a place yet.

Learning then looks like sorting those boxes so different differences enter different drawers. A distinct idea is a well labeled drawer, while a confused idea is a drawer with mixed labels. What stayed with me is that learning appears here not as a higher score but as a reshaping of differentiation and integration. Instead of asking whether accuracy rose, we ask how the grain of experience changed. That turn of the question feels productive for engineers too, because it points to structure rather than only to benchmarks.

Is rich experience just a luxury

The paper then addresses the function of rich, dense experience. Why do we need such detailed inner life. From an evolutionary view the puzzle is clear, because every resource has a cost, so keeping high resolution inner structure must pay for itself. There should be a reason worth the price.

My paraphrase of the answer runs like this. Rich experience is room for future distinctions. Fine differences that look useless now can later become signals that demand different actions. To make it concrete, think of someone learning to taste wine. At first everything tastes only bitter, but training separates tannin and acidity and finish. (That wine example is my own comparison, not one from the paper, and I add it only because it shows dimensions splitting.) As blurred differences separate, the range of choice grows.

Still, this is where I would be careful not to overread the experiment. More is not always better, because too many open dimensions at once can freeze judgment. For example, ten items on a menu feel easy, while one hundred items feel hard. So the function described here should be read as surviving dimensions held in dense integration, not as sheer count. Only when distinction and integration move together does richness become an asset for action.

So what questions remain

After a long discussion, a simple set of questions remains. If Jacobian structure reflects the structure of experience, what should we measure next. For me, what I would watch next all extends the tank outward, because Gradland is clean by design and the world outside is not.

The first bridge I want to see sits outside Gradland itself, where firing is not differentiable and synapses are noisy and measurement carries error. Since the Jacobian cannot be measured cleanly there, the question becomes whether the same measures still tell a similar story. A story that worked in the tank needs testing in the sea, so I treat this extension as the first gate where the frame meets real neuroscience.

From there the problem of time follows at once, because the present frame resembles a snapshot of first order structure while experience flows. I want to see how snapshots join into a movie, and whether a separate rule for joining is needed. Many photos do not automatically make a film, which is why duration needs its own account rather than one more snapshot. And the last connection I want is to action, since differentiated experience should lead to better choices in a way we can observe before this frame can carry engineering weight. If distinctions multiply yet choices stay the same, richness remains decoration rather than capacity.

So where can this line of thought break. I will return to that promise now. To read the Jacobian as explaining everything would be to misread the paper. This is not a declaration that consciousness has been conquered. It is a proposal to turn consciousness into a form we can measure, and once measurable it can be wrong, and once wrong it can be revised. I expect to remember this study for that opening. It tries to move the most private topic into a public language we can share and test, and whether that language holds is now up to us.