For maths, it's also wrong most of the time. Generated stuff looks right, then let Opus audit and see the disaster.
DS4 is usable iff you have a way to test the generated stuff, and to convince yourself that its production is right. With successive review-fix rounds, it's obviously way more reliable too, but that can't compensate for it's lack of rigor when reasoning.
It's very smart but neither rigorous nor careful. And that is a direct result of its architecture.
Fix your ADHD.
It certainly comes with constant fatigue.
It becomes clear that ADHD, like autism, is related to a lack of energy for the brain, and disappears when enough energy is available.
If you feel too tired to focus, if you think you might need to sleep more than others, have the need for naps and so on, go consult a sleep clinic, get tested for all kinds of conditions, and find and fix the root cause.
Ritalin won't fix the root cause and depending on how tired you are, will rather deplete your energy levels than improve your status in the long run. Ritalin gives you an illusion, not a remedy.
Causes of fatigue can be a chronic inflammatory state, some immune reactions triggered too often...
Do you have allergies? Rashes when you eat some food? Trembling hands? Some skin problems? Hearing loss or tinnitus? Are you abnormally tired after sports? Do you have abnormal cramps after sports or drinking alcohol, or when not sleeping enough? Regular teeth health issues? Do you have a sensitive vague nerve? Did you faint more than other people in the course of your life? Does your body need more time to heal? Do you feel dizziness when standing up? Have you always had the impression that you actually aren't that stable on your feet, even while actually practising a lot of sports? What about headaches and migraines?
When you work, have you specialised in some tasks that are/feel quite systematic for you? Are social interactions or other tasks that require new analysis exhausting for you? These are signs that you optimise your energy expenditure by focusing on tasks that barely require you to think with much energy.
Lack of energy is a very good reason for the brain to avoid using energy for its executive function, the function that is actually impaired in people with ADHD. It's normal, and trying to fix that feels exhausting, which is why people with ADHD avoid spending that energy.
The executive function is preliminary to focusing with other parts of the brain. If it's impaired, the brain won't even try to focus with other parts for the actual work. Two problems solved at once. Twice the energy spared.
Find the root cause of your ADHD (which often is just a symptom) and fix it.
In this paper, I analyze the relationship between ISP and DIP.
This paper makes 3 contributions:
1. A formal statement of the ownership clause for DIP, making explicit what Martin’s examples illustrate, but his DIP statement does not.
2. A proof that DIP’s ownership clause applied per client implies ISP universally at the class level, with the converse holding only under client-driven
evolution.
3. The identification of three distinct interface evolution origins—client-driven, provider-driven, and shared governance—as the conceptual framework that explains why the ISP-DIP connection was historically hard to see and why ISP retains independent value outside client-driven contexts.
For my paper about ME/CFS, I let an LLM integrate lots of findings of other scientific papers.
Then I ask the LLM to "creatively brainstorm", given all we know of ME/CFS and the newly integrated paper, to generate new hypotheses, treatment ideas or any other kind of insight it can think of.
This works really well.
Now, it's clear that I have no idea how much of this is something we would consider new and original, and how much is a kind of systematic, but not novel, easy of thinking.
What I couldn't do so far is get an LLM to generate a truly new maths theory, with new abstract concepts and dimensions and points of view. The kind that is not just a combination of existing theories and logic.
In mathematics and physics, complex numbers aren't just "imaginary" values—they are the secret language of 2D rotation. While real numbers live on a 1D line, complex numbers inhabit a 2D plane, and multiplying them acts as a bridge between dimensions.
1. The Geometry of i
To understand how we switch dimensions, look at the imaginary unit i. In a standard real-number system, you only move left or right. Adding i introduces a vertical axis.
* The 90-degree turn: Multiplying a real number by i is geometrically equivalent to a 90° counter-clockwise rotation.
* The Dimension Switch: If you start at 1 (on the x-axis) and multiply by i, you land at i (on the y-axis). You have effectively "switched" your direction from horizontal to vertical.
2. Rotation via Euler’s Formula
The most elegant link between complex numbers and rotation is Euler’s Formula:
This formula places any complex number on a unit circle in the complex plane. When you multiply a vector by e^{i\theta}, you aren't changing its length; you are simply rotating it by the angle \theta.
Why this matters:
* Algebraic Simplicity: Instead of using messy rotation matrices (which involve four separate multiplications and additions), you can rotate a point by simply multiplying two complex numbers.
* Phase in Physics: This is why complex numbers are used in electrical engineering and quantum mechanics. A "phase shift" in a wave is just a rotation in the complex plane.
3. Beyond 2D: Quaternions
If complex numbers (a + bi) handle 2D rotations by adding one imaginary dimension, what happens if we want to rotate in 3D?
To handle 3D space without hitting "Gimbal Lock" (where two axes align and you lose a degree of freedom), mathematicians use Quaternions. These extend the concept to three imaginary units: i, j, and k.
> The Rule of Four: Interestingly, to rotate smoothly in three dimensions, you actually need a four-dimensional number system.
>
Summary Table
| Number System | Dimensions | Primary Use in Rotation |
|---|---|---|
| Real Numbers | 1D | Scaling (stretching/shrinking) |
| Complex Numbers | 2D | Planar rotation, oscillations, AC circuits |
| Quaternions | 4D | 3D computer graphics, aerospace navigation |
They can be treated as vectors, but they have "superpowers" that standard vectors do not.
1. The Similarities (The 2D Map)
In a purely visual or structural sense, a complex number z = a + bi behaves exactly like a 2D vector \vec{v} = (a, b).
* Addition: Adding two complex numbers is identical to "tip-to-tail" vector addition.
* Magnitude: The "absolute value" (modulus) of a complex number |z| = \sqrt{a^2 + b^2} is the same as the length of a vector.
* Coordinates: Both represent a point on a 2D plane.
2. The Difference: Multiplication
This is where complex numbers leave standard 2D vectors in the dust.
In standard vector algebra (like what you'd use in an introductory physics class), there isn't a single, clean way to "multiply" two 2D vectors to get another 2D vector. You have the Dot Product (which gives you a single number/scalar) and the Cross Product (which actually points out of the 2D plane into the 3D world).
Complex numbers, however, can be multiplied together to produce another complex number.
The "Rotation" Secret
When you multiply two complex numbers, the math automatically handles two things at once:
* Scaling: The lengths are multiplied.
* Rotation: The angles are added.
Standard vectors cannot do this on their own; you would need to bring in a "Rotation Matrix" to force a vector to turn. A complex number just "knows" how to turn naturally through its imaginary component.
3. When to use which?
Mathematically, complex numbers form a Field, while vectors form a Vector Space.
* Use Vectors when you are dealing with forces, velocities, or any dimension higher than 2 (like 3D space).
* Use Complex Numbers when you are dealing with things that rotate, vibrate, or oscillate (like radio waves, electricity, or quantum particles).
> The Peer-to-Peer Truth: Think of a complex number as a vector with an attitude. It lives in the same 2D house, but it knows how to spin and transform itself algebraically in ways a simple (x, y) coordinate cannot.
>
DS4 is usable iff you have a way to test the generated stuff, and to convince yourself that its production is right. With successive review-fix rounds, it's obviously way more reliable too, but that can't compensate for it's lack of rigor when reasoning. It's very smart but neither rigorous nor careful. And that is a direct result of its architecture.