Every Token You Send Is a Geometry Problem. Nobody Told You What You’re Actually Paying For.

The Bill You Don’t Understand

Open any LLM pricing page.

Input: $X per million tokens. Output: $Y per million tokens. Output is always more expensive. Usually 3x to 5x more.

You accept this. You optimize around it. You write shorter prompts. You truncate outputs. You cache where you can.

But here is what nobody will say out loud.

The price difference between input and output tokens is not arbitrary. It is not a business decision. It is a direct consequence of the mathematical structure of the Transformer architecture. The geometry of attention. The sequential dependency of decoding. The memory bandwidth ceiling of modern GPUs.

You are not paying for compute. You are paying for the mathematical impossibility of parallelizing an autoregressive process.

And underneath that, you are paying for something even more fundamental. Every token processed by an LLM is a vector moving through a curved high-dimensional space. Every…