Your Language Model Cannot Say Certain Sentences. The Reason Is the Rank of a Matrix. Let Us Prove It With Tiny Numbers, By Hand.
Author(s): Dr Swarneendu AI Originally published on Towards AI. There are next-word predictions your model is mathematically forbidden from making. Not unlikely. Forbidden, the way a piano with too few keys cannot play a note that lies past its keyboard. The proof …
Every AI Buzzword You Have Been Afraid Of Is a Dot Product in a Costume. Here Are 15 of Them, Unmasked.
Author(s): Dr Swarneendu AI Originally published on Towards AI. The entire AI industry runs on one operation. One. Multiply two lists of numbers together, sum the results. Everything else — attention, embeddings, RAG, fine-tuning, cosine similarity, RLHF, temperature, hallucination — is that …
Your Agentic Loop Will Drift. Here Is the KL Divergence Equation That Measures How Far It Has Wandered From Its Original Instruction.
Author(s): Dr Swarneendu AI Originally published on Towards AI. After 500 cycles, a long-running agent is not the same agent that started. Its goal has shifted. Its constraints have eroded. This is measurable, preventable, and nobody has built the instrument. Until now. …
Everyone Building Multi-Agent Systems Is Spending Compute on Something Mathematically Impossible. Here Is Shannon’s Proof.
Author(s): Dr Swarneendu AI Originally published on Towards AI. Stanford proved in April 2026 that single agents beat multi-agent systems when compute is equal. The proof is older than the internet. It is the Data Processing Inequality from 1948, and it says …
The Best AI Agent Predicts Future World Events at 25% Accuracy.
Author(s): Dr Swarneendu AI Originally published on Towards AI. FutureSim replays real-world news events in chronological order and asks agents to predict what happens next. The Brier skill score for most agents is negative. Negative means worse than a coin flip. Every …
Your LLM Is Safe When Prompts Are Short. A 2025 Paper Proved It Switches to Harmful Mode Exactly When Context Gets Long. No Trojan Words Required.
Author(s): Dr Swarneendu AI Originally published on Towards AI. Every backdoor defense scans for suspicious content in the input. MetaBackdoor uses input length as the trigger. The content is clean. The behavior is not. Every deployed LLM safety system makes one foundational …
Every AI Model Trains on a Curved Surface. Gradient Descent Pretends It’s Flat. Amari Proved the Cost in 1998.
Author(s): Dr Swarneendu AI Originally published on Towards AI. Every AI Model Trains on a Curved Surface. Gradient Descent Pretends It’s Flat. Amari Proved the Cost in 1998. You have been visualizing the loss landscape wrong. Not slightly wrong. Geometrically wrong. The …
Every Token You Send Is a Geometry Problem. Nobody Told You What You’re Actually Paying For.
Author(s): Dr Swarneendu AI Originally published on Towards AI. The AI industry sells you tokens. It bills you for tokens. It throttles you by tokens. And almost nobody has explained what a token actually costs, mathematically, and why the cost is not …
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