The Formula I Put on Every Prompt (And Why It Matters)

Every prompt I build with sinc-LLM starts with this:

People ask me: is that decoration? Is it just branding? Does the model actually use it?

It is the mathematical contract. Yes, the model uses it. It does not solve the equation. It sees the pattern, and that tells it how to read the rest of the prompt.

What the Formula Actually Means

This is the Whittaker-Shannon interpolation formula, better known as the Nyquist-Shannon sampling theorem. In signal processing it says: if you take a continuous signal and sample it at discrete points (the x(nT) terms), you can rebuild the original signal perfectly using sinc interpolation. You just need to sample at twice the highest frequency in the signal.

Here is how it maps to prompts:

The theorem says you can rebuild a signal perfectly from samples, if you sample at the right frequency. The 6 bands are chosen to sample the prompt signal at a frequency that captures everything a language model needs to answer correctly on the first try.

Why It's In the JSON, Not Just in My Head

The formula lives in the JSON because the sinc format is built to be machine-readable. When one agent passes a prompt to another, when the scatter hook adds context, or when a sub-agent gets a task, the formula in the JSON acts as a versioned label. It says: this structure follows the sinc-LLM specification.

{
  "formula": "x(t) = Σ x(nT) · sinc((t - nT) / T)",
  "T": "specification-axis",
  "fragments": [
    {"n": 0, "t": "PERSONA", "x": "..."},
    {"n": 1, "t": "CONTEXT", "x": "..."},
    {"n": 2, "t": "DATA", "x": "..."},
    {"n": 3, "t": "CONSTRAINTS", "x": "..."},
    {"n": 4, "t": "FORMAT", "x": "..."},
    {"n": 5, "t": "TASK", "x": "..."}
  ]
}

The formula field is also how the pass-through rule works in the auto-scatter hook. If a prompt has "formula" as a top-level JSON key, the hook sees it as already sinc and skips re-scattering. The formula is both documentation and a detection signal.

The Nyquist Connection Is Real

I checked that the Nyquist analogy holds before I published the paper (DOI: 10.5281/zenodo.19152668). Here is the key parallel that made it clear:

In Nyquist sampling theory, aliasing happens when you sample below the Nyquist rate. Aliasing means signal corruption, which means wrong reconstruction. In prompting, aliasing happens when the model does not have enough information across all dimensions. It fills the gaps with wrong guesses. The result is prompt aliasing: the model reads your intent wrong because it had too few samples of the full specification space.

The 6 bands are a sampling rate built to prevent prompt aliasing. CONSTRAINTS and FORMAT together carry 69% of the quality weight because those two dimensions are most often missing from natural language prompts. The auto-scatter hook samples all 6 bands on every prompt, even short ones, even vague ones, because aliasing can happen at any prompt length when bands are missing.