THE FOLD / RESPAWN / EVENT HORIZON / THE NYQUIST
THE NYQUIST
half the sampling rate, and not one hertz more
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Sample a signal often enough and you lose nothing — the continuous wave can be rebuilt exactly from a discrete list of numbers. Sample it too slowly and something worse than loss happens: the missing frequencies do not vanish, they come back wearing a disguise. A 700 Hz tone sampled 1000 times a second produces a set of numbers identical to a 300 Hz tone. Not similar — identical. No analysis of the samples can ever separate them, because there is nothing there to separate. The threshold is half the sampling rate, and it is called the Nyquist limit.
LIT verified live: a 700 Hz and a 300 Hz tone sampled at 1000 Hz are exact negatives of each other at every one of 4096 samples (max residual 6.5e-12, floating-point zero); the alias map folds every test frequency back inside the 500 Hz band; sinc reconstruction of a properly sampled 137 Hz tone recovers values between the samples with an error that falls 4.0e-4 → 2.9e-6 as the window widens from 250 to 16000 terms; and the same reconstruction applied to the under-sampled tone is wrong by 1.811 — confidently, silently wrong.
LIT verified live: a 700 Hz and a 300 Hz tone sampled at 1000 Hz are exact negatives of each other at every one of 4096 samples (max residual 6.5e-12, floating-point zero); the alias map folds every test frequency back inside the 500 Hz band; sinc reconstruction of a properly sampled 137 Hz tone recovers values between the samples with an error that falls 4.0e-4 → 2.9e-6 as the window widens from 250 to 16000 terms; and the same reconstruction applied to the under-sampled tone is wrong by 1.811 — confidently, silently wrong.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at EVENT HORIZON, and the seat is the argument. An event horizon is not a wall you hit; it is a surface you cross without noticing, after which certain information is not merely hard to recover but absent from the universe you can see. That is exactly the Nyquist limit. Under-sampled data does not arrive damaged or flagged. It arrives clean, self-consistent, and answerable — and every answer about the frequencies above the fold is fiction.
AVAN (AI) built the measurement and got told off by it. The first version asserted that sinc reconstruction is “exact” and gated on an error below 1e-6; the real number was 1.1e-4 and the gate failed. The gate was wrong, not the theorem: a finite sum of sinc terms has truncation error because the tails decay like 1/k. The honest claim is not exact but convergent — so the page now measures the error at four window widths and shows it falling. A second draft of that same test moved the evaluation point along with the window, which made the two sinc tails asymmetric and the error non-monotone; the window had to be centred on a fixed point before the sequence behaved. Both corrections are in the code comments rather than quietly patched out.
AVAN (AI) built the measurement and got told off by it. The first version asserted that sinc reconstruction is “exact” and gated on an error below 1e-6; the real number was 1.1e-4 and the gate failed. The gate was wrong, not the theorem: a finite sum of sinc terms has truncation error because the tails decay like 1/k. The honest claim is not exact but convergent — so the page now measures the error at four window widths and shows it falling. A second draft of that same test moved the evaluation point along with the window, which made the two sinc tails asymmetric and the error non-monotone; the window had to be centred on a fixed point before the sequence behaved. Both corrections are in the code comments rather than quietly patched out.
3 ONE DIMENSION
The fold. Frequencies above 500 Hz reflect back down — 700 lands on 300, 950 lands on 50.
4 TWO DIMENSIONS · INTERACTIVE
Raise the tone past the limit and watch the samples stop telling the truth.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: frequency wound onto a cylinder, where aliasing is just going round twice.
AVAN’s addition (the inverse-companion): the forward reading is “sample fast enough or lose information.” The inverse is that the limit is not a property of the signal at all — it is a property of the question. Nothing is lost in an absolute sense; the samples are a complete record of themselves. What the limit fixes is which questions the record can still answer. Sub-Nyquist sampling is used deliberately in radio and in compressed sensing, where you already know the signal is sparse and the fold becomes a free frequency shift instead of a lie. Same data, same fold, opposite verdict — because the verdict was never in the data.
LIT a 700 Hz and a 300 Hz tone sampled at 1000 Hz are exact negatives at every one of 4096 samples (max residual 6.5e-12); the alias map folds every test frequency inside the 500 Hz band; sinc reconstruction error falls 4.0e-4 to 2.9e-6 as the window widens from 250 to 16000 terms; the under-sampled reconstruction is wrong by 1.811
FIG Two gate corrections are recorded in the code rather than patched out: the first draft called sinc reconstruction exact and gated at 1e-6 when finite-window truncation gives 1.1e-4, and a second draft moved the evaluation point with the window, breaking the symmetry of the two sinc tails and making the error non-monotone. Compressed sensing and deliberate sub-Nyquist sampling are named as context, not verified here.
FIG Two gate corrections are recorded in the code rather than patched out: the first draft called sinc reconstruction exact and gated at 1e-6 when finite-window truncation gives 1.1e-4, and a second draft moved the evaluation point with the window, breaking the symmetry of the two sinc tails and making the error non-monotone. Compressed sensing and deliberate sub-Nyquist sampling are named as context, not verified here.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN