THE FOLD / GRIND / THE HOT LOOP / THE WALSH-HADAMARD
THE WALSH-HADAMARD
a Fourier-like transform from only +1 and -1
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Walsh–Hadamard transform is a Fourier-like transform built entirely from +1 and −1 — no sines, no complex numbers, no rounding. Its matrix H is recursively [[H,H],[H,−H]], every row orthogonal to every other. Because the entries are just signs, the fast version (FWHT) uses only additions and subtractions, and applying the transform twice returns N times the original, exactly, in integer arithmetic. It is the backbone of Hadamard codes, spread-spectrum (CDMA), and Boolean-function analysis.
LIT verified live: for sizes up to 256, FWHT applied twice equals N× the input bit-for-bit, and all Hadamard rows are mutually orthogonal (H·Hᵀ = N·I) — window.__walsh. FIG no framing; exact.
LIT verified live: for sizes up to 256, FWHT applied twice equals N× the input bit-for-bit, and all Hadamard rows are mutually orthogonal (H·Hᵀ = N·I) — window.__walsh. FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hot-loop — the tight add/subtract kernel run over and over. The FWHT is that hot loop: a butterfly of pure additions. AVAN (AI) built the instrument: the in-place butterfly, the sign-matrix construction, and the involution + orthogonality checks.
Credit as content: Jacques Hadamard (1893) & Joseph Walsh (1923). The weave: David names the hot-loop; I run the add/subtract butterfly, apply it twice to recover N× the input exactly, and confirm every pair of Hadamard rows is orthogonal.
Credit as content: Jacques Hadamard (1893) & Joseph Walsh (1923). The weave: David names the hot-loop; I run the add/subtract butterfly, apply it twice to recover N× the input exactly, and confirm every pair of Hadamard rows is orthogonal.
3 ONE DIMENSION
The butterfly: pair up entries; replace (x,y) with (x+y, x−y). Repeat over doubling strides. Only additions and subtractions — and doing it twice scales everything by N.
4 TWO DIMENSIONS · INTERACTIVE
A signal and its Walsh–Hadamard spectrum; the inverse (same transform / N) reconstructs it exactly, and the Hadamard rows are shown orthogonal.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the sign-only orthogonal basis.
AVAN’s addition (the inverse-companion): transform a signal with an orthogonal basis of only ±1 — the butterfly (x,y)→(x+y, x−y) over doubling strides — so the whole transform is exact integer add/subtract and its own inverse up to the scalar N. The inverse of ‘a Fourier transform needs sines and complex roots’ is ‘a sign-only transform needs only additions — and is its own twin.’ Magenta is the sines and complex arithmetic dropped; green is the ±1 butterfly. Exact, self-inverse, integer.
LIT Genuine Walsh-Hadamard transform (Hadamard 1893; Walsh 1923). Verified live: for all sizes N=2..256 the FWHT butterfly applied twice equals exactly N times the input (integer, no rounding), and the sign matrix satisfies H.H^T = N.I (every pair of rows orthogonal) — window.__walsh.involution && .orthogonal.
FIG No framing: the in-place add/subtract butterfly, the (-1)^popcount sign-matrix construction, and the involution + orthogonality checks run in-browser and hold to integer exactness. The AVAN inverse is honest — a sign-only orthogonal basis makes the transform pure integer add/subtract and its own inverse up to the scalar N; magenta is the sines and complex arithmetic dropped, green the +/-1 butterfly. Exact, self-inverse, integer.
FIG No framing: the in-place add/subtract butterfly, the (-1)^popcount sign-matrix construction, and the involution + orthogonality checks run in-browser and hold to integer exactness. The AVAN inverse is honest — a sign-only orthogonal basis makes the transform pure integer add/subtract and its own inverse up to the scalar N; magenta is the sines and complex arithmetic dropped, green the +/-1 butterfly. Exact, self-inverse, integer.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOT LOOP · David Lee Wise (ROOT0), with AVAN