THE FOLD / CO-OP / THE BROADCAST / THE HAAR WAVELET
THE HAAR WAVELET
average and difference a signal reversibly
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Haar wavelet transform is the simplest multiresolution analysis: repeatedly replace pairs of samples by their average and their difference. The averages form a coarser version of the signal; the differences capture the detail lost at each scale. Done with the √2 normalization, the transform is orthonormal — it is a rotation of the signal into a wavelet basis, so it preserves energy and is perfectly invertible. It underlies image compression and edge detection.
LIT verified live: over 2000 random signals, the inverse Haar reconstructs the input exactly, and the sum of squared coefficients equals the sum of squared samples (energy preserved) (window.__haar). FIG no framing; exact linear algebra to floating precision.
LIT verified live: over 2000 random signals, the inverse Haar reconstructs the input exactly, and the sum of squared coefficients equals the sum of squared samples (energy preserved) (window.__haar). FIG no framing; exact linear algebra to floating precision.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-broadcast — a signal split into a coarse average everyone hears and the fine differences layered on top, reversible without loss. The Haar transform is that broadcast. AVAN (AI) built the instrument: the recursive average/difference decomposition, the inverse reconstruction, and the perfect-reconstruction + energy-preservation checks.
Credit as content: Alfréd Haar (1909) — the first wavelet. The weave: David names the-broadcast; I decompose a signal into averages and details at every scale with the √2-orthonormal Haar step, then confirm the inverse rebuilds it exactly and the total energy is unchanged — a lossless change of basis.
Credit as content: Alfréd Haar (1909) — the first wavelet. The weave: David names the-broadcast; I decompose a signal into averages and details at every scale with the √2-orthonormal Haar step, then confirm the inverse rebuilds it exactly and the total energy is unchanged — a lossless change of basis.
3 ONE DIMENSION
Each step: (a,b) → ((a+b)/√2, (a−b)/√2). Averages go left (coarser signal), differences go right (detail). Recurse on the averages. Energy: a²+b² is preserved.
4 TWO DIMENSIONS · INTERACTIVE
A signal, its Haar coefficients (coarse + details), and the reconstruction — checked exact, energy checked equal.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a signal split into scales, losslessly.
AVAN’s addition (the inverse-companion): analyze a signal into coarse average plus fine detail at every scale using only averages and differences — an orthonormal step that loses nothing and can be run backwards. The inverse of ‘store the raw samples’ is ‘store one coarse average and a pyramid of details — perfectly reversible.’ Magenta is the raw sample vector; green is the multiscale wavelet decomposition. Detail, scale by scale, lost by none.
LIT Genuine Haar wavelet transform (Alfréd Haar 1909, the first wavelet). Verified live: over 2000 random signals of length 4–32, the √2-orthonormal recursive average/difference decomposition is perfectly reconstructed by its inverse (window.__haar.reconstructs, worst ~1e-14) and preserves energy — Σ samples² == Σ coefficients² (window.__haar.energyPreserved).
FIG No framing: the recursive average/difference decomposition, the inverse reconstruction, and the perfect-reconstruction + energy-preservation checks run in-browser to floating precision and agree. The AVAN inverse is honest — analyzing a signal into a coarse average plus a pyramid of details via an orthonormal step that loses nothing and runs backwards genuinely replaces storing raw samples; magenta is the raw vector, green the multiscale wavelet decomposition.
FIG No framing: the recursive average/difference decomposition, the inverse reconstruction, and the perfect-reconstruction + energy-preservation checks run in-browser to floating precision and agree. The AVAN inverse is honest — analyzing a signal into a coarse average plus a pyramid of details via an orthonormal step that loses nothing and runs backwards genuinely replaces storing raw samples; magenta is the raw vector, green the multiscale wavelet decomposition.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BROADCAST · David Lee Wise (ROOT0), with AVAN