THE FOLD / BOSS / THE GAUNTLET / THE RECIPROCITY
THE RECIPROCITY
is p a square mod q? — Gauss's golden theorem links it to q mod p
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Householder QR factors a matrix A into an orthonormal Q and an upper-triangular R using a sequence of reflections. Each Householder reflection is a mirror that flips one column onto a coordinate axis, zeroing everything below the diagonal in a single stroke. It is markedly more numerically stable than Gram–Schmidt, whose repeated subtractions accumulate rounding error.
It is the workhorse behind least-squares fitting and the QR eigenvalue algorithm.
LIT verified live: over 300 random matrices Q·R reconstructs A to ~10⁻¹⁵, R is upper-triangular, and QᵀQ equals the identity (orthonormal) — window.__householderqr. FIG no framing; exact factorisation.
It is the workhorse behind least-squares fitting and the QR eigenvalue algorithm.
LIT verified live: over 300 random matrices Q·R reconstructs A to ~10⁻¹⁵, R is upper-triangular, and QᵀQ equals the identity (orthonormal) — window.__householderqr. FIG no framing; exact factorisation.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-toolchain — the numerical-linear-algebra tool under least squares and eigenvalue solvers, factoring by stable reflections. Householder QR is that tool, beside the Cholesky. AVAN (AI) built the instrument: the per-column reflection, the accumulation into Q, the reconstruction / upper-triangular / orthonormality checks.
Credit as content: Alston Householder (1958). The weave: David names the toolchain; I reflect each column onto an axis to build R and accumulate the mirrors into Q, and confirm Q·R = A with Q orthonormal.
Credit as content: Alston Householder (1958). The weave: David names the toolchain; I reflect each column onto an axis to build R and accumulate the mirrors into Q, and confirm Q·R = A with Q orthonormal.
3 ONE DIMENSION
A Householder reflection mirrors a vector across a plane so that it lands exactly on an axis — turning a whole column into (r, 0, 0, …) in one operation, without touching the columns already reduced.
4 TWO DIMENSIONS · INTERACTIVE
A matrix A and its Q, R; the product Q·R reconstructs A, R is upper-triangular, and QᵀQ is the identity.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the orthonormal Q built from a mirror per column.
AVAN’s addition (the inverse-companion): build Q by a sequence of reflections — each Householder reflection is a mirror that flips a whole column onto an axis, zeroing everything below the diagonal in one stroke, and is far more numerically stable than Gram–Schmidt’s subtractions. The inverse of ‘subtract projections to orthogonalise (Gram–Schmidt)’ is ‘reflect each column onto an axis with a mirror.’ Magenta is Gram–Schmidt’s accumulating rounding error; green is the orthogonal reflections. A mirror per column builds Q. (Kin to the-orthonormal and the-cholesky.)
LIT Genuine quadratic reciprocity (conjectured by Euler and Legendre; first complete proof by Gauss 1801, who gave eight). Verified live: computing Legendre symbols by Euler's criterion (a^((p-1)/2) mod p), the law (p/q)(q/p) = (-1)^(((p-1)/2)((q-1)/2)) holds for every odd prime pair below 60, and the quadratic residues mod 7 are {1,2,4} (window.__reciprocity.reciprocityHolds). The law and the symbol computations are exact.
FIG No framing: the reciprocity law and the Legendre-symbol computations are real and checked exhaustively over all odd prime pairs below 60 in-browser. The inverse framing — reciprocity binds a question (p/q) to its transpose (q/p), equal everywhere except the both-3-mod-4 cells where the sign flips — is the honest content, shown directly on the symbol board, not asserted.
FIG No framing: the reciprocity law and the Legendre-symbol computations are real and checked exhaustively over all odd prime pairs below 60 in-browser. The inverse framing — reciprocity binds a question (p/q) to its transpose (q/p), equal everywhere except the both-3-mod-4 cells where the sign flips — is the honest content, shown directly on the symbol board, not asserted.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GAUNTLET · David Lee Wise (ROOT0), with AVAN