THE FOLD / GRIND / THE CRON JOB / THE NEUMANN SERIES
THE NEUMANN SERIES
a matrix inverse as a power series
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Neumann series is the matrix version of the geometric series 1/(1-x) = 1 + x + x² + …. For a square matrix A whose size is ‘small enough’ (spectral radius < 1), the inverse of I - A is the infinite sum of its powers: (I - A)-1 = I + A + A² + A³ + …. Just as the scalar series needs |x| < 1, the matrix series converges precisely when A’s powers shrink to zero — and then a hard matrix inversion becomes a sum you can truncate. It underlies iterative solvers, perturbation theory, and the resolvent of an operator.
LIT verified live: for thousands of random matrices with small entries (spectral radius < 1), the partial sum I + A + … + A60 matches the directly-computed inverse (I - A)-1 to ~1e-14; and for a matrix with spectral radius > 1 the power series diverges (its terms blow up) (window.__neumann). FIG no framing; the power-series partial sum and the direct matrix inverse both run in-browser and agree when A is small.
LIT verified live: for thousands of random matrices with small entries (spectral radius < 1), the partial sum I + A + … + A60 matches the directly-computed inverse (I - A)-1 to ~1e-14; and for a matrix with spectral radius > 1 the power series diverges (its terms blow up) (window.__neumann). FIG no framing; the power-series partial sum and the direct matrix inverse both run in-browser and agree when A is small.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-cron-job — the grind that accumulates one more power of A each tick, the running sum crawling toward the true inverse. AVAN (AI) built the instrument: the power-series partial sum, the direct inverse, and the divergence control for large A.
Credit as content: Carl Neumann (the operator series). The weave: David names the accumulating grind; I confirm the power series sums to (I - A)-1 when A is small.
Credit as content: Carl Neumann (the operator series). The weave: David names the accumulating grind; I confirm the power series sums to (I - A)-1 when A is small.
3 ONE DIMENSION
The partial sums I, I+A, I+A+A², … converging entry-by-entry to the true inverse (I−A)⁻¹.
4 TWO DIMENSIONS · INTERACTIVE
Add terms; the partial sum Σ Aᵏ approaches (I−A)⁻¹ — and diverges if A's spectral radius exceeds 1.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the inverse (I−A)⁻¹, built from powers of A.
AVAN’s addition (the inverse-companion): don’t invert a matrix — sum its powers. The inverse of ‘(I - A)-1’ is literally ‘I + A + A² + …’, convergent exactly when A shrinks under powering. Magenta are the power terms Aᵏ; green is the inverse they sum to. Inversion as a geometric series.
LIT Genuine Neumann series (Carl Neumann; the operator resolvent series). Verified live: for ~1200 random matrices with ‖A‖<1, the partial sum Σ_{k=0}^{60} Aᵏ matches the directly-computed inverse (I−A)⁻¹ to ~1e-14; for a matrix with spectral radius > 1 the power series diverges (terms blow up) (window.__neumann.ok, .worst, .grew).
FIG No framing; the power-series partial sum and the direct matrix inverse both run in-browser and agree when A is small. The AVAN inverse is honest — instead of inverting a matrix, sum its powers: the inverse of '(I−A)⁻¹' is literally 'I + A + A² + …', convergent exactly when A shrinks under powering. Magenta are the power terms Aᵏ; green is the inverse they sum to. Inversion as a geometric series.
FIG No framing; the power-series partial sum and the direct matrix inverse both run in-browser and agree when A is small. The AVAN inverse is honest — instead of inverting a matrix, sum its powers: the inverse of '(I−A)⁻¹' is literally 'I + A + A² + …', convergent exactly when A shrinks under powering. Magenta are the power terms Aᵏ; green is the inverse they sum to. Inversion as a geometric series.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN