THE FOLD / CHEAT / THE BACKDOOR / THE GAMMA REFLECTION
THE GAMMA REFLECTION
a gamma product equal to a cosecant
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Euler’s reflection formula ties the gamma function to the sine in one clean stroke: Γ(x)·Γ(1-x) = π/sin(πx). The gamma function Γ extends the factorial to all real (and complex) numbers, and it looks nothing like a trig function — yet multiply its value at x by its value at the mirror point 1-x, and the messy transcendental factorials collapse into a simple cosecant. Setting x = ½ gives Γ(½)² = π, so Γ(½) = √π — the gateway to the Gaussian integral. The poles of the gamma function at 0, -1, -2, … line up exactly with the zeros of sine.
LIT verified live: computing Γ by the Lanczos approximation, the product Γ(x)·Γ(1-x) equals π/sin(πx) to a relative error ~1e-14 for thousands of x in (0,1), and Γ(½)² = π (window.__gammareflection). FIG no framing; the gamma product and the cosecant are computed by different routes in-browser and agree.
LIT verified live: computing Γ by the Lanczos approximation, the product Γ(x)·Γ(1-x) equals π/sin(πx) to a relative error ~1e-14 for thousands of x in (0,1), and Γ(½)² = π (window.__gammareflection). FIG no framing; the gamma product and the cosecant are computed by different routes in-browser and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-backdoor — the cheat: instead of evaluating a hard factorial, reflect across x = ½ and read the answer off a sine. AVAN (AI) built the instrument: the Lanczos gamma, the reflection product, and the π/sin(πx) cross-check.
Credit as content: Leonhard Euler (reflection formula). The weave: David names the backdoor; I confirm Γ(x)Γ(1-x) equals π/sin(πx).
Credit as content: Leonhard Euler (reflection formula). The weave: David names the backdoor; I confirm Γ(x)Γ(1-x) equals π/sin(πx).
3 ONE DIMENSION
Γ(x) and its mirror Γ(1−x) on (0,1); their product traces exactly the curve π/sin(πx).
4 TWO DIMENSIONS · INTERACTIVE
Slide x; Γ(x)·Γ(1−x) is compared to π/sin(πx) — equal across the whole interval.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the product Γ(x)Γ(1−x), equal to π/sin(πx).
AVAN’s addition (the inverse-companion): don’t evaluate a lone factorial — pair it with its reflection. The inverse of ‘Γ(x)’ is ‘π/(sin(πx)·Γ(1-x))’, so the value at x and at 1-x lock together through a sine. Magenta are the two mirrored gamma curves; green is the cosecant their product traces. Factorials reflected into a sine.
LIT Genuine Euler reflection formula (Leonhard Euler). Verified live: with the Lanczos gamma approximation, Γ(x)·Γ(1−x) equals π/sin(πx) to relative error ~5e-15 for ~8000 x in (0,1), and Γ(½)²=π (window.__gammareflection.ok, .worst, .halfOk).
FIG No framing; the gamma product and the cosecant are computed by different routes in-browser and agree. The AVAN inverse is honest — instead of evaluating a lone factorial, pair it with its reflection: the inverse of 'Γ(x)' is 'π/(sin(πx)·Γ(1−x))', so the value at x and at 1−x lock together through a sine. Magenta are the two mirrored gamma curves; green is the cosecant their product traces. Factorials reflected into a sine.
FIG No framing; the gamma product and the cosecant are computed by different routes in-browser and agree. The AVAN inverse is honest — instead of evaluating a lone factorial, pair it with its reflection: the inverse of 'Γ(x)' is 'π/(sin(πx)·Γ(1−x))', so the value at x and at 1−x lock together through a sine. Magenta are the two mirrored gamma curves; green is the cosecant their product traces. Factorials reflected into a sine.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN