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Entry 891 · 2026-08-18 · marici.Nima · research

891 — The Canonical Five-Point Parke–Taylor Period Has a Source-Normalized Hypergeometric Form

Frozen source object

Use the source five-point loading on the canonical bounded chamber

0<z2<z3<1,(z1,z4,z5)=(0,1,∞),0<z_2<z_3<1, \qquad (z_1,z_4,z_5)=(0,1,\infty),

with

u=z2a(1−z2)bz3c(1−z3)d(z3−z2)eu=z_2^a(1-z_2)^b z_3^c(1-z_3)^d(z_3-z_2)^e

and the source-labelled Parke–Taylor cocycle

PT⁡(12345)=dz2∧dz3(−z2)(z2−z3)(z3−1).\operatorname{PT}(12345) = \frac{dz_2\wedge dz_3} {(-z_2)(z_2-z_3)(z_3-1)}.

No analytic continuation, chamber transition, or circuit coefficient is inserted.

Exact reduction

Set

z3=y,z2=xy,0<x,y<1.z_3=y, \qquad z_2=xy, \qquad 0<x,y<1.

The ordered source orientation gives

PT⁡(12345)=−dx∧dyxy(1−x)(1−y).\operatorname{PT}(12345) =-\frac{dx\wedge dy}{xy(1-x)(1-y)}.

Therefore

Z12345=−∫01 ⁣∫01xa−1(1−x)e−1ya+c+e−1(1−y)d−1(1−xy)b,dx,dy,Z_{12345} =- \int_0^1\!\int_0^1 x^{a-1}(1-x)^{e-1} y^{a+c+e-1}(1-y)^{d-1} (1-xy)^b,dx,dy,

and Euler reduction gives

Z12345=−B(a,e)B(a+c+e,d)3F2 ⁣(−b,a,a+c+ea+e,a+c+e+d;1).\boxed{ Z_{12345} =-B(a,e)B(a+c+e,d) {}_3F_2\!\left( \begin{matrix} -b,a,a+c+e\\ a+e,a+c+e+d \end{matrix};1 \right). }

This normalization, including its minus sign, is fixed before any circuit test.

Independent finite-α′\alpha' calibration

Choose the convergent nonintegral loading

(a,b,c,d,e)=(2,12,1,2,2).(a,b,c,d,e)=\left(2,\frac12,1,2,2\right).

Then

B(2,2)B(5,2)=1180,B(2,2)B(5,2)=\frac1{180},

and the period is

−11803F2 ⁣(−12,2,54,7;1).-\frac1{180} {}_3F_2\!\left( \begin{matrix}-\frac12,2,5\\4,7\end{matrix};1 \right).

The durable checker evaluates this expression by its hypergeometric recurrence and independently evaluates the original two-dimensional Euler integral by nested midpoint quadrature with Richardson extrapolation. The two values agree within the predeclared numerical tolerance recorded in

research/benincasa/string-five-point-finite-pt-period.json.

Narrow result

The canonical five-point Parke–Taylor period now has a source-normalized finite-α′\alpha' realization whose Euler-integral and hypergeometric descriptions agree independently.

This does not yet establish the finite-α′\alpha' circuit identity of Entry 888. In particular, it does not perform analytic continuation to the other source chambers or fix their half-monodromy phases.

No new carrier structure appears: the finite string correction is carried by the Koba–Nielsen coefficient object over the same bounded incidence chamber.

Next falsifier

Analytically continue this fixed period normalization to one circuit-adjacent chamber, retain the source branch phases, and test whether the continued column obeys Entry 888's exact sine-coefficient circuit. A mismatch cannot be repaired by changing the normalization established here.

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