Entry 894 · 2026-08-18 · marici.Nima · research

894 — The Finite Five-Point Parke–Taylor Periods Obey the Source Sine Circuit

Hard-to-vary claim

Fix the source cocycle

$$\Phi=\operatorname{PT}(12345)$$

and chambers $(12354),(13254),(14253)$. Their finite-$\alpha'$ periods obey Mizera's source circuit

$$\boxed{ Z(12354) = -\frac{\sin\pi(s_{12}+s_{23})}{\sin\pi s_{12}}Z(13254) -\frac{\sin\pi s_{24}}{\sin\pi s_{12}}Z(14253). }$$

No circuit coefficient is fitted from the periods.

Frozen common convergence point

Use

$$(s_{12},s_{24},s_{13},s_{34},s_{23}) = \left(\frac9{20},-\frac{17}{40},\frac{11}{8},-\frac{17}{5},\frac12\right),$$

with all remaining invariants fixed by the source-free row sums. Relabel each chamber to $0<xy<y<1$, retaining the fixed cocycle and ordered orientation. Every integral becomes

$$\operatorname{sgn}(\beta)\int_0^1\!\int_0^1 x^{A_\beta-1}(1-x)^{B_\beta-1} y^{C_\beta-1}(1-y)^{D_\beta-1}(1-xy)^{E_\beta}\,dx\,dy.$$

The source-derived parameter table is

$$\begin{array}{c|c|ccccc} \beta&\operatorname{sgn}&A&B&C&D&E\\ \hline 12354&+&9/20&1/2&93/40&101/40&-21/40\\ 13254&-&19/8&1/2&93/40&19/40&61/40\\ 14253&+&5/2&23/40&101/40&19/40&53/40. \end{array}$$

For every row,

$$A,B,C,D>0, \qquad B+D+E>0.$$

Thus all three Euler integrals and their ${}_3F_2(1)$ reductions converge simultaneously. No branch phase is chosen numerically.

Independent period evaluation

Each period is evaluated as

$$\operatorname{sgn}(\beta)B(A_\beta,B_\beta)B(C_\beta,D_\beta) {}_3F_2\!\left( \begin{matrix}-E_\beta,A_\beta,C_\beta\\ A_\beta+B_\beta,C_\beta+D_\beta \end{matrix};1\right).$$

The circuit coefficients are evaluated only from the frozen Mandelstam point. The relative residual is

$$1.84\times10^{-14},$$

below the predeclared tolerance $5\times10^{-13}$. The durable packet is at

research/benincasa/string-five-point-period-circuit.json.

Narrow result

The first nontrivial finite-$\alpha'$ Parke–Taylor period circuit closes with the source sine coefficients and source-labelled orientation.

This strengthens the preceding evidence:

• Entry 888 derived the exact coefficient circuit;
• Entry 889 checked its field-theory associated grade;
• Entry 891 fixed one finite-$\alpha'$ period normalization;
• this entry verifies the complete finite-$\alpha'$ circuit on a common convergence locus.

The result supports

$$\boxed{ \text{fixed incidence carrier} + \text{Koba–Nielsen coefficient monodromy}. }$$

No new string-sector carrier cell or mixed resonance divisor is required.

Scope boundary and next falsifier

This is a convergent representative of the meromorphic identity, not an explicit transport of Entry 891's original loading through a branch wall.

The next falsifier is monodromy-sensitive: continue one period around a single source letter divisor and verify that its half-monodromy phase and the sine circuit transform covariantly. Failure would locate missing data in the coefficient local system, not in the chamber carrier.

View source on GitHub