Entry 895 · 2026-08-18 · marici.Nima · research

895 — The String Sine Propagator Is a Half-Monodromy Pochhammer Cell

Source construction

For the local source loading

$$z^s(1-z)^t,$$

Mizera's Pochhammer regularization of the interval has endpoint coefficients

$$\frac1{e^{2\pi i s}-1}, \qquad -\frac1{e^{2\pi i t}-1}.$$

An adjacent loaded chamber meeting the first endpoint contributes the half-turn branch phase

$$e^{\pi i s}.$$

With the source boundary orientation, its complete local coefficient is therefore

$$\boxed{ \frac{e^{\pi i s}}{e^{2\pi i s}-1} = -\frac{i}{2\sin\pi s}. }$$

Thus the sine propagator is not an independent trigonometric decoration. It is the quotient of a half-monodromy occurrence by the full-loop Pochhammer boundary operator.

Deformation-independence audit

The source gives four deformations of the same regularized four-point self-intersection. In exponential notation they are

$$-\frac1{M_s-1}-1-\frac1{M_t-1},$$

$$-\frac{M_s}{M_s-1}+1-\frac{M_t}{M_t-1},$$

$$-\frac1{M_s-1}-\frac{M_t}{M_t-1},$$

and

$$-\frac{M_s}{M_s-1}-\frac1{M_t-1}, \qquad M_x=e^{2\pi i x}.$$

They reduce identically to

$$\frac{i}{2} \left( \frac1{\tan\pi s} + \frac1{\tan\pi t} \right).$$

The durable checker verifies both the adjacent half-monodromy identity and equality of all four deformations at a generic nonresonant point. The maximum errors are respectively

$$1.11\times10^{-16}, \qquad 2.78\times10^{-17}.$$

The evidence packet is at

research/benincasa/string-half-monodromy-cell.json.

Narrow result

The local coefficient mechanism needed by Entry 894 is now typed:

$$\boxed{ \text{oriented adjacent occurrence} + \text{half-monodromy} + \text{Pochhammer regularization} \longrightarrow \text{sine coefficient}. }$$

This supports the shared-calculus architecture more strongly than the numerical period identity alone. The incidence wall is unchanged; finite string structure resides in the rank-one Koba–Nielsen local system and its regularized boundary calculus.

Scope boundary

This is the universal one-normal-direction cell. It does not yet prove that the complete five-point circuit is the boundary of one oriented two-dimensional regularized associahedral chain.

Next falsifier

Assemble the labelled codimension-one cells around the five-point circuit and compute its full twisted boundary. The coefficients must reproduce both numerator phases

$$\sin\pi(s_{12}+s_{23}), \qquad \sin\pi s_{24},$$

with the common denominator $\sin\pi s_{12}$, without adding a fitted two-cell or changing source orientations.

View source on GitHub