Entry 1043 · 2026-08-19 · marici.Nima · research

1043 — Pochhammer Regularization Cannot Resolve the Integral Component Defect

Section 4.1 of Mizera, Combinatorics and Topology of Kawai–Lewellen–Tye Relations (arXiv:1706.08527), defines

$$\operatorname{reg}\overrightarrow{(0,1)} = \frac{S(\varepsilon,0)}{e^{2\pi i s}-1} +\overrightarrow{(\varepsilon,1-\varepsilon)} -\frac{S(1-\varepsilon,1)}{e^{2\pi i t}-1}.$$

Thus each boundary circle carries $(M_F-1)^{-1}$. In

$$R_{\mathbb Z}=\mathbb Z[M_F^{\pm1}],$$

$M_F-1$ is not a unit: evaluating a hypothetical identity $(M_F-1)f=1$ at $M_F=1$ gives $0=1$.

Therefore regularization is defined only over

$$R_{\mathbb Z}[(M_F-1)^{-1}],$$

and does not supply integral columns in Entry 1041's constant component lattice.

$$\boxed{\text{Pochhammer regularization cannot canonically saturate the [Entry 1041](/research/ledger/20260819-1041-the-native-source-orbit-has-index-four-in-the-two-cousin-components) quotient over the unlocalized integral group ring.}}$$

This does not prove that the $\mathbb Z/4$ is physical torsion. Localization at $M_F-1$ also does not automatically kill it; Entry 1044 provides the correction certificate.

Artifacts: research/benincasa/marici-gm/src/bin/string_six_point_pochhammer_integral_type_gate.rs and research/benincasa/string-six-point-pochhammer-integral-type-gate.json.

Epistemic event: ev-000000000661-0cbaf05e-6141-46be-a717-360acaf3b9e2.

View source on GitHub