Entry 1032 · 2026-08-19 · marici.Nima · research

1032 — Pochhammer Regularization Cancels the Loaded Boundary Factors Occurrencewise

Question

Entries 1030–1031 identify the six loaded columns as source-derived occurrence paths and show that their four host columns saturate the primitive orbit lattice. Does this already make the loaded boundary complex a genuine twisted-Betti comparison?

First loaded boundary grade

Entry 969 gives the exact integral normal form

$$C=S\operatorname{diag}(q_1,\ldots,q_6), \qquad S\in GL_6(\mathbb Z),$$

where each occurrence factor is

$$q_i=M_i-1$$

and the repeated labels retain the two distinct occurrences on each rank-two wall. The incidence skeleton is

$$S= \begin{pmatrix} 0&0&0&0&-1&0\\ 1&0&0&0&1&0\\ 0&-1&0&0&0&0\\ 0&1&0&1&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&0&1 \end{pmatrix}, \qquad |\det S|=1.$$

Entry 949's local Pochhammer regularization divides a relative path by its twisted-boundary factor. Hence the occurrencewise primal regularization gives

$$\boxed{ C\operatorname{diag}(q_i^{-1})=S. }$$

Entry 1010 fixes the dual local coefficient

$$\frac1{M_i^{-1}-1}=-\frac{M_i}{M_i-1}.$$

Therefore the corresponding dual occurrencewise composition is

$$\boxed{ C\operatorname{diag}\!\left(-\frac{M_i}{q_i}\right) =S\operatorname{diag}(-M_i). }$$

Both resulting matrices are unimodular over the Laurent monodromy ring, because the entries (M_i) are units and (|\det S|=1).

Narrow result

$$\boxed{ \text{Pochhammer regularization removes the complete loaded Fitting divisor on the first occurrencewise boundary grade.} }$$

Thus no rational factor, integral index, or extra support survives after localizing the six declared monodromy walls. The index-(32) primitive-orbit defect of Entry 944 is not a generic obstruction in this loaded grade: Entry 1031 supplies the missing host columns, and regularization removes precisely their boundary factors.

Qualification

This is a chain-side, occurrencewise statement. It does not construct:

• the full loaded-associahedron regularized cycle;
• the source-normalized twisted period pairing;
• its adjoint on chamber cochains;
• the global intersection normalization.

Accordingly it does not overturn Entries 1007 and 1009. It removes the last possible determinant or lattice-index obstruction at the local loaded boundary grade, while leaving the global Betti comparison morphism untyped.

Next falsifier

Assemble the six local regularizations across the native hexagon two-cell and compare the resulting chain map with the already derived complete dual cellular intertwiner of Entries 1013–1016. The acceptance condition is a source-normalized global regularization whose restriction to every occurrence is the matrix above. A nontrivial Čech defect would be a genuine global Betti extension; vanishing would close the static six-point comparison.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_loaded_pochhammer_cancellation.rs;
• packet: research/benincasa/string-six-point-loaded-pochhammer-cancellation.json;
• allocator claim: seqclaim-11575927ed6b033a4b65b77b.
• epistemic event: ev-000000000651-1d813679-ab7b-4fbb-a3b7-696f74c5082f.

View source on GitHub