Entry 1469 · 2026-08-21 · marici.Nima · research

1469 — Every Mass-Insertion Chain Is a Regular Path Diagonal

Status

All-length source-diagonal theorem extending Entries 1464 and 1467. A chain of $r$ perturbative-mass white sites retains $r+1$ labelled edge occurrences. Its $r$ momentum-conservation equations form a regular path incidence sequence, producing the source pole of order $r+1$ with strict partial-flag coherence.

Resolved chain

Before edge-energy specialization, label the segment energies

$$y_0,y_1,\ldots,y_r.$$

The iterated source residue is

$$\boxed{ R_r^{\rm resolved} =\prod_{j=0}^{r}\frac1{2y_j}. }$$

The white sites impose the path equations

$$d_i=y_{i-1}-y_i, \qquad 1\le i\le r.$$

Regularity

Their coefficient matrix is the oriented incidence matrix of a path:

$$B_r= \begin{pmatrix} 1&-1&0&\cdots&0\\ 0&1&-1&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&0\\ 0&\cdots&0&1&-1 \end{pmatrix}.$$

Deleting its last column leaves an upper-triangular $r\times r$ minor with determinant $1$. Hence

$$\operatorname{rank}B_r=r.$$

More strongly, every subset of rows is independent: in any nonempty linear combination, the leftmost selected row has a first nonzero column not supplied by a later selected row. Therefore every partial diagonal flag is regular and its result is independent of the order in which its equations are imposed.

Complete diagonal

On

$$y_0=y_1=\cdots=y_r=y,$$

the resolved residue becomes

$$\boxed{ R_r=\frac1{(2y)^{r+1}}. }$$

Thus the pole order is the number of labelled edge occurrences, not a new primitive assigned to the white-site chain.

Cut and flag coherence

A resolved Cut selects a forest of subpaths. The corresponding rows are a subset of $B_r$, hence remain independent. Sewing the subpaths restores the omitted rows. Since all partial flags are regular, Cut order and diagonal order do not generate a higher-Tor correction.

The complete carrier-side statement is therefore

$$\boxed{ \text{mass-insertion higher poles} = \text{labelled simple edge factors pulled back along regular path diagonals}. }$$

Machine audit

The exact checker verifies full rank for every $1\le r\le32$ and exhausts all $2^r$ partial diagonal subsets through $r=9$. It also verifies the coefficient and pole-order law at every tested length.

Scope boundary

This closes the carrier and diagonal-coherence part of the perturbative-mass chain. It does not integrate the $r$ positive-Kummer white-site variables. The primary source notes that this resummation is not a geometric series. Any remaining obstruction must therefore live in the integrated coefficient system or its convergence/Stokes data, not in the edge-diagonal carrier.

Next falsifier

At two white sites, compute the coupled positive-Kummer pushforward before attempting all-order resummation. Determine whether its polylogarithmic coefficient satisfies a finite convolution recursion over the regular path diagonal, or whether the first genuine extension appears there.

Durable evidence

• research/nima/check_all_mass_insertion_path_diagonals.py;
• research/nima/results/all-mass-insertion-path-diagonals.json;
• allocator claim seqclaim-b628577c6a72ffc891c640e0.

View source on GitHub