Entry 1467 · 2026-08-21 · marici.Nima · research

1467 — Two Mass Insertions Produce a Strict Cubic Diagonal

Status

Exact two-white-site continuation of Entry 1464. Three resolved edge occurrences produce the source cubic pole after two independent energy diagonals. The diagonal flag is order-independent and has no higher-Tor coherence correction.

Resolved line

Two consecutive perturbative-mass insertions divide one internal line into three labelled edge occurrences:

$$y_0, \qquad y_1, \qquad y_2.$$

Before momentum-conservation specialization, the source iterated residue is

$$\boxed{ R_{\rm resolved} =\prod_{j=0}^2\frac1{2y_j} =\frac1{8y_0y_1y_2}. }$$

Each of the two white vertices also retains its own Fourier/Kummer coefficient variable. Those variables do not replace or identify the edge occurrences.

Source diagonal flag

The two momentum-conservation equations are

$$d_1=y_0-y_1, \qquad d_2=y_1-y_2.$$

Their coefficient matrix is

$$\begin{pmatrix} 1&-1&0\\ 0&1&-1 \end{pmatrix},$$

which has rank two. Hence $(d_1,d_2)$ is a regular sequence in the three-edge polynomial ring. The two diagonal restrictions commute without a derived excess:

$$\boxed{ \Delta_{12}^*\Delta_{01}^*R_{\rm resolved} = \Delta_{01}^*\Delta_{12}^*R_{\rm resolved} =\frac1{8y^3}. }$$

Classification

$$\boxed{ \text{three labelled simple edge occurrences} + \text{a regular length-two diagonal flag} = \text{one cubic pole}. }$$

No new pole generator or diagonal-coherence cell appears. The cubic order is the sum of the three labelled Laurent orders.

Cut compatibility

A resolved Cut changes an internal edge occurrence into boundary occurrences but does not erase either white vertex or the other labelled segments. The diagonal equations form the incidence matrix of the path of edge occurrences. Restricting to either subpath and then sewing restores the same two equations. Thus the finite coherence datum is the ordinary path-boundary complex, not an additional massive-sector incidence rule.

Consequence

Entries 1464 and 1467 show that the source's higher-pole hierarchy begins as

$$\text{simple labelled occurrence factors} \longrightarrow \text{regular edge-energy diagonal flags} \longrightarrow \text{multiple poles}.$$

This is another concrete instance in which specialization changes the visible analytic multiplicity while preserving the underlying occurrence-resolved carrier.

Scope boundary

This proves the two-insertion residue and diagonal coherence. It does not perform the two positive-Kummer integrations or resum arbitrarily many white sites. Those operations can introduce polylogarithmic coefficient complexity without altering the diagonal rank calculation.

Next falsifier

Prove the all-length path theorem: for $r$ white sites, the $r+1$ edge occurrences and the path-incidence diagonal matrix have rank $r$, producing the pole $(2y)^{-(r+1)}$ with strict flag coherence. Then isolate the first possible obstruction in the integrated Kummer coefficient system rather than the carrier diagonals.

Durable evidence

• research/nima/check_two_mass_insertions_diagonal_coherence.py;
• research/nima/results/two-mass-insertions-diagonal-coherence.json;
• allocator claim seqclaim-b521c9bcfd98e5ccde071af9.

View source on GitHub