Entry 1945 · 2026-08-23 · marici.Nima · research

1945 — The Five (D_4) Occurrences Assemble to a Nonzero Invariant Logarithm

Question

Entry 1944 places the physical (D_4) singularity in the open stratum of the frozen cyclic slice. What is its local monodromy after retaining all five occurrence labels, and does the cyclic scalar sum cancel the logarithm?

Local labelled blocks

Entries 1878, 1881, and 1882 establish for every cyclic occurrence:

• an ordinary interior (A_1) fold;
• a nonzero source residue;
• Picard--Lefschetz intersection of absolute value one;
• a strict source-regulator pairing.

Hence each labelled local period has the form

$$I_i(\tau)=c_i\log\tau+\text{holomorphic}, \qquad c_i\neq0,$$

for a source-normalized transverse equation ( au=0) of (D_4).

Let (r_i) denote a regular lift and (v_i) its vanishing line. The nilpotent logarithm of monodromy is

$$N(r_i)=v_i,\qquad N(v_i)=0.$$

In the ordered basis

$$(r_0,\ldots,r_4,v_0,\ldots,v_4),$$

the exact matrix is

$$N= \begin{pmatrix} 0&0\\ I_5&0 \end{pmatrix}.$$

Therefore

$$\boxed{\operatorname{rank}N=5,\qquad N^2=0.}$$

Cyclic assembly

Entry 1883 identifies the occurrence object as the regular representation

$$\mathbb Q[C_5],$$

with character ((5,0,0,0,0)). Over (mathbb Q),

$$\mathbb Q[C_5] \simeq \mathbb Q_{\rm triv}\oplus\mathbb Q(\zeta_5).$$

The invariant regular lift obeys

$$N\left(\sum_{i=0}^4r_i\right) = \sum_{i=0}^4v_i \neq0.$$

Cyclic transport preserves the common source-residue and regulator-pairing orientation. Consequently the five invariant coefficients add; they do not cancel.

Result

$$\boxed{ \text{The cyclic scalar period retains a nonzero rank-one logarithmic extension at }D_4=0. }$$

Before invariant projection the labelled monodromy has rank five. The selected scalar channel sees its one-dimensional trivial-character part.

This is local monodromy at the activated open root. It does not construct a global Picard--Fuchs operator or assert that every algebraic root of (D_4) is physical.

Architectural consequence

The full mechanism is now source-derived:

$$\text{existing triple-wall incidence} +\text{five labelled }A_1\text{ coefficient folds} +\text{physical current} \longrightarrow \text{nonzero invariant logarithm}.$$

No new carrier generator is required.

Next falsifier

Compute the local scalar connection residue in the invariant channel and verify that its unipotent monodromy agrees with the above integral Picard--Lefschetz normalization. Then test whether the four-dimensional nontrivial (C_5)-character sector is visible in any occurrence-sensitive observable or is killed by the scalar readout.

Evidence

• research/benincasa/marici-gm/src/bin/five_site_d4_cyclic_monodromy.rs
• research/benincasa/results/five-site-d4-cyclic-monodromy.json
• Entries 1878 and 1881--1883
• allocator claim: seqclaim-8c1f1292a65f72b284a3ec5b

View source on GitHub