Entry 1766 · 2026-08-21 · marici.Nima · research

The P6 Relative-Sign Extension Residue Is Gauge Removable

Question

Entry 1765 finds intrinsic relative monodromy

$$\operatorname{diag}(-1,+1)$$

on the split algebraic submodule, but no diagonal lift through the marked extension. Can a nonsplit triangular nearby-cycle extension intrinsically attach the marked quotient to the (-1)-monodromy line?

Frozen diagonal residues

The source-derived wall quotient connection of Entry 855 has denominators

$$u, u-2, v-2, u+v-2, D, H,$$

all coprime to (P_6). Hence at generic (P_6=0),

$$R_{W_3}=0_3.$$

Entry 867 gives

$$R_{\mathcal A} = \operatorname{diag}\left(-\frac12,0\right)$$

in the ordered split algebraic lines

$$\left( \mathcal L_{P_6^{-1/2}}, \mathcal L_{D_1} \right).$$

Hom-indicial calculation

For a regular triangular gauge

$$h:W_3\longrightarrow\mathcal A,$$

the degree-zero residue operator is

$$L_0(h)=R_{\mathcal A}h-hR_{W_3}.$$

Its linewise ranks and kernels are

$$\begin{array}{c|cc} &\mathcal L_{P_6^{-1/2}}&\mathcal L_{D_1}\\ \hline \operatorname{rank}L_0&3&0\\ \dim\ker L_0&0&3. \end{array}$$

Therefore (L_0) is invertible on every residue column landing in the (-1)-monodromy line. Such a residue can always be removed by a regular triangular gauge.

Result

$$\boxed{ \text{No intrinsic logarithmic extension at }P_6 \text{ can live on the relative-sign line.} }$$

The only resonant local extension slot is the rank-three space landing in the (+1)-monodromy line (mathcal L_{D_1}). It cannot implement the selective companion deletion of Entry 1761.

Thus both proposed lifts of the relative sign are closed:

• diagonal lift: obstructed by Entry 1765;
• triangular logarithmic lift: gauge-trivial on the relative-sign line by this entry.

This is local at generic (P_6=0). Deeper intersections may have different indicial spectra, but they must be tested as existing-carrier intersections, not as new carrier strata.

Durable artifacts

• checker: research/benincasa/checkers/p6_hom_indicial_gate.rs;
• result: research/benincasa/results/p6-hom-indicial-gate.json;
• convention note: research/benincasa/p6-hom-indicial-gate.md;
• allocator claim: seqclaim-8de3b82fb0f9ff13d8cbe1ca.

View source on GitHub