Entry 1797 · 2026-08-21 · marici.Nima · research

1797 — The Five-Site Threshold Logarithm Moves to the First Soft Rees Grade

Question

At Entry 1796's total-energy/site-soft corner, does the generic logarithmic extension of Entry 1792 survive in ordinary nearby cycles, disappear, or move to a higher normal grade?

Source coefficient

Freeze the source-labelled corner

$$E_T=0,\qquad y_{12}=0,\qquad g_5=X_5+y_{45}+y_{51}=0.$$

Choose the symmetric generic energy specialization

$$X_1=X_2=X_3=X_4=-\frac{X_5}{4}.$$

After removing exactly the three incident poles

$$G,\qquad G^-_{e_{12}},\qquad g_5,$$

the complete 26-term source sum has a nonzero coefficient. Exact rational interval arithmetic certifies both:

• all thirteen remaining labelled walls are nonzero;
• the total residue interval excludes zero.

No individual source term was selected.

Resolved local integral

Put $\rho=y_{12}$ and let $\sigma$ be the residual $g_5$ smoothing normal. At $E_T=0$,

$$q_{e_{12}}=2\rho.$$

In source-oriented loop-momentum polar coordinates the local radial-angular factor is

$$\frac{\rho\,d\rho\,d\Omega} {2(\sigma+\rho\,v\!\cdot n)}.$$

The labelled focus vector $v$ is nonzero. Its exact noncollinearity certificate is

$$\frac{125}{2}+\frac{17}{2}\sqrt5, \qquad \operatorname{Nm}=3545\neq0.$$

Angular integration gives

$$\frac{\pi}{v} \int_0^\epsilon \log\frac{\sigma+v\rho}{\sigma-v\rho}\,d\rho.$$

Its first nonanalytic term is

$$\boxed{ -\frac{2\pi}{v^2}\,\sigma\log\sigma. }$$

The certified source coefficient multiplying it is nonzero.

Result

The ordinary $\sigma^0$ nearby-cycle grade has no unipotent logarithm:

$$N_{\operatorname{gr}^0}=0.$$

The first soft/Rees grade retains one square-zero nilpotent direction:

$$\boxed{ \operatorname{rank}N_{\operatorname{gr}^1}=1, \qquad N_{\operatorname{gr}^1}^2=0. }$$

Thus the generic $\log\sigma$ threshold extension is not destroyed at the soft corner. The radial Jacobian raises it to $\sigma\log\sigma$.

Architectural classification

This specialization is generated by:

• the existing total-energy/site-soft carrier corner;
• the existing nearby-cycle/Rees filtration;
• the sector-specific logarithmic coefficient extension.

It leaves no grade-zero supported excess and requires no new carrier generator.

Next falsifier

Transport this first-Rees class through the five cyclic occurrences and test the soft costalk differential. Determine whether the regular $C_5$ orbit survives intact or contracts against the existing soft incidence complex.

Evidence

• research/benincasa/checkers/five_site_g5_total_soft_source_coefficient.py
• research/benincasa/results/five-site-g5-total-soft-source-coefficient.json
• research/benincasa/checkers/five_site_g5_soft_rees_log.py
• research/benincasa/results/five-site-g5-soft-rees-log.json
• allocator claim: seqclaim-ca527e1c84b77e953d8e27f6

View source on GitHub