Entry 1806 · 2026-08-21 · marici.Nima · research

1806 — The Complementary Soft Occurrences Produce a Pole-Twisted Logarithm

Question

What coefficient object survives at Entry 1805's remaining $y_{45}=0$ defining-edge soft corner, where $g_{15}$ and $g_{234}$ are distinct occurrences of the same fiber function?

Source gate

Freeze

$$G^-_{e_{12}}=g_5=g_{15}=g_{234}=0, \qquad y_{45}=0.$$

The source-compatible energy specialization is fixed by

$$E_T=-2y_{12}, \qquad X_5=-y_{51}, \qquad X_1+X_5=-y_{12}.$$

Exactly three OFPT terms contain all three noncommon labelled walls. After removing the four incident denominators, exact rational interval arithmetic certifies:

• all eight remaining wall labels are nonzero;
• the complete three-term coefficient is nonzero.

Thus the doubled occurrence is not killed by source cancellation.

Resolved local model

Near the soft focus, let $r$ be the radial variable and $u$ the squared angular normal to the common occurrence section. The two labelled denominators have leading forms

$$h_1+a\,ru, \qquad h_2+a\,ru, \qquad a\neq0.$$

The resolved measure obeys the exact product-variable identity

$$\int_0^Rdr\int_0^Udu\,f(ru) = \int_0^{RU}dw\, \log\frac{RU}{w}\,f(w).$$

Therefore the singular kernel is

$$\int \frac{\log(M/w)\,dw} {(h_1+aw)(h_2+aw)}.$$

If

$$K(h) \sim \frac{1}{2a}\log^2h$$

denotes its singular primitive, the two-occurrence class is the divided difference

$$\boxed{ \frac{K(h_1)-K(h_2)}{h_2-h_1}. }$$

Diagonal specialization

On the physical diagonal $h_1=h_2=h$,

$$\boxed{ \frac{K(h_1)-K(h_2)}{h_2-h_1} \longrightarrow -\frac{\log h}{a\,h}. }$$

Hence the diagonal object is a simple pole tensored with a rank-one logarithmic extension:

$$\operatorname{rank}N=1, \qquad N^2=0.$$

Architectural result

The complementary labels must remain occurrence-resolved before taking the diagonal. Their doubled section produces a divided logarithm, while physical identification produces a pole-twisted logarithm.

This is coefficient complexity generated by:

• an existing defining-edge soft carrier corner;
• occurrence resolution;
• radial/angular Rees geometry.

No new carrier generator is required.

Next falsifier

Transport this doubled-occurrence class through its free $C_5$-orbit and compute the diagonal/anti-diagonal occurrence character. Test whether the relative-sign component survives the physical occurrence trace.

Evidence

• research/benincasa/checkers/five_site_g5_complement_soft_source_coefficient.py
• research/benincasa/results/five-site-g5-complement-soft-source-coefficient.json
• research/benincasa/checkers/five_site_g5_complement_soft_divided_log.py
• research/benincasa/results/five-site-g5-complement-soft-divided-log.json
• allocator claim: seqclaim-78627145449663a3cdfb97af

View source on GitHub