Entry 1801 · 2026-08-21 · marici.Nima · research

1801 — Singleton Intersections Convert the Five-Site Logarithm into Kummer Quotients

Question

After Entry 1800's typing correction, what is the actual coefficient object at an intersection of the physical threshold with a further singleton wall?

Exact gates

For each relative label

$$d=j-i=1,2,3,4\pmod5$$

and both reflected physical sheets:

1. restrict the added singleton gradient to the $q_e$-residue sphere;
2. certify its squared norm by rational interval arithmetic;
3. impose $g_j=0$ with a source-energy compensator preserving $E_T$ and $g_i=0$;
4. remove exactly $g_i$, $g_j$, and $q_e$ from the complete 26-term source sum;
5. certify all thirteen remaining walls and the resulting residue coefficient as nonzero.

All eight labelled-sheet tests pass.

Local residue

After positive-definite Morse normalization, write the added wall as

$$g_j=h+\lambda u+\cdots, \qquad \lambda\neq0.$$

The local three-wall germ is

$$\int \frac{du\,dv} {(\tau+u^2+v^2)(h+\lambda u)}.$$

Taking the added-wall residue first gives

$$\boxed{ \operatorname{Res}_{g_j} = \frac{\pi} {\lambda\sqrt{\tau+h^2/\lambda^2}}. }$$

Its discriminant is

$$\boxed{ h^2+\lambda^2\tau=0. }$$

Result

Each of the four relative-label orbits carries a rank-one Kummer quotient:

$$T_s=-1, \qquad N=0.$$

Thus an added singleton does not produce a second logarithmic extension. It converts the wall-residue quotient of the threshold logarithm into a square-root coefficient system.

The four quadratic discriminants are derived from the existing singleton-threshold incidence and the source Morse Hessian. They are coefficient support, not new carrier generators.

Architectural consequence

This is a direct five-site example of

$$\text{existing marked carrier incidence} + \text{sector-specific change of coefficient type}$$

from unipotent logarithmic variation to semisimple Kummer variation.

Next falsifier

Assemble each Kummer quotient through its free $C_5$-orbit and compare its deck character with the occurrence and soft characters. Then pass to non-singleton region walls, whose restricted gradients can have different incidence rank.

Evidence

• research/benincasa/checkers/five_site_g5_singleton_tangent_gradients.py
• research/benincasa/results/five-site-g5-singleton-tangent-gradients.json
• research/benincasa/checkers/five_site_g5_singleton_source_coefficients.py
• research/benincasa/results/five-site-g5-singleton-source-coefficients.json
• research/benincasa/checkers/five_site_g5_singleton_kummer_quotients.py
• research/benincasa/results/five-site-g5-singleton-kummer-quotients.json
• allocator claim: seqclaim-ac00f271acc09f967faa4faa

View source on GitHub