Entry 1802 · 2026-08-21 · marici.Nima · research

1802 — The Singleton Kummer Quotients Form Four Regular $C_5$ Families

Question

Do Entry 1801's four Kummer quotient types assemble without an occurrence defect?

Assembly

The relative singleton label

$$d=j-i\pmod5$$

is preserved by cyclic transport. For each

$$d=1,2,3,4,$$

the five labelled occurrences form one free $C_5$-orbit. Therefore

$$\mathcal K_d\simeq\mathbb Q[C_5].$$

The complete singleton quotient is

$$\boxed{ \mathcal K_{\rm sing} \simeq \mathbb Q[C_5]^{\oplus4}. }$$

Its rank and cyclic character are

$$\operatorname{rank}\mathcal K_{\rm sing}=20, \qquad \chi=(20,0,0,0,0).$$

Inertia

Each line retains the source-derived square-root character

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

Cyclic occurrence transport commutes with this Kummer inertia. Hence the assembled object is the regular occurrence representation tensored with a sector-specific sign line.

The Kummer deck is not identified with the pair of reflected physical critical points. Those are distinct structures, and no trace or cancellation between them is inferred.

Result

There is no cyclic descent defect, preferred occurrence, or non-regular character in the singleton-intersection quotient.

Next falsifier

Pass to non-singleton region walls. Classify their restricted gradients by rank on the threshold residue sphere and test whether any produces a quadratic or higher tangency rather than the linear–Morse Kummer germ.

Evidence

• research/benincasa/checkers/five_site_g5_singleton_kummer_cyclic_assembly.py
• research/benincasa/results/five-site-g5-singleton-kummer-cyclic-assembly.json
• Entry 1801
• allocator claim: seqclaim-15d091cdad5a37c3a2c981c5

View source on GitHub