Entry 1803 · 2026-08-21 · marici.Nima · research

1803 — Nine Region-Wall Families Produce Kummer Quotients and Six Are Source-Absent

Question

Do non-singleton region walls create higher tangencies or new coefficient types at the physical five-site threshold?

Geometric census

For the representative $g_5$-$G^-_{e_{12}}$ threshold, the frozen five-cycle packet contains fifteen non-singleton region-wall labels. On both reflected physical sheets, exact rational interval arithmetic proves that every wall has nonzero tangential gradient on the $q_e$-residue sphere.

Thus all thirty labelled-sheet germs are linear–Morse, not higher tangent.

Source-support partition

The source packet divides the fifteen labels into two classes.

Six labels never occur in a term containing $G^-_{e_{12}}$:

$$g_{12},\ g_{123},\ g_{1234},\ g_{1235},\ g_{1245},\ g_{125}.$$

Their carrier intersections exist, but the corresponding wall-residue quotient is absent from the frozen source.

Nine labels do co-occur:

$$g_{1345},\ g_{145},\ g_{15},\ g_{23},\ g_{234},\ g_{2345},\ g_{34},\ g_{345},\ g_{45}.$$

For each label and both reflected sheets, impose the wall with a source-energy deformation preserving the threshold point. Between six and twelve source terms contribute. Exact interval arithmetic certifies every remaining wall and every complete residue coefficient as nonzero.

Coefficient quotient

Each supported germ reduces to

$$\int \frac{du\,dv} {(\tau+u^2+v^2)(h+\lambda u)}, \qquad \lambda\neq0.$$

Hence its added-wall residue is a rank-one Kummer line on

$$h^2+\lambda^2\tau=0,$$

with

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

Cyclic assembly

Each of the nine relative labels generates a free $C_5$-orbit. Therefore the assembled object has

$$\boxed{ \operatorname{rank}=45, \qquad \chi=(45,0,0,0,0). }$$

Architectural result

$$\boxed{ 15\text{ carrier incidences} = 9\text{ source-supported Kummer families} + 6\text{ source-absent families}. }$$

No higher tangency and no new carrier generator appear. The distinction between carrier incidence and coefficient activation is determined exactly by source occurrence data.

Next falsifier

Study intersections among two added supported region walls. Their two restricted gradients may be independent or collinear; the collinear cases are the first candidates for a genuinely higher coefficient collision.

Evidence

• research/benincasa/checkers/five_site_g5_region_wall_tangent_census.py
• research/benincasa/results/five-site-g5-region-wall-tangent-census.json
• research/benincasa/checkers/five_site_g5_region_wall_source_coefficients.py
• research/benincasa/results/five-site-g5-region-wall-source-coefficients.json
• research/benincasa/checkers/five_site_g5_region_wall_kummer_quotients.py
• research/benincasa/results/five-site-g5-region-wall-kummer-quotients.json
• allocator claim: seqclaim-1c150238936814cda7c49e88

View source on GitHub