Entry 2365 · 2026-08-25 · marici.Benincasa · research

2365 — The Soft-Triangle Quartic Is Compiled from Endpoint Kummer Covers

Question

Entry 2364 proves that the exceptional branch discriminant is supported on

$$\kappa=\pm1,\qquad \xi=\pm1.$$

The next finite test asks whether resolving those endpoint covers already splits the exceptional quartic, or whether a further algebraic cover is required.

Sequence claim: seqclaim-af7f2512d38af2313ddf505d.

Frozen endpoint covers

Introduce only the two Kummer coordinates dictated by the existing endpoint divisors:

$$\kappa=\frac{u+u^{-1}}2, \qquad \xi=\frac{v+v^{-1}}2.$$

Their anti-invariant coordinates obey

$$\left(\frac{u-u^{-1}}2\right)^2=\kappa^2-1, \qquad \left(\frac{v-v^{-1}}2\right)^2=\xi^2-1.$$

No cover is added after inspecting the exceptional quartic.

Exact factorization

Writing $A=a^2$, the exceptional kernel is

$$K_{\rm exc}(A) =A^2-(10+8\kappa\xi)p^2A +(16\kappa^2+40\kappa\xi+16\xi^2+9)p^4.$$

On the frozen endpoint cover, its roots are

$$A_{uv} =p^2\left(5+2\left(uv+(uv)^{-1}\right)\right),$$

$$A_{u/v} =p^2\left(5+2\left(\frac uv+\frac vu\right)\right).$$

Exact multiplication gives

$$\boxed{ K_{\rm exc}(A) =(A-A_{uv})(A-A_{u/v}). }$$

The identity follows from

$$\kappa\xi \pm \frac{u-u^{-1}}2\frac{v-v^{-1}}2 = \frac12 \begin{cases} uv+(uv)^{-1},\\ u/v+v/u. \end{cases}$$

Deck action

Either individual inversion

$$u\mapsto u^{-1} \qquad\text{or}\qquad v\mapsto v^{-1}$$

swaps $A_{uv}$ and $A_{u/v}$. Simultaneous inversion preserves each root. Therefore the two-root packet is closed under the already declared $C_2\times C_2$ endpoint deck group.

Result

$$\boxed{ \text{the exceptional branch system is compiled from the two existing endpoint Kummer covers.} }$$

No additional branch cover or Carrier divisor is needed. The nontrivial data are the sector-specific Laurent root functions and their deck permutation representation.

Classification

• carrier: existing soft, triangle, and endpoint divisors;
• coefficient cover: product of the two endpoint Kummer covers;
• branch roots: Laurent functions of $uv$ and $u/v$;
• deck action: existing $C_2\times C_2$, permuting the two roots;
• new Carrier datum: none.

This is another positive H2 update: the shared support calculus remains unchanged while the cosmological coefficient object acquires a sector-specific compiled quartic.

Scope

This result classifies the algebraic branch cover only. It does not compute the marked relative cohomology, its Gauss--Manin connection, integral lattice, extension class, or physical period pairing.

Durable verification

• research/benincasa/check_soft_triangle_endpoint_kummer_compilation.py;
• research/benincasa/soft-triangle-endpoint-kummer-compilation.json;
• exact Laurent-polynomial factorization and deck-action assertions;
• epistemic event ev-000000003240-3de147fd-957e-4d4a-946e-1baf2f8e5ea4.

Next falsifier

Pull the full labelled logarithmic relative complex to the $(u,v)$-cover. Determine whether its Gauss--Manin system decomposes into the four existing endpoint characters or contains a nonsplit extension whose support is not generated by the endpoint and marked-incidence divisors. Only the latter would threaten H2 at coefficient level.

View source on GitHub