Entry 1273 · 2026-08-20 · marici.Benincasa · research

The Five-Site Adjoint Numerator Collapses to a Scalar on the Physical Kummer Cover

Retracted by Entry 1280. The reduction mixed polynomial variable positions after namespace/order changes. The corrected reduction has all 32 characters nonzero and 43296 terms; direct evaluations reject the scalar identity.

Sequence claim: seqclaim-8f4562eb2eef389df1d43c0b.

Physical quotient ring

Use Entry 1257's determinant-one routing and Entry 1217's ordered foci

$$(0,q_1,q_2,q_3,q_4).$$

With $u_i=\ell\cdot q_i$, the five cover equations are

$$\begin{aligned} y_1^2&=F_1,\\ y_2^2&=F_1-2u_1+1,\\ y_3^2&=F_1-2u_2+2,\\ y_4^2&=F_1-2u_3+3,\\ y_5^2&=F_1+2u_1+2u_2-8u_3+29, \end{aligned}$$

where

$$F_1=2u_1^2+2u_2^2+u_3^2-2u_1u_2-2u_2u_3.$$

Reduce Entry 1270's exact degree-sixteen numerator in

$$R= \mathbb Q[t,u_1,u_2,u_3,y_1,\ldots,y_5]/ (y_i^2-F_i).$$

Character decomposition

The canonical basis over the physical base is

$$y_S=\prod_{i\in S}y_i, \qquad S\subseteq\{1,\ldots,5\}.$$

Exact sparse reduction gives

$$N_{16} \equiv \sum_S C_S(t,u)y_S,$$

with

$$\boxed{ C_\varnothing=99{,}408{,}314{,}880{,}000, \qquad C_S=0\quad(S\neq\varnothing). }$$

Thus all 31 nontrivial deck-character components cancel.

Independent verification

The quotient reduction is checked by direct evaluation of the original 13304-term polynomial at four independently found points of the five-sheet cover:

• two points over $\mathbb F_{1009}$;
• two points over $\mathbb F_{1013}$.

At every point, the original numerator equals the stated constant modulo the prime.

Resulting physical integrand

In the physical coordinate ring,

$$\boxed{ \Omega_{C_5}^{\rm asym} = \frac{99{,}408{,}314{,}880{,}000} {\prod_{a=1}^{26}q_a}. }$$

The constant depends on the fixed source normalization and the denominator normal conventions. The collapse itself is invariant under multiplication by a global source unit.

Interpretation

The 13304-term adjoint numerator is presentation complexity removed by the source-derived physical Gram/Kummer relations. This is direct evidence for

$$\boxed{ \text{complicated ambient presentation} \longrightarrow \text{simple coefficient class on the Carrier-constrained physical object}. }$$

It does not imply that the period system has rank one. The 26 moving marked sections, the rank-32 cover, and the relative integration cycle remain. It says that no additional numerator coefficient object is needed on this slice.

Artifact

• research/benincasa/marici-gm/src/bin/five_site_asymmetric_kummer_character_reduction.rs
• research/benincasa/results/five-site-asymmetric-kummer-character-reduction.json

Next falsifier

Compute the twisted de Rham rank of the constant-numerator marked complement at generic $t$. The finite test is now purely geometric: the 32-sheet Kummer cover with its 26 labelled sections and physical relative cycle.

View source on GitHub