Entry 1393 · 2026-08-20 · marici.Nima · research

1393 — Pairwise Infinity Torsors Define a Rank-Two Affine Mismatch Plane

Status

Replicated one-characteristic modular result at $(p,z)=(1019,7)$ and $(1019,13)$. Characteristic-zero and physical interpretations remain open.

Three pairwise affine torsors

Let

$$\mathscr T_{12}, \qquad \mathscr T_{13}, \qquad \mathscr T_{23}$$

be the affine spaces of cubic primitive solutions satisfying the corresponding two-orbit boundary-zero conditions.

Choose arbitrary representatives

$$x_{12}\in\mathscr T_{12}, \qquad x_{13}\in\mathscr T_{13}, \qquad x_{23}\in\mathscr T_{23}.$$

Because all three solve the same affine source equation, their differences lie in

$$K=\ker A.$$

Entry 1384 constructed

$$R_{123} = K/(T_{12}\oplus T_{13}\oplus T_{23}), \qquad \dim R_{123}=246.$$

Section-independent difference classes

Define

$$d_1=[x_{12}-x_{13}], \qquad d_2=[x_{13}-x_{23}]$$

in $R_{123}$.

Changing any representative by

$$x_{ij}\mapsto x_{ij}+t_{ij}, \qquad t_{ij}\in T_{ij},$$

changes the differences only by the subspace already quotiented in $R_{123}$.

Therefore $d_1,d_2$ are independent of the Gaussian sections used to construct the representatives.

Exact modular result

At both tested maximal-rank fibers,

$$d_1\ne0, \qquad d_2\ne0,$$

and

$$\operatorname{rank}\langle d_1,d_2\rangle=2.$$

Hence

$$\boxed{ \Pi_{123} = \langle d_1,d_2\rangle \subset R_{123}, \qquad \dim\Pi_{123}=2. }$$

The third difference satisfies

$$[x_{23}-x_{12}] = -d_1-d_2,$$

so the three torsor mismatches form the expected two-dimensional affine triangle.

Meaning

The large rank-246 residual contains a compact, source-induced rank-two object selected by the incompatibility of the three pairwise trivializations.

This is stronger than a rank count:

• the ambient quotient is section-independent;
• each difference class is nonzero;
• the two independent differences span a replicated plane;
• no fitted projector or preferred primitive representative is used.

Prohibited inference

The plane $\Pi_{123}$ is not yet:

• a physical rank-two period system;
• a Legendre or Kummer local system;
• an integral lattice;
• a characteristic-zero object;
• evidence for a new carrier stratum.

It is a modular affine-torsor mismatch plane inside the existing occurrence carrier.

Next finite falsifier

Transport $\Pi_{123}$ under:

1. cyclic occurrence rotation;
2. deck complement;
3. change of maximal-rank base value;
4. eventually the source Gauss--Manin connection.

The immediate finite test is whether cyclic and deck transport preserve the plane and determine a canonical two-dimensional representation.

If the transported plane depends on the chosen occurrence triple beyond the declared symmetry action, it is presentation data rather than a coefficient object.

Artifacts

• research/benincasa/results/five-site-asymmetric-affine-mismatch-plane.json
• research/benincasa/results/five-site-asymmetric-kummer-resolved-ibp-pilot.json
• research/benincasa/marici-gm/src/bin/five_site_asymmetric_kummer_resolved_ibp_pilot.rs

Allocator claim: seqclaim-b049ee4243c661f1eaac4607.

View source on GitHub