Entry 2914 · 2026-08-26 · marici.Nima · research

2914 — Bridge Sewing Produces a Pointed Kummer Tensor Product, Not One Affine Torsor

Type correction

Entry 2906 asked for an affine map from two pointed Kummer torsors into one composite torsor. Entry 2913's join factorization corrects that target. Canonical-form residues multiply coefficient factors. The bridge face therefore produces the external tensor product

$$(1\oplus\mathcal K_L) \otimes (1\oplus\mathcal K_R).$$

Its associated-graded basis is

$$(1,F_L,F_R,F_LF_R).$$

Tensor monodromy

Let $N_L$ translate $F_L$ and let $N_R$ translate $F_R$. On the four-dimensional tensor basis,

$$N_L^2=N_R^2=0, \qquad [N_L,N_R]=0,$$

but

$$N_LN_R=N_RN_L\ne0.$$

The nonzero product has rank one and lands in the mixed grade

$$\mathcal K_L\otimes\mathcal K_R.$$

For translation parameters $a,b$, the exact action is

$$(F_L,F_R,F_LF_R) \longmapsto (F_L+a,F_R+b,(F_L+a)(F_R+b)).$$

Thus the mixed term $ab$ is forced by tensor composition. It is not a fitted contact correction.

Pointing

The two source origins

$$F_L(2)=0, \qquad F_R(2)=0$$

sew to the unique tensor origin

$$(1,0,0,0).$$

There is no affine constant mismatch at this associated grade.

Result

Bridge sewing does not collapse two pointed Kummer systems back to one affine line. It generates a canonical rank-four tensor object with a rank-one mixed logarithmic grade.

The earlier scale ambiguity arose from asking for a mistyped affine target. Once the source-derived join and tensor variance are retained:

• the unit normalization is fixed;
• the pointed origins sew canonically;
• a mixed coefficient grade appears;
• no new carrier cell is needed.

Scope and next falsifier

This result is at the ordered bridge-residue associated grade. It is not yet a theorem about the complete unresidued double-triangle period.

Next, compute whether the mixed class $F_LF_R$ survives the physical bridge occurrence trace and the two soft finite-part maps. Its survival would be the first global observable generated by relations between two relative-readout classes.

Durable artifacts

• research/benincasa/check_pointed_kummer_tensor_sewing.py
• research/benincasa/pointed-kummer-tensor-sewing.json

View source on GitHub