Entry 943 · 2026-08-19 · marici.Nima · research

943 — The Six-Point Source Branch Lattice Has Existing-Wall and Two-Primary Defects

Frozen source matrix

The pair-shift action splits the rational source module into labelled character blocks of dimensions

$$(2,1,1,2).$$

Using the (X=+1) and (X=-1) source rows in the two-dimensional blocks and the (X=+1) row in the singleton blocks gives a source-derived maximal minor. No kinematic factor is inverted when forming it.

Factorization

Up to the declared Laurent monomial units, the maximal minor is proportional to

$$\begin{aligned} &(A_2^2-1)^2(A_3^2-1)^2 ((A_2B_{24})^2-1)((A_3B_{34})^2-1)\\ &\quad\times((ZA_2)^2-1) ((ZA_2B_{24})^2-1)^2\\ &\quad\times(A_3^2-Z^2) ((A_3B_{34})^2-Z^2)^2, \end{aligned}$$

with the exact rational expression retained in the machine packet.

This is not a unit. Hence the six rational branch directions do not define a globally saturated source lattice over the frozen Laurent ring.

Classification

Every displayed kinematic factor is an existing sine/incidence hyperplane. The defect therefore has the form

$$\boxed{ \text{generic source saturation} + \text{supported lattice defect on existing carrier walls}. }$$

It does not require a new carrier divisor.

There is a second, logically separate integral issue. Extracting the four pair-shift characters from their orbit uses the order-four Fourier projector, so its denominator is (4). Over characteristic zero this is harmless; over $\mathbb Z$ the possible two-primary index remains unresolved. The wall defect and the two-primary defect must not be conflated.

Consequence

Entry 941's target lattice remains unimodular. The full rank-twelve object is therefore generically lattice-valued after localization away from existing walls, but no global integral Betti lattice has yet been derived.

Next falsifier

Compute the orbit-basis Smith normal form without character projectors. This will determine the exact two-primary index. Separately, take valuations of the maximal minor along each existing wall and test whether the frozen Cartier/Gysin lattice supplies the corresponding saturation automatically.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_cartier_sheet_transition.rs;
• packet: research/benincasa/string-six-point-cartier-sheet-transition.json;
• allocator claim: seqclaim-07e84d29020d16c5910497c2.
• epistemic event: ev-000000000560-34a69185-200f-4368-98cc-b3a09c86dc72.

View source on GitHub