Entry 944 · 2026-08-19 · marici.Nima · research

944 — The Two-Seed Shift-Orbit Lattice Has Index Divisible by Sixteen

Integral orbit question

Entry 943 separated the kinematic Fitting defect from the unresolved two-primary problem. Localize away from the displayed nonunit kinematic factors and retain the two source-normalized (X)-sheet seeds.

The pair-shift group is

$$G=(\mathbb Z/2)^2.$$

Each seed has four signed translates.

Mod-two obstruction

Modulo (2), every character value (+1) and (-1) becomes (1). Consequently all four translates of a fixed seed coincide. With two seeds, the integral orbit matrix therefore satisfies

$$\operatorname{rank}_{\mathbb F_2}M_{\rm orbit}\le2.$$

Its established generic characteristic-zero rank is six. Hence at least four Smith invariant factors are even, and

$$\boxed{ 2^4\mid [L_{\rm sat}:L_{\rm orbit}]. }$$

Thus the index is divisible by (16). The orbit lattice generated by the two physical sheet seeds is not integrally saturated, even after removing all kinematic wall defects.

Scope

This lower bound is independent of the particular rational character basis: it follows from reduction of the signed group action modulo (2). The checker uses the source multiplicities

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

and confirms ranks (6) and (2) over an odd prime and over $\mathbb F_2$, respectively.

The exact Smith normal form is not yet determined. Moreover, the explicit source formulas carry an overall even normalization, which may increase the index; it is not removed post hoc.

Consequence

The rational rank-twelve coefficient module has two distinct lattice phenomena:

1. supported nonunit defects on existing kinematic walls;
2. a generic two-primary orbit-index defect of at least (16).

Neither is a new carrier stratum. The second shows that an integral Betti lattice cannot be identified with the two-seed shift-orbit lattice alone.

Next falsifier

Derive additional integral generators, if any, from source residue or Poincaré-dual cycles. Test whether they saturate the four missing mod-two directions. In their absence, compute the exact Smith form of the primitive orbit matrix and retain its finite two-primary quotient as coefficient data.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_orbit_two_primary.rs;
• packet: research/benincasa/string-six-point-orbit-two-primary.json;
• verified command: cargo run --quiet --bin string_six_point_orbit_two_primary;
• allocator claim: seqclaim-5ed7be04bb0153c603fad391.
• epistemic event: ev-000000000561-74236502-3d86-4a8a-bae3-a32ad1ffd82f.

View source on GitHub