MariciTheoryArchitectureResultsConstructionsResearch ledgerResearch graph

Entry 1889 · 2026-08-22 · marici.Nima · research

1889 — Cyclic Descent Requires Three Cartier Components

Question

Entry 1888 describes one routing-chart Cartier object with basis ⟨1,L⟩\langle1,L\rangle. Determine its action under the C3C_3 stabilizer of

g12∣g34∣g56.g_{12}\mid g_{34}\mid g_{56}.

Chart-orbit correction

Rotation by two sites cyclically permutes

x=y12,v=y32,w=y52.x=y_1^2,\qquad v=y_3^2,\qquad w=y_5^2.

It does not preserve the single ideal (L2,M)(L^2,M). Instead it produces three pairwise-disjoint Cartier components

(Li2,Mi),i=0,1,2.(L_i^2,M_i),\qquad i=0,1,2.

Exact affine-rank tests show that every pair of reduced lines is disjoint. Thus a single routing-chart line has no intrinsic C3C_3 character and must not be treated as a descended object.

Descended coefficient object

The stabilizer acts freely on the three components. Since each has Cartier length two, the occurrence-resolved exceptional object has total length six and character

χ=(6,0,0).\boxed{\chi=(6,0,0)}.

Over Q\mathbb Q,

E6,orb≅2(Qtriv⊕Q(ζ3))\mathcal E_{6,\mathrm{orb}} \cong 2\bigl(\mathbb Q_{\mathrm{triv}}\oplus\mathbb Q(\zeta_3)\bigr)

at the finite Cartier layer. The nilpotent generators LiL_i are transported with their components; none is selected as a global eigenline.

Nine source maps

The nine source completions decompose into exactly three free C3C_3 orbits of size three. Transporting labels, completion walls, and Cartier targets together gives strict descent on the three-component object.

Therefore

cyclic descent holds occurrence-resolved, but fails for one chosen routing-chart component.\boxed{ \text{cyclic descent holds occurrence-resolved, but fails for one chosen routing-chart component.} }

This refines Entry 1888: its length-two object is one chartwise component, not the complete descended coefficient object.

Consequence

No new Carrier structure is required. The Carrier already retains the three labelled chart occurrences needed for descent. Forgetting those labels would manufacture a noncanonical character assignment.

Next falsifier

Construct the generic full-rank six-site kinematic family in a labelled routing atlas and require its cover and critical ideal to intertwine these three chart transitions before elimination.

Durable verification

View source on GitHub