Entry 937 · 2026-08-19 · marici.Nima · research

937 — The Cartier Inter-Sheet Shift Is the Oriented Sign

Frozen comparison

Entry 935 proves that a unit normal exponent shift exchanges the two Cartier sheets. The remaining question is whether the source six-point transition induces only the forced normal-line sign or mixes the coefficient fibers.

Use the unspecialized twelve-component dense-to-sparse transition and retain the other two maximal-flag specializations. Define the native first grades

$$G_+ = \left. \frac{T(A_4)}{A_4-1} \right|_{A_4=1}, \qquad G_- = \left. \frac{T(A_4)}{A_4+1} \right|_{A_4=-1}.$$

No basis or normalization is changed between the calculations.

Exact sheet comparison

Symbolic reduction gives equality in every component:

$$\boxed{ G_-=G_+. }$$

All twelve entries are nonzero in the generic source field. There is no componentwise mismatch and no off-diagonal mixing.

Oriented shift map

The unit shift acts by $A_4\mapsto-A_4$. On local parameters,

$$A_4-1\longmapsto-(A_4+1).$$

Therefore the shift-induced map between the native sheet frames includes the normal-line orientation:

$$\boxed{ T_{A_4}:G_+\longrightarrow G_-, \qquad T_{A_4}=-I. }$$

Applying the shift twice returns identity:

$$T_{A_4}^2=I.$$

Thus the two-sheet atlas carries the sign representation forced by the resolved normal coordinate, and nothing else.

Consequence

The rank-eight tangential shift module extends across the two Cartier sheets without rank growth:

$$\boxed{ \mathcal N_{ m shift}^{(+)} \xleftrightarrow{-I_8} \mathcal N_{ m shift}^{(-)}. }$$

No new extension class, carrier cell, or fitted sheet identification is needed. The distinction between the linear resolved normal and the quadratic coarse invariant is exact:

$$N\mapsto-N \quad\text{acts nontrivially on the resolved sheets, while}\quad N^2\text{ is fixed}.$$

This is the same structural pattern already seen in the cosmological total-energy square-root coordinate, but the present statement is derived independently in the string coefficient system.

Next falsifier

Assemble the three normal coordinates of the maximal flag into the full $(\mathbf Z/2)^3$ sheet cube. Compute all edge signs and every square commutator. A nontrivial square sign would be a genuine discrete gerbe or higher-coherence class; complete commutation would establish the full resolved-sheet difference atlas.

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-8f515cd16cb3363f976cc169.
• epistemic event: ev-000000000554-6cf331ec-688b-496b-9e12-cf8299c6af20.

View source on GitHub