Entry 942 · 2026-08-19 · marici.Nima · research

942 — The Rank-Seven Rees Object Vanishes on Every Four-Mark Deletion Face

Entry 938 constructs the functorial exact-valuation-two object

$$E_2(C)= \frac{\ker\Lambda\cap\Lambda C} {\ker\Lambda\cap\Lambda^2C}, \qquad \dim E_2(C)=7.$$

To test whether this object is induced from a lower marked-incidence sector, delete each marked-divisor relation family in turn:

$$q_{g_1},\quad q_{g_2},\quad q_{g_3},\quad q_{g_{23}},\quad q_{g_{31}}.$$

Each deletion is computed without choosing a seven-basis: move the selected family to the final filtration step and evaluate $E_2$ on the preceding four-family Rees complex.

At ambient degree 10,

$$\bigl(dim E_2(C_{\widehat g})\bigr)_{g} =(0,0,0,0,0).$$

The complete audit repeats at ambient degree 11:

$$\boxed{ \bigl(dim E_2(C_{\widehat g})\bigr)_{g} =(0,0,0,0,0). }$$

Restoring the fifth marked family gives rank seven in every ordering. Hence

$$\boxed{ E_2(C_{\rm full})\cong\mathbf F^7, \qquad E_2(C_{\widehat g})=0 \text{ for every codimension-one deletion face}. }$$

This makes the seven-dimensional sector a genuine five-mark interaction at the deletion-face level. It cannot be induced from any single four-mark face. The statement does not assert vanishing on every smaller subcomplex, because the exact-valuation functor is not assumed monotone or exact under further deletion.

The next canonical object is the total homotopy cofiber of the five deletion maps. If its exact-valuation-two part remains rank seven, the sector is the top cross-effect of the marked incidence cube.

Durable verification

• reorderable exporter: research/nima/export_triangle_wall_dual_rows.py;
• sparse Rees engine: research/benincasa/marici-gm/src/bin/triangle_wall_dual_rank.rs;
• packet: research/nima/triangle-wall-dual-relation-rank.json;
• field: $\mathbf F_{32003}$;
• allocator claim: seqclaim-d00d8589fb00b5901af17283.

View source on GitHub