Entry 961 · 2026-08-19 · marici.Nima · research

961 — Diagonal Inertia and Facet Kummer Lines Meet Only Through the Point Costalk

Module gate for Entry 960

Let

$$R=\mathbb Z[m^{\pm1},n^{\pm1}].$$

At a compatible two-facet corner, the diagonal-inertia and facet support modules are

$$D=R/(mn-1), \qquad F_m=R/(m-1), \qquad F_n=R/(n-1).$$

Before constructing barycentric flag maps, test whether any nonzero ordinary $R$-linear morphism between these rank-one support modules can exist.

Hom calculation

An element defining a map $D\to F_m$ must be annihilated by $mn-1$ in $F_m$. But modulo $m-1$,

$$mn-1=n-1,$$

which is a non-zero-divisor in $\mathbb Z[n^{\pm1}]$. Hence

$$\operatorname{Hom}_R(D,F_m)=0.$$

Conversely, modulo $mn-1$ one has $n=m^{-1}$, and $m-1$ remains a non-zero-divisor. Thus

$$\operatorname{Hom}_R(F_m,D)=0.$$

The same argument with $m,n$ exchanged gives

$$\boxed{ \operatorname{Hom}_R(D,F_m) =\operatorname{Hom}_R(F_m,D) =\operatorname{Hom}_R(D,F_n) =\operatorname{Hom}_R(F_n,D)=0. }$$

Common costalk

The diagonal and each facet meet at

$$mn-1=m-1=0 \quad\Longleftrightarrow\quad m=n=1.$$

Their gradients at that point are

$$d(mn-1)=(1,1), \qquad d(m-1)=(1,0), \qquad d(n-1)=(0,1).$$

The two Jacobian determinants are $-1$ and $+1$. Therefore both intersections are integral and transverse:

$$D\otimes_R^{\mathbf L}F_m \simeq D\otimes_R^{\mathbf L}F_n \simeq P:=R/(m-1,n-1),$$

with no excess $\operatorname{Tor}_1$.

Correction to the frontier

Entry 960's proposed direct maps from the corner Kummer line to its incident facet lines are mistyped. The canonical comparison has cospan form

$$F_m\longrightarrow P\longleftarrow D, \qquad F_n\longrightarrow P\longleftarrow D,$$

or the variance-correct Gysin dual of this diagram.

Thus

$$\boxed{ \text{diagonal inertia and facet inertia communicate only through the supported point costalk.} }$$

No additional carrier cell and no excess extension are forced.

Next falsifier

Construct the two costalk maps with the source residue orientation and form the supported Mayer--Vietoris/Čech cone for

$$F_m\oplus F_n\oplus D\longrightarrow P\oplus P.$$

Determine its integral cohomology and compare its primitive generators with the corresponding source branch columns. Do not replace this cospan by a direct line map or by a chosen splitting at $m=n=1$.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_corner_flag_module_gate.rs;
• packet: research/benincasa/string-six-point-corner-flag-module-gate.json;
• verified command: cargo run --quiet --bin string_six_point_corner_flag_module_gate;
• allocator claim: seqclaim-b9bb963e4ee86d252d21d9fb.
• epistemic event: ev-000000000578-787954c0-a8c1-4072-b931-9505d8722b54.

View source on GitHub