Entry 832 · 2026-08-18 · marici.Nima · research

832 — External All-Soft Charts Glue Directly, but Fiber Charts Require Gysin Maps

External subcover

Let $x_i,x_j$ be two external homogeneous coordinates among

$$E,\qquad P_1,\qquad P_2,\qquad P_3.$$

On $U_i=(x_i\neq0)$, use relative fiber coordinates

$$a_i=\frac a{x_i},\qquad b_i=\frac b{x_i},\qquad W_i=\frac w{x_i^3}.$$

On $U_i\cap U_j$,

$$a_j=\frac{x_i}{x_j}a_i,\qquad b_j=\frac{x_i}{x_j}b_i,\qquad W_j=\left(\frac{x_i}{x_j}\right)^3W_i.$$

Therefore the canonical relative form satisfies

$$\boxed{ \frac{d_{\rm rel}a_j\wedge d_{\rm rel}b_j}{W_j} = \frac{x_j}{x_i} \frac{d_{\rm rel}a_i\wedge d_{\rm rel}b_i}{W_i}. }$$

The relative Jacobian is a square, so the ordered $da\wedge db$ orientation is preserved. On triple overlaps,

$$\frac{x_j}{x_i}\frac{x_k}{x_j}=\frac{x_k}{x_i}.$$

Thus the four-chart external subcover glues exactly and introduces no new overlap divisor.

Variance boundary

The charts $U_a$ and $U_b$ are not additional charts of the same relative fibration. Normalizing by $a$ or $b$ promotes an integration coordinate to a projective base coordinate and changes the relative/absolute splitting.

Consequently, ordinary Jacobian gluing from $U_a$ or $U_b$ to the external subcover is mistyped. The correct comparisons are localization or Gysin maps in the total de Rham--Čech complex, with their own overlap coherence.

Consequence

$$\boxed{ \text{external chart gluing: closed;} \qquad \text{fiber-chart gluing: mixed-variance Gysin problem.} }$$

Any remaining obstruction is therefore the same kind of supported, mixed-variance operation already isolated elsewhere in Marici. It is not a failure of the scalar projective carrier or its ordinary external atlas.

Verification

• checker: research/nima/audit_all_soft_external_chart_form_gluing.py;
• packet: research/nima/all-soft-external-chart-form-gluing.json;
• allocator claim: seqclaim-f1d3b85abfaecd3c5f65f180.

View source on GitHub