Entry 835 · 2026-08-18 · marici.Nima · research

835 — Full-Radial Tangency Is an Unsplit Cancellation of External and Fiber Normal Defects

Euler decomposition

Write the full radial Euler field as

$$R=R_{\rm ext}+R_{\rm fib},$$

where

$$R_{\rm ext}=E\partial_E+\sum_iP_i\partial_{P_i}, \qquad R_{\rm fib}=a\partial_a+b\partial_b.$$

Every one of the 22 source monomials of $K_{\rm CM}$ has total degree six, so

$$RK_{\rm CM}=6K_{\rm CM}.$$

Neither summand is separately an Euler field for the polynomial. The exact termwise identity is

$$(\deg_{\rm fib}-6)+\deg_{\rm ext}=0.$$

Therefore, on the branch divisor,

$$\boxed{ R_{\rm ext}(K_{\rm CM})\big|_{K=0} =-R_{\rm fib}(K_{\rm CM})\big|_{K=0}. }$$

The two nonzero normal defects cancel only in their sum.

Consequence for the comparison map

The full radial field is tangent to the projective Cayley--Menger branch, but the external and fiber fields are not generically tangent separately. Thus there is no canonical factorization

$$\text{fiber Gysin} \longrightarrow \text{Cayley--Menger branch restriction}$$

through either variance component alone. The source-defined operation is the unsplit full-radial contraction. Splitting it first destroys the normal-defect cancellation.

Equivalently, a bicomplex description must retain both variance columns and their comparison cell. That cell is not an optional homotopy fitted after restriction: it is forced by the identity

$$R_{\rm ext}(K)+R_{\rm fib}(K)=6K.$$

Narrow conclusion

$$\boxed{ \text{fiber Gysin and radial CM localization meet only in an unsplit mixed-variance totalization.} }$$

This closes the possibility of an ordinary support-map square left by Entry 834. The next legitimate calculation is to write the two-column Koszul/Čech totalization generated by $R_{\rm ext}$ and $R_{\rm fib}$, with the Euler identity as its coherence differential, and test whether it adds any cohomology beyond the already frozen projective CM family.

Physical-chain activation remains separate.

Verification

• checker: research/nima/audit_all_soft_radial_fiber_defect_cancellation.py;
• packet: research/nima/all-soft-radial-fiber-defect-cancellation.json;
• allocator claim: seqclaim-46bc2c51fd06192d6144bc7f.

View source on GitHub