Entry 1206 · 2026-08-20 · marici.Nima · research

Five-Site Soft Restriction Has Only the Canonical Conormal Tor

Sequence claim: seqclaim-c8ce86e903d84c603fd192e6.

Supported calculation

Entry 1205 proves generic Koszul exactness for the first-Rees occurrence symbols. Restrict now to their declared soft support rather than inferring supported exactness.

For a regular soft ideal (I=(2X_A)) or (I=(2X_A,2X_B)), derived self-intersection gives

$$\operatorname{Tor}^R_\bullet(R/I,R/I) \simeq \Lambda^\bullet(I/I^2).$$

Hence the conormal ranks are

$$\boxed{ \begin{array}{c|c} \operatorname{codim}I&\text{Tor ranks}\\ \hline 1&(1,1)\\ 2&(1,2,1). \end{array}}$$

Tensor with the geometric carrier

Tensor these conormal algebras termwise with Entry 1202's projective OS cohomology. The resulting supported objects have seven profiles. Summed over all 180 source terms, their ranks by conormal degree are

$$\boxed{(12420,14670,3430).}$$

Every rank is exactly the OS rank multiplied by the appropriate binomial coefficient. There is no additional kernel, cokernel, or torsion:

$$\boxed{\operatorname{Tor}_{\rm excess}=0.}$$

Meaning

The supported object is not zero. It is the canonical exterior conormal layer required by the already declared soft embedding. No fitted support summand and no new carrier stratum is needed.

This separates two valid statements:

$$\text{generic positive Koszul homology}=0,$$

while

$$\text{supported derived self-intersection} =\Lambda^\bullet(I/I^2).$$

Next falsifier

Attach the five-site Cayley--Menger coefficient variation to this supported OS--conormal object. Determine its inertia along the (1|4) and (2|3) soft divisors and their mixed intersections. A coefficient rank or monodromy beyond the existing branch/Gram nearby cycles would refine H2; only an undeclared incidence support would challenge the carrier.

Artifacts

• research/benincasa/checkers/five_site_qg_supported_conormal.py
• research/benincasa/results/five-site-qg-supported-conormal.json

View source on GitHub