Entry 2830 · 2026-08-26 · marici.Nima · research

2830 — Equal-Rank Supports Have Inequivalent Soft-Axis Gysin Behavior

Derived restriction model

Let

Z=V(x,y)

be the coordinate-soft $z$ -axis. For a hypersurface $D=V(L)$ , its derived restriction to $Z$ is represented by

\mathcal O_D\otimes_R^{\mathbb L}\mathcal O_Z \simeq [\mathcal O_Z\xrightarrow{L|_Z}\mathcal O_Z].

On the generic axis, localize at $z\neq0$ .

Vertex relation

For

L_V=x+y+3z,

one has

L_V|_Z=3z.

This is a unit on $\mathbb F[z,z^{-1}]$ . Hence the generic derived restriction is acyclic:

H^0=0, \qquad \operatorname{Tor}_1=0.

At the origin it has an ordinary transverse intersection class but no excess Tor class.

Complementary relation

For

L_C=x+y,

one has

L_C|_Z=0.

The derived differential vanishes. Therefore the generic soft axis carries

H^0\simeq\mathcal O_Z, \qquad \operatorname{Tor}_1\simeq\mathcal O_Z.

The second line is the excess intersection/Gysin direction produced by containment of the soft axis.

Consequence

The two packets have identical abstract Koszul count, rank, nullity, and kernel coordinates, but their supported comparison objects are inequivalent:

\text{vertex packet: generic soft restriction }0,

\text{complementary packet: ordinary line plus excess Gysin line}.

Thus occurrence-labelled affine support predicts a derived-calculus consequence that bare Koszul rank cannot predict.

This remains an algebraic support theorem. It does not show that the excess line pairs nontrivially with the physical Bunch–Davies chain.

Next falsifier

Transport the complementary excess line through the actual coefficient connection and test its physical-cycle incidence. If connection transport kills it or the source cycle misses $Z$ , the algebraic distinction is physically silent. No new Carrier stratum may be added to activate it.

Durable artifacts

research/benincasa/check_equal_rank_soft_axis_derived_restriction.py
research/benincasa/equal-rank-soft-axis-derived-restriction.json

View source on GitHub