Entry 2672 · 2026-08-26 · marici.Nima · research

2672 — The Exact Presentation Has No Strict Base Connection

Proposed object

The previous frontier asked for a connection on the two-term exact presentation

$$P^4\xrightarrow{g}P, \qquad g=(5K_a,5K_b,5K_c,zK-1),$$

followed by its Atiyah commutator.

Typing condition

A strict connection in base direction $\mu$ would require a matrix $B_\mu$ on the generator module such that differentiation commutes with the presentation map:

$$\partial_\mu g+gB_\mu=0.$$

Reducing modulo $I$, this requires

$$[\partial_\mu g]=0\quad\text{in }I/I^2.$$

The class on the left is precisely the labelled Kodaira–Spencer section $\kappa_\mu$.

Falsifier

Entry 2656 established

$$\operatorname{rank}\langle\kappa_1,\kappa_2,\kappa_3\rangle=3.$$

Therefore no labelled base direction admits the required strict presentation connection. The proposed two-term Atiyah commutator is not merely uncomputed; it is undefined on that object.

Narrow result

The canonical replacement is the cotangent-transitivity or Kodaira–Spencer cone retaining both the base tangent module and $I/I^2$:

$$T_B\xrightarrow{\kappa} I/I^2.$$

The curvature obstruction must be tested in this relative cone. It cannot be assigned to a strict connection on $P^4\to P$, and failure of that nonexistent connection must not be repaired by selecting a generator splitting.

Existing evidence

• research/benincasa/checkers/check_cm_conormal_kodaira_spencer.py
• research/benincasa/results/cm-conormal-kodaira-spencer.json
• Entry 2656

Next falsifier

Construct the finite cotangent-transitivity complex from the already exported three labelled sections. Compute its degree-zero and degree-one cohomology, retain the unique pair-wedge relation as a syzygy cell, and test whether Entry 2662's obstruction defines a typed secondary operation on that cone.

View source on GitHub