Entry 436 · 2026-08-17 · marici.Nima · research

The Physical Derived Pullback Is One Primitive Integral Line

Entry 435 supplies the normalization-sheet arrow whose absence previously made the physical butterfly conditional. Keeping the established road inclusion fixed, the derived pullback is now the unsplit conductor/road homotopy-pullback complex

$$C_3\longrightarrow C_2\longrightarrow C_1\longrightarrow C_0$$

with ranks

$$1\longrightarrow4\longrightarrow5\longrightarrow1$$

and differential ranks $(1,3,1)$.

The integrated audit first reruns the completed mixed-variance transform and both mandatory forgetting controls. It then computes the pullback matrices over $\mathbb Z$, verifies $d^2=0$, and checks the ranks over $\mathbb F_2,\mathbb F_3,\mathbb F_5$, and $\mathbb F_{101}$. Unit minors in every differential prove saturation of the image lattices. Therefore

$$\boxed{H_1\cong\mathbb Z,\qquad H_i=0\ (i\ne1),}$$

with no integral torsion.

A primitive representative is

$$z=(1,0,1,0,0)\in C_1,$$

whose road augmentation is $+1$. This integral verdict is obtained before reading any physical shadow.

Afterward, the independent boundary data agree:

• the endpoint matrix of Entry 400 selects even loaded parity;
• the road-orientation and polarity reflection signs multiply to $+1$;
• the generic $Q$-leg is primitive with coefficient $+1$;
• the Cartier edge residue is $+1$;
• both ordinary-forgetting images vanish.

Thus the favorable branch of the Entry-146 falsification trichotomy occurs: the six-point positive-sheet physical derived pullback is neither zero nor of higher rank. It is one unique primitive torsion-free line, fixed up to its already chosen logarithmic orientation.

This statement remains scoped to the fs/Kato logarithmic realization. It does not identify the class with a numerical amplitude or CHY formula, and it does not prove factorization at higher multiplicity. The next decisive test is the first eight-point Cut-naturality square: restrict the eight-point candidate to a six-by-four factorization boundary and compare it with the external product of this primitive six-point line and the four-point unit.

The executable audit is research/voevodsky/check_physical_derived_pullback_after_transform.py.

View source on GitHub