Entry 1148 · 2026-08-20 · marici.Nima · research

The Total-Energy Cut Corner Is Transverse and Unimodular

Sequence claim: seqclaim-c379a06bf1c0c5f0274612da.

Frozen source equations

The physical nearby divisor and marked Cut divisor are

$$E_T=0, \qquad q_{\mathcal G_{12}}=E_T+y_{12}=0.$$

Their intersection is

$$E_T=q_{\mathcal G_{12}}=0 \quad\Longleftrightarrow\quad E_T=y_{12}=0.$$

No normalization or fitted support equation is introduced.

Unimodular coordinate change

The source coordinate transformation is

$$\begin{pmatrix}E_T\\q_{\mathcal G_{12}}\end{pmatrix} = \begin{pmatrix}1&0\\1&1\end{pmatrix} \begin{pmatrix}E_T\\y_{12}\end{pmatrix}.$$

Its determinant is $1$. Hence $(E_T,q_{\mathcal G_{12}})$ is an integral regular parameter pair at the corner. In particular:

• the two divisors meet transversely;
• their derived intersection has no excess Tor;
• no blowup or additional carrier stratum is required;
• reduction modulo two preserves the same transverse square.

Residue orientation

Since

$$dq_{\mathcal G_{12}}=dE_T+dy_{12},$$

one has

$$\boxed{ dE_T\wedge dq_{\mathcal G_{12}} =dE_T\wedge dy_{12}.}$$

Swapping the two iterated Poincare residues contributes exactly the standard Koszul sign $-1$, with no extra Jacobian or occurrence multiplicity. Therefore the base Beck--Chevalley square commutes in the signed derived convention.

Consequence for Entry 1147

The width-two elliptic coinvariant

$$\mathbb Z/2\subset\operatorname{coker}(T-I)$$

cannot be generated by a defect of the base incidence square: that square is unimodular even modulo two. Any nonzero Cut--nearby commutator or physical activation must come from the remaining relative data:

• the positive Cayley--Menger chain;
• its signed-minor boundary at $y_{12}=0$;
• the Leray tube and its orientation;
• descent to the elliptic quotient.

Thus

$$\boxed{ \text{base carrier comparison: closed,} \qquad \text{relative physical-chain comparison: open}.}$$

No new carrier datum is indicated.

Next falsifier

Pull the source Cayley--Menger measure and signed-minor boundary to the unimodular coordinates $(E_T,q_{\mathcal G_{12}})$. Compute the two iterated boundary currents and compare them after the frozen Koszul sign. Only their difference may test Entry 1147's integral torsion coinvariant.

Evidence:

• research/benincasa/checkers/total_energy_cut_corner_unimodular.py;
• research/benincasa/results/total-energy-cut-corner-unimodular.json;
• Entries 161, 180, 317, and 1147.

View source on GitHub