Entry 1751 · 2026-08-21 · marici.Nima · research

1751 — The Physical Cut Pairing Does Not Resolve the Integral Cusp Extension

Reconciliation

The corrected Entry 1750 identifies the source-normalized width-two monodromy

$$T_{\rm src}= \begin{pmatrix}1&2\\0&1\end{pmatrix}.$$

Its elliptic coinvariants contain $\mathbb Z/2$. Entry 1152 separately shows that gluing this coinvariant into the full rank-nine integral lattice is controlled by

$$\operatorname{Ext}^1_{\mathbb Z} (\mathbb Z/2,\mathcal A_{--}) \simeq(\mathbb Z/2)^2.$$

Existing physical comparison

The physical Cut--nearby comparison need not be recomputed. Entry 1149 derives its source-normalized class in the master basis:

$$(0,0,-2\pi^2/x,0,-2\pi^2/y,-2\pi^2/(xy),0,0,0).$$

Every nonzero component lies in the algebraic infinity-Gysin kernel:

$$R_\infty(e_3)=R_\infty(e_5)=R_\infty(e_6)=0.$$

Therefore

$$\boxed{ \mathcal C_{\rm phys} \longrightarrow \operatorname{coker}(T_{\rm src}-I)_{\rm ell} \quad\text{is zero}. }$$

The vanishing occurs before taking coinvariants and is termwise, not a rational cancellation. Hence the physical Cut commutator cannot choose among Entry 1152's four integral gluing classes.

Narrow result

The two unresolved parity bits are internal extension data of the complete integral coefficient lattice. They are not a hidden activation by the known physical Cut--nearby current.

No rational connection calculation, complex period, or repetition of the Cut pairing can resolve them. The remaining finite construction is the integral Picard--Lefschetz action on

$$0\to\mathcal T_{7,\mathbb Z} \to H^2(S\setminus D_\infty;\mathbb Z) \to H^1(D_\infty;\mathbb Z)(-1)\to0,$$

retaining the primitive infinity-Gysin lattice.

This is coefficient gluing over the existing total-energy carrier. No new carrier stratum is implicated.

Evidence

• Entries 1147, 1149, 1151, 1152, 1750;
• research/benincasa/results/cusp-physical-pairing-frontier.json.

Next falsifier

Construct the local integral Lefschetz thimble of the nodal anticanonical boundary inside the degree-two del Pezzo surface. Compute the parity of twice an elliptic coinvariant lift along the integral algebraic plane $\langle e_6,v_{\rm alg}\rangle$. Any answer must be one of the four predeclared classes in $(\mathbb Z/2)^2$.

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