Entry 1152 · 2026-08-20 · marici.Nima · research

The Integral Cusp Ambiguity Is a Two-Bit Extension Class

Sequence claim: seqclaim-e8ae119501faedd8b4b0ffc3.

Question

Entry 1151 proves that the generic integral infinity-Gysin sequence has a primitive free algebraic kernel and free elliptic quotient. At the total-energy cusp, does Entry 1147's elliptic (mathbb Z/2) coinvariant survive as a direct summand of the full rank-nine coinvariants, or is it absorbed by an integral extension with the algebraic lattice?

Rational monodromy input

Entry 289 computes the exact total-energy residue and proves:

$$T|_{\mathcal T_7}=1, \qquad N_E=N_{\rm ell}, \qquad \operatorname{rank}N_E=1, \qquad N_E^2=0.$$

Thus the variation splits into algebraic and elliptic monodromy sectors over (\mathbb Q). In the snake sequence for (T-I), the connecting map from elliptic invariants to the free algebraic coinvariants vanishes rationally. Because the target (mathcal T_7) is torsion-free by Entry 1151, it vanishes integrally as well.

This does not force the resulting coinvariant extension to split.

Character reduction

Entry 150 shows that the elliptic boundary cohomology occupies the character of the final four-master block. The other algebraic character blocks cannot mix with it. The only allowed algebraic target is therefore

$$\mathcal A_{--} =\mathbb Z\langle e_6,v_{\rm alg}\rangle.$$

Hence the complete unresolved integral datum lies in

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

It is two parity bits, not an unbounded off-diagonal matrix.

Four possible lattices

Choose an elliptic coinvariant lift (m). The four classes have presentations

$$2m=a e_6+b v_{\rm alg}, \qquad (a,b)\in(\mathbb Z/2)^2.$$

• For ((a,b)=(0,0)), the coinvariant group is (mathcal A_{--}\oplus\mathbb Z/2); the width-two torsion remains visible.
• For each nonzero parity vector, the relation is primitive and the total group is free of rank two; the apparent elliptic (mathbb Z/2) is absorbed into the algebraic lattice extension.

The rational connection cannot distinguish these four cases.

Verdict

$$\boxed{ \text{the unresolved integral total-energy problem is exactly one class in }(\mathbb Z/2)^2.}$$

This corrects an overstrong possible inference from Entry 1147: the elliptic quotient has a width-two coinvariant, but a visible (mathbb Z/2) summand in the full rank-nine coinvariants is not established.

No carrier modification is implicated. The missing datum is an integral Betti/Gysin comparison or polarization fixing the extension class.

Next falsifier

Construct the integral Picard--Lefschetz action on

$$H^2(S\setminus D_\infty;\mathbb Z)$$

in a basis compatible with the primitive Gysin sequence. It is sufficient to compute the parity of twice one elliptic coinvariant lift along ((e_6,v_{\rm alg})). The result must be one of the four classes above; no rational reconstruction of the full connection is required.

Evidence:

• research/benincasa/checkers/total_energy_coinvariant_extension_classes.py;
• research/benincasa/results/total-energy-coinvariant-extension-classes.json;
• Entries 150, 289, 1147, and 1151.

View source on GitHub