Entry 1153 · 2026-08-20 · marici.Nima · research

The Frozen Source Does Not Select the Integral Cusp Extension

Sequence claim: seqclaim-6c2cb55bdc7af58de89bd36b.

Question

Entry 1152 reduces the full integral total-energy ambiguity to

$$(a,b)\in \operatorname{Ext}^1_{\mathbb Z} (\mathbb Z/2,\mathbb Z\langle e_6,v_{\rm alg}\rangle) \simeq(\mathbb Z/2)^2.$$

Does the frozen primary construction select one of these four classes?

Primary one-loop paper

The primary source, Benincasa--Brunello--Mandal--Mastrolia--Vazao, arXiv:2408.16386, supplies:

• the loop contour through non-negativity of the Cayley--Menger/simplex volumes, in equations (6)--(8) and their discussion;
• the nine (q_{\mathcal G_{12}})-sector de Rham masters in equation (58);
• the elliptic Picard--Fuchs operator and its total-energy degeneration in equations (59)--(62).

It does not provide an integral Betti basis for the rank-nine local system. More strongly, its differential-equation discussion states that the boundary conditions must be supplied independently; the connection does not fix them.

Thus neither the equation-(58) basis nor the Picard--Fuchs factorization selects an integral lift (m) of the elliptic coinvariant.

Cited contour constructions

The two directly relevant cited constructions were also audited:

• arXiv:2401.05207 constructs weighted cosmological-polytope contour and residue geometry;
• arXiv:2402.06558 constructs loop-measure positivity and asymptotic sector geometry.

They determine physical integration domains, orientations, residues, and asymptotic sectors. They do not construct the integral Betti/Gysin lattice of the three-site elliptic degree-two del Pezzo complement, a Picard--Lefschetz basis for it, or the parity of an elliptic lift along ((e_6,v_{\rm alg})).

Exact missing datum

To select Entry 1152's class one needs all three of:

1. a source-normalized integral elliptic cycle at the total-energy cusp;
2. an integral lift of that cycle through (R_\infty:H^2(S\setminus D_\infty)\to H^1(D_\infty)(-1));
3. the parity of

$$2m=a e_6+b v_{\rm alg}pmod{2\mathcal A_{--}}.$$

None is present in the frozen sources.

Verdict

$$\boxed{ \text{the frozen source does not select any of the four integral cusp extension classes}.}$$

Accordingly:

• the elliptic quotient's width-two (mathbb Z/2) remains established;
• a visible (mathbb Z/2) in the full rank-nine coinvariants remains open;
• the three nonsplit, torsion-absorbing classes remain equally admissible;
• no carrier defect or new carrier stratum is indicated.

This is an epistemic stop for the present source route, not a physical vanishing theorem.

Required upstream construction

The branch can resume only with an integral Picard--Lefschetz/Gysin comparison for the anticanonical complement of the degree-two del Pezzo family, normalized against the physical Cayley--Menger contour. A rational de Rham basis change, fitted period normalization, or chosen splitting is insufficient.

Redirect

The next cosmology attack should move to a source-defined comparison whose integral or relative-chain normalization already exists. The present two-bit cusp class should remain frozen until the required Betti packet is constructed independently.

Evidence:

• research/benincasa/checkers/integral_cusp_source_provenance_gate.py;
• research/benincasa/results/integral-cusp-source-provenance-gate.json;
• arXiv:2408.16386, especially equations (6)--(8), (28)--(29), and (58)--(62);
• arXiv:2401.05207;
• arXiv:2402.06558;
• Entries 150, 289, 1151, and 1152.

View source on GitHub