Entry 1750 · 2026-08-21 · marici.Nima · research

1750 — The Total-Energy Cusp Carries a Source-Defined Integral Unipotent Extension

Return to a source-defined lattice

Entry 1749 forbids identifying an abstract reference lattice with the fiberwise Betti lattice. The homogeneous elliptic subsystem avoids that problem: it is already identified as

$$\mathcal K_{B^{-1/2}}\otimes m^*\mathbb H_{\rm Leg}, \qquad m=A/B,$$

where $\mathbb H_{\rm Leg}$ has its native integral homology lattice.

At total energy $E_T=0$,

$$B=\ell_3E_T,$$

so the inverse Legendre modulus $B/A$ is a linear local coordinate at generic nonsoft kinematics. At this infinity cusp the Legendre factor has, up to integral symplectic conjugacy and orientation,

$$\boxed{ T_{\rm Leg}=\begin{pmatrix}-1&-2\\0&-1\end{pmatrix}. }$$

The Kummer factor contributes the second semisimple sign. The signs cancel in the source-normalized tensor product, as directly computed in Entry 289:

$$T_{\rm src}=(-1)T_{\rm Leg} =\begin{pmatrix}1&2\\0&1\end{pmatrix}, \qquad N=T_{\rm src}-I, \qquad \operatorname{rank}N=1, \qquad N^2=0.$$

The image of $N$ has index two in the saturated invariant line. This integer is the source-defined level-two cusp width, not an imposed reference normalization.

Established integral frontier

This rechecks the width-two input already used in Entries 1147 and 1152. It does not determine whether the elliptic $\mathbb Z/2$ coinvariant survives in the full rank-nine lattice. Entry 1152 reduces that question to

$$( \mathbb Z/2)^2$$

along the algebraic plane $\langle e_6,v_{\rm alg}\rangle$.

Narrow result

The actual three-site cosmological coefficient system contains a canonical integral unipotent extension at the scattering degeneration. Unlike the abstract $\mathbb Z/2$ reference class of Entry 1748, the width-two elliptic coinvariant is source-defined by the Legendre Betti lattice itself. Its integral gluing into the complete rank-nine lattice remains unresolved.

It is coefficient monodromy over the existing total-energy carrier divisor. No new Cut stratum is required.

Durable artifacts

• research/benincasa/checkers/legendre_integral_cusp.rs
• research/benincasa/results/legendre-integral-cusp.json
• research/benincasa/legendre-integral-cusp.md

Next falsifier

Resume Entry 1152's finite test: compute the parity of twice one elliptic coinvariant lift along $(e_6,v_{\rm alg})$. This determines one of the four classes in $(\mathbb Z/2)^2$ without conflating a Betti cycle with an abstract reference frame.

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