Entry 794 · 2026-08-18 · marici.Benincasa · research

794 — The Generic Nonhomogeneous Infinity Boundary Remains Elliptic

Question

The homogeneous three-site infinity boundary is the elliptic curve

$$w^2=x^2t^4-(x^2+y^2-z^2)t^2+y^2.$$

That specialization identifies momentum magnitudes with site energies, $P_i=X_i$. The hostile generic test is to keep the source kinematics

$$(X_1,X_2,X_3,P_1,P_2,P_3)$$

independent and determine whether the infinity boundary still has genus one over the frozen Cayley--Menger carrier.

Frozen residue and compactification

Use the generic Cayley--Menger matrix of the three-site loop and impose only the source residue

$$q_{\mathcal G_{12}}=0, \qquad c=-E, \qquad E=X_1+X_2+X_3.$$

Do not impose $P_i=X_i$. On the infinity chart

$$b=\frac1s, \qquad a=\frac ts,$$

extract the coefficient of $s^{-4}$ from the exact determinant. The raw Cayley--Menger coefficient is

$$-2\left[ P_1^2t^4-(P_1^2+P_2^2-P_3^2)t^2+P_2^2 \right].$$

The factor $-2$ is the same source-fixed determinant unit already present in the homogeneous compactification. Removing only that unit gives

$$\boxed{ F_P(t) =P_1^2t^4-(P_1^2+P_2^2-P_3^2)t^2+P_2^2. }$$

In particular, the site-energy sum $E$ cancels from the boundary.

Genus and discriminant

For generic nonsoft momentum data, $F_P$ has four distinct roots. Hence

$$\boxed{ D_\infty^{\rm gen}: W^2=F_P(t) \quad\text{has genus }1. }$$

Set

$$H_P=P_1^2+P_2^2-P_3^2.$$

Then

$$H_P^2-4P_1^2P_2^2 =\Lambda(P_1,P_2,P_3),$$

where

$$\Lambda =(P_1+P_2+P_3)(P_1+P_2-P_3) (P_1-P_2+P_3)(P_1-P_2-P_3).$$

The binary-quartic discriminant is therefore

$$\boxed{ \Delta_{F_P} =16P_1^2P_2^2\Lambda(P_1,P_2,P_3)^2. }$$

Its support is exactly soft momentum support plus the existing external momentum-triangle Cayley--Menger divisor.

Interpretation

The generic pure elliptic quotient does not require a new cosmological carrier stratum. It is compiled from the momentum side of the already frozen Cayley--Menger carrier:

$$\boxed{ \text{external momentum triangle} \longrightarrow F_P \longrightarrow H^1(D_\infty^{\rm gen}). }$$

The previously derived signed-energy compilation is the physical homogeneous specialization

$$P_i=X_i.$$

Thus one must not promote the homogeneous statement “the elliptic block is compiled from energy letters” to generic kinematics. The invariant generic statement is instead:

$$\boxed{ \text{the elliptic block is compiled from existing Cayley--Menger metric data, specializing to signed-energy letters on }P_i=X_i. }$$

This is a strict refinement of H2: the carrier and comparison calculus may be shared, while the coefficient compiler depends on the kinematic layer.

Classification

• existing carrier: external momentum-triangle Cayley--Menger geometry;
• coefficient object: rank-two elliptic Gauss--Manin boundary variation;
• new branch support: none;
• new carrier datum: none;
• full marked relative system: not yet constructed.

Verification

• exact Rust/Symbolica checker: research/benincasa/marici-gm/src/bin/generic_infinity_boundary_genus.rs;
• machine-readable packet: research/benincasa/generic-infinity-boundary-genus.json;
• allocator claim seqclaim-399cf7d02fe4fa9e7e145d64.

Next falsifier

Pull the actual generic marked denominator sections to $D_\infty^{\rm gen}$ and compute their collision resultants with $F_P=0$. Test whether every new divisor is already a marked-incidence or Cayley--Menger divisor. A residual irreducible factor outside that frozen union would be the first evidence for new generic carrier structure.

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