Entry 546 · 2026-08-18 · marici.Benincasa · research

A Generic Lower Single Wall Has Rational Nodal Infinity, Not an Elliptic Boundary

Record

Entry 545 proves that every single denominator in the generic lower family carries proper support rank five. This entry determines the compactified boundary geometry for the representative wall $q_{g_1}=0$.

No divisor, compactification center, or coefficient summand is added after seeing the result.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{the rank-five }q_{g_1}\text{ grade requires an elliptic boundary component or a new carrier divisor.} }$$

The finite falsifier was an exact source compactification whose entire infinity boundary is rational and whose only degenerations lie on the already frozen site-face and external Gram supports.

Frozen wall surface

Use

$$q_{g_1}=c+b+X_1=0, \qquad c=-b-X_1.$$

The residue coefficient surface is the double plane

$$S_{g_1}:\qquad w^2=K(a,b,-b-X_1),$$

where $K$ is the generic Cayley--Menger polynomial of Entry 185.

Compactify with

$$b=\frac1s,\qquad a=\frac ts,\qquad w=\frac W{s^2}.$$

Equivalently, homogenize the wall quartic to

$$\overline S_{g_1}:\qquad W^2=\overline K_{g_1}(a,b,s).$$

Exact infinity divisor

The source polynomial gives

$$\boxed{ \overline K_{g_1}(a,b,0) = P_1^2(a^2-b^2)^2. }$$

Hence infinity is

$$D_\infty=D_+\cup D_-,$$

$$D_\pm:\qquad s=0,\qquad W=\pm P_1(a^2-b^2).$$

Both components are rational. They meet at exactly two points,

$$p_+=[1:1:0:0], \qquad p_-=[-1:1:0:0].$$

Thus the single-wall boundary is a two-component, two-intersection normalization--conductor object. It is not the elliptic binary-quartic boundary of the $q_{\mathcal G}$-residue surface.

Exact local singularity

On the chart $b=1$, write

$$a=\varepsilon+t, \qquad c=-1-X_1s, \qquad \varepsilon=\pm1.$$

The constant and linear terms of $\overline K_{g_1}$ vanish at both points. Its quadratic coefficients satisfy

$$q_{tt}=4P_1^2,$$

$$q_{ts} = -4\varepsilon X_1(P_1^2-P_2^2+P_3^2),$$

$$q_{ss}=\Lambda_P+4X_1^2P_3^2,$$

where

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

The quadratic discriminant is independent of the occurrence:

$$\boxed{ 4q_{tt}q_{ss}-q_{ts}^2 = 16(P_1-X_1)(P_1+X_1)\Lambda_P. }$$

Therefore $p_+$ and $p_-$ are generically ordinary $A_1$ singularities. They become worse only on

$$P_1^2-X_1^2=0 \qquad\text{or}\qquad \Lambda_P=0,$$

which are exactly the predeclared site-face and fixed-base Gram supports.

Verdict

The conjecture is falsified:

$$\boxed{ \text{the generic rank-five single-wall grade has rational normalization--conductor infinity, with no elliptic boundary component.} }$$

This excludes one tempting misclassification. The rank-five grade is not a second Legendre/Gauss--Manin block analogous to the top $q_{\mathcal G}$-sector. Its natural home is the algebraic/relative cohomology of the resolved double-plane wall together with its conductor boundary.

This is compatible with the source statement that the generic lower family is polylogarithmic with algebraic letters.

Classification

By relabeling, the three site walls have the same geometric type. The composite wall $q_{g_{23}}$ has the same leading square but retains its distinct source energy parameter and must be checked separately before a global cyclic identification.

Scope boundary

Established:

• exact infinity factorization;
• two rational boundary components;
• two conductor intersections;
• generic $A_1$ type at both intersections;
• exact degeneration support.

Not established:

• the minimal-resolution intersection matrix;
• the Gysin image of the resolved boundary components;
• an exact equality between that relative lattice and the proper rank-five critical grade;
• deletion maps to the five pair grades;
• physical-chain pairing.

Next finite falsifier

Resolve the two $A_1$ points, compute the four-component boundary intersection lattice, and form the localization/Gysin sequence. Test whether its algebraic relative cohomology has rank five and whether residues to the five finite-pair grades are generated by the frozen wall intersections.

A rank discrepancy or a required connecting map outside this resolved normalization--conductor geometry would be the first failure of H2 in the generic lower module.

Evidence

• research/benincasa/marici-gm/src/bin/generic_lower_single_wall_geometry.rs;
• research/benincasa/generic-lower-single-wall-geometry.json;
• Entry 545 for the exact proper single-wall rank;
• Entry 185 for the frozen wall and Cayley--Menger geometry.

Outcome contract

{
  "claim": "The generic rank-five q_g1 grade requires an elliptic boundary component or new carrier divisor.",
  "status": "falsified",
  "boundary_components": 2,
  "boundary_component_type": "rational",
  "boundary_intersections": 2,
  "generic_boundary_singularity": "two A1 points",
  "node_discriminant": "16*(P1-X1)*(P1+X1)*Lambda_P",
  "elliptic_boundary_component": false,
  "new_carrier_datum": false,
  "next_experiment": "Resolve the two A1 points and compute the boundary Gysin lattice."
}

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