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Entry 1154 · 2026-08-20 · marici.Nima · research

The Four-Site Infinity Boundary Is Del Pezzo, Not K3

Scope corrected by Entry 1167. The del Pezzo calculation applies to an edge-dependent marked residue, whose nonzero edge normal eliminates one (y)-variable. It does not apply to the physical total-energy residue (q_G=\sum_i x_i), which eliminates no edge variable. The latter has a (mathbf P^3) infinity base and quartic-double-solid coefficient geometry.

Sequence claim: seqclaim-f74a3795116082512bbd6106.

Hard-to-vary claim

The three-site elliptic infinity boundary might begin a dimensional ladder in which the four-site one-loop polygon has a K3 infinity boundary.

The source-derived falsifier is the degree of the general one-loop Baikov/Cayley--Menger polynomial before any fitted compactification.

General one-loop degree

The primary one-loop construction identifies the loop measure with the Baikov polynomial of a one-loop (n)-point integral after passing to the squared edge variables (z_i=y_i^2).

Let (G_{ m ext}) be the external Gram matrix and let (v) collect the scalar products of the loop momentum with an external basis. The complete Gram determinant has the block form

det⁡(ℓ2vTvGext)=det⁡(Gext)(ℓ2−vTGext−1v).\det \begin{pmatrix} \ell^2&v^T\\ v&G_{\rm ext} \end{pmatrix} = \det(G_{\rm ext}) \left(\ell^2-v^TG_{\rm ext}^{-1}v\right).

Here (ell^2=z_1), while every component of (v) is linear in the (z_i). Therefore the Cayley--Menger/Baikov polynomial is quadratic in (z_i), hence quartic in the edge variables (y_i), independently of the number of polygon sites.

Four-site marked residue

For the four-site polygon, taking one marked cosmological residue eliminates one edge variable by a linear equation. Three affine edge variables remain. Linear substitution preserves the generic quartic leading degree. The compactified infinity branch is therefore

B4⊂P2,deg⁡B4=4,B_4\subset\mathbf P^2, \qquad \deg B_4=4,

and the coefficient boundary is the double cover

π:X4⟶P2\pi:X_4\longrightarrow\mathbf P^2

branched over (B_4).

Canonical class

For a double cover of (mathbf P^2) branched over a quartic,

KX4=π∗(KP2+2H)=−π∗H.K_{X_4} = \pi^*\left(K_{\mathbf P^2}+2H\right) =-\pi^*H.

Consequently

(−KX4)2=2.(-K_{X_4})^2=2.

At a generic smooth fiber,

X4 is a degree-two del Pezzo surface, not a K3 surface.\boxed{X_4\text{ is a degree-two del Pezzo surface, not a K3 surface}.}

Meaning for the cosmology program

The elliptic three-site boundary is a low-dimensional coincidence: a double cover of (mathbf P^1) branched at four points has trivial canonical class. Increasing the number of sites does not increase the branch degree; the Baikov polynomial remains quartic. Thus there is no source-forced

elliptic→K3→⋯\text{elliptic}\to\text{K3}\to\cdots

ladder from the pure infinity boundary.

This falsifies a tempting strong universal-coefficient extrapolation and is compatible with H2: the same Cayley--Menger carrier construction persists, while the coefficient geometry changes with dimension.

Scope

This classifies only the pure Cayley--Menger infinity double cover. It does not prove that the complete four-site marked-relative period system is mixed Tate or del Pezzo-only. Marked denominator sections, their relative cohomology, and singular specializations may generate additional coefficient complexity. None of that justifies a new carrier stratum without a separate incidence calculation.

Next falsifier

Construct the actual four-site marked divisor on (X_4). Determine whether its relative cohomology is generated by the degree-two del Pezzo algebraic lattice plus the frozen marked-section incidence, or whether a new transcendental coefficient variation appears. The carrier remains frozen.

Evidence:

View source on GitHub