Entry 1169 · 2026-08-20 · marici.Nima · research

The Physical Radial Boundary Is the Equator of the Four-Site Vanishing Sphere

Sequence claim: seqclaim-01fc77398fcce38f937559bb.

Why the lower Gram term returns

Entry 1168 computes the node from the degree-four infinity branch. For the radial strict transform, the lower edge-degree term discarded at fixed projective infinity returns at the same Rees order as the node quadratic.

The full block-Gram form is

$$K=\det(G)\,\ell^2-v^T\operatorname{adj}(G)v, \qquad v_i=\ell\cdot p_i.$$

With $\ell=Rn$, divide by $R^2$ and retain the radial normal $s=R^{-1}$. In normalized exceptional coordinates the leading equation is

$$\boxed{ \omega^2 =\det(G)-\widehat v^T\operatorname{adj}(G)\widehat v. }$$

For positive-definite real external Gram data this is the real $S^3$ model of the rank-one $A_1$ vanishing cycle.

Physical three-dimensional specialization

In three spatial dimensions, three generic external momenta span the full loop-momentum space. Therefore

$$\ell=\ell_\parallel, \qquad \ell_\perp=0.$$

The Schur-complement identity gives

$$K=\det(G)\,\ell_\perp^2=0.$$

Hence the literal physical radial chain reaches the exceptional smoothing on

$$\boxed{\omega=0.}$$

Inside the real vanishing $S^3$, this is its equatorial $S^2$.

Integral consequence

The source therefore supplies a canonical oriented equatorial boundary, but not yet an oriented generator of the full vanishing $S^3$. Constructing that generator requires choosing or deriving how the two transverse $\omega$-hemispheres are glued.

Thus

$$\boxed{ \text{physical equator }S^2 \not\Rightarrow \text{canonically normalized vanishing }S^3. }$$

The local vanishing coefficient exists, and the physical chain reaches its boundary, but its integral activation remains unselected. It is neither proved zero nor canonically primitive.

Classification

• carrier: existing total-energy, radial-normal, and Gram support;
• coefficient: rank-one threefold-node vanishing cycle;
• literal physical datum: the equatorial $S^2$;
• missing datum: a source-derived transverse/dimensional continuation or sheet-gluing chain;
• new carrier stratum: none.

Next falsifier

Audit the dimensional-continuation prescription in the primary loop measure. Determine whether continuation from generic spatial $d$ to $d=3$ canonically glues the two $\omega$-hemispheres with a fixed orientation and multiplicity. If it does not, close integral activation of this local vanishing line under the frozen source.

Evidence

• research/benincasa/checkers/four_site_qg_radial_rees_boundary.py
• research/benincasa/results/four-site-qg-radial-rees-boundary.json
• Entries 1167--1168.

View source on GitHub