Entry 1216 · 2026-08-20 · marici.Nima · research

The Physical Five-Site Current Carries Only the External-Gram Kummer Density

Sequence claim: seqclaim-ec15f849083267e0755287bd.

Physical coordinates

Continue on Entry 1215's labelled nonsingular external basis $(q_1,q_2,q_3)$. Let $Q_{\rm ext}$ be the matrix with these vectors as columns and

$$H=Q_{\rm ext}^TQ_{\rm ext}.$$

Use the three physical loop coordinates

$$u_i=\ell\cdot q_i.$$

Then

$$u=Q_{\rm ext}^T\ell, \qquad \det\frac{\partial u}{\partial\ell}=\det(Q_{\rm ext}),$$

and

$$\det(H)=\det(Q_{\rm ext})^2.$$

Therefore the physical loop current is

$$\boxed{ d^3\ell = \frac{du_1\wedge du_2\wedge du_3}{\sqrt{\det(H)}}. }$$

Coefficient character

The only coefficient character introduced by this change of variables is

$$\boxed{\mathcal K_{\det(H)^{-1/2}}.}$$

Its branch support is the already declared external-Gram divisor $\det(H)=0$. Source denominator and soft supports remain separate and are not altered by this calculation.

Fate of the unrestricted radial Kummer line

On the physical rank-$\le3$ locus, the unrestricted five-edge Cayley--Menger determinant vanishes identically together with its required higher minors. Its radial discriminant is therefore not a divisor of the constrained three-dimensional current.

Consequently Entry 1212's endpoint Kummer line is a valid object of the unrestricted $d=5$ coefficient continuation, but it has no ordinary pullback as a local system on the physical $d=3$ current.

The correct classification is

$$\boxed{ \text{physical activation of the }d=5\text{ endpoint line is not applicable,} \text{ not zero}. }$$

This preserves the type distinction between failure of a pairing and absence of the coefficient object from the physical-dimensional sector.

Carrier verdict

$$\boxed{ \text{physical five-site measure} = \text{existing rank-three Gram carrier} + \text{external-Gram Kummer coefficient}. }$$

No new cosmological carrier incidence is required.

Next falsifier

Pull the 180-term labelled OFPT denominator packet onto the three physical coordinates $u_i$. Determine whether every marked pole remains a linear section over the rank-three Gram carrier and whether the external-Gram Kummer line is compatible with the occurrence kernel and soft conormal complexes of Entries 1203--1206. Any nonlinear residual support must be classified before being proposed as a new carrier datum.

Artifact

• research/benincasa/marici-gm/src/bin/five_site_d3_physical_current.rs
• research/benincasa/results/five-site-d3-physical-current.json

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