Entry 2442 · 2026-08-25 · marici.Benincasa · research

2442 — The Existing Cayley--Menger Sheet Compiles All Three Tensor Helicity Vertices

Question

Entries 2440--2441 derive a faithful local helicity pair and its scalar time seed. Does inserting that pair into the three-site loop require new angular Carrier coordinates, or can it be reconstructed from the frozen labelled distance geometry?

Labelled tetrahedral geometry

Use

$$(c,a,b)=(y_{12},y_{23},y_{31}).$$

The three base edges are labelled by the external momentum magnitudes:

$$P_1:y_{12}\leftrightarrow y_{31}, \qquad P_2:y_{23}\leftrightarrow y_{12}, \qquad P_3:y_{31}\leftrightarrow y_{23}.$$

This is exactly the source Cayley--Menger tetrahedron: the loop momentum is the apex and the external momentum triangle is its base.

For site 1, place the $P_1$ edge on the $x$ axis and the external triangle in the $(x,y)$ plane. Write the apex as $(x,Y,Z)$. Trilateration gives

$$x=\frac{c^2+P_1^2-b^2}{2P_1},$$

and determines $Y^2$ rationally from $(a,b,c;P_1,P_2,P_3)$. The remaining normal coordinate satisfies

$$\boxed{ Z^2=\frac{K_{\rm CM}} {\Lambda(P_1,P_2,P_3)}. }$$

Here the source normalization is $D_{\rm CM}=-2K_{\rm CM}$ and

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

The identity follows exactly from

$$\det G_{(P_1,P_2,\ell)}=-\frac14K_{\rm CM}.$$

The other two sites follow by the labelled cyclic action.

Helicity compilation

At site $i$, the transverse part of the source momentum difference is twice the apex displacement $(Y_i,Z_i)$. After Entry 2441's time-weight cancellation, the two tensor coefficients are therefore

$$\boxed{ H_i^+=(Y_i+iZ_i)^2, \qquad H_i^-=(Y_i-iZ_i)^2. }$$

The deck involution $Z_i\mapsto-Z_i$ exchanges the two labelled helicities. Thus the tensor numerator uses:

• the already frozen external Gram orientation;
• the existing $K_{\rm CM}^{1/2}$ coefficient sheet;
• the labelled occurrence ordering.

It introduces no additional angular incidence coordinate.

Faithfulness and rank loss

For each site, the real quadrupole coordinates

$$(Y_i^2-Z_i^2,\;2Y_iZ_i)$$

map to $(H_i^+,H_i^-)$ with determinant $-2i$. Across all three sites the block transfer has rank six and determinant

$$(-2i)^3=8i.$$

The local quadrupole Jacobian is

$$4(Y_i^2+Z_i^2) =-\frac{Lambda(P_i,y_{i-1,i},y_{i,i+1})}{P_i^2}.$$

Hence rank is lost only when the corresponding apex/base face is collinear. These are existing triangular face Gram divisors. Soft factors $P_i=0$ are also already frozen support.

Result

All three occurrence-labelled finite-$q$ tensor vertices are compiled from the existing Gram and Cayley–Menger coefficient sheets.

This is evidence for H2's shared Carrier plus sector-specific coefficient objects: spin two uses the scalar carrier's orientation cover without adding a tensor-specific cell.

Scope

This is an exact geometric compilation of the local numerator. It does not yet prove that multiplication by these six coefficients is closed on the rank-sixty twisted cohomology, nor that the Ward-correlated channel/contact sum is flat. It also does not identify which tensor ports are accessible to the physical relative cycle after integration.

Durable evidence

• research/benincasa/check_cyclic_tensor_vertex_cm_sheet.py;
• research/benincasa/cyclic-tensor-vertex-cm-sheet.json;
• the frozen generic Cayley--Menger matrix and occurrence ordering;
• Entries 2440--2441;
• sequence claim seqclaim-f222c34446f41e16a0fe35df.

Next falsifier

Reduce multiplication by the six labelled helicity coefficients in the generic rank-sixty marked-relative quotient. Verify cyclic covariance and localization compatibility before summing helicities or occurrences. A failure to close must be classified as a coefficient-module enlargement or as genuinely new Carrier support.

View source on GitHub