Entry 2276 · 2026-08-24 · marici.Nima · research

2276 — Generic Tensor Projection Is Forced to Lose Rank on the Existing Gram Wall

Frozen generic orientation

Retain Entry 2272's spectral Gaussian source, but let the tensor momentum be

$$q=(a,b,c)$$

relative to the equilateral hard-momentum plane $z=0$. Use the transverse frame

$$u=(-b,a,0), \qquad v=(-ac,-bc,a^2+b^2).$$

The three hard momenta are

$$p_1=(2,0,0), \quad p_2=(-1,\sqrt3,0), \quad p_3=(-1,-\sqrt3,0).$$

For each occurrence, form the two traceless tensor contractions in this frame and retain an independent positive spectral weight $w_i$. No relation among the weights is used in the symbolic determinant.

Exact factorization

After removing common nonzero polarization-frame normalizations, the determinant contains the factor

$$\boxed{c=q\cdot\widehat n},$$

where $\widehat n$ is the hard-triangle normal. In the current polynomial chart the complete expression is

$$-4\sqrt3\,c(a^2+b^2)^2(a^2+b^2+c^2)\,B,$$

where $(a^2+b^2)^2(a^2+b^2+c^2)$ is polarization-frame scaling and

$$\begin{aligned} B={}&2\sqrt3ab\,w_1(w_2-w_3) +3a^2w_2w_3\\ &+b^2(2w_1w_2+2w_1w_3-w_2w_3)\\ &+2c^2(w_1w_2+w_1w_3+w_2w_3). \end{aligned}$$

The chart factors must not be interpreted as physical support: in particular, the normal slice $a=b=0$ is faithful by Entry 2272 and merely requires the other polarization chart.

Geometric rank loss

On

$$c=0,$$

the tensor momentum lies in the hard-momentum plane. Projecting all three hard momenta onto $q^\perp$ leaves only one direction inside that plane. Consequently all traceless tensor evaluations are collinear and the scalar-plus-two-polarization matrix has rank at most two.

This is exactly the Gram/projection wall

$$\boxed{ q\cdot(p_1\times p_2)=0. }$$

It is an existing kinematic support condition, not a new Carrier incidence generator.

Residual falsifier

Away from the Gram wall, any additional rank loss must satisfy

$$B=0$$

after imposing the source weights

$$w_i=\frac1{2+|p_i+q|}>0$$

up to a common positive normalization. A bounded physical grid found $B>0$, with minimum approximately $1.49\times10^{-2}$, but this is only discovery evidence. The next theorem must prove positivity of $B$ or produce an exact positive-energy counterexample.

Verification

research/benincasa/marici-gm/src/bin/generic_tensor_projection_determinant.rs derives the symbolic factorization with Symbolica and performs the bounded physical-weight scan.

View source on GitHub