Entry 2453 · 2026-08-25 · marici.Benincasa · research

2453 — The Tensor Numerator Preserves or Softens Every Cayley-Menger Pinch

Question

Can the finite-(q) tensor insertion create a new Landau component beyond the frozen Cayley--Menger critical scheme?

Sequence claim: seqclaim-88422b0c3c503dc2bd10c403.

Exact branch restriction

For site one, the parity-even tensor multiplier is

$$Q_1^{\rm even} =- \frac{N^2+4P_1^2K}{4P_1^2\Lambda(P)}.$$

It has no denominator in the fiber variables ((a,b,c)). Modulo the frozen Cayley--Menger branch (K=0),

$$\boxed{ Q_1^{\rm even}|_{K=0} =- \frac{N^2}{4P_1^2\Lambda(P)}. }$$

Thus it is a regular multiplier on the generic nonsoft branch.

Branch exponent and deck character

Writing (s_K^2=K), the two helicity forms split as

$$\Omega_\pm=\mathcal K\pm\mathcal T,$$

where

$$\mathcal K =- \frac{N^2+4P_1^2s_K^2} {4P_1^2\Lambda(P)s_K}, \qquad \mathcal T=\frac{N}{P_1\Lambda(P)}.$$

The reflection-symmetric physical cycle retains

$$\Omega_++\Omega_-=2\mathcal K$$

and kills the rational anti-trace. At a generic point of (K=0), where (N\ne0), the tensor trace has the same exponent (-1/2) and the same deck character as the scalar Cayley--Menger local system.

At the additional numerator-zero intersection (K=N=0), the leading pole vanishes and

$$2\mathcal K|_{N=0}=-\frac{2s_K}{\Lambda(P)}.$$

The exponent is softened to (+1/2); no new singularity is created.

Landau consequence

Landau support is fixed by the denominator/critical geometry. The tensor insertion changes neither (K) nor its critical equations. Since its only fiber effect is multiplication by a regular function, it can preserve or remove a local branch class but cannot add a critical divisor.

$$\boxed{ \operatorname{Sing}_{\rm Landau}(Q^{\rm even}\mathcal K_{\rm CM}) \subseteq \operatorname{Sing}_{\rm Landau}(\mathcal K_{\rm CM}) \cup\{P_1\Lambda(P)=0\}. }$$

The second term is already frozen soft/Gram support.

Classification

• generic Cayley--Menger exponent: unchanged;
• deck character: unchanged;
• numerator-zero collision: softened, not enhanced;
• additional Landau divisor: none;
• new Carrier support: none.

Durable evidence

• research/benincasa/check_tensor_cayley_menger_landau_exponent.py;
• research/benincasa/tensor-cayley-menger-landau-exponent.json;
• Entries 2442--2445.

Next falsifier

Compute the induced tensor map through the explicit infinity-Gysin quotient. The finite failure would be an elliptic quotient class not generated by the existing Kummer/Tate decomposition or a new pole outside the frozen energy, soft, Gram, and Cayley--Menger supports.

View source on GitHub