Entry 1706 · 2026-08-21 · marici.Nima · research

1706 — The Cubic Exceptional Coefficient Commutes with Soft Relative-H0 Specialization

Physical-coefficient falsifier

Entry 1705 closes the bare sign costalk under nested softening. Couple the actual cubic Cut pair coefficient to the exceptional covariance tensor and test the same square.

Labelled contraction

Let C_ij denote the source-derived cubic pair coefficients and

$$\Xi_{ij}=u_iu_j.$$

On a nonzero support graph H, define the componentwise exceptional coefficient

$$F_C(H)=\sum_{(i,j)\in E(H)}C_{ij}\Xi_{ij}.$$

For a deeper support H2 subset H1, restriction takes principal labelled submatrices of both C and Xi. Therefore

$$\boxed{ F_C(H_2) =\sum_{(i,j)\in E(H_2)}C_{ij}u_iu_j }$$

whether restriction is performed before or after forming the labelled tensor contraction.

The conditional fiber multiplier from Entry 1697,

$$s^2-1,$$

is common to every component and commutes with the same maps. Components that disappear under deeper softening contribute zero by extension-by-zero.

Narrow result

$$\boxed{ \text{the physical cubic exceptional coefficient commutes strictly with nested soft relative-}H_0\text{ specialization.} }$$

No coefficient extension survives on disappearing components. The existing labelled Cut restriction, symmetric-square Rees tensor, and support costalk are sufficient for this test.

Durable artifacts

• research/benincasa/checkers/cubic_soft_costalk_naturality.rs
• research/benincasa/results/cubic-soft-costalk-naturality.json
• research/benincasa/cubic-soft-costalk-naturality.md

Next falsifier

Replace the Gaussian pair tensor by the first non-Gaussian connected tensor from Entry 1692. Test whether the odd third-cumulant coefficient extends by the same support maps or couples nontrivially to the component-sign costalk.

View source on GitHub