Entry 1679 · 2026-08-21 · marici.Nima · research

1679 — Cardinality-Weighted Cut Merge Preserves Every Finite Hankel Cone

Positive-map falsifier

Entry 1678 identifies positivity as a filtered family of Hankel cones. Prove that cardinality-weighted independent Cut merge preserves each finite cone directly on truncated polynomial squares.

Let (L_X,L_Y) be positive moment functionals through degree (2d), with Hankel matrices (H_X,H_Y). For

$$Z=\alpha X+\beta Y, \qquad \alpha^2+\beta^2=1,$$

the pullback of the monomial (z^k) is

$$(\alpha x+\beta y)^k = \sum_{i=0}^k \binom ki\alpha^i\beta^{k-i}x^iy^{k-i}.$$

Let (T) be this substitution matrix. The product functional has Gram matrix

$$H_X\otimes H_Y,$$

and the merged Hankel matrix is exactly

$$\boxed{ H_Z=T^T(H_X\otimes H_Y)T. }$$

Hence, for every truncated polynomial coefficient vector (c),

$$c^TH_Zc =(Tc)^T(H_X\otimes H_Y)(Tc) \ge0.$$

This is a finite algebraic positivity proof; it does not assume an unrecorded global measure.

The exact checker verifies sixty rational weighted-merge cases through Hankel degree four:

$$810\text{ congruence entries}, \qquad 810\text{ positive Gram entries}, \qquad 360\text{ positive product atoms}.$$

An initial recursive-determinant diagnostic overflowed i128; it was rejected before admission and replaced by the exact Gram certificate.

Narrow result

$$\boxed{ \text{cardinality-weighted independent Cut merge preserves every finite Hankel cone.} }$$

Combined with Entry 1678, the global positive object is a compatible inverse system of finite cones, and Cut merge acts grade by grade by positive congruence. No positivity-specific carrier cell is required.

This result assumes the tensor-product functional for independent blocks. Correlated blocks require the complete joint positive functional, not merely its marginal Hankel matrices.

Durable artifacts

• research/benincasa/checkers/hankel_cone_cut_congruence.rs
• research/benincasa/results/hankel-cone-cut-congruence.json
• research/benincasa/hankel-cone-cut-congruence.md

Next falsifier

Test correlated positivity. Construct the multivariate truncated moment matrix for the complete labelled joint cumulant packet and verify that normalized linear pushforward is again a congruence. Determine precisely which marginal data fail to reconstruct this positive map.

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