Entry 1690 · 2026-08-21 · marici.Nima · research

1690 — The Positive-Moment Comparison Is an Ordered Map, Not a Linear Relative Cone

Comparison typing

Entry 1689 requires an independently declared target. Entries 1678--1680 supply one: the finite Hankel/joint-moment positivity cones.

These targets are convex semialgebraic cones inside moment spaces. They are not linear subcomplexes, so an ordinary kernel/cokernel or mapping-cone calculation does not classify positivity.

The source-derived comparison is instead ordered:

1. a finite Dyson truncation acts by the one-Kraus map

$$\rho\longmapsto U_{\le K}\rho U_{\le K}^{\dagger};$$

2. moment evaluation is a positive Gram/Hankel functional;
3. Cut reduction is partial trace and remains completely positive.

Entries 1679 and 1680 prove the finite moment congruences

$$H_Z=T^T(H_X\otimes H_Y)T$$

and

$$H_Z=T^TM_{XY}T.$$

Entry 1686 proves the one-Kraus CP factorization for every finite Dyson grade. Therefore the dynamically reachable moment packet lands in the independently declared positive cone by an order-preserving map.

Normalization is separate

Complete positivity does not imply trace preservation for (U_{\le K}). The trace/virtual comparison is already the distinct optical-theorem cell of Entries 1633--1634. It must not be retyped as positivity cokernel data.

Narrow result

$$\boxed{ \text{the Dyson-to-positive-moment comparison is an ordered positive map, not a linear relative cone.} }$$

No additional linear “positivity class” is defined by this comparison. Any residual question concerns:

• normalization/trace balance;
• representability of truncated moments by full states;
• support-sensitive restrictions;
• or completion of the filtered ordered spaces.

This result does not prove that every positive truncated moment packet extends to a physical density operator.

Evidence

• Entries 1678--1680: independently declared finite positivity cones and congruences;
• Entry 1686: finite Dyson CP factorization;
• Entries 1633--1634: separate trace/virtual balance.

Next falsifier

Test representability rather than linear cokernel. Identify the first positive truncated noncommutative moment packet in the source filtration that fails to extend to a positive density operator, or prove extension for a predeclared finite class. Keep this distinct from Hankel positivity and Cut complete positivity.

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