GNP arXiv v4 Question 8.20: Dicke extension cones

Gulati, Nechita and Park, Positive maps and extendibility hierarchies from copositive matrices, arXiv:2509.15201v4, Question 8.20, asks:

For mixtures of Dicke states, are the following two sets equivalent?

$$ \mathsf{PPTBExt}{n}^{(\mathcal K_r)} =\mathsf{PPTBExt}{n}^{(r-1)},\qquad r\ge2. $$

The target is exactly this versioned question. For every integer $n\ge1$, $r\ge2$ and every real upper-triangular family $Y_{ij}\ge0$, put

$$ D_{ii}=e_i\otimes e_i,\qquad D_{ij}=\frac{e_i\otimes e_j+e_j\otimes e_i}{\sqrt2}\quad(i<j),\qquad \rho=\sum_{1\le i\le j\le n}Y_{ij}|D_{ij}\rangle\langle D_{ij}|. $$

The star predicate is existence of $E\succeq0$ on $(\mathbb C^n)^{\otimes r}$ with $(I\otimes\Pi_{r-1})E(I\otimes\Pi_{r-1})=E$, every root-leaf marginal $\rho$, and $E^{T_S}\succeq0$ for every nonempty proper register subset $S$. The complete-graph predicate is existence of $F\succeq0$ on the same tensor power with $\Pi_rF\Pi_r=F$, every two-register marginal $\rho$, and $F^{T_S}\succeq0$ for every such $S$. Here $\Pi_k$ is the orthogonal projector onto $\operatorname{Sym}^k(\mathbb C^n)$, marginals are genuine unnormalized partial traces, and transposes use the standard tensor basis. The assertion is equivalence of these predicates for every $n,r,Y$, including all-zero weights, without trace normalization or invertibility.

The Library note holds the source identity and bounded source checks. The original preregistration fixes this event before execution; its tier is the first-tier short complete-proof unit.

The settling declaration is D5/S3/Quantum/Entanglement/SymmetricExtensions/DickeExtension.result. It equates the full star and complete-graph existence predicates, retaining the same matrix witness in each implication. Finite tensor indices are functions Fin r -> Fin n; the coefficient array observes only its upper triangle. The result is relevant to comparing the two hierarchies on the whole Dicke cone, rather than checking individual dimensions or selected weights.

The source poses a star-to-complete implication. The reverse implication follows from full Bose support and inclusion of the star edges. Full Bose support cannot be assumed in the forward direction: it must follow from the actual positive extension and the prescribed Dicke marginals. The bounded source checks found no full prior resolution in their inspected scope. They do not establish worldwide absence, priority or current journal wording.

The two-particle antisymmetric projector annihilates every Dicke vector. On each root edge its expectation in the prescribed marginal is zero. Actual partial-trace duality gives zero expectation for the corresponding projector placed on the $r$-register tensor product. The frozen positive zero-weight support face theorem implies its one-sided annihilation of $E$. Hence every root transposition fixes every column of $E$.

Root transpositions generate all register permutations, so these columns lie in the full symmetric subspace. The actual orthogonal permutation-average projector and Hermiticity give $\Pi_rE\Pi_r=E$. Reindexing the complement factors of the partial trace proves that all ordered edge marginals coincide. The complete graph witness is the original $E$, so each original PPT inequality is retained. Conversely full symmetric support implies the leaf-projector sandwich, and the complete edge constraints include the star constraints.

The direct frozen support input is D5/S3/QuantumStates/ZeroWeightSupportFace.zero_weight_support_face (statement identity sha256:c21f5826b1569f02708c19839f076c2ce1484bce1cbb7dbe8279a52fd19c8092). The projector inputs are in D5/S3/Quantum/Entanglement/AbsoluteSeparability/SymmetricAbsolutePPTPermutationFloor; actual marginal and transpose definitions reuse PartialTraceMutualInformation and SymmetricAbsolutePPTPureBound. No separate surrogate extension or assumed support bridge is introduced.

A mismatch of even one quantifier, Dicke normalization, tensor marginal, projector or PPT cut would invalidate the claimed correspondence. A proof with strictly positive or normalized weights, only finitely many dimensions, a full Bose premise on the star witness, or a restricted set of transpose cuts would not settle this event. None is part of the public statement. The all-zero operator, singular witnesses, $n=1$ and $r=2$ remain in scope. The theorem does not assert that every Dicke operator has either extension.

The mathematical source is DickeExtension.lean, with its canonical Blueprint. The result's statement identity is sha256:6b785387e00ba31d45b4abe703cc448050276c1f3b42b619ce0fc8aa78735f68. The Scribe's OpenProblemResolutionClaim binds exactly this frozen result to this dossier with kind Proved.

Every theorem helper and the result have proof_shape: bind-only; there is no escape witness. The helpers have actual consumers in the result's live proof closure. The terminal result uses admission_basis: open-problem-resolution under the original preregistration, rather than claiming a new general support theorem. Definitions express the actual tensor operators. Mathematical truth comes from the compiled Lean declaration and its accepted axiom closure; source fidelity and admission semantics also require independent review.

The retained Reg source contains the original telescope, actual star observation, source-law bridge, positive law, rejected-law variation, whole-family sensitivity and observational dependence. Its source-readout classification remains declared_unresolved, with no certificate: IE-C050 ClosedTruthReadout, unclassified_form, source.dictionary_or_proof_coordinate. The missing accepted admissibility/binding classification and the original theorem are disclosed through the section 3.9 linked-issue exception. This evidence is not a completed Reg audit.

• Mechanism — proved in the delivered result. Positivity converts the zero antisymmetric marginal weight into column support. Permutation generation then supplies full Bose support. All edge equalities concern the same actual matrix, so the argument never combines separately realizable witnesses.
• Whole event — proved by the compiled statement. Every $n\ge1$, $r\ge2$ and nonnegative unnormalized Dicke coefficient family is included, with zero and singular cases. No numerical extrapolation is used.
• PPT preservation — proved in the delivered result. Both implications keep the original witness; every nonempty proper cut inequality passes unchanged. The forward support argument itself does not use these inequalities or the redundant leaf-support premise. No separately retained generalization is asserted.
• Hierarchy consequence — the exact source question. The two extension cones agree on the specified Dicke cone at each common register count. This neither characterizes membership in that cone's extension sets nor makes all Dicke operators extendible.
• Adjacent questions — unresolved here. The broader comparison for arbitrary Bose marginals, Question 8.7, journal question wording and any additional theorem depending on those broader questions are outside this settlement. No downstream source theorem is claimed beyond the explicit Question 8.20 equality.

Literature completeness and worldwide novelty or priority are unverified; negative findings have exactly the bounded scope recorded in the Library note. The current journal DOI is 10.1007/s00023-026-01763-3. Only metadata, abstract and a subscription preview were acquired; journal question-body characters acquired are zero. No claim is made about that journal's question text or solution status. The source-qualified resolution is exclusively arXiv:2509.15201v4 Question 8.20. The Reg audit remains unfinished under the linked exception above; continuation := .unknown means only an unknown residual, not undecidability or inexhaustibility. Final independent reviews, remote required checks and publication are separate obligations.