Maximum purity in the 4-by-4 absolutely PPT sector

Ahiable, Kothakonda and Winter, arXiv:2608.03390v2, Conjecture 6.7, page 29:

Let $\mathcal P_{m,n}\subseteq\mathrm{APPT}{m,n}$ be the inscribed absolute PPT polytope with $2\leq m\leq n,;n>2$. Then $$\max{\lambda\in\mathrm{APPT}{m,n}}\sum{i=1}^{mn}\lambda_i^2 =\max_{\lambda\in\mathcal P_{m,n}}\sum_{i=1}^{mn}\lambda_i^2$$ and occurs at the spectra given by Eq. (44).

The exact target here is its $m=n=4$ sector. A density matrix is a complex matrix $\rho$ on $\mathbb C^4\otimes\mathbb C^4$ with $\rho\succeq0$ and $\operatorname{Tr}\rho=1$. Literal APPT requires $(U\rho U^\dagger)^\Gamma\succeq0$ for every global unitary $U$, where $\Gamma$ transposes the second tensor factor.

Corollary 6.4 and Eq. (44) give the candidate purity $(16+8)/(16+2)^2=2/27$, attained within the inscribed polytope at the spectrum $(1/6,1/18,\ldots,1/18)$. An attaining state alone supplies a lower bound. The missing direction is an upper bound for every literal APPT density matrix, followed by equality of the real purity supremum with $2/27$.

Research preregistration #17689 fixes this sector before the new upper-bound probes. The source states the general conjecture and gives numerical evidence for local dimension four. Tran, arXiv:2609.18568v1, remark following Theorem A, records the exact APPT purity gap. The existing qutrit dossier settles $m=3$ and leaves higher local dimensions open. No $4\otimes4$ upper bound was found in the searched source records or current owning library. This is a tier-3 research target and a not-found-in-searched-scope statement, not a worldwide priority certificate.

An explicit $3\otimes3$ Bell corner inside $4\otimes4$, with an identity on the seven-dimensional complement, gives two necessary real $3\times3$ boundary LMIs. The corner uses the three largest and six smallest ordered eigenvalues. It does not require a sufficient spectral characterization of APPT in local dimension four.

Let $\lambda_0\geq\cdots\geq\lambda_{15}\geq0$, with sum one, and put $t=\lambda_9$, $P=\sum_i\lambda_i^2$. The first LMI tested at $(1,1,1)$ gives $\lambda_0+\lambda_1+\lambda_2\leq\sum_{i=10}^{15}\lambda_i$; ordering and mass one then give $13t\leq1$.

If $t\leq1/18$, insert two copies of $t$ to get an ordered 18-vector. If $t\geq1/18$, delete $t$ to get an ordered 15-vector. The boundary matrices are unchanged before normalization. Positive normalization and the existing QutritN6PurityBound.certificate_bound and QutritN5PurityBound.certificate_bound give, respectively,

$$P\leq\frac{26}{400}(1+2t)^2-2t^2,\qquad P\leq\frac{23}{289}(1-t)^2+t^2.$$

The first expression is at most $2/27$ on $[0,1/18]$. For the second,

$$\frac{23}{289}(1-t)^2+t^2-\frac{2}{27} =\frac{312}{289}\left(t-\frac1{18}\right) \left(t-\frac{43}{468}\right),$$

which is nonpositive on $[1/18,1/13]$. The original certificate bodies are reused directly, rather than copied or replaced by numerical estimates. The explicit attaining state uses the existing dimension-independent LowRankRays.separableCone_one_add_two_rankOne theorem.

A PSD trace-one $4\otimes4$ matrix satisfying the literal unitary-quantified APPT condition and having real trace-square greater than $2/27$ would refute the target. Repeated eigenvalues and zero eigenvalues are included. Neither simple spectrum nor strict positivity is a hypothesis.

QuadbitMaximumPurity.result : claim has passed the canonical scoped Lean build together with QuadbitSpectralPurityBound.spectral_purity_bound. The claim includes the universal bound, real supremum and explicit diagonal attainment. Both printed axiom closures contain only propext, Classical.choice and Quot.sound; the proof introduces no additional axiom. The current scoped report covers both the mathematical root and its Reg mirror, with 91 project modules, and checks every extracted axiom closure. It reports the complete result registration as declared_validated with source-equivalence, no diagnostic, and an unknown continuation. Official required CI is checked at delivery. A mathematical and statement-fidelity review by another agent of the same model family found no gap; an earlier Oracle authentication failure yielded no Pro review.

The source-owner escape registrations for newly exposed supporting theorems remain unfinished and are disclosed in #17689. That audit issue does not supply any mathematical premise or establish a registration verdict.

Proved in Lean for the stated sector. Official required CI remains part of delivery verification; its outcome is not inferred from the local build.

Mechanism. Both necessary boundary LMIs observe only nine extremal coordinates. Inserting or deleting an interior coordinate preserves that observation, allowing certificates at adjacent total dimensions to control a new tensor-factor sector. The proved spectral estimate therefore has weaker hypotheses than literal APPT, and the final theorem discharges those hypotheses from literal APPT.

Outside scope. The other $m=4$ sectors, all $m\geq5$, a classification of every maximizer, and APPT versus absolute separability are not settled by this target. No downstream source theorem outside the stated sector is claimed as formalized.

Worldwide novelty and absence of an independent post-source settlement are not kernel facts. The earlier Oracle request failed authentication and gave no Pro result. Source provenance is recorded in the existing library notes; the mathematical theorem uses the literal density and APPT predicates and does not assume a sufficient four-dimensional Hildebrand characterization.