slug: pathak-2026-kicked-ising-negativity-refutation bibkey: pathak2026mixedstate doi: 10.48550/arXiv.2603.14292 url: https://arxiv.org/abs/2603.14292v1 triage: theorem motivation_gids:

• D5/S3/Quantum/Dynamics/KickedIsingNegativityRefutation.result

Pathak’s generic-state negativity conjecture

Problem

T. Pathak, Mixed-State Entanglement in a Minimal Model of Quantum Chaos, arXiv:2603.14292v1, Conjecture 1, p. 3, states:

2𝓔(t) = I_{A:B}^{(α)}(t), hold for generic states at all times t.

The periodic kicked Ising chain has $J=\pi /4$, $b=-\pi /4$ and arbitrary real longitudinal fields, with $U=\exp (-i{H}_K)\exp (-i{H}_I)$. Its initial state is the source’s product of Bloch-angle qubit states. “Generic” excludes both the all-transverse class and the all-longitudinal class. The target in #13296 is the exact finite-chain equality at $\alpha =1/2$ for every chain length, field, generic product state, contiguous nonempty tripartition and integer time. Refuting this specialization refutes the equality asserted for all Rényi orders; it does not refute each order separately.

Motivation

D5/S3/Quantum/Dynamics/KickedIsingNegativityRefutation.result proves ¬ claim. Its definitions retain the literal Hamiltonians, matrix exponentials, product state, partial trace, partial transpose and trace norm. The result separates an exact finite-chain assertion from an asymptotic or approximate relation.

Gap

Tier 1: the source states a named conjecture. Issue #13296 reports inspection of the three citing papers arXiv:2606.02207, arXiv:2606.11311v3 and arXiv:2608.09695 without a settlement, and no MathDB or formal-conjectures record. These are preregistration-reported literature readings, not-found-in-searched-scope, rather than an exhaustive novelty certificate. The source statement and literal model are independently checked against the primary v1 source. Repository name and conclusion-shape searches do not find a frozen owner of this model-level refutation.

Route

Inside the settling theorem, take $L=4$, $A=\{0\}$, $B=\{1\}$, $C=\{2,3\}$, $t=1$, every field equal to one and every phase zero. The angles are $(\pi /2,\pi /2,2\arctan (1/2),2\arctan (1/2))$, giving $|+⟩|+⟩|r⟩|r⟩$, where $|r⟩=(2|0⟩+|1⟩)/\sqrt{5}$. The literal Floquet operator factors as $-W^{\otimes 4}G$, where $W$ is unitary and $G$ is the periodic controlled-Z product. Subsystem local unitaries preserve both entanglement measures. Finite matrix certificates determine the spectra before those unitaries, and injectivity of the real logarithm on positive arguments gives the contradiction.

Falsifier

The reduced state’s spectrum is $\{16/25,4/25,4/25,1/25\}$. Its partial transpose has spectrum $\{23/50,17/50,17/50,-7/50\}$; both marginals are half the identity. Consequently $2\mathcal{E}(1)=\log (1024/625)$ while $I_{A:B}^{(1/2)}(1)=\log (100/81)$. The rational arguments differ. The witness is a legal contiguous partition and a generic state at positive integer time.

Evidence

The frozen settling declaration is D5/S3/Quantum/Dynamics/KickedIsingNegativityRefutation.result. All spectral identities, the Floquet reduction and local-unitary invariance used here occur on its live proof path. The module has fourteen public definitions and one public theorem; no private theorem or lemma is exported. Its axiom closure is contained in {propext, Classical.choice, Quot.sound}. The Blueprint carries a Refuted open-problem resolution claim referencing this dossier and the settling declaration. The literature locator is Library/QuantumStates/pathak2026mixedstate.md.

Triage

Refuted through the external named open-problem admission basis of CLAUDE.md §3.2, preregistered in #13296. The settling result has proof_shape: bind-only, escape_witness: none and admission_basis: open-problem-resolution (#13296; Refuted). Its certified-instance utility is the refutation of claim. Information-escape registration is paused under CLAUDE.md §3.9.

What the settlement shows

Proved inside result: genericity alone does not enforce equality of negativity and half-order Rényi mutual information. In this legal four-site state, the negative partial-transpose eigenvalue gives trace norm $32/25$, while the reduced-state square-root trace is $9/5$. The two logarithmic expressions therefore differ despite maximally mixed marginals. The final local kick and field unitaries preserve this discrepancy.

Proved inside result: the witness lies outside both solvable classes, so it supplies no counterexample to assertions restricted to either class. The source’s solvable-class early-time derivations retain their stated hypotheses; those derivations are literature results, not additional conclusions of this module.

Computed, general-order obstruction: for the periodic chain at $L=6$, $A=\{0,1\}$, $B=\{2,3\}$, $C=\{4,5\}$, $t=1$, every field one, every phase zero and initial state $|r{⟩}^{\otimes 6}$, exact arithmetic gives $2\mathcal{E}(1)=2\log (41/25)=0.98939248367221410933\dots$ and $I_{A:B}^{(2)}(1)=2\log (625/497)=0.45832324727954554394\dots$. Their difference is $2\log (20377/15625)>0$, so the general-$\alpha$ equality fails at $\alpha =2$ in this one-kick early-regime instance. The computed reduced-state and marginal purities are all $247009/390625$; the partial-transpose trace norm is $41/25$. Open: extending this discrepancy to every block size $\ge 2$ is not established by this $2/2/2$ computation.

Computed, product-of-pairs coincidence: at $t=1$ the same $L=6$ state has $2\mathcal{E}(1)=I_{A:B}^{(1/2)}(1)=2\log (41/25)$ exactly. The square-root traces of both marginals and the joint reduced state are all $41/25$. For the second tested instance, $L=9$, blocks $3/3/3$, every field one and the source’s generic parameters ${\theta }_i={\phi }_i=1$, the two measures both give $1.07073215876048042531\dots$ at $t=1$. After removing block-local kick, field and internal controlled-Z gates, these one-kick instances are products of three boundary pairs; tracing $C$ leaves two local mixed factors and one pure pair shared by $A$ and $B$. The $L=9$ numerical difference is below $5\times 10^{-80}$ at 80 decimal digits of working precision. These are the two tested product-of-pairs instances; a uniform statement for untested states or block sizes remains open.

Computed, source-parameter late-time obstruction: for the periodic chain at $L=9$, $A=\{0,1,2\}$, $B=\{3,4,5\}$, $C=\{6,7,8\}$, every field one and ${\theta }_i={\phi }_i=1$, literal $U_K{U}_I$ evolution gives:

Thus equality fails at the two computed times $t=2,3$. Open: “every $t\ge 2$” is not established by this table.

These three computed items were independently recomputed by the codex-cli implementation seat. Their candidate setups are attributed to the search seat and Claude Code orchestrator in #13296. The checks use SymPy 1.14.0 for exact $L=6$ spectra, and mpmath 1.3.0 at 60 and 80 decimal digits for $L=9$; the displayed 50-significant-digit outputs agree between precisions. The half-order numerical entropy treats eigenvalues of magnitude at most $10^{-\mathrm{dps}+8}$ as zero; normalization errors are below $2\times 10^{-80}$ at 80 digits. The calculations are computed evidence, not additional Lean proofs. The script computes partial traces and partial-transpose spectra from the states; the exact one-kick check uses their periodic controlled-Z representative, whose omitted field and kick factors are local unitaries.

Script: /Users/auric/.sshx/6655bf963c7b2d7d963c5f20/attempt-1/triage-compute.py; SHA-256: 61cf4809591718e2ab448bde137ba261c20c85da78b2f8cb98c226add7b3fa66.

Open in this module: the nearest unresolved extension is the half-order relation for generic states in the thermodynamic early-time regime. No claim about that relation follows from this finite-chain witness. No inference in the source relying on the unrestricted Conjecture 1 can use that exact equality universally. Its separate approximate, typical-state, Haar-limit and early-time claims require their own hypotheses and evidence.

ASSUMED-UNVERIFIED

The bounded literature search does not establish worldwide priority. The interpretation of the source’s tensor-product notation as the finite matrix model is explained by the public literal definitions and reviewed for fidelity; no separate infinite-chain or thermodynamic-limit construction is formalized. The result refutes the exact finite-chain half-order identity, not an asymptotic, approximate, large-block or typical-state reformulation.