Singlet padding refutes the universal precession GME threshold

Khoi-Nguyen Huynh-Vu, Lin Htoo Zaw and Valerio Scarani, Certification of genuine multipartite entanglement in spin ensembles with measurements of total angular momentum, Phys. Rev. A 109, 042402, arXiv:2311.00806v2, Conjecture 3, PDF p. 9:

Consider a spin ensemble. Perform the precession protocol with odd $K \geq 3$ on the total angular momentum of the system. If the score $P_K > \mathbf{P}_K^{\mathrm{conj}}$ is obtained, then the spin ensemble is GME.

Here $J_k=\cos(2\pi k/K)J_x+\sin(2\pi k/K)J_y$ is the sum of the particle observables, $Q_K=K^{-1}\sum_k\operatorname{pos}(J_k)$ and $P_K=\operatorname{tr}(\rho Q_K)$. The spectral weight is $(1+\operatorname{sgn}m)/2$, including one half at zero. Separability across a cut means a normalized finite convex mixture of products of density matrices. GME excludes convex mixtures over all nontrivial cuts. Preregistration: #13586.

D5/S3/Quantum/Entanglement/PrecessionUniversalThresholdRefutation.result proves the negation of the literal universal claim. Its state has constituent spins ${1/2,1/2,3/2}$: a normalized singlet projector on the pair tensored with the normalized projector onto the difference of the spin-$3/2$ endpoint basis vectors. It is a product across the pair–rest cut, yet scores $3/4>23/32$, with exact gap $1/32$. The source anchor is Library/QuantumStates/huynhvu2024universal.md.

The paper proposes a composition-independent GME threshold. The preregistration classifies the question as Tier 1 and records literature checks of arXiv:2411.03132v3, 2509.03166v4 and 2608.21217v2, with 2405.17966 and 2401.16147 also inspected, and no correction found in that searched scope. The APS full text was inaccessible in those checks; this dossier uses the verified arXiv v2 source and makes no priority claim.

The reduction to effective spins on each side of a cut must retain the spin-zero representation. A singlet-capable side has effective spin zero, although every constituent particle has strictly positive spin.

For the singlet $s$, the pair's total $J_x$ and $J_y$ annihilate $s$. Consequently the embedding $V\varphi=s\otimes\varphi$ intertwines the padded total observable and the rest observable. Finite Hermitian spectral calculus respects this intertwining. Thus, for every finite Hermitian rest family $B_k$ and every matrix $\sigma$,

$$ \operatorname{tr}\left[(|s\rangle\langle s|\otimes\sigma) \frac1K\sum_k\operatorname{pos}(P_k\otimes I+I\otimes B_k)\right] =\operatorname{tr}\left[\sigma\frac1K\sum_k\operatorname{pos}(B_k)\right] $$

whenever $P_ks=0$ and the padded observables are Hermitian. The private singlet_padding_ensemble_score identifies this family with the literal ensembleJ and ensembleQ of every spin list with two spin-1/2 particles prepended. It proves score equality for every matrix $\sigma$ and every $K$, hence for every state and every odd $K\ge3$ in the source protocol. Its coordinate equivalence only splits a padded configuration into the pair and the remaining configuration. The witness score uses this bridge on padded_precession_score → result's live path. No continuity of the sign function is assumed: the Hermitian matrices have finite spectra. pair_rest_product supplies density normalization and the non-GME cut.

The witness obeys the source's positive constituent-spin, odd-$K$, normalization and nontrivial-cut requirements. The score is the source's spectral protocol, rather than a substitute certificate. The universal claim would require this witness to be GME, contradicting its explicit one-term pair–rest decomposition.

The settling declaration is result : ¬ claim. Its only public companions are the definitions needed to state that claim. All private helpers are consumed in the refutation. The settling result and helpers have proof_shape: bind-only; the module's admission basis is open-problem-resolution (#13586; Refuted).

Registration is paused under CLAUDE.md §3.9.

Reproducible finite computation

The following command directly diagonalizes every full tensor-product observable for $K=3,5,7$, both for the GHZ state and for its singlet-padded state. Command exit code: 0. Script SHA-256: e038954f9d1e979d393cc05e57f990ad0def5144e30135655786c449751ec14c (the bytes between the heredoc delimiter lines, including the final newline).

OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 python3 - <<'PY'
import json, math
import numpy as np

def total_xy(N):
    d = 2**N
    X = np.zeros((d, d), complex)
    Y = np.zeros((d, d), complex)
    for col in range(d):
        for n in range(N):
            bit = 1 << n
            row = col ^ bit
            X[row, col] += 0.5
            Y[row, col] += 0.5j if row & bit else -0.5j
    return X, Y

def score(K, X, Y, state):
    answer = 0.0
    for k in range(K):
        angle = 2*math.pi*k/K
        values, vectors = np.linalg.eigh(math.cos(angle)*X + math.sin(angle)*Y)
        weights = np.where(values > 1e-9, 1.0, np.where(values < -1e-9, 0.0, 0.5))
        answer += float(np.sum(weights*np.abs(vectors.conj().T @ state)**2))/K
    return answer

rows = []
singlet = np.array([0, 1, -1, 0], complex)/math.sqrt(2)
PX, PY = total_xy(2)
for K in (3, 5, 7):
    X, Y = total_xy(K)
    ghz = np.zeros(2**K, complex)
    ghz[0], ghz[-1] = 1/math.sqrt(2), (-1)**((K-1)//2)/math.sqrt(2)
    padded = np.kron(singlet, ghz)
    TX = np.kron(PX, np.eye(2**K)) + np.kron(np.eye(4), X)
    TY = np.kron(PY, np.eye(2**K)) + np.kron(np.eye(4), Y)
    c = math.comb(K-1, (K-1)//2)/2**(K-1)
    threshold = 23/32 if K == 3 else (69+math.sqrt(181))/128 if K == 5 else (1+c*(K-1)/(K+1))/2
    a, b = score(K, X, Y, ghz), score(K, TX, TY, padded)
    assert abs(a-(1+c)/2) < 1e-10 and abs(a-b) < 1e-10
    assert b > threshold+1e-11 and (K+2)/2 <= 15
    rows.append(dict(K=K, dimension=2**(K+2), ghz_score=a, padded_score=b, threshold=threshold, gap=b-threshold, total_spin=(K+2)/2))
print(json.dumps(rows, indent=2))
PY

| $K$ | Padded dimension | GHZ score | Padded score | Threshold | Gap | $\sum j_n$ | | --- | --- | --- | --- | --- | --- | --- | | 3 | 32 | 0.7499999999999994 | 0.7499999999999997 | 0.71875 | 0.031249999999999667 | 2.5 | | 5 | 128 | 0.6874999999999999 | 0.6874999999999998 | 0.6441689378677633 | 0.043331062132236475 | 3.5 | | 7 | 512 | 0.6562499999999999 | 0.6562499999999999 | 0.6171875 | 0.03906249999999989 | 4.5 |

Eigenvalues within $10^{-9}$ of zero receive half weight. Formula and padding agreement are checked to $10^{-10}$. These floating-point computations support only these three values; they are not kernel proofs of Result 4's negation or of the uniform family.

What the settlement shows

• Proved in this module — $K=3$ refutation. The ${1/2,1/2,3/2}$ ensemble has a density matrix separable across a nontrivial cut, exact score $3/4$ and threshold $23/32$. result refutes Conjecture 3.
• Proved on paper; computed for $K=3,5,7$ — the GHZ family for every odd $K\ge3$. Let $L=(K-1)/2$, $c_K=2^{-(K-1)}\binom{K-1}{L}$ and $g_K=(|\uparrow^K\rangle+(-1)^L|\downarrow^K\rangle)/\sqrt2$. In the two-endpoint subspace, the diagonal entries of the positive spectral weight are $1/2$. Its off-diagonal entry is $2^{-K}\sum_{r=0}^{L}(-1)^r\binom Kr=(-1)^Lc_K/2$, using $\sum_{r=0}^{L}(-1)^r\binom Kr=(-1)^L\binom{K-1}{L}$. Precession multiplies this entry by $e^{iK(2\pi k/K)}=1$. Hence $P_K(g_K)=(1+c_K)/2$. Singlet padding preserves the score and produces a non-GME state. For $K=3$ the gap is $1/32$; for $K=5$ it is $(19-\sqrt{181})/128>0$; for $K\ge7$ it is $c_K/(K+1)>0$. The inline computation above verifies $K=3,5,7$, exit 0, SHA-256 e038954f9d1e979d393cc05e57f990ad0def5144e30135655786c449751ec14c. The all-odd family is not a separately kernel-checked theorem in this module; its Lean extension remains open.
• Proved in Lean — the arbitrary-ensemble score-preserving mechanism for equality of the unrestricted threshold and quantum supremum. singlet_padding_ensemble_score preserves the literal score of every remaining ensemble and every state, and pair_singlet_annihilation covers every linear combination of the pair's two angular-momentum components. Define $S_K^{\rm q}$ as the scores over all nonempty finite positive-spin ensembles and all their states, and $S_K^{\rm bisep}$ with states restricted to non-GME ones. Padding realizes every member of $S_K^{\rm q}$ in $S_K^{\rm bisep}$; the reverse inclusion follows from the definitions. Therefore $S_K^{\rm q}=S_K^{\rm bisep}$ and their suprema agree. This argument allows the ensemble to change and does not assert a fixed-ensemble equality. The arbitrary-ensemble operator identification and score equality are kernel checked on result's live path. The score-set argument above gives the equal-supremum consequence; no separate Lean declaration of the score sets or their suprema is exported.
• Computed — Result 4 is contradicted within its stated range. Result 4 requires odd $3\le K\le21$, $\sum j_n\le15$, and $P_K>\mathbf P_K^{\rm conj}+10^{-11}$. The computed $K=3,5,7$ ensembles satisfy every range condition, with $K+2$ spin-$1/2$ particles and $\sum j_n=(K+2)/2$. The inline script asserts the strict comparison for all three rows, exit 0, SHA-256 e038954f9d1e979d393cc05e57f990ad0def5144e30135655786c449751ec14c. The state is a pair–rest product by construction. No extension to other numerical values is asserted.
• Proved in the source or existing repository — what survives. Result 1 assumes the fixed constituent-spin sum $\sum j_n=K/2$ and uses threshold $\frac12[1+2^{-K}\binom{K-1}{(K-1)/2}]$. A singlet-bearing side leaves the rest with maximum effective spin strictly below $K/2$, where the precession score is $1/2$; this padding family has sum $(K+2)/2$, outside Result 1's hypothesis. Result 1 is a source result, not newly proved here. The existing frozen PrecessionSpinOneSeparableBound.result (#11873) proves the ${1,K/2}$ product-pure separable bound for odd $K\ge7$. The counterexample uses an effective spin-zero side and contradicts neither that theorem nor its hypotheses. Any universal maximization deduced solely from the positive-effective-spin cases must be revised; the individual positive-spin bounds survive.
• Open — corrected universal statement. Exclude spin-zero sectors on both sides of every nontrivial bipartition, or restrict the allowed ensembles to cuts admitting no singlet subspace. These restrictions remove this padding counterexample. They do not prove that the printed threshold then certifies GME; that corrected implication remains open. Positive constituent spins alone do not supply the needed restriction.

The journal full text is not independently verified; the statement and ranges use arXiv:2311.00806v2. Numerical diagonalization is floating point, with the tolerances and tested scope stated above. No model-family independence or literature-priority claim is made.