slug: volkoff-2016-pair-tunneling-ansatz-ground-state-refutation
bibkey: volkoff2016pairtunnelling
doi: null
url: https://arxiv.org/abs/1610.05807v1
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Dynamics/PairTunnelingAnsatzGroundStateRefutation.result

Volkoff’s exact-ground-state pair-tunnelling ansatz

Problem

T. J. Volkoff, arXiv:1610.05807v1, Section V.A, Eq. (20), pp. 6–7, fixes even particle number and writes:

We conjecture that for each N ≥ 4 there exists a value c_N for which |ω_+(c_N)⟩ or |ω_−(c_N)⟩ is the exact ground state.

The sector has basis $|N-k,k⟩$, indexed by $k\in \mathrm{Fin}(N+1)$, and Hamiltonian $H=a_0^{†2}{a}_1^2+a_1^{†2}{a}_0^2$. With $M=N/2$ and real $c$, the unnormalized ansatz is the coefficient vector of $(x^2+2icxy-y^2{)}^M\pm (x^2-2icxy-y^2{)}^M$, multiplied coordinatewise by $\sqrt{(N-k)!k!}$. Normalization is irrelevant for a nonzero eigenvector. An exact ground state has the least real eigenvalue among all nonzero eigenvectors in the full sector. Issue #11667 preregisters this quantified reading and its refutation at $N=8$.

Motivation

The frozen declaration D5/S3/Quantum/Dynamics/PairTunnelingAnsatzGroundStateRefutation.result refutes the assertion that this one-parameter family contains a ground state at every even $N\ge 4$. It separates exact eigenstate representation from the paper’s numerical high fidelity and metrological usefulness.

Gap

Tier 1: an explicitly stated conjecture of the 2016 source, with the bounded literature check in #11667. That check reports only arXiv v1, the four-mode follow-up arXiv:1901.00879 and the exact-solution paper arXiv:1609.05581 without a settlement of this ansatz conjecture, and no relevant formal-conjectures or repository hit. These are preregistration-reported readings, not-found-in-searched-scope; they do not establish exhaustive worldwide novelty or priority. The APS cited-by coverage is incomplete.

Route

The proof specializes the universal conjecture to $N=8$, expands the fourth powers, and uses strictly positive factorial weights to conjugate matrix action to coefficient action. The off-diagonal entries follow from two successive annihilations in one mode and two creations in the other: $\sqrt{k(k-1)}\sqrt{(N-k+1)(N-k+2)}$; shifting $k$ by two gives $\sqrt{(N-k-1)(N-k)(k+1)(k+2)}$. The adjoint supplies the reverse entry.

For the sum state, writing $t=c^2$, the eigen-equations at coordinates 0, 2 and 4 force $e=-48t-8$, $10t^2-2t-1=0$ and $12t^3+38t^2-12t-3=0$. Elimination forces $68t-26=0$, hence $t=13/34$, where the quadratic equals $-175/578$. Thus the sum state is not an eigenvector for any real parameter.

For the difference state, $c=0$ is the zero vector. Otherwise its eigen-equations force $e=-24t-18$ and $6t^2+4t-3=0$. With $t\ge 0$ this implies $t<9/20$, hence $e>-144/5$. The coefficient vector $(1,0,-4\sqrt{13},0,30,0,-4\sqrt{13},0,1)$, multiplied by the factorial weights, is a nonzero eigenvector at $-8\sqrt{13}<-144/5$. The proof also eliminates the even and odd three-term chains. Their annihilating polynomials are $\lambda ({\lambda }^2-832)({\lambda }^2-112)$ and ${\lambda }^4-904{\lambda }^2+63504$. If $\lambda <-8\sqrt{13}$, then ${\lambda }^2>832$; both polynomials are nonzero, forcing the two endpoint coordinates, and then all coordinates, to vanish. Thus every real eigenvalue is at least $-8\sqrt{13}$, and the displayed eigenvector attains this bound. The kernel-certified ground energy is exactly $-8\sqrt{13}$. A ground-state ansatz must attain that energy, which the difference state cannot do.

Falsifier

A mismatch between the polynomial coefficients, factorial weights, matrix entries or the source’s sector and sign conventions would invalidate source fidelity. The displayed formulas bind all parameters, preserve the natural-to-real casts before each square root, and use the full sector rather than a parity-restricted definition of ground state.

Evidence

Canonical Lean source: D5/S3/Quantum/Dynamics/PairTunnelingAnsatzGroundStateRefutation.lean. The authored public surface is exactly ansatzFactor, ansatzPolynomial, omega, pairTunnel, IsGroundState, claim and the one theorem result : ¬ claim. All polynomial identities, the universal real-eigenvalue lower bound and its attainment occur within the proof of result. The bound is used to identify the energy that a ground-state ansatz must attain. No authored private theorem, new axiom or native_decide is introduced. Lean also generates Nat.factorial.eq_def and a private factorial splitter when unfolding factorial; these are upstream equation auxiliaries rather than authored companion declarations.

The following independent SymPy computation constructs the same matrix, computes its spectrum, and checks the surviving $N=4,6$ sum states. The ordering of eigenvalues uses numerical sorting; the full spectrum list and the positive $N=4,6$ cases are computed readings. The $N=8$ spectral minimum is independently proved within result by the lower bound and its attainment.

uv run --with sympy python - <<'PY'
import sympy as s
x,y,c=s.symbols('x y c',real=True)
for N,t in [(4,(s.sqrt(3)-1)/2),(6,s.sqrt(6)/6),(8,None)]:
 H=s.zeros(N+1)
 for k in range(N-1): H[k,k+2]=H[k+2,k]=s.sqrt((N-k-1)*(N-k)*(k+1)*(k+2))
 spectrum=sorted(H.eigenvals().items(),key=lambda q:float(q[0]))
 print('N=',N,'spectrum=',spectrum,'minimum=',spectrum[0][0])
 if t is not None:
  P=s.Poly((x*x+2*s.I*c*x*y-y*y)**(N//2)+(x*x-2*s.I*c*x*y-y*y)**(N//2),x,y)
  v=s.Matrix([s.sqrt(s.factorial(N-k)*s.factorial(k))*P.coeff_monomial(x**(N-k)*y**k) for k in range(N+1)]).subs(c,s.sqrt(t))
  residual=(H*v-spectrum[0][0]*v).applyfunc(s.simplify)
  assert residual==s.zeros(N+1,1) and v!=s.zeros(N+1,1)
  print('c^2=',t,'nonzero=True; ground_eigen_residual=0')
PY

Computed readings: for $N=4$, the minimum is $-4\sqrt{3}$ and the sum state at $c^2=(\sqrt{3}-1)/2$ is nonzero with zero eigen-equation residual; for $N=6$, the minimum is $-6-4\sqrt{6}$ and the sum state at $c^2=\sqrt{6}/6$ is nonzero with zero residual. For $N=8$, the computed spectrum is $\{0,\pm 8\sqrt{13},\pm 4\sqrt{7},\pm (10+4\sqrt{22}),\pm (10-4\sqrt{22})\}$. The minimum $-8\sqrt{13}$ is kernel-certified within result.

Triage

theorem; resolution Refuted, under #11667. The settling theorem has proof_shape: bind-only, escape_witness: none, and admission_basis: open-problem-resolution. Utility is certified-instance with the explicit refutes edge from result to claim. There is no atom or digestion coverage edge. Information-escape registration is paused under CLAUDE.md §3.9.

What the settlement shows

• Proved within result: at $N=8$, the sum branch fails the eigen-equation itself, while the exact full-sector spectral minimum is $-8\sqrt{13}$ and a nonzero difference-branch eigenstate has strictly greater energy. Even particle number and an adjustable real parameter do not suffice for exact ground-state representation.
• Computed with the Evidence command: the specified $N=4,6$ sum states survive, and $N=8$ is the first failing even sector within $4\le N\le 8$. These positive cases are not new Lean results.
• Open: classification of all higher even sectors and a corrected exact-state ansatz with additional parameters.
• Open boundary for the source’s other results: the universal exactness conjecture cannot justify exact ground-state identification for this family. The refutation does not decide the source’s numerical fidelity, near-optimal QFI, asymptotic estimates, or independent results in other sections.

ASSUMED-UNVERIFIED

The bounded literature search is not exhaustive. The supplied search-seat APS cited-by count has incomplete interface coverage. Correspondence of the creation/annihilation action to the finite matrix is explained from the source, without a separate formal Fock-space operator construction. The full $N=8$ spectrum list and the positive $N=4,6$ cases are independent computed readings. The $N=8$ spectral minimum is proved inside the one exported settling theorem, whose public conclusion is the negation of the quantified source conjecture.