The Haldane gap conjecture for the spin-one Heisenberg antiferromagnetic chain

Claims that the pure bilinear spin-one Heisenberg chain on even periodic rings has a unique ground state for $L\ge60$ and a gap bounded below by explicit constants, so $\Delta_1\ge\log 20/784>0$ , with a corresponding lower bound for local excitations in subsequential thermodynamic limits. The companion proves Tasaki's boundary-field gap for odd open chains at $h=3/5$ and the nontrivial index $-1$ . The constants are far below the numerical value of about $0.41$ . It does NOT treat odd periodic rings, spin above one, or the bilinear-biquadratic family away from the Heisenberg point.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Haldane gap conjecture for the spin-one Heisenberg antiferromagnetic chain · Spin-one Haldane gap

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Unreleased internal OpenAI model
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Named evidence

VibeMathed record: The Haldane gap conjecture for the spin-one Heisenberg antiferromagnetic chain

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No independent mathematician has checked this yet. Theorem 1.1 was read against the posed problem: it claims a unique ground state and $\gamma_L>\frac4{105}\log\frac{80}{79}$ for every even $L\ge60$ , and $\gamma_L>\log 20/784$ for even $L\ge2304$ , so the even-periodic gap is uniformly positive. The manuscript calls this the positive even-periodic formulation; odd periodic rings are not covered. The proof rests on finite rigorous computations (interval enclosures of matrix-exponential traces and explicit integer matrix-product trial states) fed into an analytic bootstrap; the verifier was not run here. No Lean formalization. The companion was read at the level of its main theorem: a gap above $\log 10/392$ for odd open chains of $2L+1$ sites, $L\ge960$ , with endpoint field $h=3/5$ , which is Tasaki's Assumption 2 and gives Tasaki index $-1$ .

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The Haldane gap conjecture for the spin-one Heisenberg antiferromagnetic chain

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Companion: A boundary-field gap for the spin-one Heisenberg chain (Tasaki's Assumption 2, index -1)

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paper link supplied by the VibeMathed registry.

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Finite-certificate verifier for the periodic gap

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Haldane (1983), Phys. Rev. Lett. 50, 1153

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Lineage and corrections

Finite-certificate verifier for the periodic gap evidence for this event

Companion: A boundary-field gap for the spin-one Heisenberg chain (Tasaki's Assumption 2, index -1) evidence for this event

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The Haldane gap conjecture for the spin-one Heisenberg antiferromagnetic chain evidence for this event

VibeMathed record: The Haldane gap conjecture for the spin-one Heisenberg antiferromagnetic chain evidence for this event

Haldane (1983), Phys. Rev. Lett. 50, 1153 evidence for this event

The Haldane gap conjecture for the spin-one Heisenberg antiferromagnetic chain parent of this event

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