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Math-2897 · mathematics · Probability

An isolated lightest particle in the continuum O(3) spin field

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For the canonical massive continuum O(3) spin field constructed from the planar lattice model, does the two-point spectral measure have the form $\rho =Z$ $\delta _{m_1^2}+\rho _rest$, with $m_1,Z,\delta >0$ and $supp(\rho _rest)$ contained in $[(m_1+\delta )^2,infinity)$?

Short English statement

For the canonical massive continuum O(3) spin field constructed from the planar lattice model, does the two-point spectral measure have the form $\rho =Z$ $\delta _{m_1^2}+\rho _rest$, with $m_1,Z,\delta >0$ and $supp(\rho _rest)$ contained in $[(m_1+\delta )^2,infinity)$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2897-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2897

The continuum $O(3)$ spin field from the companion construction is claimed to have a lowest spectral mass represented by a positive atom, separated from further support. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

An isolated lightest particle in the continuum O$3$ spin field

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The continuum $O(3)$ spin field from the companion construction is claimed to have a lowest spectral mass represented by a positive atom, separated from further support.

Mathematical statement of the claimed result

Result 1.
Let $\phi=(\phi^1,\phi^2,\phi^3)$ be the Hermitian continuum spin field constructed from the two-dimensional nearest-neighbour $O(3)$ lattice model, using the fixed spacing and normalization of the companion construction. Its reconstructed spacetime has dimension $1+1$. With $\Delta_+(x;\mu^2)$ the positive-frequency scalar two-point distribution of mass $\mu$, write its spectral representation as \[ W_2^{ij}(x)=\delta_{ij}\int_{[0,\infty)}\Delta_+(x;\mu^2)\,\rho(d\mu^2). \] The new assertion identifies constants $m_1,Z,\delta>0$ and a positive measure $\rho_{\mathrm{rest}}$ such that \[ \rho=Z\delta_{m_1^2}+\rho_{\mathrm{rest}},\qquad \operatorname{supp}\rho_{\mathrm{rest}}\subset[(m_1+\delta)^2,\infty). \] Thus the lightest mass coupled to the spin field is an isolated atom, rather than merely a positive lower spectral edge.

Outline of the argument

Uniform finite-volume bounds from the massive construction are applied to the vector two-point spectral measure. Transfer and spectral comparisons distinguish a true isolated mass contribution from a merely positive continuous threshold, then preserve its positive residue in the continuum limit.

Formal verification

For family 215, the selected theorem gives exponential finite-free-graph $O(n)$ spin decay for $n\geq3$ and bounded ferromagnetic edge strengths. Plane limits, full $O(4)$ spectral gaps, and continuum $O(3)$ results are beyond that entry. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

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