Understanding the nature of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="normal">Δ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1600</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> re

We present a coupled-channel analysis of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>J</a:mi><a:mi>P</a:mi></a:msup><a:mo>=</a:mo><a:mn>3</a:mn><a:mo>/</a:mo><a:msup><a:mn>2</a:mn><a:mo>+</a:mo></a:msup><a:mi mathvariant="normal">Δ</a:mi></a:math>-baryon spectrum, based in the framework of Hamiltonian effective field theory (HEFT). We construct a Hamiltonian which mixes quark model-like single-particle states and two-particle meson-baryon channels, and constrain this via phase shifts and inelasticities derived from <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>π</d:mi><d:mi>N</d:mi><d:mo stretchy="false">→</d:mo><d:mi>π</d:mi><d:mi>N</d:mi></d:math> scattering data. In the same vein as Lüscher’s approach, we then connect this infinite-volume inspired Hamiltonian with finite-volume lattice quantum chromodynamics (QCD) results. Drawing on lattice correlation-matrix eigenvectors identifying the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mn>1</g:mn><g:mi>s</g:mi></g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mn>2</i:mn><i:mi>s</i:mi></i:math> states in the finite-volume <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi mathvariant="normal">Δ</k:mi><k:mo stretchy="false">(</k:mo><k:mn>3</k:mn><k:mo>/</k:mo><k:msup><k:mn>2</k:mn><k:mo>+</k:mo></k:msup><k:mo stretchy="false">)</k:mo></k:math> spectrum, and utilizing the HEFT eigenvectors describing the composition of the energy eigenstates, we resolve the structure of these states and their relation to the <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi mathvariant="normal">Δ</p:mi><p:mo stretchy="false">(</p:mo><p:mn>1600</p:mn><p:mo stretchy="false">)</p:mo></p:math> resonance. We find the dominant contributions to this resonance come from strong rescattering in the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>π</u:mi><u:mi>N</u:mi></u:math> and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>π</w:mi><w:mi mathvariant="normal">Δ</w:mi></w:math> channels. This contrasts the long-held view of a dominant quark model-like core for the <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi mathvariant="normal">Δ</z:mi><z:mo stretchy="false">(</z:mo><z:mn>1600</z:mn><z:mo stretchy="false">)</z:mo></z:math>. Further discussion of other contemporary lattice results for the <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi mathvariant="normal">Δ</eb:mi></eb:math> spectrum and <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:mi>π</hb:mi><hb:mi>N</hb:mi></hb:math> scattering states is also presented.