{ "schema_version": 1, "id": "PHYS-2026-08-14-01", "canonical_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-08-14-01/", "source_markdown_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-08-14-01.md", "metadata": { "schema_version": 1, "id": "PHYS-2026-08-14-01", "date": "2026-08-14", "updated_at": "2026-08-20", "title": "How Degrees of Freedom Survive Dimensional Reduction: Universal Clusters from Shielded Molecules to Species-Selective Traps", "summary": "Technical reconstruction of two 2026 preprints: microwave shielding generates an effective 1D molecular potential and Bose–Fermi duality, while species-selective confinement converts COM coupling and finite Feshbach range into q1D/q2D scattering parameters and STM cluster spectra.", "language": "en", "entry_kind": "daily", "status": "published", "level": "research", "user_difficulty": "unrated", "domains": [ "condensed-matter", "quantum-theory", "statistical-mechanics", "mathematical-physics" ], "estimated_minutes": 240 }, "content_markdown": "\n## The claim under pressure\n\nDimensional reduction preserves a record of every coordinate that has been removed. In the two papers studied here, that record takes two different forms. Angular fluctuations around the attractive axis of a microwave-dressed dipolar potential contribute a distance-dependent zero-point energy, producing a short-range core in an effective one-dimensional potential. Transverse oscillator modes in a species-selective trap contribute virtual closed channels, producing energy-dependent one- and two-dimensional scattering amplitudes. Neither calculation amounts to deleting coordinates from a Hamiltonian. Each calculation integrates out motion and retains its low-energy consequences.\n\nThe distinction determines the meaning of every few-body result that follows. Tingting Shi, Haitian Wang, and Xiaoling Cui derive an effective one-dimensional description for microwave-shielded polar molecules in free three-dimensional space. They benchmark the reduction against exact three-dimensional two-molecule calculations and against a three-dimensional Born–Oppenheimer problem, then predict bound states of three identical molecules with an effective two-coordinate Hamiltonian. Tingting Shi and Xiaoling Cui solve a different reduction problem for heteronuclear atoms in quasi-one- and quasi-two-dimensional traps. They calculate the confined two-body $T$ matrix with center-of-mass coupling and finite Feshbach range, extract low-dimensional scattering parameters, and insert those parameters into Skorniakov–Ter-Martirosian equations for trimers and tetramers.\n\nBoth studies use the word *universal*. Here it means that a low-energy state can be specified without a microscopic chemical potential curve or an independent short-range few-body parameter. It does not mean parameter-free. The molecular problem retains the dipolar length, microwave coupling, detuning, and polarization. The atomic problem retains the three-dimensional scattering length, resonance range, mass ratio, confinement lengths, and frequency ratio. Universality compresses microscopic information into a controlled set of scales.\n\n\"Two\n\n*Figure 1. Two reductions with the same logical structure. The left branch converts angular motion in a three-dimensional anisotropic interaction into a one-dimensional potential. The right branch converts virtual transverse and center-of-mass excitations into low-dimensional scattering parameters. All diagrams in this entry were reconstructed from the published equations and data.*\n\n## 1. The objects, the versions, and the naming problem\n\nThe first source is Shi, Wang, and Cui, [“Universal Bound States with Bose-Fermi Duality in Microwave-Shielded Ultracold Molecules”](https://arxiv.org/abs/2504.21535), read here in arXiv v4 dated January 28, 2026. Its elementary objects are diatomic molecules. A bound pair of those molecules contains four atoms and is called a tetratomic molecule; a bound state of three molecules contains six atoms and is called a hexatomic molecule. I will usually call them the two-molecule and three-molecule states because that language keeps the few-body counting visible.\n\nThe second source is Shi and Cui, [“Effective scatterings and universal clusters of heteronuclear ultracold mixtures in quasi-low dimensions”](https://arxiv.org/abs/2606.02988), read in arXiv v1 dated June 3, 2026. Its elementary objects are atoms. One light atom with two identical heavy fermions is a trimer, while one light atom with three heavy fermions is a tetramer. The two papers therefore use the word *tetramer* at different microscopic levels.\n\nThe statistics also enter in different ways. The first paper compares identical bosonic and fermionic molecules and finds an emergent Girardeau-type mapping after the molecules become effectively impenetrable along one axis. The second paper keeps the heavy atoms fermionic from the start; antisymmetry appears explicitly in the exchange terms of the few-body integral equations. Conflating these mechanisms would hide the strongest conceptual result of each paper.\n\n## 2. A scale test for any claim of effective low dimensionality\n\nSuppose a system contains a slow energy scale $E_{\\parallel}$ and an eliminated mode with gap $\\Delta_{\\perp}$. A single-channel low-dimensional description needs a hierarchy such as\n\n$$\nk_B T,\\quad |E_b|,\\quad E_{\\mathrm{collision}}\n\\ll \\Delta_{\\perp}.\n$$\n\nThe hierarchy suppresses *real* occupation of the high-energy mode. Virtual occupation remains. If the interaction couples the low and high sectors, a Feshbach projection gives\n\n$$\nH_{\\mathrm{eff}}(E)\n=PHP+PHQ\\frac{1}{E-QHQ}QHP,\n$$\n\nwhere $P$ projects onto the low sector and $Q=1-P$. The second term depends on the discarded subspace. It can generate an induced repulsion, an energy-dependent coupling, a resonance, or a nonlocal operator. The low-dimensional model earns its status only after those contributions have been included and the result has been checked against the higher-dimensional problem.\n\nThe molecular paper realizes this formula in an adiabatic coordinate-space language. The slow coordinate is the separation along the attractive $y$ axis, while the fast coordinates are small angular excursions from that axis. The atomic paper realizes the same logic in scattering language. The open channel contains two transverse ground states; excited transverse and molecular center-of-mass levels form the closed sector. The first calculation retains a position-dependent zero-point energy. The second retains a threshold expansion of the $T$ matrix.\n\n## 3. Experimental and theoretical setting\n\nMicrowave shielding has already changed the practical status of ultracold polar molecules. A near-resonant microwave field dresses rotational states and creates a long-range repulsive barrier that prevents colliding molecules from reaching chemically reactive separations. The original shielding proposal and coupled-channel analysis are given by Karman and Hutson in [Physical Review Letters 121, 163401](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.163401). Experiments have used the method to reach degenerate molecular gases, and field-linked resonances have provided an outer, anisotropic binding well on the safe side of the shield.\n\nThe most direct experimental precursor to the first paper is Chen and collaborators’ [observation of ultracold field-linked tetratomic molecules](https://www.nature.com/articles/s41586-023-06986-6). That experiment associated roughly $1.1\\times10^3$ weakly bound NaK tetramers, measured a maximum free-space lifetime of $8(2)\\,\\mathrm{ms}$, and reconstructed an anisotropic momentum-space wavefunction. Those results establish that a long-range molecular complex can be prepared and imaged. They do not establish the three-molecule state predicted by Shi, Wang, and Cui.\n\nConfinement-induced resonances provide the corresponding background for the second paper. Olshanii showed that a three-dimensional scattering length and a transverse oscillator length combine into an effective one-dimensional coupling in [Physical Review Letters 81, 938](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.81.938). Bergeman, Moore, and Olshanii later demonstrated that the resonance can be viewed as a Feshbach crossing between an open transverse channel and a bound state in closed transverse modes in [Physical Review Letters 91, 163201](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.91.163201). Species-dependent confinement complicates that picture because the pair center of mass ceases to be a spectator.\n\nThe two 2026 papers push these established mechanisms into few-body territory. Their strongest contribution lies in the intermediate step between the microscopic Hamiltonian and the cluster spectrum. Control over this intermediate step separates benchmarked predictions from untested extrapolations.\n\n---\n\n## 4. Paper I: the dressed two-molecule interaction\n\n### 4.1 Polarization convention and the long-range potential\n\nShi, Wang, and Cui use a microwave field\n\n$$\n\\mathbf E\n=E e^{\\mathrm i(kz-\\omega t)}\n\\left(\\mathbf e_+\\cos\\xi+\\mathbf e_-\\sin\\xi\\right)\n+\\mathrm{c.c.},\n$$\n\nwith $\\mathbf e_\\pm=\\mp(\\mathbf e_x\\pm\\mathrm i\\mathbf e_y)$. Their main calculation takes $\\xi=\\pi/4$, which is linear polarization along $y$. They adopt an analytic dressed interaction previously derived for elliptic polarization and extrapolate it to this endpoint. At relative detuning $\\delta_r=|\\delta|/\\Omega=0.2$, the potential is\n\n$$\n\\begin{aligned}\nV(\\mathbf r)=&\\frac{C_3}{r^3}\n\\left[3\\cos^2\\theta-1+3\\sin^2\\theta\\cos(2\\phi)\\right]\\\\\n&+\\frac{C_6}{r^6}\n\\left[\\sin^2\\theta\\sin^2(2\\phi)\n+\\sin^2(2\\theta)\\sin^4\\phi\\right],\n\\end{aligned}\n$$\n\nwhere\n\n$$\nC_3=\\frac{d^2}{48\\pi\\varepsilon_0(1+\\delta_r^2)},\n\\qquad\nC_6=\\frac{d^4}\n{128\\pi^2\\varepsilon_0^2\\Omega(1+\\delta_r^2)^{3/2}}.\n$$\n\nThe angular factors expose the geometry immediately. Along $y$, one has $\\theta=\\pi/2$ and $\\phi=\\pm\\pi/2$, so the $C_3$ term becomes $-4C_3/r^3$. Along either $x$ or $z$, it becomes $+2C_3/r^3$. The explicit $C_6$ term vanishes on all three Cartesian axes and is repulsive in generic directions. The attractive channel is therefore a narrow valley surrounded by three-dimensional repulsion.\n\nThe authors define a dipolar length using the coefficient convention of this dressed potential,\n\n$$\nl_d=\\frac{m d^2}{48\\pi\\varepsilon_0\\hbar^2},\n$$\n\nthen use $l_u=l_d/20$ and $E_u=\\hbar^2/(m l_u^2)$ as numerical units. They quote $l_d/(10^4a_0)\\simeq1.1,2.6,8.3$ for NaK, NaRb, and NaCs. For NaK, $l_u\\simeq550a_0$, and a microwave coupling $\\Omega=2\\pi\\times10\\,\\mathrm{MHz}$ corresponds to $\\hbar\\Omega/E_u\\simeq52$.\n\nThe choice $\\xi=\\pi/4$ maximizes the single-axis attraction and makes the dimensional reduction unusually clean. It also sits at the least circular end of the polarization family. Coupled-channel shielding calculations with imperfect circular polarization show that increased ellipticity can reduce shielding efficiency. The first paper acknowledges this tension: smaller $\\xi$ should improve loss suppression and add a direct $r^{-6}$ shielding core, while producing shallower states and eventually destroying the one-dimensional geometry.\n\n### 4.2 The small-angle expansion\n\nChoose either attractive direction $\\phi_0=\\pm\\pi/2$ and write\n\n$$\n\\delta\\theta=\\theta-\\frac{\\pi}{2},\n\\qquad\n\\delta\\phi=\\phi-\\phi_0.\n$$\n\nThe supplemental material expands the potential to quadratic order:\n\n$$\nV(r,\\delta\\theta,\\delta\\phi)\n=-\\frac{4C_3}{r^3}\n+\\left(\\frac{6C_3}{r^3}+\\frac{4C_6}{r^6}\\right)\n(\\delta\\theta^2+\\delta\\phi^2).\n$$\n\nThe angular part of the kinetic energy contributes\n\n$$\n-\\frac{\\hbar^2}{m r^2}\n\\left(\n\\frac{\\partial^2}{\\partial\\delta\\theta^2}\n+\\frac{\\partial^2}{\\partial\\delta\\phi^2}\n\\right).\n$$\n\nAt fixed $r$, the two angular coordinates form independent harmonic oscillators. Their combined levels are\n\n$$\n\\epsilon_\\nu(r)\n=(\\nu+1)\n\\sqrt{\\frac{4\\hbar^2}{m}\n\\left(\\frac{6C_3}{r^5}+\\frac{4C_6}{r^8}\\right)}.\n$$\n\nThe factor $r^{-2}$ in the angular kinetic energy matters. A $C_3/r^3$ angular curvature gives a zero-point scale $r^{-5/2}$, while a $C_6/r^6$ curvature gives $r^{-4}$. The latter diverges more rapidly than the attractive $r^{-3}$ channel and blocks access to the origin.\n\nThe expansion drops a term proportional to $\\delta\\theta\\,\\partial_{\\delta\\theta}$ from the spherical kinetic operator. The supplement argues that its ground-state expectation remains of constant order, while $\\langle\\partial_{\\delta\\theta}^2\\rangle$ grows as $1/r^2$ at short distance. The omission follows from power counting; no exact cancellation removes the term. Higher angular orders should matter first near the emergence of a broad state and when the polarization becomes less strongly one-dimensional.\n\n### 4.3 Adiabatic separation and the effective potential\n\nThe authors write the relative wavefunction in an adiabatic basis,\n\n$$\n\\Psi^{(2)}(\\mathbf r)\n=\\frac1r\\sum_\\nu F_\\nu(y)\n\\psi_\\nu(y;\\delta\\theta,\\delta\\phi),\n$$\n\nwhere $y=r\\sin\\phi_0$ labels the signed coordinate along the attractive axis. They retain the lowest angular channel and neglect derivative couplings between different $\\nu$. The radial envelope then obeys\n\n$$\n\\left[\n-\\frac{\\hbar^2}{m}\\frac{\\partial^2}{\\partial y^2}\n+U^{(2)}(|y|)\n\\right]F_0(y)\n=E_{1\\mathrm D}^{(2)}F_0(y),\n$$\n\nwith\n\n$$\nU^{(2)}(r)\n=-\\frac{4C_3}{r^3}\n+\\sqrt{\\frac{4\\hbar^2}{m}\n\\left(\\frac{6C_3}{r^5}+\\frac{4C_6}{r^8}\\right)}.\n$$\n\nThe approximation contains two separate assumptions. The angular wavefunction must remain localized near $y$, and transitions between adiabatic angular channels must remain weak. A small angular width supports the first condition. A large gap relative to the longitudinal kinetic matrix elements supports the second. The paper tests the combined approximation against exact three-dimensional states rather than estimating every nonadiabatic matrix element independently.\n\nAt short distance, the $C_6$ term inside the square root dominates:\n\n$$\nU^{(2)}(r)\n\\simeq-\\frac{4C_3}{r^3}\n+4\\hbar\\sqrt{\\frac{C_6}{m}}\\frac1{r^4}.\n$$\n\nSetting the derivative of this asymptote to zero gives\n\n$$\nr_m\\simeq\\frac{4\\hbar}{3C_3}\\sqrt{\\frac{C_6}{m}}\n=4\\sqrt2\\,(1+\\delta_r^2)^{1/4}l_\\Omega,\n\\qquad\nl_\\Omega=\\sqrt{\\frac{\\hbar}{m\\Omega}}.\n$$\n\nFor $\\delta_r=0.2$, the detuning factor differs from unity by roughly one percent, leading to the quoted estimate $r_m\\simeq4\\sqrt2,l_\\Omega$. The NaK example at $10\\,\\mathrm{MHz}$ gives $r_m=428a_0$. The state lives well outside chemical length scales and well inside the large dressed dipolar length.\n\n\"Scaled\n\n*Figure 2. Short-distance power counting for the effective two-molecule potential. The coefficients have been rescaled so that the asymptotic minimum lies at $r/r_m=1$. The plot shows the mechanism, not a fit to a specific molecular species. The full calculation uses the complete square root in $U^{(2)}$.*\n\n### 4.4 What the exact three-dimensional calculation contains\n\nThe two-molecule benchmark diagonalizes\n\n$$\nH^{(2)}=-\\frac{\\hbar^2}{m}\\nabla_{\\mathbf r}^2+V(\\mathbf r)\n$$\n\nin the basis\n\n$$\n\\Psi^{(2)}(\\mathbf r)\n=\\int_0^\\infty dk\\sum_{\\ell m}\nC_{\\ell m}(k)\\,\n\\phi_{\\ell k}(r)Y_{\\ell m}(\\theta,\\phi),\n$$\n\n$$\n\\phi_{\\ell k}(r)=\\sqrt{\\frac{2}{\\pi}},k j_\\ell(kr).\n$$\n\nBosonic exchange keeps even $\\ell$; fermionic exchange keeps odd $\\ell$. The microwave potential couples many $(\\ell,m)$ channels, so the two statistics generate different expansion coefficients even when their final probability densities become nearly indistinguishable. Any numerical reconstruction needs cutoffs in momentum, radial extent, $\\ell$, and $m$. The public data do not include the convergence tables or assembly code for those cutoffs.\n\nThe exact spectrum contains two low levels that enter the bound sector one by one as $\\Omega$ increases. The effective one-dimensional energies are generally shallower near threshold because the quadratic angular approximation omits higher fluctuations. At stronger coupling, the one-dimensional energies track both exact parity sectors over a wide interval.\n\nThe supplement defines a signed one-dimensional reduction of the exact wavefunction,\n\n$$\n\\widetilde\\Psi^{(2)}(y)\n=\\sqrt{\\int dx\\,dz\\,|\\Psi^{(2)}(\\mathbf r)|^2}\n\\,\\operatorname{sgn}[\\Psi^{(2)}(\\mathbf r)],\n$$\n\nand compares it with $F_0(y)$. Their overlap exceeds $97\\%$ once $\\hbar\\Omega/E_u>10$. The boson-fermion energy mismatch is measured by\n\n$$\n\\eta=\n\\frac{|E_b^{(2)}-E_f^{(2)}|}\n{|E_b^{(2)}+E_f^{(2)}|},\n$$\n\nwhich falls below $5\\%$ for $\\hbar\\Omega/E_u>2$. These two tests probe different errors. The overlap measures dimensional reduction; $\\eta$ measures the onset of statistical duality.\n\n### 4.5 The threshold region is genuinely three-dimensional\n\nThe exact bosonic state appears at a smaller critical $\\Omega$ than the exact fermionic state. Close to its emergence, the bosonic state follows the three-dimensional $s$-wave formula\n\n$$\nE_b^{(2)}\\simeq-\\frac{\\hbar^2}{m a_s^2},\n\\qquad\na_s=-\\lim_{k\\rightarrow0}\\frac{\\tan\\delta_k}{k},\n$$\n\nand crosses threshold at $a_s\\rightarrow\\infty$. The fermionic threshold state has $p$-wave character. The effective one-dimensional model cannot reproduce two different threshold laws because it gives the same impenetrable-channel spectrum to both statistics.\n\nA near-threshold state extends far beyond the narrow attractive valley. It samples the full angular manifold and resolves the difference between even and odd partial waves. Increasing $\\Omega$ shrinks the bound state toward $r_m$, raises the barrier against exchange, and confines its probability to the axial valley. The effective one-dimensional description improves as the state becomes more deeply bound.\n\nThe supplement also identifies the only plausible Efimov window. Dipolar Efimov scaling would require $l_d\\ll r\\ll|a_s|$. For the molecular parameters used here, $|a_s|>l_d=20l_u$ occurs only in the narrow interval $\\hbar\\Omega/E_u\\in(0.5,1.5)$ around the $s$-wave resonance, where $|E_b^{(2)}|\\lesssim0.004E_u$. Any Efimov states in that interval would lie close to threshold and outside the compact universal-cluster regime centered near $r_m$.\n\n## 5. Bose–Fermi duality in the molecular problem\n\nAn impenetrable one-dimensional core divides configuration space into ordered sectors. For $N$ particles, one sector is $y_10)\n=|\\Psi_g^{(2)}(y)|^2\n+\\int_0^y dy'\n\\left|\n\\Psi_g^{(2)}(y')\n\\Psi_g^{(2)}(y-y')\n\\right|^2.\n$$\n\nThe first term places the two observed particles on one bond; the convolution places an unobserved molecule between them. The correlation function therefore resolves the chain geometry without assigning labels to identical particles.\n\n### 7.3 Pair-additive approximation and the origin of extra binding\n\nThe paper compares the full zero-point surface with\n\n$$\n\\widetilde U^{(3)}\n=U^{(2)}(|y_{12}|)\n+U^{(2)}(|y_{23}|)\n+U^{(2)}(|y_{13}|).\n$$\n\nInside the ordered sector $y_1\n\n*Figure 3. The trap potential in center-of-mass and relative coordinates for a representative Li–Cr mass ratio. Equal frequencies produce aligned contours. Equal oscillator lengths, the choice used in the paper’s numerical examples, rotate the normal modes and couple collision energy to several molecular center-of-mass levels.*\n\n## 11. Quasi-two-dimensional scattering\n\n### 11.1 Open channel and threshold form\n\nIn quasi-two dimensions, the $z$ coordinates are trapped and the relative in-plane coordinate $\\boldsymbol\\rho$ is free. The incident state is\n\n$$\n|\\Psi^{(0)}\\rangle=|n_h=n_l=0;\\mathbf q\\rangle,\n$$\n\nwith energy\n\n$$\nE_{0,0;q}\n=\\frac{\\omega_h+\\omega_l}{2}\n+\\frac{q^2}{2\\mu}.\n$$\n\nThe low-energy condition $q^2/(2\\mu)\\ll\\omega_h,\\omega_l$ keeps the outgoing wave in the transverse ground channel. Its asymptotic form contains a Bessel incoming wave and a Hankel outgoing wave. The on-shell amplitude is parametrized as\n\n$$\nt_q^{(2\\mathrm D)}\n=\\frac{2\\pi/\\mu}\n{-\\ln(q^2a_{2\\mathrm D}^2)\n+\\mathrm i\\pi\n+R_{2\\mathrm D}q^2}.\n$$\n\n$a_{2\\mathrm D}$ has dimensions of length; $R_{2\\mathrm D}$ has dimensions of length squared in this convention. The logarithm fixes the two-dimensional threshold law, while $R_{2\\mathrm D}$ carries the leading energy dependence inherited from the three-dimensional resonance and virtual confinement modes.\n\n### 11.2 Auxiliary contact amplitude\n\nDefine\n\n$$\n|f\\rangle=T|\\Psi^{(0)}\\rangle.\n$$\n\nThe Lippmann–Schwinger equation becomes\n\n$$\n\\left[g^{-1}-\\delta(\\mathbf r)G_0(E)\\right]|f\\rangle\n=\\delta(\\mathbf r)|\\Psi^{(0)}\\rangle.\n$$\n\nThe contact interaction collapses the relative coordinate, leaving a function of the pair center of mass. Shi and Cui expand that function in an auxiliary molecular basis. Along the confined direction,\n\n$$\nH_M\n=-\\frac1{2M}\\frac{\\partial^2}{\\partial Z^2}\n+\\frac12M\\omega_M^2Z^2,\n$$\n\n$$\n\\omega_M\n=\\sqrt{\\frac{m_h\\omega_h^2+m_l\\omega_l^2}{M}}.\n$$\n\nThe basis vectors are $|N\\rangle_Z|\\boldsymbol\\rho=0\\rangle$. $H_M$ equals the center-of-mass trap experienced by a pair whose constituents occupy the same position. It is a basis choice for the contact pair amplitude, not a postulate that a stable closed-channel molecule already exists.\n\nThe matrix to invert is\n\n$$\n\\mathcal M(E)=g^{-1}-\\delta(\\mathbf r)\\mathcal A(E),\n$$\n\nwhere $\\mathcal A$ contains all excited single-particle oscillator channels plus a subtraction of the open-channel logarithm. The overlap entering its matrix elements is\n\n$$\nF_{N;n_h,n_l}\n=\\int dZ\\,\n\\Phi_N^*(Z)\\varphi_{n_h}^{(h)}(Z)\\varphi_{n_l}^{(l)}(Z).\n$$\n\nFinite range modifies the diagonal inverse coupling through\n\n$$\nE_N\n=\\frac{q^2}{2\\mu}\n+\\frac{\\omega_h+\\omega_l}{2}\n-(N+1/2)\\omega_M.\n$$\n\nDiagonalizing $\\mathcal M$ produces eigenvectors $|\\alpha_j\\rangle=\\sum_Nc_{jN}|N\\rangle$ and eigenvalues of the form $\\mu/(2\\pi a_s)+\\lambda_j$. The ground-channel projected inverse matrix is\n\n$$\n\\left[\n\\sum_j\n\\frac{|\\sum_Nc_{jN}F_{N;0,0}|^2}\n{\\mu/(2\\pi a_s)+\\lambda_j}\n\\right]^{-1}.\n$$\n\nExpanding this quantity at small $q$ yields the constant term $-\\ln a_{2\\mathrm D}^2$ and the quadratic term $R_{2\\mathrm D}q^2$.\n\n## 12. Ultraviolet cancellation in imaginary time\n\nThe supplement rewrites energy denominators with\n\n$$\n\\frac1x=-\\int_0^\\infty dt\\,e^{xt},\n\\qquad x<0.\n$$\n\nAt short imaginary time, the three-dimensional free propagator at contact scales as\n\n$$\nK(t;\\mathbf r=0)\n\\sim\\left(\\frac{\\mu}{2\\pi t}\\right)^{3/2}.\n$$\n\nThe same $t^{-3/2}$ divergence appears in the bare-coupling subtraction. The authors introduce a temporary short-time boundary $t_0$, isolate both singular pieces analytically, and show that they cancel. The remaining contribution includes terms proportional to\n\n$$\n2\\left(\\frac{\\mu}{2\\pi}\\right)^{3/2}\n\\left[(E-\\epsilon_N)\\sqrt{t_0}\n-\\frac{E\\epsilon_N}{3}t_0^{3/2}\\right]\\delta_{NN'},\n$$\n\nplus finite integrals from $t_0$ to infinity. A stable implementation must become independent of $t_0$ over a window where the short-time expansion is accurate and the numerical quadrature remains resolved.\n\nThe exact harmonic propagators allow the sums over transverse oscillator modes to be evaluated analytically before the remaining spatial and time integrals. The supplement also separates a long-time divergence associated with the open ground channel and cancels it against the explicit threshold subtraction. Its final Eq. (A36) is a finite expression for every matrix element $\\mathcal M_{NN'}$.\n\nAnalytic subtraction exposes cutoff independence before diagonalization. A large finite oscillator basis may appear converged while retaining a slowly drifting ultraviolet offset; the imaginary-time formulation removes that offset explicitly.\n\n## 13. Bound-state pole and the two-body closure test\n\nThe full confined two-body bound state appears when the lowest eigenvalue of $\\mathcal M(E)$ crosses zero for $E<0$. Write\n\n$$\nE_2=-\\frac{\\kappa^2}{2\\mu}.\n$$\n\nThe low-dimensional amplitude predicts the pole through\n\n$$\n\\ln(\\kappa^2a_{2\\mathrm D}^2)\n+R_{2\\mathrm D}\\kappa^2=0.\n$$\n\nAgreement between this root and the full confined root tests the truncation of the threshold expansion at $q^2$. The paper plots exact discrete roots and effective-theory curves over a broad negative-$a_s$ region and finds good agreement. The comparison validates the two-body reduction in the displayed window; it leaves three- and four-body off-shell momenta untested.\n\n## 14. Quasi-one-dimensional scattering\n\nIn quasi one dimension, each atom is trapped in the transverse plane and moves freely along $z$. The incident energy is\n\n$$\nE_{0,0;q}=\\omega_h+\\omega_l+\\frac{q^2}{2\\mu}.\n$$\n\nThe threshold amplitude is\n\n$$\nt_q^{(1\\mathrm D)}\n=-\\frac{1/\\mu}\n{a_{1\\mathrm D}-\\mathrm i/q-R_{1\\mathrm D}q^2}.\n$$\n\nAnalytic continuation $q\\to\\mathrm i\\kappa$ gives\n\n$$\na_{1\\mathrm D}-\\frac1\\kappa+R_{1\\mathrm D}\\kappa^2=0.\n$$\n\nThe molecular basis now uses the two-dimensional center-of-mass oscillator\n\n$$\nH_M\n=-\\frac{\\nabla_\\rho^2}{2M}\n+\\frac12M\\omega_M^2\\rho^2.\n$$\n\nIts states carry radial and angular quantum numbers $(N_r,M_a)$ and contain associated Laguerre polynomials. Rotational symmetry makes the contact ground-channel overlap vanish unless $M_a=0$.\n\nThe regularization differs from quasi two dimensions in one useful respect. The free one-dimensional ground-channel integral gives $\\mu/(\\mathrm i q)$ and has no ultraviolet logarithm. In the supplement’s imaginary-time representation, the corresponding $J_2$ term scales as $t^{-1/2}$ and converges at $t\\rightarrow0$. The three-dimensional contact divergence still appears in the excited-channel contribution and still cancels the bare coupling. The same short-time finite residual survives after cancellation, but the open-channel subtraction and long-time behavior have different analytic forms.\n\n
\nDimensional bookkeeping\n
\n\nIn the conventions of the paper, $R_{2\\mathrm D}$ multiplies $q^2$ inside a dimensionless denominator and therefore has dimensions of length squared. $R_{1\\mathrm D}$ multiplies $q^2$ next to $a_{1\\mathrm D}$ and therefore has dimensions of length cubed. Comparing their numerical values without restoring units would be meaningless.\n\n
\n
\n\n## 15. From two-body amplitudes to STM clusters\n\nThe low-dimensional parameters feed a Skorniakov–Ter-Martirosian equation for one light atom and $N$ identical heavy fermions, with $N=2$ for the trimer and $N=3$ for the tetramer. In a dimer-spectator frame, the amplitude\n\n$$\nf_{\\mathbf k_2,\\ldots,\\mathbf k_N}\n$$\n\ndescribes a heavy-light pair carrying momentum $-\\sum_{i=2}^N\\mathbf k_i$ and $N-1$ spectator heavy fermions. Exchanging which heavy atom belongs to the dimer generates the antisymmetric terms on the right side of the STM equation.\n\nThe kernel samples a pair energy\n\n$$\nA_{\\mathbf k_2\\ldots\\mathbf k_N}\n=E_{1+N}\n-\\sum_{i=2}^N\\epsilon^h_{\\mathbf k_i}\n-\\epsilon^d_{\\sum_{i=2}^N\\mathbf k_i},\n$$\n\nwhere $\\epsilon^h_k=k^2/(2m_h)$ and $\\epsilon^d_k=k^2/[2(m_h+m_l)]$. The two-dimensional equation contains the combination\n\n$$\n-\\ln(a_{2\\mathrm D}^2\\kappa_A^2)\n+R_{2\\mathrm D}\\kappa_A^2,\n$$\n\nwhile the one-dimensional equation contains\n\n$$\na_{1\\mathrm D}-\\kappa_A^{-1}+R_{1\\mathrm D}\\kappa_A^2,\n$$\n\nwith $\\kappa_A$ set by the negative pair energy available after spectator kinetic energies are removed. These are off-shell continuations of the fitted two-body amplitudes. A deeply bound cluster can sample momenta larger than those used in the threshold fit. Off-shell sampling sets the main uncontrolled error of the few-body stage.\n\nThe authors define the separation from the nearest breakup threshold as\n\n$$\n\\Delta_{1+N,N}=E_{1+N}-E_N.\n$$\n\n$\\Delta_{3,2}<0$ means that the trimer lies below a dimer plus a heavy atom. $\\Delta_{4,3}<0$ means that the tetramer lies below a trimer plus a heavy atom. Total binding from the free-atom threshold can be large even when $|\\Delta_{4,3}|$ is too small to resolve experimentally.\n\n## 16. Numerical systems and experimental scales\n\nThe first candidate is $^6$Li–$^{53}$Cr near the quoted $1414\\,\\mathrm G$ Feshbach resonance. It has $m_h/m_l=8.8$ and $R^*\\simeq6000a_0$. The second is $^6$Li–$^{40}$K near $155\\,\\mathrm G$, with mass ratio $6.7$ and $R^*\\simeq2400a_0$. The paper scans equal oscillator lengths $l_{\\mathrm{ho}}=1500,1000,700a_0$.\n\nAt $l_{\\mathrm{ho}}=700a_0$, the light-atom frequency is\n\n$$\n\\frac{\\omega_l}{2\\pi}\n=\\frac{\\hbar}{2\\pi m_l l_{\\mathrm{ho}}^2}\n\\simeq1.23\\,\\mathrm{MHz}\n$$\n\nfor $^6$Li. Equal oscillator length then gives approximately $0.139\\,\\mathrm{MHz}$ for $^{53}$Cr and $0.184\\,\\mathrm{MHz}$ for $^{40}$K. These are tight traps. The conversion\n\n$$\n\\frac{h\\times1\\,\\mathrm{kHz}}{k_B}\\simeq48\\,\\mathrm{nK}\n$$\n\nshows that a sub-kilohertz threshold separation can compete directly with realistic temperature and linewidth scales.\n\n### 16.1 Quasi-two-dimensional windows\n\nFor Li–Cr at $l_{\\mathrm{ho}}=700a_0$, the paper identifies $R^*/a_s\\in(-69.9,-61.8)$ as a favorable interval. Across it,\n\n$$\n\\Delta_{3,2}/h\\in(-0.28,-0.92)\\,\\mathrm{kHz},\n\\qquad\n|\\Delta_{3,2}/E_2|\\in(27\\%,10\\%),\n$$\n\nand\n\n$$\n\\Delta_{4,3}/h\\in(-0.18,-0.50)\\,\\mathrm{kHz},\n\\qquad\n|\\Delta_{4,3}/E_2|\\in(17\\%,5\\%).\n$$\n\nFor Li–K, the favorable interval is $R^*/a_s\\in(-15.3,-12.4)$, with\n\n$$\n\\Delta_{3,2}/h\\in(-0.17,-0.80)\\,\\mathrm{kHz},\n\\qquad\n|\\Delta_{3,2}/E_2|\\in(16\\%,10\\%),\n$$\n\nand\n\n$$\n\\Delta_{4,3}/h\\in(-0.10,-0.42)\\,\\mathrm{kHz},\n\\qquad\n|\\Delta_{4,3}/E_2|\\in(9\\%,6\\%).\n$$\n\nThe smaller Li–K mass ratio lowers the zero-range saturation values of the relative binding ratios, although its strongest absolute binding remains comparable to Li–Cr in the displayed quasi-two-dimensional range.\n\n### 16.2 Quasi-one-dimensional windows\n\nFor Li–Cr, the favorable interval moves to $R^*/a_s\\in(-194.0,-127.1)$ and gives\n\n$$\n\\Delta_{3,2}/h\\in(-1.79,-7.85)\\,\\mathrm{kHz},\n\\qquad\n|\\Delta_{3,2}/E_2|\\in(36\\%,20\\%),\n$$\n\n$$\n\\Delta_{4,3}/h\\in(-0.35,-1.13)\\,\\mathrm{kHz},\n\\qquad\n|\\Delta_{4,3}/E_2|\\in(7\\%,3\\%).\n$$\n\nFor Li–K, $R^*/a_s\\in(-53.3,-25.7)$ gives\n\n$$\n\\Delta_{3,2}/h\\in(-1.41,-8.81)\\,\\mathrm{kHz},\n\\qquad\n|\\Delta_{3,2}/E_2|\\in(28\\%,20\\%),\n$$\n\n$$\n\\Delta_{4,3}/h\\in(-0.21,-1.23)\\,\\mathrm{kHz},\n\\qquad\n|\\Delta_{4,3}/E_2|\\in(4\\%,3\\%).\n$$\n\nThe deepest trimer separation anywhere on the plotted curves reaches $-8.10\\,\\mathrm{kHz}$ for Li–Cr and $-10.61\\,\\mathrm{kHz}$ for Li–K in quasi one dimension, compared with roughly $-0.92$ and $-0.97\\,\\mathrm{kHz}$ in quasi two dimensions. The tetramer gains absolute binding as well, while its ratio to the dimer scale can be smaller in quasi one dimension. The trimer receives the clearest spectroscopic advantage.\n\n\"Reported\n\n*Figure 4. Magnitudes of the relative binding intervals quoted by Shi and Cui for $l_{\\mathrm{ho}}=700a_0$. The horizontal axis is logarithmic. The blue trimer intervals move by roughly one decade between q2D and q1D. Tetramer intervals remain closer to the sub-kilohertz scale.*\n\n## 17. Multiple confinement-induced resonances\n\nThe calculated $a_{2\\mathrm D}$ and $g_{1\\mathrm D}=-(\\mu a_{1\\mathrm D})^{-1}$ show several resonances as $R^*/a_s$ is scanned. Equal-frequency confinement would organize the two-body problem into a conserved center-of-mass quantum number. Species-selective confinement mixes those sectors, so several molecular center-of-mass levels acquire overlap with the open channel. Each dressed closed-channel level can cross threshold and generate a resonance.\n\nThe molecular-basis expression makes the mechanism explicit. A pole becomes strong when an eigenvalue $\\mu/(2\\pi a_s)+\\lambda_j$ approaches zero and the corresponding eigenvector has a nonzero projected overlap $\\sum_Nc_{jN}F_{N;0,0}$. A zero eigenvalue with vanishing overlap would remain dark to the chosen incident channel. Resonance count and visibility therefore depend on both the spectrum and the contact overlap.\n\nThe effective range varies more smoothly than the scattering length across much of the negative-$a_s$ regime. Narrow resonances and tight confinement amplify its contribution. Fitting a single scattering length across adjacent resonances would merge physically distinct center-of-mass channels into a misleading parameter.\n\n## 18. Reconstructing the confined-scattering and STM code\n\nThe second paper’s [Figshare archive](https://doi.org/10.6084/m9.figshare.32507691.v1) contains spreadsheets for Figs. 1–4 and no solver. A clean implementation should separate the confined two-body problem from the effective few-body problem.\n\n### 18.1 Confined two-body module\n\nThe first module generates harmonic oscillator energies and normalized basis functions for each species and for the artificial molecule. It evaluates $F_{N;n_h,n_l}$ analytically where possible and by quadrature otherwise. Selection rules should be applied before numerical integration.\n\nThe second module evaluates $\\mathcal M_{NN'}(E,q)$. It treats short imaginary time with the analytic expansion, integrates the finite interval with adaptive quadrature, and handles the long-time open-channel subtraction separately. Tests should vary $t_0$ and any long-time split $t_c$. The final matrix must remain Hermitian below threshold.\n\nThe third module projects $\\mathcal M^{-1}$ onto the open channel. It samples several momenta $q$ satisfying $q^2/(2\\mu)\\ll\\min(\\omega_h,\\omega_l)$ and fits the inverse amplitude. The fit should include a diagnostic $q^4$ term even if only $a$ and $R$ are retained. Stability under removal of the largest-$q$ point gives a direct estimate of threshold-truncation error.\n\nThe fourth module searches for the exact confined dimer pole by tracking the lowest eigenvalue of $\\mathcal M(E,0)$. The effective pole equation supplies an independent root. Storing both roots for every field and confinement value prevents a few hand-selected points from hiding a systematic drift.\n\n```python\nfor control in control_grid:\n a3d = scattering_length(control)\n\n def collision_matrix(E, q, n_cm):\n finite_propagator = regularized_imaginary_time_kernel(\n E=E,\n q=q,\n n_cm=n_cm,\n masses=masses,\n frequencies=frequencies,\n short_time_cut=t0,\n )\n inverse_contact = energy_dependent_inverse_scattering_length(\n E, a3d, R_star, reduced_mass\n )\n return inverse_contact * np.eye(n_cm) - finite_propagator\n\n inverse_t = []\n for q in threshold_momenta:\n M = collision_matrix(open_energy(q), q, n_cm)\n inverse_t.append(1 / project_open_channel(np.linalg.inv(M)))\n\n a_lowd, R_lowd, q4_diagnostic = fit_threshold_series(\n threshold_momenta, inverse_t\n )\n E2_exact = root_lowest_eigenvalue(collision_matrix, bracket=bound_window)\n E2_effective = root_effective_pole(a_lowd, R_lowd, geometry)\n```\n\n### 18.2 STM module\n\nThe STM solver takes only masses, geometry, $a$, and $R$ from the two-body stage. In quasi two dimensions, rotational decomposition converts angular dependence into Fourier channels. In quasi one dimension, the spectator momenta are signed scalars. A logarithmic or mapped momentum grid resolves both the shallow tail and the ultraviolet decay.\n\nThe trimer amplitude depends on one spectator momentum; the tetramer amplitude depends on two. Antisymmetry can be imposed by storing an ordered region and generating exchanged values with signs. Matrix-free Arnoldi iteration avoids constructing a dense tetramer kernel. For a trial $E<0$, the largest relevant eigenvalue $\\lambda_{\\max}(E)$ is computed; a bound state satisfies $\\lambda_{\\max}(E)=1$.\n\nThe momentum cutoff must be varied. In a properly universal calculation with finite effective range, the energy should stabilize before the cutoff reaches scales where the low-dimensional threshold expansion fails. Residual cutoff drift can indicate a missing higher-order shape parameter or an unresolved few-body counterterm.\n\n### 18.3 Minimum convergence record\n\nA complete numerical record would report the molecular center-of-mass cutoff, the single-particle oscillator cutoff, $t_0$, $t_c$, time-quadrature tolerance, threshold-fit interval, STM momentum cutoff, radial grid size, angular channel cutoff, and eigensolver tolerance. It would list $a_d$, $R_d$, $E_2^{\\mathrm{exact}}$, $E_2^{\\mathrm{effective}}$, $E_3$, and $E_4$ in the same table. The paper provides the physical curves and analytic regularization, but not this full convergence ledger.\n\n## 19. What the two calculations have in common\n\n| Stage | Microwave-shielded molecules | Species-selective heteronuclear atoms |\n|---|---|---|\n| High-dimensional input | Anisotropic dressed $C_3/r^3+C_6/r^6$ interaction in free 3D | 3D contact interaction, finite $R^*$, and two harmonic traps |\n| Eliminated motion | Angular excursions from the attractive axis | Transverse oscillators and molecular COM channels |\n| Retained low-energy object | Coordinate-space potentials $U^{(2)}$ and $U^{(3)}$ | Threshold amplitudes described by $a_d$ and $R_d$ |\n| Direct benchmark | Exact 3D two-molecule partial-wave calculation | Exact confined two-body $T$-matrix pole |\n| Few-body solver | Two-dimensional Jacobi grid for three identical molecules | q1D/q2D STM equations for $1+2$ and $1+3$ atoms |\n| Statistics | Girardeau sign map between impenetrable bosons and fermions | Explicit antisymmetry under heavy-fermion exchange |\n| Main extrapolation | Three molecules to a large crystalline chain | Two-body threshold fit to deeper three- and four-body kernels |\n\nThe common structure is a sequence of controlled compressions. Both papers identify a low subspace, calculate the influence of excluded motion, test the reduction in a two-body observable, and solve a harder few-body problem with the reduced description. The discarded variables return as operators: a zero-point potential in one case, an energy-dependent scattering kernel in the other.\n\n\"Evidence\n\n*Figure 5. Evidence map for the two papers. “Exact” refers to exact diagonalization within the stated microscopic or confined model and numerical cutoffs. Each move to the right adds an effective-theory assumption. The labels prevent a two-body benchmark from being misreported as a full high-dimensional three- or four-body calculation.*\n\n## 20. Three meanings of universality\n\nThe molecular states have long-range universality. Their compact scale is set by $l_\\Omega$ inside a potential whose coefficients follow from $d$, $\\Omega$, and $\\delta_r$. The repulsive core prevents access to chemistry, so no separate short-range boundary condition appears in the reduced model. Polarization remains an essential dimensionless parameter.\n\nThe heteronuclear clusters have scattering universality. The confined two-body dynamics are compressed into $a_{1\\mathrm D/2\\mathrm D}$ and $R_{1\\mathrm D/2\\mathrm D}$, after which the STM equations discard the microscopic Feshbach potential. This reduction remains quantitative only while the cluster kernel samples energies where the two-parameter threshold expansion is accurate.\n\nThe Bose–Fermi correspondence has kinematic universality. Once coincidence nodes divide a one-dimensional configuration space into ordered sectors, the energy-density mapping follows without using the detailed shape of the core. Finite transverse width and finite barrier penetration control the correction.\n\nThese meanings overlap, but they answer different questions. Long-range universality concerns sensitivity to chemical separation. Scattering universality concerns the number of low-energy constants. Kinematic universality concerns the topology of accessible configuration space.\n\n## 21. Evidence audit and unresolved errors\n\nThe molecular paper gives its strongest evidence for two molecules. The exact three-dimensional calculation, the effective energy, the reduced-wavefunction overlap, and $G_2$ all agree over a quantified coupling range. The Born–Oppenheimer problem supplies a second high-dimensional check for one light and two infinitely heavy molecules. The equal-mass three-molecule spectrum is calculated only after the one-dimensional reduction.\n\nThe heteronuclear paper gives its strongest evidence for confined two-body scattering. It derives the UV-finite matrix and compares the effective pole with the exact confined pole. The trimer and tetramer energies come from low-dimensional STM equations. The paper does not present a full confined three- or four-body calculation.\n\nThe first paper neglects higher angular orders and interchannel derivative couplings. It adopts an analytic dressed potential at $\\xi=\\pi/4$ by extrapolation. It does not compute cluster lifetimes. Its large-$N$ droplet argument uses localized pair products and a dipolar tail estimate.\n\nThe second paper truncates the low-energy amplitude at $q^2$, provides no explicit cluster error bars, and does not report lifetime, finite-temperature, trap-anharmonicity, or preparation calculations. Its numerical examples use equal oscillator lengths even though the formalism permits arbitrary frequencies. The v1 text also contains editorial residues: the introduction mentions Li–Cs where the results use Li–Cr, and the q1D captions for Figs. 3 and 4 refer to an “effective 2D model.” The formulas and surrounding discussion identify those captions as typographical errors.\n\n## 22. Public data and reproducibility\n\nThe first archive is roughly $30\\,\\mathrm{MB}$ and contains CSV/XLSX files, README descriptions, $1001\\times1001$ potential grids, two-dimensional wavefunction arrays, three-body potential surfaces, energies, overlaps, and Born–Oppenheimer curves. A reader can redraw the main plots and inspect the released arrays. Reconstructing the partial-wave Hamiltonian or the angular Hessian still requires new code.\n\nThe second archive is roughly $0.31\\,\\mathrm{MB}$ and contains spreadsheets for Figs. 1–4. It records scattering parameters and binding curves. It does not expose the imaginary-time integrator, basis cutoffs, fit windows, or STM eigensolver.\n\nThe distinction between plot reproducibility and computational reproducibility matters here. Plot reproducibility checks transcription and presentation. End-to-end reproduction tests the Hamiltonian, regularization, truncation, and convergence. Both archives support the first task and part of the second; neither supplies the complete executable pipeline.\n\n## 23. Experimental decision points\n\nThe molecular proposal needs a joint map of binding and loss as polarization changes. The ideal point maximizes the ratio of binding energy to decay width, not the binding energy alone. A useful calculation would combine the effective axial state with a coupled-channel estimate of short-range flux for several $\\xi<\\pi/4$. Three-molecule spectroscopy should search below the two-molecule-plus-molecule threshold and compare bosonic and fermionic isotopologues at matched $l_d$ and $l_\\Omega$.\n\nThe heteronuclear proposal needs threshold resolution. Radio-frequency or modulation spectroscopy must resolve $|\\Delta_{3,2}|$ or $|\\Delta_{4,3}|$, rather than the total binding from free atoms. The q1D trimer windows are the strongest targets because their separations reach several kilohertz and retain ratios near twenty percent. Tetramers sit much closer to the trimer threshold, so temperature and loss broadening can erase them even when their absolute binding is substantial.\n\nTrap calibration enters as a theoretical parameter, not a technical afterthought. A fractional error in either frequency changes the center-of-mass mixing, shifts every confinement-induced resonance, and modifies the fitted effective range. Anharmonicity can couple additional center-of-mass levels that are absent in the harmonic calculation.\n\n## 24. Derivation check: the $r^{-4}$ core\n\nTake an angular coordinate $\\alpha$ with moment of inertia $I\\sim mr^2$ and curvature\n\n$$\nK(r)\\sim\\frac{C_3}{r^3}+\\frac{C_6}{r^6}.\n$$\n\nThe zero-point energy scales as\n\n$$\nE_{\\mathrm{zp}}\\sim\\hbar\\sqrt{\\frac{K(r)}{I}}.\n$$\n\nWhen the $C_3$ curvature dominates, $E_{\\mathrm{zp}}\\propto r^{-5/2}$. When the $C_6$ curvature dominates, $E_{\\mathrm{zp}}\\propto r^{-4}$. The second power outruns the axial $-r^{-3}$ attraction and guarantees a short-distance turn upward within the quadratic regime.\n\n
\nCheck the coefficient of the asymptotic minimum\n\nUse\n\n$$\nU(r)=-4C_3r^{-3}+4\\hbar\\sqrt{C_6/m}\\,r^{-4}.\n$$\n\nThen\n\n$$\n\\frac{dU}{dr}=12C_3r^{-4}\n-16\\hbar\\sqrt{C_6/m}\\,r^{-5}=0,\n$$\n\nwhich gives $r_m=4\\hbar\\sqrt{C_6/m}/(3C_3)$. Substitution of the published $C_3$ and $C_6$ yields $4\\sqrt2(1+\\delta_r^2)^{1/4}l_\\Omega$.\n\n
\n\n## 25. Derivation check: the center-of-mass cross term\n\nInsert\n\n$$\nz_h=Z+\\frac{m_l}{M}z,\n\\qquad\nz_l=Z-\\frac{m_h}{M}z\n$$\n\ninto $m_h\\omega_h^2z_h^2/2+m_l\\omega_l^2z_l^2/2$. The coefficient of $Zz$ is\n\n$$\n\\frac{m_hm_l}{M}\\omega_h^2\n-\\frac{m_hm_l}{M}\\omega_l^2\n=\\mu(\\omega_h^2-\\omega_l^2).\n$$\n\nEqual oscillator length imposes $m_h\\omega_h=m_l\\omega_l$. For unequal masses, this relation guarantees unequal frequencies, so the cross term remains. The paper’s experimentally natural example therefore produces strong center-of-mass mixing.\n\n## 26. Derivation check: total binding versus threshold separation\n\nConsider $E_2=-10\\,\\mathrm{kHz}$, $E_3=-12\\,\\mathrm{kHz}$, and $E_4=-12.3\\,\\mathrm{kHz}$. The trimer separation is\n\n$$\n\\Delta_{3,2}=E_3-E_2=-2\\,\\mathrm{kHz},\n$$\n\nwhile the tetramer separation is\n\n$$\n\\Delta_{4,3}=E_4-E_3=-0.3\\,\\mathrm{kHz}.\n$$\n\nThe tetramer has the lowest total energy, yet its dissociation line sits only $0.3\\,\\mathrm{kHz}$ below the trimer threshold, equivalent to about $14\\,\\mathrm{nK}$. A spectrum can resolve the trimer and miss the tetramer.\n\n## 27. A falsifiable reproduction program\n\nA high-value reproduction would select three parameter points from each paper. For the molecular problem, one point should lie near the $s$-wave threshold, one near $\\hbar\\Omega/E_u=10$, and one near 52. At each point, it should report exact even- and odd-parity energies, the effective energy, reduced-wavefunction overlap, angular width, and convergence with $\\ell_{\\max}$. Those observables expose the crossover from three-dimensional threshold physics to the compact dual regime.\n\nFor the confined mixture, one point should lie far from a COM resonance, one on its flank, and one between adjacent resonances. Each point should report $a_d$, $R_d$, the $q^4$ coefficient, exact and effective dimer roots, and the maximum pair momentum sampled by the trimer and tetramer eigenvectors. Those data quantify whether the STM calculation remains inside the threshold-fit window.\n\nThe decisive extension for the first paper is a full three-dimensional equal-mass three-molecule calculation at one or two compact-state points. The decisive extension for the second is a confined three-body calculation at one q1D trimer point. Either benchmark would turn the present effective-theory evidence ladder into a closed high-dimensional loop.\n\n## 28. Source trail\n\nThe two target preprints and their supplements are [arXiv:2504.21535v4](https://arxiv.org/abs/2504.21535) and [arXiv:2606.02988v1](https://arxiv.org/abs/2606.02988). Their data are available through [Figshare 29306735](https://figshare.com/articles/dataset/Data_DipolarMolecule_rar/29306735) and [Figshare 32507691](https://doi.org/10.6084/m9.figshare.32507691.v1).\n\nThe experimental field-linked tetramer result is X.-Y. Chen and collaborators, [Nature 626, 283–287 (2024)](https://www.nature.com/articles/s41586-023-06986-6). Microwave shielding and its polarization dependence are treated by Karman and Hutson in [PRL 121, 163401](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.163401) and [PRA 100, 052704](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.052704). A recent scaling analysis of shielding universality is Dutta, Mukherjee, and Hutson, [Physical Review Research 7, 023164](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.7.023164).\n\nThe one-dimensional hard-core mapping originates with Girardeau, [Journal of Mathematical Physics 1, 516 (1960)](https://doi.org/10.1063/1.1703687). The confinement-induced resonance background is provided by [Olshanii (1998)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.81.938) and [Bergeman, Moore, and Olshanii (2003)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.91.163201). The immediate quasi-two-dimensional cluster predecessor is Liu, Shi, Zaccanti, and Cui, [Physical Review Research 6, L042004 (2024)](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.6.L042004).\n\n## 29. Final synthesis\n\nThe first paper finds a route to effective one-dimensional few-body physics without an external waveguide. The anisotropic microwave potential supplies the axis, and quantized angular motion supplies the core. Exact two-molecule calculations show where this reduction works and where three-dimensional threshold laws recover control. The three-molecule calculation then predicts a compact chain state, a Girardeau-type energy-density duality, and an extensive attractive correction for larger chains.\n\nThe second paper solves the two-body problem that an external waveguide creates when it traps two species differently. Center-of-mass and relative motion mix, several closed molecular levels become visible to the open channel, and a narrow Feshbach resonance leaves a large energy dependence. The authors retain that information in $a_d$ and $R_d$, verify the dimer pole, and predict q1D and q2D trimers and tetramers with STM equations. The strongest experimental window belongs to q1D trimers under tight confinement.\n\nThe calculations support a precise reading rule. Identify the high-dimensional benchmark, write down the information retained after projection, and mark every later result that uses the reduced object outside the benchmarked sector. 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"index": 21, "display": false, "tex": "\\phi=\\pm\\pi/2", "line": 107, "sha256": "843993855cb65dfb4f94df29a7ea9ddd3f7ff7d42eb22a517cd0f0556bfbf63c" }, { "index": 22, "display": false, "tex": "C_3", "line": 107, "sha256": "aa663e271377445dff58aff38e430135489e71cfcb674e437bd605706a8e8221" }, { "index": 23, "display": false, "tex": "-4C_3/r^3", "line": 107, "sha256": "4d311cd0166115ba504f06967064169fccfa62c3075a714c0d152f3d8cbfc3bb" }, { "index": 24, "display": false, "tex": "x", "line": 107, "sha256": "2d711642b726b04401627ca9fbac32f5c8530fb1903cc4db02258717921a4881" }, { "index": 25, "display": false, "tex": "z", "line": 107, "sha256": "594e519ae499312b29433b7dd8a97ff068defcba9755b6d5d00e84c524d67b06" }, { "index": 26, "display": false, "tex": "+2C_3/r^3", "line": 107, "sha256": "377a586c90ab5ee7d581b63eac3c8616957283b858af353aa9f626fef93d1fc1" }, { "index": 27, "display": false, "tex": "C_6", "line": 107, "sha256": "dcb4e0ba5e2088bf3466b2f36f9b984098453f30d75a447198d46cad7f9e941a" }, { "index": 28, "display": true, "tex": "l_d=\\frac{m d^2}{48\\pi\\varepsilon_0\\hbar^2},", "line": 111, "sha256": "ab01097d25b37fa45296e6af43e6ece2c392a02ec87c858905aa81c06c07ff82" }, { "index": 29, "display": false, "tex": "l_u=l_d/20", "line": 115, "sha256": "e8041363969fac25f79ec9f960472e4f1e7f6459b9210946ce3a2f93e79361d9" }, { "index": 30, "display": false, "tex": "E_u=\\hbar^2/(m l_u^2)", "line": 115, "sha256": "16b016f3866778f809572fb485d7e8a37eeeedd4724dad433aa220e63f2a68ad" }, { "index": 31, "display": false, "tex": "l_d/(10^4a_0)\\simeq1.1,2.6,8.3", "line": 115, "sha256": "7ce54c13d78277cd2eed595f73e0ada4570d66d6db54324bfee705e63586c8d5" }, { "index": 32, "display": false, "tex": "l_u\\simeq550a_0", "line": 115, "sha256": "59e984c31ccc78f8c0d0cd9a43be9a32ad152120854d1d263aa04762f86b7999" }, { "index": 33, "display": false, "tex": "\\Omega=2\\pi\\times10\\,\\mathrm{MHz}", "line": 115, "sha256": "f9421b0f90ec96ae6f7810f82ecf10ef356f6389464e284afc0d4bc592b63ed2" }, { "index": 34, 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"V(r,\\delta\\theta,\\delta\\phi)\n=-\\frac{4C_3}{r^3}\n+\\left(\\frac{6C_3}{r^3}+\\frac{4C_6}{r^6}\\right)\n(\\delta\\theta^2+\\delta\\phi^2).", "line": 131, "sha256": "fd024d11b9c94c6fda77a8f13ce06b76f998209c35cd5c70a533f9e013ecd8b0" }, { "index": 41, "display": true, "tex": "-\\frac{\\hbar^2}{m r^2}\n\\left(\n\\frac{\\partial^2}{\\partial\\delta\\theta^2}\n+\\frac{\\partial^2}{\\partial\\delta\\phi^2}\n\\right).", "line": 140, "sha256": "abc4484ecdc911ef2b624c4944d7ab19a9daada7661063b2fd18d959d17a6039" }, { "index": 42, "display": false, "tex": "r", "line": 148, "sha256": "454349e422f05297191ead13e21d3db520e5abef52055e4964b82fb213f593a1" }, { "index": 43, "display": true, "tex": "\\epsilon_\\nu(r)\n=(\\nu+1)\n\\sqrt{\\frac{4\\hbar^2}{m}\n\\left(\\frac{6C_3}{r^5}+\\frac{4C_6}{r^8}\\right)}.", "line": 150, "sha256": "d00f3ed46ed5d4986ef26dec98e100af0db24e530275150472e3b61d3d2a74a2" }, { "index": 44, "display": false, "tex": "r^{-2}", "line": 157, "sha256": 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false, "tex": "\\langle\\partial_{\\delta\\theta}^2\\rangle", "line": 159, "sha256": "783cd3ea659d1790a2b1c7c12cd985dc21e682e5e8592c55cef6ecf9d7ce690e" }, { "index": 52, "display": false, "tex": "1/r^2", "line": 159, "sha256": "e7e99afd235212530e99dae0d9556f3e1e05040abc8602ee00b378f38d9d321f" }, { "index": 53, "display": true, "tex": "\\Psi^{(2)}(\\mathbf r)\n=\\frac1r\\sum_\\nu F_\\nu(y)\n\\psi_\\nu(y;\\delta\\theta,\\delta\\phi),", "line": 165, "sha256": "4fd2ff79c45f90076e78e75d8e366059260d12c94865c00b5d73d2a91e242a01" }, { "index": 54, "display": false, "tex": "y=r\\sin\\phi_0", "line": 171, "sha256": "7cad9f6984ef2dc87b9c91d5274d158a8a1b71ca69876832d93c3857db23d814" }, { "index": 55, "display": false, "tex": "\\nu", "line": 171, "sha256": "84eb2b123d1102b8f7c00b34116df32107cffa82d46efbac7fd9543fa7459c38" }, { "index": 56, "display": true, "tex": "\\left[\n-\\frac{\\hbar^2}{m}\\frac{\\partial^2}{\\partial y^2}\n+U^{(2)}(|y|)\n\\right]F_0(y)\n=E_{1\\mathrm D}^{(2)}F_0(y),", "line": 173, "sha256": "2a632fac92bb20fcaad5a8ab9121644d7f187917febcec23e1bf85d904132ae9" }, { "index": 57, "display": true, "tex": "U^{(2)}(r)\n=-\\frac{4C_3}{r^3}\n+\\sqrt{\\frac{4\\hbar^2}{m}\n\\left(\\frac{6C_3}{r^5}+\\frac{4C_6}{r^8}\\right)}.", "line": 183, "sha256": "3d732cd6c2bf08e7f98e4a7fefb4081093882d6190332b75cda032f434d09cf1" }, { "index": 58, "display": false, "tex": "y", "line": 190, "sha256": "a1fce4363854ff888cff4b8e7875d600c2682390412a8cf79b37d0b11148b0fa" }, { "index": 59, "display": false, "tex": "C_6", "line": 192, "sha256": "dcb4e0ba5e2088bf3466b2f36f9b984098453f30d75a447198d46cad7f9e941a" }, { "index": 60, "display": true, "tex": "U^{(2)}(r)\n\\simeq-\\frac{4C_3}{r^3}\n+4\\hbar\\sqrt{\\frac{C_6}{m}}\\frac1{r^4}.", "line": 194, "sha256": "93de52d68deaf16212b270e6ee4b0b178262180980ef784364ff506be853746c" }, { "index": 61, "display": true, "tex": "r_m\\simeq\\frac{4\\hbar}{3C_3}\\sqrt{\\frac{C_6}{m}}\n=4\\sqrt2\\,(1+\\delta_r^2)^{1/4}l_\\Omega,\n\\qquad\nl_\\Omega=\\sqrt{\\frac{\\hbar}{m\\Omega}}.", "line": 202, "sha256": "16c12add61ec53bcdeb91ff41993f800c3bd76fdcab4378c87c3699fe1f3ff06" }, { "index": 62, "display": false, "tex": "\\delta_r=0.2", "line": 209, "sha256": "04f505b3a0e3dc51195db7a886ce68b167b7ebb161989b82e285c18e5048a684" }, { "index": 63, "display": false, "tex": "r_m\\simeq4\\sqrt2,l_\\Omega", "line": 209, "sha256": "caefe2da0c023beaf0a6b30ed1f20ffae4e92cb3ba8bc180c681fdb044f0adc2" }, { "index": 64, "display": false, "tex": "10\\,\\mathrm{MHz}", "line": 209, "sha256": "b7bd8cb21e3843f7b14718e7b35542ec93de3804e014e5682f4d5ea0a809d7e3" }, { "index": 65, "display": false, "tex": "r_m=428a_0", "line": 209, "sha256": "7c6406e84a411dc4ea75e368b9dc8af8aa955daee55eae5679335ed9568201c8" }, { "index": 66, "display": false, "tex": "r/r_m=1", "line": 213, "sha256": 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"tex": "|E_b^{(2)}|\\lesssim0.004E_u", "line": 272, "sha256": "61c2d5a4bd47f780bacbd77031a9312bf715343b789073050b14cffe95adc97e" }, { "index": 97, "display": false, "tex": "r_m", "line": 272, "sha256": "272cb3ca57c00592f5dff75eb6282a9ab8e453ee00c55b1f19ed4cb93276fadc" }, { "index": 98, "display": false, "tex": "N", "line": 276, "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b" }, { "index": 99, "display": false, "tex": "y_10)\n=|\\Psi_g^{(2)}(y)|^2\n+\\int_0^y dy'\n\\left|\n\\Psi_g^{(2)}(y')\n\\Psi_g^{(2)}(y-y')\n\\right|^2.", "line": 404, "sha256": "a6d80056d81bee7be08032a2401eb2b0ae200e772d8fd0f12d51f0376c8244ec" }, { "index": 147, "display": true, "tex": "\\widetilde U^{(3)}\n=U^{(2)}(|y_{12}|)\n+U^{(2)}(|y_{23}|)\n+U^{(2)}(|y_{13}|).", "line": 420, "sha256": "5eb5f38bff411b704023da5c9b3a488d89fae0baa34f5ffd0e072fa7562d5b44" }, { "index": 148, "display": false, "tex": "y_1