Tetramer Model Overview

• The tetramer model is a four-unit effective description used in fields like few-body physics, quantum magnetism, and adsorption to capture novel binding and ordering phenomena.
• It employs methods such as renormalized EFT and Faddeev–Yakubovsky equations to analyze thresholds, universal scaling, and finite-range corrections in four-body interactions.
• Its diverse applications—from ultracold gas tetramers to SU(4) plaquette models and self-assembled DNA stacks—demonstrate its role in uncovering new physics beyond dimer and trimer descriptions.

Searching arXiv for the cited tetramer-model papers and related work. arxiv_search(query="tetramer model four-body systems adsorption tetramer universal tetramer SU(4) tetramer He tetramer", max_results=10) The term tetramer model designates a modeling strategy in which the elementary object is a four-particle, four-site, or four-monomer unit. In current arXiv literature, the phrase does not denote a single formalism. It instead covers renormalized four-body EFT and Faddeev–Yakubovsky descriptions of universal bosonic bound states, $(3+1)$ and heteronuclear tetramers in ultracold gases, long-range field-linked dimer-dimer states of dipolar molecules, rigid four-sphere adsorbates in random sequential adsorption, four-spin or four-site clusters in Heisenberg, Hubbard, and SU(4) lattice Hamiltonians, and coarse-grained tetramer building blocks in DNA and eumelanin self-assembly [(Lin, 2023); (Liu et al., 2022); (Quéméner et al., 2023); (Cieśla, 2013); (Matsuda et al., 2015); (Smith et al., 2024); (Sapunkov et al., 2019)]. In all of these uses, the tetramer is the minimal four-body motif through which binding, ordering, transport, spectroscopy, or self-assembly is organized.

1. Scope and formal meanings

In the literature surveyed here, a tetramer model is best understood as a four-unit effective description whose microscopic meaning depends on the field. In few-body physics, the tetramer is a bound or resonant four-body state, often tied to a trimer threshold or to a separate four-body scale. In condensed-matter settings, it is a four-site cluster or plaquette with internal exchange or hopping structure. In adsorption and soft-matter problems, it is a rigid four-sphere or four-monomer aggregate whose geometry controls excluded volume, packing, or stacking [(Lin, 2023); (Frederico et al., 2012); (Matsuda et al., 2015); (Cieśla, 2013)].

Domain	Tetramer unit	Representative formulation
Few-body quantum physics	Four bosons, $(3+1)$ cluster, heteronuclear $H_3L$, or field-linked dimer-dimer complex	EFT, FY/AGS equations, GEM, two-channel models
Quantum magnetism and correlated electrons	Four spins or four sites	Heisenberg, extended Hubbard, SU(4) plaquette Hamiltonians
Adsorption, biomolecular, and chemical self-assembly	Four identical spheres, 4-bp duplex aggregate, planar four-monomer layer	RSA, liquid-crystal geometry, piecewise Ising model

This multiplicity is not merely terminological. It implies that the “tetramer” can be a compact cluster, a halo-like state, a resonant continuum pole, a rigid adsorbate, a singlet plaquette, or a stacked molecular layer. A recurring implication is that the four-body unit is often the smallest object for which qualitatively new structure appears: an independent four-body scale, a new optical gap, a checkerboard plaquette phase, or a left-handed twist-grain-boundary stack.

2. Universal four-body physics and renormalized few-body models

A major use of the tetramer model occurs in short-range few-body physics. In leading-order EFT for short-range bosons with large scattering length, the four-body system is built on a renormalized two- and three-body sector containing two-body contact interactions and the three-body contact interaction already required for the Efimov problem. A central issue is the appearance of deep trimers once the ultraviolet cutoff exceeds a threshold $\Lambda_t$. The four-body calculation in "Four-Body Systems at Large Cutoffs in Effective Field Theory" rewrites the kernel in terms of the three-body scattering amplitude, with the three-body subsystem resummed through

$T_{33}=K_{33}(1-K_{33})^{-1},$

so that deep trimer poles can be included or subtracted by a Cauchy principal value prescription. In that formulation, tetramer observables remain stable at large cutoff, and no four-body force is needed at leading order provided the trimer poles are handled correctly. At unitarity, there are two tetramers associated with each trimer, with

$$E_4^{(0)} = 4.60(1)B_3^{(0)}, \qquad \Gamma_4^{(0)}/2 = 0.0160(1)B_3^{(0)},$$

and

$$E_4^{(1)} = 1.0022(3)B_3^{(0)}, \qquad \Gamma_4^{(1)}/2 = 2.57(2)\times 10^{-4}B_3^{(0)}.$$

For cold $^4$He at leading order, the same framework gives $E_4^{(0)} = 526.1(5)\,\mathrm{mK}$ and $E_4^{(1)} = 128.517(1)\,\mathrm{mK}$ after subtracting deep-trimer poles (Lin, 2023).

Zero-range Faddeev–Yakubovsky analyses frame the same problem differently. "Universality in Four-Boson Systems" introduces subtracted Green’s functions with two renormalization scales, $(3+1)$0 and $(3+1)$1, so that the three-body and four-body sectors are regularized independently. In this picture, the $(3+1)$2-type $(3+1)$3 and $(3+1)$4-type $(3+1)$5 amplitudes are both essential, the tetramer spectrum depends on an independent four-body scale, and at unitarity a tetramer emerges from the atom-trimer threshold at the universal ratio $(3+1)$6 (Frederico et al., 2012). A Hamiltonian realization of the same idea appears in "Universal tetramer limit-cycle at the unitarity limit", where independent two-, three-, and four-body Gaussian short-range interactions disentangle the Efimov cycle from a distinct tetramer cycle; the universal content is encoded in a model-independent correlation between successive tetramer energies at fixed trimer energy $(3+1)$7 (Frederico et al., 2023).

Realistic helium calculations supply the finite-range benchmark for this universal sector. Using GEM with the LM2M2 potential, "Variational calculation of $(3+1)$8He tetramer ground and excited states using a realistic pair potential" finds

$(3+1)$9

with the excited tetramer only $H_3L$0 below the trimer ground-state threshold. The same work emphasizes that the short-range pair correlation for $H_3L$1 Å is essentially the same in dimer, trimer, and tetramer, and proposes the general estimate

$H_3L$2

for the first excited state of $H_3L$3He$H_3L$4 (Hiyama et al., 2011). A complementary analysis with seven realistic He-He potentials reports generalized Tjon lines among $H_3L$5, $H_3L$6, $H_3L$7, and $H_3L$8, and derives the dimerlike-pair relation

$H_3L$9

which tracks the $\Lambda_t$0–$\Lambda_t$1 correlation almost exactly (Hiyama et al., 2012).

Range corrections remain quantitatively important. For cesium near broad $\Lambda_t$2-wave Feshbach resonances, effective-range corrections to tetramer dissociation positions yield an extracted weighted average $\Lambda_t$3, consistent with the van der Waals tail of $\Lambda_t$4 (Hadizadeh et al., 2012). In realistic $\Lambda_t$5He atom-trimer scattering, the second excited tetramer is not bound but appears as a resonance below the excited trimer threshold, with

$\Lambda_t$6

for LM2M2 and

$\Lambda_t$7

for PCJS. The realistic widths are about twice the universal zero-range value, so finite-range effects are especially significant for decay properties (Deltuva, 12 Feb 2026).

3. Heteronuclear, fermionic, and field-linked tetramer constructions

Tetramer models are not confined to identical bosons. In two-dimensional mass-imbalanced fermionic mixtures, the tetramer is a universal $\Lambda_t$8 bound state of three identical heavy fermions plus one light atom. The critical heavy-light mass ratio for the ground-state tetramer to appear below the trimer threshold is

$\Lambda_t$9

and the state lies in the zero-total-angular-momentum sector $T_{33}=K_{33}(1-K_{33})^{-1},$0. The angular structure is most naturally described in the dimer-fermion frame, where $T_{33}=K_{33}(1-K_{33})^{-1},$1 and $T_{33}=K_{33}(1-K_{33})^{-1},$2 combine to give $T_{33}=K_{33}(1-K_{33})^{-1},$3. Momentum-space correlations exhibit a triangular pattern, interpreted in the paper as an interaction-induced crystal (Liu et al., 2022).

A more radical fermionic construction is the two-channel model with resonant three-body, not two-body, interaction. In "Universal $T_{33}=K_{33}(1-K_{33})^{-1},$4-wave tetramers in low-dimensional fermionic systems with three-body interaction", the basic object is a composite closed-channel fermion $T_{33}=K_{33}(1-K_{33})^{-1},$5 coupled to three distinguishable fermions. In fractional dimensions above 1D, the four-body sector can exhibit a universal $T_{33}=K_{33}(1-K_{33})^{-1},$6-wave Efimov-like effect. The tetramer energies then satisfy

$T_{33}=K_{33}(1-K_{33})^{-1},$7

and for the numerical example $T_{33}=K_{33}(1-K_{33})^{-1},$8, $T_{33}=K_{33}(1-K_{33})^{-1},$9, the universal ratio is $$E_4^{(0)} = 4.60(1)B_3^{(0)}, \qquad \Gamma_4^{(0)}/2 = 0.0160(1)B_3^{(0)},$$0. In $$E_4^{(0)} = 4.60(1)B_3^{(0)}, \qquad \Gamma_4^{(0)}/2 = 0.0160(1)B_3^{(0)},$$1, there is no Efimov-like tower, but bound tetramers still occur, with thresholds shifted upward by finite-range effects parameterized by $$E_4^{(0)} = 4.60(1)B_3^{(0)}, \qquad \Gamma_4^{(0)}/2 = 0.0160(1)B_3^{(0)},$$2 (Polkanov et al., 2024).

Heteronuclear bosonic systems provide a different threshold problem. For one light particle $$E_4^{(0)} = 4.60(1)B_3^{(0)}, \qquad \Gamma_4^{(0)}/2 = 0.0160(1)B_3^{(0)},$$3 and three heavy bosons $$E_4^{(0)} = 4.60(1)B_3^{(0)}, \qquad \Gamma_4^{(0)}/2 = 0.0160(1)B_3^{(0)},$$4, with resonant heavy-light and negligible heavy-heavy interactions, GEM calculations show that the trimer $$E_4^{(0)} = 4.60(1)B_3^{(0)}, \qquad \Gamma_4^{(0)}/2 = 0.0160(1)B_3^{(0)},$$5 and tetramer $$E_4^{(0)} = 4.60(1)B_3^{(0)}, \qquad \Gamma_4^{(0)}/2 = 0.0160(1)B_3^{(0)},$$6 cross the heavy-light dimer threshold at almost the same point. The separation

$$E_4^{(0)} = 4.60(1)B_3^{(0)}, \qquad \Gamma_4^{(0)}/2 = 0.0160(1)B_3^{(0)},$$7

decreases as the mass ratio $$E_4^{(0)} = 4.60(1)B_3^{(0)}, \qquad \Gamma_4^{(0)}/2 = 0.0160(1)B_3^{(0)},$$8 decreases and is almost coincident for $$E_4^{(0)} = 4.60(1)B_3^{(0)}, \qquad \Gamma_4^{(0)}/2 = 0.0160(1)B_3^{(0)},$$9. The paper leaves open whether remaining discrepancies with effective three-body EFT reflect numerical limitations or limitations of the dimer-atom-atom reduction near threshold (Schmickler et al., 2017).

An entirely different meaning of tetramer appears in ultracold dipolar molecules. There the tetramer is a weakly bound dimer-dimer complex formed from two molecules through a long-range field-linked state created by microwave dressing. For $$E_4^{(1)} = 1.0022(3)B_3^{(0)}, \qquad \Gamma_4^{(1)}/2 = 2.57(2)\times 10^{-4}B_3^{(0)}.$$0, the molecules are initially in $$E_4^{(1)} = 1.0022(3)B_3^{(0)}, \qquad \Gamma_4^{(1)}/2 = 2.57(2)\times 10^{-4}B_3^{(0)}.$$1, the Rabi coupling is $$E_4^{(1)} = 1.0022(3)B_3^{(0)}, \qquad \Gamma_4^{(1)}/2 = 2.57(2)\times 10^{-4}B_3^{(0)}.$$2, the dressed dipole-dipole interaction behaves as $$E_4^{(1)} = 1.0022(3)B_3^{(0)}, \qquad \Gamma_4^{(1)}/2 = 2.57(2)\times 10^{-4}B_3^{(0)}.$$3, and a scattering-length divergence and sign change occur near $$E_4^{(1)} = 1.0022(3)B_3^{(0)}, \qquad \Gamma_4^{(1)}/2 = 2.57(2)\times 10^{-4}B_3^{(0)}.$$4 MHz as the field-linked state crosses threshold. Electro-association is then achieved by ramping the field so that the lowest harmonic-oscillator-like state is adiabatically converted into the bound field-linked state. The dynamical bottleneck is an avoided crossing between $$E_4^{(1)} = 1.0022(3)B_3^{(0)}, \qquad \Gamma_4^{(1)}/2 = 2.57(2)\times 10^{-4}B_3^{(0)}.$$5 and $$E_4^{(1)} = 1.0022(3)B_3^{(0)}, \qquad \Gamma_4^{(1)}/2 = 2.57(2)\times 10^{-4}B_3^{(0)}.$$6, and the reported useful conversion window is roughly

$$E_4^{(1)} = 1.0022(3)B_3^{(0)}, \qquad \Gamma_4^{(1)}/2 = 2.57(2)\times 10^{-4}B_3^{(0)}.$$7

with an optimal value around $$E_4^{(1)} = 1.0022(3)B_3^{(0)}, \qquad \Gamma_4^{(1)}/2 = 2.57(2)\times 10^{-4}B_3^{(0)}.$$8 and an association probability of about $$E_4^{(1)} = 1.0022(3)B_3^{(0)}, \qquad \Gamma_4^{(1)}/2 = 2.57(2)\times 10^{-4}B_3^{(0)}.$$9 for $^4$0 Hz (Quéméner et al., 2023).

4. Tetramers in quantum magnetism, correlated electrons, and SU(4) plaquette physics

In magnetic and electronic models, the tetramer is typically a four-site cluster with internal couplings stronger than the couplings between clusters. For the diamond-shaped $^4$1 Heisenberg tetramer compound Cu$^4$2PO$^4$3OH, the simplest one-parameter square tetramer model predicts $^4$4 and $^4$5, but this fails to explain the observed neutron-scattering modes at $^4$6 meV and $^4$7 meV. The experimentally consistent description requires a diamond-shaped tetramer with diagonal intratetramer interactions $^4$8 and $^4$9, weak intertetramer couplings $E_4^{(0)} = 526.1(5)\,\mathrm{mK}$0 and $E_4^{(0)} = 526.1(5)\,\mathrm{mK}$1, and a structural distortion below $E_4^{(0)} = 526.1(5)\,\mathrm{mK}$2 K that splits $E_4^{(0)} = 526.1(5)\,\mathrm{mK}$3 into inequivalent $E_4^{(0)} = 526.1(5)\,\mathrm{mK}$4 and $E_4^{(0)} = 526.1(5)\,\mathrm{mK}$5. A representative parameter set is

$E_4^{(0)} = 526.1(5)\,\mathrm{mK}$6

and the two observed excitations are assigned to singlet-to-triplet-1 and singlet-to-nearly-degenerate triplet-2/triplet-3 transitions (Matsuda et al., 2015).

For CuInVO$E_4^{(0)} = 526.1(5)\,\mathrm{mK}$7, the tetramer is the unit cell of a one-dimensional spin-$E_4^{(0)} = 526.1(5)\,\mathrm{mK}$8 Heisenberg chain of coupled tetramers with Hamiltonian

$E_4^{(0)} = 526.1(5)\,\mathrm{mK}$9

DMRG reveals two multimerized VBS phases: a tetramer-singlet phase and a dimer-singlet phase. The spin gap vanishes continuously at the phase boundary, and the central charge extrapolates to $E_4^{(1)} = 128.517(1)\,\mathrm{mK}$0 at the critical point while remaining $E_4^{(1)} = 128.517(1)\,\mathrm{mK}$1 in the gapped phases. The experimental magnetization curve is reproduced only when the exchange parameters lie very close to that singlet-singlet quantum critical point (Reja et al., 2019).

Tetramerization also appears as a bond-modulation pattern in electronic models. In the $E_4^{(1)} = 128.517(1)\,\mathrm{mK}$2-filled one-dimensional extended Hubbard model for the spin-Peierls ground state of $E_4^{(1)} = 128.517(1)\,\mathrm{mK}$3, tetramerization is encoded in the four-site hopping pattern

$E_4^{(1)} = 128.517(1)\,\mathrm{mK}$4

with $E_4^{(1)} = 128.517(1)\,\mathrm{mK}$5 and $E_4^{(1)} = 128.517(1)\,\mathrm{mK}$6. Exact diagonalization at $E_4^{(1)} = 128.517(1)\,\mathrm{mK}$7 finds three low-energy optical peaks $E_4^{(1)} = 128.517(1)\,\mathrm{mK}$8, $E_4^{(1)} = 128.517(1)\,\mathrm{mK}$9, and $(3+1)$00. In weak coupling, the most characteristic tetramerization effect is the new very small Fermi-surface gap

$(3+1)$01

which is absent in the same form in the ordinary dimerized description (Yamaguchi et al., 2018).

SU(4) models generalize the same cluster logic to multicomponent Mott insulators. On a square optical superlattice with plaquette exchange $(3+1)$02 and interplaquette exchange $(3+1)$03, the isolated tetramer has a nondegenerate SU(4) singlet ground state and undergoes a level crossing at $(3+1)$04 into an $(3+1)$05 plaquette when a population-imbalance field $(3+1)$06 is applied. Extended linear flavor-wave theory then yields a phase diagram containing an SU(4)-singlet phase, a high-field $(3+1)$07 regime, a checkerboard singlet-$(3+1)$08 solid with $(3+1)$09, and an $(3+1)$10 phase. The critical coupling for the onset of $(3+1)$11 is

$(3+1)$12

and is essentially independent of $(3+1)$13 (Miyazaki et al., 2022).

On the honeycomb lattice, SU(4) tetramerization competes with an algebraic spin-orbital liquid. The tetramerized phase consists of four-site SU(4) singlet plaquettes with fourfold degeneracy, and the order parameter $(3+1)$14 distinguishes the uniform and fully tetramerized limits. Variational Monte Carlo on the projected $(3+1)$15-flux state finds that the nearest-neighbor SU(4) Heisenberg point remains robustly non-tetramerized, while a finite next-nearest exchange is needed to induce tetramerization; a rough energetic estimate places the transition around $(3+1)$16 (Lajko et al., 2013).

A related cluster phase occurs in the spin-$(3+1)$17 Heisenberg antiferromagnet on the diamond-decorated square lattice, where the dimer-tetramer phase is a product state of singlet dimers and singlet tetramers. Its ground-state manifold maps exactly onto a classical hard-dimer model on the square lattice and retains macroscopic degeneracy even under magnetic field. Near the dimer-tetramer to monomer-dimer boundary

$(3+1)$18

the low-temperature thermodynamics requires an extended monomer-dimer lattice-gas model. The residual entropy approaches $(3+1)$19 per spin, and adiabatic demagnetization exhibits an enhanced magnetocaloric effect (Karlova et al., 2024).

5. Geometric, adsorption, and self-assembled tetramer models

A distinct class of tetramer models treats the tetramer as a rigid geometric aggregate. In random sequential adsorption on a two-dimensional homogeneous collector, a tetramer is modeled as four identical spheres of radius $(3+1)$20, with monomer centers on the vertices of either a rhomboid or a square. The RSA process is irreversible, the collector has area $(3+1)$21, the time variable is

$(3+1)$22

and simulations were run to $(3+1)$23 with 100 independent runs per geometry. The maximal random coverage ratios are

$(3+1)$24

with relative error about $(3+1)$25. The low-coverage ASF fits are

$(3+1)$26

for the rhomboid and

$(3+1)$27

for the square. The effective dimensions extracted from Feder-law kinetics, $(3+1)$28 and $(3+1)$29, show that orientational degrees of freedom strongly affect adsorption on a 2D substrate (Cieśla, 2013).

In aqueous DNA self-assembly, the tetramer is the sequence $(3+1)$30-GTAC-$(3+1)$31, a self-complementary palindrome forming a 4-base-pair blunt-ended Watson/Crick duplex. Each duplex is about $(3+1)$32 Å long, with hydrophobic blunt ends that stack end-to-end into effective long-DNA-like columns of diameter $(3+1)$33 nm. These columns assemble into monolayer sheets, which in turn form a lamellar twist-grain-boundary phase in which successive layers are rotated by $(3+1)$34. The reported geometric parameters are

$(3+1)$35

and the TGB helix is concluded to be left-handed because the right-handed alternative would generate a different diffraction signature, including a second harmonic not observed experimentally (Smith et al., 2024).

In eumelanin theory, the tetramer is a planar assembly of four monomers occupying four quadrants, with stacked double tetramers forming the next structural level. The monomer species are HQ, IQ, and MQ, and the energetics are represented by a piecewise Ising model with occupation terms $(3+1)$36, in-plane couplings $(3+1)$37 and $(3+1)$38, and out-of-plane couplings $(3+1)$39, $(3+1)$40, and $(3+1)$41. The double-tetramer phase space contains

$(3+1)$42

named configurations and $(3+1)$43 non-degenerate double tetramers after symmetry reduction. Both one-step and three-step fits achieve $(3+1)$44, but the three-step decomposition—monomers $(3+1)$45 single tetramers $(3+1)$46 double tetramers—is the physically emphasized form. The average three-step coefficients are $(3+1)$47 eV, $(3+1)$48 eV, $(3+1)$49 eV, $(3+1)$50 eV, $(3+1)$51 eV, and $(3+1)$52 eV, showing that in-plane interactions dominate over stacking and that HQ-rich motifs are the most stable (Sapunkov et al., 2019).

6. Recurring structures, open disagreements, and interpretive lessons

The surveyed literature suggests two dominant architectures for tetramer models. In few-body physics, the tetramer is organized by threshold structure: trimer poles, atom-trimer or dimer-dimer channels, subtraction scales, or avoided crossings in dressed potentials. In lattice, adsorption, and self-assembly models, it is organized by internal geometry: bond inequivalence, plaquette symmetry, four-site singlets, rhomboid versus square footprints, or in-plane versus out-of-plane couplings [(Lin, 2023); (Frederico et al., 2012); (Matsuda et al., 2015); (Cieśla, 2013); (Sapunkov et al., 2019)]. This suggests that the tetramer concept is most useful when four-body organization introduces a qualitative degree of freedom absent in the dimer or trimer description.

Several objective disagreements recur. A central few-body controversy concerns whether an independent four-body scale is fundamental. Zero-range FY analyses and Hamiltonian studies at unitarity identify an independent four-body scale or four-body limit cycle [(Frederico et al., 2012); (Frederico et al., 2023)], whereas the large-cutoff EFT calculation reports that no four-body force is needed at leading order if deep-trimer poles are treated correctly (Lin, 2023). Another disagreement appears in heteronuclear near-threshold physics, where full four-body calculations favor a very large effective scaling factor for dimer-atom-atom Efimov behavior, in tension with a smaller Born–Oppenheimer estimate discussed in the same work (Schmickler et al., 2017). In realistic helium and cesium systems, zero-range universality is systematically modified by effective-range or finite-range effects, especially in widths and threshold shifts [(Hadizadeh et al., 2012); (Deltuva, 12 Feb 2026)].

A common misconception is that a tetramer must be a compact four-body molecule. The literature does not support that restriction. The $(3+1)$53He excited tetramer is a shallow atom-trimer halo-like state, the dipolar-molecule tetramer is a long-range field-linked dimer-dimer complex, the RSA tetramer is a rigid four-sphere particle, and the GTAC tetramer is a short duplex that only becomes structurally significant through hierarchical stacking and liquid-crystal ordering [(Hiyama et al., 2011); (Quéméner et al., 2023); (Cieśla, 2013); (Smith et al., 2024)]. Another misconception is that tetramerization always lowers symmetry in the same way. In SU(4) plaquette models it may yield a checkerboard singlet solid or compete unsuccessfully against an algebraic liquid; in spin chains it can define one VBS sector of a quantum critical competition; in adsorption it merely specifies the internal footprint of a rigid object [(Miyazaki et al., 2022); (Lajko et al., 2013); (Reja et al., 2019); (Cieśla, 2013)].

The present literature also indicates several forward directions. The cutoff-stable EFT treatment of four bosons is presented as a gateway to higher-order and more-body calculations, including scattering observables, loss rates, and four-nucleon systems (Lin, 2023). Electro-association of field-linked tetramers is proposed as a route toward assembling larger ultracold polyatomic molecules (Quéméner et al., 2023). The eumelanin piecewise Ising construction is explicitly designed for extension to hexamers, octomers, and triple or quadruple tetramer stacks (Sapunkov et al., 2019). Across these domains, the tetramer model functions less as a single theory than as a reusable structural principle: a four-unit building block whose internal couplings, thresholds, and geometry are rich enough to generate qualitatively new physics.