TesseraQ: Multi-Disciplinary Frameworks

• TesseraQ is a set of advanced frameworks spanning astrophysics, machine learning quantization, soft matter assembly, and quantum walk theory, each with rigorous statistical and algorithmic calibration.
• It employs methodologies such as supervised machine learning for star cataloging, progressive adaptive rounding for ultra-low-bit quantization, and Sliced-Wasserstein alignment for distributional calibration.
• The framework serves as a benchmark for diverse applications, from analyzing eclipsing quadruple-star systems and programmable tessellation in colloidal systems to constructing spectral models in quantum walks.

TesseraQ denotes a set of distinct advanced frameworks and catalogs at the intersection of astrophysics, machine learning, condensed matter, and neural network compression. The “TesseraQ” designation appears in four research communities as: (1) the TESS-era Eclipsing Quadruple Star Catalog, (2) a post-training quantization (PTQ) algorithm for ultra-low-bit LLM compression, (3) a data-driven tool for programmable tessellations in soft-matter assembly, and (4) the structure and spectral theory of 2-tessellable quantum walks. Each instance incorporates sophisticated statistical, algorithmic, or quantum-theoretic methodology and serves as a benchmark or design framework in its respective domain.

1. TesseraQ as the TESS-era Eclipsing Quadruple Star Catalog

TesseraQ is the uniform catalog of quadruple-star candidates discovered using TESS Full-Frame Image (FFI) data across Sectors 1–54. The naming derives from “TESS” and “Quadruple,” following the TGV (TESS/Goddard/VSG) series. The catalog comprises 101 eclipsing quadruple-star systems, each displaying two independent binary eclipse signatures within a single lightcurve. Of these, 100 are dynamically consistent with a 2+2 hierarchy (two gravitationally bound detached binaries), and one (TIC 37376063) is a (2+1)+2 hierarchical quintuple exhibiting tertiary eclipses associated with a triply-eclipsing inner subsystem.

Identification merged supervised machine learning and citizen science. A convolutional neural network, trained on labeled FFI light curves, extracted high-probability eclipsing binaries (EBs) from a vast candidate pool. Visual Survey Group (VSG) members then scanned these for the simultaneous presence of two or more stable eclipse periodicities, ensuring both sets originated on-target via centroid motion analysis (accepted candidates require difference image centroid shifts ≲0.2 pixel for both components).

Cataloged ephemerides include periods, primary–secondary eclipse depths, and durations (uncertainties from $\chi^2$ fits over multiple analytic templates). Noteworthy objects include TIC 219006972 (remarkably compact, with a ~168 d outer orbit and large anti-correlated eclipse time variations) and TIC 37376063 (triply-eclipsing quintuple). Periods $P_A,\,P_B$ span $0.54$–$15.97$ d and $1.79$–$1089$ d, respectively.

Dynamical analysis leverages eclipse timing variations (ETVs) both from light-travel-time effects—parameterized by the outer binary’s Keplerian elements—and by dynamical quadrupole terms reflecting three-/four-body gravitational interactions. Significant ETVs (up to $0.1$ d amplitude) are directly observed, supporting genuine four-body dynamics. The absence of period ratio clustering at simple rational values indicates random hierarchical configuration consistent with TESS/OGLE results.

The catalog’s ML-vetted sample facilitates follow-up via: high-resolution spectroscopy for mass determination; AO/speckle imaging to resolve companions; ongoing ETV monitoring; and incorporation of Gaia astrometric orbital fits, which will help disentangle light-travel and dynamical ETV signals. TesseraQ thus provides a vetted resource for population, evolutionary, and dynamical studies of high-order multiple star systems (Kostov et al., 2023).

2. TesseraQ for Ultra-Low-Bit LLM Quantization

TesseraQ is a state-of-the-art technique for post-training quantization (PTQ) of LLMs at ultra-low (2–3)-bit precision, closing much of the accuracy gap to full-precision deployment. The core innovation is block-wise reconstruction and progressive adaptive rounding (PAR) of quantized weights, coupled to fine-tuned dequantization scaling.

Given a pre-trained full-precision model, weights are initially quantized per a uniform affine mapping with fixed step-size and zero-point. For each block (multi-layer subgraph) comprising self-attention, feedforward, and normalization, TesseraQ minimizes the reconstruction loss

$$\mathcal{L}_{\rm recon}(W, \widehat{W}) = \|\mathrm{Block}(W, X) - \mathrm{Block}(\widehat{W}, X)\|_F^2$$

by treating each weight’s rounding variable as a continuous parameter (soft rounding logit $\nu_{ij}$), transitioned progressively to hard assignment via staged optimization. “Hardened” variables are clamped each iteration based on their distance to quantization boundaries, ensuring stable convergence in highly quantized (2-bit) regimes and allowing block-wise calibration on a single GPU.

Dequantization scale tuning introduces a block-dependent offset, $2 \, \sigma(\delta) \in (0,2)$, correcting systematic quantization bias without the instability caused by updating quantization steps or zero-points directly.

TesseraQ integrates seamlessly atop scale-and-clipping PTQ approaches (e.g., AWQ, OmniQuant): existing stepsize and clip transforms are applied, after which TesseraQ refines only the rounding and dequantization stages. Empirical results on LLaMA-2-7B W2A16 quantization show perplexity reductions from 14.65 (AWQ baseline) to 6.82, average accuracy improvements from 50.52% to 59.27%, and robust gains across broader quantization schemes and model sizes. Progressive Adaptive Rounding and dequantization tuning together confer the full accuracy advantage; ablation disables either results in perceptible degradation (Li et al., 2024).

3. Sliced-Wasserstein Loss for Distributional Calibration in TesseraQ

Subsequent work introduces Sliced-Wasserstein (SW) distribution alignment loss as an augmentation to the TesseraQ framework, yielding further accuracy and perplexity gains, especially at extreme quantization. The SW loss measures the average $L^1$ (Wasserstein-1) distance between projected activations from the full-precision and quantized models over $n_{\rm proj}$ random directions, effectively aligning output distributions beyond pointwise MSE minimization: $$SW(P, Q) \approx \frac{1}{n_{\rm proj}} \sum_j W_1(\operatorname{sort}(P \cdot \theta_j), \operatorname{sort}(Q \cdot \theta_j))$$ where $P, Q$ are sets of output vectors and $W_1$ is 1D Wasserstein distance.

The augmented block loss becomes

$$L_{\mathrm{block}} = (1-w_{sw}) \|Y_{fp} - Y_{q}\|_2^2 + w_{sw} \operatorname{SW}(P_{fp}, P_q)$$

summed over blocks. This dual-objective efficiently regularizes distributional collapse in the quantized model, improving zero-shot downstream task accuracy by $0.9$–$7.6$\% and reducing perplexity further (e.g., on LLaMA-2-7B, W2A16g128 by up to $2.19$\%). Calibration overhead is low (≤5\% additional time for $n_{\rm proj}=1024$), and no extra inference cost is introduced. This approach notably stabilizes gradients and compensates for activation mismatch in aggressive quantization (Cao et al., 11 Jan 2026).

4. TesseraQ in Programmable Tessellation and Soft Matter Assembly

TesseraQ also references a pattern-prediction and design protocol in programmable assembly of soft microparticles. Here, TesseraQ is a computational-experimental data framework integrating: (i) a two-step Langmuir–Blodgett double-deposition protocol for controlled stacking of PNIPAM microgel monolayers, (ii) a catalog of experimentally and simulation-validated 2D tessellation structures, (iii) a generalized Hertzian interparticle potential, and (iv) a $T=0$ thermodynamic phase map in the $(\varphi_1, \varphi_2)$ packing-fraction plane.

Experimental protocol:

• Layer 1 is deposited and immobilized as a hexagonal template.
• Layer 2, deposited orthogonally, occupies interstitial positions or constructs new motifs, with both local and global packing gradient control.

Observed structures include rectangular, honeycomb, rhomboidal, hexagonal superlattices, zig-zag/herringbone, and local dodecagonal motifs, with coordination numbers $Z=3$–$8$ and explicit primitive cell and wallpaper group assignments. Phase transitions and global minimum patterns are determined through comparison of $T=0$ free-energy densities, $f_i(\varphi)$, computed via MD using $V(r) = \varepsilon (1 - r/\sigma)^\alpha$, with experimentally-fitted $\alpha=1.9\pm0.1$. The protocol defines simple “design rules”: for force-law $\alpha < 2$, both low- and high-coordination tessellations are accessible, and each region of $(\varphi_1,\varphi_2)$ space corresponds to a unique wallpaper group and cell geometry.

TesseraQ outputs the optimal structure given user-specified ($\alpha$, $\varphi_1$, $\varphi_2$), enabling rapid rational design of 2D colloidal patterns (Grillo et al., 2019).

5. TesseraQ and 2-Tessellable Quantum Walks

In quantum walk theory, TesseraQ denotes the class of 2-tessellable (“2-tessera-quantum”) walks on graphs, as formulated by Higuchi et al. These quantum walks are defined on graphs $G=(V,E)$ admitting two tessellations $\mathcal{T}_1$, $\mathcal{T}_2$—decompositions of $V$ into cliques covering all edges. The evolution operator

$U_\theta = -e^{i\theta H_B}e^{i\theta H_A}$

acts on $\ell^2(V)$, where $H_A, H_B$ are reflections onto the polygon spans of $\mathcal{T}_1, \mathcal{T}_2$. This construction generalizes staggered and Grover quantum walks by embedding the model within the structure of bipartite multigraphs (line graphs of $G$) and a quantum Markov chain formalism.

Spectral theory exploits the double-discriminant matrix $T$ on $\ell^2(\mathcal{T}_1\sqcup\mathcal{T}_2)$, with eigenvalues $\mu\in[-1,1]$ mediating the nontrivial band structure. The quantum detailed balance (QDB) condition determines when eigenvectors derive from reversible Markov chains, with additional eigenstates constructed from the cycle space of the underlying multigraph, as classified by Betti number $b_1$.

The framework admits explicit analytic construction of eigenbases in lattices such as the kagome (line graph of the hexagonal lattice), connecting flat bands, localized states, and the interplay of nonreversibility and quantum walk localization (Higuchi et al., 2018).

6. Statistical and Design Principles, Limitations, and Future Directions

In each context, TesseraQ is underpinned by rigorous statistical calibration, algorithmic optimization, and symmetry-based classification:

• In astrophysics, completeness is constrained by ML classifier sensitivity (detached EBs are favored; contact binary completeness drops by >50% at $P<0.3$ d), and crowding/dilution effects suppress detection at low depths or high Tmag.
• In quantization, TesseraQ’s gains are maximized in the lowest-bit regimes, with block-wise and distributional calibration essential for closing the full-precision gap. Scheduling of the progressive adaptive rounding is robust; schedule fine-tuning offers diminishing returns as long as hardening slows near its end.
• In programmable soft matter, concavity of the interparticle force law ($\alpha<2$) is critical for accessing the full tessellation landscape; protocol determinism enables direct mapping from thermodynamic minima to experimental target patterns.
• In quantum walks, the reversibility and detailed balance structure of the tessellations controls the existence of localized flat-band eigenstates and the spectral decomposition.

Open directions include leveraging Gaia and high-resolution imaging for further confirmation of quadruple star system dynamics, extending TesseraQ quantization protocols to joint activation-weight ultra-low-bit settings, and exploiting the programmable design protocol in colloidal assembly for optoelectronic and functional surface engineering. Each implementation of TesseraQ offers a robust, extensible toolkit in its respective high-dimensional or high-complexity regime.