We report the implementation of radio-frequency (rf) reflectometry on selective-area-grown PbTe-Pb nanowire devices on a CdTe substrate. These nanowires are predicted to host Majorana zero modes. We demonstrate the compatibility of the rf technique, including both resistive and capacitive sensing, with these nanowires. The effect of dielectric loss from the CdTe substrate is quantitatively characterized. Furthermore, the feasibility of rf reflectometry is verified under finite magnetic fields where zero-energy modes can emerge. Our results establish the fast control of PbTe quantum devices, paving the way for their applications in topological quantum computation.

Our Native Type Universe (NTU) has been detailed through five previous papers establishing the substrate our framework's compilation pipeline targets across multiple hardware platforms. We have found in the course of that work a deeper reach this foundation makes available: negative and fractional types as native first-class constructs. James and Sabry established these dualities in 2012; Chen and Sabry later developed their categorical interpretation in compact closed categories. These dualities have practical benefit for compute modalities in our Fidelity Framework where extant general purpose compute framings lack the substrate to host them as native constructs. We see practicality with these type forms in preserving decidability and principal types through the abelian-group algebraic pattern Kennedy's dimensional types establish. The resulting isomorphisms would admit new, concise forms of resolution within our novel lowering strategy, and we sketch a notional Clef language syntax that would admit rational dimensional exponents into our algebra. We trace the implications across several problem spaces these type forms would open to our compilation and verification disciplines: Bayesian inference where fractional types would express conditioning obligations, quantum computation (and simulations) where negative types would provide the type-level adjoint, and finally adiabatic computation where the combined discipline would express Hamiltonian deformation as a reversible constraint-propagation process. The inherent structure of our NTU together with the supporting framework appears well-suited to problem spaces that current software ecosystems do not directly address, while keeping approachable development ergonomics and mature tooling aligned with operational guarantees the framework aspires to provide.

Since the redefinition of the kelvin in 2019, new methods of primary thermometry have been considered to replace the currently agreed temperature scale in the very high and low temperature limits. These new methods should provide improved uncertainties and, most importantly, a more direct link to the definition of the kelvin. We present an experimental comparison of two such primary thermometers working in the mK region: the Coulomb Blockade Thermometer (CBT) and the Shot Noise Thermometer (SNT). Both thermometers measure temperature hinging only on the natural constants $k_\mathrm{B}$, and $e$. Furthermore both of them are based on electron tunneling current and, thereby, need only electrical measurements, enhancing the practicality. CBT and the SNT are inter-compared in a range of 20 mK to 235 mK. The results show that the agreement of SNT and CBT is approximately within 2.5 % in this range. Basic measurement uncertainty is analyzed and we show that uncertainty in the measurement frequency can cause significant error to temperature measurement of the SNT at low temperatures where finite frequency plays a role.

We develop a dynamic stochastic model of markets with an externality and multiple trading frictions, and cap-and-trade as the leading application. Slow participation, limited intermediation, and heterogeneous information interact in equilibrium: agents choose costly market access, access determines residual compliance demand, intermediary constraints translate residual demand into a surrender-month premium, and the premium feeds back into access incentives. These interactions shape how effectively the market corrects the externality. We characterize access choices in closed form, prove that the equilibrium premium is unique, and show that endogenous access dampens the response to each friction in isolation, while the interaction of multiple frictions is non-additive and can amplify the price response. We quantify the model using 2.7 million EU ETS registry transactions and compliance records from 2005-2021. About 40% of operators do not trade annually, purchases concentrate in April when returns are systematically high, and operator flow predicts future returns.

Formal rigor distinguishes mathematics from other disciplines, in the sense that mathematical statements are derived from explicit axioms by logically verifiable steps. Interactive theorem provers support this by expressing definitions, theorems, and proofs in a fully formal language and verifying them mechanically. We consider the benchmark problem of formalizing all published mathematics as a machine verifiable and continuously updated corpus of mathematical knowledge. This viewpoint treats mathematics as a structured database of interdependent results and raises questions about scalability and organization of large formal libraries. As a case study, we present an ongoing formalization in categorical algebra, namely dilatations of categories, extending classical localizations and illustrating what such an implementation looks like in practice.

For fans of Gabriel's "Worse is Better" it may be ironic that C++, by way of MLIR, serves as the scaffold for compiling an ML-family language whose correctness properties are structural. A crucial intersection in our Composer compiler initiates its lowering with a fixed-point combinator that preserves the dimensional, grade, escape, and numeric-representation structure from the Program Semantic Graph. And the MLIR that's witnessed from the PSG is no passive host. Its use of static single assignment, attribute system and dialects carry that structure materially. We show that our compiler middle end uses categorical construction for lowering code with companion verification to that strata: a functor from the compilation poset to a target category, subject to the compositionality equation. The grounding of our approach comes from three sources, each on its own algebraic object: Ohori's machine-code proof theory grounds the compilation axis, parametricity grounds the content at the base, and adjoint mode logic grounds the traversal between our verification tiers. To extend the thesis we introduce compact-closed negative and fractional types, and show the type machinery can be carried with preserved structure and realized through tooling MLIR provides. More broadly, the same fixed-point primitive that preserves types through compilation also supplies proof terms that can continue to be exercised in MLIR to verify its integrity as lowering proceeds through the pipeline. We argue that this foundation is a unique additional point anticipated by our framework that includes dimensional types, Tarau's groupoid, and cellular sheaves. Throughout, the formalism is instrumented as an internal scaffold: the abstractions support the compiler's mechanics, where a developer is never required to reach for category theory in order to rely on the guarantees the compiler provides.

We study the tri-plane crossing number, that is, the minimal number of crossings in a tri-plane diagram for a bridge trisection of a knotted sphere in $S^4$. We show that every 2-knot in $S^4$ that admits a bridge trisection with at most five crossings is ribbon. As a consequence, we show that the 2-twist spin of the trefoil has crossing number 6. This is the first such computation for a non-trivial knotted surface.

We use recent results, among which modular inequalities for curves, to determine the Alexander polynomials for some classes of pencil-type conic-line arrangements. For these classes of curves we prove that the Alexander polynomial is (at least partially) combinatorial. To this end, we exemplify new techniques that are suitable for broader use, lending themselves to more general classes of curves.

Increasingly deep, confusion-limited radio surveys have pushed direct radio source-count measurements down to tens of $\mu$Jy at 1.4 GHz. Confusion-noise $P(D)$ analyses extend the statistical counts down below $1\,\mathrm{\mu Jy}$. Radio source counts have allowed for constraints on the radio-derived star formation rate density (SFRD) history through models of the backwards evolution of the local radio luminosity function, using the radio-FIR correlation, $q \propto \log(L_{\mathrm{FIR}}/L_{1.4})$, to convert radio luminosities to FIR luminosities and hence star-formation rates. Recent deep radio source counts from MeerKAT suggest a potential tension in the SFRD history between radio and UV/IR measurements at $1\lesssim z\lesssim 2$. This corresponds to a ${>}3\sigma$ discrepancy between the predicted and measured source counts near $10\,\mathrm{\mu Jy}$. We introduce a purely radio-luminosity based parameterization of the redshift evolution of the radio-FIR correlation based on changing cosmic ray losses. We find evidence (${\gtrsim}2\sigma$) that an evolution in the radio-FIR correlation consistent with a mild decrease in $q$ out to $z{\sim}2$ arising from strengthening magnetic fields can mitigate the source count tension. We additionally show that intrinsic scatter in the radio-FIR correlation is likely bounded $\sigma_q\lesssim 0.3\,\mathrm{dex}$ at these redshifts if $q$ decreases. Although we find no evidence that current radio source counts imply a breakdown in the radio-FIR correlation, future deep radio surveys from the Deep Synoptic Array (DSA) will be able to push radio source counts down to several nJy, providing stronger constraints on the allowed evolution.

The adoption of generative artificial intelligence in the public sector has been treated predominantly as a technological problem, with the expectation that productivity gains would follow from increasingly capable models. This paper argues, drawing on two auditable cases in the Brazilian Public Service, that the determining barrier to adoption observed in these units was not technological but training-related, and describes the four-layer structured pedagogical methodology developed by the author. The method was applied in two units with distinct institutional profiles: the Sectoral Internal Control Office of the Federal District Department of Health (SES/CONT) throughout 2024, and the Internal Control Unit of the Federal District Department of Economic Development, Labor and Income (UCI/SEDET) throughout 2025. In both cases, the official indicators from the Electronic Information System of the Federal District Government (SEI-GDF), verifiable by third parties, recorded substantial gains: average processing time fell by 18.2% at SES/CONT and by 50% at UCI/SEDET, with UCI also recording a 92% increase in technical-report production, the issuance of 288 formal recommendations to public managers, and the review of cases whose financial volume totaled US$ 104.3 million (the value of the cases reviewed, not an estimate of funds saved). In neither unit did internal control mechanisms identify any information-security incident, sensitive-data leakage, or formal compliance challenge from external oversight bodies during the period examined. The analysis is consistent with the hypothesis that the method is portable across agencies with distinct mandates, operates within protocols designed to comply with international and national data-protection law and with the principles of public administration, and is accessible to public entities under budget constraints, since it used free AI models.

We define the analogue of Lie-Rinehart algebras over $C^\infty$-rings. We show that given a Poisson $C^\infty$-ring $\mathcal{A}$ its module $\Omega_{\mathcal{A}}^{1}$ of $C^\infty$-Kähler differentials is (part of) a Lie-Rinehart algebra. Conversely, given a Lie-Rinehart algebra $\mathcal{M} \xrightarrow{\rho} C^\infty\mathrm{Der}(\mathcal{A})$ over a $C^\infty$-ring $\mathcal{A}$, there is a natural Poisson bracket on the $C^\infty$-ring $\mathcal{F}(\mathcal{M})$ associated with the $\mathcal{A}$-module $\mathcal{M}$ (the $C^\infty$-ring analogue of an $\mathcal{A}$-algebra freely generated by the module $\mathcal{M}$). In the case where $\mathcal{A}$ is the $C^\infty$-ring of smooth functions on a manifold $M$ and $\mathcal{M}$ is the module $\Gamma(E)$ of sections of a Lie algebroid $E \to M$, the $C^\infty$-ring $\mathcal{F}(\Gamma(E))$ is the ring of functions $C^\infty(E^\vee)$ on the total space of the vector bundle $E^\vee \to M$ dual to the vector bundle $E$.

Motivated by their applications in DNA-based storage systems, codes capable of correcting consecutive deletions have attracted significant attention. An important class of such codes consists of those that can correct multiple consecutive deletion errors, commonly referred to as multiple $b$-burst deletion-correcting codes. In this paper, we investigate the fundamental limits of multiple $b$-burst deletion-correcting codes. Specifically, we first characterize several structural properties of the associated deletion balls. Then, leveraging these properties, we derive several upper bounds and a combinatorial lower bound on the maximum size of such codes. As a consequence, our bounds improve upon the previously known results for general parameter regimes and are shown to be asymptotically optimal for certain cases.

Ballantine, Beck, and Merca defined the elementary symmetric partition map pre$_j$ that sends a partition $\lambda$ to a larger partition whose parts are the summands appearing in the evaluation of the $j$-th elementary symmetric polynomial on $\lambda$. They conjectured that pre$_j$ is injective on the set of partitions of $n$ with length $\ell \geq j$. The $\ell = j$ case was disproved by Devnani and Eyyunni; they instead conjectured the statement to be true for $\ell > j$. In this article, we answer this refined conjecture in the negative by proving that pre$_j$ is not injective on partitions of $n$ with length $2j$ for $j \geq 3$. We also prove that the analogous map prh$_j$ defined via the complete homogenous symmetric polynomial is injective on the set of all partitions.

The phase diagram of a system of interacting bosons (Cooper pairs) hoping on a one-dimensional (1D) lattice with onsite phase dissipation describing the Josephson tunneling to a nearby diffusive normal-metal electrode is studied. Starting from the system at commensurate lattice filling, it is shown by a combination of analytical techniques that the phase diagram contains two quantum phases: A dissipative Bose-Einstein condensate (D-BEC) or superconductor with long-range phase coherence, and a dissipative Mott insulator (D-Mott) or "metal" with exponentially decaying phase correlations in space and local imaginary-time correlations decaying as the local pairing correlations of the electrode. The D-Mott/metal phase can be described as a 1D array of dissipative boson puddles, weakly coupled by Josephson tunneling. The puddle size roughly corresponds to the length scale beyond which phase slips suppress phase coherence. The dissipative time-dependent Ginsburg-Landau theory phenomenologically used by Sachdev, Werner, and Troyer [Phys. Rev. Lett. {\bf 92} 237003 (2004)] for the superconductor-metal transition in quasi-1D wires is derived from this microscopic puddle picture. Thus, the criticality of the D-Mott/D-BEC transition is shown to belong to the Wilson-Fisher universality class with dynamical exponent $z\approx 2$. At small doping, the D-Mott/metal phase remains stable due to its finite compressibility, which is computed to leading order in a perturbation expansion of the dissipation strength and the inter-puddle Josephson coupling. At larger doping, using a mapping to a pseudospin chain combined with bosonization, the D-BEC/superconductor phase is the ground state for non-vanishing but arbitrarily small dissipation. Similarities and differences with deconfinement transition of an array 1D bosonic Mott insulators in anisotropic optical lattices are also discussed.

Let $G$ be an arbitrary group and let $\mathbb{F}$ be a finite field. In this paper, we determine bases for the $T_G$-ideals of graded polynomial identities of the algebra $M_2(\mathbb{F})$ for all possible $G$-gradings. The bases obtained consist of finitely many non-trivial graded identities, and are finite whenever $G$ is finite.

For positive integers $s$ and $k$, the Ramsey number $r(s,k)$ is the minimum integer $n$ such that any graph on $n$ vertices contains a clique of size $s$ or an independent set of size $k$. We prove that for any fixed $s \ge 3$ and $k$ tending to infinity, the off-diagonal Ramsey numbers satisfy \[ r(s, k) \ge \Omega \left(\frac{k^{s-1}}{(\log k)^{2s-4}} \right), \] which matches, up to polylogarithmic factors, the upper bound established over 90 years ago by Erdős and Szekeres. For $s \ge 5,$ this improves the best known lower bound of the form $r(s, k) \ge k^{\frac{s+1}{2} + o(1)}$ which was first established by Spencer in 1977 and has since only seen polylogarithmic improvements.

A lead-vacancy (PbV) center in diamond exhibits coherent emission above the liquid helium temperature, making it highly attractive for quantum network applications. Here, we report the magneto-optical and spin properties of PbV centers in diamond. We record a spin lifetime of 12 ms at 7.5 K under large off-axis magnetic field. Furthermore, we observe formation of the coherent dark state by coherent population trapping and estimate a spin dephasing time of 177 ns at 6.5 K. This work demonstrates the outstanding thermal robustness of the PbV spin compared to other group-IV centers above 4 K.

We present observational evidence, based on high-resolution imaging from HST, ALMA, and AstroSat/UVIT, that the redistribution of gas driven by the bar in the face-on spiral galaxy NGC 3351 results in suppressed star formation in its central regions. Dust and molecular gas coexist in galaxies, allowing dust lanes observed in galaxies to be used to probe the distribution of gas. In the central regions of NGC 3351, covered by the stellar bar, dust lanes are visible in the HST F438W-F814W color map, but surprisingly, these areas lack molecular gas and recent star formation. The inward orientation of the dust lane morphology towards the galaxy's center suggests that molecular gas may have once been present in this region, but was redistributed to the center due to the stellar bar's action. The direction of dust lanes, therefore, indicates the past inflows of gas toward the galaxy center, with their morphology consistently oriented inward along the bar. These findings support a scenario where the stellar bar has efficiently channeled molecular gas into the nucleus, building the central reservoir while suppressing star formation along the bar.

We show that if $G$ is a $d$-regular Vizing-class-1 graph, then the proper additive chromatic index of $G$, denoted $\eta'_p(G)$, is equal to its chromatic index. This verifies that a strengthening of the Additive Coloring Conjecture of Czerwiński et al. holds for line graphs of $d$-regular Vizing-class-1 graphs. We show that if $G$ is a $d$-regular Vizing-class-2 graph, $\eta'_{p}(G)\leq \frac{(2^{\lceil \log_2 (d+1)\rceil})^2+2}{3}$, and if $G$ is a $d$-regular Vizing-class-2 graph that admits a proper edge-coloring with a smallest color class of size $r$ and $\text{girth}(G)\geq 6r-5$, then $\eta_p'(G)\leq 2d$, among other results.

Chirality plays a crucial role in determining the structure of many systems in nature. Twisted or helical aggregates as a consequence of self-assembly can be seen in many biological and synthetic materials. Despite extensive theoretical and experimental efforts, how molecular-scale chirality gives rise to complex twisted morphologies in amphiphiles still remains unexplored. Here we study the interplay between molecular hydrophobicity, shape anisotropy and chirality using molecular dynamics simulation. Variation of relative molecular concentration and intrinsic chirality of molecules drive a sequence of twisted liquid crystalline variants of lamellar, cylindrical and vesicular phases. These structures emerge spontaneously under equilibrium conditions and are characterized by orientational correlation functions. We demonstrate that variation in molecular chirality gives rise to the development of hierarchical chiral order within the system. Further increment of chirality competes with hydrophobic interactions, leading to morphological instabilities. Our findings establish a direct link between microscopic chirality and mesoscale structure formation and their instabilities. Qualitative comparison of liquidity and pitch of the observed phase morphologies with the amount of chirality has been reported.

The demand for fast mid-wave infrared photodetectors is fueled by high-rate free-space optical communication and optical frequency comb spectroscopy. To date, only a few multi-GHz photodetectors have shown sensitive room-temperature operation in the 3.0-3.7 {\mu}m band, yet their commercial availability remains scarce. In this work, we present the remarkable response speed of an InAs/InAsSbP nBp barrier detector - a type typically not associated with high-frequency operation. A weakly reverse-biased photodiode with a diameter of 121 {\mu}m achieves a -3 dB electrical bandwidth of 2.4 GHz and -20 dB bandwidth of 8.0 GHz. This is the best result in this class of mid-infrared photodetectors confirmed optically. High signal-to-noise photodetection is also demonstrated at frequencies exceeding 19 GHz. The relatively simple device structure (devoid of cascaded structure or type-II superlattice) was realized on the mature InAs material platform, which opens new perspectives for accessible, sensitive, multi-GHz photodetectors for the 3.0-3.7 {\mu}m spectral region.

Fermionic Gaussian states describe free fermions and underlie the mean-field picture of matter, from metals to superconductors; they are also efficiently simulable on classical computers. Departures from Gaussianity -- the correlations produced by interactions -- are therefore what make a fermionic system hard to simulate classically and useful for quantum computation, analogous to the role of magic in stabilizer-based quantum computation. Yet detecting and quantifying such non-Gaussianity at scale has remained challenging. Here we introduce practical tests and witnesses of fermionic non-Gaussianity built on fermionic antiflatness, a measure derived from the two-point covariance matrix. We estimate it with two protocols -- a two-copy Bell measurement and a single-copy scheme using commuting Majorana bilinears -- that determine whether a state is Gaussian or far from it at lower measurement cost than existing approaches, using only operations native to fault-tolerant hardware. For mixed states, a purity-corrected witness certifies non-Gaussianity and remains robust under strong noise; running it on the IQM quantum processor, we find that noise can both reduce and enhance non-Gaussianity. Finally, we show that preparing pseudorandom fermionic states requires extensive non-Gaussianity. Together, these tools enable the study and certification of non-Gaussian fermionic resources on present-day quantum devices.

Thermal tuning of phononic crystals typically treats each constituent's elastic modulus as a single-valued function of temperature. Here we show that when one constituent undergoes a first-order martensitic transformation (NiTiCu), paired with a Parylene C spacer in a thin one-dimensional composite rod, the intrinsic thermal hysteresis of the transformation is transferred to the collective acoustic response, making it depend on thermal history. Transfer-matrix calculations performed separately along the heating and cooling branches reveal that the same temperature within the transformation interval can correspond to two distinct Bloch band structures and two distinct transmission spectra. As temperature is cycled, stop-band edges trace closed loops in the temperature-frequency plane, producing hysteretic phononic band structures. At a fixed probe frequency, the rod can switch between transmitting and strongly attenuating states depending solely on whether the sample was last heated or cooled, with transmission contrasts exceeding 50 dB in a six-cell structure, thereby realizing a form of acoustic memory. The filling fraction of the active segment provides an independent geometric control parameter that modifies the width of the hysteresis loops and the positions of gap closures. These predictions can be tested directly using immersion-ultrasonic measurements on NiTiCu/Parylene C composite rods.

We observed that the null strings, tensionless strings with Carrollian worldsheets, exhibit an extra gauge symmetry, \textit{Carroll-Weyl} gauge symmetry, which cannot be obtained from ultra-relativistic Carrollian limit of tensile strings. Due to the existence of this symmetry, the BMS$_3$ algebra of constraints, which is obtained as the Carrollian limit of two Virasoro algebras of the standard tensile strings, should be replaced with an BMS$_3$ algebra extended by a weight one operator. To establish further the existence and necessity of the Carroll-Weyl gauge symmetry, we carefully work through Hamiltonian analyses of constrained/gauged systems. We also discuss the extended BMS$_3$ algebra of constraints.

The Solar Neighborhood is dominated by stars smaller, colder, and fainter than the Sun: the M dwarfs. If we are to understand the context in which the Sun formed and evolved, then we must investigate the system architectures of our low-mass neighbors. We have therefore carried out the Pervasive Overview of Kompanions of Every M Dwarf in Our Neighborhood (POKEMON) speckle survey of nearby M-dwarf primaries. We created the survey with the goal of observing a volume-limited (north of -30 degrees) sample of M-dwarf primaries through M9 out to 15 pc at diffraction-limited resolution. Pre-Gaia parallax measurements yielded a catalog of 454 nearby M-dwarf primaries. However, the precise astrometry from Gaia indicated that there are additional low-mass sources within 15 pc. Here we present the POKEMON-Distance Limited Catalog (POKEMON-DLC), a supplemental catalog that consists of speckle observations for the 66 additional M-dwarf primaries identified by Gaia, increasing the number of ultracool dwarf (later than M6.5) primaries in the POKEMON catalog by a factor of 1.6. In our observations we detect four likely bound companions. After carrying out a literature search for additional companions, we update the projected separation distribution and find a peak at 7.91 au ({\sigma}log(a) = 1.1, SElog(a) = 0.10). We also update the M-dwarf stellar multiplicity and companion rates, and find values of 22.7 p/m 1.8% and 27.5 p/m 2.0%, respectively. These results emphasize the utility of Gaia for identifying low-mass, nearby sources, and we find that ensuing characterization of these sources by SPHEREx will continue to clarify the nature of the Solar Neighborhood.

We address the connection between negative electronic friction and non-Markovian effects in the nonadiabatic vibrational dynamics of molecules interacting with metal surfaces under nonequilibrium conditions. We show that a generic nonequilibrium mechanism leading to negative Markovian electronic friction, where molecular vibrations couple directly to inelastic electronic transitions, also introduces significant non-Markovian contributions to the electronic friction. To demonstrate these ideas, we investigate nonequilibrium charge transport through a molecular nanojunction containing a vibrationally coupled donor-acceptor model, where negative electronic friction reflects driving of the vibrational mode beyond standard Joule heating. By comparison to numerically exact, fully quantum hierarchical equations of motion simulations, we verify that these non-Markovian effects have a significant impact on the nonequilibrium dynamics and even on the stability of the resulting Langevin equation.

Conformal prediction is a framework for providing prediction intervals with distribution-free validity, guaranteeing predictive coverage for data drawn from any distribution. Its two main variants are full conformal prediction and split conformal prediction (also called transductive and inductive). Full conformal prediction is widely considered to be statistically more efficient (since split conformal prediction requires data splitting, and therefore can lead to wider prediction intervals due to the resulting loss in sample size), but its implementation is computationally prohibitive, as it requires the underlying model to be refit for every candidate value in the response space. Existing computational shortcuts, such as using a discrete grid of values to approximate the full conformal prediction construction, frequently lack theoretical guarantees on marginal coverage and can fail in practice. To address this limitation, we introduce a novel class of approximations to the full conformal prediction method, based on the idea of \emph{tournaments}, which enables the construction of prediction sets with a rigorous marginal coverage guarantee of $1-2\alpha$. Under stability conditions, the theoretical coverage guarantee tightens to approximately $1-\alpha$. This new framework generalizes the existing method of leave-one-out cross-conformal prediction, while allowing for flexible use of various existing approximation strategies.

We present a graph-theoretic reformulation of pitch-class set theory in which each set in $\mathbb{Z}_n$ is represented as a complete weighted graph whose edge weights are interval classes. We show that this construction is invariant under the dihedral group $D_n$, and that the full interval structure is encoded by a cyclic step composition, from which all interval data are recovered via an additivity principle. This framework yields a direct correspondence between T/I equivalence and graph isomorphism, and reinterprets Z-relation as non-isomorphic graphs with identical edge-weight multisets. We extend the model to weighted clique complexes, linking higher-order homometry to simplex-weight structure, and introduce a cent-weighted formulation enabling comparisons across different equal temperaments. Finally, we define a polynomial invariant derived from antipodal step pairings for algebraic analysis of pitch class space.

We design strategies in nonlinear geometric analysis to temper the effects of adversarial learning for sufficiently smooth data of numerical method-type dynamics in encoder-decoder methods, variational and deterministic, through the use of geometric flow regularization. We augment latent spaces with geometric flows to control structure, relying on adaptations of curvature and Ricci flow. All of our flows are solved using physics-informed learning. Traditional geometric meaning is traded for computing ability, but we maintain key geometric invariants, the primary of which are maintained, intrinsically-low structure, nontriviality due to sufficient lower bounds on curvature, distortion of volume element, that develop quality in the inference stage. We instill representations that are canonical, smooth, curvature-aware, geodesic-aware, and non-topologically void or sparse. The primary bottleneck of a Ricci curvature flow is that Ricci curvature is high order, thus expensive to compute, so we will attempt to overcome this with properly justified proxies. Our primary contributions are fourfold. We develop a loss based on Gaussian curvature using closed path circulation integration for surfaces, bypassing automatic differentiation of the Christoffel symbols through use of Stokes' theorem. We invent a new parametric flow valid under a Taylor expansion derived from the Gauss equation. We develop two strategies based on time differentiation of functionals, one with a special case of scalar curvature for conformally-changed metrics, and another with harmonic maps, their energy, and induced metrics. Our losses are diminished analytically and mostly heuristic but maintain overall integral latent structure. We showcase that curvature flows and the formulation of geometric structure in intermediary encoded settings enhance learning and overall zero-shot and adversarial fidelity.

We show that the geometry of Hertling-Manin F-manifolds $(M,\circ,e)$ provide the appropriate theoretical framework for studying the integrability of quasilinear systems of first-order evolutionary partial differential equations of the form ${\bf u}_t=X\circ {\bf u}_x$ (F-systems) under the mild assumption that the unit vector field is cyclic with respect to the operator of multiplication by the vector field $X$. This approach is very general and allows us to treat even non-regular systems that were previously beyond the scope of existing techniques. Like in the regular case the information about integrability is contained in a torsionless connection associated with the system and the integrability condition reduces to a geometric condition involving the Riemann tensor of the connection and the structure functions of the product. We prove that a locally conservative F-system is integrable and, in the analytic setting, also the converse statement, thereby providing a full characterization of integrability. Moreover, in the analytic case, we prove the existence of a family of analytic symmetries providing, in principle, the unique local analytic solution of the Cauchy problem through the generalised hodograph method.