In this paper, we evaluate some series via the WZ method, and confirm several previous conjectures. For example, we prove the following two identities conjectured by the second author: $$\sum_{k=0}^{\infty} \frac{(28k^2 + 10k + 1) \binom{2k}{k}^5}{(6k + 1)(-64)^k \binom{3k}{k} \binom{6k}{3k}} = \frac{3}π$$ and $$\sum_{k=1}^\infty \frac{d^4}{dk^4}\left(\frac{(21k-8)Γ(k+1)^2}{k^3Γ(2k+1)}\right)=\frac{1959}2ζ(6)-432ζ(3)^2. $$

Under standard assumptions, we compute the GK-dimension of Hecke eigenspaces in the mod $p$ cohomology of an inner form $D^\times$ of $\mathrm{GL}_2$ over a totally real field unramified at $p$, allowing $D$ to be a division algebra at $p$. Our arguments also apply when $D$ is a matrix algebra at $p$, in which case they give a simplified proof of a theorem of Breuil--Herzig--Hu--Morra--Schraen.

In this work, we study a single-machine scheduling problem that aims at minimizing the total cost of a schedule subject to start-time dependent costs. This framework naturally captures scenarios where costs fluctuate throughout the day, such as time-varying energy or labor prices. To model more realistic scenarios, we assume that these costs lie within a budgeted uncertainty set and propose a two-stage robust optimization approach. In a first stage, the order in which activities should be executed is decided. After a cost scenario has been revealed, the starting times for each activity are established, subject to the ordering from the first stage. We demonstrate that the proposed problem is NP-hard and not approximable, implying the complexity of its robust counterpart. Furthermore, we show that already evaluating a first-stage solution is NP-hard when the uncertainty set is discrete. We develop models and solution methods for both continuous and discrete budgeted uncertainty. In computational experiments, we compare these approaches and demonstrate the advantages of including uncertainty beforehand.

Ions and electrons accelerated to high energies during flares and coronal mass ejections at the Sun may escape the solar atmosphere and, guided by the interplanetary magnetic fields, propagate through space to near-Earth locations. These Solar Energetic Particles (SEPs) can be detected directly by spacecraft instrumentation. The highest energy SEPs may also propagate through the geomagnetic field and precipitate to low atmospheric heights, producing secondary particles including neutrons and protons that trigger the formation of cosmogenic radionuclides. The space weather effects associated with the SEP ion population (for the most part protons) consist principally of radiation risk to aviation, humans in space and spacecraft. This paper focusses on the risks to aviation and astronauts and emphasizes how the parameters of the SEP event, including fluence and spectrum, affect radiation doses. Space weather effects for events that took place in recent decades, for which a large body of measurements and models exist, are discussed. SEP events of extreme magnitudes, such as those extrapolated from radionuclide data from the distant past are then considered and first estimates of the associated radiation at aviation altitudes and in space presented. These are worst-case estimates derived within the assumption that the radionuclide spike was caused by a single SEP event and that the particle spectrum and geomagnetic conditions were the same during all events.

We establish existence and uniqueness results for asymptotically locally Euclidean (ALE) and asymptotically locally flat (ALF) gravitational instantons. In particular, we prove the existence of a unique, Ricci-flat, toric ALE and ALF gravitational instanton, for every admissible rod structure, that is smooth up to possible conical singularites. We also give an elementary proof that any toric ALE or ALF self-dual instanton is a multi-Eguchi-Hanson or multi-Taub-NUT solution.

Let $N$ be a positive integer and let $S_N$ be the set of polynomials with integer coefficients, degree less than $N$, and minimal positive integral over $[0,1]$. D. Bazzanella initiated the study of $S_N$ because of its relation to the distribution of prime numbers. Indeed, it is possible to prove that $\sum_{p^m \leq N} \log p = -\log \int_0^1 P(x) \mathrm{d} x$ for every $P \in S_N$, where the sum runs over prime numbers $p$ and positive integers $m$ such that $p^m \leq N$. For each real number $t$, let $\lfloor t \rfloor$ denote the maximal integer not exceeding $t$. The main result of this paper states that there exist infinitely many polynomials $P \in S_N$ such that $\big(x^3(1 - x)^2\big)^{\lfloor N / 6 \rfloor}$ divides $P(x)$ in $\mathbb{Z}[x]$. This improves upon a similar result of Sanna, who proved the same claim but with the lower-degree polynomial $\big(x(1-x)\big)^{\lfloor N / 3 \rfloor}$ in place of $\big(x^3(1 - x)^2\big)^{\lfloor N / 6 \rfloor}$.

In this study, we present a comprehensive analysis of the Raman active phonon modes in orthorhombic LaInO$_3$ based on a combination of polarization-angle resolved Raman spectroscopy and density functional theory calculations. By using backscattering from multiple crystallographic surface orientations and employing a full symmetry analysis, we identify and assign most of the Raman-active $Γ$-point phonons to their irreducible representations of the D$_{\rm{2h}}$ point group. A multidimensional hyperspectral fitting procedure allows us to extract the relative Raman tensor elements from the angular dependence of the scattering intensities, even for strongly overlapping modes. First-principles calculations yield the phonon dispersion along high-symmetry directions, the phonon densities of states, and atomic displacement patterns, which are found to be in good agreement with the experimental mode frequencies.

Let $G$ and $H$ be finitely generated groups. In this paper, we prove the quantitative coarse Baum--Connes conjecture for the free product $G* H$ under the assumption that the conjecture holds for both $G$ and $H$.

We are concerned in designing a suitable numerical scheme based on the equal-order hybrid high-order (HHO) method for the linear parabolic integro-differential equations. The spatial discretization is made using the equal-order HHO method and subsequently we perform the stability analysis of the corresponding semi-discrete scheme. The convergence results are presented in suitably defined Bochner norms for the semi-discrete problem. Then a second-order temporal discretization is implemented on the time domain using a Crank-Nicolson scheme where the memory term is approximated using a composite trapezoidal quadrature rule. The stability of the resultant complete discrete schemes are analyzed followed by derivation of the error estimates of order $\mathcal{O}(τ^{2}+h^{k+1})$, $k\ge 0$ is the degree of local polynomial approximation, in discrete $l^{2}(0,T;H^{1}(Ω))$ and $l^{\infty}(0,T;H^{1}(Ω))$ like norms. Numerical illustrations are performed on some polygonal meshes validating the theoretical estimates.

We study the finite element approximation of linear second-order elliptic partial differential equations in nondivergence form with highly heterogeneous diffusion and drift coefficients. A generalized Cordes condition is imposed to guarantee that a suitably renormalized version of the nondivergence-form differential operator is near the Laplacian. Based on a stabilized symmetric formulation for the gradient that enables the use of $H^1$-conforming approximation spaces, we construct a multiscale method following the methodology of the localized orthogonal decomposition with coarse basis functions tailored to the heterogeneous coefficients. We employ a novel post-processing strategy to obtain higher-order convergence rates, overcoming previous limitations imposed by the low regularity of the load functional. Numerical experiments demonstrate the performance of the method.

We introduce a notion of parity for formal morphisms between invertible objects and use it to prove a corresponding coherence theorem. Parity is conceptually similar to the sign of underlying permutations, but not defined as such. To give complete details, this work includes a thorough treatment of the free permutative category on an invertible generator, its skeletal model, known as the super integers, and an equivalence between them classified by the pair of integers $\pm$1. Our approach is organized and clarified as an application of 2-monadic algebra, particularly the concept of flexibility and the Lack model structure. The final section contains a number of examples applying the main results.

By exploiting noise as an information-bearing resource, noise-driven communication offers a promising framework for low-complexity and secure wireless system design. In this letter, the scheme of ternary noise modulation (T-NoiseMod) is proposed for noise-based wireless communication scenarios, where information is encoded into the statistical characteristics of artificial noise. Unlike conventional binary NoiseMod, which employs two variance levels, the proposed scheme introduces a third transmission state: intentional silence. By pairing two consecutive noise blocks, the signaling scheme is expanded to eight valid state combinations, enabling the transmission of three information bits per signaling interval. In our proposed scheme, the two-stage receiver is developed, consisting of mean-based silent-state detection followed by variance-based low/high classification. An analytical expression for the bit error probability (BEP) is derived for Rayleigh fading. Our computer simulation results match closely with our theoretical results and show the effects of key system parameters. Furthermore, comparisons with binary NoiseMod demonstrate the inherent trade-off between reliability and rate.

The 1-2-3 conjecture has been solved positively in 2024 for finite graphs and by extension for infinite graphs which are locally finite. The solution is non-constructive, and finding explicit solutions for large (or infinite) graphs is very hard. By exploiting the extra structure present in many non-periodic tilings, we find explicit solutions for the Chair (all three vertex placements), Non-Pinwheel, Pinwheel, Half-hex, Ammann-Beenker (two versions), Penrose Rhomb, and the Domino tilings. We prove that for any fully periodic tiling of the plane there exists a fully periodic solution, and provide an algorithm for finding such a solution. We give solutions for the fully periodic square, triangle and hexagonal lattices.

This paper is devoted to studies of the mechanical deformation of the S. aureus cell wall. The bacterium is modelled as a thin elastic membrane containing cytoplasm, which is treated as an incompressible fluid. Deformation occurs via Van der Waals interactions between the bacterium and a solid metallic surface, both with and without the influence of surface plasmon resonance (SPR). Our modelling results indicate that the excitation of surface plasmons significantly increases the effective interaction area between the bacterial membrane and the nanostructured surface. The elastic and dielectric properties of the bacterium's components are uninvestigated. Therefore, theoretical calculations are performed in wide, physically meaningful ranges. Thus, the results of studies give only a qualitative estimation. However, they are novel and, with further experiments, can solve the inverse problem of obtaining physical properties. The paper highlights the potential of SPR to enhance antibacterial strategies, inspiring further research and innovation.

Binary analysis increasingly relies on large language models (LLMs) to perform semantic reasoning over complex program behaviors. However, existing approaches largely adopt a one-pass execution paradigm, where reasoning operates over a fixed program representation constructed by static analysis tools. This formulation limits the ability to adapt exploration based on intermediate results and makes it difficult to sustain long-horizon, multi-path analysis under constrained context. We present FORGE, a system that rethinks LLM-based analysis as a feedback-driven execution process. FORGE interleaves reasoning and tool interaction through a reasoning-action-observation loop, enabling incremental exploration and evidence construction. To address the instability of long-horizon reasoning, we introduce a Dynamic Forest of Agents (FoA), a decomposed execution model that dynamically coordinates parallel exploration while bounding per-agent context. We evaluate FORGE on 3,457 real-world firmware binaries. FORGE identifies 1,274 vulnerabilities across 591 unique binaries, achieving 72.3% precision while covering a broader range of vulnerability types than prior approaches. These results demonstrate that structuring LLM-based analysis as a decomposed, feedback-driven execution system enables both scalable reasoning and high-quality outcomes in long-horizon tasks.

We propose a Hanbury Brown-Twiss interferometer for a $ν=2/5$ fractional quantum Hall edge system, in which quasiparticles tunnel between two co-propagating edge modes. In contrast to the previously studied anyonic Fabry-Pérot and Mach-Zehnder interferometers, the proposed setup relies purely on two-particle interference rather than single-particle interference. In the weak-tunneling regime, we employ a bosonized edge theory together with Keldysh perturbation theory to evaluate the cross-correlation of the tunneling currents. In the large-device limit, we obtain an analytic expression for the flux-dependent noise, whose structure closely resembles that of an electronic HBT interferometer, but with the electron charge replaced by the fractional charge $e^{\star}=e/3$ and with scaling dimensions characteristic of the fractional edge modes. In this limit, the explicit anyonic exchange phases cancel, whereas when the device size becomes comparable to the thermal length, the cross-correlation may recover a more explicit dependence on the anyonic statistical angle.

We present a minimal implementation of SU($N$) pure Yang-Mills theory in $3+1$ dimensions for digital quantum simulation, designed to enable quantum advantage. Building on the orbifold lattice simulation protocol with logarithmic scaling in the local Hilbert-space truncation, we introduce further simplified Hamiltonians. Furthermore, we test simple methods that improve the convergence to the infinite mass limit, thereby removing the requirement of a large scalar mass to obtain the Kogut-Susskind Hamiltonian. For the SU(2) theory, we can cut the resource requirement further by utilizing the embedding of $\mathrm{SU}(2)\cong\mathrm{S}^3$ into $\mathbb{R}^4$. Monte Carlo simulations of the Euclidean path integral were used to benchmark the accuracy of these new analytical improvements to the theory. These results provide further support for the noncompact-variable-based approach as a practical framework for quantum simulation of non-Abelian gauge theories.

A limited remanent polarization (Pr) in HfO2-based FeRAM remains a key obstacle to density scaling and reliability, while material and process optimizations offer only incremental improvements. This limitation fundamentally originates from the thickness-constrained switchable polarization and the intrinsic polarization ceiling of HfO2-based ferroelectrics. Here, we propose an all-ALD-grown vertical complementary FeRAM (VCF) architecture, in which the top and bottom stacked FeRAM cells maintain complementary polarization. This complementary dipole configuration converts the readout from a single-layer polarization response into a differential polarization summation, thereby amplifying the effective charge window without increasing the switching field of each individual layer or incurring area overhead. Viewed from top to bottom, an "up-down" polarization pair stores logic '1', whereas a "down-up" pair stores logic '0'. Using a complementary polarization write-read scheme, the VCF achieves an effective differential polarization above 100 uC/cm^2 and retains above 90 uC/cm^2 after 1e10 switching cycles without electrical breakdown. Robust retention (longer than 1e4 s at 85 degC) and strong disturb immunity are demonstrated, with an effective differential polarization above 80 uC/cm^2 under a V/3 scheme after 1e6 disturb pulses. Array-level operation is validated in a 5 x 5 selector-free crosspoint array. The performance enhancement of the VCF arises from the co-optimization of the all-ALD-grown process, device architecture, and operation scheme, enabling high density, a wide memory window, and strong reliability for scalable FeRAM integration.

Although modern, AI-centric datacenters heavily rely on SmartNICs, existing devices impose a hard trade-off. Commercial SmartNICs provide high bandwidth and easy software integration, but offer limited support for customization and data processing offload. In contrast, research SmartNICs often suffer from low bandwidth, limited functionality, and poor software compatibility -- to the point that many are not actual NICs in a technical sense. This gap can be closed by treating the NIC datapath as a first-class stream computation substrate with shared hardware/software abstractions for a tight co-design of infrastructure and applications. To demonstrate this, we introduce SCENIC, an open-source datacenter SmartNIC. SCENIC implements a 200G network datapath over offloaded TCP/IP and RDMA stacks, as well as a fallback path for processing arbitrary network traffic. On top of the network logic, SCENIC combines on-datapath Stream Compute Units (SCUs) for data processing and embedded ARM cores for flexible control path manipulation with direct access to GPUs and SSDs. SCENIC is fully integrated with the OS, exposing native Linux network and RDMA verb interfaces, making the programmable datapath transparent to existing applications while enabling control of, e.g., user-defined offloads and programmable congestion control. SCENIC's performance matches commercial platforms, and we show its versatility through several use cases such as offloaded collective communication and network-to-GPU hash-based data partitioning.

We present a relativistic time-dependent density functional theory (TDDFT) approach for the simulation of resonant inelastic X-ray scattering (RIXS) spectra, based on both a full four-component (4c) Dirac-Coulomb Hamiltonian and a modern atomic mean-field exact two-component (amfX2C) Hamiltonian model. The approach builds on the pseudo-wavefunction formalism and a core-valence separation scheme, enabling the efficient evaluation of couplings between two manifolds of excited states relative to a common ground state, as required for solving the Kramers-Heisenberg equation for RIXS. The relativistic formulation provides a variational description of scalar and spin-orbit relativistic effects, which are essential for accurately describing inner-shell excitations involved in RIXS processes. Its transformation to the 2c regime via the amfX2C Hamiltonian significantly reduces the computational cost while offering 4c-quality results by accounting for two-electron and exchange-correlation picture-change effects arising from the X2C transformation. In addition to two-dimensional RIXS maps, the methodology enables the direct evaluation of high-energy-resolution fluorescence detection (HERFD) and resonant X-ray emission spectra (RXES). Applications to 2p3d and 3d4f RIXS maps of selected ruthenium and uranium complexes demonstrate that the amfX2C approach reproduces reference 4c results and experimental spectra with high accuracy, capturing all key spectral features and providing reliable peak assignments.

In the combinatorial action model of contract design, a principal delegates a complex project to an agent, incentivizing a subset of actions from a ground set of $n$ actions, via a linear contract. Computing the optimal contract is a challenging problem that generally hinges on two factors: (i) the number of "critical values" - values of the linear contract parameter at which the agent's best response changes from one set to another, and (ii) the complexity of the agent's best-response problem (demand query). Prior work has used this approach to devise polynomial-time algorithms for the optimal contract problem under specific reward functions: gross substitutes, supermodular, and ultra. We develop a unified geometric framework for algorithmic contract design by establishing a fundamental link to the theory of demand types from consumer theory. Under this geometric view, bounding the number of critical values reduces to counting the best-response regions which the "contract ray" pierces. Leveraging this connection, we introduce the class of All Substitutes and Complements (ASC) functions, and show that it admits at most $O(n^2)$ critical values, strictly generalizing and unifying all previously known classes admitting poly-many critical values. We conjecture that, under some mild assumptions, ASC is the maximal such class. Turning to the demand query aspect, we develop a new technique for efficiently computing a demand query using value queries, which works in general for "succinct" demand types. Combining these structural and algorithmic results, we obtain polynomial-time algorithms for new classes of reward functions that exhibit substitutes and complements simultaneously.

We characterize which local matrix structures saturate Weyl's eigenvalue perturbation bound for graph Laplacians under geometrically constrained vertex displacements. Geometric graphs with heavy-tailed vertex noise arise across sensor networks, biological imaging, and spatial omics, yet tractable predictions for noise-induced spectral error remain limited. We study geometric graphs abstracted from biophysical systems, incorporating clearance, planarity, and identifiability constraints that govern physically realizable embeddings. Within this constrained setting, we identify witness motifs, small subgraphs in maximally noise-sensitive geometric configurations, that dominate weighted-degree and graph Laplacian spectral perturbations under tempered power-law vertex displacements. This motif decomposition reduces global spectral sensitivity to a finite catalog of local extremal structures and identifies configurations that attain Weyl-tight bounds. We then lift these constrained-graph results to general straight-line embedded graphs in arbitrary dimension via local repair operations producing a constrained surrogate graph that preserves sensitivity-relevant structure. To quantify noise-induced spectral variation in both strong-oracle and weak-oracle regimes, we introduce stochastic co-spectrality (SC) and the stochastic spectral separation index (S3I), which characterize when observed spectral distances are noise-driven and when noise parameters are separable. Together, these results provide a principled pathway from local geometric noise to global spectral error in graph Laplacian matrices, enabling estimation of spectral fragility from graph structure without exhaustive eigenvalue computation or restrictive distributional assumptions beyond moment bounds.

We extend the well-established action of the Higgs mode in $s$-wave superconductors to include quantum fluctuations (QFs). We find that already one-loop quantum corrections to the Higgs propagator shift its eigenfrequency below the superconducting energy gap $2Δ$. Consequently, the Higgs mode appears as an undamped pole below the quasiparticle continuum, leading to drastically sharper experimental signatures. We demonstrate this by calculating two characteristic fingerprints of the Higgs mode, namely in Third Harmonic Generation (THG) and inelastic Raman scattering signals. More generally, gaps measured in $s$-wave superconductors with different experimental techniques (such as scanning tunneling microscope and Raman scattering) may be different due to fluctuation corrections. Since already arbitrarily weak QFs lead to the shift and to the new pole, our results shed some light on other amplitude modes even for systems with weak QFs, including charge density waves, (anti-) ferromagnets, or cold atom fermionic condensates.

The Compton Spectrometer and Imager is a NASA satellite mission under development that will survey the entire sky in the 0.2-5 MeV range using a wide-field germanium detector array, surrounded on the sides and bottom by active shields (the Anticoincidence Subsystem, ACS). The ACS aims to suppress and monitor background events, as well as detect transient sources, such as Gamma-Ray Bursts (GRBs), through its onboard triggering algorithm. The data related to GRBs are sent to the ground and analyzed by an automated pipeline to localize the GRBs and share their positions with the community. In this work, we present a brief GRB localization method based on ACS data, utilizing deep learning (DL) techniques, which can estimate the 90\% confidence region, including cases where it is split into multiple areas. To address this, we developed a neural network classifier that predicts the GRB location as a probability distribution across the sky map following the HEALPix framework. The distribution can be used to compute the 90\% confidence regions. Future work will compare this DL-based localization approach with classical methods such as $χ^2$ fitting and Maximum Likelihood Estimation.

Permission control vulnerabilities in Non-fungible token (NFT) contracts can result in significant financial losses, as attackers may exploit these weaknesses to gain unauthorized access or circumvent critical permission checks. In this paper, we propose NFTDELTA, a framework that leverages static analysis and multi-view learning to detect permission control vulnerabilities in NFT contracts. Specifically, we extract comprehensive function Control Flow Graph (CFG) information via two views: sequence features (representing execution paths) and graph features (capturing structural control flow). These two views are then integrated to create a unified code representation. We also define three specific categories of permission control vulnerabilities and employ a custom detector to identify defects through multi-view feature similarity analysis. Our evaluation of 795 popular NFT collections identified 241 confirmed permission control vulnerabilities, comprising 214 cases of Bypass Auth Reentrancy, 15 of Weak Auth Validation, and 12 of Loose Permission Management. Manual verification demonstrates the detector's high reliability, achieving an average precision of 97.92% and an F1-score of 81.09%. Furthermore, NFTDELTA demonstrates enhanced efficiency and scalability, proving its effectiveness in securing NFT ecosystems.

In this note we explore monodromy defects for non-invertible symmetries in Maxwell theory, exploiting the conformal mapping to $AdS_{3} \times S^{1}$. With this approach we recover the spectrum of the defect conformal primaries. We also dedicate some time discussing the behaviour of Wilson/'t Hooft lines in the presence of such a monodromy defect, and highlight the following aspects of their behaviour: i) the lines can terminate on the defect, ii) lines of the unit electric (magnetic) charge may seize to be indecomposable, and can be represented as integer powers of some more elementary lines, and iii) they behave as topological objects when brought close to the defect, and this behaviour is governed by a Chern-Simons theory.

We introduce a Hybrid High-Order (HHO) method for the Schrödinger equation in the presence of a magnetic vector potential. In quantum mechanics, physical observables are invariant under continuous gauge transformations, which must be kept at the discrete level to avoid unphysical artifacts. To address this, we construct a discrete covariant gradient operator on arbitrary polyhedral meshes. We prove that the resulting discrete bilinear form guarantees gauge covariance asymptotically at the discrete level. The resulting scheme achieves optimal convergence rates and preserves a discrete Garding inequality, guaranteeing a stable ground state. The theoretical properties of the scheme are corroborated by numerical experiments, including the computation of the Fock-Darwin fundamental energy and replicating the Aharonov-Bohm effect.

In this paper, we introduce a collection of purely loxodromic free Kleinian groups, called infinite Schottky group, which are defined by a suitable collection of simple loops in a similar way as in the case for Schottky groups of finite rank. An infinite Schottky group $Γ$ admits a $Γ$-invariant connected component $Ω$ of its region of discontinuity, such that every other component is a topological disc and has trivial $Γ$-stabilizer, and $Ω/Γ$ is an infinite type Riemann surface without planar ends. Every infinite type Riemann surface $Σ_{F}$ without planar ends can be so obtained (retrosection theorem). If $G < {\rm Aut}(Σ_{F})$ acts freely and $Σ_{F}/G$ is of finite type, then we observe that it lifts to a group of automorphisms of $Ω$, for a suitable infinite Schottky uniformization of it by a infinite Schottky group $Γ$, if and only if there is a $G$-invariant collection ${\mathcal F}$ of pairwise disjoint essential simple loops on $Σ_{F}$ such that each connected component of $Σ_{F} \setminus {\mathcal F}$ is a finite planar surface, generalizing the situation for the case of Schottky groups of finite rank.

Executing a logical quantum circuit fault-tolerantly incurs a large spacetime overhead. Recent work has proposed and investigated phantom codes, defined by the property that every in-block logical $\mathrm{CNOT}$ circuit can be implemented with a physical permutation, a property that has the potential to greatly reduce the depth of compiled circuits. Here we show that phantomness comes at the cost of low encoding rate. Specifically, we prove that any binary phantom code encoding $k$ logical qubits into $n$ physical qubits with distance $d\geq 2$ obeys the bound $k\leq \log_2(n+1)$ for all $k\neq 4$. For $k=4$ we explicitly construct a nonstabiliser $(\!(8, 2^4, 2)\!)$ phantom code that violates the bound and has a transversal non-Clifford gate. We further show that, within the class of nontrivial CSS phantom codes with $k\neq 4$, there is a unique family of codes saturating this bound. In addition, we prove that this logarithmic ceiling cannot be circumvented by permitting additional local unitary gates, or by making use of subsystem codes: any subspace or subsystem code admitting a $\mathrm{SWAP}$-transversal implementation of every logical $\mathrm{CNOT}$ circuit is constrained to satisfy the same bound. These bounds follow from a general theorem relating the length of a quantum code to the structure of its automorphism group, a result which may find applications beyond phantom codes.

Regulated enterprises in the United States -- banks, telecommunications providers, large technology companies -- operate across heterogeneous systems that were rarely designed to interoperate. ERP platforms, billing engines, supply chain tools, and financial reporting infrastructure coexist within the same organization, but they do not talk to each other well. The resulting fragmentation produces familiar problems: transactions recorded in one system but unreconciled in another, asset inventories drifting from their systems of record, and audit-readiness that depends on manual effort. The PCAOB's 2024 inspection cycle put a number on the consequences: a 39% aggregate Part I.A deficiency rate across all inspected firms. This paper introduces the GERA Framework (Governed Enterprise Reconciliation Architecture) -- a vendor-neutral, four-layer data architecture that integrates deterministic cross-system reconciliation, statistical anomaly detection (baseline Z-Score with robust alternatives), governed semantic standardization, and NIST CSF 2.0-aligned security controls into a single methodology. The architecture spans four layers (ingestion, staging, core models, and semantic serving), following the multi-layer pattern now common in modern data platforms. The patterns are demonstrated through U.S. broadband operations -- where billing reconciliation, inventory aging, and governance are tightly coupled -- and draw on the author's implementation experience across three regulated enterprise environments: a regional bank, a national broadband provider, and a Fortune 500 technology company's central finance organization. This is a practitioner reference -- an architectural framework paper documenting field-tested patterns -- not a controlled experiment or benchmark study. No proprietary systems, datasets, or internal implementations are disclosed.