Abstract: Dili, the capital of Timor-Leste, is experiencing increasing freshwater demand driven by population and economic growth. It totally relies on groundwater from the Dili Intergranular Aquifer System for supply. There is very little conceptual understanding of the system and little-to-no monitoring data. Understanding the hydrostratigraphy, recharge and surface-groundwater interactions, groundwater levels and abstractions are essential for sustainable groundwater use and management. These are the aims of this study, and a numerical model was created with such purpose. The model included scenarios to assess how the aquifer could react to future increases in groundwater abstraction. Trial and error calibrated the steady-state model, and a comparison of simulated results with observed heads revealed good agreement (RMS <10%). Transient scenario simulations demonstrate that recharge (direct, river infiltration, and mountain-block processes) is a key component of the water balance and plays a critical role in aquifer sustainability under increasing groundwater abstraction. Aquifer storage is projected to decrease significantly by 2054, with the magnitude depending on the range of recharge and abstraction rates considered. The model improves conceptual hydrogeological knowledge of the basin, highlights future work needed, and provides a robust basis for sustainable groundwater management and water risk mitigation in Dili.

Abstract: In a series of recent papers, Haug and Tatum have suggested a way to resolve the Hubbletension within RH = ct cosmology. Based on the full distance ladder of Type Ia supernovae(SNe Ia), they find that the Hubble constant must be H0 = 66.8943±0.0287 km/s/Mpc. Thisvalue is close to the Planck Collaboration’s CMB-based estimate of 67.4 ±0.5 km/s/Mpc,except that their solution yields a much smaller uncertainty in the Hubble constant. TheSH0ES study by Riess et al., based on SNe Ia observations, gives a significantly higher value:H0 = 73.04 ±1.04 km/s/Mpc. The Hubble tension refers to the large discrepancy betweenthe H0 estimates obtained from the CMB method and those from SNe Ia data. Interestingly,recent JWST observations, when tied to SNe Ia, find H0 = 68.81 ±1.79. Thus, the JWSTresults lower the Hubble constant relative to the Riess study and appear to support the Haugand Tatum solution to the Hubble tension, a topic we discuss in this short note.

Abstract: Foodborne diseases and food poisoning caused by bacterial pathogens is a significant global health as well as economic concern. While synthetic compounds are widely used as preservatives to ensure food safety, growing concerns regarding their potential health risks and the rise of antimicrobial resistance have driven the search for natural alternatives. Essential oils (EOs) and their individual bioactive constituents, known as essential oil components (EOCs), have emerged as promising, eco-friendly candidates for food preservation due to their robust broad-spectrum antibacterial properties. This review provides comprehensive mechanistic insights into how individual EOCs exert their antibacterial effects, detailing the disruption of bacterial cell membranes, inhibition of vital metabolic enzymes and ATP synthesis, modulation of virulence gene expression, and the prevention and eradication of biofilms. Furthermore, the review explores the practical applications and limitations of EOCs in food systems, addressing challenges such as chemical instability, toxicity at high doses, and adverse organoleptic effects. It also highlights advanced formulation strategies, such as micro/nano-encapsulation, nano-emulsions, and chemical derivatization, which significantly enhance EOC stability, bioavailability, and overall preservative efficacy. Ultimately, understanding the multifaceted mechanisms of individual EOCs paves the way for their optimized and sustainable use, ensuring global food safety.

Abstract: The phenotypic plasticity of epithelial cells along the epithelial-mesenchymal (E-M) axis, or epithelial-mesenchymal transition (EMT), is a critical aspect of tumour progression and therapeutic resistance. During EMT, epithelial cells gradually acquire mesenchymal traits, facilitating vital functions in embryogenesis, wound healing, fibrosis, and tumour metastasis. This review article investigates the interplay between hyperglycaemia-induced metabolic stress and EMT in the context of therapeutic resistance. The study examines a complex, multifaceted network of molecular mechanisms regulating EMT, including specialised transcription factors and signalling pathways as well as growth factors, integrins, and matrix metalloproteinases in various epithelial carcinomas. Emerging findings have demonstrated the existence of EMT hybrid states along the continuum, possessing heightened metastatic potential and distinctive metabolic signatures that play critical roles in the development of therapeutic resistance in cancer cells. Hyperglycaemia has been particularly highlighted for its potential to promote EMT-driven therapeutic resistance through various interconnected mechanisms. Elevated glucose levels induce the increased production of reactive oxygen species (ROS), activation of EMT-promoting transcription factors, and a metabolic shift towards glycolysis. This hyperglycaemic stress involves upregulation of glucose transporters and glycolytic enzymes creating feed-forward loops that support drug efflux mechanisms and help maintain the mesenchymal phenotype. Clinical data also indicate that hyperglycaemia in OSCC patients is associated with more advanced tumour stages, more ex-tended hospital stays, less effective treatments, and higher rates of local recurrence and distant metastasis. Overall, these insights emphasise the urgent need for a more comprehensive understanding of the underlying mechanisms linking hyperglycaemia and EMT to the treatment resistance axis and to explore glucose control strategies that can be incorporated into cancer treatments to overcome anti-cancer therapy resistance effectively.

Abstract: In general, condition monitoring (CM) and noise, vibration and harshness (NVH) are often treated as separate disciplines, despite the fact that both rely on vibration measurements. CM relies on broadband statistical metrics such as RMS, kurtosis, and envelope analysis to detect faults. Meanwhile, NVH investigates tonal excitation mechanisms related to gear mesh frequency (GMF) and its modulation components. In this study, we investigate whether a numerical relationship can be established between classical CM indicators and physically based tonal excitation indicators derived from frequency-domain analysis. Using a controlled gearbox degradation dataset, Spearman correlation analysis was performed between broadband metrics and GMF-related tonal features, including GMF-band energy and absolute sideband energy. Results show moderate but statistically significant correlations between RMS, envelope peak amplitude, and tonal indicators, whereas kurtosis exhibits no meaningful association. Additionally, tonal amplification due to degradation is shown to be structurally localized rather than uniformly distributed across sensor locations. These findings demonstrate that broadband CM indicators partially encode tonal excitation growth, establishing a reproducible data-driven bridge between diagnostic condition monitoring and NVH excitation analysis.

Abstract:

This paper proposes a unified theoretical framework based on discrete space element dynamics. The core concept posits the existence of a conserved "spatial raw material" through which quantum virtual processes continuously generate new spatial elements, forming localized density gradients that manifest as spacetime curvature. This mechanism inherently excludes superlative effects, remains compatible with general relativity under covariance constraints, and provides a unified explanation for challenges such as dark matter, dark energy, and black hole singularities. The paper first elucidates the fundamental principle of "global covariant symmetry" and then offers an ultimate interpretation of symmetry breaking: symmetry is not "broken" but rather a local cost paid for global covariance. The core dynamics of this framework are systematically developed, with rigorous derivations of Newtonian gravitational limits, mass-energy equations, the principle of the constancy of the speed of light, the fundamental form of Maxwell's equations, and Newton's three laws from basic assumptions. Furthermore, by strictly defining k-body stable entanglement classes on discrete spacetime graphs, the symmetry group is proven to be SU(k), and the gauge group of the Standard Model—SU(3)×SU(2)×U(1)—is uniquely derived. Under the continuous limit, the Yang-Mills action, chiral fermions, Higgs field, and Einstein's gravity are obtained. The theory predicts all 28 independent parameters of the Standard Model—including gauge coupling constants, fermion mass spectra, CKM matrices, PMNS matrices, Higgs parameters, strong CP parameters, and neutrino mass squared differences—with deviations from experimental values generally below 10⁻⁴ to 10⁻⁸. These predictions constitute the "geometric periodic table" of physical constants, signifying that the 28 free parameters of the Standard Model are completely nullified. The article concludes with multiple quantitative predictions verifiable by future experiments, providing a self-consistent, comprehensive, and experimentally testable new pathway for the unification of quantum gravity and particle physics.

Abstract: Immune-mediated tissue injury is typically conceptualized as a consequence of aberrant immune activation; however, spatial patterns of lesion formation and selective tissue vulnerability across diseases suggest that immune activity alone may not determine where damage becomes established. We propose a generalizable systems framework in which regional metabolic preconditioning, defined by local perfusion dynamics, oxygen availability, and bioenergetic resilience, modulates the threshold for immune-mediated injury. In this model, tissue susceptibility emerges from the interaction between (1) immune activation intensity and (2) region-specific metabolic state. Reduced perfusion and relative hypoxia stabilize hypoxia-inducible signaling pathways, alter endothelial integrity, reprogram cellular metabolism, and amplify inflammatory responsiveness. These processes do not initiate autoimmunity but reshape the energetic and vascular landscape in which immune mechanisms operate, thereby governing spatial lesion topology and progression. We formalize this interaction as a threshold-modulation framework in which tissue injury probability is a function of both immune effector load and metabolic resilience. Applied to multiple sclerosis as a model system, this perspective integrates cerebral hypoperfusion, mitochondrial dysfunction, blood-brain barrier instability, and compartmentalized inflammation into a unified explanatory structure. The framework generates falsifiable predictions regarding perfusion-lesion coupling, metabolic biomarkers of susceptibility, and cross-disease parallels in immune-driven pathology. By positioning metabolic state as a dynamic modifier of immune injury thresholds, this model shifts emphasis from single-axis causation toward systems-level interaction, offering a conceptual template for understanding spatial selectivity and progression in immune-mediated diseases.

Abstract: Interest in non-centrosymmetric crystalline materials exhibiting second harmonic genera-tion (SHG) has increased due to their potential applications in optical sensing and bio-sensing. Saccharide-based metal complexes are particularly attractive systems, as chiral sugars can promote non-centrosymmetric crystal packing. In this work, a new lantha-num–β-D-fructose compound, [La(C₆H₁₂O₆)(H₂O)₅]Cl₃ (LaFRUCl), was synthesized by a simple and low-cost method and characterized by single-crystal X-ray diffraction. The compound crystallizes in the orthorhombic space group P2₁2₁2₁ and consists of infinite (La³⁺–fructose)ₙ chains extending along the [001] direction, forming a one-dimensional metal–organic framework. The nonlinear optical response was evaluated using the Kurtz–Perry powder technique with a Nd:YAG laser (1064 nm) and compared to a sucrose ref-erence. The measured SHG efficiency is comparable to that of previously reported alkaline earth metal–sugar analogues. While the SHG emission is significant, evaluation of the compound’s structural stability under aqueous or physiological conditions would be re-quired before considering biological applications.

Abstract: Large Language Models (LLMs) have fundamentally transformed the landscape of Natural Language Processing (NLP), subsuming and redefining tasks that were once addressed by specialized, modular pipelines. This paper surveys the role of classical and contemporary NLP within modern LLM architectures, examining how foundational techniques — tokenization, syntactic parsing, semantic representation, and discourse modeling — have been absorbed into, and continue to inform, the pre-training and fine-tuning paradigms of transformer-based models. We further investigate the critical challenge of linguistic inclusivity, focusing on low-resource and morphologically complex languages that remain underserved by dominant English-centric corpora. Drawing on recent advances in cross-lingual transfer learning, multilingual pre-training, and data augmentation, we assess the progress and persistent gaps in extending LLM capabilities to such languages. Case studies on Southeast Asian, African, and indigenous language NLP toolkits illustrate practical strategies and remaining bottlenecks. We conclude by outlining open research directions at the intersection of structural NLP and generative AI.

Abstract: Harmful algal blooms (HABs) caused by Karenia brevis (K. brevis) present a persistent ecological and public health challenge across coastal Florida. This study develops a regionally integrated machine learning framework to predict weekly K. brevis bloom occurrence using environmental data from both the Peace and Caloosahatchee Rivers, combined with coastal bloom records from Southwest Florida and Tampa Bay to enhance the spatial and temporal continuity of the response record. A Random Forest classifier was trained on a multi-decadal dataset incorporating river discharge, nutrient concentrations (total nitrogen and total phosphorus), wind forcing, sea surface temperature, salinity, and sea surface height anomalies as a proxy for Loop Current variability. The model achieved strong predictive performance on a chronologically withheld test set, with an overall accuracy of ~90%, balanced accuracy of 87.6%, and high precision and recall for bloom events. Bloom timing and persistence were captured with strong agreement during ongoing bloom periods, while non-bloom conditions were identified with low false-positive rates. Feature-response analyses indicated that bloom probability increased most sharply under moderate discharge and nutrient conditions, with diminished sensitivity at higher extremes. Learning curve analysis demonstrated robust training performance and stable generalization, with validation accuracy plateauing near 84%, suggesting a data-limited ceiling on forecast skill. By aggregating nutrient inputs across multiple watersheds and integrating spatially aligned bloom observations, this study demonstrates the utility of multi-source machine learning frameworks for regional-scale HAB prediction. The results support the development of early warning tools and provide a reproducible foundation for evaluating how combined watershed loading and physical forcing are associated with K. brevis bloom occurrence in complex estuary systems with watershed and coastal coupling.

Abstract: Background/Objectives: The aim is to assess public stigma related to people with mental health problems and to observe the influence of sociodemographic factors on it. Methods: This observational, descriptive, and cross-sectional study was conducted with a sample of 404 participants who completed a self-administered online questionnaire that included sociodemographic variables and the Community Attitudes towards Mental Illness Scale (CAMI-S, Spanish version). Non-parametric tests, multiple linear regression, and statistical power analyses were applied. Results: The mean CAMI-S total score was 84.89 (SD = 11.122), indicating a generally positive attitude towards community integration. Statistically significant associations (p-value ≤ 0.05) were found between CAMI-S scores and variables such as gender, age, place of residence, educational level, mental health disorder, and close contact with someone with mental health disorders. The regression model revealed four significant predictors of lower stigma: identifying as female (β = 2.523; p = 0.037), having a medium or higher educational level (β = 5.061; p = 0.002), experiencing a mental health diagnosis (β = 4.535; p = 0.014), and a close contact (β = 4.183; p &lt; 0.001). Conclusions: These findings underscore the need for targeted anti-stigma strategies and reinforce the role of nursing in promoting mental health inclusion.

Abstract: Cruise calls in medium-sized Mediterranean ports concentrate visitor flows along short urban connectors, intensifying congestion and localized environmental externalities. This study evaluates cruise passengers’ willingness to pay (WTP) for an electric tram linking Gaženica Port with Zadar’s historic center, an intervention designed to cut travel time and reduce on-street crowding and emissions. A two-wave, two-site, face-to-face survey was administered over two seasons at the port and in the city center. The instrument adopts a double-bounded dichotomous choice contingent valuation design with randomized starting bids calibrated via a pre-test that benchmarked prevailing transport prices. Primary WTP estimates are obtained from a binary choice model with socio-demographic and environmental covariates; inference relies on cluster-robust errors. Robustness is assessed through three complementary checks that do not require additional data: (i) a bivariate specification to accommodate within-respondent correlation between first and follow-up bids; (ii) Turnbull nonparametric bounds for the interval-censored WTP distribution; and (iii) starting-point tests via bid-set indicators and split-sample estimation. Where applicable, a spike adjustment based on “no–no at the lowest bid” responses is explored. Beyond methodological contribution, this research advances the sustainable tourism development discourse by quantifying visitors’ monetary support for low-emission urban mobility infrastructure that mitigates environmental pressures while preserving resident quality of life. The findings provide a decision-ready valuation input for port–city mobility planning in historic Mediterranean cores, aligning cruise tourism management with the broader objectives of resilient and sustainable urban destinations.

Abstract: Industrial emissions and legacy contamination from metallurgical activities can constrain sustainable land use by degrading soil quality and limiting vegetation establishment. This study combines a site-based contamination assessment with an early-stage plant tolerance screening to inform nature-based restoration planning in Central Kazakhstan. Soils were collected around three metallurgical complexes and analysed for heavy metals; exceedance relative to maximum permissible concentrations (MPC) was used to prioritise contaminants of concern. Seven locally occurring plant species were then screened in controlled Petri-dish assays using metal salt solutions (Pb, Zn, Cu, Cr, Cd and Ni), and germination percentage, germination dynamics, seedling shoot length and a growth inhibition index were quantified. Soil results showed elevated metal loads with frequent MPC exceedance, supporting the selection of these metals for biological screening. Plant responses were strongly species-specific: Brassica juncea and Medicago sativa maintained comparatively higher germination and early growth across treatments, whereas Suaeda salsa, Artemisia absinthium and Trifolium repens exhibited very low germination. These findings provide an evidence-based shortlist of candidate species for subsequent soil-based trials (including uptake and stabilisation assessment) and support risk-informed revegetation strategies for contaminated industrial landscapes.

Abstract: Background and Objectives: Bladder cancer (BCa) is characterized by high rates of re-currence and progression, underscoring the need for reliable non-invasive biomarkers. Circular RNAs (circRNAs) are covalently closed non-coding RNAs generated by back-splicing and are stable in biological fluids, including urine. Increasing evidence im-plies circRNAs in BCa pathogenesis; however, identification of clinically relevant circRNAs remains labor-intensive. This study aimed to streamline circRNA selection and identify functionally relevant urinary circRNAs in BCa. Methods: Using a database-screening ap-proach, we identified circRNAs with high predicted affinity to miR-101-3p, a tu-mor-suppressive microRNA in BCa. Candidate circRNAs were prioritized based on: (i) strong miR-101-3p binding potential; (ii) derivation from genes involved in BCa tumor-igenesis; and (iii) origination from exonic or long non-coding RNA sequences. The po-tential contribution of Argonaute-2 (Ago2) binding sites to circRNA–miRNA complex sta-bility was also evaluated. Expression levels were assessed in urine samples and BCa cell lines, and functional relevance was examined using molecular and cellular assays. Results: circCIAO1(5) and circMALAT1 fulfilled all prioritization criteria and exhibited distinct Ago2-binding site profiles. Both circRNAs were upregulated in urine from BCa patients and in aggressive BCa cell lines and showed differential expression between remission and recurrent disease. CircCIAO1(5) demonstrated higher-affinity binding to miR-101-3p, while RNA immunoprecipitation confirmed interactions of both circRNAs with miR-101-3p and Ago2. Functional assays revealed enhanced proliferation, motility, and invasion upon circRNA expression, consistent with miR-101-3p sequestration and derepression of miR-101-3p target oncogene-EZH2. Conclusions: circCIAO1(5) and circMALAT1 represent promising urinary biomarkers for BCa, illustrating the value of bioinformatics-guided circRNA discovery and significance of circRNA-mediated regulatory mechanisms in BCa biology.

Abstract: Web agents must complete long-horizon browsing tasks while controlling heterogeneous operational costs (e.g., API calls, latency, and monetary fees) and avoiding catastrophic failures (e.g., irreversible clicks, account deletion, payment submission). We formulate web interaction as a constrained MDP with a multi-dimensional cumulative cost vector and a tail-risk objective on failure penalties. We propose DCAPPO, a dual-constrained policy optimization method that (i) enforces multi-cost budgets via primal–dual Lagrangian updates with per-cost adaptive multipliers, and (ii) minimizes CVaRα_\alphaα​ of episodic failure loss using quantile regression on trajectory returns. To stabilize training under sparse success rewards, DCAPPO integrates a self-imitation buffer and a failure-aware advantage shaping that down-weights high-variance steps. We recommend evaluation on BrowserGym/WebArena-style environments with 1,200–1,800 tasks spanning 40–80 website templates, reporting (a) task success rate, (b) mean cost per success, (c) CVaR0.1_{0.1}0.1​ failure loss, and (d) constraint violation frequency. In ablations, DCAPPO isolates gains from CVaR control and per-cost dual updates, targeting a consistent reduction in tail failures under fixed cost budgets.

Abstract: Antimicrobial resistance (AMR) is a silently escalating global crisis, presenting a specific challenge for the One Health approach. Landscapes can serve as reservoirs of AMR, while synurban wildlife may act as vectors of bidirectional exchange. However, these species can also be utilised as sentinels of landscape AMR load. Herbivorous avian bioindicators, such as the Common Wood Pigeon (Columba palumbus), continuously sample the landscape during foraging and drinking, providing unbiased data on the state of AMR. This study aimed to investigate the potential of this species for assessing the impact of landscape diversity on bacterial communities and their AMR patterns. Toward this objective, two landscape units of 4-km-diameter located at an upstream and a downstream section of a river, relative to a provincial town, were compared using 16 cloacal samples per site. Heterotrophic plate count techniques resulted in 60 isolates, of which 48 were identified, and 35 were tested for AMR using the VITEK 2 Compact system. Rényi diversity profiles of landscape compositions, bacterial communities, and AMR patterns revealed that higher landscape diversity was associated with lower bacterial but higher AMR pattern diversity. Additionally, the structure of more diverse bacterial communities shifted toward Gram-negative taxa. These findings support the hypothesis that culture-based methods using Common Wood Pigeons, complemented by Rényi diversity analysis and the determination of Gram-positive/Gram-negative ratios, provide valuable data on landscape health, even with small sample sizes.

Abstract: Lipoblastomas are rare, rapidly growing benign tumors rising from embryonic white fatty cells that continue to proliferate in the postnatal period. We presented a case of a toddler with an undifferentiated myxoid neoplasm with features of a minimally differentiated lipoblastoma. Our patient was an 18-month-old female with a painless solid tumefaction in the middle third of the right leg. Histopathologically, the nodular tumor mass consisted of lipobastic cells embedded in a myxoid stroma. Immunohistochemistry showed strong diffuse positivity for vimentin, S100, CD34, CD56, NSE and rare Ki67+ cells. FOXO1 polyploidy was detected in 30% of cells by FISH. Using target RNA sequencing, we detected a fusion gene, CHCHD7-PLAG1, in the tumor sample. Sequence analysis showed that the first exons of CHCHD7 were fused to either exon 2 or exon 3 of PLAG1. Our case demonstrates that due to the histomorphologic overlaps, the molecular diagnostics is essential for confirmation of lipoblastomas.

Abstract: This paper investigates the seasonal and daily responses of the zonal‑mean O₃ mass‑mixing ratio to polar‑vortex disturbances during the boreal winter of 2023/2024, using MERRA‑2 data for the period 1 October 2023–30 April 2024. In addition to the expected latitudinal coupling during SSW events, the seasonal ozone field exhibited a pronounced zonally asymmetric distribution, referred to as the zonally asymmetric ozone oscillation (ZAOO), most evident in the lower stratosphere throughout the winter months. The seasonal behaviour of the ozone tendency was also investigated. To provide a plausible explanation for the observed features, a combination of the Quasi-biennial oscillation (QBO), dynamical transport, and photochemical processes was considered. For the first time, TEM diagnostics were applied to individual winter seasons and specific SSW events, enabling detailed examination of ozone‑tendency variability across latitude and altitude. The results provide clear quantification of the dynamical and net chemical contributions to both the seasonal (October–April) and specific SSW event ozone tendencies. These findings support systematic assessments of each intriguing winter and SSW event, offering new opportunities to identify links between SSW type and the dominant mechanisms shaping the ozone‑tendency response.

Abstract: The increasing prevalence of peri-implantitis has led to a growing clinical use of implantoplasty, a procedure involving intraoral machining of the dental implant sur-face to remove biofilm. The absence of standardized clinical protocols may contribute to premature fatigue failure of dental implants. The present study aimed to determine the influence of machining depth on the cyclic mechanical behavior of dental im-plants. A total of 250 commercially pure Grade 3 titanium dental implants were dis-tributed into four groups according to machining depth: untreated (original), 0.2 mm, 0.4 mm, and 0.6 mm wall reduction. The implant system featured an internal connec-tion with a thread height of 0.4 mm. Finite element analysis was performed for each machining depth to evaluate Von Mises stress distribution and to simulate fatigue be-havior. The numerical models were validated through experimental fatigue testing using a servohydraulic MTS Bionix testing machine under ISO 14801:2016 standard conditions. Fractographic analysis was conducted by scanning electron microscopy. The results revealed that maximum Von Mises stresses were concentrated at the junc-tion between the implant thread and the implant body. The fatigue limit of the un-treated implants was approximately 400 N. Implants subjected to 0.4 mm machining exhibited a fatigue limit of 350 N, whereas lower fatigue limits were observed for 0.2 mm (290 N) and 0.6 mm (180 N) reductions. These findings demonstrate the signifi-cant mechanical effect of thread removal. At higher applied loads, fracture occurred in the coronal region of the implant, whereas at lower loads failure shifted to the im-plant–abutment connection. Finite element predictions showed high agreement with experimental results. The findings highlight a clinically relevant criterion: implanto-plasty depth should not exceed the original thread height, as excessive wall reduction markedly compromises fatigue resistance and long-term mechanical reliability.

Abstract:

For every prime p and every integer a, the backward finite difference δp(a) := aᵖ − (a − 1)ᵖ equals the cyclotomic binary form Φp(a, a − 1) and hence the norm N_Q(ζp)/Q(a − ζp(a − 1)). For p = 3 this specialises to δ3(a) = N_Z[ω](a − ω(a − 1)), connecting the individual cubic finite difference — obtained by differencing the classical sum formula of Nicomachus of Gerasa (∼100 CE) — with the Eisenstein norm that appears in Euler's factorisation of a³ + b³.Starting from the historical identity S3(n) = Tₙ² where Tₙ = n(n + 1)/2, and applying the backward finite difference operator ∇f(n) := f(n) − f(n − 1) — formalised by Taylor (1715) and systematised by Boole (1860) — the Cubic Identity is derived: n³ = (n²/4)[(n + 1)² − (n − 1)²] = Tₙ² − Tₙ₋₁².This identity is extended to all p ≥ 1 via the Universal Faulhaber–Bernoulli Identity (UFBI): nᵖ = 1/(p+1) Σⱼ₌₀ᵖ C(p+1,j) Bⱼ⁺ δp+1−j(n), δm(n) := nᵐ − (n−1)ᵐ.The central contribution of this work is the Unified Chain Formula: ∇Tₙ² = δ3(a) = N_Z[ω](a − ω(a−1)) = Φ3(a, a−1) = N_Q(ζ3)/Q(a − ζ3(a−1)), which connects, in a single proved identity, five centuries of mathematics: Nicomachus (1st century), Boole (19th century), Euler/Eisenstein (18th century), and Gauss/cyclotomic theory (19th–20th centuries). This chain is not present as such in the existing literature; its originality lies in the explicit articulation of these connections, not in the individual equalities, each of which follows from classical results.Beyond the Unified Chain, the following new elements are introduced: (i) the Tower of Norms a³ = Σₖ₌₁ᵃ N(αk), making explicit how each perfect cube is a stack of hexagonal norms; (ii) the Cyclotomic Compatibility Index ICC(n, p), which quantifies the arithmetic obstruction to hᵖ = aᵖ + bᵖ having integer solutions; (iii) the Window Incompatibility Theorem, formalising why the hexagonal windows {a−1, a, a+1} and {b−1, b, b+1} can never merge into a single window {h−1, h, h+1} in Z[ω] for a, b ≥ 2; (iv) the Order Theorem for δm(n), providing a complete characterisation of prime divisibility of finite differences via multiplicative orders; (v) the Extreme Reduction Theorem (ERT), showing that the Order Filter eliminates every pair (a, b) with a ≥ 2 from the equation a³ + b³ = c³, reducing the problem to the case a = 1; (vi) the Fermatian Rigidity Index R(p), a quantitative measure of how far (aᵖ + bᵖ)^(1/p) is from an integer. All results are illustrated throughout by the single running example a = 6, b = 10, and the key number 91 = 7 × 13. Verified over 179,700 pairs with 50-digit precision: zero exceptions. This work does not claim to prove Fermat's Last Theorem, definitively established by Wiles [1].