A residual-set framework is introduced for analyzing additive prime conjectures, with particular emphasis on the Strong Goldbach Conjecture (SGC). For each even integer $E_n\ge 4$, the residual set $ℛ\left(E_n\right)=\{E_n-p\mid$ $p<E_n,p\in \mathbb{P}\}$ is defined, and the universal residual set ${ℛ}_{\mathbb{E}}={{{\displaystyle \cup }}^{\text{}}}_{E_n}ℛ\left(E_n\right)$ is constructed. It is shown that ${ℛ}_{\mathbb{E}}$ contains infinitely many primes. A nontrivial constructive lower bound is derived, establishing that the number of Goldbach partitions satisfies $G\left(E\right)\ge 2$ for all $E\ge 8$, and that the cumulative partition count satisfies ${\displaystyle \sum }_{E\le N}G\left(E\right)\gg \frac{N^2}{{\text{log}}^{4}N}$. An optimized deterministic algorithm is implemented to verify the SGC for even integers up to 16,000 digits. Each computed partition $E_n=p+q$ is validated using elliptic curve primality testing, and no exceptions are observed. Runtime variability observed in the empirical tests corresponds with known fluctuations in prime density and modular residue distribution. A recursive construction is formulated for generating Goldbach partitions, using residual descent and leveraging properties of the residual sets. The method extends naturally to Lemoine's Conjecture, asserting that every odd integer $n\ge 7$ can be expressed as $n=p+2q$, where $p,q\in \mathbb{P}$. A corresponding residual formulation is developed, and it is proven that at least two valid partitions exist for all $n\ge 9$. Comparative analysis with the Hardy-Littlewood and Chen estimates is provided to contextualize the cumulative growth rate. The residual-set methodology offers a deterministic, scalable, and structurally grounded approach to additive problems in prime number theory, supported by both theoretical results and large-scale computational evidence.

The Mambaka watershed is extends between latitudes 1 3°45'E and 14°15'E and longitudes 7°16'N and 6°45'N. The geology, various tectonic and structural events that have affected the Adamawa Plateau in Cameroon make it rich in multi-substance mining. The objective of this study is to map rare earth (REE) geochemical anomalies in the sediments of the watershed streams west of Mambaka, and to trace their origins and geochemical processes. Predictive maps from inverse distance interpolations (IDW), factor analysis (F1) or principal component analysis (PCA) and hierarchical bottom-up classification maps provided a better understanding of the central tendency, distribution and dispersion of REE in the samples and in the study area, based on standard deviation and variance values that generated two factors F1 (Ho-Tm-Er-Yb-Lu-Dy-Tb-Gd-Eu-Sm) and F2 (Pr-Nd-Ce-La-Sm) representing 92.44% of the total cumulative variance. The ratios Ce/Ce* > 0.78 and Eu/Eu* > 1 demonstrate positive anomalies in Ce and Eu, and clear differentiation. The normalized concentrations used to calculate fractionation ratios show that the values for LaN/YbN (0.58 to 1.34), LaN/SmN (0.61 to 0.88) and LaN/LuN (0.62 to 1.43) suggest higher fractionation in SS09 and lower fractionation in SS01. Similarly, the ratios La/Lu (61.71 to 143.46), La/Yb (9.00 to 20.72), La/Sm (4.02 to 5.83) and La/ Lu (61.71 to 143.46) confirm these higher ratios in SS09 and lower in SS01. The REE in the study area comes from hydrothermal processes based on high lineament densities at sampling points in igneous rocks with a mean ∑REE value of between 174-219 ppm.

Background: Maternity leave, whether paid or unpaid, is a critical resource that can significantly impact maternal well-being and newborn outcomes. However, its availability and utilization among mothers living in poverty remain understudied. Education is widely recognized as a key factor that increases access to both paid and unpaid leave. However, the theory of Minorities’ Diminished Returns (MDRs) posits that structural racism, segregation, and labor market discrimination limit the benefits of socioeconomic resources, such as education, for Black and Latino individuals. This suggests that the effects of education on maternity leave may not be uniform across racial and ethnic groups. Objective: This study aimed to examine the MDRs of education on access to unpaid and paid maternity leave among Black and Latino mothers compared to White mothers giving birth while living in poverty. Methods: We utilized baseline data from the Baby’s First Years Study (BFY), a longitudinal investigation of the effects of poverty on child development. The sample consisted of 1,050 mothers living in poverty who had recently given birth. Maternity leave (paid and unpaid) was assessed via self-report, and educational attainment was measured in years of schooling. Structural equation modeling (SEM) and interaction terms were employed to analyze racial and ethnic differences in the relationship between education and access to maternity leave. Results: Educational attainment was positively associated with access to unpaid maternity leave for the overall sample of mothers giving birth in poverty, but this association was weaker for Black and Latino mothers compared to non-Latino White mothers. Education did not significantly increase the likelihood of paid maternity leave, and there were no group differences for this association. Conclusion: This study highlights the urgent needs to address structural racism, labor market discrimination, and residential segregation that diminish the impact of education on living conditions for Black and Latino mothers, compared to non-Latino White mothers, even for those living under poverty. Policymakers and practitioners should develop targeted interventions to reduce racial and ethnic disparities in access to paid and unpaid maternity leave and other critical resources, particularly for new mothers living in poverty. Addressing these inequities is essential for improving maternal and newborn health outcomes and promoting social justice.

Background: Financial well-being is a key domain of overall well-being, encompassing an individual's ability to meet financial obligations, secure their financial future, and maintain a sense of financial freedom. Education is often viewed as a critical pathway to enhancing financial well-being. However, the returns of education on financial well-being are not uniform across racial, ethnic, and nativity groups. The theory of Minorities’ Diminished Returns (MDRs) suggests that the positive effects of education on outcomes such as income and financial security are weaker for marginalized groups, including Black individuals, Latinos, and immigrants. Objective: This study examines the diminished returns of education on financial well-being among Black, Latino, and immigrant populations in the United States. We aim to investigate how structural inequalities contribute to weaker financial returns on education for these groups compared to their White and native-born counterparts. Methods: We utilized data from the Understanding America Study (UAS 2014) to conduct a cross-sectional analysis of adult respondents. The study assessed financial well-being outcomes (e.g., income, savings, and financial security) and their association with educational attainment across racial, ethnic, and nativity groups. Regression models were employed to test for interaction effects between education and race/ethnicity/nativity, adjusting for sociodemographic factors such as age, gender, employment, and family structure. Results: Our analysis included 8,121 individuals. The mean age of the respondents was 48 years (SD = 16). High education was associated with higher financial well-being (B = 1.284, 95% CI: 1.157, 1.410). The interaction terms between education and immigrant status (B = -0.507, 95% CI: -0.930, -0.084), race (Black) (B = -0.770, 95% CI: -1.208, -0.331), and ethnicity (Hispanic) (B = -0.589, 95% CI: -0.969, -0.210) were all significant, suggesting that immigrant, Black, and Hispanic individuals experience diminished returns on education in terms of financial well-being, relative to US-born non-Hispanic White individuals. The significant negative interactions between education and minority statuses (Black, Hispanic, and immigrant) indicate that while education generally improves financial well-being, the magnitude of this improvement is substantially smaller for these marginalized groups. Conclusion: Understanding how education translates to financial well-being across different racial, ethnic, and nativity groups is critical for addressing persistent financial disparities.