Abstract: The 1970 Bhola Cyclone is usually narrated through catastrophic mortality, political rupture, and the subsequent expansion of cyclone preparedness in Bangladesh. Its long-term psychosocial, disability, caregiving, and intergenerational consequences remain far less visible. This critical narrative review distinguishes direct Bhola-specific evidence from contextual evidence derived from later cyclones and related disasters in Bangladesh. Direct evidence is strongest for mortality, political and institutional change, and life-course losses in adult health, education, and consumption among people exposed early in life. Bhola-specific evidence is limited or absent for prolonged grief, post-traumatic stress, depression, anxiety, disability trajectories, functional decline, gendered caregiving, migration histories, and intergenerational memory. Contextual studies nevertheless show that cyclone resilience is shaped by psychological distress, livelihood and food insecurity, inaccessible shelters and sanitation, age-related evacuation barriers, warning credibility, institutional trust, and community volunteerism. The review reframes resilience as more than survival: sustainable disaster risk reduction requires dignity, functioning, equitable access, social participation, trusted institutions, and long-term well-being. It concludes with measurable policy indicators and a research agenda linking mental health, disability inclusion, age-sensitive preparedness, social protection, accessible infrastructure, and community governance to the Sendai Framework and the Sustainable Development Goals.

Abstract: Vultures are far more than carrion-eaters. This review looks at how they are physiologically equipped to safely digest decomposing flesh, how quickly they can clear a carcass from the landscape, and how feeding habits differ from one species to the next. It also revisits the diclofenac poisoning crisis that devastated South Asia's vulture populations, along with its surprising knock-on effects for disease spread and human health, before closing with the main threats vultures face today and what can realistically be done about them. Every day in nature, large numbers of animals die, leaving carcasses behind. These carcasses are a natural part of the flow of matter and energy through ecosystems. How long a carcass remains in the environment depends on many factors, including temperature, humidity, the size of the carcass, and the presence and density of the organisms that feed on it. Organisms that consume carrion play an important role in this process, and among them, vultures are one of the most specialized bird groups when it comes to exploiting animal carcasses.

Abstract: p53-dependent signaling and the integrated stress response are the major programs aimed at fine-tuning cell metabolism and determining cell fate in the presence of stress stimuli. While activation of p53 more often results in proliferation inhibition and cell death than pro-survival adaptation, ISR can lead to distinct outcomes depending on cell context, either promoting cell survival or inducing cell death. Recent data indicate the existence of a cross-talk between these networks; however, the underlying mechanisms are still to be elucidated. Here, we have demonstrated that p53 can transcriptionally repress ATF4, one of the key players of ISR, either in the absence or presence of various metabolic stresses, such as mitochondrial dysfunction. We have shown that this effect partially relies on the p53 target p21^Cip1/Waf1 and the Rb family protein p130, which mediate p53-dependent transcriptional repression of target genes. Our findings shed light on a new possible mechanism behind the interplay of p53 and the ATF4-dependent branch of ISR, which is likely to play an essential role in cell metabolic adaptation and cell fate decisions.

Abstract: We prove that the optimal sphere packing density in R⁴ equals π²/16 ≈ 0.6169, achieved uniquely by the D₄ root lattice. The proof is local: we show every Voronoi cell of a unit-ball packing satisfies vol(V_c) ≥ 8, with equality if and only if the active neighbours form a copy of the D₄ root system scaled to nearest-neighbour distance 2. The Voronoi domain decomposes into 176 combinatorial chamber types, grouped into 22 orbits under the Weyl group W(D₄) of order 192. For each orbit representative we produce an explicit sum-of-squares witness establishing positivity of the angular volume surplus F_Ω(θ) on (0, π/3], verified by exact rational LDL^⊤ decomposition of a 6 × 6 Hessian. A uniform Taylor coefficient c₂ = 1/3 handles the neighbourhood of θ = 0. A piecewise positivity bound bridges the Taylor bound to the full angular range: on Piece I [0.1, π/3], a Lipschitz continuation with constant L ≤ 0.396 (bounded analytically from the closed-form formula (10.3)) and grid spacing h ≈ 9.5 × 10⁻⁴ gives error Lh ≤ 3.76 × 10⁻⁴ ≪ min F ≈ 0.00334, certifying F > 0; on Piece II (π/3, π/2), strict monotonicity F′ ≥ 0.10 > 0 (established from a 500-point central-difference grid using the closed-form rational formula (10.5) for the 29-vertex V-representation) gives F(θ) ≥ F(π/3) = 2 − √3 > 0. This resolves the four-dimensional sphere packing problem.

Abstract: Breast cancer is the most diagnosed cancer among women worldwide. Klotho is a protein with well-established anti-aging and tumor-suppressive functions, including a documented role in cancer biology such as breast cancer. However, the role of circulating Klotho levels as a biomarker in breast cancer patients remains unclear. We examined serum Klotho in relation to breast cancer and clinicopathological factors, including estrogen receptor (ER), progesterone receptor (PR), HER2 status, lymph node involvement, and body mass index (BMI). A total of 372 individuals were analysed: breast cancer patients (n=166), ductal carcinoma in situ (DCIS, n=24), and controls (n=182). Serum Klotho was measured by enzyme-linked immunosorbent assay (ELISA); comparisons used nonparametric Kruskal-Wallis and Mann-Whitney U tests with Bonferroni correction. Klotho was higher in breast cancer patients than controls (p < 0.01), similar between invasive and in situ disease, and correlated positively with insulin-like growth factor 1 (IGF-1) (p < 0.01), but not with BMI, ER, PR, HER2, or lymph node invasion. These findings raise the possibility that the Klotho–IGF-1 correlation reflects a negative feedback mechanism, whereby IGF-1 may promote Klotho secretion while Klotho may in turn attenuate downstream IGF-1R signalling, a mechanism that warrants further investigation as a potential regulator of IGF-1-driven tumorigenesis.

Abstract: Many scientific problems are fundamentally relational: the quantity of interest is a connection between two objects, while the information that describes similarity is available for the objects themselves. This mismatch is common in brain networks, molecular interactions, and other high-dimensional systems. We develop a covariance framework for unordered relationships that transfers object-level geometry to the relations they form while preserving endpoint identity and invariance to ordering. Building on symmetric pairwise-kernel representations, we develop new theory for the loop-free domains used in undirected networks. We establish spectral interlacing and trace-loss results after self-pairs are removed, connect the relational spectrum to regularization and risk, derive an exact inferential error for spectral truncation, and quantify how perturbations of the underlying geometry propagate to the pair covariance and estimator. Simulations show when structured borrowing improves estimation and when geometric misspecification can erode that benefit. We then apply the framework to autism neuroimaging using resting-state functional-connectivity data from the Autism Brain Imaging Data Exchange (ABIDE). The analysis considers a 116-region brain parcellation, yielding 6,670 unique functional connections, and shows that this large connectome can have a far smaller effective covariance dimension. The application also demonstrates that high explained covariance alone is not sufficient for choosing a low-rank representation when the goal is to preserve inferential accuracy. These results illustrate how biologically meaningful information defined at the brain-region level can organize dependence among connections while retaining the connection itself as the inferential unit. The framework provides a principled foundation for covariance and regularization when the statistical units are unordered relationships.

Abstract:

Ionizing radiation induces oxidative damage and disrupts plant growth and photosynthetic function. This study evaluated whether seed pre-treatment with a synthesized imatinib–zinc complex (ZnIM), with the proposed composition Zn(C₂₉H₃₁N₇O)₂, could mitigate the effects of gamma irradiation in maize (Zea mays L.). Maize seeds were pre-treated with 0.001%, 0.01%, or 0.1% (w/v) ZnIM and subsequently exposed to 150 Gy gamma irradiation. Germination, seedling growth, malondialdehyde (MDA) content, photosynthetic pigments, maximum quantum efficiency of photosystem II (Fv/Fm), and selected mature-plant traits were assessed. Irradiation reduced germination, germination energy, root length, and shoot length by 9.3%, 9.7%, 9.8%, and 12.7%, respectively. Chlorophyll a, total chlorophyll, and Fv/Fm decreased by 32.0%, 30.4%, and 6.2%, while first-week MDA content increased 2.06-fold. Plant height and 1000-grain weight declined by 11.1% and 5.4%, respectively. ZnIM partially mitigated these effects, with responses varying among concentrations. The 0.01% treatment most consistently restored germination, germination energy, and root length and improved chlorophyll a, total chlorophyll, and Fv/Fm. The 0.1% treatment produced the most sustained MDA reduction. Improvements in mature-plant traits were numerical but not statistically significant. Overall, ZnIM provided partial radioprotection, particularly during germination and early seedling development.

Abstract: In this paper, we introduce the notion of strong contractions. These contractions resemble the strong triangle inequality in the definition of strong b-metric spaces [5]. We obtain some fixed point theorems for these new contractions in the setting of metric spaces with an illustrative example. Finally, we apply our result to the Fredholm integral equation.

Abstract: In the article, a subclass of Starlike functions linked with Bernoulli,s lemniscate is studied in the unit disc. The bounds of H_3,2 Hankel determinant, the sixth coefficient |b₆| and the Zalcman functionals for the class SL* are obtained. The results are sharp.

Abstract: Wheat and rice have high economic and food importance, making the use of seeds with proven physical and physiological quality essential. This study aimed to evaluate whether reducing sampling intensity changes the results of mandatory and complementary analyses used in the seed certification system. Sampling was performed in big bags from commercial wheat and rice seed lots, using four collection intensities corresponding to 100, 75, 50 and 25% of the recommended number of primary samples. The samples were analyzed at the Flávio Farias da Rocha Seed Laboratory, Federal University of Pelotas, for physical purity, determination of other seeds, inert material, germination and viability by the tetrazolium test. The results were compared based on the tolerances established in the Rules for Seed Testing, at 5% probability. No significant differences were observed among the evaluated intensities for the studied lots and cultivars. Thus, under the conditions of this study, sampling intensity could be reduced to 25% without impairing the interpretation of laboratory results.

Abstract: Objective: To review multimodal artificial intelligence (AI) models for predicting adverse outcomes in pneumonia, with emphasis on prognosis, validation, benchmarking, explainability, and clinical implementation readiness. Materials and Methods: We conducted a PROSPERO-registered systematic review. PubMed, Scopus, IEEE Xplore, and supplementary register/citation-search sources were searched between 01 January 2000 and 27 February 2026. Eligible studies integrated at least two heterogeneous clinical data modalities within one predictive framework for pneumonia-related deterioration, mortality, escalation of care, ventilation, or related adverse outcomes. We extracted modality composition, fusion architecture, outcome alignment, validation, performance metrics, clinical benchmarking, explainability, predictive uncertainty, workflow integration, governance reporting, and risk of bias. Results: Of 10,460 identified records, 49 studies were included. The evidence base was dominated by COVID-era studies. Structured electronic health record data, laboratories, and vital signs were more common than imaging or clinical notes. Early fusion was the predominant strategy, while attention-based and graph-based fusion were rare. External validation was reported in 19/49 studies, calibration in 9/49, temporal validation in 5/49, decision curve analysis in 3/49, and prospective evaluation in 3/49. Incremental AUROC over the best unimodal baseline was computable in 19 studies and was usually modest. Discussion: The literature emphasised retrospective discrimination more than clinical utility, with limited reporting of added value, workflow, predictive uncertainty, and governance. Conclusion: Multimodal pneumonia prognosis research is active but not yet implementation-ready. Future studies should prioritise externally validated, calibrated, explainable, clinician-aligned, and workflow-aware models.

Abstract: Education in the Philippines continues to face the challenge of integrating cultural heritage into classrooms in ways that are meaningful, policy-aligned, and sustainable. Mindanao’s oral epics, folktales, and musical traditions embody resilience, identity, and collective memory, yet remain at the margins of formal curricula. This study addresses that gap by introducing the Technographic Framework for Multilingual Literature and Musical Orality in Education (TFMLMOE), which connects heritage with curriculum design through artificial intelligence as an interpretive lens. Guided by a qualitative interpretive design that employed corpus construction, sentiment analysis, hybridity mapping, and audio pattern recognition, the research examined cultural materials including the Ulahingan, Tudbulul, Darangen, folktales in Cebuano, Maranao, Chavacano, and English, as well as ritual chants and gong ensembles. The analysis was anchored in Orality-Literacy Theory, Multilingual Hybridity, Ethnomusicology, Digital Humanities and Technography, and Peace Studies with Values Integration, providing a multidimensional lens for interpretation. A total of sixty-five scholarly sources spanning 1975 to 2025 were reviewed; following Creswell and Poth’s (2018) methodological guidance, this breadth ensured saturation of themes and strengthened validity through multiple perspectives. Findings revealed ten themes like oral epics as vehicles of language and values, modern texts as sites of hybridity and identity negotiation, musical orality as collective memory, and artificial intelligence as a means of uncovering hidden cultural patterns. From these findings, the framework is articulated through four components: corpus analysis, cultural preservation, values integration, and curriculum application. It aligns with CHED mandates, MTB-MLE and OBE compliance, UNESCO principles, and the United Nations Sustainable Development Goals. While the study is limited to selected Mindanao communities, it advances scholarship and policy by revitalizing cultural heritage and positioning it as a dynamic resource for pedagogy, resilience, and cultural revitalization.

Abstract: To address the large volume of reservoir dam seepage safety assessment reports, inconsistent textual descriptions, low efficiency of manual statistics, and difficulty in maintaining consistent classification criteria, this study proposes a human-computer interactive agent collaboration framework for dam seepage safety assessment. Using a large language model as the core for semantic understanding, the framework decomposes the report review process into subtasks including data preprocessing, knowledge retrieval, semantic classification, result feedback, and manual review. Structured collaboration among multiple agents is achieved through an "entity location-safety hazard" knowledge primary key, semantic rule constraints, and the Model Context Protocol (MCP). While preserving expert control, the system transforms original texts into aggregable and traceable risk statistics, forming a closed-loop mechanism of "machine-enabled efficiency, human oversight, and knowledge accumulation". Application validation was conducted using more than 2,000 reservoir safety assessment reports, from which 2,804 severe-risk records were extracted. The results show that severe risks were mainly concentrated in key locations including the dam body, culverts, spillway, dam foundation, downstream dam, drainage facilities, and abutments. The main risk types were leakage, abnormal seepage behavior, construction quality problems, noncompliant seepage control, structural damage, and noncompliant drainage facilities. The findings indicate that the proposed framework can improve the efficiency of seepage safety assessment text processing and the consistency of risk statistics, thereby providing intelligent support for reservoir dam hazard screening and scientific decision-making.

Abstract: For a graph G, let \( \nu_{\triangle}(G) \) be the largest number of pairwise edge-disjoint triangles and let \( \tau_{\triangle}(G) \) be the smallest number of edges meeting every triangle. Tuza's conjectured that \( \tau_{\triangle}(G)\leq 2\nu_{\triangle}(G) \) for every graph. We prove the conjecture for every split graph with a specified split partition \( V(G)=C\mathbin{\dot\cup}I \) such that \( |C|=8 \), for which the triangle-active vertices of I have at most two distinct neighborhoods in C. The result allows arbitrary multiplicities of the two types. The infinite statement is reduced to 44,702 canonical cases by an exact multiplicity-truncation lemma and symmetry. Each case carries an explicit triangle cover and an explicit edge-disjoint triangle packing, checked by an independent standard-library verifier. We also give exact cover and packing reductions for arbitrary split graphs and a maximum-cut criterion that settles, at \( |C|=8 \), every instance whose independent-side neighborhood 2-shadow has at most seven edges. The unrestricted split-graph and interval-graph cases remain open.

Abstract: In this study, a new pressure stabilization (PS) system composed of a gas stable vessel (SV), PID control system and a reaction vessel (RV), etc. was used to study the dynamic characteristics of methane separation from low concentration coalbed methane com-pared with PV system. A porous medium system was constructed by sodium lignosul-fonate (SL) or calcium lignosulfonate (CL) solution which could will reduce gas-liquid interfacial tension, and enhance methane dissolution and mass transfer. At the same time, a thermodynamic accelerator of cyclopentane (CP), which effectively reduced hydration reaction conditions and improved methane storage rate, was added to the hydration reaction solution. The experiments were carried at 275.15 K and 3.0 MPa, the mass concentration of lignin was 500 ppm, the volume ratio of CP to deionized water was 1:10. The results shown that higher CH4 recovery (66.7%) and higher gas uptake (0.1139 mol) were obtained in PS + SL + CP system, but shorter t90 (154 min) and higher CH4 concentration (69.5%) in the hydrate phase were appeared under PV + SL + CP system, however shorter induction time (9 min) was found in PS + CL + CP system.

Abstract: Software-defined wireless sensor networks (SDWSNs) are deployed in mission-critical applications such as environmental monitoring, smart cities, and healthcare. However, existing protocol architectures such as SDN-WISE lack built-in mechanisms for the detection or mitigate of Denial-of-Service (DoS). Existing trust-based security solutions are rarely fully integrated into the protocol stack; nearly all of them rely on manually configured thresholds which severely undermine their effectiveness. This study integrates the Adaptive, Threshold-Free, and Automatically Weighted Trust Model (ATAW-TM) into the SDN-WISE protocol stack; the ATAW-TM model was adjusted to the resource-constraint sensor node environment by embedding three lightweight adaptations: an inverse-quadratic approximation of Gaussian-like cooperation probability that eliminates exponential function calls and mathematical libraries dependences, a quadratic approximation of second-order Rényi (replacing Shannon entropy) for entropy weight calculation that avoids logarithmic operations, and a lightweight sigmoid approximation for the aging factor that preserves monotonicity and smoothness without the use of exponential functions. To evaluate the integrated model, we developed a comprehensive evaluation framework encompassing 14 distinct DoS attack types and one data tampering attack against SDWSNs. The integrated model was evaluated through extensive Cooja-based simulations. The experimental results indicated that compared to the standard SDN-WISE, the proposed security solution successfully detects a comprehensive range of DoS attacks and one type of data tampering attack, with a negligible performance overhead (less than 5.3% additional hop delay, and 8.1% additional forwarding delay). Furthermore, simulation results show that the proposed method can adapt to variations in network traffic conditions without requiring manual threshold tuning. Overall, the proposed lightweight trust-based attack recognition model offers a practically achievable security solution for resource-constrained SDWSNs, without sacrificing protocol efficiency.

Abstract: The therapeutic landscape of cholangiocarcinoma (CCA) has undergone a paradigm shift towards precision oncology following recognition of its molecularly heterogenous landscape and rapid development of novel targeted therapeutics. Although outcomes remain poor, particularly in locally advanced or metastatic disease, comprehensive genomic profiling has identified multiple clinically actionable genomic alterations, leading to the successful implementation of biomarker-directed therapies. Targeted therapies are being rapidly developed; both to exploit additional molecular alterations that are not currently targeted in CCA and to overcome mechanisms of acquired treatment resistance. This review covers the current standard-of-care systemic treatments for localised, locally advanced and metastatic CCA with a particular emphasis on molecularly targeted therapies and immunotherapy. We discuss established precision oncology strategies targeting FGFR2 fusions and rearrangements, IDH1 mutations, ERBB2 amplification / HER2 overexpression and BRAFV600E mutations, together with the current evidence supporting immune checkpoint inhibition. Emerging therapeutic targets with the potential to further expand precision oncology in CCA are also reviewed, including homologous recombination deficiency (BRCA1/2 and PALB2), MDM2 amplification, NTRK fusions and MTAP loss. Finally, we discuss the current challenges to implementing precision oncology in CCA, including limitations in tumour tissue acquisition for comprehensive genomic profiling and difficulties performing adequately powered molecularly stratified clinical trials in this rare malignancy. We highlight how advances in molecular diagnostics, next-generation targeted therapies and innovative basket and umbrella trial designs are expected to further expand precision oncology strategies with the aim of further improving clinical outcomes for patients with CCA.

Abstract: For nearly a century, flotation kinetics has relied on deterministic first-order rate equations treating the cell as a homogeneous reactor, a paradigm that inherently fails for heterogeneous ores, especially coal, whose organic macerals, porosity, and oxidation susceptibility defy a single rate constant. Breaking from chronological cataloguing, this review proposes a three-dimensional taxonomy based on physical scale, inherent material heterogeneity, and epistemic certainty. We demonstrate that critical industrial prediction failures arise from structural mismatches between model physics and particle surface chemistry, notably time-dependent oxidation deactivation and selective maceral recovery. Six fundamental failure modes are identified, from neglected time-dependence of rate constants to the absence of a thermodynamic deactivation term, corroborated by experimental evidence from coal and base-metal flotation. Advanced microfluidic, automated mineralogical, surface-sensitive spectromicroscopic, CFD-DEM, and physics-informed machine learning tools are dismantling the black box of the flotation rate constant “k”. We introduce the Distributed Reactive Surface Kinetics (DRSK) framework, which embeds particle-scale heterogeneity into a population balance via an adaptive surface-sensitive selection function and treats kinetic uncertainty through stochastic differential equations. A comprehensive comparison table facilitates the transition from conventional models to the DRSK paradigm. We conclude with a roadmap for flotation kinetics 4.0, where digital twins, real-time froth analytics, and self-calibrating hybrid models transform this empirical discipline into a truly predictive engineering science. The framework is elaborated for coal and conventional minerals, underscoring why coal demands its own dedicated kinetic theory and how these lessons can revolutionize the processing of increasingly complex, low-grade ores and secondary resources.

Abstract: This work presents the ZPIF-USAC (Zero Pairs Interaction Functional – Unified Spectral Alignment and Coherence) framework, a comprehensive mathematical theory that introduces fractional self-fragmentation of zeros as a fundamental property of the Riemann zeta function. Unlike classical approaches that treat zeros as passive, independent entities, ZPIF-USAC reveals that each zero is an active spectral entity capable of splitting into multiple fractional components and later reuniting through natural alignment dynamics. The framework introduces three revolutionary concepts: (1) the Fractional Alignment Parameter \(\mathcal{A}_{nm}^{(\alpha)}\), which generalizes spectral alignment to arbitrary fractional orders \(\alpha\) and reveals that optimal coherence occurs at \(\alpha = 1/2\); (2) the Self-Fragmentation Coefficient \(F_n\), which describes the intrinsic ability of each zero to fragment into multiple components; and (3) the Reunification Coefficient \(R_{nm}\), which governs the tendency of fragments to reunite into coherent structures. These concepts are unified through the Fractional ZPIF-USAC Functional: \( \operatorname{ZPIF}_{SF}^{(\alpha)}(x) = \sum\limits_n \gamma_n^{2\alpha} |c_n|^2 + \sum\limits_n F_n \sum\limits_{k=1}^{M_n} \gamma_{n,k}^{2\alpha} |c_{n,k}|^2 + \sum\limits_{n \neq m} R_{nm} \, c_n c_m \) This functional represents the first unified framework that combines fractional interactions, self-fragmentation, reunification dynamics, and active spectral properties into a single mathematical structure. The framework is rigorously applied to four fundamental domains. For prime gaps, ZPIF-USAC predicts the statistical distribution through fractional alignment dynamics. For dark energy, the fractional coherence term \(\mathcal{C}^{(\alpha)}\) provides a natural model for cosmic acceleration. For neural dynamics, it describes synchronization patterns in the brain through alignment dynamics. For spacetime geometry, it induces a fractional spectral metric that bridges quantum mechanics and general relativity. Numerical simulations using the first 100 non-trivial zeta zeros confirm convergence to a unique stable state. The results demonstrate that fractional interactions of order \(\alpha = 1/2\) are the most efficient in generating spectral coherence, and that self-fragmentation provides a natural mechanism for the emergence of complex structures in the universe. This work establishes ZPIF-USAC as a new paradigm in spectral theory, number theory, and mathematical physics, revealing that fractional self-fragmentation and natural alignment are fundamental principles governing the behavior of spectral systems and their applications to prime gaps, dark energy, neural dynamics, and spacetime geometry.

Abstract: For \(q \geq 1\) and \(\gcd(a,q)=1\), consider the restricted weighted Goldbach sum \[ R_{a,q}(N):=\sum_{\substack{p_1+p_2=N\\p_1\equiv a\pmod q}}(\log p_1)(\log p_2). \] We give a fully rigorous, self-contained treatment of the restricted binary, ternary and quaternary Goldbach problems in this setting. We first prove an elementary but decisive local obstruction: if an odd prime \(\ell\) divides \(q\), then \(R_{a,q}(N)\) collapses to \(O_q(\log N)\) on a positive-density set of even \(N\), so no main term of size \(\asymp N/\varphi(q)\) can hold uniformly; for \(q=2^k\) this obstruction is absent, and we identify the correct local main term \[ M_{a,q}(N)=\frac{C_2}{\varphi(q)}S(N)N, \] Within this scope we prove a qualitative almost-all theorem, with a complete major/minor-arc derivation, showing the exceptional set has density zero. We record a conditional impossibility theorem showing that no \(X\)-independent threshold can upgrade this to an effective almost-all statement once a matching second-moment lower bound is granted, and we isolate, as honestly labelled structural cautions rather than theorems, four classical routes that fail to upgrade the almost-all theorem to unconditional finiteness. We add a fully unconditional restricted Chen-type theorem obtained from the Selberg--Chen sieve and the classical Bombieri--Vinogradov theorem, a ternary prime-anchoring transfer, and positivity of the restricted quaternary singular series through explicit local densities. As a companion study, we then prove that the variable factor \[ S_0(n):=\prod_{\ell\nmid n}\frac{\ell-1}{\ell-2} \] of the Hardy--Littlewood singular series, evaluated along shifted primes \(n=p+h\) for fixed \(h\neq 0\), converges in distribution to an explicit random Euler product \[ Y_h=\prod_{\ell>2,\ell\nmid h}\left(\frac{\ell-1}{\ell-2}\right)B_\ell \] with independent Bernoulli local factors \(P(B_\ell=1)=1/(\ell-1)\); we identify its Mellin transform as an entire function of order one, prove convergence of every integral moment, and establish that the law is non-atomic, has unbounded support, and superpolynomially decaying tails, strictly amplified relative to generic integers. We then connect the two studies: since \(S(N)=S_0(N)\) is exactly the amplitude of the restricted Goldbach main term \(M_{a,q}(N)\), the limit law furnishes a rigorous probabilistic description of how that amplitude fluctuates as \(N\) ranges over the shifted-prime sequence \(N=p+h\), a connection neither source study states. Throughout, every asserted theorem is unconditional and every numerical constant is independently certified via partial Euler products with explicit tail bounds; statements retracted at an earlier stage of this programme after failing independent verification are recorded only as open problems.