Abstract: Following the evolution from theoretical foundations to advanced deep learning algorithms, a coherent overview of reinforcement learning (RL) is proposed in this tutorial. We begin with the mathematical formalization of sequential decision-making via Markov decision processes (MDPs). In MDP, Bellman equation and Bellman optimality equation play important roles, they provide for policy evaluation and the fundamental condition for optimal behavior, respectively. The movement from these equations to practical algorithms is explored, starting with model-based dynamic programming and progressing to model-free temporal-difference (TD) learning. As a pivotal model-free algorithm, Q-learning directly implements the Bellman optimality equation through sampling. To handle high-dimensional state spaces, function approximation and deep reinforcement learning emerge, exemplified by Deep Q-Networks (DQN). Thereafter, actor-critic methods address the challenge of continuous action spaces. As a typical actor-critic scheme, the deep deterministic policy gradient (DDPG) algorithm is illustrated in detail on how it adapts the principles of optimality to continuous control by maintaining separate actor and critic networks. Finally, the tutorial concludes with a unified perspective, observing the development of RL as a logical progression from defining optimality conditions to developing scalable solution algorithms. Furthermore, future directions are summarized.

Abstract: This qualitative research examines expert advice on interventions for populations facing social vulnerability. Based on semi-structured interviews with professionals in psy-chosocial support, health, education, human geography and public policy, the study employs reflexive thematic analysis to detect common themes in how vulnerability is perceived and managed in practice. The results identify three interconnected interpre-tive clusters: first, viewing vulnerability as a product of structural factors, highlighting issues like institutional fragmentation, bureaucratic obstacles, and policy inconsisten-cies rather than individual shortcomings; second, emphasizing relational and recogni-tion processes, such as trust, active listening, and respect for personal journeys as key to meaningful engagement; and third, focusing on mediation and empowerment tactics, including institutional mediation, critical education, and digital literacy, to improve access and agency without shifting responsibility to individuals. Overall, the findings suggest that effective intervention demands integrated strategies that address struc-tural conditions, relational factors, and empowerment methods. By consolidating expert insights, the study offers empirically based guidance for practice and service organiza-tion, emphasizing the need for structurally aware, relationally grounded, and con-text-sensitive responses to current vulnerabilities.

Abstract: Background: Polysaccharide-based dynamic hydrogels are promising for wound management due to their biocompatibility, injectability, and tunable biofunctionality. The integration of therapeutic gasotransmitter donors offers a strategy to modulate the wound microenvironment. Objectives: This study aimed to develop an injectable, self-healing carbohydrate hydrogel capable of sustained hydrogen sulfide (H₂S) release for burn wound therapy, and to evaluate its physicochemical properties, in vivo efficacy, and mechanism of action. Methods: A dynamic hydrogel (ACMOD) was fabricated via Schiff-base crosslinking between oxidized dextran (OD) and carboxymethyl chitosan (CMCS), incorporating the H₂S donor ADT-OH. Rheological and recovery tests characterized its mechanical and self-healing properties. Efficacy and mechanisms were assessed in a rat full-thickness burn model, analyzing wound closure, histology, oxidative stress, macrophage polarization, angiogenesis, and collagen deposition. Results: ACMOD exhibited shear-thinning, rapid self-healing, and strong tissue adherence. Sustained H₂S release from ACMOD significantly accelerated wound closure and improved tissue regeneration compared to controls. Mechanistically, H₂S attenuated oxidative stress, promoted a pro-regenerative M2 macrophage phenotype, enhanced angiogenesis via VEGF upregulation, and fostered organized collagen deposition and extracellular matrix remodeling. Conclusions: This work demonstrates a versatile, carbohydrate-based dynamic hydrogel platform that synergizes polymer network dynamics with bioactive H₂S delivery to effectively promote burn wound healing. The findings underscore the potential of polysaccharide hydrogels with integrated gasotransmitter release for regenerative therapy and biomaterials applications.

Abstract: The construction of partial minimum spanning trees being NP-hard, several heuristic algorithms have already been formulated. Many of these heuristics (such as Kruskal's) use shortest paths to connect the components of the tree. In this work, we present an approximate construction algorithm for the minimum Steiner tree (the optimal tree for diffusion multicast). This construction is based on graph-related structures more advantageous than shortest paths. The algorithm uses connections like simple Steiner trees if necessary. These simple trees can be represented by hyperedges. A hyper metric closure can also be used.

Abstract: Objective: The relationship between social demographic factors and biomarker expression is less studied. Methods: We have reviewed 645 endometrial carcinomas with demographic information including race/ethnicity, marital status, religious belief, body mass index (BMI), and pathology staging as well as DNA mismatch repair enzyme expression (MMR) status. Statistical analysis was performed by using various programs in R-package. Results: A total of 645 hysterectomy specimens of endometrial carcinoma, including 463 low grade carcinomas (72%) and 182 high grade carcinomas (28%) were reviewed. Race/ethnicity and marital status were found significantly associated with patient’s age (p<0.01), BMI/obesity (p<0.01) and religious belief (p<0.01). Patients’ marital status was also significantly associated with tumor grade (p=0.01). MMR deficiency was statistically associated with patients’ age (p<0.01) and marital status (p=0.02) in overall endometrial carcinoma. MMR deficiency was also significantly associated with tumor grade (p<0.01), nodal metastasis (p<0.01), and FIGO stages (p<0.01) in low grade endometrial carcinoma but not in high grade endometrial carcinoma. Conclusion: Social demographic factors appear not only as risk factors for pathogenesis but also affect the tumor pathology grade, MMR expression status, clinical stages, nodal metastasis and ultimately treatment and prognosis. These correlative data also provide preliminary and incremental basis for more rigorous prospective study for MMR expression in endometrial carcinoma.

Abstract: Accurate classification of physical activity (PA) intensity is essential for exercise prescription, rehabilitation monitoring, and evaluation of guideline adherence; however, widely used wrist-worn accelerometer cut-points may substantially misclassify physiological intensity. This study evaluated absolute accelerometer thresholds during a maximal 2400m run in military office workers and examined whether individualized cut-points improve agreement with physiological intensity. Seventy-four military office workers completed the test while wearing a wrist-worn ActiGraph GT9X Link and a chest-worn Zephyr BioHarness. Participants achieved near-maximal physiological effort, with peak heart rate averaging 187 ± 11 bpm (95 ± 4.2% age-predicted HRmax). Despite this high intensity, absolute wrist-worn cut-points classified only 34.5% of participants as performing vigorous activity for most of the test. Individualized cut-points, derived from each participant’s individual reference intensity, calculated as the three highest consecutive one-minute epochs during the 2400m test, substantially improved validity. Agreement with %HRmax increased from fair (κ = 0.31), using absolute thresholds, to good (κ = 0.74), using individualized thresholds, and intraclass correlation increased from 0.52 to 0.81. These findings demonstrate that absolute cut-points markedly underestimate high-intensity activity, potentially leading to inaccurate exercise load monitoring and misinterpretation of training intensity. Individualized calibration during a standardized maximal running test provides a feasible, scalable strategy to improve the validity of intensity assessment using wearables in occupational, clinical, and sports settings.

Abstract: Being the second-largest producer of horticultural products, the sector in India is experiencing supply chain issues. Thus, the primary objective of this research is to use the resource curse and cluster theories to assess how the horticulture supply chain affects the smallholders' livelihoods. Independent (Horticulture Supply Chain Efficiency- HSCE), dependent (Smallholders' Livelihood Development- SLD), and moderating (Farmers Producer Organization Intervention- FPOI) variables are all included in the analysis. Using a combination of literature reviews, expert interviews, and focus groups, the researcher developed a preliminary research framework and measuring instruments for each latent construct. The instrument has been validated using face and language validation, followed by a pilot study and main study with 405 responses. Both SmartPLS 4.0 and SPSS 25.0 have been used for that purpose. This study found both HSCE and FPOI directly impacted SLD, explaining 65% of the variance, mostly by SC collaboration, followed by agricultural credit, SC infrastructure, and FPO. However, contrary to the theoretical part, the moderating effect was found to be negatively significant. This indicates the immaturity of FPOs to amplify these factors, which can draw the attention of policymakers to make necessary arrangements.

Abstract: Microplastics are emerging environmental contaminants capable of crossing epithelial barriers and circulating systemically, potentially affecting organisms, including humans. This study investigated the hematological and biochemical effects of subchronic oral exposure to polystyrene microplastics (PS-MPs) in male Swiss albino mice. Animals received 1 μm PS-MPs in drinking water at 0.01 mg/day for four weeks, followed by a two-week recovery period. Blood samples were collected weekly for analysis. PS-MP exposure increased white blood cell, lymphocyte, and granulocyte counts, with a reduced monocyte percentage after the first week, and a significant rise in platelet count by week six. Elevated alanine and aspartate aminotransferase activities indicated hepatic injury, while altered urea and creatinine levels suggested renal impairment. No significant recovery was observed after PS-MP withdrawal. These findings demonstrate that subchronic oral PS-MP exposure induces inflammatory responses and disrupts liver and kidney function.

Abstract: This article presents a reflective survey of research contributions that are related to functional thin film materials, photovoltaic-related architectures, and energy-oriented applications. By synthesising findings from multiple investigations focused on semiconductors, metal-oxide composite systems, nanostructured coatings, and building relevant constituents, the work concentrates on proceeding of fabrication strategies as well as structure-property interrelationships and application-driven performance metrics. Rather than giving a full review of the literature, the article combines some of the experimental observations to highlight recurrent themes such as process optimisation, interface engineering, and multifunctional material behaviour. Particular emphasis is placed on the modulation of optical, electrical, and functional performance by modest variations in deposition conditions, dopant incorporation strategies, and structural design. A cross-there theme analysis shows practical feasibility, long-term stability, and scalability as important as peak performance in determining the suitability of advanced materials for energy applications. Unlike conventional component-focused reviews, this perspective articulates a translational design logic linking materials processing decisions directly to device reliability and system-level energy performance, providing a conceptual framework for accelerating lab-to-field deployment of sustainable energy technologies. The purpose is to highlight cross-cutting translational challenges and design principles that link functional materials to device- and system-level deployment, with particular relevance to real-world and remote-environment energy applications.

Abstract: Machine learning capabilities are expanding into scientific domains at an accelerating pace. When applied to high energy physics pattern discovery, they will generate candidates faster than traditional evaluation can absorb. ML finds patterns in past data. It is inherently post hoc. Whether those patterns reflect structure or coincidence is unknowable at discovery time. This limitation applies equally to human and computational pattern finding. What differs is scale. ML candidate generation is effectively unbounded, while human evaluation capacity remains fixed. When generation rate exceeds evaluation bandwidth, binary accept or reject degenerates to random sampling. Information theoretically, the only response that preserves ranking under a finite evaluation budget is stratification. By focusing on stratification rather than binary filtering, rule adjustments can be made retroactively, thresholds tuned as results accumulate, and evaluation bandwidth focused on top ranked candidates. This paper attempts to codify those criteria, proposing seven computationally evaluable standards for stratifying ML generated patterns. The goal is not to deliver verdicts but to prioritize which candidates merit preregistration and longitudinal tracking. The framework preserves the essential paradigm: pattern plus theory equals potentially real physics. Patterns alone, however striking, remain candidates until theoretical understanding arrives. Making these criteria explicit enables prefiltering at scale while creating a collaborative resource rather than a competitive one. ML capabilities extend what physicists can search while preserving how physicists evaluate. We offer this provisional framework for community calibration, with the goal of developing validation infrastructure before the capability fully arrives.

Abstract: This research answers the knowledge gap regarding the explanation of the quantum jump of the electron. This scientific paper aims to complete Einstein’s research regarding general relativity and attempt to link general relativity to quantum laws.

Abstract: To explore the responses of soil aggregate composition and stability to different grazing intensities in alpine meadow of the Qilian Mountains, no grazing (CK) was set as the control, with four treatments including light grazing (LG), moderate grazing (MG), heavy grazing (HG) and extreme grazing (EG) established. The characteristics of soil aggregates in the 0–10 cm and 10–20 cm soil layers were determined by the dry sieving method and wet sieving method, and three stability parameters including the mean weight diameter (MWD), geometric mean diameter (GMD) and fractal dimension (D) were analyzed. Combined with environmental and biological factors, the mechanisms underlying the effects of grazing on soil aggregates structure and stability were elucidated. The results showed that: (1) Soil aggregates with the particle size of 5–10 mm were the dominant fraction in the soil structure of the alpine meadow, and this fraction changed drastically with grazing intensity. CK maintained relatively high aggregate mechanical stability but exhibited weaker resistance to water erosion compared to grazed plots. Under the CK condition, the content of water-stable aggregates with the 5–10 mm particle size decreased significantly compared with mechanical-stable aggregates (by 60.07% in the topsoil and 70.66% in the subsoil). Light and moderate grazing maintained a dynamic balance and high stability of soil structure. Heavy and extreme grazing intensified soil structure fragmentation and overall stability declined. (2) Soil aggregate stability was correlated with environmental factors. Altitude and soil bulk density were significantly positively correlated with aggregate stability (P&lt;0.001).Root biomass exerted a significant effect on the stability indices of mechanically stable aggregates in the topsoil (P&lt;0.05); high root biomass destroyed soil macroaggregates but enhanced the resistance to water erosion. Soil microbial biomass carbon (SMBC), nitrogen (SMBN), phosphorus (SMBP) were significantly positively correlated with GMD of water-stable aggregates, but negatively correlated with GMD, MWD and D of mechanical-stable aggregates, also MWD and D of water-stable aggregates. Nitrate nitrogen had a positive effect on aggregate stability, while ammonium nitrogen had a negative effect. (3) The stability of aggregate in different soil layer varied under different grazing intensity. Under LG and MG conditions, the subsoil exhibited higher aggregate stability than the topsoil, whereas the opposite pattern was observed under HG, EG and CK conditions. Therefore, from the perspective of soil structural stability and sustainable utilization, light and moderate grazing are the optimal utilization patterns for alpine meadow in the Qilian Mountains. It not only maintains the structural stability of subsoil aggregates but also balances biological cementation and physical disturbance, avoiding aggregate water stability insufficiency under no grazing and the risk of structural fragmentation under heavy or extreme grazing. The findings provide a scientific basis for rational grazing management and soil conservation in alpine meadow of the Qilian Mountains.

Abstract: Spatial light field metrics such as mean cylindrical illuminance provide essential information for qualitative lighting evaluation, yet they remain far less common in practice than horizontal illuminance. To address this gap, we present a multi-sensor prototype that simultaneously measures horizontal illuminance Eh and approximates mean cylindrical illuminance Ez from a set of vertical illuminances uniformly spaced around a cylindrical surface. The device uses a flexible PCB wrapped around a support barrel and an inertial and magnetic measurement unit for orientation tracking. The measurements enable direct calculation of the modelling factor defined in the technical standard EN 12 464 and visualization of directional light distribution using polar plots and illuminance solid. Results show that the prototype approximates mean cylindrical illuminance with high accuracy while preserving directional information, allowing the illuminance solid to be decomposed into vector and symmetric components. Compared with conventional approximation methods, the proposed multi-sensor approach reduces spatial error and yields richer data for lighting analysis. These findings indicate that multi-sensor systems can bridge the gap between theoretical spatial metrics and practical photometry and support the improved modelling evaluation and integration of qualitative lighting parameters into routine workflows.

Abstract: Gaze analysis often relies on hypothesized, subjectively defined ROIs or heatmaps: ROIs enable condition comparisons but reduce objectivity and exploration, while heatmaps avoid this, they require many pixel-wise comparisons, making differences hard to detect. Here, we propose an advanced data driven approach for analysing gaze behaviour. We use DNNs to classify conditions from gaze patterns, paired with reverse correlation to show where and how gaze differs between conditions. We test our approach on data from an experiment investigating the effects of object specific sound (e.g. church bell ringing) on gaze allocation. ROI-based analysis shows a significant difference between conditions (congruent sound, no sound, phase scrambled sound and pink noise) with more gaze allocation on sound associated objects in the congruent sound condition, however, as expected significance depends on the definition of the ROIs. Heatmaps show some not very clear qualitative differences, but none are significant after correcting for pixelwise comparisons. Our approach shows that sound alters gaze allocation in some scenes, revealing task-specific, non-trivial strategies: fixations are not always drawn to the sound source but shift away from salient features, sometime falling between salient features and the sound source. Overall, the method is objective, data-driven, and enables clear condition comparisons.

Abstract: To investigate the long-term effects of climate change on biological communities, our primary aim was to identify the most reliable indicators among available biodiversity, dominance, and evenness indices. We examined three distinct response types to climate change, represented by three taxonomic groups: Aculeata (Hymenoptera), Syrphidae (Diptera), and nocturnal macrolepidoptera (Lepidoptera). Using faunistic datasets derived from our own 3–5 decades of field surveys, we calculated 12 key indices with the vegan package in R 4.2.1. The robustness of these indices was assessed through 1000-fold bootstrap simulations and pairwise correlation analyses. Our results revealed that the Gini–Simpson, Simpson diversity, McIntosh diversity, and McIntosh evenness indices consistently demonstrated high temporal stability and strong correlations across all three climate response types. Therefore, we recommend these indices as primary climate indicators. In contrast, Chao1 estimates, Margalef Index, Menhinick Index, and the Shannon–Wiener diversity index are suitable only for analyzing specific response patterns. Meanwhile, the Berger–Parker, Buzas–Gibson indices, and Hill numbers showed high variability or limited ecological responsiveness, making them unreliable for tracking climate change impacts. Our findings underscore that selecting biodiversity indices must be tailored to the research question and the characteristics of the ecosystem in order to ensure valid and informative ecological analysis.

Abstract: We derive the inverse fine-structure constant \( \alpha^{-1} = 137.035999143 \) from first principles using information-theoretic channel capacity between an 8-dimensional octonionic computational substrate and 4-dimensional spacetime. The derivation requires zero free parameters. Beginning from seven axioms (Peano’s five plus triadic closure and computability), Hurwitz’s theorem forces the octonions as the unique normed division algebra larger than 4D. The projection from 8D to 4D operates through an eigenvector channel whose base capacity is \( 2\binom{8}{4} - 3 = 137 \), counting coordinate 4-planes with bidirectionality and triadic gauge redundancy. Four correction terms—each a named mathematical constant with a specific geometric role—refine this to \( \alpha^{-1} = 137.035999143 \), agreeing with the CODATA 2022 value 137.035999177(21) to 1.62σ. The correction hierarchy is: 1/(8π) (spherical projection through 8D geometry), γ (discrete-to-continuous impedance mismatch), ζ(3)/(137 × 20) (cubic lattice memory), and a logarithmic channel memory term x from Shannon’s theory of channels with memory. Statistical analysis shows the zero-parameter prediction achieves a Bayes factor (decisive on the Jeffreys scale) against the null hypothesis of coincidental agreement, computed under a KT- constrained prior conditioned on the empirically known neighborhood \( \alpha{-1} \) ≈ 137. The framework makes a Tier 1 falsifiable prediction: for fixed apparatus and fixed atomic species, α exhibits an altitude dependence of \( (4.60 \pm 0.15) \times 10^{-16}\,\mathrm{km}^{-1} \) , testable with current optical clock technology. Tier 2 differential measurements across species are proposed to probe the suppressed non-scalar components of the projection residual ε(Φ) without altering the Tier 1 prediction. The same algebraic engine generates all fundamental mathematical constants (e, π, ϕ, √2 , ln 2, γ, ζ(3)) as eigenvalues of discrete walk operators on the Fano plane.

Abstract:

A challenge of studying mammalian cardiac embryogenesis is the limited ability to perform experimental manipulations in animal models. The avian embryo is widely accepted as a model for mammalian heart developmental studies. In this study, we establish the methodology and protocols for studying the Japanese quail (Coturnix japonica) heart at embryonic day 10 (HH38) using the FUJIFILM VisualSonics Vevo 3100 ultrasound system equipped with a MX550D small animal cardiology transducer. These protocols were designed to measure right ventricular wall thickness, pulmonary artery diameter, and the outflow velocities of the right ventricular outflow tract (RVOT) and the pulmonary artery (PA), thereby establishing baseline parameters of the normally developing quail morphology. Quail embryos are an ideal model for cardiovascular research due to their short incubation period (16-17 days), experimental accessibility, and strong similarities to mammalian heart development. These developmental similarities include, but are not limited to, looping, chamber septation, and the development of a true four-chamber heart. High-resolution imaging modalities, including ultrasound and optical coherence tomography, enable noninvasive, real-time visualization of cardiac morphology and function throughout development. Echocardiography allows for quantitative and qualitative assessments of myocardial structure and cardiac hemodynamics. The similarity to the mammalian heart, combined with rapid embryogenesis, makes quail embryos a valuable model for investigating congenital heart defects, genetic modifications, and fundamental cardiac developmental processes. In this study, we describe reproducible incubation protocols and baseline echocardiographic parameters used to evaluate morphological and physiological changes in the developing embryonic quail heart on embryonic day 10.

Abstract: The long-term outcomes of individuals exposed to similar traumatic events often diverge dramatically: while some succumb to chronic despair, others achieve posttraumatic growth. This “resilience paradox” highlights a limitation of current trauma therapies. Although Prolonged Exposure, Cognitive Processing Therapy, and EMDR reliably reduce symptoms such as hyperarousal and intrusive memories, many patients remain existentially fragmented, reporting a loss of purpose despite substantial symptom and functional improvement. This gap suggests that standard protocols—focused on sensorimotor stabilization, narrative coherence, and functional restoration—may systematically neglect a vital fourth meta-level: the capacity for non‑identified awareness.This paper introduces the Neuro-Existential Architecture System (NEAS), a theoretical framework that hypothesizes that meaning is not merely a psychological variable but a fundamental neurobiological organizing principle structuring resilience. NEAS proposes four complementary, hierarchically organized neurobiological mechanisms: (1) hierarchical recalibration via meaning-priors, using top-down signals to reorganize the brain’s predictive hierarchy; (2) emotional criticality via limbic meta‑regulation, permitting balanced oscillation between hope (Papez system) and caution (Yakovlev system); (3) spatiotemporal coherence, extending the autorelational window to restore identity continuity; and (4) Witnessing-Space as structural meta‑stabilization, theoretically instantiated through inter‑regional gamma‑frequency binding, a candidate mechanism proposed to enable global meta‑awareness and prevent system fragmentation under stress.The NEAS clinical model operationalizes these mechanisms into a four‑level architecture (Level −1: Relational Safety; Level 0: Sensorimotor Stabilization; Level 1: Narrative Coherence; Level 2: Existential Meaning Integration). Meaning-focused work at the highest level is hypothesized to be pivotal, explicitly intended to cultivate Witnessing-Space through contemplative practice integrated with trauma‑focused processing. To begin validating this framework, we propose a multi‑site, two‑arm randomized controlled trial (N = 240) comparing NEAS‑based treatment with standard trauma‑focused cognitive‑behavioral therapy. A neuroimaging subsample (n = 80) will exploratively measure autorelational window extension, gamma synchrony, and Default Mode Network connectivity. We hypothesize that the integrated four‑level NEAS condition will yield superior functional outcomes (Sheehan Disability Scale) and greater long‑term durability compared to standard care. While the present framework is grounded in Western neuroscience and clinical contexts, its ultimate value will depend on rigorous cross‑cultural adaptation and validation. By bridging neuroscience, existential psychology, and contemplative science, NEAS aims to support a shift from trauma‑focused symptom management toward existentially grounded, neurobiologically coherent healing.

Abstract: Procyanidin C1 (PCC1), a B-type procyanidin trimer derived from natural sources, has recently garnered significant attention in preclinical models due to its potential to specifically induce apoptosis in senescent cells (senolytic activity) and extend healthspan. However, existing data reveal a pronounced pharmacological paradox: the in vitro induction of apoptosis in senescent cells typically requires high micromolar concentrations (>50 μM), whereas in vivo peak plasma concentrations following intraperitoneal (i.p.) injection in mice are distributed around the 2-15 μM range, and oral (p.o.) administration yields nanomolar exposure levels (~ 0.04 μM), often falling below this threshold. This study aims to construct an objective translational medicine evaluation framework by systematically integrating cellular pharmacodynamic thresholds, Caco-2 transmembrane transport mechanisms, and in vivo pharmacokinetic (PK) data. The analysis indicates that even with i.p. administration, systemic plasma concentrations of PCC1 struggle to sustainably reach the cytotoxicity thresholds established in vitro; its in vivo efficacy is likely derived from high accumulation and localized concentration effects within specific tissues (e.g., adipose, lymphoid tissue). Further analysis points out that high-dose application may face dual barriers of safety risks and economic costs. It is postulated that combining modern nanodelivery technologies with synthetic biology to enhance bioavailability and achieve dosage minimization represents a critical pathway for the safe, economical, and efficient clinical translation of PCC1.

Abstract: We derive an exact, practical method to update Thévenin parameters (open-circuit voltage and equivalent resistance) of a linear network under a single internal branch modification (open/short/resistance change), without recomputing the full nodal solution from scratch. The change is modeled as a rank-one perturbation of the nodal admittance matrix, and the Sherman–Morrison identity yields closed-form port updates in terms of three physically interpretable scalars: local self-coupling, port–branch coupling, and state projection across the modified branch. We discuss limiting cases (open and short), include a brief note on complex admittances (phasors/Laplace), and provide a reproducible Python check.