Abstract: The short-term stability of cell-based advanced therapy medicinal products (ATMPs) is a critical factor in their clinical handling and applicability. This study evaluates a multiparametric stability assessment approach using a GMP-manufactured ADSC-based ATMP as a model product. It consists of an ADSC suspension in Ringer’s lactate contained in polypropylene syringes. The in-use stability of the ADSC-based ATMP was evaluated over 36 h of hypothermic storage (2-8°C). Ten GMP-manufactured batches derived from a single qualified donor were analyzed immediately after preparation (T0) and after storage (T36) using validated methods for viability, sterility, mycoplasma status, endotoxin level assessment, and macroscopic appearance evaluation. Additionally, complementary quality tests using flow cytometry were applied. Cell integrity was characterized by Annexin V-FITC/7-AAD staining, whereas FSC/SSC analysis, doublet discrimination, and qualitative re-adhesion testing were used to investigate cell aggregation and post-storage recovery capacity. All tested parameters remained within predefined acceptance criteria at both time points. Mean cell viability decreased from 95.1 ± 1.8% to 88.2 ± 2.8% but remained above the acceptance threshold (≥70%) in all batches. Sterility, absence of mycoplasma, endotoxin levels, and suspension appearance met specifications. Flow cytometry demonstrated preserved membrane integrity, a small proportion of early apoptotic cells, and no changes in cell-size distribution or aggregation. These findings indicate that the investigated ADSC suspension maintains viability, structural integrity, and microbiological quality during short-term hypothermic storage under clinically relevant conditions. The study highlights the utility of complementary flow cytometry-based methods for stability assessment of fresh ATMPs.

Abstract: Jurisdictional REDD+ is widely promoted as a mechanism to align forest conservation, and climate mitigation, through standardized readiness frameworks and multilevel governance, yet a persistent dilemma remains: how can politically divergent subnational states converge on the same international climate finance outcomes? This study addresses that question by comparing two contrasting Brazilian Amazon states that both secured funding under the REDD+ Early Movers (REM) program. Methodologically, the research employs comparative process tracing, drawing on 287 documents and 74 semi-structured interviews, and applies Van Evera's evidence tests to assess the strength of causal explanations for program approval. The results reveal five necessary conditions for REM financing, formal compliance with REDD+ Readiness requirements was neither necessary nor sufficient for success; instead, informal arrangements, historical political trajectories, paradiplomatic networks, and multilevel coalitions among social actors played a decisive role. These findings lead to the conclusion that jurisdictional REDD+ operates less as a hierarchical implementation of institutional models and more as a polycentric governance process emerging from dynamic interactions across governance scales. By reconstructing the causal mechanisms that link international climate finance to subnational forest politics, this study advances theoretical debates on climate finance while offering practical insights for policy design in heterogeneous political contexts.

Abstract: This study examined associations between social relationships, depressive symptoms, and self-rated health (SRH) among adults living alone in South Korea. Data from 44,168 adults living alone who participated in the 2025 Korea Community Health Survey were analyzed using complex-samples logistic regression. Social relationships included contact frequency with family members or relatives, neighbors, and friends and neighborhood trust. Depressive symptoms were defined as a Patient Health Questionnaire-9 score ≥ 10. The weighted prevalence of depressive symptoms was 6.9%, and that of poor SRH was 22.5%. Compared with contact four or more times per week, family or relative contact less than once per month was associated with depressive symptoms (aOR = 1.50, 95% CI: 1.31–1.72) and poor SRH (aOR = 1.28, 95% CI: 1.17–1.41). Corresponding associations for infrequent friend contact were stronger (aOR = 2.02, 95% CI: 1.75–2.34; aOR = 1.77, 95% CI: 1.63–1.93, respectively). Lack of neighborhood trust was also associated with both outcomes. These findings suggest that specific social relationships, particularly friendships, are relevant to mental and perceived health among adults living alone.

Abstract: Leakage diagnosis in water distribution networks is constrained by sparse instrumentation, uncertain models, and leak concurrency. We present a physics-informed spatio-temporal graph neural network evaluated end-to-end on the open Battle of the Leakage Detection and Isolation Methods (BattLeDIM) and L-Town benchmark (782 junctions, 905 pipes, 33 pressure sensors). Message passing operates on a conductance-weighted hydraulic Laplacian, trained by self-supervised sensor masking to reconstruct nodal pressure fields and latent pipe flows, embedding Hazen-Williams energy and mass-balance residuals in the loss under learned uncertainty weighting. Pressure at withheld sensors is reconstructed to 0.29 meters. A seeded ablation shows physical constraints degrade reconstruction accuracy by 0.10 meters of head while reducing nodal continuity violations by over five orders of magnitude, rendering the inferred state hydraulically admissible. Residuals feed two complementary detectors. A level-based statistic saturates under concurrent leakage; pairing it with a two-window onset contrast raises detection from 13 to 22 of 23 ground-truth leaks (95.7 %, 95 % confidence interval 79.0–99.2 %), with median detection times of 29 hours for abrupt and 74 hours for incipient events, releasing 96 % of annual leaked volume after detection. Localization transfers poorly from simulation, with a median error of 857 meters and only three leaks meeting the joint 300 meters criterion due to signal superposition under concurrency.

Abstract: We introduce and investigate a smooth version of a bitopological gyrogroup, which we call a bitopological Lie gyrogroup. The construction combines the algebraic structure of gyrogroups with two compatible smooth topologies on the same underlying manifold. The first topology describes one smooth structure, while the second provides an independent but compatible smooth structure. We formulate the notions of bitopological gyrogroup, Lie gyrogroup, bitopological Lie gyrogroup, smooth gyrohomomorphism, smooth gyration, and bitopological Lie subgyrogroup. Several structural properties are established. In particular, we study left translations, gyrations, tangent spaces, the two induced differential structures, compatibility of the two tangent bundles, and the behavior of smooth gyrohomomorphisms. We prove that the classical Lie group situation is recovered when all gyrations are trivial. We also discuss the Einstein and M”obius gyrogroups as fundamental examples and explain how their standard smooth topology gives rise to natural bitopological variants. The paper provides a differential-geometric framework for studying nonassociative smooth group-like structures with two interacting topologies. This framework is intended as a starting point for the development of Lie theory, invariant differential geometry, and representation theory for bitopological gyrogroups.

Abstract: Malaria remains a major global public health challenge, with a substantial burden of morbidity and mortality concentrated in sub-Saharan Africa and other tropical and subtropical regions. The complex transmission process involving Plasmodium parasites, Anopheles mosquito vectors, and human hosts makes mathematical modeling an important tool for understanding transmission dynamics and evaluating control strategies. This review provides a comprehensive synthesis of mathematical modeling approaches for malaria transmission and control, including deterministic compartmental models, stochastic models, optimal control frameworks, spatial models, and cost-effectiveness analyses. Particular attention is given to major intervention strategies, including vector control, vaccination, antimalarial treatment, and other integrated control approaches. The review also examines the effects of seasonality, climate variability, drug and insecticide resistance, spatial heterogeneity, and human movement on malaria transmission and intervention outcomes. A bibliography-based keyword frequency analysis and word-cloud visualization are used to identify dominant research themes, followed by a comparative assessment of modeling approaches and intervention strategies. The findings highlight the importance of integrating multiple interventions with context-specific epidemiological and environmental conditions. Overall, mathematical models provide valuable support for optimizing malaria control strategies and addressing important challenges toward malaria elimination.

Abstract: Nondestructive evaluation (NDE) is moving from manually interpreted indications toward connected measurement systems in which sensing, imaging, automated processing, artificial intelligence (AI), and engineering decisions are coupled through a digital thread. This transition increases information availability but also introduces additional points at which measurement meaning, uncertainty, or traceability can be lost. This critical integrative review reframes NDE 4.0 as a metrological-assurance problem and proposes a six-layer architecture spanning the physical measurand, instrument, signal, information, inference, and decision layers. The architecture is combined with an uncertainty map for hybrid physical-digital inspection, a minimum digital evidence record, and a qualification V-model linking intended use to verification, representative validation, bounded autonomy, and post-deployment surveillance. A worked ultrasonic-sizing example illustrates how classical measurement contributions and model-inference uncertainty can be combined without treating AI confidence as a substitute for measurement uncertainty. The synthesis distinguishes metrological traceability from digital provenance, probability of detection from classification metrics, and probabilistic calibration from measurement uncertainty. Cross-sector analysis of aerospace, nuclear, rail, civil infrastructure, and additive manufacturing shows that increasing autonomy raises both information value and evidential burden. The central conclusion is that, for measurement-based engineering decisions, especially in higher-consequence applications, the defensibility of an NDE 4.0 system depends on an adequately traceable, uncertainty-aware, intended-use validation, and auditable sensor-to-decision chain.

Abstract: Background: Vestibular migraine (VM) and benign paroxysmal positional vertigo (BPPV) frequently present with overlapping positional symptoms, making their differential diagnosis challenging. This study aimed to characterize positional vestibular manifestations in patients with suspected VM. Methods: In this observational study, 85 adult patients presenting with positional vertigo or dizziness in whom typical BPPV could not be demonstrated were evaluated. All patients underwent detailed clinical and neurotological assessment, videonystagmography, standardized positional testing. Results: Definite VM was diagnosed in 51/85 patients (60.0%) and probable VM in 23/85 (27.1%), accounting together for 87.1% of the selected cohort. Positional nystagmus was documented in 52 patients (61.2%), of whom 38/52 (73.1%) showed at least one atypical characteristic. Pupillary Nystagmus (PNy) was detected in 61/85 patients (71.8. Among 14 patients with an initially BPPV-like positional pattern, five responded to canalith repositioning maneuvers, whereas nine had persistent or recurrent symptoms requiring diagnostic reassessment. At 3 months, 60/74 patients (81.1%) with definite or probable VM reported marked overall clinical improvement, while complete resolution of positional vertigo or dizziness occurred in 49/74 (66.2%). Positional nystagmus resolved in 36/47 patients (76.6%) who had presented with this finding at baseline. Conclusions: VM should be considered in patients with recurrent positional vestibular symptoms when positional findings are atypical, variable, or inconsistent with a single semicircular canal disorder, particularly when repositioning maneuvers are ineffective. PNy may provide an additional supportive finding, although its di-agnostic value requires validation in controlled studies. Treatment response may support the overall clinical interpretation but should not be regarded as an independent diagnostic criterion.

Abstract: Feline infectious peritonitis (FIP) remains diagnostically challenging, and rapid antemortem con-firmation is increasingly important in the era of effective antiviral therapy. This pilot diagnostic accuracy study assessed agreement between effusion RT-qPCR and cell tube block immunohisto-chemistry (CTB-IHC) in 40 cats with suspected FIP and developed an exploratory effusion-based predictive score using CTB-IHC as the reference standard. RT-qPCR and CTB-IHC showed almost perfect agreement (κ = 0.86): 30 cats were positive by both methods, eight were negative by both, and two were CTB-IHC-positive/RT-qPCR-negative. RT-qPCR showed 93.8% sensitivity (95% CI: 79.2–99.2%), 100% specificity (63.1–100%), and 95.0% accuracy (83.1–99.4%). The three-parameter FIP Predictive Score, combining globulin concentration, FIP Effusion Index, and A:G ratio, achieved an apparent AUC of 0.916 (95% CI: 0.842–0.990), with 78.1% sensitivity (60.0–90.7%) and 100% speci-ficity (63.1–100%) at ≥ 2 points. Given that the score was derived and evaluated within the same small cohort, independent external validation is essential before clinical implementation. Neverthe-less, neither a low score nor a negative molecular result should independently close investigation when FIP remains clinically suspected. These findings provide a foundation for a structured, mul-timodal effusion-based diagnostic approach integrating molecular, antigen, biochemical, and cy-tometric evidence.

Abstract: Generative artificial intelligence (GenAI) is rapidly becoming part of engineering students’ learning environments, particularly in quantitative problem solving, explanation generation, and formative feedback. The educational value of these systems, however, depends not only on output accuracy but also on whether learners can evaluate claims that appear authoritative. This paper proposes Deliberate Ambiguity Verification (DAV), an instructional framework in which electrical engineering students compare two AI-generated solutions that are mathematically plausible and professionally presented but differ in a consequential assumption, boundary condition, or physical interpretation. Students must identify the point of divergence, verify each response against an independent engineering principle, justify a decision, and report confidence before and after verification. The framework integrates epistemic vigilance, trust calibration, metacognitive monitoring, cognitive engagement, productive failure, and discipline-specific AI literacy. A randomized four-condition design is proposed for students in circuit analysis, contrasting no-AI problem solving, a single AI answer, a plausible-but-incorrect AI answer, and deliberate ambiguity. Primary outcomes are conceptual reasoning, verification behavior, confidence calibration, and AI-free transfer. The paper argues that the goal should not be generalized distrust of AI, but calibrated skepticism: the capacity to determine when an AI response should be accepted provisionally, when it requires independent verification, and when disciplinary reasoning should override the apparent authority of the system. The framework is intended to provide a testable pathway for studying epistemically responsible human-AI collaboration in engineering education.

Abstract: This study presents an axle-based, uncertainty-aware safety-envelope framework for the Automatic Braking and Stopping System (ABSS), designed to arrest rolling stock before a hazard boundary after uncontrolled movement begins. The method combines free-rolling dynamics, response delay, active braking, deterministic force sizing, axle-capacity screening, and uncertainty-adjusted distance assessment. It separates gross mass, braking demand, and verified eligible axle positions. A browser-based tool performs these calculations and configuration checks. A predefined distance serves numerical verification; operational use requires the measured detection-to-boundary distance. For illustrative 94 t/4-axle, 188 t/8-axle, and 282 t/12-axle cases, the profile yields a detection speed of 7.973 km/h. With four 10-kN elements, deterministic stopping distances are 10.211, 16.553, and 23.277 m; uncertainty-adjusted upper distances are 10.698, 17.430, and 24.636 m. Deterministic sizing requires two elements in each case, while uncertainty-adjusted sizing requires three for 282 t. Restricting that case to two eligible positions raises the upper distance to 52.417 m and leaves no feasible configuration. Sensitivity is greatest to detection speed, followed by braking force and gross mass. Thus, distance adequacy alone is insufficient: uncertainty and physical installation capacity must also be checked. The framework supports traceable screening, software verification, and subsequent component, integration, and controlled-track validation.

Abstract: Obesity is a chronic disease that is a major risk factor for serious, long-term health complications. Glucagon-like peptide-1 receptor agonists (GLP-1RAs) are a class of pharmacotherapies that are highly effective in the treatment of obesity, producing weight loss of 15-20% or greater in clinical trials. Current clinical guidance recommends continued therapy to sustain these benefits; however, barriers including cost, limited access, and adverse effects may inhibit long-term use and contribute to high discontinuation rates. Abrupt discontinuation of GLP-1RA therapy is problematic as it is associated with significant weight regain and reversal of cardiometabolic improvements. Given the challenges associated with prolonged treatment, there is a critical need to develop evidence-based strategies that support patients who reduce or discontinue GLP-1RA therapy. This paper explores the scientific premise for structured dose-reduction approaches, coupled with behavioral and psychological interventions, to promote durable weight-loss maintenance following GLP-1RA therapy.

Abstract: Composting recycles organic waste cheaply, but the product often fails fertiliser quality standards. Biochar improves both the process and the product, yet the ecotoxicological consequences of returning compost-derived biochar to composting are poorly understood. Mature but off-specification compost was pyrolysed at 550 °C, and the resulting biochar (CBC) was added at 0, 3, 6 and 9% of dry matter to pilot-scale piles of green waste and sewage sludge (9:1). After 39 days, the composts were analysed for biological stability (AT4), nutrients, heavy metals and leachate ions and tested with the Phytotoxkit (Triticum aestivum, Sinapis alba, Pisum sativum) and Ostracodtoxkit (Heterocypris incongruens) assays. CBC increased K, Mn, Mg and Cu, reduced Ca and slightly increased Cr and Pb; all metals remained below regulatory limits. Only CBC9 met the AT4 stability benchmark, yet it produced the strongest biological response (72.5% shoot inhibition in wheat; 58.7% ostracod growth inhibition). K and Mn correlated most strongly with phytotoxicity but acted as markers of biochar loading; nitrite accumulation and rising leachate salinity were the more probable drivers, whereas heavy metals and ammonium were not. Approximately 3% CBC balanced compost improvement against ecological safety, and composts meeting chemical and stability criteria may still be ecotoxic.

Abstract: We develop a topological framework for group-like algebraic systems whose multiplication is genuinely multi-valued. The central object is a hyperoperation ◦ : G × G −→ P∗(G), where P∗(G) denotes the family of all nonempty subsets of G. We focus on the compactvalued case ◦ : G × G −→ K(G), where K(G) is the hyperspace of all nonempty compact subsets of G, and impose continuity with respect to the full Vietoris topology. This provides a natural refinement of the usual upper hyperspace approach to topological hyperstructures. We formulate the notion of a Vietoris multi-valued topological group using the axioms of a polygroup, rather than using weaker identity and inverse conditions. This choice guarantees a strong algebraic connection with ordinary groups and permits the construction of fundamental quotients. We establish basic properties of continuous multi-valued translations, inversion, division, compact products, closed graphs, products, subgroups, and continuous homomorphisms. A central role is played by complete parts. We introduce Vietoris-complete multi-valued topological groups, prove that complete open neighborhoods generate natural left and right neighborhood systems, and construct canonical left and right uniform structures. We then investigate the fundamental relation and prove that, under a natural complete-part hypothesis, the fundamental quotient carries a canonical topological group structure. We also establish a singleton-collapse theorem showing that the proposed theory contains ordinary topological groups as a rigid special case. Several examples are developed, including finite hypergroups, quotient constructions, compact orbit hyperstructures, and finite-valued operations. The paper concludes with a comparison with Marty hypergroups, polygroups, and locally compact convolution hypergroups, together with open problems concerning completion, invariant measures, Lie-type structures, bitopological extensions, and multi-valued groupoids.

Abstract: Background: Digital communication tools change how hospital staff request services and follow up patients’ needs. Their value depends on how well they fit everyday work. Women over 40 may bring experience of successive technological changes, but age alone says little about professional seniority or digital confidence. Objective: To understand how Brazilian nurses over 40 perceive the effects of a digital interaction tool on their nursing work. Design: Secondary qualitative analysis informed by Braun and Clarke’s reflexive thematic analysis. Setting: A large hospital in São Paulo, Brazil. Participants: Ten Brazilian female nurses aged 41–55, selected from 20 interviews on the basis of age and explicitly recorded female identification. One additional interview requires eligibility confirmation. Methods: A second review of a pool of participants followed Braun & Clarke’s six thematic analysis phases: familiarization with data, generating initial codes, searching for themes, reviewing themes, defining and naming themes, and writing the report. Results: Four themes were developed: learning again without starting from scratch; no longer the go-between for every request; when a digital request still needs a phone call; and making work visible without losing personal contact. Participants described easier communication and greater team autonomy, alongside the effort of learning another system, duplicated requests and dependence on other departments’ responses. Comparisons with telephone communication and replacement platforms showed how they assessed successive changes in their working practices. Conclusions: Participants valued the tool when it reduced the need to relay requests and helped them organize care. Its usefulness diminished when they still had to telephone, repeat information or chase a response. Continuing support, integration between systems and clear departmental responsibilities warrant attention. These accounts do not establish an objective reduction in workload or effects specific to age or gender. The analysis remains subject to the research team’s reflexive review.

Abstract: We develop a differential-geometric framework for connectingmulti-valued topological groups with classical Lie groups and Lie algebras. A multi-valued group operation is interpreted as a hyperoperation \( \star:G\times G\longrightarrow \mathcal P^*(G), \) where \( \mathcal P^*(G) \) denotes the family of nonempty subsets of \( G \). The principal difficulty is that the usual tangent-space construction for a Lie group cannot be applied directly to an arbitrary hypergroup: the product of two points is no longer a point, and hence there is no canonical differential of the multiplication map. To overcome this obstruction we introduce the notions of smooth multi-valued topological groups, locally single-valued branches, infinitesimal-collapse structures, and branch-compatible multi-valued Lie groups. We prove that a smooth local branch defines a local Lie group and consequently a Lie algebra. We establish conditions under which different branches induce the same Lie algebra, and we construct a canonical quotient Lie group whenever the ambiguity of the multi-valued operation is generated by a normal subgroup. We prove that the tangent structure of a branch-compatible multi-valued group is a Lie algebra and that the first-order part of the multi-valued product is governed by the ordinary Lie bracket. Several examples are studied in detail. These include ordinary Lie groups viewed as degenerate multi-valued groups, quotient-induced multi-valued operations, finite-group extensions, matrix examples, a two-branch deformation of a Lie group, and a class of multi-valued structures arising from normal subgroup ambiguity. A counterexample shows why continuity alone is insufficient for the existence of a Lie algebra. Finally, we formulate a multi-valued Baker--Campbell--Hausdorff principle and describe possible directions toward a genuine theory of Lie hypergroups, multi-valued Lie groupoids, and neutrosophic Lie hypergroups.

Abstract: Program libraries guide code generation, but a search may return no code or documentation relevant to the requested functionality. In this case, abstaining may exclude solvable tasks, whereas proceeding may waste computation on tasks likely to fail. Here, we introduce the synthesis abstention card, a verification framework for deciding whether missing library information justifies declining a task before generation. The protocol combines confidence intervals for support--failure linkage, retained risk, and calls avoided with checks of transport to the target workload. These checks classify decisions as supported, non-actionable, or uncertified against prespecified requirements. We examined these requirements through controlled studies and external audits. In a small, locally trained model, we found that authoritative implementation notes reduced the probability that all three generation attempts failed from 0.910 to 0.414. In a separate controlled system, abstention avoided 50.6% of generation calls when half the tasks lacked relevant library information. Abstention also reduced failure risk to 0.129 among retained tasks, compared with 0.366 when all tasks were attempted. These benefits persisted when 75% of tasks lacked library information, although retained risk was overestimated beyond the prespecified tolerance. The external audits revealed a further limitation: generic text signals could pass transport checks while providing little failure discrimination. Together, these findings show that library signals can justify abstention only when failure discrimination, retained risk, and savings are validated for the target workload. The card provides a basis for reassessing which tasks an agent is justified in attempting as its knowledge, capabilities, and workload change.

Abstract: Three-arm binary trials may require low-dose benefit over control and additional high-dose benefit. We evaluate posterior support, false-declaration control and power when both response-probability differences must reach positive minima and at least one exceeds its target. Conditioning on the shared low-dose probability under independent binomial sampling and independent beta priors gives the posterior probability of meeting this requirement. Probabilities of complete or partial attainment and expected shortfalls describe uncertainty. For tables retaining declarations when control responder counts decrease or high-dose counts increase, maximization of false-declaration probability over all configurations failing the requirement reduces to three one-dimensional paths. Certified bounds keep the saved tables' maximal false-declaration probability below 2.5%. With illustrative thresholds and uniform priors, the joint Bayesian rule's power was at most 1.3% across 24 acceptable profiles exceeding only one target per design (250 or 386 participants). Method-independent bounds constrain fixed-sample count-based rules with the same error control. At 30% control response, 200 low-dose participants, low-dose effects between a minimum and target of 7 and 10 percentage points, and an increment above 5 points, power cannot exceed approximately 14%, regardless of other group sizes. JADE MONO-2 counts illustrate retrospective reporting. The results distinguish satisfying a requirement from reliably establishing it.

Abstract: Polyethylene (PE) and polypropylene (PP) waste form a big part of polymer waste, and their separation is complicated by similar densities and frequent combination within a single product, making PE+PP co-pyrolysis a practically relevant recycling approach. The effects of component ratio and co-pyrolysis with metal oxides at the atomistic level are insufficiently studied. Using ReaxFF-MD in AMS, with and without transition metal oxides (Fe₂O₃, NiO) pyrolysis and co-pyrolysis of PE and PP chains (C₁₂₀H₂₄₂) were investigated at PE:PP ratios of 100:0, 80:20, 50:50, 20:80, and 0:100 over 1750–3500 K. Kinetic and thermodynamic activation parameters were determined using first- and second-order kinetics based on the Arrhenius and Eyring equations. The degradation rate depended non-monotonically on the PE:PP ratio, with the 50:50 being the slowest among the mixtures, while the metal oxides lowered the decomposition threshold and reduced the activation energy 1.6–2.6-fold. Metal oxides also altered product selectivity: H₂ replaced C₂H₄ as the dominant product, with its yield rising 5–7-fold; Fe₂O₃ gave the most stable H₂ yield, while NiO gave the highest single value. The mechanisms of H₂, H₂O, and CH₄ formation were examined. These results deepen the understanding of the co-pyrolysis processes of PE and PP involving transition metal oxides.

Abstract: Primary hyperparathyroidism (PHPT) is definitively treated by parathyroidectomy (PTx), with surgical cure conventionally defined by sustained normocalcemia. However, some patients develop elevated parathyroid hormone (PTH) concentrations despite normal postoperative calcium levels. This heterogeneous condition may reflect vitamin D insufficiency, reduced calcium availability, skeletal remineralization, or renal dysfunction, but may occasionally indicate persistent parathyroid autonomy or precede biochemical recurrence. Fibroblast growth factor 23 (FGF23), an osteocyte-derived regulator of phosphate excretion and calcitriol synthesis, occupies a central position within the bone–kidney–parathyroid axis and may participate in metabolic adaptation after PTx. This narrative review examines FGF23 regulation, circulating concentrations, clinical associations, and perioperative dynamics in PHPT. A structured search of PubMed/MEDLINE, Scopus, and Web of Science identified 11 original clinical studies directly evaluating FGF23 in PHPT. Findings were discordant: some studies reported increased preoperative concentrations and a postoperative decline, whereas others found normal levels or no significant change after surgery. Differences in disease phenotype, renal function, vitamin D status, assay methodology, supplementation, and sampling time may partly explain this heterogeneity. Notably, no study related perioperative FGF23 changes to subsequent PTH trajectories, and studies of postoperative eucalcemic PTH elevation did not measure FGF23. FGF23 cannot currently be considered a diagnostic or prognostic biomarker in PHPT. Nevertheless, serial assessment alongside conventional mineral-metabolism parameters may clarify endocrine adaptation after PTx and determine whether biologically distinct postoperative phenotypes exist.