Abstract: Linear measurements are used on radiographs for various purposes, such as measuring lengths in endodontics, analysing bone loss in perio-dontal disease and making age determinations in forensic dentistry. The purpose of this study was to analyse the accuracy of periapical radio-graphs for the measurement of the contact point to the crest of the bone compared to the actual measurements of a dried skull. A dried skull had lead squares measuring 1 x 1mm attached to the contact point and the bone crest. Each site was radiographed using a parallel technique with a Rinn holder. The radiographs were taken perpendicularly to the tooth and repeated at 10° and 20° horizontal angulations. The results showed that variation in the angle of the radiograph had a significant effect on the resulting measurements, F (1.7, 26.9) = 218.265, p < 0.001. The results from this study indicate that when measuring the contact point to the crest of the bone, a shift of 20° from the perpendicular of the tooth has a significant effect on the radiograph measurement and the actual meas-urement of the contact point to the crest of the bone on the actual skull.

Abstract:

Nova Scotia, a province on Canada’s Atlantic coast, has proposed Wind West, a plan to initiate the province’s offshore wind industry. A regional offshore wind report identified eight potential development areas (PDAs), of which four were chosen. The areas were selected to avoid ecologically significant and conflict-of-use areas; however, no consideration was given to tropical cyclones (TCs) and hurricanes (intense tropical cyclones). This paper evaluates the effects of climate change and TCs on offshore wind turbines sighted on Nova Scotia’s continental shelf by analysing historical TC track data to assess the intensity and frequency of extreme wind and wave events on the continental shelf. Correlations between SSTs and extreme weather events were also examined. The findings show no clear long-term trends in TC intensity or frequency in the selected areas, although there is a clear upward trend in sea-surface temperatures (SSTs) since 1950. No strong correlation between rising SSTs and increased storm intensity or frequency within the available datasets were found, though similar studies suggest that these variables have some correlation on aggregate. While climate change is causing conditions for hurricanes to become favorable along the Scotian Shelf, current TC data shows no clear correlation with increasing intensity and frequency over time. The results are affected by the quality of the data. High uncertainty, spatial resolution, and temporal resolution leave large portions of TC tracks unmeasured. Uncertainty associated with pre- and post-1950 data makes conclusions from the results difficult. We propose a measuring buoy in each of the four selected potential development areas cost C$200,000 to develop and C$35,000 to maintain. Each buoy would have a representative radius of 50km, slightly larger than that of each of the four wind energy zones. The additional data collected would allow developers to pick appropriate design standards based on available environmental data and could additionally be used for climate change research. Currently, Nova Scotia faces many limitations developing its offshore; supplying accurate data to assess the risk from extreme weather events to offshore wind turbines is one of the first steps to ensuring success.

Abstract: This study examines the dynamics of sustainability transitions in the EU-27 during the period 2015-2024, focusing on the role of different stakeholders and the emergence of distinct convergence patterns in sustainability performance. The theoretical framework integrates sustainability transition theory, stakeholder governance, and the literature on convergence and club convergence, interpreted through the socio-technical multi-level perspective and the concept of institutional lock-in. A test model is developed based on four stakeholder-specific indices: the Government Sustainability Index (GSI), Environmental Sustainability Index (ESI), Population Sustainability Index (PSI), and Business Sustainability Index (BSI), complemented by a Composite Sustainability Index (CSI). The indices are constructed using min–max normalization of harmonized data from Eurostat, the European Environment Agency, and the Sustainable Development Report. The empirical analysis combines K-means clustering, compound annual growth rate (CAGR) calculations, and correlation analysis, complemented by a robustness module testing alternative weighting schemes, z-score normalization, and ±10% variations in index components. The results reveal four relatively stable sustainability tiers among EU member states, an S-curve-type relationship between initial sustainability tiers and subsequent growth, and a consistent hierarchy in stakeholder response speeds (ESI &gt; GSI &gt; PSI). A clear structural slowdown after 2019 is also observed. The main findings remain robust across alternative methodological specifications. The study contributes to the quantitative integration of the multi-level perspective on sustainability transitions into a stakeholder-based composite index framework for cross-country analysis within the European Union.

Abstract:

Objective: Bevacizumab (BEV) and Olaparib (OLA) have demonstrated clinical efficacy as maintenance therapies for first platinum-sensitive recurrent ovarian cancer. However, direct comparisons between these agents independent of homologous recombination deficiency (HRD) and BRCA status, remain limited and the clinical validity of OLA dose reduction has not yet been confirmed. This study aimed to compare the efficacy and safety of BEV, standard-dose OLA, and dose-reduced OLA as maintenance therapy and to evaluate the clinical utility of OLA dose reduction. Methods: This retrospective multicenter study included 101 patients with first platinum-sensitive recurrent ovarian, fallopian tube, or primary peritoneal cancer who received maintenance therapy after achieving a response to chemotherapy. Patients were classified into three groups: BEV (n = 34), standard-dose OLA (n = 31), and dose-reduced OLA (n = 36). The primary endpoint was progression-free survival (PFS), and secondary endpoints included overall survival (OS) and adverse events. Survival outcomes were evaluated using Kaplan–Meier analysis and Cox proportional hazards models. Results: Median PFS was 16 months in the BEV group, 16 months in the standard-dose OLA group, and 24 months in the dose-reduced OLA group, with significantly longer PFS in the dose-reduced OLA group (p < 0.001). In the multivariate Cox analysis, treatment remained an independent prognostic factor for PFS (HR 0.67, 95% CI 0.46–0.96, p = 0.030). Median OS was 44, 45, and 64 months, respectively, with no significant differences among groups; PFI ≥12 months was the only independent prognostic factor for OS. Grade ≥3 hematologic toxicities were more frequent in the OLA groups but were manageable. Conclusions: Dose-reduced OLA was associated with prolonged PFS while maintaining manageable toxicity, supporting its clinical validity as a maintenance option independent of HRD and BRCA status.

Abstract: This study examines whether international food price dynamics provide a reliable signal of undernourishment and human development outcomes relevant to the attainment of SDG 2 (Zero Hunger) by 2030. We apply wavelet coherence analysis to the FAO Food Price Index and the prevalence of undernourishment (SDG Indicator 2.1.1) over 2001–2023, testing statistical significance against an AR(1) red-noise null hypothesis. Hybrid ARIMA–Random Forest models generate probabilistic price forecasts through 2030. Despite strong raw coherence (R² ≈ 0.77), only 7.8% of time–frequency cells achieve statistical significance, indicating that apparent co-movement largely reflects autocorrelation rather than substantive dependence. Where significant coherence emerges, it concentrates at medium-run horizons (3–6 years), consistent with undernourishment as a habitual dietary adequacy measure linked to sustained affordability pressures affecting health, productivity, and human capital formation. Rolling correlation analysis reveals a regime shift around 2012—from negative to positive correlation—coinciding with a slowdown in progress toward reducing hunger. Price forecasts exhibit rapidly widening confidence intervals (by ±131 index points by 2030), underscoring fundamental limits to predictability. These findings caution against mechanistic inferences from global price indices to hunger and human development outcomes, redirecting policy emphasis toward domestic transmission channels and nutrition-sensitive safety nets.

Abstract: The origin of life on prebiotic Earth is one of science’s greatest mysteries. Traditional approaches to the search for answers often rely on isolated universe models that fail to account for the open, far-from-equilibrium conditions of natural systems. Such limitations might be hindering progress in understanding the thermodynamic processes that drive the emergence of life. This theoretical study presents the second installment in a series exploring a thermodynamic theory of heat transformation and its role in the development of prebiotic Earth. Building on the foundational concepts introduced in Part I, this work focuses on the dissipation of heat through chemical systems, examining chemical dissipative structures (CDSs) and their ability to store and release energy through endothermic and exothermic reactions. By modeling CDSs as heat engines, the paper investigates the interplay between entropy, order, and complexity, revealing how energy storage and transformation drive the generation of complexity. The analysis extends to the networking of gravitational dissipative structures (GDSs) covered in Part I and CDSs, demonstrating how these networks facilitate the transfer of stored energy and contribute to the development and growth of prebiotic chemosystems. The findings suggest that the dominance of endothermic synthesis reactions in CDSs plays a critical role in the generation of complexity, resilience, and mutualism, ultimately shaping the conditions for the emergence of life. This work provides a novel perspective on the thermodynamic underpinnings of prebiotic evolution, offering insights into the relationship between energy, information, and the development of complex systems.

Abstract: Rheumatoid arthritis (RA) is traditionally diagnosed once clinical joint swelling becomes apparent. However, many patients report non-localized stiffness in very early stage then preceding overt synovitis. The transitional processes linking systemic autoimmunity to localized joint inflammation remain incompletely conceptualized. This study proposed a conceptual stage-dependent transition model distinguishing a reversible inflammatory phase from a structurally consolidated phase. Tenosynovitis and joint ultrasound gray scale (GS)-negative/Power Doppler (PD)-positive synovitis were conceptualized as reversible states, whereas GS synovial hypertrophy was hypothesized to represent local immune consolidation. In this framework, “threshold” denoted a conceptual transition point rather than a fixed numerical value, at which gradual inflammatory amplification results in qualitative transition from reversible immune activation to structurally stabilized inflammation (Figure 1). A minimal dynamic representation was introduced to illustrate the theoretical plausibility of threshold-like behavior (Figure 2). Based on this model, we proposed a conceptual algorithm for ultra-early, time-limited intervention prior to structural consolidation (Figure 3). This study was hypothesis-generating and aims to provide a theoretical basis for reconsidering intervention timing in RA.

Abstract: The Cosmic Microwave Background Radiation (CMBR) harbours a cold spot in the southern celestial hemisphere — the so-called Eridanus Supervoid — whose anomalous temperature depression of ∼ 70 μK and spatial extent of roughly 300 Mpc at redshift z ≈ 1 resist explanation within a purely Gaussian, single-universe framework. In this work we develop a self-consistent theoretical programme that interprets this cold spot as a relic imprint of a conjugate parallel universe that shared a common inflationary epoch with our own before separating under a spontaneous symmetry-breaking transition. Our formalism is built upon three interlocking pillars. First, we introduce a two-dimensional imaginary time topology in which the temporal coordinate is promoted from a one-dimensional real line R¹ to a pair of complex curves C¹ and C² embedded in the R³ manifold. The two curves carry opposite orientation angles of ±2πi, yielding an affine parameter of π for the coupled system. Second, we derive the entanglement entropy of the universe pair by treating each universe as a quantum object with an imaginary-time Ket vector; the Von Neumann entropy of the resulting density matrix links macroscopic cosmological observables to quantum information-theoretic quantities. Third, we embed the birth and separation of the twin universes within Perelman’s Ricci-flow surgery framework, identifying the blow-up of a curvature singularity as the topological mechanism responsible for the universal split. In addition, the paper addresses the holomorphic structure of pre- and post-Big Bang time, the non-Gaussian cosmic-texture signatures of the cold spot, a simulation-based glitch matrix, Jacobi-field analysis of the junction conditions, and the phenomenological implications for the Mandela effect, déjà vu, and related confabulation phenomena. The paper concludes with a probabilistic treatment of conjugate-universe counting within a landscape of 10⁵⁰⁰ vacua and a omprehensive statistical analysis supporting the twin-universe conjecture.

Abstract: Kin-Selection has traditionally been viewed as a stabilizing force explaining when, and to what degree it can be adaptive for benefits to be shifted among relatives. As such, it is not usually considered a potential accelerant of evolutionary change. Yet benefits conferred on relatives — particularly from parents to offspring — may function more as forward-looking investments than simple transfers. When seen in this light, Kin-Selection emerges as a potential catalyst for rapid phenotypic change. Drawing on concepts from both evolutionary biology and economics, this article introduces the concept of “fitness capital”— traits enhanced by altruistic investment that increase an individual’s capacity to acquire resources for further investment — and shows how overlapping generations and assortative mating can generate cascading, multigenerational effects. These dynamics amplify correlations between parental altruism, offspring ability, lineage-level investment, and traits reflecting fitness capital, creating conditions for runaway increases in both altruism and capacity to invest. By recasting kin-directed altruism as a driver of cumulative social and evolutionary change, the proposed synthesis highlights a novel pathway through which populations may evolve greater investment, social complexity, and phenotypic stratification. It also indicates new perspectives on parent-offspring conflict, costly signalling, sexual selection, and Green-Beard effects.

Abstract: We generalize the concept of convolution to basis sets other than the Fourier basis. The basis set is defined by the eigenfunctions of a self-adjoint operator, which in turn defines the generalized transform. Several special cases are considered, including the scale transform and the chirplet transform, among others. We also generalize the concept of correlation between two functions. Examples are given to illustrate each result.

Abstract: Background: Cang-ai volatile oil (CAVO) is a traditional Chinese medicine with properties that soothe the liver and alleviate depression. CAVO is widely utilized in the field of antidepressant research and has surfaced as a possible treatment for depression. Depression is a common affective disorder and effective treatment methods are still limited. CAVO is effective in treating depression; however, the exact mechanism is still unclear. This study aimed to explore the likely mechanism by which CAVO reduces symptoms of depression in rats exposed to chronic unpredictable mild stress (CUMS). Methods: We established a CUMS model in Sprague–Dawley rats and administered CAVO via nebulization to evaluate its therapeutic effect. Behavioral and histology tests were conducted to evaluate brain tissue damage. We utilized metabolomics combined with proteomics to analyze the effects of CAVO. We then assessed molecular validation to further clarify the molecular mechanism of its activity. Results: In CUMS model rats, inhaling aerosolized CAVO significantly reduced brain pathology and depression-like behaviors. CAVO significantly changed serum levels of inflammatory cytokines and neurotrophic factors. Biomarkers linked to CAVO's antidepressant effects were found via metabolomics. Functional analyses highlighted key molecular players such as TrkB, and CREB, and a close association with the antidepressant action of CAVO was confirmed. Conclusion: This study reveals that CAVO reduces depression-like behaviors in CUMS rats by regulating the NT/Trk signaling pathway. These results demonstrate CAVO's therapeutic potential and lay the groundwork for future studies and the creation of depressive treatments.

Abstract: Vitamin B12 is an essential water-soluble vitamin required for critical biological processes such as DNA synthesis, erythrocyte maturation, and maintenance of nervous system integrity. Deficiency of vitamin B12 can lead to serious clinical outcomes, including megaloblastic anemia, and potentially irreversible neurological damage. Conversely, hypercobalaminemia may be associated with severe disorders, including solid neoplasms, hematological malignancies and, in some cases, may result from inappropriate supplementation or immunoglobulin–B12 macro-complexes. Although current guidelines recommend total serum vitamin B12 and holotranscobalamin (holoTC) as first-line biomarkers, total serum vitamin B12 remains the most widely used test in routine clinical practice. However, since holoTC represents the biologically active fraction of vitamin B12 available for receptor-mediated cellular uptake, it appears to provide a more reliable assessment of cobalamin status, particularly in specific clinical contexts. Compared with total vitamin B12 measurement, holoTC is assessed using a more limited number of analytical methods and the majority of available kit are aligned with the WHO reference standard, thereby improving inter-assay harmonization. This review explores literature data about the role of vitamin B12 and holoTC, discussing analytical challenges and clinical interpretation, highlights the potential advantages of holoTC over total serum B12.

Abstract: Background: At present, the gold standard for gastric cancer (GC) confirmation relies mostly on histopathology, an invasive procedure. Noninvasive detection methods using serum for large-scale screening maybe useful for the early diagnosis of GC. Helicobacter pylori (HP) infection and chronic atrophic gastritis are major GC risk factors. We recently developed a noninvasive test called DSC test—, based on patient’s age, sex, their serum PGI and PGII, anti-HP immunoglobulin (IgG), and gastrin G17 levels –predicting GC risk as low (score 0, S0) or high (score 2, S2). The comparative investigation at serum protein level of the two different patient groups detected by our DCS test (S0 and S2) may undoubtedly help to identify gastric disease-dependent proteins, resulting from bacterial infection or gastric mucosa inflammation, as well as get better insight the molecular scenario associated to pre-cancerous conditions. Methods: Mass spectrometry-based protein analysis of tryptically digested proteins was performed, followed by univariate statistical analysis for the different DSC groups from two cohort of patients (exploratory and validation). Significantly differentially abundant proteins differing more than 1.5-fold between groups were selected and validated, and their putative role(s) in gastritis and GC discussed. Results: We present data of comparative protein analysis of sera from patients at high risk to develop gastric cancer or advanced atrophy, depending on their DSC score. In particular, we used untargeted liquid chromatography-tandem mass spectrometry (LC-MS/MS)-based proteomics as semiquantitative method to profile proteins specifically associated with score 2 in sera of 80 patients. In both the exploratory and the validation cohorts, four proteins (beta-2-microglobulin, EGF-containing fibulin-like extracellular matrix protein 1, complement factor D, and Cystatin-C) were more abundant, while two (sex hormone-binding globulin and pregnancy zone protein) were less abundant in sera of S2 individuals (|fold change|≥0.6, p < 0.05, t-test). The higher presence of beta-2-microglobulin (B2M) and the lower content of pregnancy zone protein (PZP) in S2 sera were validated by immunoblotting. Conclusions: This study identified a proteomic signature differentially associated to sera of patients with a different risk to develop GC/advanced atrophy according to our DSC test. The protein marker panels presented in this work will contribute to improve GC diagnostics, once they have been transferred from a research result to a practical tool.

Abstract: Adolescents growing up in multilingual regions experience diverse educational contexts that may shape their daily routines and psychosocial environments, but their independent relevance for mental health remains unclear. South Tyrol, with its parallel German-, Italian-, and Ladin-language school systems, provides a unique setting to examine these associations. This study assessed whether school language and home–school language mismatch are associated with mental health, psychosomatic symptoms, and health-related behaviors among adolescents. We analyzed data from a population-based survey of 2005 adolescents aged 11–19 years who provided self-reported information on mental health, psychosomatic complaints, school stress, social support, digital behaviors, lifestyle, and sleep. Group comparisons by school language were conducted using general linear models and χ² tests with effect sizes. Multivariable regression analyses examined the independent association of home–school language mismatch with mental health outcomes, adjusting for sociodemographic and educational factors and further incorporating sleep-related behaviors. Mental health outcomes, psychosomatic symptoms, and most health-related behaviors showed little variation by school language, with generally small effect sizes. Home–school language mismatch was associated with slightly higher depressive symptom scores in unadjusted analyses but was not independently associated with mental health outcomes after adjustment. In contrast, weekly sleep problems emerged as the strongest correlate of depressive symptoms, accounting for a substantial proportion of explained variance. These findings indicate that adolescent mental health in this multilingual context is shaped less by the language of schooling itself than by broader behavioral and developmental factors, highlighting sleep-related behaviors as a central and modifiable target for prevention.

Abstract: Physics-informed neural networks (PINNs) have emerged as powerful tools for solving partial differential equations, but their training remains challenging due to ill-conditioned loss landscapes. While adaptive methods like Adam dominate deep learning, they exhibit instability on stiff PDEs, and second-order methods are com- putationally prohibitive. We present EPANG-Gen (Enhanced Physics-Aware Natural Gradient with Generalization), a novel optimizer that combines memory-efficient eigen- decomposition with lightweight Bayesian uncertainty quantification. EPANG-Gen in- troduces three key innovations: (1) a randomized eigenspace estimator that approx- imates Hessian curvature with O(dk) memory (k ≪ d), (2) Bayesian R-LayerNorm for per-activation uncertainty estimation, and (3) adaptive rank selection (PASA) that dynamically adjusts to problem difficulty. We evaluate EPANG-Gen on four bench- mark PDEs—Poisson 1D, Burgers’ equation, Darcy flow, and Helmholtz 2D—and on the challenging Taylor-Green vortex at Re = 100, 000, a canonical 3D turbulence problem. Results show that EPANG-Gen matches Adam’s performance on the toughest turbulent regime while eliminating the 25% catastrophic failure rate of ADOPT across 72 runs. Ablation studies confirm that eigen-preconditioning improves performance by 11–35%. The built-in uncertainty estimates provide actionable confidence metrics at negligible cost. EPANG-Gen represents the first optimizer specifically designed for geo- metric and physical AI that combines theoretical convergence guarantees with practical robustness for safety-critical applications.

Abstract: Non-dilute emulsions are emulsions where the concentration of the droplets is high enough for the neighboring droplets to interact with each other hydrodynamically but is still smaller than the packed bed concentration where the droplets are packed and deformed against each other. Thus, they cover a broad range of droplet concentration. Many emulsions encountered in industrial applications fall under this category. Non-dilute emulsions exhibit rich rheological behavior from a simple Newtonian fluid to a highly non-Newtonian fluid reflecting shear-thinning, shear-thickening, yield stress, viscoelasticity, etc. In this article, the rheology of non-dilute emulsions is re-viewed comprehensively. Emulsions of hard-sphere type droplets and deformable droplets, with and without surfactants, are covered. The mathematical models de-scribing the rheological behavior of non-dilute emulsions are discussed. The influences of electric charge and interfacial rheology on the rheological behavior of emulsions are covered in detail. The flocculation of droplets caused by different mechanisms such as depletion and bridging induced by additives and their effect on emulsion rheology are investigated thoroughly. Finally, the dynamic rheology of non-dilute emulsions is dis-cussed covering both pure oil-water interfaces and additive-laden interfaces. The mathematical models describing the dynamic rheological behavior of non-dilute emulsions are described. Based on the existing theoretical and empirical models, it is possible to a priori predict the rheology of non-dilute emulsions. However, serious gaps in the existing knowledge on non-dilute emulsion rheology remain. This review identifies the gaps in existing knowledge and points out future directions in research related to non-dilute emulsion rheology.

Abstract: Background: Mechanical thrombectomy is established for acute ischaemic stroke due to large vessel occlusion, particularly in patients with moderate to severe neurological deficits. However, the benefit–risk profile in patients presenting with mild neurological deficits despite confirmed large vessel occlusion remains uncertain.Objective: To systematically synthesise evidence on the effectiveness and safety of mechanical thrombectomy compared with best medical therapy in adults with mild acute ischaemic stroke and large vessel occlusion. Methods: Randomised controlled trials and comparative observational studies will be included. Primary outcomes will include functional outcomes at 90 days, with secondary outcomes addressing safety, reperfusion, and early neurological deterioration. Meta-analysis will be performed using random-effects models where appropriate. Risk of bias and certainty of evidence will be assessed using Cochrane and GRADE methodologies. Registration: The systematic review protocol was prospectively submitted to PROSPERO with registration granted on the 6th of March, 2026 [CRD420261323013].

Abstract: The aerodynamic performance of an aircraft wing is influenced by the angle of attack (AoA), which directly affects lift, drag, and overall efficiency. This study presents a theoretical analysis of the effect of AoA on a symmetric thin aerofoil using aerodynamic models including Thin Aerofoil Theory and Lifting-Line Theory are discussed. Results are discussed under three regimes; linear AoA range where lift increases proportionally while maintaining steady improvement in lift to drag ratio, pre-stall regime where partial flow separation takes place by reducing lift growth, near-stall where flow separation causes deterioration in aerodynamic efficiency. The study highlights the importance of wing geometry including aspect ratio, camber, and sweep angle. Understanding complex interactions between AoA, lift generation, drag forces, and wing geometry is crucial for optimizing aircraft design and improving aerodynamic performance.

Abstract:

This paper proposes a gravitational theoretical framework based on the dynamics of discrete spacetime units. The core idea is that there exists a conserved ”spacetime raw material”, and quan- tum virtual processes of matter continuously produce new spacetime units by consuming this raw material, forming local density gradients—which manifest as spacetime curvature. This mechanism naturally eliminates action-at-a-distance, is compatible with general relativity under covariance con- straints, and provides a unified explanation for dark matter, dark energy, black hole singularities, and other long-standing puzzles.First, we clarify the meta-principle of ”global common covariance”, and on this basis, give the ultimate explanation of symmetry breaking: symmetry is not ”broken”, but a local cost paid for global covariance. Then we systematically elaborate twelve core arguments of the framework, and starting from the only fundamental equation (the second-order discrete wave equation of complex fields), we rigorously and step-by-step derive the Newtonian gravity limit, mass-energy equation E = mc2 , the principle of constant speed of light, Maxwell’s equations, Newton’s three laws, Schr¨odinger equation, Dirac equation, the origin of spin-1/2, and the geometric formula of the fine-structure constant. All physical laws are derived results rather than external inputs. Finally, we present quantitative predictions that can be tested by future experiments.

Abstract: Along with some known and less known results, we discuss new insights relating combinatorics of words and the ordering of the rationals from a dynamical systems point of view, somehow continuing along the path started in [BI]. We obtain in particular a set of results that structure and enrich the correspondence between the Stern-Brocot (SB) ordering of rational numbers and the corresponding ordering of Farey-Christoffel (FC) words, a class of words that, since their appearance in literature at the end of the 18th century, have revealed numerous relationships with other fields of mathematics. Among the results obtained here is the construction of substitution rules that act on the FC words in a parallel way to the maps on the positive reals that generate the permuted SB tree both vertically and horizontally. A complete correspondence is obtained between the vertical and horizontal motions on the SB tree and the geodesic motions along scattering geodesics and the horocyclic motion along Ford circles in the upper half-plane, respectively.