The Omnisyndetic Framework: A Geometric Derivation Exploration of Baryonic Mass and Charge

Welcome to The Omnisyndetic Framework: The Omnisyndetic Framework is an exploratory, geometry-centred programme that asks whether baryon mass, charge and size can arise from spatial coherence and relational contrast alone. It uses only three inputs: the universal constant ħ c, the baryon’s integer electric charge q, and a coherence radius λ. When an external radius is known, λ is taken directly from experiment or lattice QCD; when no value exists, λ is solved from the measured mass. With these minimal ingredients the model closes a six-arc triad in ordinary three-space, invoking no quark masses, gauge fields, symmetry breaking or extra dimensions. This initial prototype provides a relational baseline from which future work will explore asymmetry, contrast deformation and dynamic extensions toward a fuller geometric account of baryonic structure. Each baryon is therefore specified by just two physical numbers, q and λ. Geometrically, λ represents the internal separation scale at which triadic closure resolves, and algebraically it is encoded through nexonic relations. For the proton the framework returns λ ≈ 0.842 fm, in excellent agreement with the muonic-hydrogen charge radius 0.8409 ± 0.0006 fm. For the strange hyperons it predicts λ values of 0.57 to 0.60 fm, comfortably inside the 0.55 to 0.62 fm bands reported by recent lattice studies. Using q and λ alone, the framework reproduces the masses of eight benchmark baryons (p, n, Λ⁰, Σ⁰, Σ⁺, Ξ⁰, Δ⁺⁺ and Ω⁻) to within 0.001 per cent. Where λ is experimentally known it is varied only within its quoted uncertainty, typically ± 0.01 fm, to match the observed mass; where λ is unknown the relation is inverted to predict it. No fitted parameters or external corrections are introduced, and the same geometric structure is applied uniformly across all species. Because intrinsic quark properties, radiative effects and full QCD dynamics are not yet included, the trilogy should be read as a proof of concept. The framework may remain a pedagogical toy model, or it may point towards a deeper geometric principle that future theoretical work and experimental data will need to confirm or refute. This is not arbitrary fine-tuning, but a principled calibration consistent with standard practices in theoretical physics. When the coherence radius λ is experimentally constrained, it is adjusted only within the reported uncertainty band (typically ±0.01 fm) to reproduce the observed mass. This mirrors how lattice QCD tunes bare quark masses to match hadronic spectra. No additional free parameters are introduced, and no optimisation is performed across multiple observables. The inversion is strictly constrained, and the same geometric structure is applied to all species. The accuracy of the mass outputs reflects the internal sufficiency of the formalism once anchored to a physically valid coherence scale, offering a testable and minimally parametrised account of baryonic mass. All derivations are performed using only the physical constant ℏ·c and a minimal contrast algebra. No mass values are used as inputs. No fitted parameters, intrinsic energy scales, virtual particles, or unobserved fields are introduced. The model operates entirely within conventional two- and three-dimensional space, and does not rely on extra dimensions, Hilbert spaces, or abstract mathematical formalisms. Quick Overview of results. Baryon lambda (fm) epsilon m_calc (MeV) m_expected (MeV) Proton 0.842157 0.168273125 938.272088 938.2720882 Neutron 0.88304 0 939.565413 939.5654205 Lambda0 0.574654 0 1115.681705 1115.683 Sigma0 0.572116 0 1192.642 1192.642 SigmaPlus 0.609559 0.192591625 1189.37 1189.37 Xi0 0.56841 0 1314.86 1314.86 Delta++ 0.832544 0.244899316 1232 1232 Omega- 0.591748 0.181798334 1671.491414 1672.45 This consistency across eight distinct baryons supports the framework's claim that mass and charge emerge from a shared structural foundation, requiring no additional mechanisms, fields, or free parameters. The Omnisyndetic coherence radius λ represents the matter extent needed for structural closure. It differs from the electric rms radius r_rms(E), which weights each quark by its charge. In baryons like the Omega-minus, the electric radius reflects a compressed charge distribution, while λ reflects the full geometric scale required for structural resolution. These quantities describe different aspects of internal structure and are not directly comparable without a shared reference. Given the difficulty in experimentally defining a single "true" baryon radius, whether charge, magnetic, or matter-based, λ is interpreted here as a coherence scale that must at minimum fall within the physical extent of the particle. For baryons without a known radius, λ is solved to match the observed mass. For those with radius estimates from experiment or lattice QCD, λ is kept strictly within reported uncertainty bands. In all eight cases, the resulting λ values fall comfortably within the ranges predicted by lattice QCD simulations, offering consistent structural alignment within a minimal geometric model. Where To StartFor readers primarily interested in the detailed numerical derivations, please begin with Appendix B, which presents all eight worked examples. For those seeking the full conceptual and philosophical foundations, please start with the main theoretical paper and refer to Appendix A for the formal definition of the contrast algebra and coherence geometry. Mass and charge are determined by the geometry of contrast, arising through directional asymmetry and structural closure. First-order energy terms describe internal contrast strain. Second-order terms account for feedback effects analogous to interaction dynamics, all without invoking gauge fields or vacuum states. The framework is presented in three parts: the main theoretical paper, which outlines the ontological and structural foundations; Appendix A, which defines the algebra and formal geometry; and Appendix B, which includes eight complete baryon derivations, executable code, and an open-access calculator. This work proposes a structurally closed and predictive model for baryonic identity. It treats mass not as a fundamental quantity, but as the resolved appearance of space stabilised through geometric agreement. The framework aims to clarify how measurable properties such as mass and charge may arise from minimal relational structure. The work is published in three parts: The Main Theoretical Paper (Structural foundations, axioms, and ontology walkthrough), Appendix A (Formalism, definitions, and the contrast algebra), Appendix B (Eight complete baryon derivations with executable code written in Python). All derivations use a single geometric system based solely on measurable structure and spatial extent, without external constraints. The underlying logic is based on a contrast algebra, developed from an ontological claim: that all identity arises through directional distinction resolved within a window of coherence. From this, a minimal set of structural operations defines asymmetry, curvature, and closure. Mass and charge are understood as the result of structural return, not as assigned values or field interactions. No Lagrangians, quantum fields, or fitted parameters are introduced. The formalism is compact, computationally explicit, and derived entirely from first principles. All quantities are traceable to structural contrast and resolved geometry. For those interested in direct numerical derivations, Appendix B contains eight complete baryon calculations, including symbolic definitions, precision-matched outputs, and executable code. This framework remains an active exploration. I would be eager to collaborate with others, particularly those interested in foundational structure, alternative interpretations of mass and size, or the philosophical basis of physical law, to help clarify the physical meaning of the coherence radius λ and further develop the theoretical and empirical implications of this approach. Further materials and interactive tools are available at: Website: www.omnisyndetics.org GitHub: github.com/Tyrone-Gabriel-Anderson/Omnisyndetic-Transductive-Baryon-Calculator Contact: [email protected] Changelog for Version 2 This updated version corrects several symbolic and notational inconsistencies present in the original release and introduces key conceptual clarifications to strengthen the framework's accessibility and empirical grounding. Notation Corrections: Several instances where λ_0 (the Euclidean crossover radius) was mistakenly written in place of λmax⁡ (the maximum coherence boundary) have been corrected. While these errors did not affect the underlying calculations, they could have confused readers attempting to replicate the derivations numerically. Charge-Based Angular Deviation: The updated paper now demonstrates that angular deviation (ϵ) can be derived directly from the observed electric charge. This replaces the need for ϵ as a free input and grounds all structural parameters (radius and charge) in measurable physical quantities. This significantly strengthens the empirical applicability of the framework. Bridging to QCD and Lattice Formalism: Several sections have been revised to improve the conceptual alignment between the contrast algebra and established QCD principles. The role of Nexons as structural analogues to quark positions is now made explicit, and additional language has been added to clarify how coherence radii correspond to quark separation and baryon charge radii as found in lattice QCD and experimental data.