A Conjectured Topological Field Invariant Ki(M) for Multi-Scale Quantization Conditions EN_FR_ES MULTILINGUEL

A Conjectured Topological Field Invariant Ki(M)for Multi-Scale Quantization Conditions From the Coulomb field of the electron to quark confinement:a unified topological framework with testable predictions "Epistemic Status of Manifold Assignments" Abstract We propose a conjectured topological invariant Ki(M) = 1 − 1/χeff(M) where χeff(M) = (1+b1)(1+εw12), with b1 the first Betti number, w12 the Stiefel–Whitney obstruction, and ε a single calibration constant. We conjecture that four distinct physical scales — the electron electromagnetic field, proton three-quark configuration, quark confinement, and the gluon field — correspond to four topological manifolds (Cylinder, Torus T², Klein bottle K², and a Klein variant with b1=8 = dim su(3)) spanning 23 orders of magnitude. The constant ε = 1/5 is shown to be the unique rational value (among fractions with denominator ≤ 10) satisfying Kie/Kiq = 9/13 exactly. This yields a primary falsifiable prediction: Kie/Kiq = 0.6923 ± 3%, constant over Q² ∈ [1, 10⁴] GeV², testable at the Electron-Ion Collider (EIC, ~2029). A secondary prediction — Kiπ/Kiq = 12/13 = 0.923 — falls within the range of existing Drell-Yan data from Fermilab E615 (σDY(π⁻N)/σDY(pp) ≈ 0.89–0.95 at xF=0.2–0.8). A topological entropy Stop(M) is defined whose value at the Torus (proton) predicts an effective temperature scale of 151.5 MeV, comparable to the QCD crossover temperature Tc ≈ 155 MeV from lattice QCD. We present this work as a mathematical conjecture with empirical motivations; the manifold assignments remain postulated rather than derived from first principles. The framework's primary value is the falsifiable EIC prediction and the systematic Ki table for Standard Model particles. Table of Contents Physical Motivation and Discovery History Formal Definition of Ki(M) 2.2 Epistemic Status of Manifold Assignments (new v4) Uniqueness of ε = 1/5 Ki★ Table for Standard Model Particles Primary Prediction: Ki_e/Ki_q at EIC Secondary Prediction: Ki_π/Ki_q and Drell-Yan Topological Entropy and Hagedorn Temperature Honest Assessment: What Is and Is Not Derived Open Questions and Future Work References Important epistemic note The manifold assignments (electron↔Cylinder, quark↔Klein K², etc.) are postulated correspondences, not derived from the QCD Lagrangian or from a homology calculation on gauge field configurations. This is the primary weakness of the framework, openly acknowledged here and addressed in Section 8. Idiomas preprints y figures EN_ES_FR