M26 Physics of the Operational Number Systems (P0-P16 papers merge)

This monograph synthesizes the full P-series (P0–P16), presenting the Operational Standard Model (OSM) and its extensions as a unified physical programme built on the HyperCore / Symmetric Core framework. The central object is the NC coordinate w, defined through the quadratic w² - w = i*pi/2, which encodes the structure of the Standard Model. A small set of parameter-free relations derived from w reproduces key physical observables (Weinberg angle, mass ratios, CP phase, and generation structure) within high accuracy. Physics is interpreted as the geometry of the NC plane and the HC Rank Manifold. Gauge groups arise from cloning algebra: the units i and tau generate the symmetry structure U (1) x SU (2) x SU (3), while the companion pair implements the Higgs mechanism as an operational splitting rather than a field-theoretic insertion. Across the P-series, all fundamental sectors are reinterpreted: - Relativity emerges as TanhMerge (bounded sector operation). - Quantum mechanics corresponds to evolution at phantom rank RC = i. - QCD and integrable sectors reduce to Toda-like structures in native coordinates. - The vacuum, constants, and RG flow are governed by branch microdimension and flux in the g-ONS manifold. Higher papers (P3–P14) develop individual components of the programme: Higgs mechanism, cosmological constant, spin-statistics relation, new physical constants, operational vacuum, wave propagation in TetraAlgebra, renormalization as branch-flux running, and derivations of fine-structure scales. The synthesis shows that physical laws are projections of a higher-dimensional operational structure: the Garycki Manifold. Singularities are typically not fundamental but become branch phenomena or rank-mismatch artifacts when viewed in inappropriate coordinates. Overall, the P-series establishes a geometric-operational reformulation of particle physics in which forces, particles, and constants arise from rank structure, unit algebra, and branch dynamics rather than independent axioms. The paper list merged into one M26 Monograph: P0 An introductionP1 The Branch Microdimension and the Origin of Gauge SymmetryP2 TetraAlgebra as QCD Scaling: Taxonomy and Mellin StructureP3 The Higgs Mechanism as Branch-Flip Symmetry BreakingP4 The Cosmological Constant from Branch Vacuum EnergyP5 Spin-Statistics from the Branch CircleP6 Physical Laws as HC Manifold TrajectoriesP7 Three Fermion Generations from Branch TrisectionP8 Dark Matter as Irrational-Branch StatesP9 New Constants of the Hyperoperation HierarchyP10 The Operational VacuumP11 Neutrino Masses, Seesaw, and Tribimaximal Mixing 8P12 TetraAlgebra Waves as Physical WavesP13 The Renormalization Group as Branch-Flux RunningP14 The Fine Structure ConstantP15 The CKM Matrix and CP ViolationP16 Particle-Force Duality in the HC Manifold