M19a The HyperSum Science: The f-Merge Hierarchy, the Trigonometric Symmetric Core, the theta-Family, and the Amult Principle

This monograph develops the HyperSum Science: a unified theory of n‑ary aggregation operations across the full Symmetric Core hierarchy, with particular emphasis on the Trigonometric Symmetric Core and its role as the transition layer between multiplication, trigonometry, and higher hyperoperations. The work establishes the f‑Merge framework, showing that classical aggregation laws—addition, multiplication, power means, trigonometric addition, relativistic composition—are all instances of conjugated addition in an appropriate NC coordinate. Beyond this, the monograph identifies ApowMerge as the first operation that transcends the f‑Merge paradigm by using multiplication rather than addition in NC space, marking a genuine increase in operational rank. A central contribution is the systematic analysis of TrigMerge families (Sin, Cos, Tan, Tanh, Sinh, Cosh), their group structures, explicit algebraic formulas, and Hermit unit equations. These structures are shown to generate Chebyshev dynamics, the Lexem crate lattice, and the full discrete symmetry hierarchy underlying the cascade. The appearance of specific primes (notably 2, 5, and 7) in Hermit discriminants is identified as an intrinsic arithmetic feature of the trigonometric layer, not a numerical coincidence. The monograph introduces the θ‑Merge family, a continuous deformation interpolating between pure multiplication (θ = 0) and pure trigonometry (θ = π/2). Within this family, the carry cocycle emerges as the key structural invariant controlling deviation from multiplicativity. This culminates in the Amult Principle, which explains how arithmetic carry, branch structure, and nonlinearity arise naturally once θ ≠ 0. M19a also reformulates the Riemann Hypothesis in Merge language. Multiple equivalent RH criteria are derived using TanMerge, TanhMerge, and HalbMerge applied to zero pairs, all reducing to the NC‑quadratic condition Im(s² − s) = 0. The NC quadratic family is organised into a spectral ladder that unifies the LP boundary, Hermit units, the golden‑ratio pole, and the Riemann zeros within a single algebraic geometry. M19a thus provides the mathematical backbone of the Trigonometric Core: the complete operational, spectral, and arithmetic infrastructure upon which the physical interpretations of M19b and the RH synthesis of M19c are built.