The Integer Traceability Chain: From SU(5) embedding to cosmological predictions in seven exact links.

The Integer Traceability Chain: From SU(5) embedding to cosmological predictions in seven exact links. This paper is part of the HOWL research archive—a collection of physics papers exploring integer fraction derivations across multiple domains using exact arithmetic and automated comparison. Abstract The beta unification program (PHYS-25) found that the integers 13, 19, 20, and 22 — extracted from the gauge coupling beta coefficients — predict seven cosmological observables at sub-percent precision. This paper derives each integer from the gauge group alone, with no measurement, through a chain of seven exact links. The chain starts at the SU(5) embedding condition Tr(Y²) = k₁ Tr(T₃²), which gives the GUT normalization factor k₁ = 3/5 from explicit traces over one SM generation. From k₁, the Dynkin index coefficients (2/5, 2/3, 1/3) for vector-like fermion pairs follow. Applied to the Cabibbo Doublet (3,2,1/6), these produce Δb = (1/15, 1, 1/3) and modified betas b' = (25/6, −13/6, −20/3). The numerator magnitudes |6b₂'| = 13 and |3b₃'| = 20 are the integers that enter every cosmological formula. Every link is verified EXACT in Fraction arithmetic. The chain is Level 1 — it holds in any universe with the same gauge group. A structural finding: the cosmological integers 13 and 20 come from the SU(2) and SU(3) Dynkin coefficients C₂ = 2/3 and C₃ = 1/3, which do not contain the GUT normalization factor k₁. The cosmological program is independent of the GUT normalization convention. Falsification Criteria All papers in this archive are subject to falsification through direct comparison to published experimental measurements. Each derived value is tested against independent data with explicit PASS/FAIL criteria. Any derived value that fails its comparison is documented and published alongside the successes. Research Context This archive documents an ongoing research program in integer fraction physics. The methodology is: derive values from gauge group integers using exact fraction arithmetic, compare to published measurements, and document all results including failures. The archive spans multiple physics domains connected through the soliton boundary framework described in the constituent papers. Package Contents manuscript.md: The complete derivation and supporting analysis. README.md: Navigation, dependencies, and citation (Registry: HOWL-PHYS-26-2026). Dependencies: HOWL-PHYS-1-2026, HOWL-PHYS-13-2026, HOWL-PHYS-21-2026, HOWL-PHYS-24-2026, HOWL-PHYS-25-2026 Motto: Derive. Compare. Publish.Status: Complete