Rank-one trilinear ternary semirings: classification, universal localization, and coefficient spectra

We classify all $R$-trilinear commutative $\Gamma$-indexed ternary products on the rank-one regular semimodule $R$. Each product is uniquely of the form $[a,b,c]_{\gamma}=\lambda_{\gamma}abc$. Hence these systems are equivalent to commutative semirings with marked coefficients. This classification yields functorial base change and a universal localization theorem. Under the weaker hypothesis that the coefficients generate the unit ideal, rather than requiring one coefficient to be invertible, we identify ternary ideals and primes and prove \[
\SpecG\TT(R,\Lambda)\cong\Spec(\Sat^{-1}R). \] We construct the associated functorial locally ternary-semiringed spectrum and identify its stalks. We also characterize common binary envelopes, prove that coefficient saturation is a categorical reflection, and show exactly why the symmetric numerator--denominator rule collapses every label to the untwisted product. Examples over $\mathbb Z$, $\mathbb N$, $k[x]$, and $\mathbb Z/12\mathbb Z$ separate the hypotheses and exhibit explicit local behaviour.