Inner Groupoids: Equation Expansion and Induced Groupoid Structures from Root-Ratio Coefficient Permutations

For one-variable polynomial equations over fields of characteristic zero, this paper defines inner groupoids on ordered nonzero root tuples. A root permutation determines a cyclic sequence of adjacent root ratios; an edge-ratio permutation and a base root determine a generated ordered root tuple; the monic Vieta product determines a generated equation. The parameter space carries a right regular action groupoid, and the root-generation and equation-generation maps induce history-lifted groupoids and coarse value-compression groupoids. The layered construction is defined by the indexed recursion \(I_{k+1}=I_k^3\) and the value recursion \(R_{k+1}=R_kR_kR_k^{-1}\). It yields layerwise history groupoids, persistence correspondences, exponent-kernel collision records, fixed-degree equation spaces, reduced history spaces, and the closed value universe \(R_\infty=R_0\Lambda(\alpha)\). The Galois record construction attaches root–equation–field–Galois data to history nodes and proves that structural layered Galois groups are quotients of the initial Galois group under the stated stability hypotheses. The multivariable construction is obtained from ordinary projection fibres and descends to the base exactly when the generated coefficients are Galois invariant. The generalization layer gives value-equivalence and history-equivalence criteria for graph edge systems, weighted Laurent actions, commutative algebraic groups, configuration spaces, arithmetic specializations, and finite-subset dynamics. No irreducibility, full symmetric Galois group, or inverse Galois realization conclusion follows without additional hypotheses. --- Keywords inner groupoid; root-ratio generation; edge-ratio permutation; Vieta map; history-lifted groupoid; coarse value compression; layered expansion; Galois conservativity; projection fibre; descent criterion