Exact Finite-Amplitude Permutation-Quotient Cross-Host Consequence Dynamics in Canonical ACRF-4: Orbit Completeness, Stratified Host-Boundary State, and a Nine-Dimensional Generic Quotient

This preprint derives an exact finite-amplitude quotient dynamics for a two-host partition of canonical ACRF-4.

For S={1,2} and C={3,4}, the full current-plus-retained-boundary state is decomposed under the internal relabeling group S2 x S2. The invariant record

Q = (E, UU^T, VV^T, c^2, cUV^T)

is proved to separate internal-label orbits exactly. Because the canonical ACRF-4 update is permutation-equivariant, the full nonlinear dynamics descends uniquely to the quotient for every finite horizon. Every permutation-invariant receiver-host consequence therefore factors through the quotient, giving a receiver-conditioned finite-amplitude law depending on the constituted receiver, constituted source/complement, and their shared retained boundary.

The paper also proves that receiver-odd asymmetries, although silent to first order at symmetric base states for every finite horizon, re-enter invariant receiver dynamics at second order after one update. This establishes why a purely first-order host representation is insufficient at finite amplitude.

The original 16-scalar polynomial invariant embedding reduces to 15 after the fixed edge-budget constraint. The eleven odd-sector polynomial invariants are shown to form a minimal homogeneous generating set. Geometrically, however, the exact orbit quotient has generic dimension nine. A global nine-scalar canonical representative is constructed. On the generic chart d_S d_C != 0, the quotient takes the interpretable 3+3+3 form

W_S = (m_S, p_S, D_S),
W_C = (m_C, p_C, D_C),
B_SC = (A, B, C),

so the shared boundary contributes three independent generic degrees of freedom. Globally the quotient is stratified because symmetry-restoration sets carry nontrivial stabilizers.

A two-step numerical observability audit performed during the derivation reported rank 9, with smallest singular value approximately 1.21e-2. This supports, but does not replace, a proof of generic consequence-minimality.

Claim boundary: exact within canonical finite ACRF-4 are orbit separation, nonlinear quotient descent at every finite horizon, receiver consequence sufficiency, the nine-dimensional generic quotient, and the generic three-degree-of-freedom boundary reduction. Not yet proved is generic consequence-minimality in the stronger observability sense. No universal Registration Theory cross-registration law, multi-host cocycle, sparse primitive locality, spacetime reconstruction, Lorentz structure, or universal physical speed is claimed.