A reversible spin interaction separates identical operational states
Two classical spin preparations have identical first and cross moments, yet the Hamiltonian interaction H=J u_z v_z gives them different later probabilities for an allowed product measurement. The hidden dynamics is smooth, complete and reversible. It therefore supplies an explicit failure of descent to C125’s retained operational state, rather than the reversible gate required by B70.
Q01 mechanical test, 2026-09-13. Keep C125’s unit orientations u,v, all joint Borel preparations and affine local/product terminal measurements. Their operational state is (1,E[u],E[v],E[u v^T]). Introduce fixed classical angular momentum magnitudes S_A,S_B>0, with action units, and J>0 with energy units. These are supplied mechanical parameters, not a selected Planck constant. Source coverage and written review are in B72.
1. The Hamiltonian flow
On S^2 times S^2 use the classical spin Poisson brackets
\[\{u_i,u_j\}=S_A^{-1}\epsilon_{ijk}u_k,\quad \{v_i,v_j\}=S_B^{-1}\epsilon_{ijk}v_k,\quad \{u_i,v_j\}=0,\qquad H=J u_z v_z. \tag{1}\]
The convention is dot f={f,H}. In particular,
\[\dot u_x=-\Omega_A u_y v_z,\quad \dot u_y=\Omega_A u_x v_z,\quad \dot u_z=0, \qquad\Omega_A=J/S_A,\] \[\dot v_x=-\Omega_B v_y u_z,\quad \dot v_y=\Omega_B v_x u_z,\quad \dot v_z=0, \qquad\Omega_B=J/S_B. \tag{2}\]
The sphere norms and H are conserved. This smooth vector field on a compact phase space has a complete flow with inverse obtained by changing t to -t. It is an interaction: each rotation frequency depends on the other spin. Time is the specified Hamiltonian evolution time; Omega has inverse-time units.
2. An exact same-moment pair
Let e_x,e_y,e_z be the standard unit vectors and prepare
\[\mu_A=\delta_{e_y}(du)\, \tfrac12(\delta_{e_z}+\delta_{-e_z})(dv),\qquad \mu_B=\delta_{e_y}(du)\, \tfrac12(\delta_{e_x}+\delta_{-e_x})(dv). \tag{3}\]
Both have E[u]=e_y, E[v]=0 and E[u v^T]=0. They are therefore exactly equivalent for all C125 terminal tests, including mixed and coarse-grained product tests. Their microscopic energies are also identically zero.
Since u_z=0 remains zero, v is constant. For mu_A, write v=sign e_z:
\[u(t)=(-\mathrm{sign}\sin(\Omega_A t), \cos(\Omega_A t),0),\qquad E_{\mu_A}[u_x(t)v_z(t)]=-\sin(\Omega_A t). \tag{4}\]
For mu_B, v_z=0, so u(t)=e_y and E_muB[u_x(t)v_z(t)]=0. The allowed product effect e(u,v)=(1+u_x)(1+v_z)/4 consequently gives
\[p_A(t)=\tfrac14[1-\sin(\Omega_A t)],\qquad p_B(t)=\tfrac14. \tag{5}\]
For every sufficiently small positive t these differ. No map, linear or otherwise, from the retained initial operational state to the retained state at such a time can represent this Hamiltonian on all admitted preparations. Both evolved preparations still yield valid minimal-composite states; the failure is single-valued evolution on the quotient, not state positivity.
3. The missing moment and the implication for reconstruction
Equations (2) expose the additional information immediately:
\[\frac{d}{dt}E[u_xv_z]=-\Omega_A E[u_yv_z^2]. \tag{6}\]
The right-hand side equals -Omega_A for mu_A and zero for mu_B at t=0. C125 retained E[u_i v_j] but not this moment quadratic in v. Adding the interaction to the allowed transformation toolbox therefore changes the operational equivalence relation: its precomposed terminal measurement now distinguishes the two preparations.
This is consistent with B70’s theorem. That theorem assumes a reversible linear action on the fixed operational tensor state space. Here the classical interaction has a reversible action on hidden measures but fails to induce even a single-valued action on that fixed quotient. Physical interaction and operational reversible closure are distinct premises.
The decisive remaining choice is to justify a preparation/effect restriction stable under physical interactions, or to enlarge the retained state space. A separate argument must show that such a repair remains finite-capacity and locally tomographic before importing B70. Exact Hamiltonian reversibility alone does not establish that repair. The fixed S_A,S_B supply action units to this example; none of the moment calculations forces a nonzero universal value.
The finite-repair decision is now complete: C128 rules out finite observable enlargement for this interaction and all Borel preparations. The open repair must change a model premise or admit infinite observable dimension; STATE owns its strategic follow-through.