Reversible interaction excludes the minimal orientation composite
C125 cannot retain its separable operational state space and admit a continuous reversible interaction beyond local rotations. This is an application of Theorem 1 of de la Torre, Masanes, Short and Müller, not a new reconstruction theorem. Their Theorem 2 selects quantum composition for identical qubits with local tomography and operational closure. The remaining physical question is why a mechanical interaction must descend reversibly to this restricted operational description. No action scale follows from this source audit.
Q01 third milestone, 2026-09-13. B70 records selected primary passages and the unaudited proof dependencies. C129/B77 subsequently accepts equations (8)–(14) and the tensor-space restriction; the later proof remains unaudited.
Assumptions already present in C125
The local state coordinate r lies in the unit ball in R^3. Write
\[\rho(r)=\tfrac12(I+r\cdot\sigma),\qquad E=aI+b\cdot\sigma.\]
Then tr(E rho)=a+b dot r. The local effect constraints in C125 are precisely positivity and normalization of qubit effects. This is an invertible change of coordinates, with dimensionless Pauli matrices; no global positive cone or entangled preparation has been assumed. For n components the same tensor basis maps C125’s moment coordinates to Hermitian matrices. Its joint hidden measures map exactly to convex mixtures of product density matrices.
| Source hypothesis (PDF pp. 1–2) | C125 and the proposed extension |
|---|---|
| Local qubit states and effects | Ball and affine directional effects already supply these |
| Local tomography and product preparations/tests | Already supplied by tensor moments and product tests |
| All local unitary conjugations | Already supplied by SO(3) rotations on each ball |
| Connected linear reversible group larger than the local group | New operational interaction requirement; affine mixture compatibility extends the maps linearly to the unnormalized span |
| Identical systems, spectator extension and relabelling | Arbitrary n and permutations already supplied; any new gate must remain admissible under these operations |
| Ancilla preparation, joint evolution, measurement and discard | Must be imposed when claiming the full quantum measurement/channel conclusion; C125 specified terminal tests |
An interaction of hidden orientations is not automatically an interaction on these moments. Two hidden preparations with the same moment tensor must remain operationally equivalent after evolution; the induced map must also have an allowed inverse. These are the precise descent and reversibility requirements.
The source theorem and the exclusion
Theorem 1 assumes a connected linear group containing local unitaries and valid probabilities for every product preparation followed by a group element and a product test. If that group is larger than the local group, its action on a pair with fixed product spectators contains an entangling unitary or its partial-transpose conjugate (source p. 2).
The first branch takes some pure product preparation to an entangled state, which is outside C125’s minimal composite. In the second branch choose a product input whose partial transpose is a product input entangled by the unitary. The output is a partial transpose of a pure entangled two-qubit state. In Schmidt coordinates its eigenvalues include the opposite pair plus/minus s_1 s_2, with both Schmidt coefficients positive. It is therefore not positive semidefinite and cannot be a separable density matrix either. Product spectators do not change either obstruction. These are direct applications of the source’s two branches; neither requires initially admitting entangled states as a hypothesis.
For larger admissible composites, Theorem 2 uses interchangeable copies and three-system consistency to exclude the partial-transpose alternative. The source then invokes entangling-gate universality and operational closure to obtain all quantum preparations, measurements and completely positive trace-preserving maps. Positivity of all resulting tests excludes further states/effects. The broad conclusion includes these closure assumptions and applies to finite numbers of identical qubits, not arbitrary local state spaces.
Decision, evidence and remaining physical premise
Accept the source-backed exclusion of a reversible interacting extension preserving C125’s minimal state space. C125 itself remains a valid model with local dynamics. Adding interaction as a premise is substantive: it changes the allowed composite, rather than repairing a coordinate choice. Theorem 1’s technical Lie-algebra proof and the imported gate-universality results were not independently audited. This milestone checks their statements, the hypothesis map and the conditional consequence; it claims no novelty.
The source fixes probability structure. It supplies no mechanical Hamiltonian, no identification of its transformation parameter with physical time/energy, and no universal action normalization. G03’s stochastic relaxation interaction also supplies no reversible operational gate of this kind.
This completes the selected Q01 dependency decision. Full proof auditing and a mechanical justification of exact reversible descent remain supporting work. After comparing this third Q01 milestone with the gap track, select G04’s operator-identification test: construct G03’s finite-volume Hermitian representative and isolate the extra clock/action and physical-Hamiltonian identifications. Stop at that transfer decision, rather than another coupling variant. STATE owns the queue.