Ask when retained information is sufficient to predict future outputs autonomously. Explain the completion criterion and why keeping observations need not keep every dynamically relevant distinction.
A retained observation describes what can be read now. A predictive state must also contain enough information to determine the statistics or values of future observations under the admitted dynamics. These tasks are different. The latest alpha draft includes an exact criterion for when a linear completed record supports an autonomous update.
This issue is central to DTT. Inner generation may retain phase or memory not present in the current outer record. If that hidden information later affects a readable output, the outer record alone is not a closed state description. Calling it complete does not remove the dependence.
Let H be a finite-dimensional carrier, P the orthogonal projection onto retained information, Q=I−P the hidden sector, and M the completed update. We seek an update T acting only on retained records such that PM=TP.
Multiplying this identity by Q shows the necessary condition PMQ=0. Conversely, if PMQ vanishes, PM=PMP and the required map is T=PM restricted to the retained subspace. This proves necessity, sufficiency, and uniqueness. The criterion asks whether hidden input can leak into the next visible output.
Write a state as (x,y), with x retained and y hidden. If M maps it to (ax+by,cx+dy), the next record is ax+by. An autonomous rule using x alone exists exactly when b=0. Otherwise two states with the same x and different y have different next records.
The best retained-only linear approximation uses T(x)=ax, but its error on hidden changes is controlled by b. In the general Hilbert-space model, the exact worst-case residual is the operator norm of PMQ. This turns a qualitative claim of missing memory into a quantitative diagnostic.
The instantaneous imaginary part of a quaternion has three real components and forgets its scalar part. A later circular phase action can mix the scalar part into a visible imaginary component. Two quaternions with the same present three-dimensional readout can therefore have different later readouts.
A second suitably delayed observation can recover the missing scalar component when the phase sine is nonzero. The combined observations then need four real dimensions to retain all that information. This does not contradict the rank-three instantaneous detector theorem. It shows that the number of presently readable directions and the dimension of an autonomous predictive state answer different questions.
For finite-dimensional unitary M, exact autonomy is equivalent to P commuting with M. However, when M is a product of inner operations, the criterion constrains the completed product rather than every prefix. An inner rotation can mix retained and hidden directions, and a subsequent inverse rotation can cancel that mixing.
For example, with P selecting the first coordinate, a rotation U₁ by θ has leakage |sin θ|. Taking U₂=U₁⁻¹ makes the complete update U₂U₁=I, with zero leakage. This fits the DTT distinction between unfinished internal work and a completed observable return, while leaving the actual completion rule to be physically justified.
The projection P in this theorem is a linear information map. A quantum measurement on density matrices requires an instrument. The operation ρ↦PρP is trace decreasing and represents one outcome; normalizing it is postselection. Retaining both outcomes gives a trace-preserving measurement channel. These are not interchangeable descriptions.
A practical SMM/DTT construction should therefore identify its retained state, full update, and actual observation instrument separately. If autonomy fails, it can retain additional memory, restrict the operation language, or model the reduced process with history dependence. The exact leakage norm provides a useful target for approximate completion, but neither the criterion nor a cancellation example derives the six nested stages or the physical coupling normalization.
KEEP THE CONNECTION
Scope of the argument. The theorem concerns finite-dimensional linear completion; quantum measurement claims additionally require a specified instrument.
Draft file: smm_dtt_alpha0_137_v3.pdf
Working manuscript. Article-level references identify the consulted sections and the scope of the interpretation. No public URL was supplied for this exact manuscript version.
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