For a finite linear update M and retained projection P, an autonomous record update exists exactly when PM(I−P)=0. When this fails, retaining more state or a memory-dependent description restores the missing information.
Observation and prediction ask different questions
A thermometer displays one number, but that number alone need not predict its next reading: the surrounding environment and internal response can matter. Similarly, a completed source record can omit phase or memory that later influences an observable. Calling the record complete must therefore specify whether it is complete for a present test or for all admitted future tests.
In SMM/DTT, inner generation can carry distinctions absent from the outer display. This gives a concrete reason to separate the generative state, a retained observation, and the minimal state needed for prediction. Their dimensions need not agree. A single source interpretation does not eliminate this information requirement.
The exact finite linear criterion
Let P be the orthogonal projection onto retained information, Q=I−P its hidden complement, and M the completed linear update. Seek a map T acting on retained records so that PM=TP. Multiplying by Q proves the necessary condition PMQ=0, because PQ=0. Conversely, if PMQ=0, then PM=PMP, so the restriction T=PM|PV works.
The condition is exact and operational: hidden variations cannot change the next visible result. It applies to this supplied linear model. A quantum measurement on density matrices is a different object and requires an instrument; the projection P here is an information map rather than a normalized postselection operation.
A two-coordinate counterexample and its error
Let M(x,y)=(ax+by,cx+dy), while P retains x. The next displayed value is ax+by. Two states with the same x but different y produce different outputs whenever b≠0. Therefore an autonomous rule for x alone exists precisely when b=0.
For the operator-norm statement, restriction to hidden inputs gives the lower bound ‖PMQ‖ for every T. Choosing T=PMP on the retained sector attains it. In the two-coordinate example this residual is |b|. The obstruction is thus measurable in principle: prepare two hidden variations with identical retained inputs and compare their later outputs under the same completed operation.
A three-dimensional display can need four-dimensional memory
Write a quaternion q=a+xe₁+ye₂+ze₃ and retain only its imaginary part (x,y,z). Left multiplication by cosθ+e₁sinθ gives a later first imaginary component x′=a sinθ+x cosθ. The initially invisible scalar a has become visible.
At θ=π/2, x′=a. One delayed observation can therefore recover the missing scalar when the phase operation and the present imaginary components are known. A rank-three instantaneous readout is entirely consistent with a four-real-dimensional predictive carrier. The result does not enlarge physical space by one dimension; it enlarges the memory needed to predict the admitted experiment.
This is why the spinor readout and other quotient constructions must be assessed against their full operation language.
A completed cycle can close after leaking internally
Let P retain the first coordinate of a real plane and let U₁ be a rotation through θ. Its retained output depends on the hidden input with magnitude |sinθ|. Follow it by U₂=U₁−1. The completed product is the identity, so PU₂U₁Q=0 even though PU₁Q generally does not vanish.
This is a simple realization of the distinction between unfinished generative work and an autonomous completed update. It does not establish that every physical process performs an inverse or returns to the same state. The source model must determine the actual sequence and certify when its completed product becomes available.
Eliminating a hidden mode produces an exact memory term
Consider observed x and hidden y governed by ẋ=−ax+by and ẏ=−λy+kx, with λ>0. Multiplying the second equation by exp(λt), integrating, and substituting gives:
+bk∫₀ᵗe−λ(t−s)x(s) ds.
The reduced equation contains both a memory kernel and the hidden initial preparation. Neither term is added by hand after the fact; both follow from the enlarged equations. For a>0, the full two-variable system is asymptotically stable when aλ−bk>0. A past-dependent integral alone does not establish relaxation, entropy production, or a temporal arrow.
Memory is a target for source predictions
The enlarged example uses one parameter t. Its memory therefore does not prove the existence of a second independently measurable time. Conversely, representing the reduced process with an auxiliary variable does not make the generative interpretation meaningless; it identifies the comparison model that the interpretation must address.
A substantive SMM/DTT advance would derive the allowed kernel, its coefficients, or a relation between memory and another observable from the same nested source dynamics. One could then compare multiple preparations with a fixed prediction. The immediate design rule is already clear: retain enough information for future tests, restrict the tests explicitly, or use a history-dependent state description.
Continue with the articles
- Retained observations versus predictive states
A readable record need not contain enough information to predict its next update. An exact leakage criterion identifies when a linear completion is autonomous and measures the error when hidden information returns.
- When two histories count as the same observation
Completion identifies histories that leave the same record, while observation identifies records that available tests cannot distinguish. Their combined quotient is consistent only with the declared operations and contexts.
- Test languages and observable quotients
A test language determines which records an observer can distinguish. Enlarging that language refines the observable quotient and reveals when apparently different instruments retain the same information.
- Eliminating hidden variables creates memory
Eliminating a relaxing hidden sector yields an exact memory equation for the retained variables. The resulting kernel and initial-state term preserve causal dependence while exposing information absent from the current readout.
- When memory can be represented by one time
Finite auxiliary states can generate memory while evolving in one laboratory time. Rational-kernel realization identifies when that reduction is exact and which additional conditions make the auxiliary system a physical reservoir.
- Temporal asymmetry: reading the published model
The temporal-asymmetry model connects projected dynamics with memory and path dependence. A solvable hidden-mode example distinguishes those effects from dissipative stability and from evidence for a second observable clock.