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.
The present visible state may not determine its own future when a hidden variable continues to act on it. Solving the hidden equation makes that missing information explicit as a memory kernel and an initial-state term. The calculation is exact for the supplied coupled equations and gives SMM/DTT a precise reduced-dynamics example without deriving the hidden relaxation law from its ontology.
Write the observable evolution as q̇=F(q)+Kr and the hidden evolution as ṙ=−Λr−Kᵀ∇Ψ(q). Here K couples retained and hidden variables, Λ is a stable hidden relaxation operator, and Ψ is the selected structural functional. Both equations depend on the present enlarged state. The observable q alone is generally insufficient because the same q can coexist with different hidden r. Eliminating r exposes that dependence rather than removing its physical effect.
Variation of constants gives a freely decaying hidden initial state plus a convolution of the past observable gradient. Substituting into q̇ yields an exact Volterra equation. The memory kernel is K exp[−Λ(t−s)]Kᵀ, and the term K exp(−Λt)r₀ records the initially hidden preparation. It cannot be discarded before its decay is justified. No short-memory approximation is needed for this algebraic elimination.
For one observable and one hidden mode, let Ψ=kq²/2, K=g and Λ=λ>0. The memory term becomes −g²k∫₀ᵗe^(−λ(t−s))q(s)ds. Two preparations with the same q₀ but different r₀ have initial derivatives differing by gΔr₀. An experiment measuring only q₀ would therefore see apparently nonunique evolution unless it retains a history or specifies how r₀ was prepared. The exact kernel provides a systematic way to include that missing information.
DTT interprets r as a hidden generative sector and q as a completed readout. The elimination calculation itself is a standard dynamical mechanism and does not uniquely select that interpretation. Its value for the programme is diagnostic: a proposed completion map should say which variables disappear, which couplings survive, and which initial conditions remain relevant. A future autonomous source law could calculate K and Λ and thereby predict a specific memory response. Without that step, the article demonstrates a conditional mechanism rather than a unique temporal ontology.
The convolution integrates only over earlier times s≤t. Its nonlocality is temporal in the sense that a reduced state needs its past, not in the sense that later observations alter prior causes. A hidden initial state supplies another ordinary causal contribution. This distinction helps explain why eliminating variables can produce an equation that looks unfamiliar while remaining compatible with a well-posed enlarged initial-value problem. For DTT the physical question is whether the required hidden state is a generative completion variable, an ordinary environment, or an effective combination of both; the Volterra form alone does not decide that ontology.
A reduced equation can lose predictive information because several larger states share one observed state. That does not make projection alone a universal source of thermodynamic entropy. Positive hidden relaxation, a chosen initial condition or reservoir, and a condition on the reversible flow do essential work in this model. The structural functional Ψ is not automatically Boltzmann, Gibbs or von Neumann entropy. Connecting its decay to a laboratory second law requires an operational coarse-graining, thermodynamic variables and calibrated coefficients. This qualification preserves the useful mechanism while preventing a model-level monotonicity theorem from being mistaken for an unconditional derivation of time’s arrow.
A finite closed Hamiltonian system preserves information globally and can display recurrences. Replacing some of its degrees of freedom by a permanently relaxing operator requires a specified approximation, reservoir or limiting regime. The exactness claimed for hidden-variable elimination is exactness relative to the supplied coupled equations, including their positive relaxation operator. It is not an independent proof that those equations follow from a conservative temporal action. A source-based derivation should track where the discarded information resides and over which interval the reduced dissipative description remains accurate.
KEEP THE CONNECTION
Scope of the argument. Exact reduction of the stated linear hidden-sector equations; relaxation and coupling are supplied assumptions.
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.
Source register →SMONAD
Share this page, or keep a clear reference to the work you are reading.
Use this reference for the explanation on this page. “n.d.” means its first publication date is not specified; the update date is recorded separately.
