Exact versus approximate reduction

Exact versus approximate reduction

It is worth collecting, in one place, what has been proved exactly and where an approximation would have to enter.

Exact. Theorem 2.3 (the quotient and the closed-range criterion), Proposition 3.2 and Corollary 3.3 (functional descent), Theorem 3.4 (stable observability of a linear functional, i.e. boundedness of the descended functional in the inherited norm iff $a\in\operatorname{Ran}K^{*}$), Theorem 4.4 (the truncation error, which is an exact evaluation of an approximation’s cost), Theorem 5.2 (descent iff invariance), Theorem 6.2 (the projected identity), Theorem 6.4 (closure iff $PLQ=0$), Theorems 7.4 and 7.6 (minimal completion and Volterra closure), Proposition 7.7 (finite computation of $N_{Q}$), Theorem 8.2 (nonlinear projectability), Theorem 9.2 (Schur complement), Proposition 10.2 and Theorem 10.5 (envelope identity and total conservation). Apart from the explicitly identified rank-$r$ approximant in the Eckart–Young–Mirsky result, these identities introduce no approximation, limiting procedure, ensemble, or small parameter; the theorem itself evaluates the truncation error exactly.

Where approximation would enter. Four places, and it is important that they are four and not one.

1. Rank truncation. Replacing $K$ by $K_{r}$ costs exactly $\sigma_{r+1}$ in operator norm (Theorem 4.4). The cost is quantified, the reduction is not exact, and it is completely unrelated to $\ker K$.

2. Regularization of unstable inversion. When $\operatorname{Ran}K$ is not closed, recovering the source from the readout requires a regularization scheme (Tikhonov, spectral cutoff), with a bias–variance trade-off governed by the spectral behaviour of $K^{*}K$ near $0$ [4]. When $K$ is compact, that behaviour is encoded in the discrete sequence $\sigma_{n}\to0$ and the trade-off is classically described by the rate of decay of the $\sigma_{n}$; for a general bounded operator with nonclosed range there need be no discrete singular sequence at all, $K^{*}K$ may have continuous spectrum accumulating at $0$, and it is the spectral measure of $K^{*}K$ near the origin, not a sequence, that governs the scheme. This is in any case an approximation of the inverse, not of $K$.

3. Truncating the memory. Approximating (6.8) by an autonomous equation, dropping $\eta_{z_{0}}$ and $\mathcal{K}$, or replacing $\mathcal{K}$ by a delta function (the Markov approximation), is legitimate only under an additional hypothesis. The property actually needed is exponential stability of the unresolved propagator: constants $M\ge1$, $\alpha>0$ with $\lVert e^{\tau QLQ}\rVert_{\mathrm{op}}\le M e^{-\alpha\tau}$, so that $\lVert\mathcal{K}(\tau)\rVert\le M\lVert PLQ\rVert\lVert QLP\rVert e^{-\alpha\tau}$ and the convolution has an effective memory time of order $1/\alpha$; this is then normally accompanied by a quantitative separation of timescales, $\alpha^{-1}$ small compared with the timescale on which $x$ varies (governed by $\lVert PLP\rVert$ and the coupling). These are prerequisites, not a proof. Exponential decay of $e^{\tau QLQ}$ alone does not establish that replacing $\mathcal{K}$ by a delta function is controlled: that step additionally requires quantitative control of the kernel’s moments, at minimum $\int_{0}^{\infty}\lVert\mathcal{K}(\tau)\rVert\,d\tau$ and $\int_{0}^{\infty}\tau\lVert\mathcal{K}(\tau)\rVert\,d\tau$, the second being what multiplies the error made in replacing $x(t-\tau)$ by $x(t)$, together with a bound on the variation of $x$ over the memory timescale, and then an explicit error estimate or a limiting argument ($\alpha\to\infty$ at fixed $\int\mathcal{K}$, say) carried out in the concrete model at hand. We supply none of this here, and it should not be assumed to be automatic. Note also that a large norm $\lVert QLQ\rVert$ implies nothing of the sort by itself, the norm bounds $\lVert e^{\tau QLQ}\rVert\le e^{\tau\lVert QLQ\rVert}$ from above only, and a large-norm generator may have purely oscillatory or even growing propagator. What matters is the location of the spectrum (or, in infinite dimensions, the growth bound of the semigroup), not the size of the generator. We have made no such hypothesis, and Theorem 6.4 shows that without $PLQ=0$ such a truncation is never exact.

4. Loop corrections. (9.4) and (10.3) are classical stationary eliminations (Remark 9.5). A quantum treatment produces determinant and loop terms.

The physical reading. What Sections 5–7 establish is this. A model may have a completely coherent observable theory, an honest Hilbert space of observable states, an honest algebra of observable functionals, an honest exact quotient, and still fail to be autonomous on that observable state. When it does fail, the exact projected law records precisely three things beyond the naive autonomous term $PLP\,x$: dependence on unresolved initial data (the term $\eta_{z_{0}}$), history dependence through the memory kernel $\mathcal{K}$, and, if one insists on restoring a genuine initial-value problem, the additional state information $E_{\mathrm{dyn}}$ needed to close it. Eliminating a field sector instead of a state sector produces the same structure in field-theoretic guise: an effective operator $A-BC^{-1}B^{*}$, whose correction may be nonlocal when the composed operator $BC^{-1}B^{*}$ retains the nonlocal Green-operator directions, and a covariant effective stress-energy $T^{\mathrm{eff}}$ defined by variation.

None of this is evidence that our universe has hidden source variables, memory, or any particular effective correction. The theorems are conditional: if a model declares a source/readout architecture, then these are the necessary and sufficient conditions for descent, and these are the exact reduction identities when descent fails. A memory kernel is not a physical discovery; it is a formula. It becomes physics only when the model that produced $L$ and $P$ is itself physics.