Paper 05Canonical2026-07-15

Observable Quotients and Exact Projected Dynamics: Closure, Memory, Minimal Dynamical Completion, and Effective Field Operators

When observable dynamics close autonomously, and the exact price when they do not

DOI: 10.5281/zenodo.21371251 View on Zenodo

Authority role

Develops the observable and dynamical layer: an induced observable evolution exists exactly when the dynamics preserve the readout kernel; otherwise the exact projected law carries an unresolved-initial-state term and a memory kernel, with a minimal dynamical completion measuring exactly what must be restored.

Summary

The dynamical layer of the framework. A bounded observation operator always yields an exact observable quotient; the paper keeps sharply apart exact loss, unstable inversion, controlled low-rank approximation, and failure of autonomous dynamics. An induced observable evolution exists if and only if the source semigroup preserves the readout kernel; when it fails, the exact projected equation carries a deterministic unresolved-initial-state term and an exact memory kernel — no stochastic, Markovian, or timescale approximation anywhere. The minimal dynamical completion is constructed and computed by a Kalman-type observability rank in finite dimensions, and sector elimination yields Schur-complement effective operators and a covariant effective stress-energy with total — not sectorwise — conservation.

Notes

Reading notes

The dynamical layer. A bounded observation operator $K: \mathcal{H}_{\mathrm{src}} \to \mathcal{H}_{\mathrm{obs}}$ always produces an exact observable quotient $\mathcal{H}_K = \mathcal{H}_{\mathrm{src}}/\ker K$ , canonically bijective onto $\mathrm{Ran}\,K$ — with bounded inverse in the inherited norm exactly when the range is closed. The paper insists on keeping four phenomena apart: (D1) exact loss ( $\ker K \neq \{0\}$ ), (D2) unstable inversion (nonclosed range), (D3) controlled low-rank approximation ( $\lVert K - K_r \rVert = \sigma_{r+1}$ ), and (D4) failure of autonomous dynamics. Compactness supplies (D3) and by itself implies neither (D1) nor (D2); (D4) implies (D1) but not conversely.

Theorem — Semigroup descent and autonomous closure

An induced evolution on the observable state exists if and only if the source semigroup preserves the readout kernel: $U(t)\ker K \subseteq \ker K$ . For an orthogonal resolved/unresolved splitting with bounded generator $L$ , autonomous first-order closure valid for every source initial state is equivalent to $PLQ = 0$ .

When closure fails, the exact projected equation (a Mori–Zwanzig-type identity, derived purely by variation of constants) carries two extra terms: a deterministic unresolved-initial-state term $\eta_{z_0}(t)$ — explicitly not a stochastic force — and an exact memory kernel $\mathcal{K}(\tau) = PLQ\,e^{\tau QLQ}QLP$ — explicitly not a friction coefficient. No ensemble, Markov, or timescale approximation appears anywhere.

Theorem — Minimal dynamical completion

The unresolved response map descends to an injective map on $E_{\mathrm{dyn}} = \mathcal{H}_Q/N_Q$ , the coarsest extension of the resolved state determining every resolved response; every sufficient extension maps onto it. In finite dimensions its size is the rank of a Kalman-type observability matrix. Minimality is relative to the declared target — directions in $N_Q$ are not "physically absent" in any absolute sense.

The same architecture is carried into field theory: eliminating a quadratic sector yields the exact Schur-complement effective operator $A - BC^{-1}B^*$ (possibly, but not automatically, nonlocal), and stationary elimination in a diffeomorphism-invariant action yields a covariant effective stress-energy defined as a genuine variational response, with total — not sectorwise — covariant conservation and an explicit exchange current between sectors.

Cite this paper

Rodgers, Jeremy. (2026). Observable Quotients and Exact Projected Dynamics: Closure, Memory, Minimal Dynamical Completion, and Effective Field Operators. https://doi.org/10.5281/zenodo.21371251