Introduction

1 Introduction

An error estimate for a quantum amplitude becomes a physical transport estimate only after the relevant derivatives, source coordinates and current constitution have been retained. This distinction is especially important for a coherently driven continuum. A small omitted occupation does not eliminate its interference current; a small omitted forcing does not eliminate response generated by feedback; and an accurately resolved electronic transition does not determine the current of a quantum source. These are analytic questions even before an experiment or a measurement record is specified.

We develop a common set of quantitative interfaces for these questions. The complete wave remains a vector on all admitted positions and internal degrees of freedom. Its error is controlled in a form that dominates the physical kinetic derivatives, then converted to the full coherent current. The response may be driven by an unbounded coupling, contain genuine bound poles and a continuum, and receive a time-dependent vector input from another interacting block. We retain the input of the coupled evolution throughout.

There are three main results. First, an ordered inverse lift converts temporally regular form-dual forcing into a first-form response under absolutely continuous common-form work bounds. It avoids a weighted inverse-square-root derivative that is not uniformly controlled in dimension. Second, an operator spectral-window estimate controls a varying Hilbert-space input. The proof uses interval means, spectral derivatives and orthogonal output sectors; it does not replace the vector input by a scalar supremum. Ordered resolvent identities then transfer genuinely noncommuting physical-energy weights, while explicit far-frequency dressings retain both endpoints. Third, a compact-profile electron–oscillator model supplies an analytic Coulomb spectral calculation, an actual-source inverse bound and a complete finite-time current calculation with source feedback.

Several complementary results make these interfaces usable. A spatial force estimate retains covariant strain, curvature and first-order transport. A temporal common-domain estimate instead accepts a residual without a spatial form norm and retains its right clock. An exact positive quadratic invariant supplies a comparison form even through an inverted trap interval. The appendices preserve a spatial forcing compression with arbitrary source coefficients and the causal high-sector feedback that such compression must not erase. Each has its own input hypotheses; choosing the smallest coefficient from incompatible estimates would not be a valid composition.

1.1 Relation to established methods and prior work

The functional-analytic ingredients have substantial precedents. Kato's smoothing and factorized-resolvent framework [6] and its inhomogeneous development [7] motivate the relation between response and resolvents. Our fixed positive imaginary part is an earned finite-resolution coefficient, not a claim of a uniform limiting-absorption estimate. Common-form propagation belongs to the classical evolution theory of Kisyński [8]; we give the argument for the precise absolutely continuous relative-work assumptions used here. Common operator-domain regularity, used in the separate temporal graph theorem, is discussed in [9]. We keep these two kinds of domains distinct.

The oscillator invariant is classical [12, 13]; its contribution here is an explicit physical-current metric with the centroid and mass factors preserved. The Coulomb functions and their normalization are standard [10]; the spectral calculation is included to make the forcing measure and source-weighted bounds reproducible. Isolated-band constructions such as [11] require their own symbol classes and gap hypotheses. Neither a finite column corrector nor a frozen internal gap is asserted here to supply such a band reduction of the complete position/source parent.

The current October research editions of the measurement programme include equilibrium and nonequilibrium record constructions [4, 5]. Those works ask which original laws yield outcome weights and whether a distinct receiver preserves an earlier actual event. Their apparatus constants and preparation premises are not inputs to the estimates proved here. The new companion manuscripts treat retained-archive preparation (P1) [1], effective repeated records (P2) [2], and reference-compatible flows and whole-history stability (P3) [3]. The present response and current estimates enter such arguments only when their model, entrance and flow premises agree. The proofs below are independent of those companions; the specific flow application in Section 11.2 uses the revised P3 theorem explicitly.

These new manuscripts remain distinct from both the current October website editions and the preserved September editions. No integration into one apparatus is asserted. In particular, assumptions of the pilot-medium and massive-configuration constitutions are not interchangeable. The contribution here is the proved combination of response, derivative and complete-current estimates under stated hypotheses, rather than a priority claim for their established individual ingredients.

1.2 Conventions and the structure of the results

We use $i\partial_t\Psi=H\Psi$ with $H$ in rate units, unless reduced-mass atomic units are explicitly declared. Hilbert-space norms include every physical source coordinate; a finite passive internal reference can be included by tensoring the identity. A source wave or its quantum covariance is not an assumption about an actual configuration law. Form inequalities always refer to the complete stated parent.

Section 2 first supplies the current and coordinate conversion. Sections 3–7 prove the form-dual, spectral-window and ordered-resolvent results. Section 8 develops the alternative spatial-force, invariant and temporal-domain estimates. Sections 9–11 give the hydrogenic example. Appendices A and B preserve the full-source spatial compression and feedback mechanisms.

The hydrogenic example has $\Omega=.01$, $|g|\leq10^{-6}$ and a radius-$40$ compact dipole profile in its specified units. Its low-energy physical resolvent sandwich is below $4.05\times10^{-9}$ under the stated source weight. A separate all-frequency, stronger-source-weight estimate gives, for the declared entrance and $T=100$, integrated electron and source current differences below $.046$ and $.004$ in their respective coordinate lengths. The comparison uses the actual projected wave, including feedback. It is not a second independently flowing wave, and these numbers are not probabilities.