# BT7b Passive Aperture / Operator-Norm Theorem Stub

## Status

Status: accepted narrow finite-dimensional exact-linear-response/operator-norm theorem language after Pro theorem-status review `pro_bt7b_operator_norm_theorem_status_review_2026_06_10`, route `bt7b_finite_dimensional_operator_norm_theorem_language_ready`.

Claim class: finite-dimensional fixed-response amplitude/operator-norm theorem. Continuous scalar Helmholtz / Green-function derivation, Born-linearized remainder control, intensity/routed-power conversion, bench interpretation, broader boundary grammar, Layer-3 interpretation, and Tau-X mission interpretation route through later review and promotion gates.

Public boundary: this note is dimensionless and operator-level. It uses synthetic apertures, abstract measures, route labels, and proof objects only; protected layouts, RF/EM geometries, material recipes, constants, tolerances, waveform searches, value registers, and notebooks remain outside scope.

## Finite-Dimensional Synthetic Statement

Let `A_N = {1, ..., N}` be a fixed aperture grid with fixed positive weights `w_i`. Let `H_delta = C^N` with weighted norm

```text
||delta n||_{w,2}^2 = sum_i w_i |delta n_i|^2.
```

Fix before optimization:

- incident field `u_in`;
- incident-power normalization;
- wavelength/regime label;
- baseline response `u_0`;
- admissible passive/small-signal class `C_adm`;
- bounded target functional `L`;
- target normalization and target support;
- response vector or linear response operator `T_N : H_delta -> C`.

Assume the finite-dimensional response is exactly linear with the fixed response operator `T_N`, so that

```text
L(u_delta) - L(u_0) = T_N(delta n)
```

for every admissible perturbation in the stated theorem class. Equivalently, the theorem's amplitude-change object may be defined directly as `T_N(delta n)`.

For perturbations `delta n in C_adm` with `E := ||delta n||_{w,2}^2`, the finite-dimensional amplitude-improvement theorem is:

```text
|L(u_delta) - L(u_0)| = |T_N(delta n)| <= ||T_N||_{H_delta*} ||delta n||_{w,2}.
```

Therefore, if `E = ||delta n||_{w,2}^2`,

```text
amplitude_improvement(E) <= ||T_N||_{H_delta*} * sqrt(E).
```

For absolute target amplitude, use the triangle-inequality corollary:

```text
|L(u_delta)| <= |L(u_0)| + ||T_N||_{H_delta*} * sqrt(E).
```

The operator inequality is the theorem object. The square-root curve is an `L2`-budget corollary after the norm, aperture, target, and response operator have been fixed. The finite-dimensional statement is accepted in this narrow exact-linear-response scope; a Born-linearized scalar Helmholtz/Green-function version with explicit bounded response and remainder envelope is the next review target.

Weighted-norm convention: if `T_N(delta)=sum_i conj(a_i) delta_i`, then `||T_N||_{H_delta*}^2=sum_i |a_i|^2 / w_i`. If instead `T_N(delta)=<t,delta>_w`, then `||T_N||_{H_delta*}=||t||_{w,2}`. The verifier currently exercises the unweighted special case.

## Continuous Or Born-Linearized Review Target

A physical scalar-aperture theorem needs one of the following constructions:

- a continuous scalar Helmholtz/Green-function lemma with a bounded response operator `T : L2(A,mu) -> C`;
- a Born/small-signal theorem with

```text
L(u_delta) - L(u_0) = T(delta n) + R(delta n)
```

and a declared remainder envelope, for example

```text
|R(delta n)| <= beta ||delta n||_2^2
```

inside a stated validity regime;
- or a deliberately finite-dimensional theorem whose continuum limit is held for a separate review.

The next formal BT7b target is the Born-linearized scalar Helmholtz/Green-function theorem. The response operator is not assumed to be physical until the scalar regime, function spaces, admissible perturbation class, and target functional make it bounded.

## Metric And Energy Policy

Amplitude is the first theorem metric. Intensity and routed-power diagnostics require conversion lemmas or separate theorem targets.

The theorem uses:

```text
E := ||delta n||_{w,2}^2.
```

Legacy Appendix C language that uses integrated index contrast is background/audit context. An `L1`-like physical cost may feed this theorem only through a declared conversion lemma under fixed aperture, fixed weights/measure, bounded contrast, and fixed discretization. Without that conversion, the route is:

```text
legacy_l1_energy_claimed_as_l2
```

Grid refinement, changed weights, target relocation, or target shrinkage changes the theorem object and routes out of the fixed-aperture statement.

## Admissibility Notes

The Hilbert-ball optimizer in the synthetic verifier is a review harness. Physical passive perturbations may be real, sign-constrained, phase-only, lossy-only, bounded in magnitude, or otherwise restricted. The upper bound still holds over admissible subsets when `T` is bounded, but saturation/equality behavior should not be promoted unless the admissible class is the full complex Hilbert ball or a separate admissibility proof supports saturation.

Preferred first target functionals:

- normalized mode overlap;
- target-region average;
- bounded detector functional.

Point evaluation requires a declared field space/regularity and a target placement that keeps evaluation bounded.
