# BT7b Born-Linearized Scalar Helmholtz / Green-Function Theorem V1

Date: 2026-07-11

Status: post-review local theorem candidate; fixed-point/PDE status, bounded-domain Helmholtz instantiation, and route fixtures patched after external model formal triage.

Claim class: `bt7b_born_helmholtz_v1_theorem_candidate`.

Controlling predecessor: the accepted finite-dimensional fixed-response amplitude/operator-norm theorem in `open-theorem-working-notes/bt7b-passive-aperture-theorem-stub.md`.

## Purpose

This note constructs the first physical operator behind BT7b. It derives a bounded first-Born target-amplitude functional from a fixed scalar Helmholtz baseline and pairs it with an explicit quadratic remainder envelope.

The result is an amplitude theorem over a fixed aperture, fixed incident field, fixed baseline resolvent, fixed target functional, and fixed `L2` Helmholtz-contrast budget. Intensity, routed power, resonant storage, active gain, nonlinear response, near-field/evanescent classes, changing apertures, and changing targets use separate routes.

## Fixed Objects

Freeze before selecting or scoring a perturbation:

- a finite-measure aperture `(A, mu)`;
- a scalar-wave regime and fixed frequency/wavenumber label;
- a baseline scalar Helmholtz problem and baseline field `u_0`;
- a Banach field space `Y`;
- a bounded baseline Green/resolvent map `G_0 : L2(A,mu) -> Y`;
- the a.e. restriction map `r_A : Y -> L_infinity(A,mu)`, bounded in the declared field norm;
- a bounded complex target functional `L : Y -> C`;
- an admissible contrast class `C_adm subset L2(A,mu)`;
- the perturbation variable, norm, aperture support, target support, and incident normalization;
- a small-signal radius `rho > 0`.

For the abstract theorem, define the perturbed field by the fixed-point equation

```text
u_eta := (I-B_eta)^(-1)u_0,
B_eta u := G_0(eta * r_A u).
```

Equivalently, `u_eta` is the unique solution in `Y` of

```text
u_eta = u_0 + G_0(eta * r_A u_eta).
```

Here `eta` is the scalar Helmholtz contrast perturbation supported on `A`.
Identification of this fixed point with the solution of a particular
Helmholtz boundary-value or outgoing problem requires the corresponding
baseline-resolvent equivalence hypothesis. The bounded-domain instantiation
below supplies one such realization. Signs can be absorbed into `eta` or
`G_0` when the admissible class is closed under the declared sign convention;
the norm bounds are unchanged.

Define

```text
C_G := ||G_0||_{L2(A)->Y},
C_A := ||r_A||_{Y->L_infinity(A)},
gamma := C_G C_A.
```

Assume

```text
gamma rho < 1.
```

## Lemma 0: A Concrete Bounded-Domain Helmholtz Instantiation

Let `D subset R^d`, `d in {2,3}`, be a bounded `C^{1,1}` domain, and let
`A subset D` be measurable with finite measure. Let the fixed baseline
Dirichlet Helmholtz operator, for a frozen `n_0 in L_infinity(D)`, be

```text
A_0 u := -Delta u - k^2 n_0^2 u,
A_0 : H^2(D) intersection H_0^1(D) -> L2(D).
```

Assume the fixed frequency is non-resonant so that `A_0` is bijective with bounded
inverse `R_0`. Let `E_A : L2(A) -> L2(D)` be extension by zero and set

```text
Y   := H^2(D) intersection H_0^1(D),
G_0 := R_0 E_A,
r_A := a.e. restriction from Y to A.
```

Then `G_0 : L2(A) -> Y` is bounded. Sobolev embedding on a bounded
`C^{1,1}` domain in `d <= 3` gives a bounded map

```text
Y -> L_infinity(D),
```

so `r_A : Y -> L_infinity(A)` is bounded. For any baseline source
`f in L2(D)`, let `u_0=R_0 f`. If `gamma ||eta||_2 < 1`, the fixed point
`u_eta=(I-B_eta)^(-1)u_0` is equivalent to the unique strong solution of

```text
A_0 u_eta = f + E_A(eta * r_A u_eta)
```

with the declared Dirichlet boundary condition.

### Proof

Boundedness of `G_0` follows from bounded invertibility of `A_0` and bounded
extension by zero. The Sobolev embedding gives the restriction bound. Since
`eta in L2(A)` and `r_A u in L_infinity(A)`, the product lies in `L2(A)`.
Applying `A_0` to the fixed-point equation gives the displayed Helmholtz
problem; applying `R_0` to that problem gives the fixed-point equation.
Lemma 2 supplies uniqueness.

This lemma records one non-vacuous Helmholtz realization. Exterior/outgoing,
boundary-target, and dimension `d >= 4` versions require their own localized
resolvent and regularity hypotheses.

## Lemma 1: Contrast Multiplication And Green Map

For `eta in L2(A,mu)`, define

```text
B_eta u := G_0(eta * r_A u).
```

Then `B_eta : Y -> Y` is bounded and

```text
||B_eta||_{Y->Y} <= gamma ||eta||_2.
```

### Proof

For every `u in Y`,

```text
||eta * r_A u||_2
  <= ||eta||_2 ||r_A u||_infinity
  <= C_A ||eta||_2 ||u||_Y.
```

Applying the Green-map bound gives

```text
||B_eta u||_Y
  <= C_G C_A ||eta||_2 ||u||_Y.
```

The product is measurable and belongs to `L2(A)` by the multiplication pairing
`L2(A) times L_infinity(A) -> L2(A)`.

## Lemma 2: Small-Contrast Well-Posedness

If `||eta||_2 <= rho` and `gamma rho < 1`, then `I-B_eta` is invertible on `Y`,

```text
u_eta = (I-B_eta)^(-1) u_0,
```

and

```text
||(I-B_eta)^(-1)|| <= 1 / (1-gamma ||eta||_2)
                    <= 1 / (1-gamma rho).
```

### Proof

Lemma 1 gives `||B_eta|| < 1`. The Neumann series converges in operator norm.

## Lemma 3: Bounded First-Born Target Operator

Define the first-Born target-amplitude response

```text
T(eta) := L(B_eta u_0)
        = L(G_0(eta * r_A u_0)).
```

Then `T : L2(A,mu) -> C` is bounded and

```text
||T|| <= ||L|| C_G ||r_A u_0||_infinity
      <= ||L|| C_G C_A ||u_0||_Y.
```

Because `L2(A,mu)` is a Hilbert space, there is a unique Riesz kernel `K in L2(A,mu)`. Using the convention that the inner product is linear in its second argument,

```text
T(eta) = integral_A conjugate(K(x)) eta(x) dmu(x),
||K||_2 = ||T||.
```

When `G_0` and `L` admit an integral or adjoint-field representation, `K` is the corresponding baseline-field times target/adjoint Green-field kernel. The theorem requires only the bounded operator and its Riesz representative.

## Theorem: Born Response With Explicit Remainder

Under the fixed objects and assumptions above, every admissible `eta` with `||eta||_2 <= rho` has

```text
L(u_eta) - L(u_0) = T(eta) + R(eta),
```

where

```text
|T(eta)| <= ||K||_2 ||eta||_2
```

and

```text
|R(eta)|
  <= ||L|| ||u_0||_Y
     * gamma^2 ||eta||_2^2 / (1-gamma ||eta||_2)
  <= beta_rho ||eta||_2^2,
```

with

```text
beta_rho := ||L|| ||u_0||_Y gamma^2 / (1-gamma rho).
```

The first remainder bound is pointwise in `||eta||_2`; `beta_rho` is the
uniform worst-case constant over the declared contrast ball.

Consequently, for the contrast budget

```text
E := ||eta||_2^2 <= rho^2,
```

the target-amplitude change obeys

```text
|L(u_eta)-L(u_0)| <= ||K||_2 sqrt(E) + beta_rho E.
```

For absolute target amplitude,

```text
|L(u_eta)| <= |L(u_0)| + ||K||_2 sqrt(E) + beta_rho E.
```

### Proof

The fixed-point identity gives

```text
u_eta-u_0
  = B_eta(I-B_eta)^(-1)u_0
  = B_eta u_0 + B_eta^2(I-B_eta)^(-1)u_0.
```

Apply `L`. The first term is `T(eta)`. The second is `R(eta)`. Lemmas 1 and 2 give

```text
|R(eta)|
  <= ||L|| ||B_eta||^2 ||(I-B_eta)^(-1)|| ||u_0||_Y
  <= ||L|| ||u_0||_Y
     * gamma^2 ||eta||_2^2 / (1-gamma ||eta||_2).
```

The Riesz-kernel bound is Cauchy-Schwarz. Substituting `E=||eta||_2^2` proves the budget form.

## Perturbation Variable And Refractive-Index Conversion

The theorem's continuum `L2` variable is the Helmholtz contrast `eta`, because it enters the fixed-point equation linearly.

For a refractive-index perturbation `delta n`, a common scalar model gives

```text
eta(delta n) = k^2[(n_0+delta n)^2-n_0^2]
             = 2 k^2 n_0 delta n + k^2 (delta n)^2.
```

The linear index response uses `2 k^2 n_0 delta n`. The quadratic conversion term requires a declared `L4`, `L_infinity`, finite-dimensional, or other admissible-class bound. Continuum `L2` control alone does not imply a quadratic `L2` bound for `(delta n)^2`.

Therefore:

```text
refractive_index_without_conversion_certificate
  -> bt7b_contrast_conversion_defer
```

A later index-based corollary may enter only after its conversion constant and admissible class are frozen.

For example, if `n_0 in L_infinity(A)` and the admissible class freezes
`||delta n||_infinity <= M`, then

```text
||eta(delta n)||_2
  <= k^2 (2||n_0||_infinity + M) ||delta n||_2.
```

This supplies a valid conversion constant and a contrast-ball radius for an
index corollary. An `L4` certificate controlling `||(delta n)^2||_2` is another
valid route.

## Physical Admissibility

The analytic upper bound holds on any subset of the `L2` ball. Physical passivity must still be declared through the admissible class. Examples include real-valued contrast, sign constraints, bounded loss, fixed support, bounded magnitude, or fabrication classes.

The theorem does not assert optimizer saturation for a restricted physical class. Equality in the Hilbert-ball bound does not imply that a passive material can realize the Riesz optimizer.

## Target Functional Policy

Preferred bounded targets are:

- normalized mode overlap;
- target-region average;
- bounded detector functional;
- another declared element of `Y*`.

Point evaluation requires boundedness in the declared `Y` norm and target
placement inside the region where the declared regularity holds, excluding
singular, boundary-undefined, or undeclared near-field behavior. Otherwise:

```text
unbounded_point_target -> bt7b_unbounded_target_functional_defer
```

## Route Surface

| Condition | Route |
|---|---|
| All fixed-object, boundedness, small-contrast, linear-response, and remainder checks pass | `bt7b_born_helmholtz_certificate_pass` |
| `gamma rho >= 1` or tested contrast exceeds `rho` | `bt7b_born_validity_envelope_defer` |
| Target functional is unbounded in the declared field space | `bt7b_unbounded_target_functional_defer` |
| Baseline resolvent/Green map is unbounded, nonexistent, or pole proximity lacks a reviewed finite bound | `bt7b_resonant_or_pole_proximity_route` |
| Aperture, incident normalization, target, or baseline object changes | `bt7b_changed_object_no_go` |
| Support or norm changes | `bt7b_support_or_norm_mismatch_no_go` |
| Hidden active gain or source enters | `bt7b_hidden_gain_or_source_no_go` |
| Field-dependent/nonlinear response enters the theorem class | `bt7b_non_born_or_nonlinear_route` |
| Index perturbation lacks a contrast-conversion certificate | `bt7b_contrast_conversion_defer` |
| First-order response exceeds its operator certificate | `bt7b_linear_certificate_fail` |
| Remainder exceeds the declared quadratic envelope, or the total first-order-plus-remainder envelope fails after the linear certificate has passed | `bt7b_remainder_certificate_fail` |
| Intensity or routed power is reported without a conversion theorem | `bt7b_amplitude_only_conversion_defer` |
| Near-field/evanescent behavior lies outside the declared scalar regime | `bt7b_near_field_or_evanescent_route` |
| Required constants or declarations are incomplete | `bt7b_certificate_incomplete_defer` |

A reviewed finite `C_G` remains admissible even when large. The theorem and
certificate are frozen per `(baseline, frequency)` pair; no uniformity across
frequency or continuation through a pole is claimed. A large `C_G` shrinks the
allowed radius through `gamma rho < 1` and enlarges the remainder constant.

## Scope Boundary

This V1 object establishes a scalar contrast-space fixed-point theorem with a
derived remainder envelope and records one bounded-domain Helmholtz
instantiation. It does not extend that instantiation to arbitrary outgoing
problems, convert amplitude to intensity or routed power, certify a hardware
geometry, cover unresolved resonant storage or nonlinear media, prove physical
optimizer attainability, or provide Layer-3 or Tau-X status.

Next gate:

```text
bt7b_born_helmholtz_v1_postreview_patch_confirmation
```
