# RESULTS — E4-H160

All numbers below are produced by `code/h160_investigation_v2.py`; raw output in
`results/h160_results.csv` and `results/h160_run_log.txt`. Nothing here is measured.

## T1 — Geometry: illuminated bandwidth and naive floor

Parameters: pitch 2.00 µm, θ₀ = 20°, λ₀ = 684.0 nm, first order. `B_eff = c/(Nλ₀)` (normal-incidence
form; the cos θ₀ factor cancels in this geometry).

| N | D (mm) | B_eff (GHz) | floor c/2B_eff (µm) | floor/D |
|---|---|---|---|---|
| 256 | 0.51 | 1712.0 | 87.56 | 0.171 |
| 512 | 1.02 | 856.0 | 175.11 | 0.171 |
| 1024 | 2.05 | 428.0 | 350.23 | 0.171 |
| 2048 | 4.10 | 214.0 | 700.46 | 0.171 |
| 4096 | 8.19 | 107.0 | 1400.91 | 0.171 |
| 8192 | 16.38 | 53.5 | 2801.83 | 0.171 |

**Readings.**
- `B_eff ∝ 1/N` and the floor `∝ N`: the floor scales with the *number of elements*, so at fixed
  pitch it grows linearly with aperture. This confirms the dossier's (R2) and contradicts the
  original's §9 sentence.
- `floor/D = 0.171` is constant — the floor is a fixed fraction of the aperture, which is the
  characteristic signature of a dwell/envelope limit rather than a bandwidth limit.
- The original's alternative `Nλ/2` form and `c/(2B_eff)` differ only by the O(1) constant:
  `Nλ/2 = 0.171 D` here, identical to `c/(2B_eff)`. The two forms are **the same law**, which
  resolves the dossier's noted inconsistency.

## T2 — Cramér–Rao bound: point versus extended target

N = 2048 (`B_eff` = 214 GHz), 4096 samples, peak SNR 20 dB. `A(f)` is the Gaussian main-lobe
illumination envelope for a point target on the beam axis; the extended target is modelled as always
partly illuminated, so `A(f) = 1` across the whole chirp.

| B (GHz) | B/B_eff | CRB point (ps) | CRB extended (ps) | point/ext |
|---|---|---|---|---|
| 10 | 0.05 | 0.0086 | 0.0086 | 1.00 |
| 50 | 0.23 | 0.0018 | 0.0017 | 1.02 |
| 100 | 0.47 | 0.0009 | 0.0009 | 1.07 |
| 214 | 1.00 | 0.0005 | 0.0004 | 1.33 |
| 500 | 2.34 | 0.0005 | 0.0002 | 3.12 |
| 1000 | 4.67 | 0.0008 | 0.0001 | 8.76 |
| 2000 | 9.35 | 0.0011 | 0.0000 | 24.78 |

**Readings — this is the central result.**

1. **Below `B_eff` the two coincide** (ratio 1.00–1.07): every frequency sample carries signal, and
   bandwidth is the only thing that matters.
2. **The point-target bound saturates**, then *worsens* (0.0005 → 0.0011 ps between B_eff and
   9.35 B_eff). The original claim says extra bandwidth is merely useless; the calculation says it is
   mildly **harmful** at fixed sample count and fixed peak power, because the extra samples add noise
   where `|A|` is small. That is a refinement, not a refutation.
3. **The extended-target bound keeps improving as `1/B`** (0.0086 → 0.0000), reaching a factor **24.8**
   better than the point-target bound at 9.35 B_eff.

**Therefore M1 (fundamental bound) is rejected and M2 (information bound) is supported.** The floor
exists only while the target is envelope-limited. A target subtending more than the beamwidth breaks it.

## T3 — Monte-Carlo attainment check

N = 2048, 120 trials per point, peak SNR 20 dB, target at 1 m (inside `c/(2Δf)` at every B tested).

| B (GHz) | estimator | RMSE (ps) | CRB (ps) | RMSE/CRB |
|---|---|---|---|---|
| 50 | matched | 0.0134 | 0.0175 | 0.76 |
| 50 | unweighted | 0.0114 | 0.0175 | 0.65 |
| 100 | matched | 0.0069 | 0.0092 | 0.75 |
| 100 | unweighted | 0.0059 | 0.0092 | 0.65 |

**This test is only partially successful, and the failure must be stated.**

- Both estimators track the bound's *trend*: halving the CRB between 50 and 100 GHz produces a
  corresponding halving of RMSE.
- But **RMSE/CRB ≈ 0.75 is below 1**, which a valid bound cannot permit. Either the CRB normalization
  is too large by a constant (most likely: a factor in how the complex noise variance enters `J`), or
  the interpolated peak estimator has a small bias. I did **not** resolve which.
- **Consequence:** the *scaling* in T2 is unaffected (it is a ratio of bounds computed identically), but
  the absolute claim "this estimator attains the bound" is **not established**.
- At B ≥ 500 GHz the estimate wrapped on the delay grid (error ≈ `1/Δf`), so those points were removed
  rather than reported.

## Summary of what the numbers support

| Claim | Verdict | Evidence |
|---|---|---|
| `B_eff = c/(Nλ)` | Confirmed by derivation | T1 |
| Floor grows with aperture | Confirmed | T1, `floor/D` constant |
| `Nλ/2` and `c/(2B_eff)` are the same law | Confirmed | T1 |
| Floor is fundamental | **Rejected** | T2, 24.8× at extended target |
| Extra bandwidth useless above B_eff | **Corrected**: mildly harmful for a point target | T2 |
| Estimator attains the bound | **Not established** | T3, ratio 0.75 |