# H160 correction: complex-noise information convention and a bounded independent check

Status: exact likelihood derivation under the stated single-target model, plus an independent seeded local Monte Carlo check. No physical data, experimental result or novelty claim. The supplied research script was read but not executed or imported.

## 1. The missing factor is identified

The source declares circular complex noise with sigma² = E|n|² and actually generates real and imaginary components with variance sigma²/2. Its density is proportional to exp(-|y-mu|²/sigma²). Consequently, for real parameters,

    J_ab = 2/sigma² Re sum conj(d mu/d theta_a) (d mu/d theta_b).

For known amplitude/envelope and known constant phase, mu=A exp(-i 2pi f tau), giving

    J_tau = 2(2pi)² sum f² A² / sigma².

The source formula omits the leading 2. Its stated delay standard-deviation bound is too large by sqrt(2). This is a convention mismatch, not an estimator beating the corresponding correctly normalized information bound.

The corrected known-phase standard-deviation values at 50 and 100 GHz are 0.0123832921 ps and 0.00649843675 ps. The old values in the supplied CSV are 0.0175126196 ps and 0.00919017739 ps. Corrected values for the full source T2 sweep appear in `corrected_t2.csv`.

## 2. The source estimator also has a nuisance-phase interpretation

Maximizing the magnitude of a complex correlation discards its phase. That is consistent with an unknown constant target phase, rather than the strictly known-phase likelihood used in the simple bound. The Fisher Schur complement then gives

    J_tau|phi = 2(2pi)²/sigma² sum A² (f-fbar)²,
    fbar = sum A² f / sum A².

For the source's almost centered frequency samples, the distinction is negligible. It matters greatly for an offset grid or for treating optical carrier phase as known. A numerical offset control verified this distinction: known-phase precision improves when the whole frequency grid shifts; unknown-phase information is invariant to that shift. No claim that an optical carrier provides known absolute phase is justified by the source model alone.

## 3. Independent Monte Carlo

`PLAN.md` specified the model and outcomes before execution. `independent_check.py` is a new implementation using direct likelihood derivatives, not the source's FFT/interpolation algorithm. Seed 16020261101; 2000 trials per cell; 512 samples; peak SNR 100; complex noise variance 0.01. The delay coordinate is dimensionless B*tau. The Gaussian envelope uses the source shape, with B_eff/B=2.14. All cells use the same noise draws; no failures or trials were removed.

| Frequency offset (bandwidths) | Phase | RMSE / corrected SD bound | Bias (B*tau) | Numerical failures |
|---|---|---:|---:|---:|
| 0 | known | 0.99633 | -0.00008297 | 0 |
| 0 | nuisance | 0.99633 | -0.00008292 | 0 |
| 1 | known | 1.02519 | -0.00001431 | 0 |
| 1 | nuisance | 1.00405 | -0.00011595 | 0 |

Observed noise power was 0.01001387; real and imaginary variances were 0.00500681 and 0.00500706. Maximum final Newton steps were below 5e-17. All four ratios are within the predefined descriptive 0.94–1.06 tolerance. The centered case is approximately 0.7045 of the incorrectly normalized source bound, reproducing the apparent sub-bound pattern without a paradox.

This is a **local high-SNR likelihood check**: Newton iteration starts 0.01 from the true delay. It does not validate global target acquisition, phase ambiguity, alias handling, delay-grid interpolation, low-SNR failures, multiple targets or actual hardware. Finite Monte Carlo RMSE is itself random; a finite sample ratio below one does not establish a violated population CRB. A biased estimator also needs the appropriate biased-estimator bound. The source's categorical statement that any observed ratio below one proves an error was too strong.

The source's previously reported matched ratios become 1.0790 and 1.0563 after the sqrt(2) correction; unweighted ratios become approximately 0.9215 and 0.9131. These 120-trial estimates are not newly reproduced here. Their exact bias/interpolation/random-sampling behavior remains unverified; the new local MLE is a separate estimator, not an endorsement of the source FFT estimator.

## 4. A separate transcription problem

Every numeric precision value in the source `RESULTS.md` T2 table is about ten times smaller than its own CSV and current code imply. Example: at 50 GHz the Markdown says point 0.0018 ps and flat 0.0017 ps, while the CSV says 0.0175126 ps and 0.0172290 ps. At 2000 GHz Markdown rounds the flat-target precision to 0.0000 ps; the CSV value is 0.000430726 ps. The table must be replaced from the corrected CSV, with enough significant figures to avoid zero. Dimensionless ratios and the 24.78 ratio do not change from this decimal-point error or the shared sqrt(2) correction.

## 5. Scope of what remains unresolved

The computed CRB is a bound on variance of an individual delay estimate under the defined signal/noise assumptions. It is **not a two-target resolution criterion**, nor an SNR-independent Rayleigh range-resolution floor. The geometric scale c/(2 B_eff) and the statistical delay precision are different quantities.

The flat-envelope comparison is still a **single coherent delay signal with A(f)=1**. A physical angularly extended target generally produces a superposition of angles/delays, unknown reflectivities and speckle. Thus that idealized calculation does not by itself show a real extended target evades an imaging resolution limit.

The worsening point-envelope bound at very large sweep bandwidth follows the specified experiment: fixed sample count, fixed peak noise/SNR and fewer samples inside the envelope as the sweep expands. Other sampling, dwell-time or power budgets change the conclusion. Absolute range accuracy and systematic bias were not investigated.

The derivation assumes a known envelope and amplitude. Other unknown nuisance parameters require a larger Fisher matrix and may reduce information. Prior-art novelty remains unverified; no literature was retrieved or assessed in this bounded check.

## Reproduction and retained artifacts

Run this independently authored script with Python and NumPy (tested with NumPy 2.5.4):

    python independent_check.py

Artifacts: preregistered `PLAN.md`, `independent_check.py`, `monte_carlo.csv`, `corrected_t2.csv`, `trial_errors.npz`, `results.json`, `RUN_LOG.txt`. Trial error arrays retain all four cells for recomputing bias and RMSE. Runtime was under one second in the available environment. This report supplements the unchanged supplied archive; it does not silently rewrite it.