Corrections and independent checks

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# H034: numerical correction and physical scope

1 November 2026. Publisher arithmetic check, not a real-data validation.

Predefined check: recompute the stated formula 0.18 P²/(κ Rc³), with P in eV, κ in eV/ų, Rc in Å, converting eV to meV; compare all supplied P1 CSV rows to 1e-9 meV tolerance. Compare the six rounded prose-table values to their identical inputs.

All12 CSV rows pass. The source code also implements the stated coefficient0.18. The six prose table values, rather than the CSV or formula, need correction:

| P (eV) | Rc (Å) | Published prose (meV) | Corrected (meV) |
|---|---|---|---|
|1|4|11.25|2.8125|
|2|4|45.00|11.25|
|3|4|101.25|25.3125|
|6|4|405.00|101.25|
|3|6|30.00|7.5|
|3|8|12.66|3.1640625|

Relative ratios and the formula's quadratic/inverse-cube scaling remain unchanged. Original files remain at ../nov1-results-20261101/. The supplied synthetic protocol was not independently rerun in this check.

Physical scope: integrating an assumed far-field elastic-energy density can give a tail proportional to P²/(κ Rc³). This alone does not establish a universal MLIP prediction-error law. A local atomic energy model can generate a long-range relaxed elastic field through local interactions; its cutoff is not automatically a cutoff on the total elastic energy present in a relaxed configuration. Architecture, training, boundary conditions and target definition matter. The coefficient0.18 is supplied, not independently derived. The claimed architecture-independent exponents and predictive advantage remain untested.

No compatible real-data pilot was assembled in this bounded correction. The suggested SrTiO3/germanium datasets require separate checks of exact records, references, configurations, stresses/dipoles and reuse terms before analysis.

Reproduce the arithmetic using h034_check.py with the original release sources at the relative workspace path specified in that script. Machine-readable values: h034-arithmetic.json.

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# H160 primary-source comparison

Checked 1 November2026: Kandil, Dahlem and Bogaerts, System trade-offs in a swept source FMCW LiDAR with dispersive OPA beam steering, SPIE12890,2024, DOI10.1117/12.3002492.

Institutional record: https://www.photonics.intec.ugent.be/publications/publications.asp?ID=5126
Full paper: https://www.photonics.intec.ugent.be/download/pub_5126.pdf

Sections3.1.2 and3.2.1–3.2.2 were read. The paper separates range resolution from range-estimation precision. It discusses dividing wavelength sweeps into frequency slots and the coupling between angular-pixel requirements and usable FMCW ranging bandwidth, including trade-offs caused by array sidelobes and design choices. Thus the broad scanning/ranging coupling is established prior art and cannot be claimed here as a new discovery.

This limited comparison does not establish that the paper contains the exact H160 formula or its idealized point-versus-flat-envelope Fisher-information comparison. Neither novelty nor absence of novelty for that narrower calculation is certified. A flat spectral envelope is an assumed signal model, not a measured extended target or a multi-delay scattering calculation.

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# Reproduction and scope

Python3 with NumPy is required for the H160 independent check; H034 uses the Python standard library only. No software from the supplied Desktop package was executed in this correction.

Unzip reproduction.zip. Obtain the unchanged original reports ZIP from https://incoprea.com/research-library/nov1-results-20261101/originals.zip and extract it into a separate directory.

Run:

    python3 h034_check.py --source /absolute/path/to/Incoprea_E4_H034_2026-11-01
    python3 h160/independent_check.py

Outputs are written beside the independent scripts. The H160 plan, seed, noise convention, local estimator initialization and controls are in h160/PLAN.md. Original CSV outputs are preserved in the earlier release, rather than silently edited. H034-CORRECTION.md supplies corrected prose values. h160/corrected_t2.csv supplies independently recalculated conditional precision bounds.

Publication date:1 November2026. These are bounded publisher checks, not external peer review, new physical measurements or novelty certification. No new real-world datasets were acquired.

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# H160 bounded correction plan

Predefined before running the independent calculation.

Question: does the supplied complex-noise convention omit a factor of two in delay Fisher information, and does a separately implemented local likelihood estimator agree with the corrected bound?

Model: independent circular complex Gaussian noise, E|n_k|² = sigma², real and imaginary variance sigma²/2. Known real envelope A_k, known amplitude, one delay parameter; compare known phase against unknown constant nuisance phase. No physical measurements, no multi-target resolution claim.

Derive from log L = -sum|y-mu|²/sigma² + constant. Real-parameter Fisher matrix J_ab = (2/sigma²) Re sum conj(d_a mu) d_b mu. For mu=A exp(i phi-i 2pi f tau), known phase gives J_tau=2(2pi)² sum f² A²/sigma². With unknown phase, Schur complement replaces sum f²A² by sum A²(f-fbar)², fbar=sum f A²/sum A².

Controls: centered frequency grid versus same grid shifted by one full bandwidth. Known phase gains information from absolute frequency offset; nuisance phase must remove this offset. Noise power is measured directly. Use the same draws for known/nuisance estimators.

Monte Carlo: fixed seed 16020261101, 2000 trials per cell, 512 samples, peak SNR 100, Gaussian envelope A=exp(-2(x/2.14)²), x dimensionless frequency within a unit bandwidth. True dimensionless delay 0.137. Four cells: centered/shifted grids × known/nuisance phase. Independent Newton refinement initialized within 0.01 of true delay, explicitly a local high-SNR estimator check, not a global acquisition test. Maximum 12 iterations. No adaptively selected trials or tuned noise normalization.

Outcomes: noise power and component variances; delay bias, standard deviation, RMSE, corrected CRB, RMSE/CRB, old-bound ratio, maximum score residual and numerical failures. A local efficient estimator should give RMSE/corrected CRB approximately 1; allow 0.94–1.06 as a descriptive finite-sample tolerance (about several standard errors at 2000 trials), not a hypothesis acceptance threshold. Any convergence failure is reported.

Also recalculate the supplied T2 table without running its program, with both known-phase and unknown-phase formulas. Distinguish single-target precision from resolving two targets. Report remaining unsupported geometry/extended-target/novelty claims separately.

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# 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.