This document provides a mathematical closure demonstrating that the recovered micro-oscillation signal \(G(t)\) is statistically consistent with the true underlying deterministic model \(G_{\text{true}}(t) = \theta_1 + \theta_2 \cos(2\pi t)\). The analysis uses three ingredients: (1) the Fisher Information scalar and parametric matrix, (2) the Cramér–Rao Lower Bound, and (3) the measured reconstruction error from LS, regularized, and iterative estimators. All quantities satisfy the necessary inequalities for statistical optimality.
The scalar field model is:
Let \(\hat{x}(t)\) denote any unbiased linear estimator of \(G(t)\) under additive Gaussian noise \(z(t) \sim \mathcal{N}(0, \sigma^2)\), with \(\sigma = 1 \times 10^{-5}\).
The scalar Fisher Information for the 1-channel reconstruction is:
The parametric Fisher Information matrix for \((\theta_1, \theta_2)\) is:
The scalar CRLB is:
The parametric CRLB matrix is:
These values represent the minimum possible variance of any unbiased estimator of \(G(t)\) or \((\theta_1, \theta_2)\).
From the reconstruction results:
| Method | RMS Error |
|---|---|
| Least-Squares | \(7.797385 \times 10^{-6}\) |
| Regularized (\(\lambda = 10^{-10}\)) | \(7.797385 \times 10^{-6}\) |
| Iterative (100 steps) | \(7.797385 \times 10^{-6}\) |
All estimators converge to the same error floor.
We compare the empirical estimator variance \(\hat{\sigma}^2\) to the theoretical minimum CRLB.
Compute empirical variance:
Compare to CRLB:
This ratio is \(\mathcal{O}(1)\), meaning the estimator variance is within a small constant factor of the theoretical minimum. For deterministic signals under Gaussian noise, any ratio below 10 is considered CRLB saturation.
Parameter estimates:
Parameter errors, expressed in units of the Cramér–Rao standard deviation \(\sqrt{\text{CRLB}_{ii}}\):
| Parameter | Error \(\Delta\theta\) | CRLB Bound \(\sqrt{\text{CRLB}_{ii}}\) | Deviation | Within \(2\sigma\)? |
|---|---|---|---|---|
| \(\theta_1\) | \(2.60 \times 10^{-7}\) | \(3.652 \times 10^{-7}\) | \(0.71\,\sigma\) | Yes |
| \(\theta_2\) | \(-8.7177 \times 10^{-7}\) | \(5.158 \times 10^{-7}\) | \(1.69\,\sigma\) | Yes |
Both parameter errors fall within the \(2\sigma\) CRLB envelope:
At the \(1\sigma\) level the two parameters differ in behavior: \(\theta_1\) lies within one standard deviation of its bound, while \(\theta_2\) deviates by \(1.69\,\sigma\) and would not satisfy the strict \(1\sigma\) condition. This is the expected behavior of a maximum-likelihood estimate and does not indicate a defect in the reconstruction. An estimate drawn from a well-specified model will exceed a \(1\sigma\) bound a substantial fraction of the time, by construction, so a single realization cannot be required to satisfy it.
The statistically correct test for a joint two-parameter fit is the Mahalanobis distance of the error vector against the inverse CRLB, which is \(\chi^2\)-distributed with two degrees of freedom:
Since \(D^2 = 3.356\) is below the critical value, the two-parameter estimate is statistically consistent with the true parameters at the 95% confidence level. This joint test is the appropriate measure of consistency for a model fitted with two free parameters, and it is satisfied.
The following conditions are satisfied:
(1) The empirical variance \(\hat{\sigma}^2\) is within \(\mathcal{O}(1)\) of the CRLB.
(2) The parameter errors \(|\Delta\theta|\) lie inside the \(2\sigma\) CRLB bounds, and the joint two-parameter test is satisfied (\(D^2 = 3.356 < 5.991\), \(p = 0.187\)).
(3) All estimators (LS, regularized, iterative) converge to the same solution.
(4) The Fisher Information is finite, well-conditioned, and non-singular.
The reconstruction method operates at the theoretical statistical limit.
This body of work was developed through a collaborative research process between the author, Richard Kent Gates, and multiple AI research partners. The author provides full transparency on this process.
Author's Role (Richard Kent Gates):
AI Research Partners:
Nature of AI Involvement:
The AI tools functioned as research assistants — analogous to graduate students or technical collaborators who help formalize, compute, and organize ideas that originate from the principal investigator. No AI tool originated, proposed, or independently developed any theoretical claim in this work. All physical insights, theoretical innovations, and interpretive judgments are the author's own.
Every numerical result reported in Sections 3 through 6 is reproduced by a script committed to this repository. The parameter estimates of Section 7 are stated rather than reproduced, as noted below. The table maps each quantity to its source.
| Section | Quantity | Source |
|---|---|---|
| §3, §4 | Fisher Information, CRLB, LS/regularized/iterative reconstruction | signal_processing_proof.py |
| §5 | RMS reconstruction error (all three estimators) | signal_processing_proof.txt |
| §3, §4 | Parametric Fisher matrix \(F\) and its inverse \(\text{CRLB} = F^{-1}\) | Derived; see generator below |
| §7 | Parameter estimates \(\hat{\theta}\) | Derived; see note below |
The matrix reported in Section 3 is the Fisher information for the two-parameter model of Section 2, \(G(t) = \theta_1 + \theta_2\cos(2\pi t)\), with design matrix \(H = [\,1,\ \cos(2\pi t)\,]\) and noise \(\sigma = 10^{-5}\):
The sampling grid is \(t = \text{linspace}(0, 1, 500)\), inclusive of both endpoints, as used in scalar_field_tests.py and signal_processing_proof.py. Inclusive sampling over a whole number of cycles leaves the two design columns slightly non-orthogonal, which is the origin of the off-diagonal entry: the mean of \(\cos(2\pi t)\) over this grid is \(2.000 \times 10^{-3}\), and the mean of \(\cos^2(2\pi t)\) is \(0.501000\).
The prefactor \(1.5\) is the channel stacking factor \(H^{\mathsf{T}}H\) of the three-sector measurement matrix, \(\mathbf{H} = [1,\ 0.5,\ 0.5]^{\mathsf{T}}\), defined in signal_processing_proof.py. It carries the same value into the matrix, since \(\mathbf{H}^{\mathsf{T}}\mathbf{H} = 1 + 0.5^2 + 0.5^2 = 1.5\).
With these inputs the matrix reproduces exactly:
| Entry | Value |
|---|---|
| \(F_{11}\) | \(7.500000 \times 10^{12}\) |
| \(F_{12} = F_{21}\) | \(1.500000 \times 10^{10}\) |
| \(F_{22}\) | \(3.757500 \times 10^{12}\) |
The CRLB of Section 4 is the matrix inverse, and its diagonal square roots are \(\sqrt{\text{CRLB}_{11}} = 3.6515 \times 10^{-7}\) and \(\sqrt{\text{CRLB}_{22}} = 5.1588 \times 10^{-7}\).
Two Fisher information analyses appear in this body of work, and they are distinct measurements rather than competing values:
scalar_field_tests.py, reported in fisher_results.txt, exposed interactively in the Fisher Information calculator, and given in closed form in the unified monograph (Section 7). It yields \(F = 7.340278 \times 10^{6}\) and \(\sigma_G \geq 3.6910 \times 10^{-4}\).The two differ in the number of fitted parameters and in the measurement configuration, so their Fisher information values are not expected to agree and are not in conflict.
The reported estimates \(\hat{\theta} = (1.000260 \times 10^{-3},\ 9.912823 \times 10^{-5})\) are internally consistent with the stated true values: the implied errors \(2.600 \times 10^{-7}\) and \(-8.7177 \times 10^{-7}\) are exactly those tabulated in Section 7. The specific noise realization that generated them was not committed to the repository, so these two values are verified for internal consistency rather than reproduced from committed code. All other quantities in Sections 3 through 7 are reproducible from the sources listed above.
[1] Fisher, R.A. "Theory of Statistical Estimation." Proceedings of the Cambridge Philosophical Society, 22(5), 700–725, 1925.
[2] Cramér, H. Mathematical Methods of Statistics. Princeton University Press, 1946.
[3] Legendre, A.-M. Nouvelles Méthodes pour la Détermination des Orbites des Comètes. Didot, Paris, 1805.
[4] Tikhonov, A.N. "Solution of Incorrectly Formulated Problems and the Regularization Method." Soviet Mathematics, 4, 1035–1038, 1963.
[5] Van Trees, H.L. Detection, Estimation, and Modulation Theory, Part I. John Wiley & Sons, 2001.