Methodology & Approach

Mathematical Framework

TRP centers on three pieces: divergence (ΔZ*), alignment (Charlie Index), and Bayesian posterior regime‑change detection. Together, these identify unusual periods and quantify agreement across models.

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ΔZ* Divergence Metric

What it means (in plain English): ΔZtells us how much the different models disagree at each point in time. If ΔZ is high, it means the models are “seeing” something different, like scientists disagreeing about what happened in the past.

Analogy: Think of ΔZ* like the “spread” in a group’s answers to a quiz: if everyone gives the same answer, the spread is zero; if answers are all over the place, the spread is high.

Formal Definition: Suppose we have $N$ datasets, $X_1(t), X_2(t), ..., X_N(t)$, all sampled on a common time grid. The generalized divergence metric ΔZ* is:

$$ \Delta Z^*(t) = \left( \sum_{i=1}^N w_i |X_i(t) - \bar{X}(t)|^p \right)^{1/p} $$ where: - $X_i(t)$ is the value from dataset $i$ at time $t$ - $\bar{X}(t)$ is the mean across datasets at time $t$ - $w_i$ are weights (typically uniform) - $N$ is the number of datasets - $p$ is the norm order (commonly $p=2$; other norms such as $p=1$ or $p=\infty$ can be explored for robustness)

Properties: - ΔZ* is always non-negative. - Sensitive to outliers if $p$ is large; $p=2$ (Euclidean) is standard, $p=1$ (Manhattan) is more robust to outliers, $p=\infty$ gives the maximum deviation.

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Charlie Index (Formal Alignment Proxy)

What it means (in plain English): The Charlie Index measures how “in sync” the models are. A high score means the models are moving together; a low score means they’re out of step.

Analogy: It’s like a “synchrony score” for a dance group; if everyone moves together, the score is high.

Formal Definition: The formal Charlie Index $C(t)$ at time $t$ is:

$$ C(t) = 1 - \frac{\sum_{i$w_{ij} \ge 0$ is the weight for dataset pair $(i,j)$, normalized so $\sum_{i (default: uniform weighting; this reflects equal trust in all datasets, but non-uniform weights can be used for sensitivity analyses or to emphasize more reliable records) - $D_{\max}^{(i,j)} = \max_t |X_i(t) - X_j(t)|$ is the maximum possible difference for pair $(i,j)$ across the analyzed period. - The sum is over all unique dataset pairs ($i).

Properties: - $0 \leq C(t) \leq 1$ - $C(t) = 1$ → perfect alignment (all datasets identical at $t$) - $C(t) = 0$ → maximal disagreement (pairwise differences equal to historical maxima) - Robust to missing data: pairs with missing values are omitted and weights renormalized. - Weighting allows sensitivity analysis (default: uniform; e.g., if one dataset is less reliable, its pairwise weights can be reduced). - Version: v1.0 (see pseudocode in project docs for implementation).

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Bayesian Posterior Regime Change Detection

What it means (in plain English): This method combines the evidence from ΔZ* and Charlie Index features (the likelihood) with prior knowledge of known events to estimate the probability of a regime change at each time.

Analogy: It’s like combining a weather forecast (data) with what you know about the season (prior knowledge) to guess if it’s likely to snow, except here we’re predicting sudden changes in Earth’s magnetic field.

Formal Definition: This method estimates the chance that a big change (a “regime change”) happened at a certain time, based on both the data and what we already know about past events (e.g., known geomagnetic excursions).

$$ P(\text{regime change at } t | \text{data}) = \frac{P(\text{data} | t) \cdot P_0(t)}{\int P(\text{data} | t') P_0(t') dt'} $$ where $P_0(t)$ is the prior (e.g., known events), and $P(\text{data} | t)$ is the likelihood derived from ΔZ* and Charlie Index features.

> Callout: When ΔZ* is high and the Charlie Index is low, it’s a sign that something unusual is happening in Earth’s magnetic field, possibly a regime change.

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Figures

Below are key figures referenced in the methodology:

Posterior regime-change probability density (Laschamp peak highlighted).

ΔZ* divergence for all datasets (expand for full width).

Reproducibility

You can reproduce the TRP analysis and figures using the following scripts and notebooks from this repository:

- Data ingest and preprocessing script - ΔZ* and Charlie Index calculation script - Bayesian posterior regime-change notebook

All code and data are versioned for transparency and reproducibility. For details, see the project README and documentation.

Limitations & Ongoing Work

- $X'(t)$ definition is operational, not axiomatic. - Paleoclimate comparisons show null correlation. - Next step: cross-check with cosmogenic nuclide records.