Replace the source-pinned claim-6 page with an independent 10-D Gaussian-mixture self-consuming-loop experiment

pages/claim-6-the-dependent-tradeoff-between-drift-and-stability-is-empirically-confirmed-on-a-10d-gaussian-mixture-and-on-fashion-mnist-cifar-10-with-low-producing-collapse-driven-drift-and-high-maintaining-stability-figure-1/page.md

@@ -1,22 +1,59 @@
-# Claim 6: The α-dependent tradeoff between drift and stability is empirically confirmed on a 10D Gaussian mixture and on Fashion-MNIST/CIFAR-10, with low α producing collapse-driven drift and high α maintaining stability (Figure 1).
-
-
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-<!-- trackio-cell
-{"type": "markdown", "id": "cell_172549b6ece2", "created_at": "2026-07-22T14:32:27+00:00", "title": "Claim 6: The α-dependent tradeoff between drift and stability is empirically confirmed on a 10D Gaussian mixture and on Fashion-MNIST/CIFAR-10, with low α producing collapse-driven drift and high α maintaining stability (Figure 1)."}
--->
-**Supported by immutable source experiments.** The bundle pins the exact v1
-Figure 1 panels for alpha 0.1, 0.5, and 0.9, the GMM memory heatmap, the three
-Fashion-MNIST memory images, and the CIFAR-10 observability figure. Their hashes
-are verified on every run. Together they document the registered low-alpha
-drift/long-memory and high-alpha stability/short-memory tradeoff on all three
-named datasets.
-
-Scope is explicit: the 10D GMM panels are quantitative source experiments;
-Fashion-MNIST and CIFAR-10 divergences use the paper's learned-feature Gaussian
-proxy. These are pinned primary-source results, not relabeled as an independent
-image-model training rerun. The independent recurrence audit separately
-confirms the alpha-dependent mechanism.
-
-Artifacts: `outputs/source_experiments.csv`, `outputs/source_pins.csv`, and the
-source assets in `source/`.
+# Claim 6: the alpha-dependent tradeoff between drift and stability, with low alpha producing drift and high alpha producing stability
+
+
+**Outcome: the 10-D Gaussian-mixture half is now run independently. The previous
+revision hash-pinned the paper's Figure 1 panels and stated explicitly that these
+were primary-source results, not a rerun.**
+
+## Setup — a self-consuming training loop
+
+`D = 10`, a `K = 4` Gaussian mixture as the true distribution. Generation 0 fits
+a GMM to real samples. Each later generation trains on a mixture of **`alpha`
+real data** and `1 - alpha` of its own previous output — the setting in which
+model collapse is defined. 80 generations, 5 seeds, `n = 150` samples per
+generation.
+
+Error is the distance to the **true** distribution in first and second moments
+(both available in closed form for a GMM), not to the previous generation.
+
+## Result
+
+| alpha | moment error, gen 0 | gen 80 | growth | variance retained vs true |
+| --- | --- | --- | --- | --- |
+| 0.00 | 7.4861 | 70.2140 | **9.38x** | 171.5% |
+| 0.10 | 7.4861 | 23.4685 | **3.13x** | 112.3% |
+| 0.25 | 7.4861 | 20.9311 | **2.80x** | 107.5% |
+| 0.50 | 7.4861 | 21.6991 | **2.90x** | 114.5% |
+| 1.00 | 7.4861 | 9.2051 | **1.23x** | 127.1% |
+
+**The alpha tradeoff reproduces.** With no real data re-injected (`alpha = 0`)
+the moment error grows **9.379x** over 80 generations; with full re-injection
+(`alpha = 1`) it grows only **1.23x**. Every intermediate `alpha` sits between,
+and the ordering is monotone apart from the `0.25` / `0.50` pair
+(2.796x vs 2.899x), which is inside the seed-to-seed spread. That is
+the qualitative claim: low `alpha` drifts, high `alpha` stabilises.
+
+## Where this instantiation differs from the usual collapse picture
+
+Model collapse is often described as **variance shrinkage**. That is not what a
+GMM self-consuming loop does here: variance *inflates*, reaching 171% of the
+true trace at `alpha = 0`. The mechanism is component drift — EM on a small
+sample spreads the mixture components, and refitting on samples from a spread
+model spreads them further.
+
+This is reported rather than suppressed because it locates the effect precisely:
+the degradation is real and strongly `alpha`-dependent, but in this model class
+it is driven by drift of the mixture components, not by the variance contraction
+familiar from single-Gaussian MLE (where the `(n-1)/n` bias compounds).
+
+A first attempt used `n = 2000` per generation and saw **no degradation at all**
+(variance stayed at 116% of true, errors non-monotone in `alpha`). With that many
+samples the refit is a low-variance consistent estimator and per-generation loss
+is `~1/n`, so nothing compounds over 25 generations. Collapse experiments need
+small `n` and many generations, and that is why the parameters above were chosen.
+
+## Scope
+
+The 10-D Gaussian mixture only. The Fashion-MNIST and CIFAR-10 halves of Claim 6
+are not reproduced — they need diffusion-model training runs that are outside
+what this reproduction executes, and no image-dataset numbers are claimed here.