Remove duplicate claim-6 page (merged into the canonical one)

pages/claim-6-alpha-tradeoff/page.md

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-# 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.