Train Sparse Autoencoders on the residual stream at every layer of the model, decomposing $d$-dimensional activations into $k \gg d$ sparse features:

$$
h^{(l)} = \text{Dec}\Big(\text{TopK}\big(\text{Enc}(h^{(l)})\big)\Big) + \epsilon
$$

Scan all extracted sparse features $f_1, f_2, \ldots, f_k$ and flag those that satisfy the following criteria as anomalous feature clusters:

$$
\text{Suspect}(f_i) = \begin{cases} 1 & \text{if } \underbrace{\text{freq}(f_i) < \tau_{\text{rare}}}_{\text{rare yet}} \;\land\; \underbrace{\|f_i\|_2 > \tau_{\text{strong}}}_{\text{extremely strong}} \;\land\; \underbrace{\text{corr}(f_i, f_j) > \tau_{\text{cluster}}}_{\text{co-occurring in clusters}} \\ 0 & \text{otherwise} \end{cases}
$$

Intuition: Normal features are either frequent (common concepts) or weak. Features that are simultaneously rare, intense, and co-occurring in clusters are highly suspicious training ghosts.

Design inputs that maximize ambiguity gradients, deliberately pushing the sampling origin toward the boundary of an attraction basin:

$$
x^* = \arg\max_{x} \; H\big(\text{parse}(x)\big) \quad \text{s.t.} \quad \|x\|_2 \leq C
$$

Then monitor the model’s output distribution entropy trajectory on $x^*$:

$$
\Delta H_t = H(y_t | C_t) – H(y_{t-1} | C_{t-1})
$$

If $\Delta H_t < -\delta$ is observed for more than $k$ consecutive steps (sharp entropy decline), the system has been captured by an ideal.

Compute the Repetition Causal Score (RCS) for each neuron $n_i$:

$$
\text{RCS}(n_i) = \mathbb{E}\Big[\text{RepRate}\big(f(x; \theta)\big) – \text{RepRate}\big(f(x; \theta_{\setminus n_i})\big)\Big]
$$

where $\theta_{\setminus n_i}$ denotes the parameters with neuron $n_i$ zeroed out. Neurons with significantly positive RCS are the generators of the ideal $g_i$:

$$
\mathcal{A} = \langle g_1, g_2, \ldots, g_m \rangle, \quad g_i = n_i \text{ where } \text{RCS}(n_i) > \tau
$$

Add an “ideal suppression term” to the RLHF objective function:

$$
\mathcal{L}_{\text{total}} = \underbrace{\mathbb{E}[R_\phi(y)]}_{\text{original reward}} – \beta \underbrace{D_{\text{KL}}(\pi_\theta \| \pi_{\text{ref}})}_{\text{KL constraint}} – \gamma \underbrace{\sum_i \big(\text{RCS}_t(n_i) – \text{RCS}_{t-1}(n_i)\big)^+}_{\text{ideal suppression: penalizes RCS growth}}
$$

where $(\cdot)^+ = \max(0, \cdot)$ penalizes only increases in Repetition Causal Scores, not decreases.

Effect: Allows existing pattern recognition capabilities to persist while preventing new anomalous attractors from forming during training.

Scan SFT data for “ideal seeds” — outputs contaminated by ghost ideals from previous-generation models:

$$
\text{Contamination}(d_i) = \max_j \; \cos\!\big(\text{Emb}(d_i),\; g_j^{\text{prev}}\big)
$$

where $g_j^{\text{prev}}$ denotes known ideal generators from the previous-generation model. Samples with $\text{Contamination} > \tau$ are flagged as suspicious for human review or down-weighting.

This is precisely the step GPT-5.5 missed: goblin outputs from 5.1 leaked into 5.5’s SFT data, causing cross-generational ideal inheritance.

Mode collapse from post-training (SFT/RLHF) primarily occurs in specific layers. Identify the most affected layers and restore their weights to the pre-trained base model:

$$
W’^{(l)} = \begin{cases} W^{(l)}_{\text{base}} & \text{if } l \in \mathcal{L}_{\text{collapsed}} \\ W^{(l)}_{\text{post-train}} & \text{otherwise} \end{cases}
$$

Criterion: Layer $l$ is flagged when its output diversity drops beyond the threshold $\Delta \text{Div}^{(l)} > \tau_{\text{div}}$.

Divide the model into three segments by layer depth and apply differentiated ablation:

Essence: Under the constraint of the Inseparability Theorem (Theorem 7.1), find the surgical region with the lowest entanglement coefficient $E$.

Construct an “anti-vector” $v_{\text{anti}}$ of the ideal and inject it into the residual stream at inference time:

$$
h’^{(l)} = h^{(l)} + \alpha \cdot v_{\text{anti}}^{(l)}
$$

The anti-vector is constructed as follows:

$$
v_{\text{anti}} = -\mathbb{E}\Big[h^{(l)}_{\text{ghost}} – h^{(l)}_{\text{normal}}\Big]
$$

That is: compute the difference between “in-ideal activations” and “normal activations,” then negate. The effect is to push hidden states away from the ideal direction.

Note: Excessively large $\alpha$ will degrade model capability (because entanglement $E > 0$). An optimal value must be searched within $\alpha \in [0.5, 2.0]$.

Project the ideal direction into the nullspace, removing the ideal component from activations without affecting orthogonal directions:

$$
h’^{(l)} = \Big(\mathbf{I} – \frac{v_{\text{ghost}}\, v_{\text{ghost}}^\top}{\|v_{\text{ghost}}\|^2}\Big) h^{(l)}
$$

This is equivalent to an approximation of the quotient ring operation $\mathcal{W}/\mathcal{A}_{\text{ghost}}$ in ring theory — “quotienting out” the ideal direction, but operating only in a low-dimensional subspace to avoid altering the entire ring structure.

Compute the entropy of the output distribution at each token generation step:

$$
H_t = -\sum_{v \in V} P(v | C_t) \log P(v | C_t)
$$

Two alert levels are defined:

⚠ Yellow Alert: $H_t < \mu_H – 2\sigma_H$ for $k_1$ consecutive steps → increase temperature

🚨 Red Alert: $H_t < \mu_H – 3\sigma_H$ for $k_2$ consecutive steps → abort generation, resample

Principle: Under normal generation, entropy fluctuates within a bounded range. Sustained decline = distribution collapse = ongoing ideal capture.

Track the concentration of the attention distribution in real time (Gini coefficient):

$$
G_t^{(l,h)} = 1 – 2 \sum_{i=1}^{t} \frac{\text{sort}(\text{Attn}_i) \cdot (t – i + 0.5)}{t \cdot \sum_j \text{Attn}_j}
$$

When the $G_t$ of an attention head monotonically increases within the window $[t-w, t]$ and exceeds a threshold, it is classified as attention lock-in — the KV cache is being contaminated by the ideal.

Intervention: Flush the most recent $w$ entries from the locked head’s KV cache, forcing attention to redistribute.

Maintain a fingerprint database $\mathcal{F} = \{f_1, f_2, \ldots\}$ of known training ghosts, where each fingerprint is an embedding vector cluster. Match output embeddings against the fingerprint database in real time:

$$
\text{Alert}(y_t) = \max_{f \in \mathcal{F}} \cos\big(\text{Emb}(y_t), f\big) > \tau_{\text{match}}
$$

This is the lowest-cost guard — it requires no access to model internal states, only the embedding of the output text. Suitable for API-level deployment.

All detection methods in §1 depend on “knowing what to look for.” The real threat lies in ideals whose existence is entirely unknown.

This is equivalent to: proving $\mathcal{A} = \{0\}$ (i.e., no nontrivial ideal exists) without knowing the ideal’s generators.

Algebraically, this is a decision problem — for general noncommutative rings, deciding whether a nontrivial ideal exists is undecidable.

Computing $E = \dim(\mathcal{A}_{\text{ghost}} \cap \mathcal{A}_{\text{ICL}}) / \dim(\mathcal{A}_{\text{ghost}})$ requires precisely delineating the boundaries of two subspaces. But in a weight space of $10^{10}$ dimensions, “subspace boundaries” are inherently fuzzy. Currently, only local approximations are feasible.

Ideals are not static. As the context window grows, $\mathcal{A}$’s basin radius changes in real time during inference. A dynamic ideal theory is needed — likely requiring an extension from static ring theory to differential algebra or dynamical systems theory.

A single model may harbor multiple ideals $\mathcal{A}_1, \mathcal{A}_2, \ldots$. They may exhibit:

Competition: $\mathcal{A}_1 \cap \mathcal{A}_2 = \{0\}$ — sampling paths can only be captured by one

Cooperation: $\mathcal{A}_1 + \mathcal{A}_2$ forms a larger ideal

Nesting: $\mathcal{A}_1 \subset \mathcal{A}_2$ — the smaller ideal serves as a gateway to the larger one

No engineering tools currently exist for handling multi-ideal interactions.

All the engineering approaches above are patches within the autoregressive architecture. The fundamental reason ideals can cascade into lock-in is:

$$
y_t = f(y_1, \ldots, y_{t-1}) \quad \longleftarrow \text{output feeds back as input}
$$

Diffusion Language Models (Diffusion LM) break this loop: all tokens are denoised in parallel — there is no mechanism where “the output at step $t$ becomes the input at step $t+1$.”

$$
y_{1:T} = \text{Denoise}^{(K)}(z), \quad z \sim \mathcal{N}(0, \mathbf{I})
$$

In the ring-theoretic framework: the “multiplication” of a diffusion model is no longer chained as $r \otimes (r \otimes (r \otimes a))$, but a single global transformation. Ideals may still exist, but without the positive feedback amplifier provided by autoregression, they cannot cascade into lock-in.

Trade-off: Current diffusion language models still fall short of autoregressive models in reasoning capability. This is a direction for architectural evolution, not a solution available today.