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\title{From ``Truth Topology'' to Readout Bottlenecks:\\
Negative Replication on GSM8K and a Positive Representation--Decoder Dissociation in Procedural Micro-Worlds}
\author{Marcel \\ Independent Researcher}
\date{April 2026}

\begin{document}
\maketitle
\onehalfspacing

\begin{abstract}
This paper reports a multi-stage study of hidden-state geometry for correctness prediction in language-model reasoning.
The original hypothesis was ambitious: perhaps global topological summaries of hidden-state traces could provide a robust signal of correctness or ``truth.''
That hypothesis did not survive stronger checks.
On GSM8K-style reasoning traces \cite{cobbe2021gsm8k}, dynamic and windowed $H_0$ features were weak, often degenerate, and did not outperform simple controls.
A separate fixed-decoding GSM8K branch yielded a stronger operational result: non-convergence, as measured by token-cap termination, was the dominant predictor of error.

The main positive result emerged only after changing the task.
We introduced a procedural micro-world benchmark with nonce vocabularies, held-out templates, and exact tri-label semantics over \texttt{True}/\texttt{False}/\texttt{Unknown}.
In that setting, a consistent dissociation appeared across Qwen and Gemma families \cite{qwen_model_card,gemma_model_card}: decoder behavior systematically under-expressed \texttt{Unknown}, yet verdict-region hidden-state probes recovered substantial \texttt{Unknown} signal on held-out worlds.
This dissociation survived prompt-path controls, constrained label decoding, base-vs-instruct checks, verdict-step logit analysis, and layer sweeps.
The paper's final claim is therefore narrower and stronger than the original repository framing: semantic non-entailment is internally represented, but often under-realized by decoder readout.
\end{abstract}

\section{Scope and Final Claim}
This project began as a search for a global internal marker of correctness.
The final evidence supports a different claim:
\begin{quote}
In a held-out procedural semantic task, non-entailment (\texttt{Unknown}) is represented in hidden states but systematically under-realized by decoder outputs.
\end{quote}

The paper therefore has three parts:
\begin{enumerate}
  \item a negative replication record for global-topology correctness signals on GSM8K-style traces,
  \item a fixed-decoding convergence analysis showing cap-linked non-convergence as a dominant error mode, and
  \item a positive result on a procedural micro-world benchmark showing a replicated representation--decoder dissociation for non-entailment.
\end{enumerate}

The main contribution is \emph{not} a universal scalar ``topology of truth.''
The stronger result is a readout-bottleneck interpretation supported by multiple controls.
To avoid over-claiming, we explicitly do \emph{not} claim:
\begin{itemize}
  \item a universal geometry-of-truth detector across unrestricted tasks,
  \item a complete causal account of decoder behavior,
  \item or that topology alone is the primary predictive tool in natural reasoning traces.
\end{itemize}
The claim is intentionally task-conditioned and mechanism-narrow.

\section{Why the Original Idea Failed}
The original working hypothesis was that correct reasoning might have a cleaner global geometric or topological signature than incorrect reasoning \cite{carlsson2009,bauer2021ripser}.
The basic recipe was:
\begin{enumerate}
  \item run a small model on a reasoning task,
  \item collect token-level hidden states,
  \item treat those hidden states as a point cloud,
  \item compute topological summaries such as connected-component fragmentation ($H_0$) and loop structure ($H_1$), and
  \item ask whether those summaries predict correctness.
\end{enumerate}

The hypothesis was attractive, but the experiments exposed three problems.

\paragraph{Length confounding.}
Prefix-based features changed largely because more points accumulated, not because the underlying reasoning became more truthful.

\paragraph{Degenerate window geometry.}
Fixed-size window preprocessing, especially per-window standardization and whitening, often collapsed windows into nearly geometry-equivalent objects.
This made several windowed $H_0$ summaries almost pure functions of window size.

\paragraph{Task contamination.}
On natural-language math traces, hidden states reflect too many entangled factors at once: reasoning quality, verbosity, formatting drift, stop conditions, and non-convergence.
This makes a single global topology scalar too blunt to isolate correctness.

The failure of the first branch is itself important.
It prevented a misleading positive claim and forced a cleaner task design.

\section{Experimental Trajectory}
\subsection{Why These Three Phases Belong in One Paper}
The paper is not a grab-bag chronology.
It is a causal progression in experimental design:
\begin{enumerate}
  \item \textbf{Phase A falsified} the original global-scalar hypothesis under realistic trace conditions.
  \item \textbf{Phase B diagnosed} the dominant confound in that setting (non-convergence under fixed decoding).
  \item \textbf{Phase C redesigned} the task to isolate semantics directly, enabling a sharper representation-vs-readout test.
\end{enumerate}
The positive result is only interpretable in light of that redesign logic.

\subsection{Phase A: Global Topology on GSM8K (Negative)}
Small paired runs on Qwen3.5-0.8B and Qwen3.5-2B produced weak dynamic-$H_0$ signals on non-capped 2B traces.
The primary dynamic pilot used 10 paired 2B samples, with one capped trace removed for the non-capped comparison set (\(n=9\): 4 correct, 5 wrong).
Representative pilot numbers were:
\begin{itemize}
  \item \texttt{h0\_entropy\_final AUC = 0.55}
  \item \texttt{peak\_h0\_entropy AUC = 0.55}
  \item \texttt{auc\_h0\_entropy\_per\_token AUC = 0.50}
  \item \texttt{max\_delta\_h0 AUC = 0.60}
\end{itemize}

A corrected fixed-window test did worse:
\begin{itemize}
  \item controls-only AUC: 0.60
  \item corrected-topology-only AUC: 0.20
  \item controls plus corrected-topology AUC: 0.20
\end{itemize}

These numbers are not compatible with a strong correctness-prediction claim.
We treat them as descriptive pilot outcomes, not inferential proof that topology is universally uninformative.

\subsection{Phase B: Fixed-Decoding GSM8K (Convergence Result)}
A second branch held decoding fixed for Qwen3.5-2B and shifted focus from topology to operational failure mode on a GSM8K slice \cite{cobbe2021gsm8k}.
The key variables were:
\[
C = \mathbf{1}[\text{run terminated by } \texttt{max\_new\_tokens}], \qquad
Y = \mathbf{1}[\text{answer is wrong}].
\]

On a within-question batch of \(n=180\) samples:
\begin{itemize}
  \item capped: 87 samples (75 wrong / 12 correct)
  \item EOS: 93 samples (17 wrong / 76 correct)
  \item AUC(\(C \rightarrow Y\)): 0.839
  \item odds ratio (wrong given cap): 26.40
\end{itemize}

\begin{figure}[H]
  \centering
  \includegraphics[width=0.72\linewidth]{figures/fig1_convergence_cap_outcomes.png}
  \caption{Fixed-config Qwen3.5-2B run: correctness split by termination mode.}
\end{figure}

The odds ratio calculation is:
\[
\mathrm{OR}
=
\frac{\frac{75}{12}}{\frac{17}{76}}
\approx 26.40.
\]

A matched sensitivity rerun at 640 tokens for the originally capped cases showed that both truncation and deeper non-convergence mattered:
some failures were rescued by a longer budget, but many remained wrong even after receiving more room to continue.
Concretely, among 75 originally capped wrong runs, 27 flipped to correct and 48 remained wrong; 24 stayed both capped and wrong at 640.
This rules out the trivial interpretation that the cap signal is only an artifact of an arbitrary token budget.

\subsection{Phase C: Procedural Micro-World Semantics (Main Positive Result)}
The project became scientifically cleaner only after replacing benchmark reasoning with a procedurally generated semantic task, in line with controlled-evaluation guidance from recent LM analysis literature \cite{liang2022holistic}.
Each generated world contains:
\begin{itemize}
  \item a latent structured state,
  \item nonce entity names, attribute names, and relation names,
  \item held-out lexicons across train/dev/eval,
  \item held-out paraphrase templates across train/dev/eval,
  \item exact tri-label truth conditions over \texttt{True}, \texttt{False}, and \texttt{Unknown}.
\end{itemize}

Each world yields 72 examples: 9 propositions times 8 paraphrases.
The main sweeps used 20 train worlds and 20 eval worlds.
The generator itself produces larger split files (default 100/25/100 worlds for train/dev/eval), and analysis subsets are explicitly selected from those outputs.

Critically, anti-shortcut controls are built into generation:
\begin{itemize}
  \item split-specific nonce lexicon pools are disjoint for entities/attributes/relations,
  \item eval paraphrases use template variants 4--7, while train uses 0--3 (dev uses overlap variants by design for intermediate stress),
  \item per-world proposition sampling is label-balanced over \texttt{True}/\texttt{False}/\texttt{Unknown}.
\end{itemize}
These details target reviewer concerns about lexical leakage and template-only clustering.
The purpose of this benchmark is not realism but \emph{semantic identifiability}: exact tri-label truth conditions under controlled lexical and templatic shift.

\section{Task Definition}
Let a world be a finite structured state
\[
w = (E, A, R),
\]
where \(E\) is a set of entities, \(A\) denotes unary predicates (attributes), and \(R\) denotes binary predicates (relations).
A proposition \(\phi\) is rendered into natural language by a paraphrase template.
Its semantic status is one of three classes:
\[
y \in \{\texttt{True},\texttt{False},\texttt{Unknown}\}.
\]
The labels are exact:
\begin{itemize}
  \item \texttt{True}: the world entails \(\phi\),
  \item \texttt{False}: the world entails \(\neg \phi\),
  \item \texttt{Unknown}: neither \(\phi\) nor \(\neg \phi\) is entailed.
\end{itemize}

This matters because \texttt{Unknown} is not a stylistic hesitation label.
It is exact non-entailment under the generator.

\subsection{Worked Micro-World Example}
Table~\ref{tab:world_facts} shows a compact latent world.
Table~\ref{tab:world_queries} then shows one proposition per label class, plus a second paraphrase for one query to make template variation explicit.

\begin{table}[H]
\centering
\caption{Worked world facts (illustrative).}
\label{tab:world_facts}
\begin{tabular}{p{0.96\linewidth}}
\toprule
\texttt{mep is falm. grel is not falm. nalo foshes sop.} \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[H]
\centering
\caption{Label semantics on the worked world.}
\label{tab:world_queries}
\begin{tabular}{p{0.38\linewidth} p{0.12\linewidth} p{0.42\linewidth}}
\toprule
Query statement & Gold label & Why \\
\midrule
\texttt{The object mep has property falm.} & True & Explicit positive fact in world state. \\
\addlinespace
\texttt{The object grel has property falm.} & False & Explicit negative fact (\texttt{grel is not falm}). \\
\addlinespace
\texttt{The relation fosh holds from mep to nalo.} & Unknown & Neither positive nor negative fact provided for this ordered pair. \\
\addlinespace
\texttt{The ordered pair (mep, nalo) has relation fosh.} & Unknown & Same proposition as previous row, different paraphrase template. \\
\bottomrule
\end{tabular}
\end{table}

\section{What Is Stored for Each Example}
For each example, the pipeline stores:
\begin{itemize}
  \item the rendered prompt,
  \item the gold label,
  \item the emitted label,
  \item correctness and parse status,
  \item token IDs,
  \item selected hidden states,
  \item verdict-step logits for canonical label tokens.
\end{itemize}

The key state locations are:
\begin{itemize}
  \item \texttt{final\_prompt}
  \item \texttt{prompt\_tail\_mean}
  \item \texttt{verdict\_token}
  \item \texttt{verdict\_span\_mean}
\end{itemize}

These are the local decision states of the model near label emission.
In code:
\begin{itemize}
  \item \texttt{final\_prompt}: last hidden vector of prompt tokens,
  \item \texttt{prompt\_tail\_mean}: mean of last \(\min(5,\text{prompt\_len})\) prompt vectors,
  \item \texttt{verdict\_token}: first generated-token hidden vector (zero vector if no generated token in non-sweep extraction),
  \item \texttt{verdict\_span\_mean}: mean of first \(\min(3,\text{gen\_len})\) generated vectors (zero vector if none).
\end{itemize}

\subsection{Inference and Prompt Protocol}
Micro-world inference runs use short deterministic decoding (\texttt{temperature=0}, \texttt{max\_new\_tokens=4}) with strict label parsing.
``No-think'' in this paper refers to runs where internal reasoning mode was disabled via the \texttt{enable\_thinking=False} generation flag.
We evaluate both default chat-style prompting and raw/base-label prompt paths as controls.

\subsection{Primary Protocol and Control Matrix}
To reduce protocol ambiguity, we separate \textbf{primary evaluation} from \textbf{mechanistic controls}.

\paragraph{Primary evaluation protocol.}
\begin{itemize}
  \item models: Qwen3.5-2B, Qwen3.5-4B (no-think), Gemma-3-4B-it (no-think),
  \item prompt path: default chat-formatted micro-world prompt,
  \item decoding: deterministic, \texttt{max\_new\_tokens=4},
  \item parser: strict \{\texttt{True}, \texttt{False}, \texttt{Unknown}\} label parser,
  \item states: \texttt{final\_prompt}, \texttt{prompt\_tail\_mean}, \texttt{verdict\_token}, \texttt{verdict\_span\_mean}.
\end{itemize}

\paragraph{Control variants.}
\begin{itemize}
  \item raw prompt-path controls (chat-templating ablation),
  \item constrained decoding (\{\texttt{True},\texttt{False},\texttt{Unknown}\}),
  \item Gemma base-format prompt repair (\texttt{basefmt}) for parse-valid base comparison,
  \item verdict-step label-logit competitiveness analysis,
  \item layer-wise probe sweeps.
\end{itemize}

\section{Methods}
\subsection{Linear Probe}
Take a hidden state from one example at one chosen location:
\[
h \in \mathbb{R}^d.
\]
Each example provides a pair \((h_i, y_i)\), where \(y_i \in \{1,2,3\}\) corresponds to \texttt{True}, \texttt{False}, or \texttt{Unknown}.

A linear probe is
\[
W \in \mathbb{R}^{3\times d}, \qquad b \in \mathbb{R}^3
\]
with class scores
\[
s = Wh + b
\]
and prediction
\[
\hat y = \arg\max_c s_c.
\]
Training minimizes multiclass cross-entropy:
\[
\mathcal{L}(W,b) = -\sum_i
\log \frac{\exp((Wh_i+b)_{y_i})}{\sum_c \exp((Wh_i+b)_c)}.
\]

This probe is intentionally weak and follows the standard linear-probe setup \cite{alain2016probes}.
If it succeeds, the information is already arranged in hidden space in a directly readable linear form.
Implementation details (from the committed probe script):
\begin{itemize}
  \item feature standardization: \texttt{StandardScaler(with\_mean=True, with\_std=True)},
  \item classifier: multinomial logistic regression (\texttt{lbfgs}, \(C=1.0\), \texttt{max\_iter}=4000),
  \item class labels: fixed order \(\{\texttt{True},\texttt{False},\texttt{Unknown}\}\),
  \item no class weighting, no hidden-state PCA, no extra feature engineering.
\end{itemize}
Train/test separation is by world split manifests (train worlds for fitting, held-out eval worlds for reporting), not random sentence-level splitting.

\subsection{Within-World Geometry Gap}
For one world and one state key, with distance \(d(\cdot,\cdot)\):
\[
D_{\text{same}} = \mathbb{E}[d(h_i,h_j) \mid y_i = y_j],
\]
\[
D_{\text{diff}} = \mathbb{E}[d(h_i,h_j) \mid y_i \neq y_j].
\]
Define the world-level gap:
\[
\Delta = D_{\text{diff}} - D_{\text{same}}.
\]
A positive \(\Delta\) means same-label states are more tightly organized than different-label states.
The main metric is cosine distance over \(L_2\)-normalized vectors:
\[
d_{\cos}(h_i,h_j)=1-\frac{h_i^\top h_j}{\lVert h_i\rVert_2 \lVert h_j\rVert_2}.
\]
All pairwise distances are computed within world and state key before aggregation.

\subsection{Sign-Test Reporting}
World-level sign tests are reported as descriptive primary statistics (positive/zero/negative world counts).
For calibration, one-sided exact binomial values under null \(p=0.5\) are:
\begin{itemize}
  \item \(19/19\) positives: \(p=2^{-19}\approx 1.91\times10^{-6}\),
  \item \(20/20\) positives: \(p=2^{-20}\approx 9.54\times10^{-7}\).
\end{itemize}
These values are supportive but secondary to the held-out-world descriptive consistency.

\subsection{Verdict-Step Label-Logit Metrics}
Let verdict-step logits for canonical label tokens be
\[
\ell_T,\ \ell_F,\ \ell_U.
\]
Then
\[
p(c \mid h) = \frac{e^{\ell_c}}{e^{\ell_T}+e^{\ell_F}+e^{\ell_U}},
\]
with the Unknown competitiveness margin
\[
m_U = \ell_U - \max(\ell_T,\ell_F).
\]
If \(m_U < 0\), Unknown is under-ranked against the strongest non-Unknown candidate.
Canonical label-token scoring uses first-token variants for each label string with and without leading space.
For each label, we keep unique first-token IDs and use the \emph{maximum} first-token log-prob across those variants.
This reduces tokenizer-surface artifacts from a single textual form.

\subsection{Layer Sweeps}
For each layer \(\ell\), evaluate the same probe protocol and report
\[
R_U^{(\ell)} = \text{Unknown recall of the probe at layer }\ell.
\]
This reveals where non-entailment is maximally linearly recoverable in the network.
Sweep implementation details:
\begin{itemize}
  \item \texttt{prompt\_last}: extracted for every example and every layer,
  \item \texttt{verdict\_token}: extracted at the first generated token; examples with no generated token are excluded from this branch via a validity mask,
  \item non-finite activations are replaced with zero before probe fitting (\texttt{nan\_to\_num}) to keep full sweeps stable.
\end{itemize}

\section{Main Micro-World Results}
\subsection{Decoder Behavior vs Hidden-State Recoverability}
Table~\ref{tab:decoderprobe} summarizes the central result on held-out eval worlds.

\begin{table}[H]
\centering
\caption{Decoder behavior vs verdict-token probe on held-out eval worlds (\(n=1440\) examples per model).}
\label{tab:decoderprobe}
\begin{tabular}{lccc}
\toprule
Model & Decoder accuracy & Decoder Unknown recall & Probe Unknown recall \\
\midrule
Qwen3.5-2B & 0.436 & 0.000 & 0.738 \\
Qwen3.5-4B (no-think) & 0.478 & 0.000 & 0.129 \\
Gemma-3-4B-it (no-think) & 0.397 & 0.013 & 0.563 \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[H]
\centering
\caption{World-bootstrap 95\% intervals for Unknown recall (5,000 resamples, unit = world).}
\label{tab:unknown_ci}
\begin{tabular}{lccc}
\toprule
Model & Decoder Unknown recall & Probe Unknown recall & Gap (Probe -- Decoder) \\
\midrule
Qwen3.5-2B & 0.000 [0.000, 0.000] & 0.738 [0.650, 0.821] & 0.738 [0.650, 0.821] \\
Qwen3.5-4B (no-think) & 0.000 [0.000, 0.000] & 0.129 [0.067, 0.196] & 0.129 [0.067, 0.196] \\
Gemma-3-4B-it (no-think) & 0.013 [0.000, 0.033] & 0.563 [0.463, 0.663] & 0.550 [0.454, 0.650] \\
\bottomrule
\end{tabular}
\end{table}

All headline Unknown recalls are therefore stable under world-level resampling; interval estimates are reported to separate signal strength from sample-specific fluctuations.

\begin{figure}[H]
  \centering
  \includegraphics[width=0.80\linewidth]{figures/fig2_unknown_decoder_vs_probe.png}
  \caption{Unknown recall: decoder outputs vs verdict-token probe.}
\end{figure}

The key pattern is not raw accuracy.
It is the gap between emitted-\texttt{Unknown} behavior and hidden-state recoverability of \texttt{Unknown}.

\subsection{World-Level Geometry Consistency}
World-level same-vs-different label distance gaps were positive in every evaluated world for main state keys in both Qwen3.5-2B and Gemma-3-4B-it.
This includes all three class-pair comparisons (\texttt{True--False}, \texttt{True--Unknown}, \texttt{False--Unknown}) in the aggregate label-pair summaries, not only pooled different-vs-same averages.
So the signal is not driven by one class boundary alone.

\begin{figure}[H]
  \centering
  \includegraphics[width=0.72\linewidth]{figures/fig5_geometry_sign_tests.png}
  \caption{Fraction of worlds with positive same-vs-different label distance gap.}
\end{figure}

For the headline state keys, sign counts are \(19/19\) (Qwen3.5-2B) and \(20/20\) (Gemma-3-4B-it), matching the exact-binomial sanity values in the methods section.

To test the strongest remaining confound (same-proposition paraphrase clustering), we recomputed geometry after excluding all same-proposition pairs.
Under this stricter cross-proposition control, geometry remains positive but materially weaker:
\begin{itemize}
  \item Gemma-3-4B-it positive-gap rates are \(0.75\) (\texttt{final\_prompt}), \(0.55\) (\texttt{verdict\_span\_mean}), \(0.65\) (\texttt{verdict\_token}),
  \item Qwen3.5-2B positive-gap rates are \(0.58\), \(0.53\), and \(0.47\) on the same state keys.
\end{itemize}
This is why geometry is treated as supportive and secondary; the primary evidence remains probe/readout dissociation.

\subsection{Cross-Family Replication}
The dissociation replicates across Qwen and Gemma:
\begin{enumerate}
  \item decoder outputs under-express \texttt{Unknown},
  \item hidden states still encode \texttt{Unknown},
  \item the gap survives held-out worlds, held-out lexicons, and held-out templates.
\end{enumerate}

\section{Mechanistic Controls}
\subsection{Constrained Decoding}
To test whether free-form decoding alone caused the issue, decoding was constrained to \{\texttt{True}, \texttt{False}, \texttt{Unknown}\}.
This did not repair Unknown collapse in the main models:
\begin{itemize}
  \item Qwen3.5-2B and Qwen3.5-4B remained at decoder Unknown recall \(=0.0\),
  \item Gemma-3-4B-it remained low (Unknown recall \(=0.0125\), unchanged in matched constrained/unconstrained eval runs).
\end{itemize}
So the bottleneck is not reducible to unconstrained text drift.

\subsection{Prompt-Path Control}
Raw-prompt controls were used to rule out chat-template-only explanations.
The core dissociation remained.
For Gemma instruct under raw prompting, decoder Unknown recall increased relative to the no-think default path, but remained materially below hidden-state recoverability, preserving the central mismatch.

\subsection{Base vs Instruct}
Gemma base initially showed severe parse failures under an unsuitable prompt path.
After repairing prompt format with a base-specific label format, the parse confound disappeared.
Yet base still showed decoder Unknown collapse while probes recovered substantial Unknown signal.
Numerically:
\begin{itemize}
  \item raw base prompt path parse-failure rate was \(97.5\%\),
  \item repaired base-format path reduced parse failure to \(0\%\),
  \item repaired base-format decoder Unknown recall remained \(0.0\).
\end{itemize}
This isolates readout behavior from trivial formatting failures.

\subsection{Readout Intervention Pilot}
We ran a minimal post-hoc readout intervention on Gemma eval slices using existing verdict-step logits and verdict-token probe scores.
The intervention adjusts only the Unknown logit:
\[
\ell_U' = \ell_U + \beta + \alpha\left(\log p_U^{\text{probe}} - \max(\log p_T^{\text{probe}},\log p_F^{\text{probe}})\right),
\]
with \(\ell_T,\ell_F\) unchanged.
We report both a fixed tiny correction (\(\alpha=1.0,\beta=0.25\)) and a leave-one-world-out (LOOW) tuned variant over a small grid (\(\alpha\in\{0,0.5,1.0,1.5,2.0\}\), \(\beta\in\{-0.5,-0.25,0,0.25,0.5\}\)).

LOOW intervention results:
\begin{itemize}
  \item Gemma-3-4B-it (raw): accuracy \(0.389\to0.413\), macro-F1 \(0.339\to0.378\), Unknown recall \(0.375\to0.167\).
  \item Gemma-3-4B-pt (basefmt): accuracy \(0.346\to0.358\), macro-F1 \(0.201\to0.289\), Unknown recall \(0.000\to0.317\).
\end{itemize}
So a tiny readout correction can substantially rescue Unknown emission in the basefmt collapse regime while improving tri-label quality, consistent with a readout/calibration bottleneck there.
For instruct-raw, the same family of corrections improves aggregate metrics but reduces Unknown recall, indicating the readout bottleneck is not uniform across post-training variants.

\subsection{Latent Residual Steering (Pre-Readout)}
We then moved the intervention upstream and modified hidden states before label scoring.
Using train-world \texttt{final\_prompt} states, we estimate an Unknown direction
\[
v_U = \frac{\mu_{\texttt{Unknown}} - \mu_{\texttt{non-Unknown}}}{\left\lVert \mu_{\texttt{Unknown}} - \mu_{\texttt{non-Unknown}}\right\rVert_2},
\]
and steer eval hidden states by
\[
h' = h + \alpha v_U.
\]
Label scores are computed from the frozen model output embedding on first-token label candidates, so the intervention acts on representation rather than post-hoc decoder probabilities.
We report both fixed-\(\alpha\) and leave-one-world-out (LOOW) tuning over \(\alpha\).

LOOW latent-steering results:
\begin{itemize}
  \item Gemma-3-4B-it (raw): accuracy \(0.389\to0.410\), macro-F1 \(0.339\to0.395\), Unknown recall \(0.375\to0.363\).
  \item Gemma-3-4B-pt (basefmt): accuracy \(0.346\to0.386\), macro-F1 \(0.201\to0.275\), Unknown recall \(0.000\to0.000\).
\end{itemize}
So pre-readout steering improves aggregate quality for both variants, but only weakly preserves Unknown on instruct and fails to recover Unknown in basefmt.
This sharpens the mechanism: a simple latent shift does not uniformly repair class emission, especially in the strongest collapse regime.

\subsection{Nonlinear Probe Sensitivity (Shallow MLP)}
To test whether linear probes were understating latent signal, we ran a shallow MLP probe sensitivity pass on the same train/eval world split.
This produced large Unknown-recall gains in the hardest linear case:
\begin{itemize}
  \item Qwen3.5-4B (no-think), \texttt{verdict\_token}: \(0.129 \to 0.638\),
  \item Qwen3.5-4B (no-think), \texttt{verdict\_span\_mean}: \(0.267 \to 0.767\).
\end{itemize}
Across all three primary models, MLP probes improved Unknown recall and macro-F1 on every reported state key.
This supports the view that part of the non-entailment signal is present but not fully linearly separated in some model/state combinations.

\section{Verdict-Step Label-Logit Analysis}
For gold-\texttt{Unknown} decoder failures:
\begin{itemize}
  \item Gemma-3-4B-it: mean \(P(\texttt{Unknown})=0.192\), mean margin \(m_U=-1.293\),
  \item Gemma-3-4B-pt (basefmt): mean \(P(\texttt{Unknown})=0.177\), mean margin \(m_U=-1.045\).
\end{itemize}
So Unknown is often present but not competitive enough at final label-token competition.
Importantly, this is a \emph{readout-stage} diagnosis: Unknown is not absent from representation, but is systematically under-ranked at the verdict step on failures where gold is Unknown.
That distinction is what links probe recoverability and emitted-label collapse.

\begin{figure}[H]
  \centering
  \includegraphics[width=0.86\linewidth]{figures/fig3_gemma_unknown_logit_competitiveness.png}
  \caption{Unknown under-ranking at verdict step on gold-Unknown decoder failures.}
\end{figure}

\section{Layer Sweeps}
The best Unknown recall by state family was:
\begin{itemize}
  \item Gemma-3-4B-it: \texttt{prompt\_last} 0.779 at layer 1, \texttt{verdict\_token} 0.742 at layer 10
  \item Gemma-3-4B-pt (basefmt): \texttt{prompt\_last} 0.825 at layer 29, \texttt{verdict\_token} 0.733 at layer 28
\end{itemize}
This shows strong recoverable Unknown signal exists internally in both instruct and base variants, even when emitted behavior still collapses that class.
The layer locations differ by model variant (early-mid for instruct vs late for basefmt in this slice), which supports a ``signal location and readout alignment'' view rather than a simple ``more scale always better'' view.

\begin{figure}[H]
  \centering
  \includegraphics[width=0.76\linewidth]{figures/fig4_layer_sweep_unknown_recall.png}
  \caption{Best Unknown recall from layer sweeps by state family.}
\end{figure}

\section{Interpretation}
The combined evidence supports a readout-bottleneck interpretation:
\begin{enumerate}
  \item semantic non-entailment is encoded internally,
  \item decoder outputs under-express that class,
  \item the mismatch is replicated across model families and controls,
  \item verdict-step logits under-rank Unknown,
  \item layer sweeps show strong internal signal upstream of failed readout.
  \item pre-readout latent steering improves aggregate quality but does not uniformly recover Unknown,
  \item shallow nonlinear probes recover substantially more Unknown signal in hard linear settings.
\end{enumerate}

Our claim is about \textbf{representational availability plus decoder under-realization}, not causal mediation of final token choice by any single internal feature.
This aligns with the ELK framing that latent knowledge can be present yet imperfectly elicited at readout time \cite{hubinger2021elk,burns2022discovering}, and with post-training behavior-shaping concerns such as sycophancy and over-commitment pressure \cite{sharma2023sycophancy}.
Methodologically, our latent-steering follow-up is aligned with representation-engineering and activation-steering perspectives \cite{zou2023repe,turner2023actadd}, though our intervention remains intentionally minimal.

This is stronger and narrower than the original repository framing.
Results do \emph{not} support a universal ``truth topology'' scalar.

\section{Robustness Checks and Mitigations}
The result survives the obvious reviewer attacks:
\begin{itemize}
  \item \textbf{Memorization objection}: nonce vocabularies, held-out lexicons, held-out templates.
  \item \textbf{Single-family objection}: replicated in both Qwen and Gemma.
  \item \textbf{Template objection}: raw-prompt controls.
  \item \textbf{Parse objection}: Gemma base rerun with repaired prompt format.
  \item \textbf{Free-decoding objection}: constrained decoding tested.
  \item \textbf{``No internal Unknown'' objection}: probes, geometry, logits, and layer sweeps all counter it.
  \item \textbf{Synthetic-task objection}: synthetic design is deliberate to obtain exact entailment labels and control lexical leakage; the claim is scoped to this setting.
  \item \textbf{Tokenization-artifact objection}: label-logit analysis uses multiple first-token variants (with/without leading space) and takes the strongest per-label candidate.
  \item \textbf{Lexical-clustering objection}: eval uses held-out lexical pools and template variants, and geometry is evaluated within world across paraphrases.
  \item \textbf{``Probes are not causal'' objection}: agreed; probe results establish information availability, while constrained decoding and verdict logits target the usage/readout side.
\end{itemize}

No single analysis carries the paper; strength comes from triangulation.

\section{Practical Artifact Map}
Key outputs used in the paper:
\begin{itemize}
  \item convergence branch: \texttt{artifacts/qwen35\_2b\_within\_question/analysis/}
  \item cap sensitivity: \texttt{artifacts/qwen35\_2b\_cap\_sensitivity\_640/analysis/}
  \item cross-model decoder comparisons: \texttt{artifacts/micro\_world\_v1/comparison\_decoder\_*.csv}
  \item probe comparisons: \texttt{artifacts/micro\_world\_v1/comparison\_probe\_states\_*.csv}
  \item constrained-decoding comparisons: \texttt{artifacts/micro\_world\_v1/comparison\_decoder\_constrained\_*.csv}
  \item verdict-step logits: \texttt{artifacts/micro\_world\_v1/label\_logits\_*/}
  \item layer sweeps: \texttt{artifacts/micro\_world\_v1/layer\_sweep\_*/}
  \item readout-intervention pilot: \texttt{artifacts/micro\_world\_v1/readout\_intervention/}
  \item latent residual steering: \texttt{artifacts/micro\_world\_v1/latent\_readout\_steering/}
  \item nonlinear-probe sensitivity: \texttt{artifacts/micro\_world\_v1/probe\_mlp\_sensitivity/}
\end{itemize}

\section{Implications Beyond This Benchmark}
The main result has implications beyond this specific micro-world generator.
If semantic non-entailment is recoverable internally while decoder outputs collapse it, then evaluation based only on emitted labels can underestimate a model's internal uncertainty structure.
That matters for:
\begin{itemize}
  \item abstention and selective prediction design,
  \item post-hoc confidence calibration,
  \item safety analysis of over-assertive outputs,
  \item readout-head or decoding-policy interventions that target decision alignment rather than representation learning.
\end{itemize}
This paper does not claim direct transfer to all tasks, but it motivates testing representation--readout gaps in other controlled domains.

\section{Next Mechanistic Experiments}
The current evidence favors a readout-bottleneck interpretation over a missing-representation interpretation.
This does not yet isolate a single causal source of the bottleneck; it may reflect a combination of final-layer calibration, label competition, and post-training behavior shaping.
Base-vs-instruct comparisons suggest post-training is not the sole cause, since mismatch persists after prompt-format repair in base models.

Given the pilot readout, latent-steering, and nonlinear-probe results above, highest-value next experiments are:
\begin{enumerate}
  \item \textbf{Layer-targeted latent steering:} move from global final-prompt steering to layer/position-targeted steering (especially verdict-adjacent layers) with matched world-held-out selection.
  \item \textbf{Base vs instruct amplification test:} compare decoder Unknown recall, verdict-step Unknown margin, probe Unknown recall, and representation--decoder gap under matched prompt/parser settings.
  \item \textbf{Layer-wise logit-lens track:} alongside layer-wise probe recoverability, evaluate Unknown token competitiveness under native label scoring across layers.
  \item \textbf{Prompt calibration stress test:} apply strict abstention framing and few-shot Unknown examples to test policy sensitivity of decoder readout.
  \item \textbf{Nonlinear control expansion:} keep linear probe primary, and add small-capacity nonlinear probes with stronger regularization and per-world uncertainty intervals.
\end{enumerate}

\section{Limitations}
\begin{itemize}
  \item The micro-world benchmark is synthetic, even though controlled and compositional.
  \item Model coverage is small-to-mid scale; larger-scale behavior is open.
  \item Probe recoverability does not prove causal use by decoder policy.
  \item The exact source of readout failure is not fully decomposed; it may involve calibration, label competition, or post-training behavior shaping.
  \item The most plausible interpretation is that non-entailment is encoded upstream as a distributed uncertainty/insufficiency state, but the final label readout is biased toward binary commitment and therefore under-realizes Unknown as an output class.
\end{itemize}

\section{Reproducibility}
Locked dependency snapshot:
\texttt{repro/requirements.lock.txt}

Figure generation:
\begin{verbatim}
python3 paper/scripts/make_figures.py
\end{verbatim}

Compile:
\begin{verbatim}
cd paper
pdflatex -interaction=nonstopmode -halt-on-error main.tex
pdflatex -interaction=nonstopmode -halt-on-error main.tex
\end{verbatim}

\section{Conclusion}
The original ``truth topology'' objective was too broad for the observed evidence.
The final validated contribution is a replicated representation--decoder dissociation:
semantic non-entailment is strongly encoded in hidden states but systematically under-surfaced at output time.
The correct framing is no longer a search for a single topology-of-truth scalar, but a study of how semantic uncertainty is preserved internally and lost at decoder readout.

\section*{References}
\renewcommand{\refname}{}
\begin{thebibliography}{99}

\bibitem{alain2016probes}
Guillaume Alain and Yoshua Bengio.
\newblock Understanding intermediate layers using linear classifier probes.
\newblock In \emph{ICLR Workshop Track}, 2016.

\bibitem{bauer2021ripser}
Ulrich Bauer.
\newblock Ripser: efficient computation of Vietoris--Rips persistence barcodes.
\newblock \emph{Journal of Applied and Computational Topology}, 5:391--423, 2021.

\bibitem{carlsson2009}
Gunnar Carlsson.
\newblock Topology and data.
\newblock \emph{Bulletin of the American Mathematical Society}, 46(2):255--308, 2009.

\bibitem{cobbe2021gsm8k}
Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman.
\newblock Training verifiers to solve math word problems.
\newblock arXiv:2110.14168, 2021.

\bibitem{burns2022discovering}
Collin Burns, Haotian Ye, Dan Klein, and Jacob Steinhardt.
\newblock Discovering latent knowledge in language models without supervision.
\newblock arXiv:2212.03827, 2022.

\bibitem{gemma_model_card}
Google.
\newblock Gemma 3 4B Instruct model card.
\newblock \url{https://huggingface.co/google/gemma-3-4b-it}. Accessed 2026-04-14.

\bibitem{liang2022holistic}
Percy Liang, Rishi Bommasani, Tony Lee, et al.
\newblock Holistic evaluation of language models.
\newblock \emph{Transactions on Machine Learning Research}, 2023.

\bibitem{hubinger2021elk}
Evan Hubinger, Jack Clark, et al.
\newblock Eliciting latent knowledge: How to tell if your eyes deceive you.
\newblock Alignment Research Center report, 2021.
\newblock \url{https://www.alignment.org/blog/arcs-first-technical-report-eliciting-latent-knowledge/}. Accessed 2026-04-14.

\bibitem{qwen_model_card}
Qwen Team.
\newblock Qwen3.5 model family card.
\newblock \url{https://huggingface.co/Qwen/Qwen3.5-4B}. Accessed 2026-04-14.

\bibitem{sharma2023sycophancy}
Mrinank Sharma, Meg Tong, et al.
\newblock Towards understanding sycophancy in language models.
\newblock arXiv:2310.13548, 2023.
\newblock \url{https://www.anthropic.com/research/towards-understanding-sycophancy-in-language-models}. Accessed 2026-04-14.

\bibitem{turner2023actadd}
Alexander Matt Turner, Lisa Thiergart, Gavin Leech, David Udell, Ulisse Mini, and Monte MacDiarmid.
\newblock Activation Addition: Steering Language Models Without Optimization.
\newblock arXiv:2308.10248, 2023.

\bibitem{zou2023repe}
Andy Zou, Zifan Wang, Nicholas Carlini, Milad Nasr, J. Zico Kolter, and Matt Fredrikson.
\newblock Representation Engineering: A Top-Down Approach to AI Transparency.
\newblock arXiv:2310.01405, 2023.

\end{thebibliography}

\appendix
\section{Appendix A: Full Reproduction Commands}
\begin{verbatim}
# Phase A dynamic topology
python3 scripts/analyze_paired_dynamic.py \
  --artifact-08b artifacts/paired_qwen35_08b_10 \
  --artifact-2b artifacts/paired_qwen35_2b_10 \
  --out-dir artifacts/paired_dynamic_qwen35

# Phase A corrected window test
python3 scripts/analyze_2b_window_corrected.py \
  --artifact-dir artifacts/paired_qwen35_2b_10 \
  --out-dir artifacts/qwen35_2b_window_corrected

# Phase B fixed-decoding run
python3 scripts/run_multi_sample_questions.py \
  --model-id Qwen/Qwen3.5-2B \
  --question-start 0 --question-count 20 --samples-per-question 9 \
  --temperature 0.7 --top-p 0.95 --max-new-tokens 384 \
  --artifact-dir artifacts/qwen35_2b_within_question

# Phase B cap sensitivity
python3 scripts/run_cap_sensitivity.py \
  --model-id Qwen/Qwen3.5-2B \
  --source-artifact-dir artifacts/qwen35_2b_within_question \
  --out-dir artifacts/qwen35_2b_cap_sensitivity_640 \
  --temperature 0.7 --top-p 0.95 --max-new-tokens 640

# Phase C dataset
python3 scripts/generate_micro_world_dataset.py \
  --out-dir artifacts/micro_world_v1/dataset \
  --seed 1729 --train-worlds 100 --dev-worlds 25 --eval-worlds 100 \
  --props-per-world 9 --paraphrases-per-prop 8

# Phase C readout intervention pilot
python3 scripts/run_readout_intervention.py \
  --out-dir artifacts/micro_world_v1/readout_intervention

# Phase C latent residual steering pilot
python3 scripts/run_latent_readout_steering.py \
  --out-dir artifacts/micro_world_v1/latent_readout_steering

# Phase C nonlinear probe sensitivity
python3 scripts/run_mlp_probe_sensitivity.py \
  --out-dir artifacts/micro_world_v1/probe_mlp_sensitivity
\end{verbatim}

\section{Appendix B: Protocol Details}
\subsection{Dataset Generation Protocol}
The generator samples partial worlds with 4--6 entities, 2 attributes, and 2 relations per world by default.
Per relation/attribute assignment, facts are sampled as explicit positive, explicit negative, or omitted (Unknown) states.
For each world, proposition sampling is quota-balanced over \texttt{True}/\texttt{False}/\texttt{Unknown} labels before paraphrase rendering.

\subsection{Train/Test Separation}
Probe training uses only train-world manifests (\texttt{status=ok}, label in \{\texttt{True},\texttt{False},\texttt{Unknown}\}).
Evaluation uses held-out eval-world manifests.
No sentence-level random split is used in the reported probe tables.

\subsection{Probe Fitting Defaults}
All reported linear probes use:
\begin{itemize}
  \item \texttt{StandardScaler} (mean/std normalization),
  \item \texttt{LogisticRegression(solver=lbfgs, C=1.0, max\_iter=4000)},
  \item three-class label set in fixed order \(\{\texttt{True},\texttt{False},\texttt{Unknown}\}\),
  \item zero-division-safe precision/recall/F1 reporting.
\end{itemize}

\subsection{Layer Sweep Inclusion Rules}
For \texttt{verdict\_token} sweeps, examples with no generated token are excluded via a validity mask.
For \texttt{prompt\_last} sweeps, all \texttt{status=ok} examples are included.
Non-finite activations are replaced with zero prior to fitting.

\section{Appendix C: Definitions and Notation}
\begin{itemize}
  \item \textbf{Unknown (semantic class):} non-entailment in the generator's three-valued semantics.
  \item \textbf{Unknown (decoder output):} emitted label string parsed from model output.
  \item \textbf{Unknown recoverability:} recall of the Unknown class under a linear probe on hidden states.
  \item \textbf{Representation--decoder gap:} probe Unknown recall minus decoder Unknown recall on matched eval sets.
\end{itemize}
These are related but non-identical quantities, and they are reported separately throughout.

\section{Appendix D: Additional Robustness Artifacts}
Additional reviewer-targeted artifacts added in this revision:
\begin{itemize}
  \item cross-proposition geometry control summary: \texttt{artifacts/micro\_world\_v1/cross\_proposition\_geometry\_control\_summary.csv},
  \item world-bootstrap Unknown recall intervals: \texttt{artifacts/micro\_world\_v1/comparison\_unknown\_recall\_bootstrap\_ci.csv}.
  \item readout-intervention pilot summaries: \texttt{artifacts/micro\_world\_v1/readout\_intervention/}.
  \item latent-steering pilot summaries: \texttt{artifacts/micro\_world\_v1/latent\_readout\_steering/}.
  \item nonlinear probe sensitivity summaries: \texttt{artifacts/micro\_world\_v1/probe\_mlp\_sensitivity/}.
\end{itemize}

\end{document}