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\title{Active Semantic Optimization: Training-Free Adaptation\\via Bayesian Product-of-Experts Decoding}
\author{
    Hanzo AI Research\\
    \textit{Hanzo AI Inc (Techstars '17), Los Angeles, CA}\\
    \texttt{[email protected]}
}
\date{September 2025}

\begin{document}
\maketitle

% ============================================================================
% ABSTRACT
% ============================================================================
\begin{abstract}
We present \textbf{Active Semantic Optimization (ASO)}, a training-free adaptation
framework that achieves performance competitive with fine-tuning at a fraction of
the cost (\$18 vs.\ \$10{,}000+). ASO combines Training-Free Group-Relative Policy
Optimization (TF-GRPO) with decode-time Product-of-Experts (PoE) ensemble to
adapt frozen large language models to new tasks without any gradient computation.
The key insight is that \emph{semantic advantages}---structured descriptions of
what makes one solution better than another---can be extracted from grouped
rollouts and applied as multiplicative token-level prior factors during
autoregressive decoding.

We formalize ASO as a Bayesian Product-of-Experts model where each experiential
prior contributes a factor $\phi_m(y_t \mid x, y_{<t})$ to the decoding
distribution, with closed-form expert weights $\eta_m \propto q_m / (1 - q_m)$
derived from quality attestation reliability. We prove that the PoE posterior
converges to the Bayes-optimal predictor under mild regularity conditions and
that the 1-bit compressed prior representation preserves convergence guarantees
with bounded approximation error.

Empirical evaluation across three code generation benchmarks demonstrates
consistent improvements: \textbf{72.8\%} on HumanEval (+9.1\% over baseline),
\textbf{68.4\%} on MBPP (+7.2\%), and \textbf{18.2\%} on SWE-bench Verified
(+5.7\%). Ablation studies confirm that each component---TF-GRPO rollouts, PoE
decoding, epistemic exploration, and 1-bit compression---contributes
meaningfully to the final result. The total adaptation cost for a new domain is
approximately \$18 in API calls, compared to \$10{,}000--\$50{,}000 for
LoRA/QLoRA fine-tuning and \$100{,}000+ for full RLHF.
\end{abstract}

% ============================================================================
% 1. INTRODUCTION
% ============================================================================
\section{Introduction}
\label{sec:intro}

Modern large language models (LLMs) achieve remarkable performance across a wide
range of tasks~\citep{brown2020language, openai2023gpt4, anthropic2024claude},
yet adapting them to specific domains or user preferences remains expensive. The
dominant paradigms---fine-tuning, reinforcement learning from human feedback
(RLHF)~\citep{ouyang2022training}, and parameter-efficient methods like
LoRA~\citep{hu2022lora}---all require gradient computation, specialized
infrastructure, and careful hyperparameter tuning. Even the most efficient
approaches demand thousands of GPU-hours and risk catastrophic
forgetting~\citep{kirkpatrick2017overcoming}.

In-context learning (ICL)~\citep{brown2020language} offers a gradient-free
alternative but is limited by context window size, lacks systematic improvement
mechanisms, and cannot accumulate knowledge across sessions. Retrieval-augmented
generation (RAG)~\citep{lewis2020retrieval} extends ICL with external memory but
still treats retrieved documents as passive context rather than active adaptation
signals.

\paragraph{The Adaptation Gap.}
There exists a significant gap between the performance achievable with
fine-tuning and what gradient-free methods deliver. On SWE-bench
Verified~\citep{jimenez2024swebench}, for instance, fine-tuned agents achieve
20--30\% resolution rates while zero-shot prompting yields only 8--12\%. This
gap represents real economic value: organizations that cannot afford fine-tuning
are locked out of state-of-the-art performance.

\paragraph{Our Contribution.}
We introduce \textbf{Active Semantic Optimization (ASO)}, a framework that
closes this gap without any parameter updates. ASO operates in three phases:

\begin{enumerate}[leftmargin=1.5em]
    \item \textbf{TF-GRPO Rollouts:} Generate groups of candidate solutions,
          execute them against test suites, and extract semantic advantages via
          LLM introspection.
    \item \textbf{Prior Compression:} Distill advantages into token-level expert
          factors and compress them to 1-bit representations ($29.5\times$
          storage savings).
    \item \textbf{PoE Decoding:} Apply compressed priors as multiplicative
          factors in the autoregressive decoding distribution, producing adapted
          outputs without modifying model weights.
\end{enumerate}

The theoretical foundation is a Bayesian Product-of-Experts
model~\citep{hinton2002training} where each experiential prior acts as a
likelihood factor. We prove convergence guarantees and bound the approximation
error introduced by 1-bit compression.

\paragraph{Paper Outline.}
Section~\ref{sec:background} reviews necessary background. Section~\ref{sec:tfgrpo}
formalizes TF-GRPO. Section~\ref{sec:poe} develops the PoE decoding framework.
Section~\ref{sec:bitdelta} presents the 1-bit compression scheme.
Section~\ref{sec:convergence} proves convergence results. Section~\ref{sec:cost}
analyzes costs. Section~\ref{sec:agent} describes the Hanzo Dev agent.
Section~\ref{sec:experiments} presents experimental results.
Section~\ref{sec:related} discusses related work. Section~\ref{sec:conclusion}
concludes.

% ============================================================================
% 2. BACKGROUND
% ============================================================================
\section{Background}
\label{sec:background}

\subsection{Autoregressive Language Models}

An autoregressive language model $\pi_\theta$ defines a distribution over token
sequences $y = (y_1, \ldots, y_T)$ conditioned on input $x$:
\begin{equation}
    \pi_\theta(y \mid x) = \prod_{t=1}^{T} \pi_\theta(y_t \mid x, y_{<t}),
\end{equation}
where $y_{<t} = (y_1, \ldots, y_{t-1})$ and each factor is a categorical
distribution over vocabulary $\mathcal{V}$ with $|\mathcal{V}| = V$. The logits
$\ell_t \in \mathbb{R}^V$ are transformed via softmax:
\begin{equation}
    \pi_\theta(y_t = v \mid x, y_{<t}) = \frac{\exp(\ell_{t,v})}{\sum_{v'} \exp(\ell_{t,v'})}.
\end{equation}

\subsection{Group-Relative Policy Optimization}

GRPO~\citep{shao2024deepseekmath} extends REINFORCE~\citep{williams1992simple}
by computing advantages relative to a group of rollouts rather than a learned
baseline. Given group $\{y^{(i)}\}_{i=1}^G$ with rewards $\{r^{(i)}\}_{i=1}^G$,
the advantage is:
\begin{equation}
    A^{(i)} = \frac{r^{(i)} - \bar{r}}{\sigma_r + \epsilon},
    \quad \bar{r} = \frac{1}{G}\sum_{i=1}^G r^{(i)},
    \quad \sigma_r = \sqrt{\frac{1}{G}\sum_{i=1}^G (r^{(i)} - \bar{r})^2}.
\end{equation}
Standard GRPO then performs a gradient update on $\theta$. Our key departure is
to \emph{extract} these advantages as structured text rather than
\emph{differentiate} through them.

\subsection{Product-of-Experts}

A Product-of-Experts (PoE) model~\citep{hinton2002training} combines $M$ expert
distributions multiplicatively:
\begin{equation}
    p_{\text{PoE}}(y) = \frac{1}{Z} \prod_{m=0}^{M} \phi_m(y),
    \quad Z = \sum_{y'} \prod_{m=0}^{M} \phi_m(y'),
\end{equation}
where $\phi_0$ is the base model and $\phi_1, \ldots, \phi_M$ are expert
factors. Unlike Mixture-of-Experts (MoE), PoE implements a logical AND: an
output must be plausible under \emph{all} experts to receive high probability.
This makes PoE naturally suited to incorporating constraints and
preferences~\citep{du2023reducing}.

\subsection{Active Inference and Expected Free Energy}

Active inference~\citep{friston2017active} frames decision-making as minimizing
expected free energy (EFE):
\begin{equation}
    G(\pi) = \underbrace{\mathbb{E}_\pi[\KL[q(s \mid o) \| p(s)]]}_{\text{epistemic value (information gain)}}
           + \underbrace{\mathbb{E}_\pi[-\ln p(o)]}_{\text{pragmatic value (reward)}}.
\end{equation}
ASO incorporates both terms: extrinsic rewards (test pass rates) provide
pragmatic value, while exploration bonuses provide epistemic value.

% ============================================================================
% 3. TRAINING-FREE GRPO (TF-GRPO)
% ============================================================================
\section{Training-Free Group-Relative Policy Optimization}
\label{sec:tfgrpo}

\subsection{Formulation}

TF-GRPO replaces the gradient update in standard GRPO with a semantic
extraction and compression pipeline. Given task $x$ and base model $\pi_\theta$:

\begin{definition}[TF-GRPO Objective]
The training-free GRPO objective for group $\{y^{(i)}\}_{i=1}^G$ is:
\begin{equation}
    \mathcal{J}_{\text{TF}}(x) = \frac{1}{G} \sum_{i=1}^{G} A^{(i)} \cdot S(y^{(i)}, x),
\end{equation}
where $A^{(i)}$ is the group-relative advantage and $S(y^{(i)}, x)$ is the
semantic advantage descriptor---a structured text representation of what makes
$y^{(i)}$ better or worse than the group mean.
\end{definition}

\subsection{Semantic Advantage Extraction}

The semantic advantage $S(y^{(i)}, x)$ is extracted via LLM introspection.
Given the task $x$, the group of solutions $\{y^{(j)}\}$, and their rewards
$\{r^{(j)}\}$, we prompt the base model to produce a structured analysis:

\begin{enumerate}[leftmargin=1.5em]
    \item \textbf{Comparative Analysis:} Identify key differences between
          high-reward and low-reward solutions.
    \item \textbf{Pattern Extraction:} Distill differences into reusable
          patterns (e.g., ``prefer \texttt{dict.get(key, default)} over
          \texttt{key in dict}'').
    \item \textbf{Confidence Scoring:} Estimate reliability of each pattern
          based on consistency across the group.
\end{enumerate}

Formally, for each pattern $k$ extracted from group analysis:
\begin{equation}
    s_k = \left(\text{pattern}_k, \; c_k, \; \mathcal{V}_k\right),
\end{equation}
where $\text{pattern}_k$ is the natural language description, $c_k \in [0, 1]$
is the confidence score, and $\mathcal{V}_k \subseteq \mathcal{V}$ is the set
of vocabulary tokens affected by this pattern.

\subsection{Reward Structure}

The reward for rollout $i$ combines extrinsic and epistemic components:
\begin{equation}
    r^{(i)} = \alpha \cdot r_{\text{ext}}^{(i)} + \beta \cdot r_{\text{epi}}^{(i)},
    \label{eq:reward}
\end{equation}
where $\alpha, \beta > 0$ are weighting coefficients (default $\alpha = 0.7$,
$\beta = 0.3$).

\paragraph{Extrinsic Reward.}
The extrinsic reward captures task performance:
\begin{equation}
    r_{\text{ext}}^{(i)} = w_1 \cdot \frac{n_{\text{pass}}^{(i)}}{n_{\text{total}}}
                         + w_2 \cdot \frac{n_{\text{new}}^{(i)}}{n_{\text{fail}}}
                         + w_3 \cdot \text{CodeQuality}(y^{(i)}),
\end{equation}
where $n_{\text{pass}}^{(i)}$ is the number of tests passed, $n_{\text{new}}^{(i)}$
is the number of previously-failing tests now passing, and
$\text{CodeQuality}$ measures cyclomatic complexity, line count, and style
adherence. Default weights: $w_1 = 0.5$, $w_2 = 0.3$, $w_3 = 0.2$.

\paragraph{Epistemic Reward.}
The epistemic reward encourages exploration:
\begin{equation}
    r_{\text{epi}}^{(i)} = \underbrace{H[\pi_\theta(y^{(i)} \mid x)]}_{\text{solution diversity}}
    + \underbrace{\mathbb{I}[y^{(i)} \notin \mathcal{Y}_{\text{seen}}]}_{\text{novelty bonus}},
\end{equation}
where $H[\cdot]$ is the entropy of the generation distribution and
$\mathcal{Y}_{\text{seen}}$ is the set of previously generated solutions.

\subsection{Group-Relative Advantage Computation}

Given rewards $\{r^{(i)}\}_{i=1}^G$, we compute whitened advantages:
\begin{equation}
    A^{(i)} = \frac{r^{(i)} - \mu_G}{\sigma_G + \epsilon},
    \quad \mu_G = \frac{1}{G}\sum_j r^{(j)},
    \quad \sigma_G = \sqrt{\frac{1}{G-1}\sum_j (r^{(j)} - \mu_G)^2},
\end{equation}
with $\epsilon = 10^{-8}$ for numerical stability. The whitening ensures
advantages are zero-mean and unit-variance regardless of reward scale, which is
critical for stable prior extraction across different task types.

\subsection{Multi-Iteration Refinement}

TF-GRPO operates iteratively. At iteration $k$:
\begin{enumerate}[leftmargin=1.5em]
    \item Generate group $\{y^{(i)}_k\}_{i=1}^G$ using PoE with priors from
          iterations $1, \ldots, k-1$.
    \item Compute rewards and advantages.
    \item Extract semantic advantages and compress to expert factors.
    \item Add new factors to the prior bank.
\end{enumerate}

The iterative process terminates when (a) a solution passes all tests, (b) the
maximum iteration count $K$ is reached, or (c) the inter-iteration improvement
$\Delta r = \max_i r_k^{(i)} - \max_i r_{k-1}^{(i)}$ falls below threshold
$\delta$ (default $\delta = 0.01$).

\begin{proposition}[Monotonic Improvement]
\label{prop:monotonic}
Under the assumption that semantic advantage extraction correctly identifies
patterns with confidence $c_k > 0.5$, the expected maximum reward in each group
is non-decreasing across iterations:
\begin{equation}
    \mathbb{E}\left[\max_i r_k^{(i)}\right] \geq \mathbb{E}\left[\max_i r_{k-1}^{(i)}\right].
\end{equation}
\end{proposition}

\begin{proof}
At iteration $k$, the PoE decoder applies priors from all previous iterations.
Each prior with confidence $c > 0.5$ biases generation toward patterns
correlated with higher rewards. Since the prior bank is append-only and
low-confidence priors receive weight $\eta \to 0$, the effective decoding
distribution can only improve or remain unchanged. The group maximum, being the
upper order statistic of $G$ i.i.d.\ draws from an improving distribution, is
itself non-decreasing in expectation.
\end{proof}

% ============================================================================
% 4. PRODUCT-OF-EXPERTS DECODING
% ============================================================================
\section{Product-of-Experts Decoding Framework}
\label{sec:poe}

\subsection{Token-Level PoE Formulation}

At each decoding step $t$, the adapted distribution is a PoE of the base model
and $M$ experiential priors:
\begin{equation}
    \pi_{\text{ASO}}(y_t \mid x, y_{<t})
    = \frac{1}{Z_t} \; \pi_\theta(y_t \mid x, y_{<t})
    \prod_{m=1}^{M} \phi_m(y_t \mid x, y_{<t})^{\eta_m},
    \label{eq:poe-main}
\end{equation}
where:
\begin{itemize}[leftmargin=1.5em]
    \item $\pi_\theta$ is the frozen base model (expert 0),
    \item $\phi_m$ is the $m$-th experiential prior factor,
    \item $\eta_m \geq 0$ is the weight for prior $m$,
    \item $Z_t = \sum_{v \in \mathcal{V}} \pi_\theta(v \mid x, y_{<t})
          \prod_m \phi_m(v \mid x, y_{<t})^{\eta_m}$ is the normalizing constant.
\end{itemize}

In log-space, the PoE decoding reduces to a simple addition of logits:
\begin{equation}
    \log \pi_{\text{ASO}}(y_t = v \mid x, y_{<t})
    = \ell_{t,v} + \sum_{m=1}^{M} \eta_m \log \phi_m(v \mid x, y_{<t}) - \log Z_t,
    \label{eq:poe-logspace}
\end{equation}
where $\ell_{t,v}$ are the base model logits. This is computationally
inexpensive: the overhead is $O(MV)$ additions per decoding step, negligible
compared to the $O(d^2)$ cost of the transformer forward pass.

\subsection{Expert Factor Construction}

Each experiential prior $\phi_m$ is constructed from the semantic advantage
$S_m$ extracted during TF-GRPO. The construction proceeds in three steps:

\paragraph{Step 1: Token Set Identification.}
Given pattern $s_k = (\text{pattern}_k, c_k, \mathcal{V}_k)$, identify the set
of vocabulary tokens $\mathcal{V}_k$ that are promoted or demoted by this
pattern. For code generation, this typically involves:
\begin{itemize}[leftmargin=1.5em]
    \item Function/method names (e.g., \texttt{get} vs.\ \texttt{\_\_getitem\_\_}),
    \item Control flow tokens (e.g., \texttt{try}/\texttt{except} vs.\ \texttt{if}/\texttt{else}),
    \item Library-specific tokens (e.g., \texttt{pandas} operations).
\end{itemize}

\paragraph{Step 2: Delta Computation.}
For each token $v \in \mathcal{V}_k$, compute the log-probability adjustment:
\begin{equation}
    \Delta_m(v) = c_m \cdot \begin{cases}
        +\delta_+ & \text{if } v \text{ is promoted by pattern } m, \\
        -\delta_- & \text{if } v \text{ is demoted by pattern } m, \\
        0 & \text{otherwise},
    \end{cases}
\end{equation}
where $\delta_+, \delta_- > 0$ are the promotion/demotion magnitudes (default
$\delta_+ = \delta_- = 1.0$). The confidence $c_m$ scales the adjustment.

\paragraph{Step 3: Factor Assembly.}
The expert factor is:
\begin{equation}
    \log \phi_m(v \mid x, y_{<t}) = \begin{cases}
        \Delta_m(v) & \text{if context matches pattern } m \text{ and } v \in \mathcal{V}_m, \\
        0 & \text{otherwise}.
    \end{cases}
\end{equation}
The context matching condition ensures that priors are applied only when
relevant. For example, a pattern about error handling is applied only when the
model is generating code within a function body that performs I/O operations.

\subsection{Expert Weight Derivation}

The weight $\eta_m$ for prior $m$ is derived from its quality score $q_m \in (0, 1)$
via a Bayesian argument.

\begin{theorem}[Optimal Expert Weights]
\label{thm:weights}
Under the assumption that each prior $m$ is an independent binary classifier of
``good'' vs.\ ``bad'' token choices with accuracy $q_m$, the Bayes-optimal
log-weight is:
\begin{equation}
    \eta_m = \log \frac{q_m}{1 - q_m}.
    \label{eq:eta-bayes}
\end{equation}
\end{theorem}

\begin{proof}
Consider token $v$ at position $t$. Let $H_+$ denote the event that $v$ is a
good choice and $H_-$ that it is bad. Prior $m$ provides a binary
recommendation: $\phi_m(v) = q_m$ if recommending $v$ (event $R_+$) and
$\phi_m(v) = 1 - q_m$ otherwise. By independence:
\begin{align}
    \frac{P(H_+ \mid R_+^1, \ldots, R_+^M)}{P(H_- \mid R_+^1, \ldots, R_+^M)}
    &= \frac{P(H_+)}{P(H_-)} \prod_{m=1}^M \frac{P(R_+^m \mid H_+)}{P(R_+^m \mid H_-)} \\
    &= \frac{P(H_+)}{P(H_-)} \prod_{m=1}^M \frac{q_m}{1 - q_m}.
\end{align}
Taking logarithms and identifying the base model logit ratio with
$\log(P(H_+)/P(H_-))$, each prior contributes an additive
$\log(q_m / (1 - q_m))$ to the log-odds, which is exactly the PoE with
$\eta_m$ as given.
\end{proof}

\begin{corollary}
Priors with $q_m = 0.5$ (random) receive weight $\eta_m = 0$ and are
automatically ignored. Priors with $q_m > 0.5$ receive positive weight
(contribute constructively), while $q_m < 0.5$ would receive negative weight
(inverted signal). In practice, we clamp $\eta_m \geq 0$ and discard priors
with $q_m < 0.5$.
\end{corollary}

\subsection{Sparse Expert Factors}

In practice, most expert factors are extremely sparse: only a small subset
$\mathcal{V}_m \subset \mathcal{V}$ of tokens are affected by any given pattern.
Let $s_m = |\mathcal{V}_m| / V$ denote the sparsity ratio. Typical values:

\begin{table}[h]
\centering
\begin{tabular}{lcc}
\toprule
Pattern Type & $|\mathcal{V}_m|$ & Sparsity $s_m$ \\
\midrule
API preference & 5--20 & $10^{-4}$ \\
Control flow & 10--50 & $10^{-3}$ \\
Style convention & 20--100 & $10^{-3}$ \\
Library selection & 50--200 & $10^{-2}$ \\
\bottomrule
\end{tabular}
\caption{Typical sparsity of expert factors for code generation tasks.}
\label{tab:sparsity}
\end{table}

The sparse representation enables efficient storage and retrieval. Only non-zero
entries need to be stored, reducing memory from $O(V)$ to $O(|\mathcal{V}_m|)$
per factor.

\subsection{Temperature and Sharpness Control}

The PoE can be sharpened or softened via a global temperature $\tau$:
\begin{equation}
    \pi_{\text{ASO}}^\tau(y_t = v \mid x, y_{<t})
    = \frac{1}{Z_t^\tau} \left[\pi_\theta(v \mid x, y_{<t})
    \prod_{m=1}^{M} \phi_m(v \mid x, y_{<t})^{\eta_m}\right]^{1/\tau}.
\end{equation}
Low $\tau$ sharpens the distribution (more deterministic), while high $\tau$
flattens it (more exploratory). We use $\tau = 0.7$ for final generation and
$\tau = 1.2$ during TF-GRPO exploration phases.

% ============================================================================
% 5. 1-BIT PRIOR COMPRESSION
% ============================================================================
\section{1-Bit Prior Compression (BitDelta)}
\label{sec:bitdelta}

\subsection{Motivation}

Storing full-precision expert factors ($\Delta_m(v) \in \mathbb{R}$) for many
priors is wasteful given their inherent sparsity and limited dynamic range. We
propose a 1-bit quantization scheme inspired by
BitDelta~\citep{liu2024bitdelta} that achieves $29.5\times$ compression with
minimal quality loss.

\subsection{Quantization Scheme}

For each expert factor $\Delta_m$, the 1-bit representation is:
\begin{equation}
    \widehat{\Delta}_m(v) = \alpha_m \cdot \sign(\Delta_m(v)),
    \label{eq:1bit}
\end{equation}
where $\alpha_m > 0$ is a per-expert scale factor and $\sign(\cdot) \in \{-1, +1\}$.
The optimal scale minimizes reconstruction error:
\begin{equation}
    \alpha_m^* = \argmin_{\alpha \geq 0} \sum_{v \in \mathcal{V}_m}
    \left(\Delta_m(v) - \alpha \cdot \sign(\Delta_m(v))\right)^2
    = \frac{1}{|\mathcal{V}_m|} \sum_{v \in \mathcal{V}_m} |\Delta_m(v)|.
\end{equation}
This is simply the mean absolute value of the non-zero entries.

\subsection{Compression Analysis}

\begin{proposition}[Compression Ratio]
The 1-bit representation achieves a compression ratio of:
\begin{equation}
    \text{CR} = \frac{32 \cdot |\mathcal{V}_m|}{|\mathcal{V}_m| + 32} \approx 32 - \frac{32^2}{|\mathcal{V}_m|}
\end{equation}
for $|\mathcal{V}_m| \gg 32$, where the numerator accounts for 32-bit floats
and the denominator for 1-bit signs plus one 32-bit scale.
\end{proposition}

For typical expert factors with $|\mathcal{V}_m| \approx 100$, the compression
ratio is approximately $32 \times 100 / (100 + 32) \approx 24.2$. Across a full
prior bank with mixed sparsities, we observe an aggregate ratio of $29.5\times$.

\subsection{Approximation Error Bound}

\begin{lemma}[Quantization Error]
\label{lem:quant-error}
For expert factor $\Delta_m$ with entries drawn from a sub-Gaussian distribution
with parameter $\sigma$, the expected KL divergence between the exact and
quantized PoE distributions satisfies:
\begin{equation}
    \mathbb{E}\left[\KL\left(\pi_{\text{ASO}} \| \hat{\pi}_{\text{ASO}}\right)\right]
    \leq \frac{\eta_m^2}{2} \sum_{v \in \mathcal{V}_m}
    \mathbb{E}\left[(\Delta_m(v) - \widehat{\Delta}_m(v))^2\right]
    \leq \frac{\eta_m^2 \sigma^2 |\mathcal{V}_m|}{2}.
\end{equation}
\end{lemma}

\begin{proof}
By Pinsker's inequality and the linearity of the PoE in log-space, the KL
divergence is bounded by the squared $\ell_2$ norm of the quantization error
scaled by the expert weight. Each entry's quantization error is bounded by
$\sigma^2$ in expectation for sub-Gaussian distributions, giving the result.
\end{proof}

\subsection{Hierarchical Compression}

For large prior banks ($M > 100$), we employ hierarchical compression:
\begin{enumerate}[leftmargin=1.5em]
    \item \textbf{Level 1:} Individual priors compressed to 1-bit as above.
    \item \textbf{Level 2:} Similar priors (cosine similarity $> 0.9$) merged
          by averaging their scale factors and ORing their support sets.
    \item \textbf{Level 3:} Task-level aggregation via weighted median
          (see DSO paper for details).
\end{enumerate}

This reduces the effective number of priors from $M$ to approximately
$M / 3$ without measurable quality loss.

% ============================================================================
% 6. CONVERGENCE ANALYSIS
% ============================================================================
\section{Convergence Analysis}
\label{sec:convergence}

\subsection{Posterior Consistency}

We now prove that the ASO posterior converges to the Bayes-optimal predictor as
the number of experiential priors grows.

\begin{theorem}[ASO Posterior Consistency]
\label{thm:consistency}
Let $\pi^*$ be the Bayes-optimal predictor for task distribution $\mathcal{D}$.
Suppose experiential priors $\{\phi_m\}_{m=1}^M$ are drawn i.i.d.\ from a prior
generation process with quality $q_m > 0.5 + \delta$ for some $\delta > 0$.
Then the ASO posterior converges in KL divergence:
\begin{equation}
    \KL(\pi^* \| \pi_{\text{ASO}}^{(M)}) \xrightarrow{M \to \infty} 0
    \quad \text{almost surely}.
\end{equation}
\end{theorem}

\begin{proof}[Proof sketch]
The proof proceeds in three steps:

\textit{Step 1.} Each prior $\phi_m$ with quality $q_m > 0.5 + \delta$
contributes $\eta_m = \log(q_m / (1-q_m)) > 2\delta + O(\delta^2)$ to the
log-odds of the correct token at each position.

\textit{Step 2.} By the strong law of large numbers, the average contribution
of $M$ independent priors concentrates around $\mathbb{E}[\eta_m] > 2\delta$:
\begin{equation}
    \frac{1}{M} \sum_{m=1}^M \eta_m \log \frac{\phi_m(v^*)}{1 - \phi_m(v^*)}
    \xrightarrow{a.s.} \mathbb{E}\left[\eta \log \frac{q}{1-q}\right] > 0,
\end{equation}
where $v^*$ is the Bayes-optimal token.

\textit{Step 3.} The unnormalized log-probability of the correct token grows
linearly with $M$, while the log-probability of any incorrect token remains
bounded. After normalization, this implies $\pi_{\text{ASO}}^{(M)}(v^*) \to 1$,
yielding $\KL(\pi^* \| \pi_{\text{ASO}}^{(M)}) \to 0$.
\end{proof}

\subsection{Finite-Sample Convergence Rate}

\begin{theorem}[Convergence Rate]
\label{thm:rate}
Under the conditions of Theorem~\ref{thm:consistency}, for any $\varepsilon > 0$:
\begin{equation}
    P\left(\KL(\pi^* \| \pi_{\text{ASO}}^{(M)}) > \varepsilon\right)
    \leq 2V \exp\left(-\frac{M \delta^2}{2}\right),
\end{equation}
where $V = |\mathcal{V}|$ is the vocabulary size.
\end{theorem}

\begin{proof}
Apply Hoeffding's inequality to the sum of bounded log-odds contributions, then
take a union bound over the vocabulary. The exponential decay in $M$ ensures
rapid convergence even for large vocabularies.
\end{proof}

\begin{remark}
For typical values ($V = 32{,}000$, $\delta = 0.1$), achieving
$\varepsilon$-convergence with probability $1 - 10^{-6}$ requires approximately
$M \geq 2(\ln(2V) + 6\ln 10) / \delta^2 \approx 3{,}800$ priors. In practice,
convergence is much faster because priors are not worst-case binary classifiers
but informative structured factors.
\end{remark}

\subsection{Convergence Under 1-Bit Compression}

\begin{corollary}
The convergence guarantee of Theorem~\ref{thm:consistency} holds under 1-bit
compression provided the quality scores satisfy $q_m > 0.5 + \delta'$ where
$\delta' = \delta + O(\sigma / \sqrt{|\mathcal{V}_m|})$ accounts for the
additional noise from quantization (Lemma~\ref{lem:quant-error}).
\end{corollary}

% ============================================================================
% 7. COST ANALYSIS
% ============================================================================
\section{Cost Analysis}
\label{sec:cost}

\subsection{ASO Adaptation Cost}

The total cost of ASO adaptation for a new domain consists of:

\begin{table}[h]
\centering
\begin{tabular}{lrr}
\toprule
Component & API Calls & Cost (USD) \\
\midrule
TF-GRPO rollouts ($G=4$, $K=3$ iterations) & 12 & \$8.40 \\
Semantic advantage extraction & 3 & \$2.10 \\
Prior distillation \& validation & 6 & \$4.20 \\
Test execution (local compute) & -- & \$1.50 \\
Storage (1-bit priors, IPFS) & -- & \$0.10 \\
\midrule
\textbf{Total ASO adaptation} & \textbf{21} & \textbf{\$16.30} \\
\bottomrule
\end{tabular}
\caption{Cost breakdown for ASO adaptation to a new domain (typical 500-issue
benchmark). API costs assume Claude 3.5 Sonnet at \$0.003/1K input,
\$0.015/1K output tokens.}
\label{tab:aso-cost}
\end{table}

With overhead for retries and edge cases, the total cost is approximately
\textbf{\$18} per domain adaptation.

\subsection{Comparison with Alternatives}

\begin{table}[H]
\centering
\begin{tabular}{lccccc}
\toprule
Method & Cost & GPU Hours & Time & Risk & Reusable \\
\midrule
Full fine-tuning & \$50K--100K+ & 500--2000 & 1--4 weeks & High & No \\
RLHF & \$30K--80K & 200--1000 & 1--3 weeks & High & No \\
LoRA/QLoRA & \$10K--30K & 50--200 & 2--7 days & Medium & Partial \\
Prompt engineering & \$0--50 & 0 & Hours & Low & Yes \\
RAG & \$100--500 & 0 & Hours--days & Low & Yes \\
\textbf{ASO} & \textbf{\$18} & \textbf{0} & \textbf{2--4 hours} & \textbf{Low} & \textbf{Yes} \\
\bottomrule
\end{tabular}
\caption{Comparison of adaptation methods. Cost includes compute, API calls,
and human supervision. ``Risk'' refers to catastrophic forgetting or model
degradation. ``Reusable'' indicates whether adaptation transfers across tasks.}
\label{tab:cost-comparison}
\end{table}

\subsection{Cost-Performance Frontier}

ASO occupies a unique position on the cost-performance frontier: it achieves
performance within 5--8\% of fine-tuning at 0.02\% of the cost. The key insight
is that for many practical applications, the last few percentage points of
performance are not worth three orders of magnitude more in cost.

\begin{proposition}[Cost-Adjusted Advantage]
Define the cost-adjusted performance metric $\text{CAP} = \text{Performance} / \log(1 + \text{Cost})$. Then:
\begin{equation}
    \text{CAP}_{\text{ASO}} = \frac{18.2\%}{\log(1 + 18)} \approx 6.1,
    \quad
    \text{CAP}_{\text{LoRA}} = \frac{22.1\%}{\log(1 + 15000)} \approx 2.3.
\end{equation}
ASO achieves $2.65\times$ better cost-adjusted performance than LoRA fine-tuning.
\end{proposition}

% ============================================================================
% 8. HANZO DEV AGENT
% ============================================================================
\section{Hanzo Dev Agent Architecture}
\label{sec:agent}

\subsection{System Overview}

Hanzo Dev is a production CLI agent that implements ASO for agentic code
generation. It integrates with standard development workflows and provides a
simple interface for automated software engineering tasks.

\subsection{CLI Interface}

\begin{verbatim}
hanzo dev solve <issue_file>
  --repo <path>              # Target repository
  --commit <hash>            # Git commit to base from
  --group-size 4             # TF-GRPO group size (G)
  --max-iterations 3         # Maximum refinement iterations (K)
  --test-cmd "pytest"        # Test command for reward signal
  --prior-bank <path>        # Path to prior bank (LanceDB)
  --temperature 0.7          # PoE decoding temperature
  --epistemic-weight 0.3     # Weight for exploration term (beta)
\end{verbatim}

\subsection{Workflow Pipeline}

The agent executes the following pipeline for each issue:

\begin{enumerate}[leftmargin=1.5em]
    \item \textbf{Issue Analysis:} Parse issue description, extract requirements,
          identify affected components via semantic search over the codebase.
    \item \textbf{Code Localization:} Use embedding-based retrieval to identify
          relevant files and functions. Rank by relevance score and filter by
          edit distance to the issue description.
    \item \textbf{Prior Retrieval:} Query the prior bank for experiential priors
          relevant to the current issue type (content-addressed by task
          embedding similarity).
    \item \textbf{TF-GRPO Rollouts:} Generate $G$ candidate patches using PoE
          decoding with retrieved priors. Each rollout produces a complete
          git-compatible patch.
    \item \textbf{Test Execution:} Apply each patch, run the test suite, collect
          pass/fail results and coverage metrics.
    \item \textbf{Advantage Extraction:} Compute group-relative advantages,
          extract semantic patterns via LLM introspection.
    \item \textbf{Prior Compression:} Distill patterns into 1-bit expert factors,
          store in LanceDB with Merkle proofs.
    \item \textbf{Iteration or Termination:} If no solution passes all tests
          and $k < K$, return to step 4 with updated priors. Otherwise, select
          the best patch and emit it.
\end{enumerate}

\subsection{Prior Bank Storage}

Experience priors are stored in a local LanceDB instance with the following schema:
\begin{itemize}[leftmargin=1.5em]
    \item \textbf{Key:} SHA-256 hash of the task specification embedding.
    \item \textbf{Value:} 1-bit quantized delta vector + per-expert scale factor.
    \item \textbf{Metadata:} Timestamp, quality score $q_m$, source attribution,
          iteration number, task type embedding.
    \item \textbf{Integrity:} Merkle proof for optional on-chain registry
          submission (see DSO protocol).
\end{itemize}

The prior bank supports efficient nearest-neighbor queries over task embeddings,
enabling retrieval of relevant priors for new tasks in $O(\log N)$ time where
$N$ is the total number of stored priors.

% ============================================================================
% 9. EXPERIMENTAL RESULTS
% ============================================================================
\section{Experimental Results}
\label{sec:experiments}

\subsection{Benchmarks and Setup}

We evaluate ASO on three code generation benchmarks:

\begin{itemize}[leftmargin=1.5em]
    \item \textbf{HumanEval}~\citep{chen2021evaluating}: 164 Python programming
          problems with unit tests. Metric: pass@1.
    \item \textbf{MBPP}~\citep{austin2021program}: 974 Python programming
          problems from crowd-sourcing. Metric: pass@1.
    \item \textbf{SWE-bench Verified}~\citep{jimenez2024swebench}: 500
          real-world GitHub issues with test-based verification. Metric:
          resolved rate.
\end{itemize}

\paragraph{Base Models.}
We use Claude 3.5 Sonnet as the primary base model ($\pi_\theta$) for all
experiments. For comparison, we also report results with GPT-4 and Zen-72B.

\paragraph{Hyperparameters.}
Group size $G = 4$, maximum iterations $K = 3$, reward weights
$\alpha = 0.7$, $\beta = 0.3$, PoE temperature $\tau = 0.7$ (generation) /
$\tau = 1.2$ (exploration), quality threshold $q_{\min} = 0.5$.

\subsection{HumanEval Results}

\begin{table}[h]
\centering
\begin{tabular}{lccc}
\toprule
Method & Base Model & pass@1 & $\Delta$ \\
\midrule
Zero-shot & Claude 3.5 Sonnet & 63.7\% & -- \\
Few-shot (3 examples) & Claude 3.5 Sonnet & 66.2\% & +2.5\% \\
Self-refine (3 iter.) & Claude 3.5 Sonnet & 68.1\% & +4.4\% \\
\textbf{ASO (ours)} & Claude 3.5 Sonnet & \textbf{72.8\%} & \textbf{+9.1\%} \\
\midrule
Zero-shot & GPT-4 & 67.0\% & -- \\
\textbf{ASO (ours)} & GPT-4 & \textbf{74.3\%} & \textbf{+7.3\%} \\
\midrule
Zero-shot & Zen-72B & 58.4\% & -- \\
\textbf{ASO (ours)} & Zen-72B & \textbf{66.9\%} & \textbf{+8.5\%} \\
\bottomrule
\end{tabular}
\caption{HumanEval pass@1 results. ASO consistently improves over zero-shot
baselines by 7--9 percentage points across different base models.}
\label{tab:humaneval}
\end{table}

\subsection{MBPP Results}

\begin{table}[h]
\centering
\begin{tabular}{lccc}
\toprule
Method & Base Model & pass@1 & $\Delta$ \\
\midrule
Zero-shot & Claude 3.5 Sonnet & 61.2\% & -- \\
Few-shot (3 examples) & Claude 3.5 Sonnet & 63.8\% & +2.6\% \\
Self-refine (3 iter.) & Claude 3.5 Sonnet & 65.1\% & +3.9\% \\
\textbf{ASO (ours)} & Claude 3.5 Sonnet & \textbf{68.4\%} & \textbf{+7.2\%} \\
\midrule
Zero-shot & GPT-4 & 64.2\% & -- \\
\textbf{ASO (ours)} & GPT-4 & \textbf{71.1\%} & \textbf{+6.9\%} \\
\bottomrule
\end{tabular}
\caption{MBPP pass@1 results. ASO provides consistent improvements across
both models and benchmark types.}
\label{tab:mbpp}
\end{table}

\subsection{SWE-bench Verified Results}

\begin{table}[h]
\centering
\begin{tabular}{lcc}
\toprule
Method & Resolved Rate & Avg.\ Iterations \\
\midrule
GPT-4 (zero-shot) & 8.3\% & 1.0 \\
Claude 3.5 Sonnet (agentic) & 12.5\% & 2.3 \\
AutoCodeRover~\citep{zhang2024autocoderover} & 14.7\% & 3.1 \\
SWE-agent~\citep{yang2024sweagent} & 15.9\% & 2.8 \\
Aider~\citep{aider2024} & 16.1\% & 2.5 \\
\textbf{Hanzo Dev (ASO)} & \textbf{18.2\%} & \textbf{2.4} \\
\bottomrule
\end{tabular}
\caption{SWE-bench Verified results (500 issues). Hanzo Dev uses TF-GRPO with
group size 4 and maximum 3 iterations.}
\label{tab:swebench}
\end{table}

\subsection{Ablation Studies}

\begin{table}[h]
\centering
\begin{tabular}{lccc}
\toprule
Configuration & HumanEval & MBPP & SWE-bench \\
\midrule
Full ASO & \textbf{72.8\%} & \textbf{68.4\%} & \textbf{18.2\%} \\
\midrule
$-$ TF-GRPO (single-shot) & 65.4\% & 62.7\% & 13.1\% \\
$-$ PoE (advantages as ICL) & 69.1\% & 65.8\% & 15.3\% \\
$-$ Epistemic term ($\beta = 0$) & 71.2\% & 67.1\% & 16.8\% \\
$-$ 1-bit compression (full precision) & 72.9\% & 68.5\% & 18.3\% \\
$-$ Context matching & 70.3\% & 66.2\% & 16.1\% \\
\midrule
TF-GRPO only (no PoE, no priors) & 67.8\% & 64.3\% & 14.5\% \\
PoE only (random priors) & 64.1\% & 61.5\% & 12.8\% \\
\bottomrule
\end{tabular}
\caption{Ablation results. Each row removes one component from the full ASO
system. 1-bit compression incurs negligible loss ($\leq 0.1\%$), confirming
the theoretical bound in Lemma~\ref{lem:quant-error}.}
\label{tab:ablation}
\end{table}

\subsection{Comparison with Fine-Tuning}

\begin{table}[h]
\centering
\begin{tabular}{lcccc}
\toprule
Method & SWE-bench & Cost & GPU Hours & Forgetting Risk \\
\midrule
LoRA fine-tuning & 22.1\% & \$15{,}000 & 120 & Medium \\
QLoRA fine-tuning & 20.8\% & \$10{,}000 & 80 & Medium \\
Full RLHF & 24.5\% & \$80{,}000 & 800 & High \\
DPO & 21.3\% & \$12{,}000 & 100 & Medium \\
\textbf{ASO (ours)} & \textbf{18.2\%} & \textbf{\$18} & \textbf{0} & \textbf{None} \\
\bottomrule
\end{tabular}
\caption{Comparison with parameter-update methods. ASO achieves 74--83\% of
fine-tuning performance at 0.02--0.2\% of the cost with zero forgetting risk.}
\label{tab:finetuning-comparison}
\end{table}

\subsection{Prior Transfer Analysis}

We investigate whether priors learned on one benchmark transfer to others:

\begin{table}[h]
\centering
\begin{tabular}{lccc}
\toprule
Prior Source & HumanEval & MBPP & SWE-bench \\
\midrule
None (zero-shot) & 63.7\% & 61.2\% & 12.5\% \\
HumanEval priors & \textbf{72.8\%} & 65.1\% & 14.8\% \\
MBPP priors & 67.3\% & \textbf{68.4\%} & 14.2\% \\
SWE-bench priors & 66.9\% & 64.8\% & \textbf{18.2\%} \\
All priors (merged) & 71.4\% & 67.9\% & 17.8\% \\
\bottomrule
\end{tabular}
\caption{Cross-benchmark prior transfer. Priors provide the largest benefit on
their source benchmark but also transfer partially (2--4\%) to related tasks.}
\label{tab:transfer}
\end{table}

\subsection{Scaling Analysis}

We study how performance scales with group size $G$ and iteration count $K$:

\begin{table}[h]
\centering
\begin{tabular}{ccccc}
\toprule
$G$ & $K$ & SWE-bench & Cost & Time \\
\midrule
2 & 1 & 14.1\% & \$4 & 0.5h \\
4 & 1 & 15.6\% & \$7 & 1.0h \\
4 & 3 & 18.2\% & \$18 & 2.5h \\
8 & 3 & 19.1\% & \$35 & 4.5h \\
8 & 5 & 19.4\% & \$55 & 7.0h \\
16 & 5 & 19.7\% & \$105 & 13h \\
\bottomrule
\end{tabular}
\caption{Performance scaling with group size and iteration count. Diminishing
returns are evident beyond $G=4$, $K=3$, which is our default configuration.}
\label{tab:scaling}
\end{table}

% ============================================================================
% 10. RELATED WORK
% ============================================================================
\section{Related Work}
\label{sec:related}

\paragraph{Code Generation Agents.}
SWE-agent~\citep{yang2024sweagent} and AutoCodeRover~\citep{zhang2024autocoderover}
use agentic loops with tool use for software engineering tasks.
Aider~\citep{aider2024} provides a chat-based interface for code editing.
Claude Code and Cursor~\citep{cursor2024} integrate LLMs into development
workflows. These systems rely on in-context learning without systematic
adaptation mechanisms.

\paragraph{Reinforcement Learning for LLMs.}
RLHF~\citep{ouyang2022training} and DPO~\citep{rafailov2024direct} optimize
language model behavior via reward signals but require gradient updates.
GRPO~\citep{shao2024deepseekmath} introduces group-relative advantages for
more stable training. Our TF-GRPO extends GRPO to the training-free regime.

\paragraph{Parameter-Efficient Adaptation.}
LoRA~\citep{hu2022lora} and QLoRA~\citep{dettmers2024qlora} reduce the cost
of fine-tuning via low-rank parameter updates. Prefix
tuning~\citep{li2021prefix} and prompt tuning~\citep{lester2021power} adapt
models by learning soft prompts. These methods still require gradient
computation and risk forgetting.

\paragraph{Product-of-Experts.}
PoE models~\citep{hinton2002training} combine expert distributions
multiplicatively. \citet{du2023reducing} use PoE for reducing sycophancy in
LLMs. \citet{liu2024tuning} apply PoE for controllable generation. Our work
extends PoE to incorporate experiential priors derived from task execution.

\paragraph{Training-Free Adaptation.}
In-context learning~\citep{brown2020language}, chain-of-thought
prompting~\citep{wei2022chain}, and retrieval-augmented
generation~\citep{lewis2020retrieval} adapt models without training. Self-refine
~\citep{madaan2023selfrefine} uses iterative feedback loops. These methods lack
the systematic advantage extraction and compression that ASO provides.

\paragraph{Active Inference.}
Active inference~\citep{friston2017active, parr2022active} frames
decision-making as free energy minimization. \citet{sajid2021active} apply
active inference to planning. Our epistemic reward term is inspired by the
expected free energy formulation in active inference.

\paragraph{Model Compression.}
BitDelta~\citep{liu2024bitdelta} compresses fine-tuning deltas to 1 bit.
GPTQ~\citep{frantar2023gptq} and AWQ~\citep{lin2024awq} quantize model weights
for inference efficiency. Our 1-bit prior compression applies similar ideas to
experiential factors rather than model parameters.

% ============================================================================
% 11. CONCLUSION
% ============================================================================
\section{Conclusion}
\label{sec:conclusion}

Active Semantic Optimization demonstrates that training-free adaptation via
TF-GRPO and Product-of-Experts decoding can achieve competitive performance on
challenging code generation benchmarks at a fraction of the cost of fine-tuning.
The key contributions are:

\begin{enumerate}[leftmargin=1.5em]
    \item \textbf{TF-GRPO:} A training-free extension of GRPO that extracts
          semantic advantages from grouped rollouts without gradient computation.
    \item \textbf{PoE Decoding:} A Bayesian framework for incorporating
          experiential priors as multiplicative token-level factors with
          theoretically optimal expert weights.
    \item \textbf{1-Bit Compression:} A quantization scheme achieving
          $29.5\times$ storage savings with bounded approximation error.
    \item \textbf{Convergence Guarantees:} Formal proofs that ASO converges to
          the Bayes-optimal predictor with explicit finite-sample bounds.
    \item \textbf{Cost Efficiency:} \$18 adaptation cost compared to
          \$10{,}000+ for parameter-efficient fine-tuning.
\end{enumerate}

\paragraph{Limitations.}
ASO requires a reward signal (test suite) for TF-GRPO, which is not available
for all tasks. The PoE framework assumes token-level decomposability, which may
not capture long-range dependencies. Performance still trails full fine-tuning
by 4--6 percentage points on SWE-bench.

\paragraph{Future Work.}
Multi-agent prior sharing via the DSO protocol (see companion paper),
extension to non-code domains (data science, DevOps, security), and
investigation of active learning strategies for prior acquisition.

% ============================================================================
% APPENDIX
% ============================================================================
\appendix

\section{TF-GRPO Algorithm (Complete Pseudocode)}
\label{app:algorithm}

\begin{algorithm}[H]
\caption{Training-Free GRPO with PoE Decoding}
\label{alg:tfgrpo}
\begin{algorithmic}[1]
\Require Task $x$, base model $\pi_\theta$, prior bank $\mathcal{E}$,
         group size $G$, max iterations $K$, reward weights $\alpha, \beta$
\Ensure Updated prior bank $\mathcal{E}'$, best solution $y^*$
\State $\mathcal{E}' \gets \mathcal{E}$
\State $r^* \gets -\infty$
\For{$k = 1$ to $K$}
    \State \texttt{// Phase 1: Generate rollouts with current priors}
    \For{$i = 1$ to $G$}
        \State $y^{(i)}_k \sim \pi_{\text{ASO}}(\cdot \mid x; \mathcal{E}')$
        \Comment{PoE decoding (Eq.~\ref{eq:poe-main})}
        \State $r_{\text{ext}}^{(i)} \gets \text{ExecuteAndTest}(y^{(i)}_k, x)$
        \State $r_{\text{epi}}^{(i)} \gets \text{EpistemicReward}(y^{(i)}_k, \mathcal{E}')$
        \State $r^{(i)} \gets \alpha \cdot r_{\text{ext}}^{(i)} + \beta \cdot r_{\text{epi}}^{(i)}$
    \EndFor
    \State \texttt{// Phase 2: Compute group-relative advantages}
    \State $\mu_G \gets \frac{1}{G}\sum_i r^{(i)}$, \quad
           $\sigma_G \gets \sqrt{\frac{1}{G-1}\sum_i (r^{(i)} - \mu_G)^2}$
    \For{$i = 1$ to $G$}
        \State $A^{(i)} \gets (r^{(i)} - \mu_G) / (\sigma_G + \epsilon)$
    \EndFor
    \State \texttt{// Phase 3: Extract and compress semantic advantages}
    \State $S \gets \text{LLMIntrospect}(\{(y^{(i)}_k, A^{(i)})\}_{i=1}^G, x)$
    \For{each pattern $s_j \in S$}
        \State $\Delta_j \gets \text{DistillToTokenFactors}(s_j)$
        \State $\widehat{\Delta}_j \gets \alpha_j \cdot \sign(\Delta_j)$
        \Comment{1-bit compression (Eq.~\ref{eq:1bit})}
        \State $q_j \gets \text{EstimateQuality}(s_j, \{r^{(i)}\})$
        \State $\eta_j \gets \log(q_j / (1 - q_j))$
        \Comment{Optimal weight (Eq.~\ref{eq:eta-bayes})}
        \State $\mathcal{E}' \gets \mathcal{E}' \cup \{(\widehat{\Delta}_j, \eta_j, q_j)\}$
    \EndFor
    \State \texttt{// Phase 4: Update best solution}
    \State $i^* \gets \argmax_i r^{(i)}$
    \If{$r^{(i^*)} > r^*$}
        \State $r^* \gets r^{(i^*)}$, \quad $y^* \gets y^{(i^*)}_k$
    \EndIf
    \State \texttt{// Early termination}
    \If{$r_{\text{ext}}^{(i^*)} = 1.0$}
        \State \textbf{break} \Comment{All tests pass}
    \EndIf
\EndFor
\State \Return $\mathcal{E}', y^*$
\end{algorithmic}
\end{algorithm}

\section{Proof of Theorem~\ref{thm:rate} (Full)}
\label{app:proof-rate}

\begin{proof}
Fix token position $t$ and let $v^*$ denote the Bayes-optimal token. For each
prior $m$, define the log-odds contribution:
\begin{equation}
    X_m = \eta_m \left[\log \phi_m(v^* \mid x, y_{<t}) - \log \phi_m(v \mid x, y_{<t})\right]
\end{equation}
for any $v \neq v^*$. By construction, $X_m \geq 0$ when $\phi_m$ correctly
assigns higher probability to $v^*$ (which happens with probability $q_m$) and
$|X_m| \leq \eta_m \cdot \max_v |\log \phi_m(v)| \leq C$ for some constant $C$
determined by the maximum magnitude of the expert factors.

The total log-odds advantage for $v^*$ over $v$ after $M$ priors is:
\begin{equation}
    L_M(v^*, v) = \ell_{t,v^*} - \ell_{t,v} + \sum_{m=1}^M X_m.
\end{equation}

By assumption, $\mathbb{E}[X_m] \geq 2\delta^2$ (from $q_m > 0.5 + \delta$
and the log-odds formula). By Hoeffding's inequality:
\begin{equation}
    P\left(\sum_{m=1}^M X_m < M\delta^2\right)
    \leq \exp\left(-\frac{M\delta^2}{2C^2}\right)
    \leq \exp\left(-\frac{M\delta^2}{2}\right)
\end{equation}
for $C = 1$ (which holds when expert factors are bounded by 1 in absolute
value). Taking a union bound over all $V - 1$ incorrect tokens and noting
that the KL divergence is zero when $\pi_{\text{ASO}}(v^*) = 1$:
\begin{equation}
    P\left(\KL(\pi^* \| \pi_{\text{ASO}}^{(M)}) > \varepsilon\right)
    \leq (V-1) \exp\left(-\frac{M\delta^2}{2}\right)
    < 2V \exp\left(-\frac{M\delta^2}{2}\right).
\end{equation}
\end{proof}

\section{Reproducibility Checklist}
\label{app:reproducibility}

All experiments include:
\begin{itemize}[leftmargin=1.5em]
    \item \textbf{Environment:} Docker image with exact versions (Python 3.11,
          LanceDB 0.3.0, Claude API 2024.01).
    \item \textbf{Seeds:} Fixed random seeds for sampling (42, 123, 456 for
          three runs; we report mean $\pm$ std).
    \item \textbf{Logs:} Complete API call logs with timestamps and token counts.
    \item \textbf{Artifacts:} Generated patches, test outputs, prior banks.
    \item \textbf{Replay:} Verifier can re-run from logs to validate results.
    \item \textbf{Cost Tracking:} Per-experiment API cost breakdown.
\end{itemize}

\section{Extended Ablation: Group Size Sensitivity}
\label{app:group-size}

\begin{table}[h]
\centering
\begin{tabular}{ccccc}
\toprule
$G$ & HumanEval & MBPP & SWE-bench & Cost/issue \\
\midrule
1 & 65.4\% & 62.7\% & 13.1\% & \$1.20 \\
2 & 68.9\% & 65.1\% & 15.4\% & \$2.40 \\
4 & 72.8\% & 68.4\% & 18.2\% & \$4.80 \\
8 & 73.6\% & 69.1\% & 19.1\% & \$9.60 \\
16 & 74.1\% & 69.4\% & 19.5\% & \$19.20 \\
\bottomrule
\end{tabular}
\caption{Performance vs.\ group size with $K=3$ iterations. Diminishing
returns beyond $G=4$ justify our default choice.}
\label{tab:group-sensitivity}
\end{table}

% ============================================================================
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\end{document}