Imbuing Large Language Models with Bidirectional Logic for Robust Chain Repair

Autoregressive chain-of-thought (CoT) reasoning in large language models (LLMs) is fundamentally forward-directed: each step conditions only on prior tokens. This unidirectional inductive bias renders even capable models susceptible to error snowballing, wherein a single logical or arithmetic mistake in an early step irreversibly corrupts the entire reasoning chain. We introduce Teleological Reasoning Infilling (\TRI{}), a training framework that endows decoder-only transformers with a native \emph{goal-conditioned bridging} capability. The key insight is to reframe erroneous reasoning segments as fill-in-the-middle (FIM) tasks: given a verified prefix premise $P$, a verified downstream milestone $S$, and the original query $Q$, the model must synthesise the logical bridge $M$ that connects $P$ to $S$ rigorously and completely. To achieve this with standard causal architectures, we introduce a Prefix-Suffix-Middle (PSM) sequence rearrangement with three non-overlapping sentinel tokens, enabling $M$ to attend to both $P$ and $S$ without any structural modification to the self-attention mechanism. Training proceeds in two stages: (i) Supervised Fine-Tuning (SFT) on symbolically verified $(P, S, M)$ triples extracted from formal mathematics corpora, and (ii) Direct Preference Optimisation (DPO) with a deterministic symbolic verifier (Lean 4 / Python) as the sole reward oracle, eliminating LLM-judge sycophancy. At inference, TRI operates as a surgical repair module within a dual-system loop: a causal draft model generates an initial trace, the verifier pinpoints failures, and TRI infills only the damaged segment, leaving verified sections intact. Comprehensive experiments on three benchmarks demonstrate that TRI achieves state-of-the-art performance across all tasks, while reducing per-problem token expenditure by 31.2%.

Synthetic Hallucinations, Real Gains: Hard Negatives from Frontier Models for FIM Hallucination Mitigation

Small open-source code models that power IDE autocomplete still emit hallucinated Fill-in-the-Middle (FIM) completions: syntactically natural calls to methods, parameters, variables, and imports that do not exist in the surrounding project. Existing mitigations either require per-language execution sandboxes that do not apply at mid-keystroke or preference-optimisation pipelines that need large human-labelled corpora. We propose an execution-free alternative: use frontier code models to synthesise plausible-but-wrong completions as hard negatives, then leverage the contrast between these synthetic hallucinations and the ground-truth developer edit as a supervised fine-tuning signal. Our pipeline scrapes multilingual FIM contexts from public GitHub across eight languages and asks a panel of three frontier generators to produce one hard negative per context for each of four hallucination types drawn from the Delulu taxonomy, a Docker-verified multilingual FIM hallucination benchmark, yielding a paired chosen/rejected dataset. Fine-tuning Qwen2.5-Coder-7B-Instruct on a 100K-row curated subset lifts Delulu exact match by +18.8 points and edit similarity by +0.22 on every language and every type, while also improving every HumanEval-Infilling split and every SAFIM subset. The same recipe at 3B lifts Delulu by +12.8 EM with a small, characterised general-FIM trade-off. Five-axis ablations (size, type mix, language coverage, base-model family, and a difficulty-aware fool rate) plus a head-to-head SFT vs. DPO/ORPO comparison map which design choices drive the gain. We release the full pipeline source code -- generation, fool-rate LLM judging, curation, and the FIM fine-tuning recipe -- so that the experiments in this paper can be reproduced end-to end on any permissively licensed corpus.