arXiv subjects

Isaac Llorente-Saguer

Publications and source records attributed to Isaac Llorente-Saguer.

4 recordsLinked to original sources

arXiv2026

Harmful Intent as a Geometrically Recoverable Feature of LLM Residual Streams

Aligned language models refuse harmful instructions, but the representations through which they recognise such instructions are less well characterised than the behaviours they produce. Harmful intent is linearly separable from residual-stream activations across 12 models spanning four architectural families (Qwen2.5, Qwen3.5, Llama-3.2, Gemma-3) and three alignment variants (base, instruction-tuned, abliterated), with parameter scales from 0.5B to 1.3B and a within-family scale extension to 9B on Qwen3.5. A direction fitted from 100 labelled examples per class via Soft-AUC optimisation reaches mean effective AUROC 0.982 and TPR@1\%FPR 0.797, generalises to three held-out harm benchmarks and a hard-benign control, and matches its instruction-tuned counterpart within $\pm 0.003$ AUROC in abliterated variants from which the refusal mechanism has been removed. The supervised strategies all exceed AUROC 0.96, but their TPR@1\%FPR varies by more than ten times the AUROC gap; a deployed 9B safety classifier shows the same pattern at AUROC 0.94 and TPR 0.30, motivating low-FPR reporting as a default in safety-adjacent detection evaluation. Geometric measurements refine the picture. The recovered direction is concentrated within each extraction protocol but protocol-dependent across them: two pooling choices applied to the same chat-templated activations at the same residual-stream layer (max-pool over content tokens versus last-token at the post-instruction position) recover harm directions $73^\circ$ apart, and projecting one out leaves detection under either max-pool extraction essentially intact. Probing identifies a protocol-specific direction rather than a unique computational feature.

cs.LG↗

arXiv2026

The Geometry of Harmful Intent: Training-Free Anomaly Detection via Angular Deviation in LLM Residual Streams

We present LatentBiopsy, a training-free method for detecting harmful prompts by analysing the geometry of residual-stream activations in large language models. Given 200 safe normative prompts, LatentBiopsy computes the leading principal component of their activations at a target layer and characterises new prompts by their radial deviation angle $θ$ from this reference direction. The anomaly score is the negative log-likelihood of $θ$ under a Gaussian fit to the normative distribution, flagging deviations symmetrically regardless of orientation. No harmful examples are required for training. We evaluate two complete model triplets from the Qwen3.5-0.8B and Qwen2.5-0.5B families: base, instruction-tuned, and \emph{abliterated} (refusal direction surgically removed via orthogonalisation). Across all six variants, LatentBiopsy achieves AUROC $\geq$0.937 for harmful-vs-normative detection and AUROC = 1.000 for discriminating harmful from benign-aggressive prompts (XSTest), with sub-millisecond per-query overhead. Three empirical findings emerge. First, geometry survives refusal ablation: both abliterated variants achieve AUROC at most 0.015 below their instruction-tuned counterparts, establishing a geometric dissociation between harmful-intent representation and the downstream generative refusal mechanism. Second, harmful prompts exhibit a near-degenerate angular distribution ($σ_θ\approx 0.03$ rad), an order of magnitude tighter than the normative distribution ($σ_θ\approx 0.27$ rad), preserved across all alignment stages including abliteration. Third, the two families exhibit opposite ring orientations at the same depth: harmful prompts occupy the outer ring in Qwen3.5-0.8B but the inner ring in Qwen2.5-0.5B, directly motivating the direction-agnostic scoring rule.

cs.LG↗

arXiv2026

A Lock-Free, Fully GPU-Resident Architecture for the Verification of Goldbach's Conjecture

We present a fully device-resident, multi-GPU architecture for the large-scale computational verification of Goldbach's conjecture. In prior work, a segmented double-sieve eliminated monolithic VRAM bottlenecks but remained constrained by host-side sieve construction and PCIe transfer latency. In this work, we migrate the entire segment generation pipeline to the GPU using highly optimised L1 shared-memory tiling, achieving near-zero host-device communication during the critical verification path. To fully leverage heterogeneous multi-GPU clusters, we introduce an asynchronous, lock-free work-stealing pool that replaces static workload partitioning with atomic segment claiming, enabling $99.7$% parallel efficiency at 2 GPUs and $98.6$% at $4$ GPUs. We further implement strict mathematical overflow guards guaranteeing the soundness of the 64-bit verification pipeline up to its theoretical ceiling of $1.84 \times 10^{19}$. On the same hardware, the new architecture achieves a $45.6\times$ algorithmic speedup over its host-coupled predecessor at N = $10^{10}$. End-to-end, the framework verifies Goldbach's conjecture up to $10^{12}$ in $36.5$ seconds on a single NVIDIA RTX 5090, and up to $10^{13}$ in $133.5$ seconds on a four-GPU system. All code is open-source and reproducible on commodity hardware.

cs.MS↗

arXiv2026

GoldbachGPU: An Open Source GPU-Accelerated Framework for Verification of Goldbach's Conjecture

We present GoldbachGPU, an open-source framework for large-scale computational verification of Goldbach's conjecture using commodity GPU hardware. Prior GPU-based approaches reported a hard memory ceiling near 10^11 due to monolithic prime-table allocation. We show that this limitation is architectural rather than fundamental: a dense bit-packed prime representation provides a 16x reduction in memory footprint, and a segmented double-sieve design removes the VRAM ceiling entirely. By inverting the verification loop and combining a GPU fast-path with a multi-phase primality oracle, the framework achieves exhaustive verification up to 10^12 on a single NVIDIA RTX 3070 (8 GB VRAM), with no counterexamples found. Each segment requires 14 MB of VRAM, yielding O(N) wall-clock time and O(1) memory in N. A rigorous CPU fallback guarantees mathematical completeness, though it was never invoked in practice. An arbitrary-precision checker using GMP and OpenMP extends single-number verification to 10^10000 via a synchronised batch-search strategy. The segmented architecture also exhibits clean multi-GPU scaling on data-centre hardware (tested on 8 x H100). All code is open-source, documented, and reproducible on both commodity and high-end hardware.

cs.MS↗