Isaac Llorente-Saguer

Isaac Llorente-Saguer

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.

Isaac Llorente-Saguer

Harmful intent is geometrically recoverable from large language model residual streams: as a linear direction in most layers, and as angular deviation in layers where projection methods fail. Across 12 models spanning four architectural families (Qwen2.5, Qwen3.5, Llama-3.2, Gemma-3) and three alignment variants (base, instruction-tuned, abliterated), under single-turn, English evaluation, we characterise this geometry through six direction-finding strategies. Three succeed: a soft-AUC-optimised linear direction reaches mean AUROC 0.98 and TPR@1\%FPR 0.80; a class-mean probe reaches 0.98 and 0.71 at <1ms fitting cost; a supervised angular-deviation strategy reaches AUROC 0.96 and TPR of 0.61 along a representationally distinct direction ($73^\circ$ from projection-based solutions), uniquely sustaining detection in middle layers where projection methods collapse. Detection remains stable across alignment variants, including abliterated models from which refusal has been surgically removed: harmful intent and refusal behaviour are functionally dissociated features of the representation. A direction fitted on AdvBench transfers to held-out HarmBench and JailbreakBench with worst-case AUROC 0.96. The same picture holds at scale: across Qwen3.5 from 0.8B to 9B parameters, AUROC remains $\geq$0.98 and cross-variant transfer stays within 0.018 of own-direction performance This is consistent with a simple account: models acquire a linearly decodable representation of harmful intent as part of general language understanding, and alignment then shapes what they do with such inputs without reorganising the upstream recognition signal. As a practical consequence, AUROC in the 0.97+ regime can substantially overestimate operational detectability; TPR@$1\%$FPR should accompany AUROC in safety-adjacent evaluation.

Isaac Llorente-Saguer

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.