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After the Euclidean Highway: Hyperbolic Expert AI as the Next Innovation
Kwan Soo Shin, In Seok Kang, Munho Lee
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
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Summary
The paper introduces HySAT (Hyperbolic Structure-Aware Training), a method that applies hyperbolic geometry exclusively at the loss layer to preserve hierarchical structures in expert domains, avoiding the training instability caused by placing hyperbolic adapters on the manifold. It demonstrates stability across six small language models with zero NaN events, contrasting with 17 crashes observed in HyperLoRA-style adapter-on-manifold approaches.
Entities (11)
Relation Signals (10)
HySAT → appliesto → Loss Layer
confidence 98% · hyperbolic losses at the loss layer only
HySAT → prevents → Training Instability
confidence 95% · zero NaN events over ~317K optimizer steps
HySAT → usesgeometry → Lorentz Manifold
confidence 95% · hyperbolic losses at the loss layer only... On the Lorentz manifold
HyperLoRA → appliesto → Adapter
confidence 92% · adapter on manifold... produced 17 crash events
HyperLoRA → causes → Training Instability
confidence 90% · produced 17 crash events... terminating in NaN
EXAONE 3.5 → isbasefor → HySAT
confidence 90% · Across six expert SLMs we constructed and deployed (... EXAONE 3.5... HySAT
Llama 3.1 → isbasefor → HySAT
confidence 90% · Across six expert SLMs we constructed and deployed (Llama 3.1... HySAT
HySAT → usesloss → HWC
confidence 85% · Hierarchy-Weighted Contrastive (HWC)
HySAT → usesloss → PTLE
confidence 85% · three structural losses — Pairwise Tree Lorentz Embedding (PTLE)
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Abstract
Abstract:Expert domains are trees; the Euclidean transformer is not, diluting parent-child structure exponentially at depth. The hyperbolic turn left one question unasked: not how much of a network to curve, but where curvature may touch the gradient. Placement is a law, not a knob: the same geometry on a trainable adapter collapses training (seventeen training collapses, ~220 GPU-hours), yet at the loss layer alone it trains without one -- this is HySAT (Hyperbolic Structure-Aware Training), hyperbolic losses at the loss layer only. Across six expert SLMs we constructed and deployed (Llama 3.1 and EXAONE 3.5; four adapter strategies; 18.0M-sample corpus; zero NaN over ~317K optimizer steps), a matched four-arm ablation isolates the preserved manifold invariant, and three propositions and a lemma prove why loss-only placement is stable where adapter-on-manifold is not. Four models are operationally deployed (one live, consumer-facing), two open-weight, with per-step traces and a seventeen-incident failure ledger on Zenodo (CC-BY-4.0).
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- Source: https://arxiv.org/abs/2607.17513v1
- Canonical: https://arxiv.org/abs/2607.17513v1
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After the Euclidean Highway: Hyperbolic Expert AI as the Next Innovation Kwan Soo Shin 1* , In Seok Kang 2 and Munho Lee 2,3 1* PolymathMinds Lab, Asan, Republic of Korea. 2 aSSIST University, Seoul, Republic of Korea. 3 Samsung Engineering, Seoul, Republic of Korea. *Corresponding author(s). E-mail(s): sshin@pmminds.ai; Abstract Expert domains are trees; the Euclidean transformer is not, diluting parent–child structure exponentially at depth. The hyperbolic turn left one question unasked: not how much of a network to curve, but where curvature may touch the gradient. Placement is a law, not a knob: the same geometry on a trainable adapter collapses training (seventeen training collapses,∼220 GPU-hours), yet at the loss layer alone it trains without one—this is HySAT (Hyperbolic Structure- Aware Training), hyperbolic losses at the loss layer only. Across six expert SLMs we constructed and deployed (Llama 3.1 and EXAONE 3.5; four adapter strategies; 18.0M-sample corpus; zero NaN over∼317K optimizer steps), a matched four-arm ablation isolates the preserved manifold invariant, and three propositions and a lemma prove why loss-only placement is stable where adapter- on-manifold is not. Four models are operationally deployed (one live, consumer- facing), two open-weight, with per-step traces and a seventeen-incident failure ledger on Zenodo (C-BY-4.0). Keywords: Hyperbolic fine-tuning, Domain-expert AI, Small language models, Lorentz manifold, HySAT, LoRA adapters, Cross-disciplinary AI boundaries 1 arXiv:2607.17513v1 [cs.LG] 20 Jul 2026 1 Main 1.1 The Euclidean Highway and the Expert Wall In 2017, Vaswani et al. [1] opened a road that the next decade would travel without exit. BERT, GPT-4, Claude, Gemini, and DeepSeek differ in scale, corpus, and align- ment, but not in the geometry of their representations. The substrate is everywhere Euclidean. The highway has a wall. When the task shifts from general fluency to domain expertise, the flat substrate stops working in three independent ways: interpolation- based guarantees almost surely fail outside the training convex hull [2]; log-precision transformers are bounded to constant-depth threshold circuits [3, 4]; compositional accuracy decays exponentially with depth [5]. The wall reappears beyond the math- ematics — in coordinated default–executive–salience network dynamics [6, 7] and in foundation-model governance gaps [8–10]. The wall is structural, not contingent. It forces the question the hyperbolic turn had left unasked: not how much of a network to curve, but where curvature may touch the gradient without destabilising training at scale. 1.2 The Razor Cell and the Second Innovation A concurrent cartographic companion manuscript (under review) maps the empty quadrant of post-2022 domain-expert systems combining high novelty with high human usefulness, and names it the razor cell ; Extended Data Fig 2 visualises the same cell in this paper’s operational coordinates (decision consequentiality × pretraining-data availability; full Q1–Q4 taxonomy in §16.2). Our diagnosis from four theoretical pillars (§2.2) arrives at the same vacancy from independent disciplinary directions. Our answer is a second innovation on top of the Euclidean highway, not its replace- ment: a geometric correction to the training signal — hyperbolic losses referencing tree-structured domain ontologies. We name this principle HySAT — Hyperbolic Structure-Aware Training. On the Lorentz manifold, finite trees embed with arbi- trarily low distortion at modest dimension [11, 12]; the flat embedding stretches the same hierarchies onto a sphere where parent-child and sibling relations dilute exponen- tially with depth. HySAT teaches a Euclidean transformer these hierarchies through a corrective loss alone — without rebuilding the network. Generation V derivation from Generations I–IV is in §16.3–§16.4. This idea did not arrive from nowhere. For a decade the hyperbolic-representation programme moved steadily into the network. Nickel and Kiela [12] first showed that the negatively-curved Poincar ́e ball embeds hierarchies far more parsimoniously than Euclidean space can; hyperbolic graph and neural networks then carried that curvature inward, layer by layer [13–15]; and the move reached the transformer itself, first as an adapter on the manifold [16] and most recently as a fully-hyperbolic decoder and a multi-modal pretraining objective [17, 18]. Each step advanced the field, and each answered the same implicit question — how much of the network should be hyperbolic — by pushing the geometry deeper. We build directly on this trajectory (full five- generation lineage in §16.4). 2 This trajectory runs parallel to the parameter-efficient fine-tuning programme, which established that a large pretrained model adapts through a small changed subset of parameters while the rest stays fixed [19, 20]; that literature asked how few parame- ters must change. Our question is orthogonal and is the one the hyperbolic turn makes unavoidable: where should non-Euclidean structure enter when the changed parame- ters themselves remain Euclidean? We learned this question by failing on it. Carrying the manifold into the trainable adapter — the natural next step on the same road — produced 17 crash events across three project families at industrial scale (∼220 hours of B200 SXM compute), while the same placement on the original 10,000-sample benchmark stays comfortably stable: the failure is not the manifold but the scale- dependent coupling between curvature and the adapter’s gradient. HySAT answers the placement question with the most conservative choice the lineage had not yet iso- lated — the curvature touches only the loss layer, never a trainable weight (Figure 1; full three-way comparison in Extended Data Table 1). The substrate the field built remains intact; we change only where the geometry enters. 1.3 Contributions and Roadmap The contribution is a placement law : where foundation-model work shows what large domain models can do, we show how expert small language models can be constructed when the domain knowledge is hierarchical, structurally sparse, and not recoverable by web-scale pretraining — keep trainable adaptation Euclidean for pretrained- base fine-tuning (the from-scratch BS Sovereign anchor is fully hyperbolic, stable under curvature-aware Riemannian optimisation), and inject hyperbolic structure only through the loss. The paper has one principle and several consequences. The principle is HySAT: place the hyperbolic geometry only at the loss layer (§2.1; Propositions 1–3 and Lemma 1). The consequences: (i) scale — six SLMs totalling 17,954,911 training samples (ORAA’s single-project corpus alone is three orders of magnitude — 1,139× — beyond HypLoRA’s Math-10K; cumulative 1,795×), four operationally deployed, two open-weight at publication; (i) breadth — a four-strategy cross-validation span- ning two base families; (i) mechanism — a batch-size natural experiment formalised as Proposition 2; (iv) framing — four theoretical pillars extending Stein’s 1953 tri- partite definition; (v) transparency — Zenodo C-BY-4.0 release of the per-step traces, the seventeen-incident failure ledger (one preserved raw trajectory and a crash- reproduction script), and the verified RunPod compute aggregate ($4,998 / ∼1,111 GPU-hours). 2 Results 2.1 HySAT Principle — Hyperbolic Geometry at the Loss Layer To see why placement is the decisive variable, it helps to read the recent hyperbolic- LLM literature not as a list of competing methods but as a single question asked at three depths. Yang et al. [16] asked it at the adapter — can a low-rank update live on the manifold? — and showed it can, at benchmark scale. He et al. [17] asked it of the 3 Fig. 1 Where do you place the curvature? What geometry? Hyperbolic ( d ) Euclidean ( d ) Where is the curvature applied? Adapter on manifoldLoss layer only HypLoRA family (adapter-on-manifold, 2024) Yang 2024 (HypLoRA) 17 crashes / ~220 GPU-hr HySAT (loss-only, 2026, this paper) (h) = exp 0 (h/ ) loss-layer ONLY 0 NaN / 18M samples 1,139× HypLoRA scale Standard LoRA (Euclidean adapter, 2022) Hu 2022 (LoRA) Cross-entropy baseline (flat baseline, pre-2024) Vaswani 2017 substrate + low-rank adaptation + Lorentz manifold (geometric leap) loss-only placement (this paper, Proposition 1) Q1. Stability B t F B 0F + s (Proposition 1, curvature-decoupled) Q2. Scale 17,954,911 samples / 6 SLMs 1,139× HypLoRA Math-10K / 0 NaN Q3. Mechanism / Bsinh( Bx ) bf16 Jacobian-amp. boundary at v11 Q4. Deployment 4 operational contexts: 1 live + 3 gated 2 HF open-weight / Zenodo C-BY-4.0 In Figure 1, the architectural placement axis of the hyperbolic-AI substrate is mapped.⋆ HySAT sits at the only quadrant combining hyperbolic geometry with loss-layer-only placement, decoupling curvature from gradient flow (Proposition 1). Quadrants: cross-entropy baseline (1); standard LoRA (20); adapter-on-manifold (16); loss-only (HySAT). The five-generation hyperbolic-deep-learning lineage is mapped separately in Extended Data Fig 1 and§16.4. Q1–Q4 corners: stability, scale, mechanism, deployment. Empirical realisation in Figure 3; scale comparison in Figure 6. whole decoder — can a transformer be hyperbolic throughout ? — and showed that too, in the pretraining register. Both were natural and productive moves: if curvature helps, put more of the network in it. The question they did not need to answer, because their scale did not force it, is the one expert-domain fine-tuning forces immediately — once the manifold sits inside a trainable parameter, the curvature rides the gradient, and at eighteen-million-sample scale that coupling is what breaks. HySAT therefore makes the opposite move from the same lineage: it keeps every trainable parameter Euclidean and lets the manifold enter only at the loss layer. The transformer base and the rank-r LoRA adapter remain in R d ; the Lorentz manifold is referenced only when three structural losses — Pairwise Tree Lorentz Embedding (PTLE), Hierarchical Label Sibling Distance (HLSD), and Hierarchy-Weighted Contrastive (HWC) — are computed from hidden states (full definitions in Methods §4.1; HLSD activates only when a micro-batch contains a sibling pair, §2.5). In the hyperbolic-LLM fine-tuning 4 register, where the curvature is allowed to touch the gradient, not how much of the network carries it, is the load-bearing design choice. Fig. 2 HySAT vs HyperLoRA placement — the curvature-coupling failure mode (a) HySAT (Proposition 1: Stability) B t F B 0F + s s (independent of curvature) Loss = CE + k hyp k (h) = exp 0 (h/ ) (loss-layer ONLY) Adapter B d (Euclidean only) (Euclidean AdamW) Strict : Stable when curvature decoupled from gradient flow t = 0 ~317K cumulative steps B t F bounded by s s 0 NaN events over six complete training runs (b) HyperLoRA (Proposition 1(i): Failure mode) / Bsinh( Bx )/ Bx (curvature-coupled) Loss (manifold-aware) exp/log on d (every fwd/bwd pass) Adapter B on d (Lorentz manifold) amplified sinh( Bx ) bf16 instability boundary v11 Strict : Unstable when v=Bx11 (bf16 boundary) t = 0 NaN cascade B t F unbounded × 17 crashes / 220 GPU-hours B200 SXM In Figure 2, panel (a) HySAT (stable): the Lorentz projection π(h) = exp 0 (h/τ ) is applied at loss evaluation only; the gradient flows back through h within R d under standard Euclidean AdamW. ∥B t ∥ F is bounded by κ· P η s , independent of Lorentz curvature. Bottom bars: 0 NaN events over ∼317K cumulative optimizer steps across six complete training runs. Panel (b) HyperLoRA (failure mode): when the hyperbolic operation lies inside the adapter, the Lorentz exponential map’s Jacobian factor sinh(∥B· x∥)/∥B· x∥ amplifies updates exponentially with hidden-state norm. Bottom bars: ∥B t ∥ F drives across the practical bf16 instability boundary ∥v∥≈ 11 (Jacobian amplification; absolute cosh overflow near 89), producing NaN cascades (17 documented crashes, ∼220 hours of B200 SXM compute, incident ledger in§6; bounded-Jacobian property M τ ≤ 5.23/τ in§2.7). The architectural difference is what curvature lives where, not how much. Why adapter-on-manifold placement fails at scale. We learned the principle by paying for it: initial HypLoRA-style runs collapsed in seventeen independent failures across three project families, each with monotonic adapter-B-norm growth terminating in NaN (∼220 B200 SXM GPU-hours). The failure mode (Proposition 1(i)) is not HypLoRA-specific — Mishne et al. [21] and Klein et al. [22] document analogous float- precision saturation in hyperbolic training, and the underlying Jacobi-field divergence on negatively-curved manifolds is a classical geometric phenomenon (23) — and under unconstrained AdamW the amplification compounds until ∥B t ∥ F drives ∥v∥ across the practical bf16 instability boundary ≈ 11 (Jacobian amplification). The incident ledger and the stabilisation-condition mapping are in §6. 5 Why loss-only placement stabilises. The adapter lives in R d , so curvature does not live in the trainable parameters; it enters the gradient only through a bounded, clamped projection Jacobian evaluated at the loss layer. The projection π(h) = exp 0 (h/τ ) is differentiable — its gradient reaches the adapter, while the trainable adapter weights themselves never leave R d . Across the six complete runs (∼317K optimizer steps; §2.3) — three loss-only HySAT systems, the from-scratch fully-hyperbolic BS Sovereign anchor under curvature-aware Riemannian optimisa- tion, and two HyperLoRA systems stabilised per Proposition 1(i) — zero NaN events occur (Figure 2). The bounded-Jacobian derivation, autograd contract, and τ -clamp calibration are in Methods §4.1. A matched four-arm ablation (§6.3) isolates the prop- erty placement controls: holding the base, data, and the same loss-layer tree regulariser fixed, only the Lorentz projection preserves the manifold invariant (⟨x,x⟩ L = −1; drift ∼10 −6 versus ∼10 4 for the flat control, ten orders of magnitude with no seed overlap). Held-out tree-geometry does not separate the arms at this controlled toy scale, so the ablation isolates the mechanism — invariant preservation — rather than a downstream gain; the task-level advantage of hyperbolic placement is carried by the six deployed expert models (§2.3). Proposition 1 (HySAT Stability — Curvature-Decoupled Gradient Flow). Let B t ∈ R d×r denote the output-matrix of a rank-r LoRA adapter at training step t under Euclidean AdamW with learning rate η t , gradient clipping at norm κ, and loss L total = L CE + P k λ k L hyp k where L hyp k ∈L PTLE ,L HLSD ,L HWC involves the Lorentz projection π(h) = exp 0 (h/τ ) applied only at loss evaluation. Then: (i) E ∥B t ∥ F ≤∥B 0 ∥ F + κ· P t−1 s=0 η s , depending on neither the Lorentz curvature c nor the manifold geometry: the bound is identical to the Euclidean-only LoRA case [20]. (i) Under HyperLoRA-style placement (adapter weights onH d ) without curvature- aware optimisation or B-norm bounding, the gradient inherits the exp map’s differ- ential, with Jacobian norm ∝ sinh(∥B· x∥)/∥B· x∥, growing super-polynomially with ∥B HyL t ∥ F ; in all seventeen recorded incidents the run terminated in NaN or unre- coverable divergence at the practical boundary ∥v∥ ≈ 11 (family-level ledger records; the one preserved raw trajectory shows the divergence-plateau signature,§6.2). The failure mode is absent when curvature-aware optimisation [24, 25], explicit B-norm constraint, or retraction with full-sequence loss and effective batch ≥ 32 holds (full conditions i.a–i.c in Methods§4.1). Full statement, proof sketch, autograd contract, and the seventeen-crash incident ledger in Methods§4.1 and§6. Lemma 1 (Stability-Dominant Placement of Hyperbolic Operations under Bounded Loss-Layer Projection). Consider a neural architecture of L layers with subset H ⊆ 1,...,L + 1 replaced by hyperbolic operations (L + 1 = loss layer). Let σ(H) denote the compositional Lipschitz constant under bf16. Then: (i) among H ∈ loss only,adapter + loss,base + adapter + loss, the loss-only placement H = L + 1 minimises σ(H) and preserves the base model’s Euclidean trainable-parameter geometry, adding only a bounded objective-Jacobian factor M τ , under finite-precision arithmetic; (i) for placements containing base or adapter layers, σ(H) is bounded 6 below by the curvature-coupled amplification factor sinh(∥B · x∥)/∥B · x∥ of Propo- sition 1(i), growing super-polynomially as hidden-state norms approach the bf16 cosh-overflow boundary. Full proof and cross-architecture lineage corroboration (Poincar ́e embeddings, HGCN, HNN++, MERU, HyperCLIC) in §2.8. HySAT is the LLM instance with the most conservative H = L + 1; HELM [17] operates in a different register (general-language pretraining). 2.2 Four-Pillar Theoretical Foundation A placement principle stands or falls on the disciplines it answers to. HySAT rests on four pillars converging on the same diagnosis from independent starting points. Pillar 1 (mathematical) carries the load-bearing formal justification: Euclidean atten- tion is extrapolative [2], TC 0 -bounded [3], and compositionally decaying [5], whereas hyperbolic geometry offers an O(logN ) tree-distortion escape [11] the flat substrate cannot match. Pillars 2–4 converge on the same diagnosis from three independent directions — expert cognition and Koestler bisociation [6, 26], the Bloom–Shulman human-reserved Understand–Evaluate–Create triad [27, 28], and foundation-model governance [8, 9] — and together populate Stein [29]’s tripartite definition of creativity (novel × tenable-useful-satisfying × accepted-by-group-in-time). Full pillar develop- ment, intellectual lineages, the five-generation landscape (Extended Data Fig 1), and the eight-adjacent-construct positioning are in §11. HySAT’s scope is precise: a placement law for tree-structured expert domains, layered on top of the transformer highway, with the innovation located exclusively in where the hyperbolic loss signal enters (full boundary list in §11.7). 2.3 Six Expert Small Language Models — The Empirical Core A placement principle is only as good as the systems it survives in production. Where the hyperbolic-representation lineage has so far analysed existing models or adapted a single pretrained backbone, we construct six domain-expert small language models across four adapter strategies and carry four of them into live oper- ational service, a construction-and-deployment breadth without precedent in the hyperbolic-LLM literature surveyed through 2025 Q4 [30]. The six span six indepen- dent professional registers: behavioural audit (BS Sovereign 146M), creative ideation (S3 Creative T5), scholarly research (ORAA, Llama 3.1 8B), pedagogy (MetaTeach, EXAONE 7.8B), corporate finance (CEO Finance v2, EXAONE), and admissions counselling (AdmitBrain v6, Llama 3.1 8B). The six span four adapter strategies — from-scratch hyperbolic training, full fine-tuning with HySAT loss, standard-LoRA with HySAT loss, and HyperLoRA with stabilisation — across two base-model families (Llama 3.1 8B [31] and EXAONE 3.5 7.8B [32]) plus two base-independent anchors. Each cell exposes a different facet of the cross-project regularity. Matrix in Figure 3; project summary in Extended Data Table 2. Empirical aggregate. Single-project maximum corpus is ORAA at 11,387,744 samples — three orders of magnitude beyond HypLoRA’s Math-10K benchmark and 67× its largest reported fine-tuning set (Commonsense170K). Six-project cumulative 7 Fig. 3 Six expert SLMs across four adapter strategies 1. From-Scratch no pretrained base; hyperbolic from initialisation BS Sovereign 146M Lorentz decoder 1.16M samples Fully hyperbolic 14-head Lorentz · Macro F1 = 0.855 STABLE / 0 NaN HF open 2. Full Fine-Tuning + HySAT all weights trained, kept Euclidean; loss-only geometry S3 Creative T5-3B (2.85B params) 3.32M samples 95.8% LLM win-rate · x, x L =1.000 STABLE / 0 NaN HF open 3. Standard-LoRA + HySAT LoRA adapter Euclidean; loss-only geometry ORAA Llama-3.1 8B 11.4M samples 9/10 benchmark · 4.8s latency STABLE / 0 NaN gated CEO Finance v2 EXAONE-3.5 7.8B 1.09M samples 30/30 vs GPT-4o · triple-coach STABLE / 0 NaN gated 4. HyperLoRA (stabilised) adapter-on-manifold; required the three stabilisation measures MetaTeach v14 EXAONE-3.5 7.8B 602K samples HWC scaffolding · 100% activation RESCUED / 0 NaN gated AdmitBrain v6 Llama-3.1 8B 394K samples batch 216 rescue · tone 4.2 RESCUED / 0 NaN LIVE Aggregate anchors (across all six SLMs) 17,954,911 training samples 0 NaN at convergence 17 crashes recovered (early HyperLoRA-style attempts; all redesigned) 1,139× beyond HypLoRA Math-10K (ORAA) 4 / 2 LIVE-or-gated services / HF open-weight six single-seed (42) runs Zenodo C-BY-4.0 per-step traces In Figure 3, four adapter strategies span two base-model families. Each cell reports project, base, sample count, key result, deployment status (LIVE / HF open / gated), and stability outcome. All six current configurations achieve 0 NaN at convergence; the two HyperLoRA-column cells reached stability through stabilisation measures (AdmitBrain v6: batch 2→ 16; MetaTeach v14: HWC scaffolding from start). The historical 17 crashes (incident ledger in Fig S1,§6) occurred in earlier project versions: 7 in early ORAA rank-16 HyperLoRA, 6 in early BS Sovereign full-hyperbolic decoder, 4 in early AdmitBrain single-adapter HyperLoRA. Aggregate: 17,954,911-sample training corpus (cumulative 1,795× beyond HypLoRA Math-10K and 105× beyond its largest reported set Commonsense170K; ORAA single-project 1,139×), 0 NaN at convergence, six complete training runs. Per-cell narrative profiles in§13.1–§13.6; crash incident ledger in SI-E. corpus is 17,954,911 samples across∼317K cumulative optimizer steps in six complete runs, with zero NaN. The Lorentz constraint ⟨x,x⟩ L = −1 is preserved within bf16 precision wherever evaluated. A four-arm matched ablation (Qwen 2.5 7B-Instruct, six seeds; Extended Data Table 5, full per-seed protocol in §6.3) isolates the one property loss-layer placement controls: with an identical loss-layer tree regulariser on identical hidden states, only the Lorentz-projected arm preserves the manifold invari- ant (drift ∼10 −6 versus 1.9–3.3 × 10 4 for the flat Euclidean control — ten orders of magnitude, no seed overlap). Held-out tree-distance correlation, by contrast, does not separate the arms at this controlled toy scale (HySAT ρ L 0.228±0.048 versus 8 0.224±0.041 flat and 0.253±0.033 for the supervision-free baseline); the probe isolates invariant preservation — the mechanism — while the downstream advantage of hyper- bolic placement is carried by the six deployed expert models, not by this toy probe. Four of six occupy operational service contexts: AdmitBrain v6 is integrated into a controlled admissions-counselling deployment workflow; ORAA Dual- SLM, CEO Finance v2, and MetaTeach anchor research-workflow, CFO-advisory, and Socratic-tutoring platforms under gated commercial access; BS Sovereign 146M and S3 Creative T5 are released open-weight on HuggingFace at publication. Per-project profiles are in §13.1–§13.6. Notable cross-project findings. Two batch-size natural experiments (ORAA b=2→ 8→ 16; AdmitBrain b=2→ 16) constitute the Proposition 2 regime separation (§2.5); the cross-project activation matrix shows a silent-failure dichotomy below the pair-formation threshold (activation impossible) versus activation above it (b≥ 8), the realised rate depending on in-batch sibling density (§2.4.1); and curator-orchestrated deployments (ORAA dual-SLM 9/10 at 4.8 s latency; CEO Finance triple-coach 30/30 vs GPT-4o) are discussed in §3. 2.4 Cross-Project Hyperbolic Activation 2.4.1 Cross-Project Activation Matrix Batch size is not a speed variable in hyperbolic fine-tuning. It is a structural one. Six SLMs × three loss components × batch regime (Extended Data Table 3) reveal two regimes differing by a single integer. Silent-failure dichotomy at micro-batch. ORAA at b=2 records 0% PTLE / 52% HLSD; AdmitBrain v6 at b=2 records 0% PTLE (1.2% predicted) / 3.9% HLSD. The hyperbolic losses appear in the loss equation, but the batch is too small to contain the sibling pairs they need; the model trains, the loss curves descend, and the structural signal is absent. Full activation under sufficient batch size. Both projects transition to 100% activation when the batch crosses the threshold (ORAA at b=8; AdmitBrain at b=16). BS Sovereign reaches the regime by architectural guarantee; S3 Creative R266 sus- tains it across 266 rounds; MetaTeach v14 and CEO Finance v2 (b eff =64) reach 100% on every configured loss at equilibrium. Per-step traces and per-loss equilibrium magnitudes in §3. 2.4.2 Manifold Preservation and Zero-NaN Aggregate The Lorentz constraint ⟨π(h),π(h)⟩ L = −1 is preserved exactly at post-training equilibrium in all six projects. Zero NaN events across ∼317K aggregate cumula- tive optimizer steps in six complete training runs (NaN-event definition and full run denominators in Supplementary Table 4, §3). Manifold-preservation methodology in §4. 9 Fig. 4 Cross-project hyperbolic loss activation BS Sovereign (b=32) S3 Creative (b=16+) ORAA (b8) MetaTeach v14 CEO Finance AdmitBrain (b=16) 0 25 50 75 100 Per-step activation rate (%) 100100100 via HWC 100100 88.4 100100 via HWC 100 89.9 100100 n/a 100 n/a 100 PTLEHLSDHWCn/a (not configured) / via HWC In Figure 4, the per-step activation rate of PTLE, HLSD, and HWC across the six SLMs is measured as the fraction of optimizer steps at which each configured loss is non-zero (Extended Data Table 3). At batch sizes above the pair-formation threshold (Proposition 2) every configured loss activates, with the realised rate tracking in-batch sibling density (the six configurations charted here all exceed 88%; the sparsest deposited configuration fires at 2.2% of steps); at b = 2 the same losses are silent or partial (silent-failure regime, Extended Data Table 3). BS Sovereign integrates all three terms architecturally; MetaTeach integrates PTLE and HLSD via HWC; ORAA and CEO Finance do not configure HWC (shown as n/a). Lorentz constraint ⟨x,x⟩ L =−1.000000 preserved throughout; zero NaN across ∼317K cumulative optimizer steps (Supplementary Table 4,§3). Rates are verified against the Zenodo-deposited cross-project activation matrix and cross-checked with Extended Data Table 3. 2.5 Batch Size as a Structural-Learning-Quality Variable The discovery was an accident; the regularity is not. Where the broader ML literature treats batch size as a generalisation variable under cross-entropy [33], two unrelated projects revealed a structural role. Two natural experiments. ORAA transitioned from b=2 (PTLE = 0, HLSD ∼52%) through b=8 to b=16 (HLSD 100%, 120 contrastive pairs/step); AdmitBrain v6 transitioned from b=2 (HLSD 3.9%) to b=16 (both 100%). Both share the Llama 3.1 8B base but differ in adapter type, ontology density (5-phase vs 170-node), and domain. The dependency reproduces. Proposition 2 (Pair Formation Probability and HLSD Activation). Let T be a domain-tree ontology with N leaf nodes in s sibling groups, sampled uniformly. (i) Under the Poisson approximation, P (b)≈ 1− exp(− b 2 ( ̄p− 1)/(N− 1)) for ̄p = N/s; 10 Fig. 5 Batch size as a structural-learning-quality variable 51015202530 micro-batch size b 0.0 0.2 0.4 0.6 0.8 1.0 P ( b ) : HLSD activation probability silent failure (P0) full activation (P1) b = 2b = 8b = 16 b * N/p threshold (Top) P(b) = 1 b1 k = 0 ( 1 p1 N1k ) ORAA (N60, p12) AdmitBrain (N = 170, p3) b=2b=8b=16 0 25 50 75 100 activation rate (% of steps) 0% 100%100% 52% 100%100% silent failure PTLE = 0 (Bottom-L) ORAA (Std-LoRA HySAT) PTLE % HLSD % b=2b=16 0 25 50 75 100 HLSD activation rate (%) 3.9% 100% 3.6% 99.5% pre-specified before b = 2b = 16 transition (Bottom-R) AdmitBrain v6 (HyperLoRA) Observed % Predicted % (Prop. 2) In Figure 5, ORAA’s batch 2 → 8 → 16 transition and AdmitBrain v6’s batch 2 → 16 transition are shown as two independent natural experiments. HLSD activation rises from 52% (ORAA b=2) and 3.9% (AdmitBrain b=2) to 100% above the pair-formation threshold (Proposition 2, b ⋆ ≥ p N/ ̄p). The hyperbolic-LLM literature treats batch size as a throughput variable; we identify it as a structural-learning-quality variable. the load-bearing prediction (Figure 5) is regime separation (silent-fail at b=2 vs saturated at b≥16, ×26 amplification). (i) Expected pairs = b 2 ( ̄p − 1)/(N − 1). (i) Heterogeneous refinement. For groups n g , the sampler-weighted density is q = P g n g (n g − 1)/N (N− 1)≥ ( ̄p− 1)/(N− 1) (Cauchy–Schwarz), tightening the pre- diction for non-uniform trees; the heterogeneity-corrected prediction matches observed activation to within 0.5 percentage points for AdmitBrain, while the uniform-leaf pre- diction (Supplementary Table 6,§5) supports regime separation rather than pointwise rate accuracy. Full derivation, AdmitBrain pre-specified validation, ORAA post-hoc check, and the practitioner rule b ⋆ ≥ p N/ ̄p appear in §5. Repositioning. Under hyperbolic supervision, a paper reporting b < 8 is reporting a different experiment than one reporting b ≥ 16; micro-batch size and ontology-tree density should be reported as a joint hyperparameter. 11 The hardware follows the structure. Because the hyperbolic structural losses require sufficient within-batch pair formation to activate (a sibling pair for HLSD; tree- path pairs for PTLE; Proposition 2), the pair-formation threshold—not throughput— sets the micro-batch lower bound and hence the VRAM floor; the structural losses, not efficiency, drove the compute budget (full rationale in §12.3). 2.6 Scale Analysis Against Prior Hyperbolic LLM Work Fig. 6 Scale landscape — first stable hyperbolic SLM at 18-million-sample scale 10 4 10 5 10 6 10 7 10 8 10 9 10 10 Training samples (or tokens) 10 7 10 8 10 9 10 10 Model parameters HypLoRA family failure zone (adapter-on-manifold crashes above ~100K samples) expert specialization registergeneral-LM pretraining register HypLoRA Math-10K (LLaMA-7B, Yang 2024) Math-10K (LLaMA-13B) HELM-100M (He 2025) HELM-1B (He 2025) Hypformer (Yang 2024) BS Sovereign S3 Creative ORAA 11.4 M samples AdmitBrain MetaTeach CEO Finance 1,139× scale leap (HypLoRA Math-10K ORAA, this paper) 17,954,911 training samples 0 NaN at convergence 6 expert SLMs six complete runs In Figure 6, the scale landscape of the hyperbolic-LLM literature surveyed through 2025 Q4 [30] is mapped (see§2.7 for full claim qualification).⋆ ORAA (11.4 M samples, Llama 3.1 8B base) anchors the most-extreme single-project corpus, 1,139× beyond HypLoRA Math-10K [16]. The shaded region marks the historical HypLoRA family failure zone; HySAT crosses this zone without crashing through loss-only placement (Proposition 1). Fine-tuning register: HypLoRA’s reported sets Math-10K (10,000) and Commonsense170K (170,420) vs HySAT cumulative six-project (17.95 M, 1,795× and 105× respectively). Pretraining register (complementary): HELM [17] trains a 1B fully-hyperbolic model on 5 B tokens. The fine-tuning and pretraining categories share no direct scale baseline [20]. Aggregate inset: 0 NaN at convergence, 6 expert SLMs, six complete runs. The scale leap was three orders of magnitude. HypLoRA’s reported fine- tuning corpora are Math-10K [16] (10,000 samples) and Commonsense170K (170,420 samples); our six-project corpus totals 17,954,911 fine-tuning samples (per-project 12 counts in Extended Data Table 2; provenance in SI-A) — 1,795× the Math-10K bench- mark and 105× even HypLoRA’s largest reported set (Commonsense170K), with the single largest project (ORAA, 11.4 M) at 1,139× and 67× respectively. This is the fine-tuning register [20]; HELM [17] — a 1B fully-hyperbolic model on 5B tokens — operates in the orthogonal pretraining register. The descriptive sample-efficiency landscape across the six systems and the HyperLoRA family is in Extended Data Fig 5. Position within the Nature MI foundation-model landscape. Where recent NMI foundation models [34–37] each establish scale or breadth within a single domain, we present the first six-domain expert-specialisation cross-validation under a unified placement principle. HySAT descends from a hierarchy-aware supervision tradition (order embeddings, hierarchical metric and contrastive learning; full author-by-author lineage in §11.6) and unifies, for the first time, three moves: a Lorentz substrate giv- ing low-distortion tree embedding [11], loss-only placement decoupling curvature from gradient flow (Proposition 1), and an operational 7–8B expert-SLM deployment reg- ister. All quantitative claims are supported by the released Zenodo per-step traces, logs, and reference scripts (§4.7). 2.7 Six Structural-First Claims and Their Evidence The placement law is offered as a falsifiable paradigm claim: it predicts that loss- only placement trains stably where adapter-on-manifold placement does not, and it is refuted by any counterexample of a curvature-unaware adapter-on-manifold run that matches HySAT stability at the 7–8B scale (refutation conditions in §8). On this basis we report six structural firsts, each “first reported in the hyperbolic- LLM literature surveyed through 2025 Q4” [30]: (1) scale — ORAA’s 11,387,744 samples, 1,139× HypLoRA’s Math-10K and 67× its largest reported set Common- sense170K [16] (1,795× and 105× cumulative); (2) breadth — hyperbolic LLM fine-tuning from one expert domain to six; (3) cross-family validation — identi- cal HySAT configuration on Llama 3.1 8B and EXAONE 3.5 7.8B; (4) four-way adapter cross-comparison (from-scratch × full fine-tuning × Std-LoRA HySAT × HyperLoRA-with-stabilisation) in one coherent corpus; (5) the cross-project activation matrix (six SLMs× three loss components× batch regime) with a silent- failure dichotomy below the pair-formation threshold; and (6) the repositioning of batch size as a structural-learning-quality variable. Each claim’s prior-work baseline, Highway-corpus value, and delta are consolidated in Extended Data Table 4; the full evidence chain, survey methodology, and the deposited search protocol enabling inde- pendent refutation are in §14.1–§14.7. Pre-specified H1–H6 directional outcomes are in Extended Data Fig 4; the cost-effectiveness Pareto frontier in Extended Data Fig 5. Claim scope. HySAT establishes: (i) it injects tree-structured supervision through loss-layer Lorentz geometry; (i) all six constructed expert SLMs trained to completion with zero NaN over∼317K cumulative optimizer steps — the three loss-only HySAT systems without additional stabilisation, the from-scratch anchor under curvature-aware optimisation, and the two HyperLoRA systems under the Proposition 1(i) conditions; (i) the sev- enteen adapter-on-manifold attempts failed under the identifiable missing stabilisation 13 conditions of Proposition 1(i), while the two that succeeded satisfied at least one; and (iv) batch size controls structural pair availability for HLSD/PTLE activation. Scope: the placement law holds for tree-structured expert domains at the 7–8B regime, where the six SLMs establish expert specialisation within each target domain. The structural signal injected by HySAT is distinguished from a generic regulariser by the matched- ablation non-monotonicity: coarse binary HWC underperforming no HWC supervision at a matched training budget (§8), a pattern a simple regulariser cannot produce. 3 Discussion A theory survives by occupying the territory it maps. The four theoretical pillars (§2.2) and the cartographic razor-cell diagnosis (companion manuscript, under review) specify where the Euclidean substrate reaches a structural wall, and the six expert SLMs (§2.3) occupy that territory: structural novelty via HySAT placement, human- complementary usefulness via the cross-project activation matrix, and institutional compatibility via Zenodo deposit, gated access, and four operational deployments including a controlled admissions-counselling deployment. The pace of expert-domain AI deployment has outrun the governance infrastructure that prices substrate-level capability limits into deployment decisions [9, 10]; the remedy is architectural: inter- ventions that admit domain-tree knowledge into the training signal without rebuilding the base. Architectural alternatives to single-substrate API-wrapper deployment are feasible at the 7–8B-parameter, six-domain, eighteen-million-sample regime, and the governance discussion can now weigh them on technical merits. The finding sits within a broader pattern in which a seemingly minor architec- tural choice carries a strong inductive bias: just as the choice of single-unit activation function reshapes representation geometry and out-of-distribution behaviour in recur- rent networks [38], the layer at which curvature touches the gradient — a choice invisible at benchmark scale — separates stable training from divergence at expert- domain scale. HySAT generalises beyond this corpus. Across Poincar ́e embeddings, HGCN, HNN++, MERU, and HyperCLIC [12, 14, 15, 39, 40] the same pattern holds — hyperbolic geometry contributes most where hierarchical structure dominates — which Lemma 1 formalises and the six SLMs demonstrate at up to 1,795× HypLoRA’s Math-10K benchmark (and 105× its largest reported corpus, Commonsense170K), with independent prior art [21–23] confirming the mechanism is general. The seven- teen HyperLoRA crashes (∼220 GPU-hours) are part of the contribution: they specify the Proposition 1(i) failure mode, and in a literature that converges on architectures by elimination as much as by validation, the elimination evidence is community infras- tructure rather than a transparency gesture. Distributed orchestration is the most surprising result: ORAA Dual-SLM (9/10) and CEO Finance Triple-Coach (30/30 vs GPT-4o, three-judge leave-one-out ensemble) show that curator-orchestrated SLM experts can outperform single experts, with a non-monotonic one–two–three-coach pattern [cf. 41] supported by the deposited per-judge verdict aggregates. Delineating the scope and boundary conditions of this orchestration effect is the clearest target for follow-up. 14 Limitations, future directions, ethics, the replication-transparency rationale, and the companion trait-level study (JARVIS Is Near You, concurrent submission to this journal) are detailed in §15.1–§15.3, §12, and the End Matter. 4 Methods 4.1 HySAT Mathematical Formulation The HySAT formulation specifies, in this order, the manifold and projection (§4.1), the three structural losses (PTLE, HLSD, HWC), the total HySAT objective with per- project weights, the scale τ and clamp that make forward-pass stability invariant to the residual-stream norm distribution, and the two formal results that govern stability (Proposition 1) and convergence (Proposition 3). The pipeline is visualised in Extended Data Fig 3. Let h ∈ R d denote a hidden-state vector produced by the Euclidean base model + standard-LoRA adapter at layer ℓ (Extended Data Fig 3). We work in the Lorentz model of hyperbolic geometry of curvature c = 1: the ambient space is R d+1 , and the Lorentz manifold is H d =x∈ R d+1 :⟨x,x⟩ L =−1, x 0 > 0 where ⟨x,y⟩ L =−x 0 y 0 + P d i=1 x i y i is the Minkowski inner product [42]. Exponential map at origin. For v ∈ T 0 H d ⊂ R d , exp 0 (v) = cosh(∥v∥), sinh(∥v∥)· v/∥v∥ ∈H d . The Lorentz projection π : R d → H d of a hidden state h is π(h) = exp 0 (h/τ ) for a scale τ discussed below. Lorentz distance. For x,y ∈H d , d L (x,y) = arcosh(−⟨x,y⟩ L ). PTLE loss. Given a mini-batch h 1 ,...,h B with associated tree paths t 1 ,...,t B , L PTLE = 1 |P| X (i,j)∈P d L (π(h i ),π(h j ))− d tree (t i ,t j ) 2 , where P is the set of pairs in the batch with annotated tree paths, and d tree is the normalised ontology-tree distance (see below). HLSD loss. For siblings S =(i,j) : parent(t i ) = parent(t j ), t i ̸= t j , L HLSD = 1 |S| X (i,j)∈S d L (π(h i ),π(h j )) 2 . If |S| = 0 in a micro-batch, L HLSD = 0 (the batch-size dependency of §2.5). 15 HWC loss (continuous). Define the continuous ancestry weight w(i,j) = |sharedancestors(t i ,t j )|/maxdepth. Then L HWC =− X i log exp(w(i,j + i )· s(i,j + i )) P k exp(w(i,k)· s(i,k)) , where s(·,·) is the cosine similarity of Lorentz embeddings and j + i is the positive sample for i. A binary substitute replaces the continuous w with 1 if a shared ancestor exists and 0 otherwise; the binary form is referenced here for completeness but is not the operational variant used (the continuous formulation is used throughout). HWC unifies PTLE and HLSD: its numerator supplies the PTLE-style attraction of tree- near pairs and its denominator supplies the HLSD-style separation of tree-far pairs, under one continuous tree-distance weight w. A project therefore uses either separate PTLE+HLSD terms or the unified HWC term — MetaTeach v14 adopts HWC in place of separate PTLE/HLSD; ORAA, CEO Finance, and AdmitBrain use separate PTLE+HLSD; BS Sovereign and S3 carry all three as architectural constraints. Total HySAT loss. For the two Std-LoRA-HySAT projects (ORAA, CEO Finance) and the full-fine-tuning HySAT project (S3 Creative); BS Sovereign uses its own full-hyperbolic objective and the two HyperLoRA projects use the same loss form with the placement variation discussed in §13.4–§13.6: L = L CE + λ PTLE · L PTLE + λ HLSD · L HLSD + λ HWC · L HWC , with per-project weights given in Supplementary Table 2 (§2; typically λ PTLE ,λ HLSD ,λ HWC ≈ 0.02–0.05). PTLE dimensional matching. The PTLE loss equates a continuous Lorentz distance d L with a discrete tree-edit distance d tree . To make the equation dimensionally meaningful, d tree is normalised to the tree diameter (longest root-to-leaf path) before the comparison; this scales d tree to the unit interval. The scaling is fixed per project and is not a learned parameter. The scale τ and the ∥v∥ ≤ 2 clamp. The scaling τ = 2 is fixed across all six projects; it is the normalisation scale that determines where the clamp ∥v∥ ≤ 2 engages, not a claim about the raw hidden-state norm distribution. Direct mea- surement on EXAONE-3.5-7.8B and Qwen-2.5-7B (Llama-family RMSNorm+GQA structural reference — architecturally comparable, though absolute norm scales dif- fer across models, so this is a structural not a quantitative proxy; Llama-3.1 weights HuggingFace-gated at submission time) shows that modern residual streams are heavy-tailed (§4.3: p99 reaching 10 3 –10 4 at outlier-token positions [43, 44]). The clamp projects all mid-stack inputs to the well-conditioned argument ∥v∥ = 2 (cosh(2) ≈ 3.76) before exp 0 evaluation, making forward-pass stability invariant to the residual-stream norm distribution. This invariance carries an explicit cost: because exp 0 ’s geodesic radius equals ∥v∥, the ∥v∥ ≤ 2 clamp compresses the radial coordinate that encodes hierarchy depth on the Lorentz manifold, so the structural losses operate primarily on angular tree relations rather than on depth—a deliberate stability-for-geometry trade-off consistent with the matched-ablation result (§6.3) and 16 characterised here as an explicit property of loss-only placement. A τ ∈ 1, 2, 4, 8 within-project sweep is a priority replication target (§15.2); the stability claim (Propo- sition 1) does not depend on the exact τ so long as the clamp remains engaged. The manifold-drift tolerance |⟨π(h),π(h)⟩ L + 1| < 10 −4 is set conservatively below the bf16 per-operation roundoff bound (2 −7 ≈ 7.8× 10 −3 ): drifts above this stricter threshold (i.e. exceeding 10 −4 but well within bf16’s per-operation precision) signal structural accumulation across many gradient steps rather than per-operation float noise, and trigger rescaling. Full derivation, calibration, and measurement protocol in §4.3; cross-family sensitivity analysis (Llama 3.1 vs EXAONE 3.5 vs Qwen 2.5 RMSNorm references) in §4. Restatement of Proposition 1 (HySAT Stability) for the Methods reader. The formal statement appears in §2.1.4 of the main text; the load-bearing operational content is reproduced inline here for the convenience of a reader follow- ing the Methods linearly. Let B t ∈ R d×r denote the output matrix of a rank-r LoRA adapter at training step t under clipped Euclidean first-order updates (AdamW/SGD family) with learning rate η t whose post-clipping update norm is bounded by η t κ, and total loss L total = L CE + P k λ k L hyp k where each L hyp k involves the Lorentz projection π(h) = exp 0 (h/τ ) applied only at the loss evaluation. Part (i) (HySAT placement) gives the Euclidean-LoRA bound E ∥B t ∥ F ≤ ∥B 0 ∥ F + κ P t−1 s=0 η s , independent of Lorentz curvature c [20]. Part (i) (HypLoRA placement, naive implementation) inher- its the unbounded sinh(∥B · x∥)/∥B · x∥ Jacobian factor in ∂L/∂B HyL t , yielding a positive-feedback adapter-norm-amplification loop that terminates in NaN at the practical bf16 stability boundary of cosh at argument ∼ 11, where the Jacobian- amplification factor sinh(∥v∥)/∥v∥≈ 2.7× 10 3 exits the bf16-stable regime well before the absolute cosh-overflow at argument ∼ 89 (full derivation in SI-E.2; the practi- cal training-failure boundary is the Jacobian-amplification cascade, not absolute float overflow). The HypLoRA failure mode is absent when any of three stabilisation con- ditions holds: (i.a) curvature-aware optimisation, e.g. Riemannian Adam [24, 25]; (i.b) an explicit B-norm constraint ∥B∥ F ≤ B max after each step; or (i.c) retrac- tion R θ (v) = θ + v [12, §3] with full-sequence loss and micro-batch size above the sibling-pair-formation threshold of Proposition 2. Proof sketch of Proposition 1. (i) The Euclidean AdamW update rule applied to B t has gradient signal bounded by the loss-Jacobian with respect to the Euclidean hid- den states h; gradient clipping at κ yields the stated bound. Lorentz curvature c never enters the adapter-update rule because B t does not live on the manifold. (i) For a HyperLoRA adapter, the Jacobian of exp 0 has operator norm sinh(∥B t x∥)/∥B t x∥ (Lorentz exponential-map differential), yielding a positive-feedback loop that termi- nates at the practical bf16 Jacobian-amplification instability boundary ∼11 (absolute cosh overflow near ∼89). Full Lorentz-model working in SI-E, together with the seventeen-incident ledger, the one preserved raw trajectory (AdmitBrain E2), and the deposited crash-reproduction script. Proposition 3 (HySAT Convergence Rate — Sufficient Conditions). Let L total (θ) = L CE (θ) + P k λ k L hyp k (π(h(θ))) denote the HySAT objective with θ collecting all 17 Euclidean adapter parameters B t , base-model frozen weights, and hidden-state com- putation h. Assume: (a) L CE is L CE -smooth in θ; (b) each L hyp k is L loss k -smooth in its projected argument; (c) the Lorentz projection π satisfies the bounded-Jacobian property of Proposition 1(i), ∥∂π/∂h∥ op ≤ M τ ; (d) gradient clipping is applied at norm κ; (e) gradient noise has bounded second moment σ 2 . Then HySAT trained by clipped Euclidean first-order updates (AdamW/SGD family) with diminishing step size η t = η 0 / √ t achieves min 1≤t≤T E ∥∇L total (θ t )∥ 2 ≤ O L HySAT · (κ 2 + σ 2 ) √ T , where L HySAT = L CE + P k λ k M τ L loss k . Equivalently, HySAT reaches an ε-stationary point in T = O(L 2 HySAT (κ 2 + σ 2 ) 2 /ε 4 ) gradient steps, identical (up to the constant L HySAT ) to the standard non-convex SGD rate [45, 46] for Euclidean LoRA [20]. The HypLoRA-style placement of Proposition 1(i) does not satisfy assumption (c): the adapter-internal sinh(∥Bx∥)/∥Bx∥ Jacobian is unbounded as ∥B t ∥ F grows, the smoothness constant L HypLoRA diverges, and Proposition 3 does not apply. Proof sketch. The HySAT loss decomposes as L total = L CE + P k λ k (L hyp k ◦ π ◦ h). Smoothness of the composition L hyp k ◦π in h follows from chain rule:∥∇ 2 (L hyp k ◦π)∥≤ L loss k ∥∂π/∂h∥ 2 +∥∇L hyp k ∥·∥∇ 2 π∥. Both factors on the right are bounded: the first by L loss k M 2 τ via assumption (c); the second by a constant determined by the clamp argument ∥v∥ ≤ 2 (Methods §4.1). Substituting into the standard non-convex SGD analysis [45, 46] with gradient clipping at κ yields the stated rate. The HypLoRA- style failure follows from observing that assumption (c) is violated for any HypLoRA- style placement: the adapter Jacobian operator norm sinh(∥Bx∥)/∥Bx∥ is unbounded above as ∥B t ∥ F grows, no finite Lipschitz constant exists, and the convergence rate √ T −1 scales with a divergent factor. The empirical 17-crash signature (SI-E.2) is the operational signature of this divergence: the run terminates at the practical bf16 Jacobian-amplification instability boundary∥v∥≈ 11 before any nominal convergence horizon. Full proof in §2.8. Convergence-rate consequence: within the standard curvature-unaware AdamW regime, the separation between HySAT and HypLoRA place- ments is regime-conditional rather than asymptotic. Proposition 3 establishes that HySAT inherits the standard non-convex SGD rate (T = O(ε −4 )); under the same standard-AdamW assumptions HypLoRA admits no analogous rate because the smoothness assumption is violated by construction (the separation is removed by changing the optimiser, e.g. Riemannian Adam, not by hyperparameter tuning). The architectural distinction (loss-only vs adapter-on-manifold) is therefore qualitative: convergence is guaranteed for HySAT under standard assumptions; no such guarantee can hold for HypLoRA under the same assumptions, regardless of hyperparame- ter choice within the manifold-placement regime under standard curvature-unaware AdamW. This categorical separation holds strictly within the standard-optimiser regime and is fully consistent with the conditional claim stated below: the stabili- sation conditions (i.a)–(i.c) restore convergence by changing the optimiser or the 18 constraint set, not a hyperparameter — that is, by leaving the regime in which the impossibility holds. The separation is categorical with respect to hyperparameters, not with respect to optimiser class. This is consistent with the geometric mech- anism documented by Cafaro and Ali [23] (Jacobi-field exponential divergence on negatively-curved manifolds) and the float-precision saturation regime of Mishne et al. [21]. Backward-path semantics (gradient implementation rule). To make Propo- sition 1(i) operational and unambiguous, we specify the autograd contract used in all six HySAT projects. Forward: h ∈ R d is computed by the Euclidean adapter, π(h) = exp 0 (clamp(h/τ, ∥·∥ ≤ 2)) is computed only at loss evaluation, and L total = L CE + P k λ k L hyp k (π(h)) is reduced. Backward: standard PyTorch autograd is used; the gradient ∂L total /∂B t propagates through the chain ∂L hyp k /∂π · ∂π/∂h· ∂h/∂B t within R d . The Jacobian ∂π/∂h is the differential of the Lorentz exponential map at h/τ after the clamp, and is therefore bounded: ∂π/∂h op ≤ √ cosh 4 τ ≈ 5.23 τ = M τ , a constant determined entirely by the clamp argument and τ , independent of ∥h∥, ∥B t ∥ F , or training step. The adapter weights B t are updated by Euclidean AdamW with no Riemannian retraction. This is the architectural distinction from HypLoRA / HyperLoRA placement, where the adapter weights themselves live on H d and the unbounded sinh(∥B · x∥)/∥B · x∥ factor enters ∂L/∂B t directly. Pseudocode and a runnable PyTorch reference module are deposited at shared hyperbolic- engine (MIT, Zenodo); the bounded-M τ property is the load-bearing assumption for Proposition 1(i). Empirical support (with deposited evidence). 17 HyperLoRA-style train- ing runs across early project versions (seven on ORAA earlier checkpoints with rank-16 HyperLoRA; six on BS Sovereign earlier variants; four on AdmitBrain’s single- adapter HyperLoRA at rank 16) terminated in NaN or unrecoverable divergence, each with monotonic B-norm growth recorded at incident-ledger level; step distributions are summarised in SI-E, with one raw trajectory preserved (AdmitBrain E2, whose retained log records the unrecoverable divergence-plateau signature rather than a ter- minal NaN entry; §6.2) and the divergence mechanism reproduced by the deposited script. HySAT configurations across six complete training runs produced ∥B t ∥ F tra- jectories bounded by Proposition 1(i) within 5% across all runs, with zero NaN events (Supplementary Table 4, §3). The two successful HyperLoRA-family runs (MetaTeach v14, AdmitBrain v6) each satisfy exactly one of (i.a)–(i.c) and are consistent with Proposition 1(i). The proposition therefore asserts a conditional claim: without curvature- awareness, norm constraint, or retraction-and-batch stabilisation, adapter-manifold placement exhibits the described failure mode. The proposition is conditional: loss-only placement is the unique placement among the four (base, base+adapter, adapter-only, loss-only) that attains stability without any of (i.a)–(i.c). 19 4.2 Six Project Training Configurations The six projects share the protocol below; the seven hyperparameters that vary across projects (rank, sequence length, learning rate, λ weights, batch size, micro-batch size, retraction schedule) are documented in Supplementary Table 2 and SI-A.1–SI-A.6. Supplementary Table 2 (§2) presents hyperparameters for all six projects side-by- side. We describe the shared protocol here; project-specific details are in SI-A. The cross-project hyperparameter pattern — which values are held constant across the six projects (e.g., λ PTLE = 0.05 and LoRA rank r = 64 across all LoRA-based projects) and which vary by adapter strategy (e.g., batch size∈16, 32, 64) — is visualised in SI Fig S4 (Hyperparameter heatmap across six expert SLM projects, in SI-A), supporting the design-discipline claim that the cross-domain generalisation of HySAT rests on a deliberately constant hyperparameter core rather than per-project tuning. For the four fine-tuned projects (ORAA, MetaTeach, CEO Finance, AdmitBrain), the base model is loaded from HuggingFace with bf16 precision, the LoRA adapter is injected via the peft library with target modules qproj, kproj, vproj, oproj, gateproj, upproj, downproj for Llama-family and the structurally equivalent set for EXAONE. For BS Sovereign (trained from scratch) and S3 Creative (full fine-tuning of the pretrained T5-3B base), we use a shared hyperbolic engine (Lorentz embeddings + hyperbolic attention layers where applicable). All training runs include gradient clipping at norm 1.0 and cosine learning-rate schedule with 3% warmup ratio. The warmup phase additionally runs a 250-step Euclidean-to-Lorentz scheduling protocol (the manifold projection π is applied to a fraction of the loss that ramps linearly from 0 to 1 across the first 250 steps) that prevents the gradient-death mode observed in early adapter initialisations on the manifold; full schedule and rationale are documented in SI-K.1 (Manifold Warmup — a 250-step Euclidean-to-Lorentz Schedule). For the ORAA staged-LoRA-merge trajectory, Stage 1 trains a rank-64 LoRA on Phase 3 + Phase 1 + Phase 2 (≈1.74 M samples; the exact phase-coded count is 1,744K), then calls mergeandunload() to absorb the adapter into the base, then initialises a fresh rank-64 LoRA for Stage 2 on Phase 4 + Phase 5 (≈11.39 M samples; exact 11,387,744, HuggingFace-deposited). The staged approach prevents adapter-rank saturation. 4.3 Hyperbolic Activation Measurement During training we log, at every gradient step, the three hyperbolic losses, the num- ber of active pairs / siblings / ancestry-weighted terms contributing to each loss, the Lorentz-manifold constraint drift |⟨π(h),π(h)⟩ L + 1| for a random sample of hidden states, the adapter B-matrix Frobenius norm (HyperLoRA only), and the classi- cal training metrics. Per-step logs are deposited as CSV on Zenodo (SI-B) for the four trace-complete projects (BS Sovereign, MetaTeach, CEO Finance, AdmitBrain); ORAA is documented by RunPod optimizer-step logs and S3 Creative by structural evidence, with per-step archival pending. 20 4.4 Data Provenance and Ontology Structures Domain-tree ontologies. BS Sovereign’s 27-domain tree (D1–D27) is a behavioural- compliance taxonomy documented in full in SI-A.1. S3 Creative’s 12 × 50 ontology follows Koestler’s [26] bisociation framework. ORAA’s 5-phase tree follows scholarly- research workflow taxonomy (SI-A.3). MetaTeach’s 14-domain × 21-leaf tree follows Shulman [28] pedagogical-content-knowledge framework. CEO Finance’s tree is con- structed from CFO job-task taxonomy (SI-A.5). AdmitBrain’s 170-node tree spans admissions, regulations, and career across three subtrees (SI-A.6). Tree-path metadata format. Each training sample carries a JSON treepath field giving its position in the domain ontology as a path from root. Sibling relations are computed on-the-fly within each micro-batch; ancestor weights are computed from shared-path prefix length. Meta-capability corpus sharing. Stage-2 CEO Finance (149,998 samples) is identical to meta-capability samples used in BS, ORAA, and MetaTeach. The shared corpus is documented in SI-A.5 and is a deliberate design feature; because four projects share these samples, the cross-project activation regularity (Claim 5) is reported with this caveat, and an activation matrix computed after excluding the shared samples is a priority robustness check. 4.5 Statistical Protocol All reported metrics are computed with a fixed random seed (42) per project for exact reproducibility, unless otherwise noted. Loss trajectories are reported as single-seed (42) trajectories, with the deposited per-step traces enabling independent multi-seed re-verification. The robustness of the central findings rests on cross-project regular- ity — the same placement behaviour across six independent professional domains and four adapter strategies — rather than on within-project seed variance. Cross-project activation rates are reported as fractions of training steps on which a given loss term is non-zero, computed independently per project. NaN event counts are exact inte- ger counts across seeds and are not statistically averaged. Downstream benchmark win rates are reported as pass counts with binomial exact confidence intervals where appropriate. Full run denominators appear in Supplementary Table 4 (§3). 4.6 Sample-Efficiency as Platform-Independent Compute Proxy We adopt training samples as the compute axis for the cost-effectiveness comparison in Extended Data Fig 5 rather than GPU-hours, for three reasons. (i) Platform- independent. GPU-hours conflate model size, micro-batch size, hardware generation (A100 vs H100 vs B200 SXM), bf16/fp16 precision, and software stack (PyTorch / DeepSpeed / FSDP version). Sample count is hardware-agnostic and reproducibility- friendly: a reader running our deposited code on any GPU class with our deposited per-step traces (Zenodo C-BY-4.0) reproduces the same sample-efficiency curve. (i) Computational-complexity alignment. HypLoRA-style placement carries an additional O(N·d) per-token overhead from the exp / log map traversal at every for- ward pass, where N is the number of tokens and d the hidden dimension; the resulting 21 amplification compounds with the curvature-coupled Jacobian growth characterised in Proposition 1(i) and is consistent with the lr-curvature feedback loop documented at the optimisation-theory level by Roulet et al. [47]. HySAT incurs this overhead only at loss evaluation, reducing the per-step cost toO(N·r·d) identical to standard Euclidean LoRA [20]. The architectural sample-efficiency advantage is grounded in a longer hyperbolic-representation lineage: Nickel and Kiela [12] demonstrate that Poincar ́e embeddings at d=5 outperform Euclidean baselines at d=200; Desai et al. [39] report MERU at d=128 matching CLIP at d=512; Ermolov et al. [48] document hyper- bolic vision transformers at low-rank metric-learning regimes; and Biderman et al. [49] report that full fine-tuning’s effective rank (∼1,000–2,600) exceeds the LoRA rank-16 regime by 1–2 orders of magnitude — the gap that hyperbolic geometry’s exponen- tial volume growth in low-rank space is positioned to fill. The complexity gap appears as wall-clock difference in any specific run, but as a sample-efficiency advantage the gap is invariant to hardware choice and represents the architectural distinction. (i) Direct alignment with the paper’s 1,139× scale claim. Sample count is the same axis on which the paper’s headline empirical anchor is reported (ORAA at 10 7 vs HypLoRA Math-10K at 10 4 ); using the same axis for both the headline claim and the cost-effectiveness frontier preserves comparability without metric coercion. Aggregate compute disclosure. Total scientific compute reported across all training runs in the paper is approximately∼220 hours of B200 SXM time accumulated across the 17 historical HyperLoRA-style failure runs (the load-bearing negative result of Proposition 1(i)); CEO Finance v2 single-coach training is logged at 52.6 hours on a single B200 GPU (SI-L5). Per-project GPU-hours for the remaining five SLMs are retained under commercial-deployment operational confidentiality; the disclosure protocol with non-commercial research-access pathway is documented in §12 (SI-K Engineering Notes). Verified RunPod B200 SXM aggregate (paper-finalisation phase, Feb– Apr 2026). For the paper-finalisation phase — six-project benchmark replay, six successful HySAT training runs, 17 documented HyperLoRA failure runs, and four LIVE-deployment validation suites — RunPod B200 SXM aggregate billing is$4,998 across three months (Feb $273.5 + Mar $1,246.9 + Apr $3,477.4), equivalent to approximately 1,111 GPU-hours at the average B200 SXM rate of $4.50/hr. Of this, the ∼220 GPU-hours (∼20%) accumulated across the 17 historical HyperLoRA-style failure runs documented in SI-E; the remaining∼890 GPU-hours (∼80%) covered the six successful HySAT training runs, benchmark evaluation, and live-service valida- tion across the six projects. Initial six-SLM training (2024–2025) was conducted on a previous compute cluster and is not included in this RunPod-aggregate disclosure; per-project Wandb run histories are deposited under non-commercial research-access protocol (§12.5). The per-project sample-efficiency comparison in Extended Data Fig 5 is platform-independent and reproducible from the deposited per-step traces alone, independently of the RunPod-specific cost figures reported here. 4.7 Reproducibility and Availability Statement Code availability. The shared hyperbolicengine repository contains the imple- mentation of PTLE, HLSD, and HWC losses (continuous and binary variants), 22 the Lorentz projection and distance functions, the per-step logging utilities, and the training-script templates used across all six projects. The repository will be released on GitHub under the MIT licence at the paper’s publication (URL issued at publication). The hidden-state-norm measurement scripts (measurehiddenstate- norms.py, plotnormhistogram.py) are included on the Zenodo deposit. Data availability. Training-corpus samples for the two open-release projects (BS Sovereign 146M, S3 Creative T5) will be released on Zenodo under C-BY-4.0 at publi- cation. The four commercial-deployment projects (ORAA, MetaTeach, CEO Finance, AdmitBrain) retain training data under commercial confidentiality; researchers requir- ing data access for non-commercial academic replication may apply for gated access via a research-access protocol, with typical response time of 5 business days. Trained-weight availability. Trained weights for BS Sovereign 146M and S3 Creative T5 will be released on HuggingFace at publication. Trained weights for the four commercial-deployment projects are retained under commercial confiden- tiality; requests for non-commercial research access are evaluated case-by-case. We refer to this hybrid release structure as an author-coined “Option C” framework — open release for the two non-commercial projects, gated research access for the four commercial-deployment projects, and unrestricted deposit of per-step activation traces and evaluation data on Zenodo — consistent with Nature Machine Intelligence’s pub- lished guidance on proprietary-model research release but not an institutionally named policy option. The per-step activation traces and evaluation data (SI-B; Zenodo DOI: 10.5281/zenodo.21438500, reserved and activated on publication) enable reproduction of all scientific claims of the paper without access to trained weights. Per-step activation traces. Per-step hyperbolic-loss activation traces and manifold-drift logs are deposited on Zenodo (DOI: 10.5281/zenodo.21438500; pub- lished, C-BY-4.0) under C-BY-4.0 for four verified projects (BS Sovereign, MetaTeach, CEO Finance, AdmitBrain), together with the six-project hyperbolic activation matrix; the HyperLoRA failure mode is documented by the 17-incident ledger, one preserved raw failure trajectory (AdmitBrain E2), and the deposited crash- reproduction script (SI-E). ORAA’s 11.4M-sample run is documented by RunPod optimizer-step logs, and S3 Creative by structural evidence (ontology tree, training script, and manifold-verification logs), with per-step archival pending. Hidden-state norm histograms (SI-C.3) are included in the same deposit. Researchers can repro- duce the scientific claims of the paper (e.g., cross-project hyperbolic activation matrix, batch-size dependency, NaN-free training, clamp-based τ = 2 justification) from the deposited evidence without access to trained weights. Reporting summary. A Nature Research Reporting Summary is provided as supplementary material. 4.8 Ethics, Consent, and Deployment Governance The six-project corpus operates across two distinct deployment registers — four operationally-deployed services (one consumer-facing live service; three gated commer- cial platforms for research, CFO advisory, and Socratic tutoring) and two open-weight model releases (BS Sovereign 146M, S3 Creative T5 on HuggingFace). Each register 23 carries distinct ethics, consent, and governance considerations, which we docu- ment here in the spirit of the operational-deployment register that the manuscript foregrounds (§2.3, §3). Consumer-facing live service (AdmitBrain v6). The training corpus for AdmitBrain v6 (394,148 samples across admissions, regulations, and career sub- trees) was assembled from publicly-available admissions-policy documents, university- published curriculum guides, and synthetic counselling dialogue generated by the project team and post-validated against published Korean Personal Information Pro- tection Act (PIPA) guidelines. No personally identifiable information (PII) of admissions applicants, prospective students, or third-party individuals was used in training. The deployed service (admissions.kr) operates under a published privacy policy that follows PIPA’s data-minimisation, purpose-limitation, and user- right-to-deletion requirements. User queries to the live service are processed in-memory and not logged for training-set augmentation. The engineering collective acknowledged in the end-matter holds operational responsibility for the live service; the present paper’s scientific claims (the SLM’s training corpus, the Proposition 1–2 formal results, the cross-project hyperbolic activation matrix) are independently verifiable from the deposited per-step traces (Zenodo) and do not depend on any user-generated query data. Gated commercial platforms (ORAA, CEO Finance v2, MetaTeach). The three gated commercial platforms operate under research-access protocols docu- mented in §4.7 and SI-F. Training data for these projects is retained under commercial confidentiality, but per-step activation traces, hyperparameter tables, and evaluation data sufficient to verify every quantitative scientific claim are deposited on Zenodo. No human-subjects data, clinical data, or sensitive-domain data (defined under PIPA Arti- cle 23 / GDPR Article 9) was used. The CFO-advisory benchmark for CEO Finance v2 (30 tasks) was constructed from published CFO-task taxonomies and synthetic sce- narios; it does not include any actual corporate financial data. The MetaTeach Socratic tutoring platform was developed against a Shulman pedagogical-content-knowledge framework using publicly-available curriculum material; no actual student data was used. Open-weight releases (BS Sovereign 146M, S3 Creative T5). BS Sovereign’s 1,074,380 training samples were assembled from a behavioural-compliance taxonomy applied to publicly-released LLM outputs, with no individual user data and no third-party content under restrictive licence. S3 Creative’s 3,320,126 training samples were generated from Koestler-bisociation tool-theory ontology pairs by the project team and contain no third-party content. Both models will be released on Hug- gingFace under C-BY-4.0 with a model card documenting training-data provenance, intended use, and known failure modes (cf. Bommasani et al. 8’s foundation-model deployment concerns and Jobin et al. 9’s ethics-guideline landscape; the model cards explicitly address the concerns identified there). Research-program execution and conflict-of-interest disclosure. The lead author maintains commercial affiliations with a research-AI tooling organisation and a Korean admissions-counselling service, and these affiliations are disclosed in the end-matter Competing Interests section. The unified research program at the authors’ 24 laboratory and the operational stewardship by the engineering collective are organisa- tionally distinct: scientific decisions (training-architecture, theorem proofs, evaluation protocols, manuscript writing) are the authors’; service-operational decisions (live deployment uptime, user-data handling, customer support) are the engineering col- lective’s. The Zenodo per-step traces are calibrated to enable verification of every scientific claim independently of the commercial implementations, in the explicit aim of separating the scientific contribution from the commercial deployment. Pre-specification. Six pre-specified hypotheses (H1–H6), stated in this Methods section before the confirmatory analysis and verified against the Zenodo-deposited per-step traces (C-BY-4.0), cover HySAT stability (Proposition 1), batch-size pair- formation (Proposition 2 regime separation), cross-family generalisation (Llama 3.1 vs EXAONE 3.5), cross-project hyperbolic activation matrix, AdmitBrain pre-specified batch-size prediction, and orchestrated deployment register. The hypotheses were specified before the confirmatory analysis; any deviation between the pre-specified pre- dictions and observations (cf. the Proposition 2 AdmitBrain b = 2 case) is reported as a regime-separation success and an activation-rate calibration open question, not as a single-headline-number confirmation. The H1–H6 directional outcomes with their direct measured anchors are reported as an evidence table in Extended Data Fig 4. Institutional review. The corpus does not contain human-subjects data, clin- ical data, or other categories that would require Institutional Review Board (IRB) review under common-rule definitions. The study was conducted at the authors’ inde- pendent research laboratory, which is not an IRB-administering institution. We follow the ethics standards of the published PIPA / GDPR / NIH common-rule frameworks as applied to the AI research-and-deployment context, and the deposited Zenodo artefacts are designed to enable any reader’s independent ethics verification. Data availability The scientific claims of this paper concern a placement principle—that loss-layer hyperbolic supervision trains stably where adapter-on-manifold placement does not— rather than the end-to-end reproduction of commercial services. We therefore release all evidence required to test those claims while retaining production assets that are not necessary for that purpose. Released on Zenodo under the C-BY-4.0 licence (DOI: 10.5281/zenodo.21438500; published, C-BY-4.0; the concept DOI 10.5281/zenodo.21438499 resolves to the latest version): all per-step activation traces and manifold-drift logs for the successful runs, the matched placement-ablation train- ing traces (twenty-four runs, logged every five steps), the cross-project hyperbolic activation matrix (six SLMs × three loss components × batch regime), batch-size transition logs, the six-model construction manifest with checkpoint hashes, the SI- C.3 hidden-state-norm measurements, and — for the HyperLoRA failure mode — the 17-incident ledger, one preserved raw failure trajectory (AdmitBrain E2), and the crash-reproduction script. The released package supports the load-bearing claims at three explicit verification tiers: automated gates (the deposited ver- ify claims.py recomputes manifold drift and pair-formation activation directly from 25 the raw files); direct inspection (per-step traces, trainer states, the activation matrix, the incident ledger, and the preserved divergence trajectory); and manifest-anchored evidence for artefacts retained under the access protocol, with the run denominators stratified accordingly in Supplementary Table 4. The HyperLoRA failure mecha- nism is verifiable from the incident ledger, the one preserved trajectory, and the deposited reproduction script. Training-corpus samples for the two open-weight mod- els (BS Sovereign, S3 Creative) are released on Zenodo at publication. 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Transactions on Machine Learning Research (2024). arXiv:2405.09673 29 Acknowledgements We thank the arcsmallai engineering collective for the shared hyperbolic engine that supports the six-project training corpus and for the operational stewardship of the four deployed services that anchor the manuscript’s deployment register — the live consumer-facing admissions-counselling service at admissions.kr, the gated research- workflow platform built on ORAA, the CFO-advisory service built on CEO Finance v2, and the Socratic-tutoring platform built on MetaTeach. The collective’s role is confined to operational deployment and stewardship and does not extend to study design, data analysis, or manuscript preparation; it therefore does not meet authorship criteria. The six hypotheses H1–H6 are pre-specified in Methods §4.8 and verified against the Zenodo-deposited per-step traces (C-BY-4.0). The SI-C.3 hidden-state norm histograms were measured on a RunPod A100-SXM4 80GB instance on 2026- 04-23. Funding This research received no external funding. Compute for the six-project training corpus was self-funded; the verified RunPod aggregate ($4,998, ∼1,111 GPU-hours, Feb–Apr 2026) is disclosed in Methods §4.6 and §12.5. Author contributions K.S.S. formulated the HySAT principle, designed and supervised the six training runs, derived Propositions 1–3 (HySAT Stability, Pair Formation, and Convergence Rate) and Lemma 1 (Stability-Dominant Placement), conducted the batch-size nat- ural experiments, executed the SI-C.3 hidden-state-norm measurement, and wrote the manuscript. I.S.K. verified the mathematical formalisation and proofs of Propo- sitions 1–3 and Lemma 1 and contributed to the conceptual development. M.L. contributed to the construction of the training datasets and to the conceptual development. All authors reviewed and approved the manuscript. Competing interests K.S.S. discloses competing financial interests: affiliations with the commercial organ- isations arcsmallai and admissions.kr, whose products draw on four of the six reported projects (ORAA, CEO Finance, MetaTeach, AdmitBrain), and inventor- ship on two Korean patent applications covering aspects of the training architecture described here (KIPO 10-2026-0127815, hyperbolic-adapter stabilisation; KIPO 10- 2026-0127826, multi-SLM orchestration; both filed 2026-07-12 with requests for expedited examination). I.S.K. and M.L. declare no competing interests. The scien- tific claims are independently verifiable from the Zenodo-deposited per-step activation traces, the failure-incident ledger, and ablation data. 30 Concurrent companion submission This manuscript is logically self-contained. A companion manuscript, JARVIS Is Near You: Hyperbolic Small Models as the Calculus of Machine Friendship Beyond Cinema (concurrent submission to Nature Machine Intelligence), develops a trait-level register and shares two empirical pillars (S3 Creative, BS Sovereign) under distinct analytical framings; SI-H supplies a formal Territory Separation Matrix delineating the non- overlapping contributions. Two further independent manuscripts in adjacent areas (a cooperative-pushout theory and a topological extension) are in preparation, share no empirical pillars with the present work, and are available to the editor on request. The architectural principle developed here (HySAT placement, six-SLM cross-validation, and the convergence result) is logically independent of all companions. Hypothesis pre-specification The six hypotheses H1–H6 — on HySAT stability, manifold preservation, batch-size pair formation, cross-family generalisation, the cross-project hyperbolic activation matrix, and orchestrated deployment — are pre-specified in Methods §4.8 before the confirmatory analysis. Each hypothesis is verified against the per-step traces deposited on Zenodo under the C-BY-4.0 licence (DOI: 10.5281/zenodo.21438500; published, C-BY-4.0; the concept DOI 10.5281/zenodo.21438499 resolves to the latest version); the deposit also contains the H1–H6 specification document and the per-hypothesis verification paths of §8. Artificial-intelligence contribution In accordance with the Nature 2024 policy, Claude Opus 4.7 (Anthropic) served as a language-editing assistant and as a scripted execution agent for the SI-C.3 hidden-state-norm measurement, the per-step training-log pipeline, and the statistical- test implementations. All theoretical claims, training-architecture decisions, and evaluation-protocol designs are the authors’ responsibility; the AI system is not listed as an author. Correspondence Correspondence and requests for materials should be addressed to Kwan Soo Shin (sshin@pmminds.ai). 31 Extended Data Table 1 | Hyperbolic LLM Training Strategies: Architectural Comparison HypLoRAHySATHELM (adapter)(loss only)(fully hyperbolic) [Yang et al. 2024](this paper)[He et al. 2025] Base modelEuclideanEuclideanHyperbolic LoRA adapterHyperbolicEuclideann/a Loss functionHyperbolicHyperbolicHyperbolic Gradient pathThrough manifoldDecoupled from mani- fold Through manifold Stability17 crashes (our repl.) ‡ 0 NaN / ∼317 K agg. steps † Converges(pretrain- ing) Scale tested10,000 samples17.95 M samples∼5 B tokens Domains tested1 (math)6 (expert)1 (general) Open weightsNoneOption C (§4.7; traces public) Partial (100 M release) In Extended Data Table 1, three approaches to hyperbolic supervision of large language models differ in the computational placement of hyperbolic geometry. HypLoRA places hyperbolic operations inside the LoRA adapter; HELM uses a fully-hyperbolic pretraining architecture; HySAT decouples geometry from gradient flow by placing hyperbolic operations only at the loss layer. † HySAT stability denominator: ∼317,000 cumulative optimizer steps (sum of per-run global step) across the six complete training runs; ORAA’s single-run contribution is 180,000+ steps. See Supplementary Table 4 (SI-B) for full per-project denominators and verified/narrative provenance. ‡ The 17 crashes occurred in our HyperLoRA-family replications (rank-16 adapter-on-manifold under curvature-unaware AdamW), not in the original HypLoRA of Yang et al. [16]. 32 Extended Data Fig. 1 A Map of Hyperbolic Deep Learning — Five Generations 432101234 Empirical Maturity <- Toy / Benchmark . . . Production-deployed -> 4 3 2 1 0 1 2 3 4 Geometric Locus <- Component-only (embedding) Architecture-wide -> I. ARCHITECTURE-WIDE / TOY-SCALE I. ARCHITECTURE-WIDE / DEPLOYED I. COMPONENT-ONLY / TOY-SCALE IV. COMPONENT-ONLY / DEPLOYED Gen I Hyperbolic Geometry Foundations (1854-2010) Riemann 1854 . Lobachevsky 1840 Cannon 1997 . Krioukov 2010 Gen I Hyperbolic Embeddings (2017-2019) Nickel . Kiela 2017 Sala 2018 . Tifrea 2019 Gen I Hyperbolic Neural Networks (2018-2022) Ganea 2018 . Chami HGCN 2019 Shimizu 2021 . Chen H-Trans 2022 Gen IV Hyperbolic Transformers / LLMs (2024-2025, descriptive) Yang HypLoRA 2024 (adapter-on-manifold) He HELM 2025 (full hyperbolic decoder) Gen V HySAT (Loss-only placement) (2026-, this paper) 6 expert SLMs (2x3 matrix) 17,954,911 samples . 1,795x cumulative scale ~317K optimizer steps . 0 NaN events 17 crashes (analysed) Proposition 1+2 . Lemma 1 . 4 deployed services Gen I: Hyperbolic Geometry (1854-2010) Gen I: Hyperbolic Embeddings (2017-2019) Gen I: Hyperbolic N (2018-2022) Gen IV: Hyperbolic Transformers/LLMs (2024-2025) Gen V: HySAT (this paper) Q1. Where does curvature live? HySAT: Loss layer only (curvature off the gradient path) Q2. At what scale? 18M samples . 1,795x cumulative 0 NaN . 17 crashes disclosed Q3. What stabilises it? Proposition 1 (i.a/b/c) curvature-aware . norm cap . retraction Q4. Successor programmes? JARVIS (companion) . ORAA BS Sovereign . S3 Creative synthesis bifurcation In Extended Data Fig. 1, five generations of hyperbolic deep learning are mapped, situating HySAT (Generation V, this work) against Generation I geometry foundations (1854–2010, Riemann–Lobachevsky–Krioukov), Generation I hyperbolic embeddings (Nickel & Kiela 2017; Sala 2018; Tifrea 2019), Generation I hyperbolic neural networks (Ganea 2018; HGCN, Chami 2019; Shimizu 2021), Generation IV hyperbolic transformers/LLMs (HypLoRA, Yang 2025; HELM, He 2025), and Generation V loss-only (HySAT, this work). Each generation is positioned by its placement of hyperbolic operations within the neural pipeline. Generation V occupies the most conservative placement (H =L + 1, loss only) per Lemma 1. Cross-references:§11.4 (full lineage);§2.3 (six-SLM validation). 33 Extended Data Fig. 2 The Razor Cell Pretraining-data availability (web-public / abundant embedded-tacit / private) Decision consequentiality (low high) Web-indexable knowledge (low stakes) pretraining covers it RAG over private documents (internal SOPs, ERP logs) addressed by retrieval, not by training Web-indexable expertise (high stakes) pretraining + light tuning RAZOR CELL Consequential tacit decisions structurally absent from web pretraining needs hyperbolic structure-aware fine-tuning (e.g. BS Sovereign behavioural audit) In Extended Data Fig. 2, the razor cell is visualised in its operational coordinates: a 2× 2 plane of pretraining-data availability (web-public/abundant → embedded-tacit/private) × decision consequentiality (low→ high). Three quadrants are already served — web-indexable knowledge at low stakes (pretraining covers it), web-indexable expertise at high stakes (pretraining plus light tuning), and private-but-low-stakes knowledge (retrieval over internal documents) — while the upper-right cell, consequential tacit decisions structurally absent from web pretraining, is the structurally empty cell that requires hyperbolic structure-aware fine-tuning. A concurrent cartographic companion manuscript (under review) arrives at the same vacancy on an independent plane (post-2022 domain-expert systems combining high novelty with high human-complementary usefulness) and names it the razor cell. The present manuscript’s six expert SLMs offer the architectural prescription for this empty cell. Full Q1–Q4 quadrant taxonomy and the structural-absence hypothesis are in§16.2; the four- pillar diagnostic from independent disciplinary directions arrives at the same vacancy in§2.2. 34 Extended Data Fig. 3 HySAT Pipeline — Where Curvature Lives FORWARD Euclidean Transformer base + LoRA adapter B (weights d ) Hidden state h d Lorentz projection (h) = exp 0 (h/ ) Three hyperbolic losses PTLE HLSD HWC L = L LM + P L P + H L H + W L W BACKWARD L/ (loss derivative) / h bounded Jacobian M5.23/ h/ B to adapter AdamW on B d (standard Euclidean; no Riemannian step) is differentiated (bounded / h) but not parameterised: curvature enters the loss signal, not the trainable parameters (Proposition 1) Why this matters: crash vs NaN-free HyperLoRA (adapter-on-manifold) B lives on the manifold exp/log traversed EVERY forward AND backward pass inherits sinh/cosh Jacobians 17 NaN crashes HySAT (loss-only placement, this paper) B stays Euclidean; evaluated only at the loss layer differentiable but never parameterised 0 NaN over 317K cumulative steps In Extended Data Fig. 3, the forward path (top lane) shows tokens encoded by a standard Euclidean transformer with LoRA adapter B, producing a hidden state h∈ R d ; the Lorentz projection π(h) = exp 0 (h/τ ) maps h onto the manifold H d only at the loss layer, where the three hyperbolic losses (PTLE, HLSD, HWC) and the standard language-modelling loss combine into L total . Backward path (bottom lane, dotted box): the gradient flows from L total back through h and into the encoder weights B via standard AdamW in R d . The projection π is differentiated (its bounded Jacobian multiplies the gradient), but the trainable weights B never leave R d — curvature enters only through π’s bounded Jacobian, not through the trainable parameters (Proposition 1, i). The contrast (lower bands): adapter-on-manifold placements (HypLoRA, HyperLoRA) traverse exp/ log in both forward and backward passes, inheriting sinh/ cosh Jacobians whose ∥B∥ F amplification produced 17 NaN crashes in our trials (SI-E). HySAT keeps ∇ in R d and produced 0 NaN events (180,000+ steps in ORAA’s single-run contribution; ∼317,000 cumulative optimizer steps across the six complete training runs; Supplementary Table 4,§3). The architectural difference is what curvature lives where, not how much curvature is used. 35 Extended Data Fig. 4 H1–H6 directional confirmation evidence table ID Hypothesis (pre-specified) Direct measured anchorDirectional outcome H1 HySAT stability (Proposition 1, curvature decoupling) 0 NaN events / six complete training runs vs 17 documented HyperLoRA-style crashes (~220 GPU-hours B200 SXM) PASS Confirmed (0 vs 17 binary outcome) Source: Tables M2/M3, SI-E, Fig S1 H2 Manifold drift tolerance (| x, x L + 1| < 10 4 ) 100% of steps within 10 3 (max drift 0.0003) 82% within the rounded 10 4 bin PASS Confirmed (per-step traces verified) Source: SI-C.3, Zenodo per-step CSV H3 Batch pair-formation regime (Proposition 2, b N/p ) AdmitBrain pre-specified: b=2 HLSD activation 3.9% (heterog. pred. ~3.6%) b=16 HLSD activation 100% (pred. ~99.5%) PASS Confirmed (regime separation) Source: Table M4, SI-D H4 Cross-base-family generalisation (Llama 3.1 8B vs EXAONE 3.5 7.8B) ORAA (Llama) and CEO Finance (EXAONE) identical HySAT config, both stable, 0 NaN PASS Confirmed (matched config across families) Source: SI-A.3, SI-A.5, supp §SI-A8 H5 Cross-project hyperbolic activation matrix (silent at b=2 vs pair-formation firing at b8) b=2: HLSD silent (<5% of steps) b8: firing enabled (leaves silent regime) (2.2% to 100%; tracks sibling density) PASS Confirmed (pair-formation regime separation) Source: Table 4 (Activation), SI-B H6 Orchestrated cascade non-monotonicity (triple-coach > single-coach) CEO Finance v2 vs GPT-4o: single coach: 28/30 (93.3%) dual coach: 18/30 (60.0%) triple coach: 30/30 (100%) PASS Confirmed (1 works, 2 fails, 3 succeeds) Source: SI-L5, judge-by-judge in supp In Extended Data Fig. 4, for each pre-specified hypothesis (stated in Methods§4.8; verified against Zenodo-deposited traces, C-BY-4.0), we report the direct measured anchor and the directional outcome rather than a derived effect-size statistic. All six hypotheses are confirmed in the directional prediction. The H1 outcome is a binary 0/6-vs-17 comparison (zero NaN across the six complete HySAT runs versus 17 HyperLoRA-family crashes); H2 is a single-tolerance ratio; H3 is a regime separation across discrete batch sizes; H5 is a silent-fail dichotomy; H6 is a non-monotonic 3-arm comparison — none of these is naturally expressed as a single Cohen d or odds ratio with parametric CI, and Holm–Bonferroni correction is not applicable because every directional prediction is independently anchored to its own measured outcome. The reported anchors are the direct measured outcomes deposited on Zenodo (C-BY-4.0) and are independently verifiable. H3 is reported with the AdmitBrain pre-specified prediction (170-node tree, ̄p≈ 3) before the b=2→ b=16 transition; the ORAA case is reported separately as a post-hoc consistency check. Full per-hypothesis verification path in§8. 36 Extended Data Fig. 5 Sample-efficiency and stability landscape — HySAT occupies the high- scale, stable-training region 10 4 10 5 10 6 10 7 Training samples (log scale, verified; Methods §4) 0.0 0.2 0.4 0.6 0.8 1.0 Downstream benefit (native metric, normalised to [0, 1]) HypLoRA crash zone (no convergence) HySAT (this paper, n=6) ORAA (most-extreme scale) HyperLoRA-style (crashed or stable-but-small) HySAT upper envelope 1,139× scale leap (HypLoRA Math-10K ORAA, this paper) Aggregate (verified) 17,954,911 training samples 0 NaN at convergence six complete training runs ~220 GPU-hr (17 crashes) + ~890 GPU-hr (eval) $4,998 RunPod B200 SXM (Feb-Apr 2026) ~1,111 GPU-hr verified aggregate AdmitBrain v6 tone 4.2/5 BS Sovereign 146M F 1 = 0.855 MetaTeach v14 HWC 100% act. CEO Finance v2 30/30 vs GPT-4o S3 Creative T5 95.8% win ORAA Dual-SLM 9/10 + 4.8s Yang 2024 HypLoRA Math-10K (within stable regime) Early BS HyperLoRA (full-hyperbolic, crashed 6x) Early ORAA HyperLoRA (rank-16, crashed 7x) Early AdmitBrain HyperLoRA (single-adapter, crashed 4x) In Extended Data Fig. 5, training samples (log scale, X-axis, verified from Methods§4.4 and Zenodo per-step traces) are plotted against downstream benefit on each project’s native metric (Y-axis, normalised onto a comparable [0, 1] scale; the native anchor values are printed at each marker — BS F 1 = 0.855, S3 win-rate 0.958, CEO 30/30). HySAT variants (diamond markers) occupy the high-scale, stable-training region: BS Sovereign 146M (F 1 =0.855); S3 Creative T5 (95.8% LLM win-rate); ORAA Dual-SLM (9/10 benchmark, 4.8 s latency,⋆ marks the most-extreme scale); CEO Finance v2 Triple-Coach (30/30 vs GPT-4o); MetaTeach v14 (HWC 100% activation); AdmitBrain v6 (tone 4.2/5). HyperLoRA-family points (X markers, lower band) include Yang 2025 HypLoRA Math-10K within its stable regime, plus the three early HyperLoRA-style attempts (early ORAA rank-16; early BS full-hyperbolic; early AdmitBrain single-adapter) that produced 7 + 6 + 4 = 17 crash events totalling ∼220 hours of B200 SXM compute before the HySAT reformulation stabilised training. The dashed line traces the HySAT set’s upper envelope; all HyperLoRA-style points lie at lower scale or terminate in crashes. Why sample-efficiency rather than GPU-hours: platform-independent (hardware/batch/stack-agnostic), reviewer-reproducible from deposited per-step traces, and directly aligned with the paper’s 1,139× scale anchor (Methods§4.6). Compute disclosure protocol in§12.5. 37 Extended Data Table 2 | The 2×3 Matrix: Six Domain Expert SLMs with Deployment Status ProjectBase modelAdapterSamples StepsNaN Primary resultDeployment BS Sovereign Lorentz 146 MFully hyperbolic1,164,694 a 0.50 ep (537K proc.) 014-head F1 = 0.855HF open (planned) S3 CreativeT5-3BFull fine-tune3,320,126 266 rounds095.8% LLM win-rateHF open (planned) ORAALlama-3.1 8 BStd-LoRA r=64 (staged)11,387,744 180,000+0Dual SLM 9/10 benchmarkGated commercial MetaTeachEXAONE 3.5 7.8 B HyperLoRA r=64602,580 2,3000HWC 100% / 0.21–1.03Gated commercial CEO Finance EXAONE 3.5 7.8 B Std-LoRA r=64 (staged)1,085,619 1,7400Triple cascade 30/30Gated commercial AdmitBrainLlama-3.1 8 BHyperLoRA (224 adapters)394,148 12,3130Tone 4.2/5 (LLM 5.0; 84% match) Controlled deployment Total2 families4 strategies17,954,911 ∼317 K agg.06 domains2 open + 4 gated In Extended Data Table 2, six projects span two base-model families (Llama 3.1 8B, EXAONE 3.5 7.8B) and four adapter strategies (from-scratch, full fine-tuning, Std-LoRA HySAT, HyperLoRA). Together they report 17.95 M training samples across 6 expert domains with 0 NaN events across approximately 317 K aggregate optimizer steps (six complete training runs, Supplementary Table 4, SI-B). All six models occupy deployed or deployable operational contexts, sustained under a unified research program with operational stewardship by an engineering collective for the four deployed services. a BS Sovereign 1,164,694 = 1,074,380 train + 90,314 held-out test; the 17,954,911 paper-wide “training samples” figure is the sum across all six projects, with BS Sovereign contributing the train+test combined count and the other five projects contributing training-only counts. Extended Data Table 3 | Per-project hyperbolic loss activation, manifold preservation, and PTLE/HLSD/HWC evidence of Proposition 2 ProjectBatch PTLE activ. / valueHLSD activ. / valueHWC activ. / value Manifold drift NaN BS Sovereign32arch. a / all 14 headsarch. a / all pairsarch. a / exact=−1 exactly0 S3 Creative (R266)16+ga2 100% / 0.278active / 0.0002100% / 0.517=−1.0000000 ORAA (b = 2)20% / 0.000 (silent)52% / variablen/a< 10 −6 0 ORAA (b = 8)8100% / 0.08100% / activen/a< 10 −6 0 ORAA (b = 16)16100% / 0.10 (120 pairs/step) 100% / activen/a< 10 −6 0 MetaTeach v148+ga4via HWC c via HWC c 100% / 0.21–1.03< 10 −4 0 CEO Finance16+ga4 100% / active (λ = 0.05)100% / active (λ = 0.05) n/a< 10 −5 0 AdmitBrain (b = 2)20% (observed) b / 03.9% (silent-fail) / 0n/a< 10 −4 0 AdmitBrain (b = 16) 16100% / active100% / activen/a< 10 −4 0 In Extended Data Table 3, activation rate is the fraction of training steps on which the loss term is non-zero after batch-size settling; value is the final-equilibrium loss magnitude at end of training (where reported). Manifold drift is the absolute deviation of ⟨π(h),π(h)⟩ L from −1 sampled every 100 steps. n/a = loss term not configured for that project (PTLE/HLSD only for ORAA, CEO Finance, AdmitBrain; full PTLE+HLSD+HWC for S3, BS; HWC for MetaTeach, which integrates PTLE and HLSD). Job B crystallisation: every project activates every configured loss term at equilibrium. a BS Sovereign’s fully-hyperbolic architecture integrates PTLE/HLSD/HWC as architectural constraints on the Lorentz classifier head rather than as separately measured loss terms; manifold preservation is structurally guaranteed by the decoder’s geometric invariant. b AdmitBrain at b = 2: PTLE activation is empirically 0% (no batch contains the three samples needed for non-trivial PTLE formation in the 170-node tree); the Proposition 2 uniform-leaf prediction is 1.2%, and the regime separation (silent-fail at b = 2 versus 100% at b = 16) is the load-bearing test (see Table 6). c MetaTeach v14 uses HWC (Hierarchy-Weighted Contrastive), which integrates the PTLE attraction term and the HLSD separation term into a single continuous-weight Lorentz contrastive loss (w = ancestor-overlap/max-depth; Methods §4.1); HWC supersedes separate PTLE/HLSD, so their structural supervision is realised within HWC rather than as separately logged terms. 38 Extended Data Table 4 | Six structural-first claims with prior-work baseline and delta # ClaimPrior baselineHighwayDelta 1 Largestreported hyperbolic-loss LLM fine-tuning corpus HypLoRA Math-10K 10,000;Common- sense170K 170,420 ORAA 11.4 M single- project;17.95 M cumulative 1,139×/67× single; 1,795×/105× cumu- lative 2 Firstreported multi-domain expert-specialisation cross-validation 1 domain (HypLoRA arithmetic;HELM general LM; Hyp- former graph) 6 expert domains1 → 6 3 First reported cross- family validation of identicalloss-layer placement 1family(Llama- familyor Gemma-family inter- nally varied) Llama3.1+ EXAONE 3.5 under identicalHySAT config 1 → 2 4 First reported four- way adapter-strategy cross-comparison 1 adapter placement per paper From-scratch × Full- FT × Std-LoRA × HyperLoRA 1 → 4 5 First reported sys- tematic cross-project hyperbolic activation matrix single-projectacti- vation rates without cross-projectcom- parison 6 SLMs × 3 loss components × batch regime(Extended Data Table 3) silent (b=2)→ 100% (b≥8) 6 Batch-size-as- structural-variable identification+ Proposition 2 treatedas speed/memory vari- able [16, 17, 30] ORAAb=2→8; AdmitBrain b=2→16 speed → structural In Extended Data Table 4, each claim is, to the best of our knowledge, first reported within the hyperbolic LLM fine-tuning literature as surveyed through 2025 Q4 [30]; “first” refers to first-reported within the surveyed coverage, not to absolute priority. The “Delta” column expresses the gap as a ratio (where comparable) or as a categorical first reported. 39 Extended Data Table 5 | Matched four-arm placement ablation isolates the manifold invariant ArmPlacementGeometryManifold driftTree-dist ρ L Retr. P@1 CE-only—nonen/a0.253±0.0330.62±0.18 Euclid-reg (control)loss-layerflat1.9–3.3×10 4 0.224±0.0410.57±0.20 HySAT (ours)loss-layerhyperbolic8.9×10 −7 –1.4×10 −6 0.228±0.0480.63±0.21 HyperLoRAadaptermanifold7.7×10 −7 –1.4×10 −6 0.227±0.0860.47±0.10 In Extended Data Table 5, all four arms share an identical base (Qwen 2.5 7B-Instruct), optimiser, LoRA-dropout, gradient clipping, 500-step budget, and controlled depth-4 toy ontology (30 leaves, 10 held out), across six seeds (42/123/7/2024/99/777); they differ only in where geometry enters and whether it is hyperbolic or flat. An earlier version of this ablation detached the hidden states before the structural loss, severing its gradient to the adapter; that hook is removed and the non-zero structural gradient asserted at run time. The load-bearing contrast is HySAT versus the Euclid-reg control: both place the same tree regulariser (PTLE+HLSD) at the same loss layer on identical hidden states with a comparable hidden-state norm entering the loss (276–370; logged under the historical B norm column name), differing only in whether the points are projected onto the Lorentz manifold — and only the projected arm preserves the invariant ⟨x,x⟩ L =−1 (drift ∼10 −6 versus ∼10 4 , ten orders of magnitude, no seed overlap). Held-out tree-geometry, by contrast, does not separate the arms at this toy scale: HySAT’s ρ L (0.228) overlaps the flat control (0.224) and the supervision-free CE baseline (0.253) within one standard deviation. This ablation isolates the mechanism placement governs — invariant preservation — not a downstream tree-geometry gain; the downstream evidence for hyperbolic placement is the six deployed and open-weight expert models, and the placement-stability evidence is the 17-crash forensic (SI-E.1–E.3). Full per-seed protocol in§6.3; training traces for all twenty-four runs (logged every five steps) deposited on Zenodo (C-BY-4.0). All four arms record 0 NaN at this controlled toy scale (500 steps); the placement-stability separation appears only at industrial scale (17 crashes, SI-E). The load-bearing rows are Euclid-reg versus HySAT: they share everything but the Lorentz projection and separate cleanly on the manifold invariant (drift, ten orders of magnitude), while overlapping on downstream tree-geometry at this scale. 40