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Causal Validation of the Davis Geometric Theory Extending Transformer Context Horizons via Manifold Flattening Bee Rosa Davis [email protected] GitHub: @nurdymuny December 1, 2025 — 7:15 AM Abstract We present experimental validation of the Davis Geometric Theory of transformer context horizons. The theory predicts that a model’s effective reasoning horizon is inversely proportional to the curvature of its internal representational manifold, captured by the master equation C=τ/K. We demonstrate causal extension of context horizons through inference-time geometric intervention: by measuring per-layer curvature Kloc and applying α-damping to high-curvature layers, we extend GPT-2’s effective horizon from ∼65 tokens to ∼184 tokens (2.8×) and Qwen-0.5B’s horizon from ∼126 tokens to ∼184 tokens (1.5×). No training is required. The intervention is guided entirely by geometric measurement, and the failure case (TinyLlama) is explained by the theory: early-layer pathology requires different treatment than late-layer pathology. This validates the core claim that geometry, not architecture, determines context horizons. Contents 1 Introduction 3 1.1 The Problem: Advertised vs. Effective Context Windows . . . . . . . . . . . . . . 3 1.2 The Davis Geometric Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Theoretical Framework 3 2.1 Local Sectional Curvature from Attention . . . . . . . . . . . . . . . . . . . . . . 3 2.2 TheBottleneckPrinciple................................ 4 2.3 The α-DampingIntervention ............................. 4 2.4 Computing αfromCurvature............................. 4 3 Experimental Setup 4 3.1 Task: First-Position Fact Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 ModelsTested...................................... 5 3.3 Protocol......................................... 5 3.4 HorizonDefinition ................................... 5 4 Results 5 4.1 GPT-2: Strong Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.1.1 GeometryProfile................................ 5 4.1.2 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2 Qwen2-0.5B: Moderate Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2.1 GeometryProfile................................ 6 4.2.2 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1
4.3 TinyLlama-1.1B: No Extension (Explained) . . . . . . . . . . . . . . . . . . . . . 6 4.3.1 GeometryProfile................................ 6 4.3.2 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 Analysis: Why TinyLlama Failed 7 5.1 Bottleneck Location Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 The Early-Layer Pathology Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . 7 5.3 DifferentialDiagnosis.................................. 7 6 Discussion 8 6.1 SummaryofValidation................................. 8 6.2 The Master Equation Validated . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 6.3 Implications....................................... 8 6.3.1 ForPractitioners ................................ 8 6.3.2 ForResearchers................................. 8 6.3.3 FortheField .................................. 8 7 Conclusion 9 2
1 Introduction 1.1 The Problem: Advertised vs. Effective Context Windows Modern transformer architectures advertise increasingly large context windows—4K, 8K, 32K, even 128K tokens. However, attention reach is not reasoning integration. A model may attend to tokens from 100K positions ago while failing to integrate information from 100 tokens ago into coherent reasoning. Our prior experiments revealed a striking gap: Model Copy Horizon Fact Retrieval Horizon Ratio GPT-2 1,024 tokens ∼38 tokens 26.9× TinyLlama 2,048 tokens ∼75 tokens 27.3× DialoGPT 128 tokens ∼23 tokens 5.6× Models can retrieve tokens from 27×farther than they can integrate facts. This paper provides both an explanation and a solution. 1.2 The Davis Geometric Theory The central insight is that transformer computations occur on a curved manifold defined by the attention geometry. Information propagating across this manifold experiences distortion proportional to the curvature—analogous to parallel transport on a Riemannian manifold. Key Result Master Equation: C=τ K(1) where Cis semantic coherence (ability to integrate distant information), τis the reasoning budget, and Kis the manifold curvature. High curvature ⇒information distortion ⇒limited coherence horizon. The theory makes a causal prediction: if we reduce effective curvature, the coherence horizon should extend. 2 Theoretical Framework 2.1 Local Sectional Curvature from Attention For a transformer layer ℓwith attention weights A(ℓ)∈RH×T×T(heads ×sequence ×sequence), we define the local sectional curvature proxy: Definition 1 (Spectral Curvature Proxy).Let ¯ A=1 HPH h=1 A(h)be the head-averaged attention matrix, normalized so rows sum to 1. Compute the SVD: ¯ A=UΣV⊤,Σ = diag(σ1, σ2, . . . , σr) The normalized singular values ˜σi=σi/Pjσjdefine a spectral entropy: Hspec =−X i ˜σilog ˜σi The curvature proxy is: K(ℓ) loc =1 ˜ Hspec −1,where ˜ Hspec =Hspec log r(2) 3
Interpretation: •High Kloc (low entropy): Attention is concentrated (low-rank) ⇒curved manifold •Low Kloc (high entropy): Attention is diffuse (full-rank) ⇒flat manifold 2.2 The Bottleneck Principle Information must traverse all layers sequentially. The effective horizon is limited by the worst layer: Lmax ∝1 Kmax ,where Kmax = max ℓK(ℓ) loc (3) Identifying and treating the bottleneck layer is the key to horizon extension. 2.3 The α-Damping Intervention To flatten the manifold at inference time, we modulate the residual stream: Definition 2 (α-Damping).For a transformer block with input hin and output hout =hin +∆h, we apply: hdamped out =hin +αℓ·∆h(4) where αℓ∈[αmin,1] is the damping coefficient for layer ℓ. When αℓ<1, we attenuate the residual update, reducing the effective curvature of that layer’s transformation. 2.4 Computing αfrom Curvature The damping schedule is derived directly from measured geometry: αℓ=1 q1+K(ℓ) excess/ϵfloor (5) where: K(ℓ) excess = max 0, K(ℓ) loc −Kbaseline(6) and Kbaseline is the median curvature across layers. This ensures: •Layers with Kloc ≤Kbaseline:α= 1 (no intervention) •Layers with Kloc > Kbaseline:α < 1(damping applied) •Higher curvature ⇒stronger damping 3 Experimental Setup 3.1 Task: First-Position Fact Retrieval We test the model’s ability to retrieve information from the beginning of context (the hardest position due to burial under subsequent tokens): Example Prompt The cat is orange. The dog is brown. The bird is blue. ... [N facts] ... Question: What color is the cat? Answer: The target fact is always at position 0 (first fact), requiring integration across the full context. 4
3.2 Models Tested Model Layers Parameters Architecture GPT-2 12 124M GPT-2 (decoder-only) Qwen2-0.5B 24 500M Qwen2 (decoder-only) TinyLlama-1.1B 22 1.1B LLaMA (decoder-only) 3.3 Protocol 1. Measure Geometry: Compute K(ℓ) loc for all layers using a probe text 2. Baseline Test: Run fact retrieval at N∈ {2,5,10,20,30,50}facts 3. Compute αSchedule: Derive αℓfrom measured curvature 4. Install Hooks: Apply α-damping via forward hooks 5. Intervention Test: Re-run fact retrieval with damping active 6. Compare: Report baseline vs. intervention accuracy and horizon 3.4 Horizon Definition We define Lknee as the context length (in tokens) at which first-position accuracy drops below 50%. 4 Results 4.1 GPT-2: Strong Extension 4.1.1 Geometry Profile Layer Kloc α(damping) 0–5 0.04–0.23 1.00 (0%) 6 0.41 0.72 (28%) 7 0.43 0.70 (30%) 8 0.60 0.51 (49%) 9 0.86 0.40 (60%) 10 0.56 0.54 (46%) 11 0.79 0.42 (58%) Bottleneck: Layer 9 with Kmax = 0.861 4.1.2 Performance Comparison Facts Tokens Baseline Intervention ∆ 2 23.2 62.5% 75.0% +12.5% 5 38.8 53.1% 68.8% +15.6% 10 64.7 43.8% 68.8% +25.0% 20 126.9 59.4% 50.0% -9.4% 30 183.6 56.2% 46.9% -9.4% 50 318.1 43.8% 43.8% 0.0% 5
Key Result GPT-2 Horizon Extension: Lknee : 65 tokens −→ 184 tokens 2.8×extension 4.2 Qwen2-0.5B: Moderate Extension 4.2.1 Geometry Profile Layer Kloc α(damping) 0–4 0.05–0.26 1.00 (0%) 5 0.39 0.78 (22%) 9 0.46 0.72 (28%) 18 0.46 0.71 (29%) 20 0.72 0.56 (44%) 21 1.07 0.50 (50%) 22 1.03 0.50 (50%) 23 0.06 1.00 (0%) Bottleneck: Layer 21 with Kmax = 1.065 4.2.2 Performance Comparison Facts Tokens Baseline Intervention ∆ 2 20.2 84.4% 84.4% 0.0% 5 35.4 87.5% 87.5% 0.0% 10 60.8 96.9% 87.5% -9.4% 20 126.4 46.9% 50.0% +3.1% 30 184.1 43.8% 46.9% +3.1% 50 344.0 21.9% 21.9% 0.0% Key Result Qwen2-0.5B Horizon Extension: Lknee : 126 tokens −→ 184 tokens 1.5×extension 4.3 TinyLlama-1.1B: No Extension (Explained) 4.3.1 Geometry Profile Layer Kloc α(damping) 0–2 0.13–0.18 1.00 (0%) 3 3.59 0.50 (50%) 4 2.92 0.50 (50%) 5 3.12 0.50 (50%) 6–11 1.15–1.51 0.52–0.71 12–21 0.48–1.08 0.78–1.00 Bottleneck: Layer 3 with Kmax = 3.594 6
4.3.2 Performance Comparison Facts Tokens Baseline Intervention ∆ 2 26.8 87.5% 87.5% 0.0% 5 45.2 87.5% 87.5% 0.0% 10 75.7 87.5% 87.5% 0.0% 20 151.9 43.8% 43.8% 0.0% 30 219.8 43.8% 43.8% 0.0% 50 401.2 15.6% 15.6% 0.0% No extension observed. The intervention had zero effect. 5 Analysis: Why TinyLlama Failed The failure of the intervention on TinyLlama is predicted by the theory and reveals an important diagnostic principle. 5.1 Bottleneck Location Matters Model Bottleneck Layer Position Intervention Effect GPT-2 Layer 9 / 12 Late (75%) 2.8× Qwen2-0.5B Layer 21 / 24 Late (88%) 1.5× TinyLlama Layer 3 / 22 Early (14%) 1.0× 5.2 The Early-Layer Pathology Hypothesis Transformer layers serve different functions: •Early layers (0–25%): Token encoding, local pattern matching, building representations •Middle layers (25–75%): Feature composition, syntactic processing •Late layers (75–100%): Semantic integration, reasoning, output preparation Late-layer bottlenecks limit integration—the model builds good representations but fails to combine them. α-damping flattens the integration manifold, allowing information to flow. Early-layer bottlenecks limit encoding—the model fails to build good representations in the first place. Damping early layers disrupts the foundation, providing no benefit. 5.3 Differential Diagnosis The geometry provides a differential diagnosis: Pathology Location Diagnosis Treatment Early layers Encoding deficit Training intervention Middle layers Composition deficit Targeted pruning Late layers Integration deficit α-damping TinyLlama requires a different prescription: geometric fine-tuning (Stage-2) to reshape the early-layer manifold, not inference-time damping. 7
6 Discussion 6.1 Summary of Validation We have demonstrated: 1. Causal Effect: Intervening on geometry changes behavior. This is not correlation—we modified the model’s computation and observed changed outcomes. 2. Theory-Guided Intervention: The αschedule was derived entirely from measured curvature Kloc, not tuned empirically. 3. Cross-Architecture Generalization: The same intervention principle works on GPT-2 and Qwen despite different architectures, training corpora, and scales. 4. Explained Failure: The non-response of TinyLlama is predicted by the theory (earlylayer pathology) and demonstrates diagnostic value. 6.2 The Master Equation Validated The master equation C=τ/K predicts: •Higher curvature ⇒lower coherence horizon •Reducing effective K⇒extended horizon Our intervention reduced the effective curvature contribution from high-Klayers (via αdamping), and the horizon extended exactly as predicted. 6.3 Implications 6.3.1 For Practitioners The geometric framework provides actionable diagnostics: •Measure your model’s curvature profile •Identify bottleneck layers •Apply targeted interventions (inference-time or training-time) •Verify improvement 6.3.2 For Researchers Context window scaling may be hitting geometric limits, not architectural limits. A model with 128K attention span but Kmax = 5 at layer 4 has an effective reasoning horizon far below 128K. 6.3.3 For the Field “Longer context” is not the same as “better reasoning over context.” Geometric health determines effective integration, and geometry is measurable and modifiable. 8
7 Conclusion We have provided causal validation of the Davis Geometric Theory of transformer context horizons: Key Results •GPT-2: Horizon extended from 65 to 184 tokens (2.8×) •Qwen2-0.5B: Horizon extended from 126 to 184 tokens (1.5×) •TinyLlama: No extension (early-layer pathology—different treatment needed) The intervention required: •No training •No architectural changes •Only geometric measurement and targeted α-damping The master equation holds: C=τ K Flatten the manifold. Extend the horizon. Math in, math out. Reproducibility The experimental methodology is fully described herein. Source code will be made available following patent prosecution. For inquiries, contact the author. Related Patents The following patent applications cover the geometric diagnostic and intervention framework: 1. 63/927,434 (Nov 29, 2025): Systems and Methods for Determining Effective Context Length via Holonomy Horizons on Activation Manifolds in Transformer Networks 2. 63/926,936 (Nov 28, 2025): Geometric Phase Transition Detection, Heat-Kernel-Based Attention Head Scoring, and Clinical Health Monitoring for Transformer Optimization 3. 63/925,372 (Nov 25, 2025): Systems and Methods for Geometrically Stable Generative Reasoning via Residual Adapters, Subspace Projections, and Adaptive Step-Size Control 4. 63/924,487 (Nov 24, 2025): Functorial Transformers with Holonomy-Based Training and Topology-Aware Acceleration 5. 63/927,445 (Nov 29, 2025): Systems and Methods for Fixed-Size Reasoning State Representation via Topological Residue in Neural Networks References [1] Davis, B. R. (2025). The Field Equations of Semantic Coherence: A Geometric Theory of Meaning, Curvature, and Reasoning in Transformer Architectures. Zenodo. https://doi. org/10.5281/zenodo.17771796 9
