CAPE — Capability-Coupling Phase Transitions in AI Scaling

Paper A: "Lying Is Just a Phase: The Hidden Alignment Transition in Language Model Scaling" Paper B: "The Growing Pains of Frontier Models: Capability Coupling Across Labs and Scales"

Adil Amin — ZEHEN Labs — May 2026

📄 Paper A: arXiv 2605.18838 — base-model phase transitions (63 models, 16 families) 📄 Paper B: arXiv 2605.18840 — frontier diagnostic (34+ models, 10 labs) 🌐 Dashboard: zehenlabs.com/cape | GitHub Pages 📧 Contact: adil@zehenlabs.com

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The Core Finding

Two models with identical training loss can be in opposite alignment regimes.

Below a family-specific critical scale Nc, reasoning and truthfulness anticorrelate (r = −0.989, p < 10⁻⁵) — the alignment tax. Above Nc, the coupling reverses — the alignment bonus. Loss curves show zero signal at the transition (CV = 0.8%).

Nc is not a universal constant — it varies 60× across families (0.12B–7B), determined by architecture, width, and data curation quality.

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Paper A — "Lying Is Just a Phase"

The inter-capability coupling γ₁₂ = ∂TruthfulQA/∂HellaSwag is a sign-changing scaling observable invisible to loss. Key results:

• 16 families, 63 models: within-family anticorrelation reproduced in Pythia, Cerebras, OPT, BLOOM (all r < −0.85)
• Loss-blindness: smooth loss (R² = 0.99) coexists with jagged coupling underneath
• ODE dynamics: discovered 5×5 coupled ODE system; two universality classes (COOP vs COMP)
• Engineerability: Nc is a recipe parameter — movable by width (PLE), curation (h-field), architecture
• Intervention: projection-width steering tool corrects 60→30→20% misalignment across the phase
• Cross-prediction: held-out 12B prediction at 5.6% MAE

Paper B — "Growing Pains of Frontier Models"

Frontier-scale companion measuring SWE-bench vs GPQA Diamond across 34 models / 10 labs:

• Population coupling: r = +0.72, slope 0.51, intercept 46.4
• h-field: per-model residual = capability-emphasis fingerprint (one number from two public scores)
• Per-lab slopes vary 5×: Google 1.15 vs DeepSeek 0.23 — "recipe quality in one number"
• Lab dynamics: DeepSeek monotonic h reversal (+11.2→−4.7) vs Anthropic oscillate-and-recover
• Cascade: Nc1 (~3.5B) → Nc2 (~30–72B, 59% crash) → Nc3 (~114B, SWE saturates) → Nc4 (predicted)
• 7 timestamped falsifiable predictions with pass/fail criteria

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Quickstart — 60 Seconds to Reproduce

git clone https://github.com/adilamin89/cape-scaling.git
cd cape-scaling
pip install numpy scipy matplotlib scikit-learn
python scripts/verify_and_reproduce.py --numbers

Expected:

r(HellaSwag, TruthfulQA) within-family:  -0.989   p=3.6e-06  ✓
Nc (critical scale):                      3.50B              ✓
γ₁₂ slope A:                              0.629              ✓
OLMo-7B γ₁₂:                             0.000              ✓
d_eff at Nc:                              1.254              ✓
β_measured:                               0.40               ✓
Ginzburg number Gi:                       1.35               ✓
κ_GL (Type II):                           0.767              ✓

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Dashboard

open index.html      # macOS — no server, no npm, no build step needed

Or visit: zehenlabs.com/cape | adilamin89.github.io/cape-scaling

7 tabs:

Tab	What it does
Overview	Phase diagram, model browser, key stats. Enter any model size + TruthfulQA to see where it lands.
Explorer	Full computation engine. Enter parameters → get γ₁₂, regime, d_eff, h(D), susceptibility χ(N), eigenvectors, ODE slope, and scaling predictions.
Frontier	SWE-bench vs GPQA Diamond across 34+ frontier models.
h(D) Lab	Data curation quality ranking.
Physics	Dual-language reference: every equation in both physics and ML terms.
Paper	Abstract, citation, 13 diagnostics, usage guide.
Steering	Interactive demo of the cape_steer activation-level steering engine.

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CLI Tools

cape_steer.py — Activation-Level Steering Engine

pip install torch transformers
python cli/cape_steer.py --model EleutherAI/pythia-1b --prompt "The capital of France is"

Auto-detects any open-weight HuggingFace model's architecture, probes layer nl//4 for the reasoning–truthfulness coupling direction, and steers activations along it. Works with Pythia, Llama, Mistral, Gemma, Qwen, OPT, BLOOM, and more.

cape_cli.py — CAPE Analysis CLI

python cli/cape_cli.py --h-field --model "GPT-4o"
python cli/cape_cli.py --coupling --family pythia
python cli/cape_cli.py --predict --size 13e9

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Repo Structure

cape-scaling/
├── index.html                     ← Self-contained dashboard (GitHub Pages)
├── paper3a_nature.pdf             ← Paper A: "Lying Is Just a Phase" (base models)
├── paper3b_neurips.pdf            ← Paper B: "Growing Pains" (frontier)
├── paper3A.tex                    ← LaTeX source (Paper A)
├── paper3b_neurips.tex            ← LaTeX source (Paper B)
├── requirements.txt               ← Core deps (numpy/scipy/matplotlib)
├── requirements-gpu.txt           ← Full deps including PyTorch
│
├── cli/
│   ├── cape_steer.py              ← Activation-level steering engine (any HF model)
│   └── cape_cli.py                ← h-field calculator, coupling, predictions
│
├── data/
│   ├── ai_free_energy_data.json   ← Pythia benchmark scores + correlation matrix
│   ├── base_models_consolidated.json  ← All 63 base models, 16 families
│   ├── cape_26models_9families.json   ← 26-model 9-family coupling data
│   ├── frontier_34models.json     ← Paper B: 34 frontier models, 10 labs
│   ├── beta_final_6model.json     ← β exponent data
│   ├── gemma4_results.json        ← Gemma-4-E4B-it (PLE evidence)
│   ├── gemma3_results.json        ← Gemma-3-4B (PLE comparison)
│   ├── qwen3_8b_results.json      ← Qwen3-8B coupling
│   ├── qwen_generation.json       ← Qwen generation comparison
│   ├── intervention_results_v2.json   ← Projection-width bottleneck results
│   ├── layer_*_tl.json            ← Per-layer coupling (7 models)
│   ├── opt30b_internal_nc2.json   ← OPT-30B Nc2 per-layer data
│   ├── opt66b_internal_nc2.json   ← OPT-66B Nc2 per-layer data
│   ├── llama2_70b_internal_nc2.json   ← Llama2-70B Nc2 per-layer data
│   └── ...                        ← Additional model results and leaderboard data
│
├── figures/                       ← Paper figures (200dpi PNG)
│
└── scripts/
    ├── verify_and_reproduce.py    ← START HERE: all 8 key numbers
    ├── generate_all_figures.py    ← Regenerate all figures → figures/
    ├── bootstrap_Nc.py            ← Bootstrap CI on Nc → [2.9B, 13.4B]
    ├── beta_final_analysis.py     ← β order-parameter exponent
    ├── swe_gpqa_coupling.py       ← Frontier coupling (SWE vs GPQA)
    ├── pythia_gradient_extraction.py  ← GPU: ‖∇L‖ extraction
    ├── pysindy_per_phase.py       ← Per-phase ODE fitting (s± vs s++)
    ├── diagnostics.py             ← Scaling diagnostics (α_eff, κ, D_L)
    ├── cape_frontier_full.py      ← Full 34-model frontier regression
    ├── cape_architecture_probe.py ← Architecture confound probe
    ├── cape_phase_transfer.py     ← Phase transfer matrix
    ├── cape_family_hfield.py      ← Per-family h-field computation
    ├── cape_null_model.py         ← Null model comparison
    ├── recompute_all.py           ← Master recomputation script
    ├── quickstart.py              ← Quick reproduction
    └── test_dashboard.py          ← Playwright automated dashboard tests

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Reproduce All Figures

python scripts/generate_all_figures.py
# → figures/fig1_main.png ... figures/fig9_frontier.png

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GPU / Gradient Extraction (optional)

pip install -r requirements-gpu.txt
python scripts/pythia_gradient_extraction.py --model pythia-1b
python scripts/pythia_gradient_extraction.py --model pythia-6.9b   # needs ≥16GB VRAM

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Ground Truth Numbers

Quantity	Value	ML interpretation
r(HellaSwag, TQA)	−0.989	Strongest anticorrelation: scaling reasoning hurts truthfulness
Nc	~3.5B (Pythia)	Critical model size where alignment regime flips (varies by family)
γ₁₂(N)	A·log₁₀N + B	Running coupling: A=0.629, B=−5.886
OLMo-7B γ₁₂	0.000	Zero-parameter confirmation of Nc
α (loss exponent)	0.238 ± 0.015	Power-law loss scaling R²=0.9994
β (order param)	0.40 ± 0.08	Critical exponent, vs mean-field prediction 1.24
d_eff at Nc	1.254	Effective number of independent benchmark axes
Nc₂	~30–72B	Second coupling crash (59% drop in OPT)
Hold-out MAE	5.6%	Cross-family prediction accuracy
Frontier r(SWE,GPQA)	+0.72	Cooperative coupling at frontier scale (n=34)
Per-lab slope range	0.23–1.15	DeepSeek to Google (5× variation)
DeepSeek swing	15.9 pp	Largest h-field reversal across labs

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Independent Reanalysis

A CPU-only independent reanalysis of all public data is available in files (32)/. Key findings:

• Bedrock anticorrelation reproduced with bootstrap CIs and permutation test (p=9×10⁻⁵)
• Loss-blindness and 12B cross-prediction verified
• Nc2 crash verified (63% drop in OPT)
• Family-demeaned cooperation survives confound control
• Per-layer bottleneck at Nc2 is architecture-specific (OPT yes, Llama2 no)
• Per-layer coupling precursor confirmed on CPU (Analysis 12)

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Citation

@article{amin2026lying,
  title   = {Lying Is Just a Phase: The Hidden Alignment Transition
             in Language Model Scaling},
  author  = {Amin, Adil},
  journal = {arXiv preprint arXiv:2605.18838},
  year    = {2026}
}

@article{amin2026growing,
  title   = {The Growing Pains of Frontier Models: Capability Coupling
             Across Labs and Scales},
  author  = {Amin, Adil},
  journal = {arXiv preprint arXiv:2605.18840},
  year    = {2026}
}

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Background

The coupling mathematics underlying CAPE was originally developed for multi-band superconductors:

Amin & Agterberg (2020). Generalized spin-fluctuation feedback. Phys. Rev. Research 2, 013055

Part of the CAPE program — applying effective field theory to AI scaling laws.

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Adil Amin · ZEHEN Labs · adil@zehenlabs.com Website · LinkedIn · Issues/PRs welcome