All 18 Modalities
One Sheffer kernel. 18 modalities. Every benchmark, exactly as measured.
Six compete head-to-head with stochastic AI. Twelve more are kernel-native capabilities with no mainstream machine learning equivalent. Precise figures, no rounding up.
The 6 -- compete with stochastic AI
Head-to-head benchmarks against established stochastic baselines.
| Modality | Key result | vs. |
|---|---|---|
| Neural Networks | 97.78% MNIST accuracy | ReLU 96.89% / GELU 97.03% / Swish 96.69% |
| Symbolic Regression | 43W / 31T / 26L on 100 AI Feynman equations | PySR 1.5.10, matched wall-clock |
| Symbolic Regression (high-dim) | 22W / 0L at 5+ variables -- sweeps every case | PySR; Sheffer mean R2 0.92+, PySR 0.67-0.84 |
| Reinforcement Learning | 7/8 episodes stabilized, mean return -16.50 | LQR: 3/8, -24.71 -- 21-parameter policy |
| Neuro-Symbolic Execution | Heating R2 0.9963 / Cooling R2 0.9944 | Ridge: 0.9335 / 0.9313 |
| Neuro-Symbolic Execution | Concrete R2 0.8777 | Ridge: 0.4342 |
| Generative Models | 5/5 UCI datasets beat RealNVP | Canonical MAF-split benchmark |
| Generative Models | 72,102-79,051 samples/sec at batch 1024 | Quadro P1000 (two-gen-old GPU) |
| Generative Models | 1,064/1,064 bit-exact Python/Rust | Zero drift across 96 coupling layers |
| Bayesian Optimization | 6W / 7L full suite; 4W / 0L held-out | scikit-optimize gp_minimize |
| Bayesian Optimization | 11.5x lower regret on constrained Rosenbrock | scikit-optimize; closed-form Constrained EI |
| Bayesian Optimization | 5-20x faster across all problems | CPU only |
The 12 -- kernel-native, no mainstream equivalent
Capabilities enabled by the Sheffer kernel's monotone closed form, algebraic closure, and commitment discipline -- not portable to ReLU, GELU, or attention stacks.
| Modality | Key result | vs. | |
|---|---|---|---|
| Compositional Programming | Residual ~1e-7 at corners (AND/OR/XOR/NAND/NOR) | float32 noise floor | Deep-dive |
| Compositional Programming -- add | Max delta ~7.5e-7 on 1,000 probes | float32 ULP limit | Deep-dive |
| Verifiable Inference | 0.0 max delta Python/Rust, 10,000 inputs | no stochastic equivalent | Deep-dive |
| Verifiable Inference | 128-byte proof, 2.7ms verify | no stochastic equivalent | Deep-dive |
| Compositional Dynamical Inference | NRMSE 0.048 at N=64 -- beats 0.05 target | Neural ODE: never below 0.05 within N=512 | Deep-dive |
| Compositional Dynamical Inference | Lorenz-63 explained variance >0.9999 | single observed coordinate | Deep-dive |
| Rigorous Enclosure Engine | 0 violations in 1e7 probes | sampled confidence bands | |
| Exact Expectation Engine | 2056-3349x speedup vs 1e9-point MC | Monte Carlo UQ | |
| Exact Expectation Engine | Exact Sobol agrees with SALib N=8192 | seed-dependent MC sensitivity | |
| Behavioral Equivalence | 200k rewrite variants, 0 mismatches | byte hash != function identity | |
| Exact Inversion | Round-trip worst 3.04e-10 on 1e6 probes | ReLU preimage NP-hard | |
| Distribution Fingerprints | Power 0.367 vs RBF-MMD 0.316 | stochastic MMD baselines | |
| Conformal Prediction | Coverage within +/- 1% at three levels | posterior sampling calibration | |
| Deterministic Spectral Engine | ECG AFib pooled test AUC 0.8637 | stochastic spectral pipelines | |
| Atum Tensor | 14/16 cells at R2 >= 0.99 | stochastic tensor regression | |
| ASM (Algebraic Shadow Models) | 98.21% label retention at 40.11% params | teacher model footprint |
See it applied to financial model risk
The Financial MRM use case shows how these benchmarks translate into an SR 11-7-ready deployment.
View Financial MRM Use Case