omnibias.core
The backend-agnostic mathematical core. No framework dependencies — every backend imports these symbols, which is what makes results bit-identical.
import omnibias.core as oc
Polynomial coefficients
The recurrences behind the closed-form derivative tower.
| Symbol | Meaning |
|---|---|
sigmoid_polynomial_coeffs | Eulerian recurrence: σⁿ(z) = Pₙ(sigmoid(z)) |
tanh_polynomial_coeffs | Legendre-style recurrence: tanhⁿ(z) = Tₙ(tanh(z)) |
hermite_coeffs | probabilist's Hermite coefficients for the Gaussian tower |
Bell polynomials / Faà di Bruno
The combinatorics of differentiating compositions — the engine under jets.
| Symbol | Meaning |
|---|---|
bell_partial | partial (incomplete) Bell polynomial |
bell_complete | complete Bell polynomial |
bell_number | Bell number |
faa_di_bruno_terms | the Faà di Bruno decomposition terms |
Multi-index combinatorics
Bookkeeping for multivariate jets.
multi_indices · multiply_table · multi_index_factorial ·
index_position · num_multi_indices
The activation protocol
| Symbol | Meaning |
|---|---|
ActivationSpec | generic activation descriptor backends specialize |
NthDerivativeFn, TensorFn, TensorT | typing aliases for the protocol |
Moments, information, probability helpers
Backend-agnostic statistical utilities used by the GLM / uncertainty surfaces:
- moments:
central_moments_from_cumulants,cumulants_from_raw_moments,raw_moments_from_cumulants,gaussian_central_moments,delta_method_central_moments,second_order_delta, … - information:
binary_entropy,has_cumulant_tower,is_log_partition_activation. - probability:
cdf_normalization,dkw_epsilon,is_cdf_activation.
The rigorous & formal layers
omnibias.core.verified (intervals, affine, Taylor models, Lohner flow) and
omnibias.core.proof (certificate format + Lean bridge) are part of core but
documented in the architecture section:
certified register and
formal loop.
See also
- The pure-Python core — the design rationale.
- Closed-form derivatives.