The pure-Python core

omnibias-core is the single source of truth for the math. It contains no torch / jax / tensorflow / keras imports, which is what makes the backends bit-identical and the verified substrate backend-free.

What lives here

Module	Responsibility
omnibias.core.polynomials	coefficient generators: sigmoid_polynomial_coeffs, tanh_polynomial_coeffs, hermite_coeffs
omnibias.core.bell	Bell polynomials / Faà di Bruno combinatorics
omnibias.core.spec	the ActivationSpec dataclass (metadata shared by all backends)
omnibias.core.multi_index	multi-index ordering + Cauchy-product table for multivariate jets
omnibias.core.verified	interval / affine / Taylor-model arithmetic and validated numerics
omnibias.core.proof	the hash-sealed certificate format and the Lean bridge

The derivative-tower contract

This is the contract every backend honors:

• All polynomial coefficients come from omnibias.core.polynomials. Backends import them; they do not reimplement them.
• An activation is described by an ActivationSpec. Backends specialize the tensor type but share the metadata.
• A fast-path kernel computes σⁿ(z) directly. For n < 0 it raises ValueError; genuinely unimplemented orders raise NotImplementedError.
• OperatorBlock dispatches on the op tag: identity | grad | laplacian | derivative | band | integral.

from omnibias.core import ActivationSpec
from omnibias.core.polynomials import tanh_polynomial_coeffs

coeffs = tanh_polynomial_coeffs(5)   # exact rational/integer coefficients

Why pure Python?

A single definition means:

• Bit-identical backends — the same coefficients produce the same bits.
• A backend-free verified register — interval and Taylor-model arithmetic can use the exact same coefficients without depending on torch or jax.
• Auditability — the math is small, pure, and readable in one place.

The ActivationSpec

ActivationSpec(
    name="tanh",
    forward=...,             # σ(z)
    derivative=...,          # σ'(z) if known
    fastpath=...,            # σⁿ(z) for supported orders
    riccati_polynomial=...,  # P with σ'(z) = P(σ(z)), or None
    noise_model="...",       # GLM family whose log-partition is σ, or "none"
    operator_role="...",     # role of the K=2 bias-collapse unit
)

Backends wrap this with their tensor type but never change its meaning.

Continue

• Backends — how torch / jax / keras specialize the core.
• Certified register — the verified and proof modules in depth.