Skip to main content

Exactness & scope

omnibias is deliberate about what kind of "exact" each result is. This is the canonical reference for those distinctions — read it before quoting a guarantee.

Three kinds of exactness​

  1. Closed-form exact. Computed from an analytic identity, not by differentiating a graph. Example: σⁿ(z) for the Riccati class, and the Laplacian / Hessian of a one-layer field. These are exact up to floating round-off in the final arithmetic.

  2. Autodiff-exact. Computed by exact (forward- or reverse-mode) differentiation of an analytic expression. Example: the metric derivatives in omnibias-geometry are exact forward-mode autodiff of the analytic metric — we label them honestly as autodiff-exact, not closed-form.

  3. Numerical. A grid, sample, or iterative approximation with a controlled error. Example: omnibias-fractional is non-local and grid-based — it is explicitly not closed form.

The rule we never break

A result is labelled with the strongest honest category — never a stronger one. If a page says "closed form," it means an analytic identity, not "autodiff that happens to be exact."

What is closed-form​

  • σⁿ(z) for the Riccati class (sigmoid, tanh, softplus, exp, gaussian, and the trigonometric / hyperbolic members) at the orders listed in the Stability matrix.
  • The gradient, Laplacian, and poly-Laplacian Δᵏ of a one-layer scalar field.
  • The one-layer parameter Hessian / Fisher / KFAC factors.

What is autodiff-exact​

  • Multi-layer field derivatives via the jet machinery (exact composition, but built from the closed-form tower plus exact combinatorics).
  • Metric derivatives, Christoffel symbols, and curvature in geometry, which differentiate an analytic metric.

What is numerical​

  • Fractional calculus (Grünwald–Letnikov, Riemann–Liouville, Caputo, spectral) on uniform / periodic grids.
  • Lattice Monte Carlo drivers and validated (interval) dynamics, where the output is a rigorous enclosure rather than a point value.

The rigorous register​

Where exactness is not analytically available, omnibias offers rigorous enclosures instead of unverified floats: outward-rounded interval, affine, and Taylor-model arithmetic that provably contain the true value. An enclosure is a different — and often stronger — guarantee than a point estimate. See the Certified register.

Honest boundaries on hard problems​

omnibias includes scaffolding aimed at hard mathematical problems (for example, spectral-gap and residual certificates). Be precise about what that is:

Proof-preparation, not a solver

The certified and formal stacks are proof-preparation infrastructure. They produce machine-checkable certificates for finite, rational obligations and honestly leave the infinite analytic obligations as open. They do not claim to solve open problems in mathematics, and the formal loop can never set theorem_prover_verified without a genuine Lean-kernel pass on a finite obligation. Asserting a grand claim without that pass blocks the verdict.

Why this matters​

Scientific users do not adopt numerics they cannot characterize. By telling you exactly which of the three categories a result belongs to — and by returning enclosures where point exactness is unavailable — omnibias makes its numbers usable in review-critical and safety-critical contexts.

See also​

Last updated on