omnibias-qcalculus

Calculus on a geometric grid. Exact q-polynomial coefficients approach ordinary derivatives.

Static poster · Narrow-screen animation · How this visual is computed

A q-parameter and algebraic coefficients enter; q-numbers, Jackson derivatives and q-integrals leave. Multiplicative sampling supports calculations on a geometric grid, distinct from an additive finite-difference stencil.

The animation uses computed outputs to explain this package. Frame transitions are illustrative unless a training step is explicitly identified; it is not a performance comparison.

API reference · Source · Tests · Talk to Derivon

The mathematical connection

The defining limit here is q → 1, which recovers ordinary calculus. It is distinct from both bias collapse (normalized nearby shifts) and temperature collapse (sharpening soft alternatives). Neither founding mechanism should be substituted for the q-calculus operator definition or its domain restrictions.

Run this README

The examples use omnibias-qcalculus on Python >=3.10. Their installed-wheel profile selects runtime features, not an editable workspace. Install the prepared prerelease from PyPI:

python -m pip install --pre "omnibias-qcalculus==0.1.0a2"

For local development before publication, build and test the coordinated wheelhouse using the release guide. The package's wheel profile executes the examples below outside the source checkout.

Existing published consumers may need historical primitive versions; see the compatibility policy.

Why this package exists

Some discrete and multiplicative-scale problems are expressed more naturally by q-differences than by ordinary shifts. Qcalculus exposes that deformation explicitly and connects it to the ordinary derivative as q approaches one. Exact polynomial operations make the relation easy to inspect without numerical differencing.

What you can build

• q-brackets, factorials, binomials and polynomial transforms.
• Jackson derivatives and antiderivatives.
• q-exponential families, series bounds and optional tensor realizations.

Use qcalculus for multiplicative sampling, q-series experiments and time-scale or symbolic consumers that need this register. Its q→1 limit is a separate mechanism from bias collapse and temperature hardening. Select the register that represents the mathematical problem rather than treating the parameters as interchangeable temperatures.

A working example

from fractions import Fraction
from omnibias.qcalculus import q_derivative_poly

# Coefficients are ordered from constant term upward: f(x) = x**2.
assert q_derivative_poly([0, 0, 1], Fraction(1, 2)) == (Fraction(0), Fraction(3, 2))
assert q_derivative_poly([0, 0, 1], Fraction(1)) == (Fraction(0), Fraction(2))

Choose the right contract

Numeric series require a supported q-domain and truncation/convergence controls. Near q=1, a direct quotient may be poorly conditioned; prefer the explicit limit or polynomial path where available. Exact rational coefficients do not make arbitrary floating-point series evaluations exact.

Explore and validate

The API guide contains the generated module/export inventory. Use it to find the focused implementation rather than guessing a symbol from another package. The capability map connects the primitives to larger scientific workflows.

From the main repository, run the package’s regression suite:

uv run pytest packages/omnibias-qcalculus/tests -q

License

Apache-2.0. See LICENSE and the licensing policy.