Sigma-C Framework v5.0.1

Susceptibility-based scale selection across physical, computational, and data-driven systems.

Sigma-C Framework now ships in two interoperable layers:

• v4 disciplined-reader kernel (sigma_c_v4/): the implementation of the foundation paper Operational scale selection: axioms, spectral concentration, and a regime trichotomy (Zenodo DOI 10.5281/zenodo.20548818, JSP submission JOSS-S-26-00346, under review). Every output either cites a theorem from the paper or admits it cannot.
• v1–v3 multi-adapter applications stack (sigma_c/): the existing domain adapters (Quantum, Finance, Climate, Seismic, Magnetic, GPU, ML, Number Theory, Protein, Edge, LLM cost), public API since v1.0, unchanged in v4 except for additive integration with the v4 kernel.

The two layers consume the same susceptibility-peak idea. The kernel is strict and paper-anchored; the adapters are domain-specific and operationally framed. Choose whichever fits the question.

What is sigma_c?

For any system observed at variable resolution sigma, the susceptibility-peak scale-selection functional is

sigma_c[O] := argmax_sigma |dO / d log sigma|

In words: where does the observable O respond most sharply to a multiplicative change of resolution? The location of the peak, sigma_c, is the characteristic scale; the sharpness of the peak, kappa, quantifies how pronounced the transition is.

Under explicit named hypotheses (positive-noise transfer operator with discrete spectral gap, single-mode faithful observable), sigma_c factors as sigma_c = rho_star * tau with tau = -T_* / log |lambda_2 / lambda_1| the system-intrinsic relaxation scale and rho_star a probe-shape constant. The foundation paper formalises this and proves three load-bearing results: a compatibility theorem, a cross-observable concentration theorem, and a regime trichotomy (single-mode / multi-mode / spectrally-flat).

Install

pip install sigma-c-framework

Optional extras for domain integrations:

pip install sigma-c-framework[quantum]   # Qiskit, Cirq, PennyLane
pip install sigma-c-framework[gpu]       # CuPy
pip install sigma-c-framework[finance]   # QuantLib
pip install sigma-c-framework[all]       # everything

Quick start

v4 disciplined kernel

import numpy as np
from sigma_c_v4 import analyze, gamma_k, Framework

# Bare exponential decay -> regime I, sigma_c = relaxation time
sigma = np.geomspace(0.1, 100, 400)
O = np.exp(-sigma / 5.0)
result = analyze(sigma, O, window="bare")

print(result.summary())
# sigma_c       : 5.003
# tau           : 5.003
# rho_star      : 1   [analytic:bare, analytic]
# regime        : geom=I_geom
# Theorem backing:
#   * paper Def 2.2 (def:sigmac)
#   * paper Prop 4.1 (prop:structural-reduction)
#   * paper Thm 8.3 (thm:trichotomy-geometric)
#   * paper Thm 6.1 (thm:cross-obs-concentration)

result.card("out.png")  # FUSE-style light-theme card

Power-law data (no characteristic scale) returns sigma_c = None honestly, not NaN:

result = analyze(sigma, sigma**0.6, window="bare")
assert result.sigma_c is None        # regime III, bottom value
assert result.regime.geometric == "III_geom"

v3 multi-adapter applications

import numpy as np
from sigma_c import Universe

temperatures = np.linspace(1.5, 3.5, 50)
magnetization = np.where(
    temperatures < 2.269,
    np.abs(2.269 - temperatures)**0.125,
    0.01 * np.random.randn(50),
)
mag = Universe.magnetic()
result = mag.compute_susceptibility(temperatures, magnetization)
print(f"Detected Tc: {result['sigma_c']:.3f}, kappa = {result['kappa']:.1f}")

Examples

Documentation

• sigma_c_v4/README.md — v4 kernel scope, modules, what is and is not in scope.
• THEOREM_MAP.md — label → paper-number binding, camera-ready safe; v4 docstrings cite labels and resolve numbers here.
• docs/01_getting_started.md — quick-start for the v3 adapters.
• docs/02_core_concepts.md — chi, sigma_c, kappa, observable discovery.
• docs/04_contraction_geometry.md — v3 contraction geometry (D, gamma).
• docs/05_type_classification.md — v3 four-type classification (D / O / S / R), separate from the v4 trichotomy.
• docs/06_api_reference.md — v3 API.
• docs/EXTENDING_DOMAINS.md — write your own domain adapter.

Version history

Foundation paper

Wurm, M. C. (2026). Operational scale selection: axioms, spectral concentration, and a regime trichotomy. Zenodo. doi:10.5281/zenodo.20548818 Manuscript JOSS-S-26-00346 under review at Journal of Statistical Physics (Springer).

For the applied validation on real quantum hardware:

Wurm, M. C. (2026). Operational scale detection in quantum magnetism. AVS Quantum Science 8 (1), 013804. doi:10.1116/5.0312410

License

Sigma-C Framework is offered under a dual license:

1. Open Source / Non-Commercial: GNU Affero General Public License v3.0 or later. Derivative works must also be licensed under AGPL-3.0-or-later.
2. Commercial: for proprietary use without AGPL obligations, see license_COMMERCIAL.txt or contact nfo@forgottenforge.xyz.

See LICENSE.txt for the dual-license notice.

Contributing

Issue reports and pull requests are welcome. Before opening a PR please read CONTRIBUTING.md and the CODE_OF_CONDUCT.md. Security disclosures go through SECURITY.md.

Acknowledgements

Sigma-C Framework is developed at ForgottenForge by M. C. Wurm.