Scale Graphs I: Minimum-Production Renormalization and a Strict Lyapunov Identity
Abstract
We isolate the finite-dimensional algebraic core of the Onsager–Prigogine scale-graph framework. Each edge of a finite scale graph carries a nonnegative quadratic production polynomial P_e(λ) = a_e λ² − 2J_e λ + Y_e, P_e(λ) ≥ 0, which is equivalent to the Onsager bound J_e² ≤ a_e Y_e and to an exact Prigogine square-completion P_e(λ) = a_e (λ − p_e)² + κ_e, with κ_e ≥ 0. Eliminating intermediate vertices by minimum production gives a canonical dyadic renormalization map. The main result is not merely monotonicity of production under decimation, but an exact strict Lyapunov identity: for every eliminated vertex, the production drop is a computable square measuring the local failure of Prigogine closure. In current variables it is the mismatch square (q₁ − q₂)²/(a₁ + a₂), and it is dominated by the edgewise alignment defects. This yields dyadic summability and a Cesaro n^(−1/2) rate for the mean square-root defect along renormalization orbits. The final section reports the numerical evidence relevant to this algebraic paper only. In the archived shell ensemble, the Onsager inequality was tested on 25,200 pairs with zero violations, residual production was nonnegative on 570/570 samples, and the observed shell RG tangent eigenvalue was recomputed from the raw RG-pair table as δ_u^shell = 1.4187394, 95% CI [1.4002, 1.4366], with spectral-gap ratio approximately 3.95. The leading eigenvalue is real; a subleading complex-conjugate pair has modulus 0.2574. The shadowing diagnostic has R² = 0.0021 and is not used as a confirmation of this eigenvalue because the shipped v32 certificate documents a resolution limit for the N = 22 shell testbed. The numerical files and the Python reproduction script are archived in the accompanying reproducibility record, DOI: 10.6084/m9.figshare.32942288.
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BibTeX
@article{caffagni2026scale,
title = {{Scale Graphs I: Minimum-Production Renormalization and a Strict Lyapunov Identity}},
author = {Andrea Caffagni},
journal = {Figshare},
year = {2026},
month = jul,
doi = {10.6084/m9.figshare.32942279},
url = {https://figshare.com/articles/preprint/Scale_Graphs_I_Minimum-Production_Renormalization_and_a_Strict_Lyapunov_Identity/32942279},
note = {CC BY},
}
RIS
TY - JOUR
TI - Scale Graphs I: Minimum-Production Renormalization and a Strict Lyapunov Identity
AU - Andrea Caffagni
PY - 2026
DA - 2026/07/08
JO - Figshare
DO - 10.6084/m9.figshare.32942279
UR - https://figshare.com/articles/preprint/Scale_Graphs_I_Minimum-Production_Renormalization_and_a_Strict_Lyapunov_Identity/32942279
AB - We isolate the finite-dimensional algebraic core of the Onsager–Prigogine scale-graph framework. Each edge of a finite scale graph carries a nonnegative quadratic production polynomial P_e(λ) = a_e λ² − 2J_e λ + Y_e, P_e(λ) ≥ 0, which is equivalent to the Onsager bound J_e² ≤ a_e Y_e and to an exact Prigogine square-completion P_e(λ) = a_e (λ − p_e)² + κ_e, with κ_e ≥ 0. Eliminating intermediate vertices by minimum production gives a canonical dyadic renormalization map. The main result is not merely monotonicity of production under decimation, but an exact strict Lyapunov identity: for every eliminated vertex, the production drop is a computable square measuring the local failure of Prigogine closure. In current variables it is the mismatch square (q₁ − q₂)²/(a₁ + a₂), and it is dominated by the edgewise alignment defects. This yields dyadic summability and a Cesaro n^(−1/2) rate for the mean square-root defect along renormalization orbits. The final section reports the numerical evidence relevant to this algebraic paper only. In the archived shell ensemble, the Onsager inequality was tested on 25,200 pairs with zero violations, residual production was nonnegative on 570/570 samples, and the observed shell RG tangent eigenvalue was recomputed from the raw RG-pair table as δ_u^shell = 1.4187394, 95% CI [1.4002, 1.4366], with spectral-gap ratio approximately 3.95. The leading eigenvalue is real; a subleading complex-conjugate pair has modulus 0.2574. The shadowing diagnostic has R² = 0.0021 and is not used as a confirmation of this eigenvalue because the shipped v32 certificate documents a resolution limit for the N = 22 shell testbed. The numerical files and the Python reproduction script are archived in the accompanying reproducibility record, DOI: 10.6084/m9.figshare.32942288.
ER -