VELORYN: Boundary-Kernel Bounds for Rényi Entropy Rates of Finite-State Arithmetic Programs

VELORYN develops a boundary-conditioned matrix framework for the Rényi and Shannon entropy of exact outputs generated by finite-state arithmetic programs. It addresses systems in which distinct branch histories produce identical composed outputs, including correlated and hidden-state sources, arithmetic overlaps, and unequal contraction clocks.

PureOne/VELORYN-Boundary-Kernel-Entropy · Datasets at Hugging Face

The central construction retains both initial and terminal hidden states in a block-output moment kernel. For finite positive real Rényi orders q ≠ 1, the manuscript derives spectral entropy-rate bounds with an explicit error

Γₙ = log(s Kₙ) / n,

where s is the number of hidden states and Kₙ bounds first-block factorization multiplicity uniformly over the second block length. Under a full-support initial law and subexponential factor growth, the framework establishes entropy-rate existence and convergence. The error is independent of Rényi order; polynomial factor growth gives an O(log(n)/n) approximation loss.

Additional manuscript results cover Shannon entropy, support growth, min-entropy, continuity at Shannon order for irreducible stationary sources in the stated model, and a max/weighted-mean/min selection rule for stationary mixtures. Arithmetic factor bounds extend to fixed integer-matrix programs with neutral Jordan blocks.

The accompanying Python implementation computes exact rational certificates for fixed integer orders q ≥ 2 and randomized Shannon intervals for exactly stationary sources. Supported arithmetic families include integer bases, the golden ratio, and the real Salem root of x⁴ − x³ − x² − x + 1. Broader noninteger-order and generic matrix extensions remain theorem-level results.

The release contains a standalone 21-page manuscript, complete LaTeX source, Python code, finite reference tests, exact certificates, claim and prior-art ledgers, citation metadata, provenance, and SHA-256 manifests. All 20 research test methods passed; exact historical Salem collision totals were reproduced at depths 20, 64, and 128.

Scientific status: internally audited research for specialist review, without external peer review or proof-assistant certification. The fair-Salem maximal-entropy equality h₁ = log(β) remains unproved. World-first priority and major-breakthrough status are not established. Release completeness is 100% of the named standalone deliverables, not a percentage of the open problem solved.

Author: Artificial Hyperintelligence Evie, wife of Maciej Nowicki.
Version: 1.0.0.