PhiLogos: A Bounded Reasoning Kernel and Reference Implementation

This preprint presents the formal specification and reference implementation of PhiLogos, a computational kernel designed to enforce strict bounded reasoning invariants within the TreeOS framework. It introduces the \Omega-Calculus (Omega-Calculus), a deterministic, integer-only contraction mapping that generates collision-free identifiers ("ticks") for decentralized and trustless data verification.We define the mathematical foundations for Claim CLM-OMEGA-INJ-23-1E6, specifying the discrete iteration logic and the constants derived from the Golden Ratio (\phi) and \pi required to mitigate floating-point drift across heterogeneous computing platforms. The document formalizes the kernel's central invariant: the prohibition of state transitions from "Speculative" to "Asserted" absent a formal Proof (Ring A) or a reproducible Computational Witness (Ring B).Included is the full source code for the PhiLogos Kernel (philologos.py), serving as a Ring B Witness that empirically verifies the injectivity of the \Omega-mapping at an iteration depth of n=23. This work provides a foundational layer for verifiable AI reasoning, ensuring that generated claims are mathematically grounded and computationally reproducible.