K-Machine and Participant-Indexed Computational Model Value is categorically distinct from information — a non-separable joint state of Participant, Context, and Content that cannot be embedded in any information-theoretic structure without destruction. This categorical fact demands a computational response: any machine that correctly computes value must have a participant dimension that the Turing Machine does not have. This paper introduces that machine — the K-Machine. The K-Machine extends the Turing Machine by one primitive dimension — WHO — alongside the three classical dimensions of WHAT, WHERE, and HOW. The Recovery Theorem establishes that anonymous global computation is recovered exactly when the participant dimension is set to the trivial unit object. Every Turing-computable function is K-Machine computable. The inclusion is strict: participant-indexed computation subsumes classical computation, not the reverse. The geometric content is formalised through differential geometry: participants as points on a smooth manifold, context as tangent space, interactions as geodesics. The Krama Context Clock provides the local temporal structure. Four structural consequences follow: structural parallelism, local finality, Emergent Participant Intelligence, and security from the holonomy of the participant manifold. Companion paper to: Value, Information, and the Cartesian Degeneracy (CC2, Zenodo, https://doi.org/10.5281/zenodo.19148908) and PIC: Operational Semantics of Participant-Indexed Computation (CC4, Sarva Labs, March 2026).