Emergence of Spatial Structure From a Logical Substrate

This work shows how the kinematic framework of a quantum geometry follows from a logical substrate, and bounds by two no-go theorems what cannot follow from it. It does not propose a competing theory of quantum gravity: the group, the state space and the unit of horizon counting, one bit per face, are obtained here from the substrate. The value of the Immirzi parameter requires a horizon, hence the next levels. The minimal grain has four configurations, realised as two binary distinctions. A toggle and a conditional operation do not commute. Their commutators generate a Lie algebra of dimension three, isomorphic to that of rotations, in which the three dimensions of space are read (an interpretation). Under an information criterion, the grain has four legs. On an arena chosen by comparison, not by classification, the grains tile space with identical tetrahedra, whose angles close exactly around every edge: space is flat, and the tiling is bipartite. Under stated premises, the signature of spacetime cannot be derived at this level, no root of minus one is chosen, and there is no dynamics.The concept of the programme is posited before any computation, The seven reading rules formulated knowing their targets follow from it, and so does the Born rule. A single coupling governs the network. Beyond a threshold it orders into a vacuum, and the balance between the complexity of its exchanges and its capital, the entropy of its configurations, which measures durability, places the coupling in a zone just above the threshold. This vacuum is not critical: the continuum is an effective description of it, isotropic up to fourth order. On this vacuum, the three pairings of the grain are read as three families, and, under a postulate on its modes, a node with one extra leg gives the sixteen states of a generation. The level fixes ranges, not values. The eighteen computed mass ratios and mixing angles fall within their ranges, up to the measurement error; these are agreements, not predictions. With the measured mass splitting, the sum of neutrino masses lies between 0.063 and 0.070 electronvolt, straddling the current cosmological bound, and the violation of charge–parity symmetry by leptons has the sign opposite to that of the quarks, a refutable consequence of the framework. The compromise between accumulating and multiplying links fixes the form of the gauge action with no free parameter, and the coupling at the grain falls within the value required by the measured couplings run to unification. Fine values settle at the next levels. Seven structures are missing at this level; a single new choice at the next level would lift them, together with results already acquired; that it is the only remedy is a conjecture. Every statement carries its status.