Timeless Two-Dimensional Formation, an Executed Three-Dimensional Transition, and the Birth of Physical Time: A Reproducible Finite-Pregeometry Theorem

This paper gives a connected finite-pregeometry construction in which a one-dimensional relational parent forms its first filled two-cell without physical elapsed time, undergoes an explicitly executed two-dimensional-to-three-dimensional transition within the model, and thereby acquires both the first irreversible physical ordinal in the constructed history and the physical capacity to produce, distinguish, and retain information. The result closes the five-step Goal-3 chain of the Quantum Genesis programme under the explicitly stated owner-authorized adopted working laws. The first transition is exact. Five integer source events are allocated over seven relations of an open eight-vertex path. Among the 462 weak compositions, exactly 42 satisfy the two-sided hinge condition, and every firing configuration has one unique hinge. The selected adjacent relations retain their lineage; one transverse relation is created between their outer vertices; and the resulting three-cycle is filled by a two-simplex. The relation-count Gram rank changes from one to two, and the Cayley–Menger relation-count area diagnostic changes from zero to \(3/16\). A direct dependency audit then proves that this selected Stage-2 construction uses no physical duration, clock, observer/record capacity, measurement channel, pointer/readout, or durable accessible event/measurement record. Its ordered construction remains exact after all temporal language and \(d\lambda/dt\) terms are deleted. Construction order is therefore retained without being renamed elapsed physical time. The second transition is governed by the owner-authorized adopted working edge-dual restriction (EDR), a route-specific gain law (GAIN), and fold-to-cone activation (FCA). The gain reaches its first fold exactly at \(x_*=\tfrac12\ln3\). FCA acts directly on the lineage-fixed triangle, creates a new apex and its three incident relations, creates three side faces, fills the resulting tetrahedron, and then holds the fired sector. The output has \(f=(4,6,4,1)\), nonzero \(C_3\), and the original triangle literally survives as a boundary face. Physical time begins here in the bounded sense proved by the construction: FCA supplies the first nonempty irreversible ordering between an uncommitted sector and a committed, retained three-cell. This is a first physical ordinal, not a calibrated duration. Independently, the post-transition apex and response packet satisfy a positive observer-capacity predicate and carry positive Holevo information. The construction can therefore produce, distinguish, and retain a result without requiring that an accessible first observation has already been performed. Exact finite and matrix computations reproduce every load-bearing identity. The result ends at time onset and observation/record capacity; it does not claim a calibrated clock, a completed observer engine, smooth spacetime, or empirical cosmology.