independent
plain-language theorem explainer
The independent definition supplies the baseline spatial semantics in which an LNAL program evolves a field of voxels by applying the single-voxel operator to each element in isolation. Researchers building the classical-to-discrete fluid bridge cite it as the starting point before neighbor coupling is added. The definition is realized by a direct structural assignment that maps the voxel operator across the entire field.
Claim. The independent spatial semantics is the structure whose step operator sends a program $P$ and a field of voxels to the field obtained by applying the single-voxel evolution operator to every voxel independently.
background
SpatialSemantics is the minimal interface that supplies a step function taking an LNAL program and an LNALField (an array of register-auxiliary pairs) to a successor field. The companion voxelStep definition executes one LNAL instruction on a single voxel by initializing a local state, running the program step, and extracting the updated registers. The module is deliberately minimal: it provides spatial execution over an LNALField without any neighbor graph or inter-voxel coupling, as stated in the module header for milestone M2.
proof idea
The definition is a direct one-line wrapper that populates the step field of SpatialSemantics by composing the field-wide map operation with the already-defined voxelStep operator.
why it matters
This definition anchors the LNAL semantics milestone and is invoked by the lemma that the listen-noop program leaves any field unchanged. It feeds downstream into the RS-NS bridge specification, Noether-derived momentum conservation, and the phi-ladder mass-to-light results in astrophysics. It corresponds to the initial uncoupled stage of the classical bridge before the eight-tick octave or spatial coupling is introduced.
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