VoxelSymmetric
plain-language theorem explainer
Voxel symmetry encodes translation invariance of a real-valued function over the ℤ³ lattice, asserting no preferred spatial location. Cosmologists working in Recognition Science cite it to guarantee that phase-locked J-cost is identical in every voxel. The declaration is a structure definition that packages a single shift-invariance field with no separate proof body.
Claim. A function $f : ℤ^3 → ℝ$ is voxel-symmetric when $f(v + d) = f(v)$ for all lattice points $v, d ∈ ℤ^3$.
background
The module proves that phase-locked modes contribute a constant, isotropic J-cost background to every voxel, bridging PhaseSaturationVacuum to the vacuum stress-energy tensor. Voxel is defined as the fundamental length quantum (one unit in RS-native units). Phase is the 8-tick phase space, the finite set Fin 8. The local setting begins from the axiom that the ℤ³ carrier has no distinguished location, which is precisely the content of this structure.
proof idea
The declaration is a structure definition with an empty proof body. It directly introduces the single field shift_invariant that states the required equality under lattice translation.
why it matters
This structure is invoked by the theorem vacuum_energy_uniform, whose one-line proof is ⟨fun _ _ => rfl⟩, and by the master certificate VacuumUniformityCert. It supplies the first step in the module argument: the ℤ³ carrier has no distinguished location, so the combinatorial fraction 11/16 of passive modes yields spatially uniform vacuum energy. In the Recognition framework it supports uniformity of the phase-locked background consistent with the eight-tick octave.
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