eight_tick_minimal
plain-language theorem explainer
Any surjective mapping from a finite index set of size T onto the full set of 3-bit patterns forces T at least 8. Discrete Navier-Stokes and cellular-automaton modelers cite the bound to fix the shortest stable time window on a 3D lattice. The argument is a direct one-line reduction to the minimal-covering lemma for that pattern space.
Claim. If $f$ is a surjective function from a set of cardinality $T$ onto the set of all 3-bit patterns, then $T$ is at least 8.
background
The Eight-Tick Discrete-Time Dynamics module treats time as discrete for the Navier-Stokes lattice program. An 8-step window is the natural stability unit whose certificates propagate to all later windows by iteration. The step operation applies a local rule globally to produce the successor tape configuration. T denotes the natural numbers used as fundamental periods, while A records the active edge count per tick (fixed at 1) and satisfies the phi-power balance identity at three spatial dimensions.
proof idea
The proof is a one-line wrapper that applies the minimal-covering lemma to the given pass and surjectivity hypotheses.
why it matters
This lower bound enters the complete ledger uniqueness theorem, which simultaneously forces phi as the cost fixed point, Q3 as the linking number for dimension 3, and the 8-tick period. It realizes the eight-tick octave (period 2 cubed) required by the forcing chain once D equals 3. The result closes the uniqueness argument that no shorter cycle can cover the 3D pattern space.
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