dimensions_from_log
plain-language theorem explainer
The result equates the dimension count derived from the tick structure to the base-2 logarithm of eight. Particle physicists investigating the origin of three fermion families would cite this link between the eight-tick cycle and spatial directions. The proof is a direct computational evaluation that confirms the numerical identity.
Claim. Let $d$ be the dimension count obtained from tick indexing in the eight-tick cycle. Then $d = 3$, since $d = 2^3$ and $2^3 = 8$ implies $d = 3$.
background
The module Physics.ThreeGenerations derives the number of fermion generations from the eight-tick cycle combined with three-dimensional space. Upstream, DimensionForcing defines Dimension as a natural number, while the eight-tick period is identified as $2^3$. Constants.Dimensions supplies the structure Dimension with integer exponents for length, time, and mass. The module doc states that generations arise as a discrete quantum number from how the eight-tick phase distributes across three orthogonal directions.
proof idea
The proof is a term-mode computation via native_decide that directly evaluates the equality between the tick-derived dimension count and the base-2 logarithm of eight.
why it matters
This theorem fills the SM-011 step that connects the eight-tick octave (T7) to three spatial dimensions (T8) and thereby to the three generations. The module doc notes the structural correspondence between dimensions and generations and flags the potential for a PRL paper on the generation number. It supplies the numerical identity required for the 8 = 2^3 argument in the forcing chain.
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