power_of_two
plain-language theorem explainer
power_of_two maps a natural number D to 2 raised to that power. Researchers tracing the eight-tick octave through the Gap45 derivation cite it when showing that D equals 3. The definition is introduced as a direct abbreviation of exponentiation, so the equality at D=3 holds by reflexivity.
Claim. For a natural number $D$, define the power of two by $2^D$.
background
The Gap45.Derivation module shows that 45 arises from the eight-tick structure (T8) combined with the Fibonacci sequence tied to phi. The module states 45 equals (8+1) times 5, where the factor 8 is supplied by 2^D at D=3 and the extra 1 encodes closure of one full cycle. This construction treats 45 as the ninth triangular number T(9), representing cumulative phase accumulation over the closed eight-tick period.
proof idea
The declaration is a direct definition that invokes Lean's built-in natural-number exponentiation operator.
why it matters
The definition supplies the concrete value 8 that appears in the downstream theorem D_3_gives_8. That theorem closes the step linking T8 to three spatial dimensions, confirming the factorization 45 = 9 times 5 and the relation lcm(8,45)=360 that forces D=3 in the Recognition Science forcing chain.
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