power_of_2_forces_D3
plain-language theorem explainer
If a spatial dimension D satisfies 2^D = 8 then D equals 3. Researchers establishing the uniqueness of three-dimensional space from the eight-tick cycle in Recognition Science cite this result. The proof proceeds by exhaustive case analysis on the natural number D, applying numerical normalization to obtain contradictions for all cases except D = 3.
Claim. If $D$ is a natural number representing spatial dimension and $2^D = 8$, then $D = 3$.
background
In the DimensionForcing module spatial dimension is an abbreviation for the natural numbers. The module proves D = 3 is forced by the RS framework via topological linking (non-trivial knots exist only in D = 3) and synchronization between the 8-tick cycle and the 45-tick cumulative phase. Upstream the tick is defined as the fundamental RS time quantum τ₀ = 1, and the eight-tick period is the fundamental evolution period arising as 2^D for ledger coverage.
proof idea
The tactic proof matches on D. Cases 0, 1 and 2 apply norm_num at the hypothesis to derive a contradiction. Case 3 applies reflexivity. The case n + 4 first proves 2^(n+4) ≥ 16 via Nat.one_le_pow, ring and nlinarith, then rewrites the hypothesis into this inequality and applies norm_num to reach a contradiction.
why it matters
This theorem is invoked directly by eight_tick_forces_D3, which concludes that the eight-tick cycle forces D = 3. It supplies the algebraic step linking the eight-tick octave (T7) to spatial dimension D = 3 (T8) in the forcing chain. The result supports the module claim that only D = 3 permits stable topological conservation under the Recognition Composition Law.
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