three96_factorization
plain-language theorem explainer
396 factors as 2 squared times 3 squared times 11 in the natural numbers. Researchers examining Ramanujan's 1914 series for 1 over pi would cite this result to isolate the passive field edge count of 11 inside the denominator 396. The factorization confirms that 396 equals 4 times 9 times E_passive where E_passive equals 11 for dimension 3. The proof is a direct numerical normalization that verifies the equality without hypotheses.
Claim. The natural number 396 admits the prime factorization $396 = 2^{2} × 3^{2} × 11$.
background
In Recognition Science the passive field edges for a given dimension d are defined by passive_field_edges d := cube_edges d - active_edges_per_tick. For d = 3 this evaluates to 11 and is abbreviated E_passive in both the Masses.Anchor and Physics.MassTopology modules. The module places the factorization inside Ramanujan's series 1/π = (2√2/9801) Σ (4n)! (1103 + 26390n) / ((n!)^4 × 396^{4n}), noting that 396 = 4 × 9 × 11 isolates the topological integer 11 that arises from the recognition geometry of the 8-tick octave.
proof idea
The proof is a one-line wrapper that applies the norm_num tactic to verify the numerical equality 396 = 4 × 9 × 11 directly.
why it matters
This theorem supplies the explicit arithmetic link between the Ramanujan denominator 396 and the passive edge count E_passive = 11 forced by D = 3 in the Recognition Science chain. It supports the module's explanation that RS topological integers appear in rapidly convergent series for 1/π, consistent with the eight-tick structure that yields π as the recognition-circle constant. No downstream theorems are recorded, but the result closes the basic factorization step required before discussing the Heegner-number connection to 163.
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