factor_passive_sq_in_9801
plain-language theorem explainer
The theorem states that 9801 equals 81 times the square of the number of passive field edges when the dimension is three. Researchers linking Ramanujan's 1/π series to Recognition Science topological integers would cite this factorization to account for the factor of 11 squared in the denominator. The proof is a one-line wrapper that substitutes the value of passive field edges at D=3 and evaluates the arithmetic.
Claim. $9801 = 81 × (E_{passive})^2$, where $E_{passive}$ is the number of passive field edges in three dimensions.
background
The module deciphers Ramanujan's 1914 series for 1/π, whose denominators contain the integers 396, 9801, and 1103. Recognition Science accounts for these via topological integers forced by the simplicial structure on the three-dimensional cube. Passive field edges are defined as total cube edges minus active edges per tick; the upstream lemma passive_edges_at_D3 shows this quantity equals 11 when the dimension is three.
proof idea
The proof applies the lemma passive_edges_at_D3 to replace passive field edges D by 11, then uses norm_num to verify the resulting numerical identity 9801 = 81 × 11².
why it matters
This result supplies the explicit factorization 9801 = (9 × 11)² inside the RamanujanBridge module, confirming that the passive field edges of 11 appear squared in the denominator of the series. It aligns with the Recognition Science account of π arising from the eight-tick octave in three dimensions. The module notes that 1103 remains prime and is not claimed to be an RS topological integer.
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