eleven_not_div_1103
plain-language theorem explainer
The declaration establishes that 11 does not divide 1103. Researchers examining Ramanujan's 1914 series for 1/π cite the fact to keep the prime 1103 separate from the factor 11 that enters the denominators via 396 and 9801. The proof assumes a hypothetical multiplier k and applies the omega tactic to obtain an immediate arithmetic contradiction.
Claim. $11$ does not divide $1103$.
background
The module situates Ramanujan's series 1/π = (2√2/9801) Σ (4n)!(1103 + 26390n) / ((n!)^4 396^{4n}) inside Recognition Science. Here 396 = 2² × 3² × 11 and 9801 = 99² = (9 × 11)², so the RS topological integer 11 (passive field edges of Q₃) appears explicitly in the denominators. The module notes that π itself is forced by the eight-tick octave structure, making the presence of these integers natural. 1103 is stated to be prime and to arise from the class-number theory of Q(√−163) rather than from any product of RS integers.
proof idea
The term-mode proof introduces the divisor assumption ⟨k, hk⟩ and hands the resulting equation directly to the omega tactic, which discharges the goal by arithmetic normalization.
why it matters
The result closes a small but necessary gap in the RamanujanBridge account: it confirms that 1103 does not factor through the RS integer 11, preserving the claim that 1103 enters the series via Heegner number d = 163. The declaration therefore supports the module's separation between denominators built from E_passive and the numerator constant that is independent of it. No downstream theorems are recorded, so the lemma functions as a local arithmetic fact within the factorization lemmas of the same file.
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