factor_11_in_396
plain-language theorem explainer
The declaration establishes that 396 factors as 4 times 9 times the passive field edges at dimension 3. Analysts of Ramanujan's 1/π series cite this to embed the RS topological integer 11 directly into the denominator. The proof reduces in one simplification step to the already-established value of passive edges at D=3.
Claim. $396 = 4 × 9 × E_{passive}(3)$ where $E_{passive}(d)$ is the number of passive field edges in dimension $d$, obtained by subtracting active edges per tick from the total cube edges, and this quantity equals 11 when $d=3$.
background
Recognition Science counts passive field edges as total cube edges minus active edges per tick. For dimension 3 this count is fixed at 11 by the upstream lemma passive_edges_at_D3. The module situates this integer inside Ramanujan's 1914 series for 1/π, whose denominator contains 396 raised to the fourth power, and notes that 396 = 4 × 9 × 11 while 9801 = (9 × 11)².
proof idea
The proof is a one-line wrapper that applies the lemma passive_edges_at_D3 via the simp tactic with that single rewrite rule.
why it matters
This factorization places the RS-derived count of 11 passive edges (forced by the eight-tick octave and D=3) inside the classical Ramanujan series. It supports the claim that topological integers from the Recognition framework appear in efficient π approximations. The module links the factor to the Heegner number 163 context but explicitly does not claim 1103 itself is an RS integer.
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