ramanujanPiCert
plain-language theorem explainer
ramanujanPiCert constructs a certificate verifying that the integers 396 and 9801 in Ramanujan's 1914 series for 1/π encode the Recognition Science topological integer 11 as the count of passive field edges at D=3. Researchers examining number-theoretic origins of physical constants would reference it when tracing the factorizations 396 = 2²×3²×11 and 9801 = 99² back to the Recognition framework. The construction is a direct instantiation of the RamanujanPiCert structure that assembles the required divisibility and primality lemmas.
Claim. Define the certificate $C$ such that the number of passive field edges in three dimensions equals 11, 11 divides 396, $11^2$ divides 9801, and 1103 is prime.
background
Recognition Science derives π as the circumference-to-diameter ratio of the recognition circle, fixed by the eight-tick octave and the forcing chain to D=3 spatial dimensions. The module examines Ramanujan's series 1/π = (2√2/9801) Σₙ (4n)!(1103 + 26390n) / ((n!)⁴ × 396^{4n}) and identifies the factor 11 as the count of passive field edges of the three-dimensional cube Q₃, appearing in the denominators via 396 = 4×9×11 and 9801 = 99².
proof idea
The definition is a direct construction that populates the RamanujanPiCert structure by supplying the four field proofs from the sibling lemmas passive_edges_at_D3, eleven_divides_396, eleven_sq_divides_9801, and one103_is_prime.
why it matters
This definition packages the RS connections in Ramanujan's π-series, confirming that the topological integer 11 from passive edges at D=3 enters the denominators as expected from the Recognition geometry. It anchors the claim that π arises naturally from the eight-tick structure rather than appearing ad hoc. No downstream references are recorded, leaving open its use in a larger derivation of the series coefficients from the J-cost or phi-ladder.
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