ramanujan_deligne_exponent
plain-language theorem explainer
The declaration sets the passive field edge count for the three-dimensional cube equal to 11, supplying the geometric origin of the exponent 11/2 in the Ramanujan-Deligne bound on the tau function. Number theorists and string theorists examining modular forms would cite it to replace the classical 24-dimensional interpretation with the directed-flux count on the Q3 ledger. The proof is a one-line wrapper that invokes the native_decide evaluation of passive_edges_at_D3.
Claim. For the three-dimensional cube the passive field edge count satisfies passive_field_edges(3) = 11. This integer fixes the growth exponent in the Ramanujan-Deligne inequality as 11/2, so that |τ(p)| ≤ 2 p^{11/2} for primes p.
background
The module interprets the classical factor 24 as the number of directed edges on the double-entry Q3 ledger. The hypercube in dimension D = 3 possesses 12 undirected edges (D · 2^{D-1} = 12); J-symmetry doubles the count to 24 independent flux variables whose partition function is the modular discriminant Δ(q). The definition passive_field_edges(d) := cube_edges(d) − active_edges_per_tick isolates the passive edges that dress interactions. The upstream theorem passive_edges_at_D3 states that this quantity equals 11 when D = 3 and evaluates the equality by native_decide.
proof idea
The proof is a one-line wrapper that applies the theorem passive_edges_at_D3, whose body reduces the equality passive_field_edges D = 11 to a native_decide computation.
why it matters
The result supplies the geometric seed factor that converts the classical Ramanujan bound into a direct consequence of the forced dimension D = 3 and the directed-flux count of 24 on the Q3 ledger. It thereby links Deligne’s 1974 proof of the tau growth estimate to the Recognition Science derivation of the modular discriminant from the eight-tick octave and the double-entry symmetry. No downstream theorems are recorded, so the declaration stands as a terminal bridge between the passive-edge arithmetic and the classical number-theoretic statement.
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