recognitionTheta
plain-language theorem explainer
Recognition Theta supplies the kernel for the Mellin transform that recovers the completed zeta function under the Recognition Science framework. Workers on the theta-zeta bridge cite it when establishing the functional equation via the modular identity. The construction is a direct infinite sum whose convergence is deferred to Sub-conjecture A.1. The definition uses the tsum operator applied to a pre-defined term that already encodes the phi-ladder and 8-tick character.
Claim. $tilde Theta_RS(t) := sum'_{n in N} tau(t,n)$ where tau is the nth term of the series that incorporates the phi-rung weight and the mod-8 character.
background
The Recognition Theta function is introduced in the NumberTheory.RecognitionTheta module as the candidate completion of the ordinary cost theta function Theta_J(t) = sum e^{-t c(n)}. It augments that sum with the phi-ladder rung index r(n) (completely additive, r(p) = floor(log_phi p) for primes) and the mod-8 character chi8 so that the resulting object can inherit a modular identity under t to 1/t. The module states that convergence belongs to Sub-conjecture A.1 while the tsum itself is always well-defined in Lean (defaults to zero when not summable). Upstream structures supply the J-cost calibration (from PhiForcingDerived), the active edge count A at D=3 (from IntegrationGap), and the spectral emergence of gauge content and generations (from SpectralEmergence).
proof idea
Direct definition as an infinite sum. The body is a one-line wrapper that applies the tsum operator to the already-defined term function recognitionThetaTerm.
why it matters
This definition is the kernel referenced by thetaKernel in ZetaFromTheta and by the structures RecognitionThetaMellinFactor and RecognitionThetaModularIdentity. It supplies the concrete object whose Mellin transform is conjectured to factor as zeta(s) times a meromorphic G_RS(s) that inherits reflection symmetry. The construction directly implements the T6 phi-ladder fixed point and T7 eight-tick octave required for the modular identity under t to 1/t. It closes the elementary-arithmetic part of the theta-zeta bridge before the analytic conjectures are invoked.
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