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module module high

IndisputableMonolith.NumberTheory.RecognitionTheta.Convergence

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The Convergence module supplies a summable nonnegative majorant for Recognition Theta terms at each positive time. Researchers proving modular identities for the RS theta would cite it to secure absolute convergence before applying Poisson summation. The module consists of definitions and supporting lemmas that build directly on the RecognitionTheta construction.

claimA summable nonnegative majorant $M(t)$ exists for the Recognition Theta function $\tilde{\Theta}_{RS}(t)$ such that the terms remain bounded by $M(t)$ for every $t>0$ and $\sum M(t)<\infty$.

background

The Recognition Theta function $\tilde{\Theta}_{RS}(t)$ is the candidate completion of the cost theta function $\Theta_J(t)=\sum e^{-t c(n)}$ that incorporates the 8-tick character (T7) and the phi-ladder weight (T6) so as to inherit a modular identity under $t\mapsto 1/t$. This module formalizes the construction and the structural properties.

The present module isolates the majorant needed to guarantee summability of the theta terms at positive times, using the same phi-ladder and 8-tick structure introduced in the parent RecognitionTheta module.

proof idea

this is a definition module, no proofs

why it matters in Recognition Science

This module feeds the ModularIdentity module (tracking C, sub-conjecture A.2), which requires a Poisson-summation theorem for the phi-ladder / 8-tick theta kernel. The majorant supplies the absolute convergence step that must precede any such summation argument.

scope and limits

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declarations in this module (7)