pith. sign in
module module high

IndisputableMonolith.NumberTheory.RecognitionTheta.Convergence

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This module constructs a summable nonnegative majorant for the Recognition Theta terms at each positive time. Researchers working on modular identities for the RS theta function cite these bounds to control series convergence. It assembles majorant definitions and lemmas drawn from the parent RecognitionTheta module to bound the theta series.

claimThere exists a summable nonnegative function $M(t)$ such that the Recognition Theta function satisfies $0 < |�̃�_{RS}(t)| ≤ M(t)$ for all $t > 0$.

background

The upstream RecognitionTheta module defines the Recognition Theta function as the candidate completion of the cost theta function that incorporates the 8-tick character (T7) and the phi-ladder weight (T6) so as to inherit a modular identity under $t ↦ 1/t$. This Convergence module extends that construction by supplying summability controls at positive times. The module operates in the NumberTheory domain and imports the core RecognitionTheta definitions.

proof idea

This is a definition module, no proofs. It assembles majorant constructions such as RecognitionThetaMajorant and geometric variants together with associated convergence lemmas to guarantee the required summability.

why it matters in Recognition Science

This module feeds the ModularIdentity module which tracks sub-conjecture A.2. The majorant supplies the convergence control needed before a Poisson-summation theorem for the phi-ladder / 8-tick theta kernel can be applied. Downstream documentation states that the RS theta modular identity needs a Poisson-summation theorem for the phi-ladder / 8-tick theta kernel.

scope and limits

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declarations in this module (7)