IndisputableMonolith.NumberTheory.RecognitionTheta.ModularIdentity
This module supplies candidate modular prefactor data for the Recognition Theta identity. Researchers advancing sub-conjecture A.3 in the Mellin factor development cite it to furnish explicit prefactors. It builds on the summability comparison theorem imported from the Convergence module by packaging the prefactor candidates and their equivalences.
claimCandidate modular prefactor data for the Recognition Theta identity, consisting of structures and equivalences that supply an explicit nonzero function $G : ℂ → ℂ$ satisfying the required modular properties.
background
This module belongs to the NumberTheory.RecognitionTheta namespace, which tracks sub-conjectures A.1 through A.3. It imports the Convergence module whose doc-comment states: 'The current RecognitionThetaConvergence statement asks for summability of the Recognition Theta term for every t > 0. This module proves the general comparison theorem needed to discharge it: if the terms admit a summable nonnegative majorant for each positive t, then A.1 follows.' The local setting introduces definitions for candidate prefactor data that support the Mellin identity construction.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
This module feeds the MellinFactor module that tracks sub-conjecture A.3. The downstream doc-comment states: 'The existing RecognitionThetaMellinFactor structure in RecognitionTheta.lean is only a placeholder: it asks for a nonzero function G : ℂ → ℂ and leaves the actual Mellin identity as True. This module makes that explicit by inhabiting the current structure and then'. It supplies the concrete prefactor data required to replace the placeholder.
scope and limits
- Does not prove summability of the Recognition Theta term.
- Does not discharge the full Mellin identity.
- Does not supply numerical checks of the prefactor values.
- Does not address the eight-tick octave or spatial dimension forcing.