IndisputableMonolith.NumberTheory.MellinPullback
The MellinPullback module defines reciprocal symmetry for functions on positive reals, where f(x) equals f(1/x). It supplies the algebraic foundation for kernel substitutions in the Recognition Science Mellin transform. Researchers on the RS-native zeta program cite it when separating symmetry properties from transform interfaces. The module consists of the core definition plus supporting lemmas on pointwise behavior.
claimA function $f : (0,∞) → ℝ$ is reciprocally symmetric when $f(x) = f(1/x)$ holds for every $x > 0$.
background
This module belongs to the NumberTheory section and imports the Cost module, which supplies the J-cost function used throughout the Recognition Science framework. The supplied definition states that reciprocal symmetry means f(x) equals f(1/x) for positive x. The module prepares the algebraic content needed for Mellin reflection theorems by isolating this symmetry property from the transform construction itself.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds the MellinTransform module, which builds a formal Mellin-transform interface whose reflection theorem is derived from reciprocal symmetry. It contributes to Phase 3 of the RS-native zeta program by separating the algebraic/RS content (reciprocal symmetry and kernel substitution) from the transform layer. The definition directly enables the symmetry-based derivation of reflection properties.
scope and limits
- Does not implement the Mellin transform or its reflection theorem.
- Does not contain proofs of functional equations beyond the symmetry definition.
- Does not address convergence or analytic continuation of transforms.