OnePointPickPositive
OnePointPickPositive encodes the one-point Pick positivity inequality for an abstract conformal chart from the right half-strip into the unit disk and the xi-sensor Cayley field X. Researchers pursuing the Recognition Science route to the Riemann Hypothesis via Schur-Pick theory cite it as the N=1 principal minor of the full Pick matrix. The declaration is a direct definition of the required non-negative ratio without any reduction steps or lemmas.
claimLet $φ$ be any conformal chart from the right half-strip into the open unit disk and let $X$ be the Cayley transform of the logarithmic derivative of the Riemann xi function. One-point Pick positivity asserts that $0 ≤ (1 - |X(s)|^2) / (1 - |φ(s)|^2)$ holds for every $s$ in the right half-strip.
background
The module develops an unconditional approach to the Riemann Hypothesis that targets Pick/Schur positivity for the xi-sensor Cayley field without assuming bounded defect cost. XiCayleyField is the abstract map $X(s) = (2(ξ'/ξ)(s) - 1)/(2(ξ'/ξ)(s) + 1)$. HalfStripDiskChart is a structure supplying a map φ that sends every point of the right half-strip strictly inside the unit disk. RightHalfStrip is the region of positive real part with imaginary part confined to a vertical strip; the inequality is the N=1 principal minor of the associated Pick matrix.
proof idea
As a definition the declaration states the inequality directly. No lemmas are applied and no tactics are used; the body is simply the universal quantification over the right half-strip of the displayed non-negative ratio.
why it matters in Recognition Science
The definition supplies the one-point component required by the structure FinitePickPositive and is the hypothesis of the theorem schur_of_onePointPickPositive that derives the pointwise Schur bound on the right half-strip. It advances the Recognition Science forcing chain by furnishing the algebraic positivity condition needed to exclude poles of ξ'/ξ. The module records that the remaining analytic step, showing the Schur bound rules out such poles, is still open.
scope and limits
- Does not establish positivity of finite Pick matrices beyond the single-point case.
- Does not derive the Schur bound on the right half-strip.
- Does not prove absence of poles for ξ'/ξ.
- Does not assume or use any bound on defect cost.
Lean usage
theorem use_one_point (chart : HalfStripDiskChart) (X : XiCayleyField) (h : OnePointPickPositive chart X) : SchurOnRightHalfStrip X := schur_of_onePointPickPositive chart X hformal statement (Lean)
90def OnePointPickPositive (chart : HalfStripDiskChart)
91 (X : XiCayleyField) : Prop :=
proof body
Definition body.
92 ∀ s : ℂ, RightHalfStrip s →
93 0 ≤ (1 - ‖X s‖ ^ 2) / (1 - ‖chart.φ s‖ ^ 2)
94
95/-- Full finite Pick positivity, stated as an interface. The one-point field is
96included explicitly because it is the part we use theorem-grade here. -/