IndisputableMonolith.NumberTheory.CostOperatorRegularity
CostOperatorRegularity supplies the dense domain, potential bounds, summability, essential self-adjointness, and regularity chain for the cost operator on StateSpace. Researchers building the algebraic skeleton of a Hilbert-Pólya candidate operator would cite these results when checking domain and trace-class properties. The module is organized as a collection of sibling definitions and lemmas that import Cost and HilbertPolyaCandidate without introducing new proof tactics.
claimThe dense domain satisfies $D = StateSpace$ where $StateSpace = MultIndex →_0 ℝ$. Additional objects include $CostPotentialBound$, $WeightSquareSummable$, $EssentialSelfAdjointness$, $CompactResolvent$, $TraceClassHeatKernel$, and the composite $regularity_chain$.
background
The module sits inside the Recognition Science treatment of the Hilbert-Pólya conjecture. The upstream HilbertPolyaCandidate module states that it constructs the algebraic skeleton of a candidate operator whose eigenvalues would be the imaginary parts of the non-trivial zeta zeros. CostOperatorRegularity imports the Cost module for cost-related operators and records that StateSpace already encodes finite support, so the full space serves as its own dense subspace.
proof idea
This is a definition module, no proofs. It declares the listed sibling objects (denseDomain through regularity_chain) to organize the regularity statements needed downstream.
why it matters in Recognition Science
The module supplies the regularity infrastructure required by the Hilbert-Pólya candidate construction. It thereby supports the broader Recognition framework goal of realizing a self-adjoint operator whose spectrum encodes the zeta zeros, without yet addressing the forcing chain or phi-ladder constants.
scope and limits
- Does not derive explicit eigenvalues of the candidate operator.
- Does not invoke the Recognition Composition Law or T5-T8 forcing steps.
- Does not establish numerical bounds inside the alpha interval (137.030, 137.039).
- Does not connect the cost operator to the phi-ladder mass formula.
depends on (2)
declarations in this module (13)
-
def
denseDomain -
theorem
denseDomain_nonempty -
theorem
denseDomain_mem -
structure
CostPotentialBound -
def
CostPotentialLinearGrowth -
def
WeightSquareSummable -
def
WeightDecayPolynomial -
theorem
weight_polynomial_decay_summable -
structure
EssentialSelfAdjointness -
structure
CompactResolvent -
structure
TraceClassHeatKernel -
theorem
regularity_chain -
theorem
cost_operator_regularity_certificate