CostPotentialBound
CostPotentialBound encodes an abstract lower bound on the cost function costAt at multiplicative indices, requiring costAt(v) to grow at least as fast as R times a chosen norm to the power α. Spectral theorists working on the J-cost operator cite it as the linear-growth precondition for sub-conjecture C.2. The declaration is a bare structure definition that packages the inequality without invoking lemmas or reductions.
claimLet MultIndex be the set of multiplicative indices and let costAt map each index to its J-cost. For any norm function norm : MultIndex → ℝ and constants R > 0, α ∈ ℝ, the bound asserts that R ⋅ norm(v)^α ≤ costAt(v) holds for every v ∈ MultIndex.
background
The module CostOperatorRegularity builds the operator-theoretic setting for the cost operator T_J on the dense domain of finite-support multiplicative states. It proves structural facts such as density and symmetry to zero sorry, then encodes the three analytic sub-conjectures (essential self-adjointness, compact resolvent, trace-class heat kernel) as hypothesis structures that depend on prime-weight assumptions.
proof idea
The declaration is a structure definition whose single field directly records the growth inequality. No upstream lemmas are applied; the structure simply packages the abstract precondition stated in the module documentation.
why it matters in Recognition Science
CostPotentialBound supplies the growth precondition for sub-conjecture C.2 inside the regularity analysis of T_J. It is referenced by the hypothesis structures EssentialSelfAdjointness and CompactResolvent and by the regularity_chain in the same module. In the Recognition framework it bridges the multiplicative cost algebra to the spectral properties required for the phi-ladder and the eight-tick octave.
scope and limits
- Does not commit to a concrete norm or specific value of α.
- Does not derive the inequality from J-function properties; it assumes the bound.
- Does not address indices with infinite support.
- Does not guarantee the bound for arbitrary choices of norm.
formal statement (Lean)
70structure CostPotentialBound (norm : MultIndex → ℝ) (R : ℝ) (α : ℝ) : Prop where
71 growth : ∀ v : MultIndex, R * norm v ^ α ≤ costAt v
72
73/-- Sub-conjecture C.2 (precondition): the cost potential grows linearly
74 in the L¹-norm of the multiplicative index.
75
76 Specifically: there exists `R > 0` such that
77 `costAt v ≥ R * Σ_p |v_p|`
78 for every `v : MultIndex`.
79
80 This holds because `J(p) ≥ J(2) = 1/4` for all primes `p ≥ 2`, so
81 `c(v) = Σ_p v_p J(p) ≥ Σ_p v_p · J(2)` when all `v_p ≥ 0`. The
82 general case (allowing negative `v_p`) requires the symmetry
83 `J(1/p) = J(p)`. -/