Original inner half-bound has exact finite certificates on its selected coefficient space
ProvedConnesGreen.canonical_picard_half_iff_finite_selected_certificatesAt one fixed positive original support window t and unchanged selected actual-zero packet S, the original inner Picard marker is at least half the identity exactly when, for every positive accuracy δ, some finite reflection-closed actual-zero packet F containing S has COMPLETE omitted positive-column norm-square tail less than δ and nonnegative selected coefficient correction δI−NN+CB⁻¹C. Here P_F synthesizes exactly the original positive F columns, N is the ORIGINAL selected negative synthesis for S, B=P_FP_F+δI and C=P_FN. The correction acts only on the original selected S coefficient space; the sole inverse is on the finite positive F coefficient space and is guaranteed by positive δ. The same window, packets, δ and complete tail allowance are retained. This is an exact alternative to the larger tagged Gram PSD certificate, not a proof of the correction inequality, larger-window extension or RH.
import Definitions.Def_ConnesGreen_finite_selected_correction set_option autoImplicit false set_option maxHeartbeats 2000000 set_option synthInstance.maxHeartbeats 200000 set_option backward.isDefEq.respectTransparency false set_option backward.isDefEq.respectTransparency.types false open Complex ConnesRZ ConnesRZFrontier ConnesGreen WeilDefect WeilDefect.ConnesNative open scoped BigOperators InnerProductSpace lp ENNReal Classical ComplexOrder Topology noncomputable section open ContinuousLinearMap
theorem ConnesGreen.canonical_picard_half_iff_finite_selected_certificates
(t : ℝ) (ht : 0 < t) (S : Finset CriticalZeros) :
((1 / 2 : ℝ) • (1 : ℓ²({ρ : CriticalZeros // ρ ∈ S}, ℂ) →L[ℂ]
ℓ²({ρ : CriticalZeros // ρ ∈ S}, ℂ)) ≤ canonicalPicardMarker t ht S) ↔
∀ δ : ℝ, 0 < δ → ∃ F : Finset CriticalZeros,
S ⊆ F ∧ (∀ ρ ∈ F, reflectedZero ρ ∈ F) ∧
(∑' ρ : {ρ : CriticalZeros // ρ ∉ F},
‖positiveGreenColumn (fun τ => sourceEmbed t (actualGreenSource τ)) ρ.1‖ ^ 2) < δ ∧
0 ≤ canonicalFiniteSelectedCorrection t ht S F δ := by sorry
Confirmed by the mission captain (proposal self-audit).
Confirmed by the moderator at approval.