D-041 — Task 2.9-D1: the Krein-space frame and the deficiency-convergence theorem (with the D-040 Δ-constraint)

Date: 2026-08-19 | Session: 008 | Task: 2.9 (deliverable D1 of D-035) Status: complete (frame + theorem statements + proof skeletons + lemma status map). D-054 note (2026-08-19, task 4.3b): (1.12) must be read on compacts avoiding the poles of F = z²ξ/ξ′ (Suzuki's printed "every compact K ⊂ ℂ" is the standard abuse); the §4 reduction's (⇐) is a skeleton (L3/L4; D-051: L4 = (★)) — presentations of the reduction must carry this split (D-052 corrected). Bookkeeping items → task 4.5d. Sources: Suzuki 2606.09096 §7 read in full (local text); D-026 (R-003), D-034 (literature), D-035 (design), D-040 (E-022/E-022b).

0. What this file delivers

D1 as ranked in D-035 §5: the unconditional Krein-space frame, the exact deficiency-convergence ⟺ strong-resolvent-convergence statement, and the reduction "limit formula (1.12) ⟺ the Krein-space operator D has real spectrum", with every lemma's status named. NEW vs D-035: the D-040 numerical constraint is folded in as a necessary-condition filter (§5), and Suzuki's §7.5 limit extension is identified as the ψ = π/2 one (§4) — fixing the target of the θ(a) flow.

1. The frame (unconditional)

Objects (Suzuki, conventions of E-009…E-022):

• Screw function g (real, even, continuous; g(0) = 0); ĝ(z) = z⁻²ξ′(½−iz)/ξ(½−iz) [Suzuki 2023, (1.2)/(1.8)].
• Finite a: T_a = A_a − λ (any λ < λ_a; numerics λ = −1); ℋ(T_a) is a HILBERT space whenever T_a > 0 (measured λ_a > 0 for a ≤ 12, D-019 F1/E-019; under RH λ = 0 works ∀a). D_a = i d/dx minimal on L²_0(−a,a)-based domain; deficiency indices (1,1); self-adjoint extensions D̄{a,θ}, θ ∈ [0,2π); spec(D̄{a,θ}) = zeros of W(a,θ;z) = (z−i)V₊(z) + e^{iθ}(z+i)V₋(z) (Thm 1.5) — all real for every θ at every a with T_a > 0 (self-adjointness in a Hilbert space). [PROVED — Suzuki Thm 1.5 + λ_a positivity]
• a = ∞ WITHOUT RH: Q_W on C∞_c(ℝ) is an indefinite Hermitian form; negative index κ = number of off-line zero pairs (Bombieri [Bom00]; Weil positivity ⟺ κ = 0 ⟺ RH). ℋ(A∞) is a KREIN space (Pontryagin space if κ < ∞; κ finite or not is UNKNOWN — wall W-1).
• D∞ := i d/dx on the quotient/completion, closed J-symmetric, deficiency indices (1,1) in the Krein sense; J-self-adjoint extensions D̄_ψ. Under RH (κ = 0): J = I, all self-adjoint. [FRAME: standard Krein–Langer extension theory applies; the κ = ∞ case needs the generalized (Krein–Langer) deficiency theory — flag as L0, see §6.]
• Embedding: ℋ(A_a) ↪ ℋ(A∞) (Suzuki §7.6 argument: [v] = 0 ⟹ v = 0 on C∞_c(−a,a)). CAUTION: the published argument uses [14]'s de Branges isomorphism, proved under RH. Unconditional version needs the injectivity of v ↦ [v] directly from Paley–Wiener + the screw structure. Lemma L1 (open, expected true; finite-rank-isotropy bounds it).

2. The limit operator D and its spectrum (unconditional reading)

Under RH (Suzuki §7.2–7.5, [14] Thm 1.1): B = de Branges space, U: ℋ(A∞) → B isometric; M = multiplication by z, deficiency (1,1); extensions M_ψ, ψ ∈ [0,π); M_{π/2} has purely discrete spectrum Γ (zeros of ξ(½−iz)) with eigenfunction ONB F_γ; D := U⁻¹M_{π/2}U is the translation-group generator (§7.5: τ_t = e^{itD} unitary, Stone). Unconditional reading: without RH the same formulas define a de Branges–Krein space B (indefinite inner product, negative squares at off-line γ-pairs) and D is J-self-adjoint with spec(D) = Γ (with multiplicities m_γ), masses = m_γ = 1/K_B(γ,γ). Hence: spec(D) ⊂ ℝ ⟺ Γ ⊂ ℝ ⟺ RH. [Definitional + Suzuki §7; the indefinite version is the Krein–Langer/de Branges–Krein correspondence — standard modulo L0.]

3. THEOREM D1 (deficiency convergence ⟺ strong resolvent convergence)

Setting: H_a := ℋ(T_a) ↪ H := ℋ(A∞) (L1); P_a orthogonal projection (Krein: J-orthogonal) onto H_a; D_{a,θ} extended by 0 on H_a^⊥ as a self-adjoint RELATION R_{a,θ} ⊂ H ⊕ H.

Theorem D1. Assume L0, L1. Let θ(a) → θ_∞ and z₀ ∈ ℂ∖ℝ a common regular point (z₀ ∈ ρ(R_{a,θ(a)}) eventually and z₀ ∈ ρ(D)). TFAE: (i) v_±(a, z₀) → v_±(∞, z₀) in the strong (Hilbert-majorant) topology of H, where v_±(a,z₀) = deficiency elements of D_a at z₀; (ii) the Weyl/Q-functions converge: Q_a(z) → Q(z) on ℂ∖ℝ; (iii) R_{a,θ(a)} → D (the θ_∞ = π/2 extension) in strong resolvent sense: (R_{a,θ(a)} − z)⁻¹P_a f → (D − z)⁻¹f for all f ∈ H, z ∈ ℂ∖ℝ. Proof engine (Hilbert case classical; Krein case = Krein–Langer): Krein's resolvent formula for defect-(1,1) operators writes (R_{a,θ} − z)⁻¹ = (R_{a,ref} − z)⁻¹ + rank-one[v(a,z), Q_a(z), θ]; every coefficient is an explicit continuous function of the deficiency elements, the Q-function, and θ. (i) ⟹ the rank-one terms converge in norm; the reference resolvents converge strongly on the nested subspaces (D_a ↗ D∞ by construction: D_a = D∞|H_a, Suzuki §7.6). (iii) ⟹ (ii) ⟹ (i) by standard identifications (Q_a(z) = ⟨v(a,z), v(a,z₀)⟩-type Gram entries). ∎ (Assembly = standard; the only non-classical inputs are L0/L1 and J-self-adjointness bookkeeping.)

Content: Suzuki's §7.5 expectation ("natural to consider D_{a,θ(a)} → D in strong resolvent sense") is reduced to the SINGLE checkable hypothesis (i) — norm convergence of the deficiency elements in the LIMIT Krein space. Combined with D-026: (★) (tail cancellation) is the analytic form of (i); the WH factorization of ĝ is the tool that would prove (i) (Gohberg–Feldman finite-interval asymptotics), and the factorization is RH-gated (D-035 §2). The full logical chain:

(★) ⟹ D1(i) ⟹ strong resolvent conv. ⟹ spectral measure conv. μ_a ⇒ Σ_γ m_γδ_γ ⟹ Cor 1.6's (1.12) [all ⟹ proved or standard-modulo-L0/L1] (1.12) ⟹ RH (Suzuki Cor 1.6) ⟺ spec(D) ⊂ ℝ.

4. The reduction theorem (D1's criterion-3-shaped statement)

Theorem (reduction). The limit formula (1.12) holds for some choice of θ(a), ϕ(a,z) ⟺ spec(D) ⊂ ℝ (i.e. Γ ⊂ ℝ). (⇒) (1.12) ⟹ RH (Suzuki Cor 1.6, [PROVED]) ⟹ Γ ⊂ ℝ ⟹ spec(D) ⊂ ℝ (§2). (⇐) spec(D) ⊂ ℝ ⟺ RH; under RH: A_a > 0 ∀a (Yoshida direction), λ = 0 admissible; ĝ's poles all real ⟹ ĝ admits the canonical WH factorization (Gohberg–Feldman class inclusion = Lemma L3) ⟹ finite-interval deficiency solutions converge (Krein 1958/62 asymptotics = Lemma L4) ⟹ D1(i) ⟹ resolvent convergence ⟹ spectral measures converge ⟹ (1.12) via W = F·M (R-003) with M → e^{−ϕ} (the (★) step, now a THEOREM under RH). Status: (⇐) is a proof SKELETON with lemmas L3 (symbol class inclusion — meromorphic symbols with real poles, tempered growth) and L4 (GF theory for this symbol class) named; both are literature assembly, not new obstruction. The paywall W-2 (D-035) sits in the CONVERSE direction of usage: nothing here forces real spectrum from finite-a data alone.

5. The D-040 constraint (necessary conditions, measured)

Strong resolvent convergence (Hilbert case) ⟹ weak convergence of spectral measures on compacts (Kato/Weidmann; Krein case with care). For μ_a = Σ_k w_kδ_{z_k} ⇒ Σ_γ m_γδ_γ it is NECESSARY that (N1) valley depth Δ(a) → ∞ (K_a(z,z)/K_a(γ,γ) → ∞ for z ∉ Γ); (N2) in-window residual mass μ_a(ℝ∖Γ^ε ∩ [0,Z]) → 0 for every ε, Z (mass may escape to ±∞ but not linger near the lattice); (N3) θ(a) → the π/2-extension (§2: the limit operator is the ψ = π/2 extension M_{π/2}). Measured (E-022/E-022b, D-040): at FIXED θ = π: Δ(a) ≈ 3.33 − 4.76/a SATURATES (violates N1 asymptotically); in-window residual ≈ 48 (a=4) → 58 (a=12) INCREASING (violates N2); valley masses ≈ 1.00 already (target m_γ = 1 attained per-valley). Conclusion [NUMERICAL-VALIDATED]: the fixed-θ=π family cannot converge to D; the θ(a) flow is obligatory, and its target is the π/2-extension. This converts the open θ(a) problem (D-023) into a sharp requirement: find θ(a) → (π/2-extension) such that Δ(a) → ∞. The valley depth Δ(a;θ) is the cheap probe (E-022b machinery, minutes per point). → motivates E-024 (θ-sweep/flow numerics).

6. Lemma status map

7. Sanity gates

Davenport–Heilbronn/Epstein analogues: off-line zeros ⟹ κ > 0, spec(D) non-real; the frame survives, §4's (⇐) fails at L3 (poles off ℝ — no canonical factorization). Consistent: no part of D1 proves or disproves RH; D1 is a REDUCTION (criterion 3).

8. What D1 buys

• Suzuki's §7.5 heuristic is now a precise theorem-pair (D1 + §4) with named lemmas; the analytic content is isolated in (★)/L3/L4.
• The numerical probe (Δ(a)) is promoted to a NECESSARY CONDITION monitor; fixed-θ saturation is a theorem-shaped obstruction for the naive family, redirecting effort to the θ(a) flow (E-024).
• Open convention item: finite-a θ ↔ B-side ψ mapping (needed to state "θ(a) → π/2" in the W-convention; E-024 will pin it empirically by scanning θ).