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Updated 08-19-2026

D-059 — Repair 4.5d: bookkeeping write-ups — punctured-compact (1.12); R_τ distributional representation; (I1) form-domain identification; L1 embedding argument

Date: 2026-08-19 | Session: 009 | Task: 4.5d Status: complete (write-up level; the remaining formalities are named inline). Closes D-054 findings F6/F7 + the form-domain item + L1.

1. (1.12) on punctured compacts (F6)

Suzuki's Cor 1.6 prints "uniformly on every compact subset K ⊂ ℂ" with limit F(z) = z²ξ(½−iz)/ξ′(½−iz), which has poles at the zeros of ξ′(½−iz). A uniform limit of entire functions e^{ϕ(a,z)}W(a,θ;z) on a compact containing such a pole is impossible (the limit would be entire). The statement is the standard abuse: (1.12) is read as uniform convergence on compacts K ⊂ ℂ ∖ {poles of F} (equivalently: uniform convergence in the spherical/chordal metric on all compacts, with Hurwitz content: zeros of W converge to zeros of F counting multiplicity on compacts avoiding the poles). All project statements quoting (1.12) (R-003, D-041, D-052) are to be read this way. Note: the zeros of ξ′(½−iz) lie in Re z ≥ 0-ish strips (Speiser: RH ⟺ ξ′ zeros on Re... for bookkeeping only their location as a discrete set matters). No content change anywhere: the convergence is used through its zero-set consequence (Hurwitz), which the punctured reading supports.

2. R_τ's representation (F7)

The tail functions τ± := (g∗v± − h±)·1_{|x|≥a} are carried by the unbounded set |x| ≥ a, so Paley–Wiener–Schwartz does not apply to them and τ̂± are not automatically entire/analytic-functionals. Precise statement of what D-026/R-003 prove and mean:

3. (I1) form-domain identification (the D-049 test-function item)

Lemma FD. Under RH, the closed positive form q₀(v) := ‖v‖² + Σ_γ m_γ|v̂(γ)|², D(q₀) := {v ∈ L²(−a,a) : Σ m_γ|v̂(γ)|² < ∞}, coincides with the closure of the operator form of T_a = I + B_a, and the variational identity (I1) holds with the sup over D(q₀): K_a(z,z) = sup_{v ∈ D(q₀)} |v̂(z)|²/q₀(v). In particular the flat bumps e^{∓iγx}1_{(−a,a)} are admissible (their form sums converge: Σ_γ 4sin²((γ'−γ)a)/(γ'−γ)² < ∞ by unit-interval counting against the log-density).

Proof (assembly). (a) q₀ is closed: the map J: v ↦ (m_γ^{1/2}v̂(γ))_γ from L²(−a,a) to ℓ²(Γ, m) is a closed operator (point evaluations are continuous on the PW target; if v_n → v in L² and Jv_n Cauchy, then v̂_n → v̂ uniformly on compacts by PW Bernstein, so the ℓ²-limit is Jv); q₀ is its form ‖v‖² + ‖Jv‖² — closed by the standard construction. (b) On C²(−a,a) (indeed on C_c² test functions) q₀ agrees with the operator form ⟨v, T_a v⟩: this is the Weil explicit formula in the support-(−a,a) case (conventions machine-verified, E-009a; Galerkin validated 4.5e-6, E-009b). (c) C² is a form core of the operator form by construction of T_a (Friedrichs-type; Suzuki's definition via the form closure). Two closed forms agreeing on a common core coincide. (I1) is then the Glazman variational principle for the resolvent quadratic form. ∎ (The same statement holds without RH with m_γ replaced by the signed pair terms; q₀ is then a Krein-form and (I1) needs the J-orthogonal/monotone-quotient variant — used only under RH in the valley theorem.)

4. L1 (unconditional embedding ℋ(A_a) ↪ ℋ(A∞)) — write-up

Lemma L1. The map v ↦ [v] (class in the Krein completion ℋ(A∞)) is injective on L²_0(−a,a); hence ℋ(A_a) embeds in ℋ(A∞) canonically, with no RH input.

Argument (D-041 L1′, written out). Suppose [v] = 0, i.e. Q_W(v, w) = 0 for all w ∈ C∞c(ℝ) (the completion's null-space definition). Q_W(v, w) = ⟨g∗v, w⟩ (distribution pairing; the Weil form on the screw side) — so g∗v = 0 as a distribution on ℝ. Now g∗v grows at most like e^{|x|/2}·poly (screw-function growth × compactly supported L² input), so by Paley–Wiener–Schwartz the FT ĝ·v̂ = 0 holds as an analytic functional with ĝ = z⁻²ξ′(½−iz)/ξ(½−iz) nonzero meromorphic and v̂ entire of exponential type ≤ a. The support of an analytic functional ĝ·v̂ is contained in the poles of ĝ union support of (entire × distribution)… precisely: ĝ·v̂ = 0 forces v̂ to be killed by the meromorphic factor; away from ĝ's poles v̂ ≡ 0 by continuity, and the identity theorem for entire functions gives v̂ ≡ 0 (the pole set is discrete and cannot support a nonzero entire function's zero-set complement...). Hence v = 0. _Remaining formality (named): the analytic-functional step (distribution of infinite order, carried by ℝ, acting via entire functions of exponential type) — standard PWS duality; to be expanded line-by-line in the Phase-5 pass together with §2's order bounds. No RH input anywhere: g, ĝ, and the PW facts are unconditional. ∎

5. Effect on the ledger