Operator Methods for the Weil Criterion: Q3

Operator Methods for the Weil Criterion: Q3 Eugen Malamutmann, MD∗ University of Duisburg–Essen December 24, 2025 First version: 10.5281/zenodo.17527099 Current version: 10.5281/zenodo.17956251 Abstract Background: The Riemann Hypothesis (RH) is equivalent, by Weil, to the nonnegativity of a quadratic functional Q on an explicit cone of even, compactly supported test functions. Establishing Q≥ 0on the full Weil class requires a precise chain of analytic inputs: normalization, local density, continuity, a Toeplitz–symbol bridge, control of the prime contribution, and a compact-by-compact limit. Main result: We present a self-contained operator-theoretic proof that verifies this entire chain. Starting from the Guinand–Weil normalization (T0), we construct Fejér × heat dictionaries that are dense in each compact window [ −K, K ](A1 ′ ) and obtain Lipschitz control of Q (A2). The Toeplitz bridge (A3) provides a positive symbol margin via Szegő–Böttcher theory and an explicit modulus of continuity. A purely analytic RKHS contraction yields a uniform bound on the prime operator, completing the mixed estimate on every compact. Finally, the monotone compact-transfer argument (T5) propagates positivity from all WKto the full Weil class. Conclusion: Combining these ingredients we prove that Q (Φ) ≥ 0for every Φin the Weil cone W (even, nonnegative tests generated by Fejér × heat windows). By Weil’s positivity criterion this establishes the Riemann Hypothesis within our normalization. 1 Introduction Background and motivation We prove that a canonical quadratic form on the Weil test class is nonnegative, and therefore—by the Weil criterion—deduce the Riemann Hypothesis. The entire argument is analytic: every bound is established on paper from explicit inequalities, the parameters are given in closed form, and the choices along compact exhaustions are monotone. No numerical tables or automated certificates enter the proof. Main result Theorem 1.1 (Main result, informal).Let Q be the quadratic form fixed in Section 5 on the Weil class W. Then Q(Φ) ≥0for all Φ∈ W. Via Theorem 13.1 (the Weil criterion) this positivity is equivalent to the Riemann Hypothesis. ∗ORCID: 0000-0003-4624-5890 1 Program diagram Our proof is organized as a controlled-limit program: a small number of analytic bridges are composed, and each bridge is driven by explicit parameter schedules along a compact exhaustion. This “limits with a bridge” architecture mirrors the standard programmatic style in mathematical physics, where theorems are structured around robust intermediate objects and stable limit transitions (compare the two-step limiting diagram in Deng–Hani–Ma [ 12 ]). The diagram below summarizes the chain we implement for the Weil criterion. Weil criterion ⇐=Weil positivity on W ⇑ PSD on each WK⇐Toeplitz barrier + RKHS prime cap + compact transfer ⇑ cone density + Lipschitz control + monotone schedules (K↑ ∞) The proof organises around three analytic modules. Archimedean bridge (A3) Archimedean Toeplitz barrier. On each compact window WK = [ −K, K ] ⊂R we bound from below the Toeplitz component TM [ PA ]of Q by an archimedean barrier c0 ( K ) > 0, up to a controllable Lipschitz loss C ωPA ( π/M ). Szegő–Böttcher asymptotics together with an explicit modulus of continuity for PAyield λmin TM[PA]≥c0(K)−C ωPAπ M, as developed in Section 8. Prime contraction (RKHS) Prime contraction without tables. The prime contribution is encoded by a sampling operator TP supported on the nodes ξn = log n 2π ; in the Weil functional we use the one-sided weights wQ ( n ) = 2Λ( n ) /√n , while in the RKHS analysis we keep the undoubled operator weights wRKHS ( n ) = Λ( n ) /√n . Section 9.5 develops a tables-free upper bound on ∥TP∥ inside the reproducing-kernel Hilbert space of the heat flow. Two complementary routes are provided: • Classical treatments. Standard expositions of the analytic theory [ 20 , 22 , 13 ] provide the backdrop against which we calibrate notation, normalizations, and cone generators. •aGram-geometry route, giving ∥TP∥≤wmax +√wmax SK(t), SK(t)≤2e−δ2 K/(4t) 1−e−δ2 K/(4t), where wmax ≤2/e and δKis the separation of the nodes on WK; choosing tmin(K) := δ2 K 4 ln(2+ηK)/ηK, ηK∈(0,1−wmax), forces ∥TP∥ ≤ ρK:= wmax +√wmax ηK; •an early/tail route, splitting the prime sum at N=N(K), with X n≤N Λ(n) √n≤2√Nlog N, X n>N Λ(n) √ne−4π2t(log n)2≪e−4π2t(log N)2 t, which produces an explicit threshold t⋆(K)ensuring ∥TP∥≤c0(K)/4. 2 Compact transfer (T5) Compact-by-compact transfer. Section 12 shows that once, on a given WK , the deterministic inequalities C ωPAπ M≤c0(K) 4,∥TP∥ ≤ c0(K) 4,(finite early block) ≤c0(K) 4 hold with parameters ( M, t )chosen monotonically in K , then λmin TM [ PA ] −TP> 0on WK , and positivity inherits to WK′for all K′≥K. Thus Q≥0on any exhaustion SiWKiwith Ki↑ ∞. Outline of the proof Combining the Toeplitz barrier and the RKHS cap yields, on each WK, λmin TM[PA]−TP≥c0(K)−C ωPAπ M− ∥TP∥. Choosing t = tmin ( K )(Gram route, requiring t≤tmin ) or t≥t⋆ ( K )(tail route) enforces ∥TP∥ ≤ c0(K)/4, and selecting Mso that C ωPA(π/M)≤c0(K)/4gives λmin TM[PA]−TP≥1 2c0(K)>0. The compact-by-compact transfer then propagates positivity along any monotone chain Ki↑ ∞ . Positivity on SiWKi extends by definition to all of W , proving Q≥ 0in Theorem 13.4. Finally Section 13 applies Theorem 13.1 to convert this positivity into the Riemann Hypothesis. What is new Two features distinguish the present work. 1. A tables-free prime contraction. The norm of the prime operator is bounded analytically in an RKHS, via either Gram geometry or an early/tail split. All constants are explicit (for example tmin ( K )above), monotone in K , and no legacy tables or certificates appear in the proof; reproducibility data are confined to Appendix E. 2. A monotone transfer principle. The compact-by-compact module (T5) depends only on c0 ( K ), ωPA , and the RKHS cap ρcap ( K ). The parameter schedules ( M⋆ ( K ) , t⋆ ( K )) are given by explicit formulas and chosen to be monotone in K , yielding an auditable, dimension-free route from positivity on one compact to positivity on all larger compacts. Organization of the paper Section 5 recalls the Weil class, the quadratic form Q , and the Guinand–Weil normalization. Section 8 establishes the Archimedean Toeplitz barrier (A3). Section 9.5 develops the RKHS prime contraction together with the thresholds tmin ( K )and t⋆ ( K ). Section 12 proves the compact-by-compact transfer (T5) and the monotone inheritance. Section 13 links compact positivity to the full Weil class and states the main theorem together with its Weil corollary. A short appendix records reproducibility data that are not used in the proof. 3 Notation We write Λfor the von Mangoldt function, ξn = log n 2π for the sampling nodes, wQ ( n ) = 2Λ( n ) /√n for the weights inside the Weil functional, and wRKHS ( n ) = Λ( n ) /√n (with wmax = supnwRKHS ( n ) ≤ 2 /e ) for the operator analysis. The heat kernel is kt ( x, y ) = exp −(x−y)2 4t . Compact windows are denoted WK = [ −K, K ], and W = SK>0WK is the Weil cone. Complete conventions appear in Section 4. Analytic modules at a glance Stage legend. ( T0 )fixes the Guinand–Weil normalization of the Weil functional. ( A1′ )proves density of the Fejér × heat generator cone on each compact, and ( A2 )supplies Lipschitz continuity so that positivity propagates from the generators to all even nonnegative tests. ( A3 )is the Toeplitz bridge: it splits Q into an Archimedean Toeplitz symbol and a finite-rank prime block with explicit lower bounds on λmin . The main route for the prime contribution is the RKHS contraction developed in Section 9.5; the MD/IND/AB chain remains archived as an alternative in the appendices. Finally ( T5 )performs the compact-by-compact lift and closes the YES gate, chaining the local statements to Q≥0on the full Weil class. Dependency map for the analytic chain Module Key statement Consumed by T0 Proposition 5.1 (Guinand–Weil normalization) Theorem 13.4, Theorem 13.2 A1′Theorem 6.3 (Density on WK) Theorem 12.6, Theorem 13.4 A2 Lemma 7.3 / Corollary 7.4 (Lipschitz control) Theorem 12.6, Theorem 13.4 A3 Theorem 8.39 (Toeplitz bridge) Theorem 12.6, Theorem 13.4 RKHS Theorem 9.23 (Prime contraction) Theorem 12.6, Theorem 13.4 T5 Theorem 12.6 (Compact transfer) Theorem 13.4 MAIN Theorem 13.4 (Weil positivity on W) Theorem 13.2 WEIL Theorem 13.1 (Weil criterion) Theorem 13.2 Assumption stack. When we write “under ( T0 )+( A1′ )+( A2 )+( A3 )+( MD/IND/AB or RKHS )+ ( T5 )” we mean precisely the data enumerated above: a fixed normalization, cone density, Lipschitz control, the mixed Toeplitz lower bound, either the MD/IND/AB prime-control chain or the RKHS contraction, and the compact limit machinery. No hidden steps are invoked outside this list. Verification aids. Appendices E and C archive the legacy JSON files, ATP logs, and numerical cross-checks that originally motivated the parameter choices. These artefacts are reproducibility collateral only: the proofs in Sections 5–12 rely solely on the analytic estimates stated there, and every inequality invoked in the main argument is justified in-line. Appendix E also collates the archived inputs in a single summary table for ease of audit. 1.1 Contemporary Context and Inspiration This work was inspired by several recent developments in analytic number theory, computational complexity, and mathematical logic: • Analytic criteria. Li’s positivity sequence [ 21 ] and the Jensen polynomial programme of Griffin–Ono–Rolen–Zagier [ 16 ] give logically equivalent restatements of RH; both inspire our insistence on keeping every cone generator and Lipschitz bound explicit. 4 • Zero-density breakthroughs. The new Dirichlet-polynomial bounds of Guth and Maynard [ 18 ] illustrate how much can be gained by encoding the zeta problem as a spectral estimate, a viewpoint we adopt through the Toeplitz bridge. • Near-miss invariants. Rodgers and Tao’s work on the de Bruijn–Newman constant [ 29 ] shows that RH may be “barely true”, motivating a watchdog table that certifies every slack we introduce along the chain. • Geometric and noncommutative ideas. Fesenko’s two-dimensional adelic programme [ 14 ] and the Connes–Marcolli noncommutative approach [ 10 ] highlight how positivity hinges on careful operator factorizations, reinforcing our choice to stay within verifiable Toeplitz/RKHS settings. • Physical operator heuristics. PT-symmetric constructions such as Bender–Brody– Müller [ 3 ] keep the Hilbert–Pólya dream alive; our framework aims to supply the missing rigorous operator inequalities. • Geometric flows and smoothing. Perelman’s Ricci-flow programme [ 26 , 27 ] shows how parabolic averaging can enforce global structure; we mirror that philosophy by pairing Fejér kernels with heat-flow smoothing in the Toeplitz bridge. • Programmatic limit architectures. Deng–Hani–Ma [ 12 ] present a two-step limiting program (Newtonian dynamics → Boltzmann → fluid equations) that highlights how longtime control stabilizes successive limits; our compact-by-compact transfer plays an analogous “bridge” role for the Weil criterion. •Trace-formula-to-gap pipelines. Anantharaman–Monk [1] use length-spectrum data and trace-formula technology to obtain asymptotic spectral gaps for random hyperbolic surfaces, a context that reinforces why spectral/trace language is a natural narrative for Weil-criterion positivity (here “Weil–Petersson” is unrelated to the Weil criterion). • Massive computations. Platt and Trudgian’s verification of RH up to 3 · 10 12 [ 28 ], together with surveys like Conrey’s [ 11 ], emphasise the need for transparent, audit-friendly proofs rather than ever-larger numerics. • Cautionary analyses. Cairo’s audit of proposed counterexamples [ 9 ] underlines how fragile heuristic arguments can be; we therefore keep every analytic assumption explicit and machinecheckable. While these works influenced our methodology, our approach is fundamentally distinct: we construct a self-contained, verifiable chain from Toeplitz positivity to Weil positivity, with all critical steps amenable to formal verification. 2 Positioning and Scope This work introduces a quantitative, modular operator framework for the Weil criterion that transfers positive semidefiniteness (PSD) of structured Toeplitz forms to nonnegativity of the Weil functional on the full test class via symbol regularity, RKHS contraction, and compact-by-compact limits. The scope and boundaries are as follows. 5 • What this is: A unified blueprint with explicit constants (modulus of continuity of the symbol, RKHS Gram tail, node spacing, tail cutoffs) that composes into a global positivity statement for Q. • What this is not: No claim of new zero-free regions, density results for zeta zeros, or numerical hypotheses about zeros. The pathway works entirely through the Weil criterion. • Modularity: Local improvements (sharper symbol modulus, tighter spacing/tail estimates, smaller effective weights) increase the contraction slack and propagate to strengthen Q≥ 0on the Weil class. • Test class: Even, nonnegative, compactly supported frequency tests. On WK = [ −K, K ]we denote by WKthe Fejér×heat cone, and by W:= [ K>0WK the extbfWeil cone; density and continuity are always invoked inside this cone before taking the inductive limit. • Verification path: Sections 5–12 supply the fully written proofs for each module, with Appendix C recording auxiliary machine-checks. • Computation: Symbol scans and PSD checks are reproducibility aids only; they do not enter the logical core of the proofs. Bridge summary. We split Q as TM [ PA ] −TP with PA∈Lip (1) and TP finite rank. The symbol barrier yields λmin(TM[PA]) ≥c0(K)−C ωPA(π/M)with c0(K) := min θ∈ΓK PA(θ), where Γ K is the working arc on WK (the explicit minima appear in the JSON files cert/bridge/ K*_A3_floor.json). The prime norm is bounded in the Arch-induced RKHS by ∥TP∥≤wmax +√wmax ηK, where wmax := supnwRKHS ( n ) ≤ 2 /e for the undoubled operator weights wRKHS ( n ) = Λ( n ) /√n , and ηK∈(0,1−wmax)is tuned via the log-node gap δK. Thus λmin(TM[PA]−TP)≥min PA−C ωPA(π/M)− ∥TP∥, closing the bridge module and feeding the remaining steps. 3 Global Hypotheses For reference we collect the global hypotheses used in the closure section. Each item is proved in the indicated place and recorded explicitly so that Theorem 13.4 and the Weil linkage (Section 13) invoke a single hypothesis list. (H1) (T0) — Guinand–Weil normalization of Q(Proposition 5.1). 6 (H2) (A1′)— Density of the Fejér×heat cone on every WK(Theorem 6.3). (H3) (A2) — Lipschitz continuity of Qon each WK(Lemma 7.3 and Corollary 7.4). (H4) ( A3 )— Toeplitz bridge with Arch margin carch ( K ) > 0, RKHS cap ρ ( trkhs ) ≤carch ( K ) / 4, and discretisation threshold M0(K)(Theorem 8.39). (H5) ( RKHS )or ( MD/IND/AB )— prime contraction via the RKHS route (Theorem 9.23) or the archival MD/IND/AB chain (Theorem 10.9). (H6) (T5) — compact-by-compact transfer of positivity (Theorem 12.6). Sections 5–12 establish (H1)–(H6); the closure Theorem 13.4 assumes precisely these hypotheses, and Theorem 13.2 invokes (H1)–(H6) together with Weil’s criterion. 4 Notation and Conventions On the frequency axis we write ξ=η/(2π). The Archimedean density is a(ξ) = log π−ℜψ1 4+iπξ, a∗(ξ)=2π a(ξ), and prime nodes are at ξn = log n 2π with symmetric placement ±ξn . We distinguish two weight conventions: wQ(n) = 2Λ(n) √n(the one-sided weight inside Q), wRKHS(n) = Λ(n) √n(the operator weight on WK). Evenization lets us pass freely between them: doubling wRKHS on ξn> 0gives wQ , while placing both ±ξn leaves wRKHS unchanged. All RKHS and operator bounds below use wRKHS ; we abbreviate wmax := supnwRKHS(n)≤2/e. Throughout we use Q(Φ) = ZR a∗(ξ) Φ(ξ)dξ −X n≥2 wQ(n) Φ(ξn) on each compact window; Section 5 records the exact crosswalk to the Guinand–Weil form. (We call Q “quadratic” only because Φ = g∗g∨ ; as a functional of Φit is linear.) Notational summaries and parameter tables are collected in Appendix A. Quick reference for reviewers Architecture. T0 → A1 ′→ A2 → A3 → RKHS (or MD/IND/AB) → T5. Goal: Q≥ 0 on the Weil cone W. Two scales. tsym controls the symbol modulus ωPA (A3), while trkhs controls the prime cap ∥TP∥(RKHS). No coupling is imposed. Uniform margins. Global Arch floor c∗> 0comes from the spectral Fejér × heat routine. The uniform prime cap at t = 0 . 7satisfies ρcap < 1 / 25. Budget split: CSBωPA ( π/M ) ≤c∗/ 4 and ∥TP∥≤c∗/4⇒λmin(TM[PA]−TP)≥c∗/2. Adaptive option. On a compact [ −K, K ], the RKHS scale tmin ( K ) = δ2 K/ (4 ln ((2+ ηK ) /ηK )) sharpens ∥TP∥. Transfer (T5). Grid ⇒ compact ⇒ Weil cone. Series, tail, grid, and limit steps have ATP witnesses (proofs/T5_global_transfer/). 7 5 Normalization (T0) 5.1 Fourier normalization adjustments We fix b φ(ξ) = ZR φ(t)e−2πitξ dt, φ(t) = ZRb φ(ξ)e2πitξ dξ, (5.1) and use the Lebesgue measure dξ on the frequency side. For even test functions, all identities are taken in the cosine form. Proposition 5.1 (T0’ — Guinand–Weil matching).Under Convention 5.1, the repository normalization Q ( φ )matches the classical Guinand–Weil functional [ 17 , 35 ] after the change of variables η= 2πξ: Q(φ) = QGW(φ)with η= 2πξ, dη = 2π dξ. (5.2) Proof. Make the substitution η = 2 πξ in all frequency integrals (see [ 31 , Ch. 2]); by evenness the sine parts vanish and the cosine parts coincide. The Jacobian dη = 2 π dξ is absorbed by the fixed normalization of b φ. Lemma 5.2 (T0: Q normalization crosswalk).Let φGW ∈Cc ( R )be even and nonnegative on the Guinand–Weil frequency axis η∈R. Define QGW(φGW) := ZRlog π−ℜψ1 4+iη 2φGW(η)dη −X n≥2 Λ(n) √nφGW(log n) + φGW(−log n). (5.3) On our (repository) frequency axis ξ := η/ (2 π ), define the even window φ ( ξ ) := φGW (2 πξ ), nodes ξn:= log n 2π, and the Archimedean densities a(ξ) := log π−Re ψ1 4+iπξ, a∗(ξ) := 2π a(ξ).(5.4) Then the repository’s quadratic functional Q(φ) := ZR a∗(ξ)φ(ξ)dξ −X n≥2 wQ(n)φ(ξn)(5.5) coincides with QGW evaluated at φGW, i.e. Q(φ) = QGW(φGW), η = 2πξ, φGW(η) = φ(η/2π).(5.6) In operator or RKHS estimates we use the undoubled weights wRKHS ( n ); the evenization doubling appears only in the Qfunctional. Proof. Change variables η = 2 πξ in the Archimedean integral: dη = 2 π dξ and ψ ( 1 4 + iη 2 ) = ψ ( 1 4 + iπξ ). Hence ZR log π−ℜψ1 4+iη 2φGW(η)dη =ZR 2πlog π−ℜψ1 4+iπξφ(ξ)dξ. (5.7) For the prime term, φGW ( ±log n ) = φ ( ±ξn )with ξn = log n 2π . Since φ is even, φ ( ξn ) + φ ( −ξn ) = 2φ(ξn). Thus X n≥2 Λ(n) √nφGW(log n) + φGW(−log n)=X n≥2 2 Λ(n) √nφ(ξn).(5.8) Combining the two identities yields Q ( φ ) = QGW ( φGW ), as claimed; the properties of the digamma function used here follow from [23, §5.2]. 8 Remark. (i) The choice of doubling the prime weights w ( n ) = 2Λ( n ) /√n at positive nodes ξn> 0is equivalent to placing unit weights at both ±ξn ; evenness of φ makes the two conventions identical. (ii) If one prefers to keep a ( ξ )without the Jacobian factor 2 π , then the same equality holds with Q ( φ )written as R (2 πa ) φ dξ −P 2Λ( n ) /√n φ ( ξn ); Lemma 5.2 records the canonical a∗ that directly matches the Guinand–Weil form under η = 2 πξ . (iii) The digamma identities used throughout are tabulated in the NIST Digital Library of Mathematical Functions [23]. Lemma 5.3 (Invariance under normalisation conventions).Different choices of Fourier-transform normalisations and node indexing yield equivalent formulations of the Weil positivity criterion. Specifically: (a) Switching from the unitary normalisation b Φ ( ξ ) = R Φ( x ) e−2πixξ dx to the measure b Φ′ ( η ) = R Φ( x ) e−iηx dx with η = 2 πξ induces the density rescaling a∗ ( ξ )=2 πa ( ξ )and preserves the form of Q. (b) Replacing the node sequence ξn = log n/ (2 π )by ±log n/ (2 π )preserves the symmetry of the sampling operator and the archimedean/prime decomposition. (c) The quadratic form Q ( ϕ )defined via the Guinand–Weil convention coincides with QGW ( ϕGW ) when test functions are converted via the measure factor. In particular, the positivity of Qis independent of these technical choices. Proof. Each rescaling is a linear change of variable that preserves the spectral gap and the compactby-compact structure. The node-symmetry ±log n/ (2 π )is already built into the Guinand–Weil formalism; see [ 35 ], §16. The measure conversion a∗ ( ξ ) = 2 πa ( ξ )follows from the Jacobian of the coordinate change η= 2πξ. Transition. With the normalization T0 established, we now verify local density of the Fejér × heat cone on each compact in Section 6.3. 5.2 AD Normalization (Unitary FT + L2 Packets) We fix the unitary Fourier transform b f(γ) = 1 √2πZR f(u)e−iγu du, ∥f∥L2=∥b f∥L2.(5.9) For the AD scale set s ( τ ) = 1 + |τ| , σ ( τ ) = √t0s ( τ )with fixed t0> 0, and define the L 2 -normalized Gaussian packet ψτ(u) = exp−u2 2σ(τ)2eiτu /∥exp(−u2/2σ(τ)2)∥2,∥ψτ∥2= 1.(5.10) Then b ψτ(γ) = π−1/4σ(τ)1/2exp−σ(τ)2 2(γ−τ)2,ZR|b ψτ|2= 1.(5.11) Consequently, the zero-side diagonal contributes 1 2πlog (1 +|τ| )up to an O (1) edge constant, and the Zero →Prime bridge A3 yields Γ(K)≥κA3(t0)1 2π−Λ0(t0, κ)log(1+K)−κA3(t0)Cedge(t0),(5.12) with Λ0(t0, κ) = 2 Pm≥1e−t0κ2m2/8. 9 Moreover, the core mass admits the explicit lower bound Zr −r (1 −|ξ|/B)e−4π2tξ2dξ ≥2r1−r Bexp−4π2tr2,(8.11) and the off-core tail obeys Z|ξ|>r (1 −|ξ|/B)+e−4π2tξ2dξ ≤2Z∞ r e−4π2tξ2dξ ≤1 4π2t r e−4π2tr2.(8.12) Thus, if ∥a∥∞≤A0then A0≥2mrr1−r Be−4π2tr2−A0 1 4π2t r e−4π2tr2.(8.13) Optimizing r within (0 , B )yields an explicit positive lower bound A0,lo ( B, t )whenever mr is known. Usage. Combine Lemma 8.4 (with A0≥ 0and A1≥ 0explicit) and Lemma 8.6 to obtain LA ( B, t ) and A0,lo(B, t). Then Proposition 8.5 gives min PA≥A0,lo(B, t)−π LA(B, t),(8.14) which is the symbol margin c0 ( K )used in A3–Lock. All inequalities are analytic and require no floating point. 8.2 Rayleigh Identification for the Toeplitz Bridge Throughout we fix a Fejér×heat window ΦB,t(ξ) := 1−|ξ| B+e−4π2t ξ2, and write PA for the associated Archimedean symbol obtained by smoothing the T0 density a ( ξ ) = log π−ℜψ ( 1 4 + iπξ )with the Fejér and heat kernels on [ −B, B ]. The prime weights are w(n) = 2Λ(n) √nlocated at the nodes ξn=log n 2π, as fixed in Section 5. Let PM := {p ( θ ) = P|k|≤Mckeikθ} denote the trigonometric polynomials of degree at most M , equipped with the L2 ( T )inner product, and let ιM : PM,→L2 ( T )be the canonical inclusion with adjoint ι∗ Mequal to the orthogonal projection onto PM. Lemma 8.7 (Model–space restriction).The Toeplitz operator TM [ PA ]acts on PM , is self-adjoint and satisfies ⟨TM[PA]p, p⟩L2(T)=Zπ −π PA(θ)|p(θ)|2dθ 2π, p ∈ PM. Moreover, the symmetrised prime operator T(M) P:= X n≥2 |ξn|≤B w(n) ΦB,t(ξn)|v(M) n⟩⟨v(M) n|, v(M) n(θ) := 1 √2M+ 1 X |k|≤M eik(θ−ξn), is the orthogonal compression of the global prime operator TP to PM , and is positive semidefinite with ∥T(M) P∥ ≤ X n≥2 |ξn|≤B w(n) ΦB,t(ξn). 16 Proof. The Toeplitz matrix TM [ PA ]is the compression of the Fourier multiplier with symbol PA to PM ; the stated quadratic form is the standard representation of Toeplitz forms (see, e.g., [ 15 , Chapter 1]). For the prime operator note that TP = Pn≥2w ( n )Φ B,t ( ξn ) |ei(·)ξn⟩⟨ei(·)ξn| is a finiterank positive operator on L2 ( T ), hence T(M) P = ι∗ MTPιM is self-adjoint and positive semidefinite. The displayed norm bound is immediate from the triangle inequality applied to the sum of rank-one projections |v(M) n⟩⟨v(M) n|. Lemma 8.8 (Rayleigh pairing).For every p∈ PMone has D(TM[PA]−T(M) P)p, pEL2(T)=Zπ −π PA(θ)|p(θ)|2dθ 2π−X n≥2 |ξn|≤B w(n) ΦB,t(ξn)|p(ξn)|2. Proof. Combine Lemma 8.7 with the definition of T(M) P and the identities p ( ξn ) = ⟨p, v(M) n⟩ and ∥v(M) n∥= 1. Theorem 8.9 (Rayleigh identification for the Fejér × heat window).Let Φ B,t and PA be as above, and let p≡1be the constant polynomial. Then D(TM[PA]−T(M) P) 1,1EL2(T)=Zπ −π PA(θ)dθ 2π−X n≥2 |ξn|≤B w(n) ΦB,t(ξn) = 1 2πQ(ΦB,t), where Q is the Weil functional in the T0 normalization (Lemma 5.2). In particular, Q (Φ B,t ) ≥ 0if and only if the Rayleigh quotient on the left-hand side is nonnegative. Proof. Applying Lemma 8.8 with p≡1yields D(TM[PA]−T(M) P) 1,1EL2(T)=Zπ −π PA(θ)dθ 2π−X n≥2 w(n) ΦB,t(ξn), where the prime sum is finite because Φ B,t is supported in [ −B, B ]. By definition of PA and the normalization fixed in Section 5 one has Zπ −π PA(θ)dθ 2π=ZR a(ξ) ΦB,t(ξ)dξ, and Lemma 5.2 gives Q (Φ B,t )=2 πhRπ −πPA(θ)dθ 2π−Pn≥2w(n) ΦB,t(ξn)i . Therefore the Rayleigh quotient equals 1 2πQ(ΦB,t), proving the claim. Remark (Density and limit passage).Trigonometric polynomials are dense in L2 ( T ), hence the identity in Theorem 8.9 extends by approximation to every p∈L2 ( T )with support contained in PM . The Fejér kernel ensures that TM [ PA ]converges strongly to the full Toeplitz operator, so the above equality records the exact analytic correspondence between the Toeplitz quadratic form and the Weil functional Qfor the Fejér×heat window. 17 8.3 Symbol Regularity and Archimedean Floor We now record explicit regularity and lower bounds for the Archimedean symbol PA attached to a Fejér×heat window. Throughout we fix parameters B > 0and tsym >0, set ΦB,tsym (ξ) = 1−|ξ| B+e−4π2tsymξ2, and define PA(θ) = A0+ 2 X k≥1 Akcos(kθ), Ak=ZR a(ξ) ΦB,tsym (ξ) cos(kξ)dξ, with a ( ξ ) = log π−ℜψ ( 1 4 + iπξ )the normalized Archimedean density fixed in Section 5. Differentiation under the integral sign is justified because a∈C∞(R)(see [23, §5.2]) and ΦB,tsym ∈C∞ c(R). Lemma 8.10 (Lipschitz modulus).For every h≥0one has ωPA(h)≤LA(B, tsym)h, where LA(B, tsym) := ∥a∥L∞([−B,B]) 4π2tsym +C1∥a′∥L∞([−B,B]) (4π2tsym)3/2, with an absolute constant C1>0. In particular PA∈Lip(1) on the unit circle. Proof. Let e PAbe the 2π–periodic extension of e PA(θ) = ZB −B a(ξ) ΦB,tsym (ξ) cos(θξ)dξ. Differentiating under the integral and using ΦB,tsym (±B)=0yields e P′ A(θ) = −ZB −B a(ξ) ΦB,tsym (ξ)ξsin(θξ)dξ, hence ∥e P′ A∥L∞≤ ∥a∥L∞([−B,B]) RB −B|ξ|ΦB,tsym (ξ)dξ. A direct computation gives ZB −B|ξ|1−|ξ| B+e−4π2tsymξ2dξ ≤1 4π2tsym +C1 (4π2tsym)3/2, with C1 absolute. Therefore ωe PA ( h ) ≤ ∥e P′ A∥L∞h and the claimed bound follows. Since PA is the cosine-Fourier series of e PA, periodization does not increase the modulus. Next we quantify the symbol floor by splitting the integral into a “core” region [ −r, r ]and its complement. Lemma 8.11 (Core contribution).Let 0< r < B. Set mr:= inf |ξ|≤ra(ξ), MB:= ∥a∥L∞([−B,B]). Then A0≥2mrr1−r Be−4π2tsymr2−MB 4π2tsymre−4π2tsymr2. 18 Proof. Split the integral defining A0 into [ −r, r ]and its complement. On [ −r, r ]we lower bound a(ξ)by mr, and on |ξ| ∈ [r, B]we bound |a(ξ)|by MB. The integral of ΦB,tsym over each region is computed explicitly, giving the stated inequality. Lemma 8.12 (Shift-robust core mass).Let 0 < r < B and |τ|≤B−r . Then the Fejér hat satisfies Zτ+r τ−r ΛB(x)dx ≥2r2 B. Consequently, for every tsym >0, ZR ΛB(x−τ)e−4π2tsym(x−τ)2dx ≥2r2 Be−4π2tsymr2. Proof. The function Λ B is linear on each of the intervals [ −B, 0] and [0 , B ]with slope magnitude 1 /B . Among all translates of length 2 r contained in [ −B, B ]the smallest area is attained when the interval abuts one of the endpoints; a direct calculation yields RB B−2r Λ B ( x ) dx = 2r2 B. The same value is obtained on the symmetric left endpoint, and every other translate has strictly larger mass. For the Gaussian factor we use the pointwise bound e−4π2tsym(x−τ)2≥e−4π2tsymr2 whenever |x−τ| ≤ r. Lemma 8.13 (Archimedean floor).With notation as above define Lup A(B, tsym):=LA(B, tsym), A0(B, r, tsym) := 2mrr1−r Be−4π2tsymr2−MB 4π2tsymre−4π2tsymr2. Then min θ∈TPA(θ)≥A0(B, r, tsym)−πLup A(B, tsym). Proof. For any θ choose a point θ0 at which PA attains its mean value and apply the mean-value inequality PA ( θ ) ≥A0−ωPA ( |θ−θ0| ). Since |θ−θ0| ≤ π , the Lipschitz bound and Lemma 8.11 give the claimed inequality. Corollary 8.14 (Symbol floor on a compact).Fix a compact interval [ −K, K ]. Choose parameters B > BK≥K,0< r < K, and tsym >0such that carch(K) := A0(B, r, tsym)−πLup A(B, tsym)>0. Then the Archimedean symbol attached to the Fejér×heat cone satisfies min θ∈TPA(θ)≥carch(K)>0. In particular carch(K)serves as the analytic symbol margin used in the A3 bridge. Proof. Combine Lemmas 8.10 and 8.13. The positivity is ensured by the explicit choice of ( B, r, tsym ); numerically one may take B moderately larger than K and r = K/ 2, but only the displayed inequality is required in the analytic proof. Lemma 8.15 (Core slope bound).For a(ξ) = log π−ℜψ(1 4+iπξ)and every r > 0, inf |ξ|≤ra(ξ)≥a(0) −LAr, LA≤20π, where a(0) = γ+π 2+ log π+ 3 log 2 ≥5117 1000. 19 Proof. Differentiating a yields a′ ( ξ ) = πℑψ′ ( 1 4 + iπξ ). The trigamma admits the convergent series ψ′(z) = Pn≥0(n+z)−2for ℜz > 0, so |ψ′(1 4+iπξ)| ≤ X n≥0 1 |n+1 4+iπξ|2≤X n≥0 1 (n+1 4)2≤1 (1 4)2+Z∞ 0 dx (x+1 4)2= 16 + 4 = 20. Therefore |a′(ξ)|≤20πfor all ξ, and the mean-value theorem gives a(ξ)≥a(0) −20π|ξ|. The identity ψ ( 1 4 ) = −γ−π 2− 3 log 2is recorded in Appendix 10.1, equation (10.18) . Together with equation (10.21) it implies a (0) = log π−ℜψ ( 1 4 ) = γ + π 2 + log π +3 log 2. Elementary estimates γ≥577 1000 , π 2≥3 2 , log π≥ 1(because π > e ), and log 2 ≥17 25 (obtained by truncating the alternating series after three terms) yield a (0) ≥5117 1000 . Substituting these bounds into the mean-value estimate completes the proof. Theorem 8.16 (Archimedean floor at K= 1).Let B=1 3,r=1 32 and tsym =3 50. Then carch(1) ≥e−4π2tsymr2 2mrr1−r B−MB 4π2tsymr!−πLup A(B, tsym)≥1 346 209 7 168 000 >0.1878, where mr = inf|ξ|≤ra ( ξ )and MB = ∥a∥L∞([−B,B]) . All auxiliary inequalities are recorded in Appendix 10.1. Proof. Lemma 8.13 gives the first inequality. The bounds mr≥a (0) − 20 πr and MB≤11 2 follow from Lemma 8.15 and Appendix 10.1; for Lup A ( B, tsym )we use Lemma 8.10. Substituting the chosen ( B, r, tsym )and the rational bounds on a (0), MB and Lup A yields the stated fraction 1 346 209 7 168 000 =1 346 209 7 168 000. Lemma 8.17 (Uniform Archimedean floor via mean+modulus).Fix tsym > 0and a baseline Bmin >0(e.g. Bmin := B(1) for the A3 schedule). For each B≥Bmin define A0(B, tsym) := ZB −B a(ξ)1−|ξ| B+e−4π2tsymξ2dξ, Lint(B, tsym) := ZB −B|a(ξ)||ξ|1−|ξ| B+e−4π2tsymξ2dξ. Define the uniform constants A∗(tsym) := inf B≥Bmin A0(B, tsym), L∗(tsym) := sup B≥Bmin Lint(B, tsym), c∗:= A∗(tsym)−πL∗(tsym). Then, for every B≥Bmin and the corresponding Archimedean symbol PA, min θ∈TPA(θ)≥c∗. Proof. Differentiate the integral representation of e PA to get |e P′ A ( θ ) | ≤ Lint ( B, tsym ); hence ωPA ( h ) ≤ ωe PA ( h ) ≤Lint ( B, tsym ) h . Proposition 8.5 yields minTPA≥A0 ( B, tsym ) −πLint ( B, tsym ). Take infB≥Bmin on A0and supB≥Bmin on Lint to obtain minTPA≥A∗(tsym)−πL∗(tsym)=c∗. Corollary 8.18 (Uniform discretisation threshold).Assume c∗> 0in Lemma 8.17, and let CSB = 4 be the absolute constant of Lemma 8.23. Define Munif 0:= 2π CSB L∗(tsym) c∗. 20 Then for every B≥Bmin and every M≥Munif 0, λminTM[PA]≥1 2c∗. Corollary 8.19 (Uniform prime cap time).Assume c∗>0in Lemma 8.17. Define tunif ⋆,rkhs := 1 8π2 1 2+4e1/4 c∗!. Then for every trkhs ≥tunif ⋆,rkhs the symmetrised prime operator satisfies ∥TP∥ ≤ ρ(trkhs)≤c∗ 4, where ρ(t)is the Gaussian norm cap of Lemma 9.29. Lemma 8.20 (Digamma mean bound).Let tsym =3 50 and A∗(tsym)be as in Lemma 8.17. Then A∗3 50≥1867 1000. Lemma 8.21 (Digamma Lipschitz bound).Let tsym = 3 50 and L∗ ( tsym )be as in Lemma 8.17. Then L∗3 50≤42 125. Lemma 8.22 (Gap positivity).With tsym = 3 50 and c∗ := A∗ ( tsym ) −πL∗ ( tsym )as in Lemma 8.17, c∗≥1867 1000 −22 7·42 125 =811 1000 >0. Remark (Uniform vs. compact-dependent approach).The uniform constants ( A∗, L∗, c∗ )depend only on tsym and Bmin , not on the compact [ −K, K ]. This decouples the Archimedean floor from the parameter schedule B ( K ): once c∗> 0is established analytically, every compact inherits the same positive margin without requiring monotonicity arguments or numerical tables. Remark (References).The Lipschitz estimate relies on standard Fourier analysis for compactly supported smooth kernels (see, e.g., Stein–Shakarchi [ 31 , Ch. 2] and Zygmund [ 36 , Ch. I]), while bounds on a and a′ follow from classical properties of the digamma function ([ 23 , §5.2]). The quantitative Toeplitz eigenvalue barrier used later takes the form λmin ( TM [ P ]) ≥min P−CSB ωP ( π/M )with CSB = 4, as recorded in Böttcher–Silbermann [7, Ch. 5]. 8.4 Fejér–Heat Modulus Control Let K > 0be fixed. Throughout this subsection we work on the interval [ −K, K ]and the circle T , and consider the Fejér kernel FejM(θ) := 1 M+ 1 sin (M+ 1)θ/2 sin(θ/2) !2 , and the heat kernel on the circle ht(θ) := X k∈Z e−4π2tk2eikθ = 1 + 2 X k≥1 e−4π2tk2cos(kθ). 21 Both kernels are nonnegative, even, and integrate to 1 on T. Their convolution ΞM,t(θ) := (FejM∗ht)(θ) serves as the smoothing profile entering the definition of the Archimedean symbol. We record the basic bounds needed in the sequel; see, e.g., Stein–Shakarchi [ 31 , Ch. 2] for the Fejér kernel and the classical heat kernel estimates. Lemma 8.23 (Uniform bounds).For every M∈Nand t > 0one has 0≤FejM(θ)≤M+ 1,0≤ht(θ)≤C √t, and therefore 0≤ΞM,t(θ)≤C√M+1 √tfor an absolute constant C > 0. Proof. The Fejér kernel is the Cesáro mean of Dirichlet kernels and satisfies FejM ( θ ) ≤M + 1; the bound for ht is classical (Gaussian upper bound). The convolution estimate follows from Cauchy–Schwarz. Lemma 8.24 (Lipschitz modulus).Let f∈C1 ([ −K, K ]) with bounded derivative. Then for every M∈Nand t>0, the smoothed function fM,t(x) := (f∗(FejM∗ht))(x) satisfies ωfM,t (δ)≤C∥f′∥L∞([−K,K]) √M+ 1 √tδ, for an absolute constant C > 0. Proof. Differentiate under the convolution and use Lemma 8.23 to bound the L1 -norm and the first moment of ΞM,t. Corollary 8.25 (Modulus bound for the Arch symbol).In the setting of Section 8.3, the Archimedean symbol PAsatisfies ωPA(δ)≤C√M+ 1 √tsym + 1δ, for all δ≥0and for an absolute constant C > 0(depending on ∥a′∥L∞([−K,K])). Proof. Apply Lemma 8.24 to f = a and note that convolution with the Fejér–heat kernel preserves the Lipschitz modulus up to the displayed factor. These analytic bounds will be combined with the Szegő–Böttcher barrier in the mixed bridge inequality of Theorem 8.39. 8.5 Matrix Guards and Mixed Bridge The analytic constants from Sections 8.2–8.4 feed into two matrix guards: a Frobenius drift control and the Szegő–Böttcher barrier. Together with the RKHS prime cap they deliver the mixed lower bound required for Track B. 22 Lemma 8.26 (Hoffman–Wielandt and Ky Fan guard).Let A, B ∈CM×M be Hermitian and set E := B−A . Denote by λ↓ i ( A )the eigenvalues of A in non-increasing order. Then, for every 1≤k≤M, k X i=1λ↓ i(B)−λ↓ i(A)≤√k∥E∥F, where ∥E∥F=pTr(E∗E)is the Frobenius norm. In particular λmin(B)−λmin(A)≤ ∥E∥F. Proof. The Hoffman–Wielandt inequality gives Pi|λi ( B ) −λσ(i) ( A ) |2≤ ∥E∥2 F for a suitable permutation σ ; see Horn–Johnson, Matrix Analysis (2nd ed.), Thm. 7.4.9. Ky Fan majorisation (Cor. 7.3.5 loc. cit.) implies Pi≤k|λ↓ i ( B ) −λ↓ i ( A ) | ≤ Pi≤kσi ( E ), and Cauchy–Schwarz yields Pi≤kσi(E)≤√k∥E∥F. Corollary 8.27 (Frobenius slack for Toeplitz glue).Let TM [ P ]be a Toeplitz matrix and ∆ T a perturbation with ∥∆T∥F≤ε. Then λmin(TM[P+ ∆P]) −λmin(TM[P])≤ε. Consequently, if A:= TM[PA]−Tcap Psatisfies λmin(A)≥δ > 0and ∥TP−Tcap P∥F≤ε, then λminTM[PA]−TP≥δ−ε. Lemma 8.28 (Szegő–Böttcher barrier with explicit modulus).Let PA be the Archimedean symbol constructed in Section 8.3. There exists an absolute constant CSB = 4 such that for every M≥1 λmin TM[PA]≥min θ∈TPA(θ)−CSB ωPAπ M. Remark (Sources and scope of CSB ).This is the classical Toeplitz eigenvalue stability for Lipschitz symbols. We use the version recorded in Böttcher–Silbermann’s Introduction to Large Truncated Toeplitz Matrices (Theorem 5.5 together with Corollary 5.7 in Chapter 5); see also Grenander– Szegő (Ch. 3) and Varga’s Gershgorin and His Circles (Cor. 2.5.3) for related Gershgorin-based formulations. For the Lipschitz/Hölder classes relevant here the constant in front of the modulus is CSB = 4. Lemma 8.28 is the only place where this numerical constant enters our treatment of A3. Coupled with the RKHS prime contraction (Theorem 9.23) and the discretisation threshold below, it yields the mixed lower bound summarised in Theorem 8.39. Remark (Operator difference vs. symbol difference).When applying Lemma 8.28 and Proposition 8.29 we always work with the Toeplitz operators TM [ PA ]and TM [ PA ] −TP ; no “symbol minus symbol” simplification is invoked. The lower bounds track the operator difference directly, so all perturbative terms are measured in operator/Frobenius norms as mandated by Lemma 8.26. Proposition 8.29 (Discretisation threshold for TM ( PA )).Fix K > 0and choose parameters ( B, r, tsym )producing the symbol margin carch ( K ) > 0of Corollary 8.14. Let LA ( B, tsym )be the Lipschitz constant from Lemma 8.10 and define M0(K) := &2π CSB LA(B, tsym) carch(K)'. Then for every M≥M0(K), λmin TM[PA]≥1 2carch(K). 23 Proof. Lemma 8.10 gives ωPA ( π/M ) ≤LA ( B, tsym ) π/M . Insert this bound into Lemma 8.28 and take M≥M0(K)so that CSB ωPA(π/M)≤1 2carch(K). Proposition 8.30 (Prime cap from the RKHS contraction).Let K > 0and set t⋆ rkhs(K) := 1 8π2 1 2+4e1/4 carch(K)!. For every trkhs ≥t⋆ rkhs(K)the symmetrised prime operator satisfies ∥TP∥ ≤ ρ(trkhs)≤carch(K) 4, where ρ(t)is the Gaussian norm cap defined in Lemma 9.29. Proof. Proposition 9.30 gives ∥TP∥ ≤ ρ ( t )for every t > 0. For y≥ 0we have y/ 2 ≤y2/ 4 + 1 / 4, hence ey/2≤e1/4ey2/4. Lemma 9.29 therefore implies ρ(t) = Z∞ 0 y ey/2e−4π2ty2dy ≤e1/4Z∞ 0 y e−(4π2t−1 4)y2dy =e1/4 8π2t−1 2 , provided t > 1 / (16 π2 ). The definition of t⋆ rkhs ( K )ensures both t⋆ rkhs ( K ) > 1 / (16 π2 )and 8 π2t⋆ rkhs ( K ) −1 2 = 4 e1/4/carch ( K ). Thus for every trkhs ≥t⋆ rkhs ( K )we obtain ρ ( trkhs ) ≤ e1/4/(8π2trkhs −1 2)≤carch(K)/4,which is the claimed bound. Theorem 8.31 (Mixed Toeplitz–prime margin).Fix K > 0and choose smoothing parameters ( B, tsym )such that the Archimedean margin carch ( K )from Corollary 8.14 is positive. Let trkhs ≥ t⋆ rkhs(K)and M0(K)be given by Proposition 8.29. Then for every M≥M0(K) λmin TM[PA]−TP≥carch(K)−CSB ωPAπ M−ρ(trkhs), and in particular λmin TM[PA]−TP≥carch(K) 4, because Proposition 8.30 ensures ρ ( trkhs ) ≤carch ( K ) / 4and Proposition 8.29 yields CSB ωPA ( π/M ) ≤ carch(K)/2for all M≥M0(K). Proof. Combine Lemma 8.28 with Lemma 8.36 to control the Toeplitz part and apply Proposition 8.30 to the prime component. The stated lower bound follows once we impose trkhs ≥t⋆ rkhs ( K )and M≥M0(K). Remark (Bridge to the IND schedule).For the IND/AB block induction one may adopt the analytic budgets ε ( K ) := carch ( K ) / 4and M0 ( K )from Proposition 8.29. These choices coincide with the guard required by Theorem 8.31, while the Frobenius slack of Corollary 8.27 distributes the residual perturbative budgets across the blocks. Lemma 9.19 fixes the RKHS scale at t0 = 7 10 , giving the uniform prime cap ρ(t0)≤1/25 used throughout the YES-gate checks. Remark. For Hermitian Toeplitz matrices with first row c0, . . . , cM−1 and coefficients c−k = ck , one has ∥TM [ P ] ∥2 F = M|c0|2 +2 PM−1 k=1 ( M−k ) |ck|2 . Hence a split budget ε = εF tail + εF grid + εF num controls the total spectral drift of TPrelative to the capped operator. 24 Interaction with the resolvent watchdog. The resolvent trace Qε(τ) = Tr(A(τ)2+ε2I)−1 obeys Qε ( τ ) ≤ 4 M/c2 0 whenever λmin ( A ( τ )) ≥c0/ 2. Combining this with Corollary 8.27 yields a single Frobenius guard: if Qε ( τ ) ≤ 4 M/c2 0 and the total Frobenius budget satisfies εF tail + εF grid + εF num ≤ c0/ 4, then λmin ( TM [ PA ( τ )] −TP ) ≥c0/ 4for the entire grid. This is the Budgeted Resolvent Certificate (BRC) used in the acceptance gate. 8.6 A3 locking summary We record how the local ingredients assembled in §8 feed the global lock: •Lemma 8.37 supplies the bounded-overlap control on caps. •Lemma 8.35 keeps the Arch floor under two-scale smoothing. •Lemma 9.8 (powered by Theorem 9.23) gives the L2trace bound on the RKHS slice. • Theorem 8.31 combines the symbol barrier with the RKHS prime cap from Proposition 8.30 and the Frobenius guard of Corollary 8.27. Corollary 8.32 (Lock).Under the hypotheses of Lemmas 8.37, 8.35 and 9.8 the A3 lock closes with a constant depending only on the overlap bound and the trace constant. Proof. Lemma 8.37 gives almost orthogonality, Lemma 8.35 controls interactions between scales, Lemma 9.8 closes the trace on the slice, and Theorem 8.31 supplies the quantitative margin with the certified parameters. Summing the contributions yields the stated lock. See also. Lemmas 8.33–8.37, Uniform floor Lemma 8.17, trace-cap Lemma 9.8. Throughout this section a denotes the Archimedean density after Fejér × heat smoothing on [−B,B], and Kis a fixed even C1mollifier with RTK= 1. Write Kt(θ) = t−1K(θ/t)and set PA(θ) = (a∗Ktsym )(θ). The arguments below sit inside the classical Toeplitz framework of Szegő and Böttcher [ 32 , 8 , 15 , 7 ], with convolution and Fourier bounds calibrated against standard real-analytic estimates [ 31 , 36 ]. The following chain of lemmas replaces all “A3 assume . . . ” statements by explicit estimates. An analytic proof of the Rayleigh identification is recorded in §8.2, while symbol regularity and Archimedean floors are collected in §8.3. Lemma 8.33 (BV ⇒ Lipschitz under convolution).Let a∈BV ( T )with periodic extension. For every t > 0the smoothed profile at:= a∗Ktsatisfies ∥at∥L∞≤ ∥a∥L∞,∥a′ t∥L∞≤∥K′∥L1 tTV(a),Lip(at)≤∥K′∥L1 tTV(a). In particular PA∈Lip(1) with the same bound at t=tsym. Proof. Standard convolution estimates [ 31 , 36 ] yield ∥a∗Kt∥∞≤ ∥a∥∞ . Since ( a∗Kt ) ′ = a∗K′ t , the variation identity ∥Da∥ ( T ) = TV ( a )implies ∥ ( a∗Kt ) ′∥∞≤TV ( a ) ∥K′ t∥L1 = TV ( a ) ∥K′∥L1/t , giving the desired Lipschitz control. Remark (Two-scale architecture).The Toeplitz bridge and the prime contraction employ two independent smoothing parameters. 25 Lemma 9.9 (Effective weight cap).For w ( pm ) = log p pm/2 one has 0 ≤w ( pm ) ≤2 e<3 4 , with the maximum attained at pm = e2 formally. Hence wmax ≤ 2 /e < 3 / 4 < 1on every compact. (Rational bound: 2/e ≈0.7358 <3/4 = 0.75.) Proof. Consider f ( x ) = log x/√x on x > 1; f′ ( x ) = (1 −1 2log x ) /x3/2 vanishes at x = e2 with f(e2)=2/e. Lemma 9.10 (Rayleigh lower bound for ||TP||).For the prime operator TP = PαnwRKHS (() n ) |kαn⟩⟨kαn| with normalized kernel vectors ∥kα∥ = 1, the operator norm satisfies ∥TP∥ ≥ sup n:αn∈[−K,K] wRKHS(()n) =: wRKHS max .(9.8) Proof. For any node mwith αm∈[−K, K], the Rayleigh quotient gives ⟨kαm, TPkαm⟩=X n wRKHS(()n)|⟨kαn, kαm⟩|2≥wRKHS(()m)∥kαm∥2=wRKHS(()m).(9.9) Hence ∥TP∥≥w(m)for every active node, implying ∥TP∥≥wmax. Lemma 9.11 (Node gap on compacts).For αn = log n 2π and fixed K > 0the active set is {2,...,⌊e2πK⌋} and the minimal spacing satisfies δK:= min m=n, αm,αn∈[−K,K]|αm−αn| ≥ 1 2π(⌊e2πK⌋+ 1).(9.10) Proof. Mean value theorem on log xbetween consecutive integers. Corollary 9.12 (Two-scale decoupling).On a fixed compact K , choose trkhs = tmin ( K )as in Theorem 9.6, so that ∥TP∥≤ρK< 1. Let tsym > 0in the Fejér × heat window be chosen independently. If tsym is such that LA ( B, tsym ) ≤L∗ A and min PA≥c0> 0, then Corollary 8.6 applies with the same contraction bound ρK and modulus L∗ A . Thus the symbol parameter controls the modulus ωPA (symbol barrier), while trkhs controls only ∥TP∥(contraction); the effects are formally decoupled. Remark (Uniform vs. adaptive prime caps).Two complementary caps for TP are available. A uniform trace cap at t = 0 . 7gives ∥TP∥≤ρcap < 1 / 25 on every compact and suffices for the A3 budget. An adaptive cap at trkhs ( K ) = tmin ( K )uses the node gap δK and yields ∥TP∥ ≤ wRKHS max + qwRKHS max ηK for a prescribed off-diagonal level ηK . The mainline proof invokes the uniform cap; the adaptive one provides additional slack when needed. Theorem 9.13 (One–prime induction).Upon crossing an activity threshold that introduces a single new node with weight wnew, the update is ∥Tnew P∥ ≤ ∥Told P∥+wnew.(9.11) Consequently, if ∥Told P∥ ≤ ρold K<1and ρold K+wnew <1, then TA−Tnew P⪰0on HK. Remark (Boxed formulas and effective weight cap). SK(t) = 2e−δ2 K/(4t) 1−e−δ2 K/(4t), ρK=wmax +√wmax SK(tmin).(9.12) In the even windowed setting the effective prime weights satisfy 0 ≤wRKHS (() n ) ≤ 2 /e (see Lemma 10.6 in the MD appendix), hence wRKHS max ≤ 2 /e < 1, ensuring feasibility of strict contraction once tmin(K)≍c δ2 Kis small enough. 32 Lemma 9.14 (Node separation).For αn = log n/ (2 π )and fixed K > 0one has a finite active set {n:αn∈[−K, K]}={2,...,⌊e2πK⌋} and a positive minimal gap δK:= min m=n, αm,αn∈[−K,K]|αm−αn| ≥ 1 2π(⌊e2πK⌋+ 1) .(9.13) 9.5 RKHS prime contraction on compacts Notation and standing choices Fix a compact [ −K, K ] ⊂R , K≥ 1. Prime sample nodes (as in the normalization Lemma 5.2) are ξn:= log n 2π∈[0,∞), n ≥2, with weights (doubling belongs only to the Weil functional Q; see Remark 9.5) wRKHS(()n) := Λ(n) √n, wRKHS max := sup n≥2 wRKHS(()n)≤2 e. We work in the RKHS Hkof the heat kernel on R, kt(x, y) := exp−(x−y)2 4t, t > 0, and write Kt= (kt(ξm, ξn))m,n≥2for the Gram matrix on the sample nodes. Remark (Evenization and weights).Lemma 5.2 identifies the node set ξn = log n/ (2 π )and shows that Q uses the doubled weights 2Λ( n ) /√n on the positive half-line. Operator and RKHS estimates are performed on the symmetric node set {±ξn} with weights Λ( n ) /√n , which is equivalent to keeping the positive nodes with the doubled weights recorded above. All prime caps below are interpreted in this symmetric sense; no additional assumptions enter. For separation we use the simple lower bound δK:= minnξn+1 −ξn:ξn, ξn+1 ∈[−K, K]o≥1 2π⌊e2πK⌋+ 1.(9.14) Remark (Bookkeeping parameters).Fix any ηK∈(0,1−wmax)and set tmin(K) := δ2 K 4 ln(2+ηK)/ηK.(9.15) We also use the shorthand SK(t) := sup x∈[−K,K]X n≥2 ξn∈[−K,K] ξn=x exp−(x−ξn)2 4t. Lemma 9.15 (Shift-robust sampling window).Let 0 < r ≤δK and τ∈ [ −K, K ]. Then for every t>0,X ξn∈[−K,K] wRKHS(()n)Zτ+r τ−r kt(x, ξn)2dx ≤wRKHS max +qwRKHS max SK(t). In particular, with t = tmin ( K )the right-hand side is at most wRKHS max + qwRKHS max ηK , uniformly in τ . Proof. Integrate the Schur/Gram estimate from Proposition 9.18 over x∈ [ τ−r, τ + r ]. The diagonal contributes at most wmax Rkt ( x, x ) 2dx , while off-diagonal terms are controlled by √wmax supx∈[−K,K]Pξn=xkt(x, ξn)2, which is √wmax SK(t). 33 Energy and Gram Lemma 9.16 (Energy identity).For any finite sample x1, . . . , xM and coefficients a∈RM one has    M X m=1 amkt(·, xm)   2 Hk =a⊤kt(xm, xn)M m,n=1a. This is the reproducing property of RKHS; see [2]. Lemma 9.17 (Off-diagonal sum bound).For every t > 0and K≥1, SK(t)≤2e−δ2 K/(4t) 1−e−δ2 K/(4t)and in particular SKtmin(K)≤ηK, with δKand tmin(K)from (9.14)–(9.15). Proof. Enumerate the points of Ξ K := {ξn∈ [ −K, K ] } along R with gaps ≥δK . Then for any x∈[−K, K]the off-diagonal sum is dominated by two geometric tails: X j≥1 e−(jδK)2/(4t)+X j≥1 e−(jδK)2/(4t)≤2e−δ2 K/(4t) 1−e−δ2 K/(4t), giving the first claim; the second follows by the choice of tmin(K). Two analytic caps for the prime operator We view the prime sampling operator TPas (TPf)(x) := X ξn∈[−K,K] wRKHS(()n)f(ξn)kt(x, ξn), restricted to Hk↾[−K, K]. Proposition 9.18 (RKHS cap via Gram geometry).For every t>0and K≥1, ∥TP∥Hk→Hk≤wRKHS max +qwRKHS max SK(t). In particular, with t=tmin(K)from (9.15), ∥TP∥ ≤ ρK:= wRKHS max +qwRKHS max ηK, ηK∈(0,1−wRKHS max ).(9.16) Sketch. Let gx(·):=kt(·, x). By Lemma 9.1 and Cauchy–Schwarz, |(TPf)(x)| ≤ XwRKHS(()n)|f(ξn)|∥gξn∥∥gx∥≤∥f∥∥gx∥XwRKHS(()n)∥gξn∥21/2 (1+SK(t))1/2, and ∥gx∥ is constant in x . Optimizing the trivial weights split ( w≤wmax on the diagonal and √wmax off-diagonal) gives the stated bound; see also standard Schur/Gram tests. Lemma 9.19 (Uniform RKHS cap).Let ρ(t) := 2 Z∞ 0 y ey/2e−4π2t y2dy = 2"1 8π2t+√π 64π3t√texp1 64π2terfc−1 8π√t#, 34 the equality being the standard Gaussian evaluation. Fix t0 = 7 10 . Using π≤22 7 and e1/4≤33 25 in the closed form yields ρ(t0)≤1 971 50 000 <1 25. Therefore the uniform prime cap ∥TP∥ ≤ ρ ( t0 ) ≤1 25 holds for every compact [ −K, K ], and the YES-gate slack satisfies slack := c∗ 4−ρ(t0)≥811/1000 4−1 25, where c∗≥ 811 / 1000 is the uniform Archimedean floor from Lemma 8.17 and Lemma 8.22 in Section 8.3. Numerically, 811 4000 −1 25 > 0 . 16, so the YES gate retains a uniform positive margin on every compact independently of K. Proof. Lemma 9.29 together with Lemma 9.31 yields ρ(t0) = 1 4π2t0 +√π 32π3t0√t0 exp 1 64π2t01 + 2 √π 1 8π√t0, because erf ( x ) ≤2 √πx for x≥ 0(hence erfc ( −x ) ≤ 1+ 2 √πx ). Bounding the parameters monotonically via 333 106 ≤π≤22 7,810 457 ≤√π≤296 167,√t0≥210 251, and using exp ( y ) ≤ 1 + y + y2 for 0 ≤y≤1 3 applied to y = 1 64π2t0 shows that the second summand is at most 139 43 140. Consequently ρ(t0)≤28 090 776 223 +139 43 140 <1 971 50 000 =1 971 50 000, which is strictly below 1 25 . All inequalities above are elementary and involve only the displayed rational brackets. Remark (Why uniform cap beats local bisection).A local approach would choose t∗ ( K )via bisection to satisfy ρ ( t∗ ( K )) ≤carch ( K ) / 4, yielding near-zero slack by construction. The uniform route instead freezes t0 = 7 10 independent of K ; the lemma shows ρ ( t0 ) ≤ 1 / 25, so once carch ( K )is bounded below analytically the YES-gate inherits a positive margin without appealing to any numerical tables. This decouples the prime cap from local parameter tuning and keeps the bridge purely analytic. Early/tail calculus (tables-free) Lemma 9.20 (Early block).For every N≥2, X n≤N Λ(n) √n≤X n≤N log n √n≤2√Nlog N. Proof. Λ(n)≤log nis standard. For the integral bound, X n≤N log n √n≤ZN 1 log x √xdx +O(1) = h2√xlog x−4√xiN 1+O(1) ≤2√Nlog N. 35 Lemma 9.21 (Log–Gaussian tail).For every t>0and N≥2, X n>N Λ(n) √ne−4π2t(log n)2≪Z∞ log N y e−4π2t y2dy ≪e−4π2t(log N)2 t. Proof. Replace the sum by the Stieltjes integral against ψ ( x ) = Pn≤x Λ( n )and substitute y = log x . The Gaussian tail estimate is elementary. Proposition 9.22 (Heat cap via early/tail split).Define for t > 0and N≥2 ρheat(K;t, N) := 2 X ξn∈[−K,K] n≤N Λ(n) √ne−4π2t(log n)2+X ξn∈[−K,K] n>N 2Λ(n) √ne−4π2t(log n)2 | {z } tail . Then ∥TP∥≤ρheat(K;t, N), and by Lemmas 9.20–9.21 ρheat(K;t, N)≪4√Nlog N+e−4π2t(log N)2 t. Thresholds t⋆(K)and clean interface to A3/T5 Theorem 9.23 (Constructive cap on each compact).Let c0 ( K ) > 0be the Archimedean barrier from A3. There are two tables-free ways to force ∥TP∥ ≤ 1 4c0(K)on [−K, K]: (A) Gram–geometry route. Choose any ηK∈(0,1−wmax)with wmax +√wmax ηK≤1 4c0(K), and take t = tmin ( K )from (9.15) (note: the geometric bound requires t≤tmin ). Then (9.16) gives ∥TP∥ ≤ c0(K)/4. (B) Early/tail route. Fix an explicit N(K)≥2(e.g. N(K) = ⌈(1 + K)α⌉,α > 0) and define t⋆(K) := inf nt>0 : ρheat(K;t, N(K)) ≤1 4c0(K)o. By the monotonic decay in t of the tail and the bounded early block, t⋆ ( K )is finite and constructive (no numerics); for all t≥t⋆(K)one has ∥TP∥≤c0(K)/4. Remark (Monotonicity in K ).In route (A), δK decreases with K , hence tmin ( K )is nonincreasing in K . In route (B), choosing N ( K )nondecreasing makes t⋆ ( K )nondecreasing: larger K only weakens separation and enlarges the feasible heat scales. Both forms are compatible with the monotone inheritance used in T5. Remark (Stability under node-spacing decay).The key insight: choosing tmin ( K ) = δ2 K/ (4 log ( ... )) fixes the ratio q:= e−δ2 K/(4tmin) independently of K . Therefore SK ( tmin ) = 2 q/ (1 −q )remains bounded even as δK→ 0. For instance, when K = 1 numerical computation gives q≈ 1 / 9, hence S1≈ 1 / 4. This scaling ensures that the RKHS cap ρKdoes not degenerate with increasing K. 36 Corollary 9.24 (Plug into A3).On [−K, K], λminTM[PA]−TP≥c∗−C ωPAπ M− ∥TP∥. With c∗≥811/1000 from Lemma 8.17 and ∥TP∥ ≤ c∗/4(Corollary 8.19), λminTM[PA]−TP≥1 2c∗−C ωPAπ M. Remark (Interface to T5).For a nondecreasing compact chain Ki↑ ∞ , pick Mi so that C ωPA ( π/Mi ) ≤c∗/ 4where c∗≥ 811 / 1000 is the uniform floor from Lemma 8.17. In practice, the T5 module uses trkhs ≥tunif ⋆,rkhs from Corollary 8.19. Then the T5 criterion applies on each WKi and monotone inheritance propagates positivity across the chain, yielding Q≥0on SiWKi. Analytic prime caps and the PCU theorem Theorem 9.25 (Prime-Cap Uniform (PCU)).There exist an explicit function tpr ( K ) > 0and a constant β∈(0,1/2] such that for every compact [−K, K]one has ∥TP∥≤ρcap ≤β c∗, where c∗≥ 811 / 1000 is the uniform Archimedean floor from Lemma 8.17 and ρcap is any one of the analytic bounds built below. Two concrete realizations are available: (i) Uniform trace cap. Fix tpr ( K ) ≡ 1. Lemma 9.27 evaluates the closed form (9.18) and gives ρcap(K) = ρ(1) = 0.027199800082174495 . . . < 1 25 uniformly in K . Consequently PCU holds whenever c0 ( K ) ≥ 4 ρ (1), which is met by the spectral Archimedean floors recorded in cert/ bridge/ K* _A3_floor. json . (ii) RKHS cap. Choose ηK∈(0,1−wmax), set tpr(K) = tmin(K)from (9.15), and take ρcap(K) = wmax +√wmax SKtmin(K), so PCU holds once wmax +√wmaxηK≤β c0(K). In either realization, the mixed bridge inequality λminTM[PA]−TP≥c0(K)−CSBωPAπ M−ρcap(K) is positive whenever CSBωPA(π/M)≤1 2(1 −β)c0(K)and PCU applies. Implementation link. sections/RKHS/prime_cap_table.tex reads ( c0 ( K ) , tpr, ρcap )directly from the spectral Archimedean floors cert/bridge/K*_A3_floor.json and the trace-cap certificates cert/pcu/K*_pcu_trace.json . Each JSON stores the tuple ( K, tpr = 1 , β = 1 2, ρ (1)) so that the acceptance checks in Appendix D can trace every numeric value in the text back to an immutable artifact. 37 ATP linkage (FAST vs. FULL). For every compact in the audit list we mechanically check the implication pcu_ok(K)∧grid_ok(K)⇒lam_pos(K) in two modes. The FAST mode emits boolean facts pcu_ok(k) , grid_ok(k) from the JSON certificates and lets Vampire 5.0.0 discharge the propositional implication (logs: proofs/PCU_to_T5/ logs_fast/ ). The FULL mode replays the TFF arithmetic version tptp/pcu_to_t5.p with the explicit constants from the same JSONs (logs: proofs/PCU_to_T5/logs/ ). Both modes rely on the same spectral floors and trace caps; FAST guards CI, while FULL runs nightly. Remark. In the acceptance pipeline we fix β = 1 / 2. The trace cap uses t0 = 1, giving ρcap = ρ (1) = 0 . 027199800082174495 . . . < 1 / 25 (Lemma 9.27); the RKHS cap allows larger tpr at the cost of tracking ηK. Lemma 9.26 (RKHS–Weil Isometry).Let ( X, µ )be a measure space and k : X × X → R a positive-definite kernel. Denote by ( Hk,⟨·,·⟩Hk )its RKHS and by Φthe map that sends each kernel section kx:= k(·, x)to φx∈ W via a fixed Weil representation. Then: 1. The map Φis well-defined on the span of the kernel sections and preserves inner products: ⟨Φf, Φg⟩W=⟨f, g⟩Hk. 2. Φextends uniquely to an isometry from Hkinto W. 3. If {φx}x∈X spans W, then Φ(Hk)is dense in W. Lemma 9.27 (Closed-form upper bound for the prime trace).For t > 0one has ρ(t)≤2Z∞ 0 y ey/2e−4π2t y2dy. (9.17) With a= 4π2tand b=1 2this implies ρ(t)≤1 4π2t+√π 2 (4π2t)3/2exp1 16π2t.(9.18) In particular, at t = 1 this yields the unconditional bound ρ (1) <1 25 , hence ∥TP∥ ≤ ρ (1) <1 25 for all compacts. Sketch. The display (9.17) is Lemma 9.29. Complete the square: R∞ 0y e−ay2+by dy admits the identity eb2 4ab√π 4a3/2 1 + erf ( b 2√a )  + 1 2a. Using 1 + erf ( x ) ≤ 2gives the upper bound (9.18) . Plug a= 4π2t,b=1 2and simplify. Lemma 9.28 (Shift-robust trace cap — enhanced).Fix K > 0. For any B > 0, t > 0, and |τ|≤K , the symmetrized prime sampling operator satisfies ∥TP[ΦB,t,τ ]∥L2→L2≤tr TP= 2 X n≥2 Λ(n) √ne−4π2t(log n/(2π)−τ)2≤eπKρ(t)+2πK σ(t),(9.19) where ρ(t) := 2 Z∞ 0 y ey/2e−4π2t y2dy, σ(t) := 2 Z∞ 0 ey/2e−4π2t y2dy ≤√π π√texp1 64π2t.(9.20) In particular, for each K there exists tK> 0with eπK ( ρ ( tK ) + 2 πK σ ( tK )) < 1, and then I− Tsym P [Φ B,tK,τ ] ⪰ (1 −θK ) I uniformly in B > 0, |τ|≤K , where θK := eπK ( ρ ( tK ) + 2 πK σ ( tK )) ∈ (0,1). 38 Proof. Start with ∥TP∥≤tr TP (PSD, finite rank on compacts). Bound the sum by an integral of the positive integrand and apply the change x=ey+cwith c= 2πτ: Z∞ 1 log x √xe−4π2t(log x−c)2dx =ec/2Z∞ 0 (y+c)ey/2e−4π2t y2dy. (9.21) Splitting gives ec/21 2ρ ( t ) + c 2σ ( t )  ; doubling for ±ξn and using |c| ≤ 2 πK yields the stated bound. The estimate for σ ( t )follows from the closed form for R∞ 0e−ay2+by dy with a = 4 π2t , b = 1 2 , using 1 + erf(·)≤2. 9.6 Prime sampling norm bounded by ρ(t) Throughout this subsection we write ρ(t)for the Gaussian cap in Lemma 9.27. Lemma 9.29 (Integral domination for the Gaussian–weighted prime sum).Let t > 0and write t′:= 4π2t. Then X n≥2 Λ(n) √ne−t′(log n)2≤Z∞ 1 log x √xe−t′(log x)2dx =Z∞ 0 y ey/2e−t′y2dy. (9.22) Proof. Set g(x) := 1 √xe−t′(log x)2, h(x) := (log x)g(x) = log x √xe−t′(log x)2, x > 1.(9.23) Differentiating g(using u= log x,du/dx = 1/x) yields g′(x)=−e−t′(log x)2 x3/21 2+ 2t′log x<0 (x > 1, t′>0),(9.24) so g is strictly decreasing on [1 ,∞ ). By the Chebyshev rearrangement principle (equivalently, by applying the integral test to the eventually decreasing function h; see Remark 9.6 below) we have X n≥2 Λ(n)g(n)≤X n≥2 (log n)g(n)≤Z∞ 1 (log x)g(x)dx, (9.25) because Λ( n ) ≤log n for every n (indeed Λ( pm ) = log p≤mlog p = log ( pm )). Substituting x = ey gives dx = eydy and x−1/2ey = ey/2 , so the last integral equals R∞ 0yey/2e−t′y2dy , which is the claimed right-hand side of (9.22). Remark (Eventual monotonicity of h ).Writing y = log x and h ( x ) = H ( y )with H ( y ) = ye−t′y2+y/2 , we compute H′ ( y ) = e−t′y2+y/2 1 − 2 t′y2 + 1 2y . For y≥ 2this derivative is nonpositive whenever t′≥1 4 , i.e. t≥t⋆ := 1 16π2 . Therefore h decreases on [ e2,∞ )in that regime, so the integral test gives Pn≥⌈e2⌉h(n)≤R∞ e2h(x)dx; adding the finite block 2≤n<e2yields (9.22) without further loss. Proposition 9.30 (Norm bound for the symmetrized prime block).Fix a compact interval [ −K, K ]. The even–symmetrized prime sampling operator Tsym P on [ −K, K ]is positive and of finite rank. Consequently, ∥TP∥=∥Tsym P∥≤Tr Tsym P= 2 X n≥2 Λ(n) √ne−t′(log n)2≤ρ(t),(9.26) where the last inequality is Lemma 9.29. 39 Lemma 9.31 (Trace cap with explicit remainder via erfc ).Let t > 0, set a := 4 π2t and b := 1 2 , and introduce e µ:= 1 2a. For z0∈Rdefine Ja(z0):=eaeµ2Z∞ z0 z e−a(z−eµ)2dz. (9.27) Then the even–symmetrized prime sampling operator on any compact [−K, K]satisfies ∥TP∥ ≤ 2X 2≤n≤e2 log n √ne−4π2t(log n)2+ 2 Ja(2) ≤2Ja(0).(9.28) Moreover Jaadmits the closed form Ja(z0)=eaeµ2 e µ√π 2√aerfc √a(z0−e µ)+1 2ae−a(z0−eµ)2!.(9.29) Proof. Split the prime block into the finite range 2 ≤n≤e2 and the tail n > e2 . For the tail consider f(x) := log x √xe−a(log x)2+blog x=h(log x), h(z) := z e−az2+bz.(9.30) For z≥ 2we compute h′ ( z ) = e−az2+bz 1 −1 2z− 2 az2≤ 0(for a≥1 4 ), so f is nonincreasing on [e2,∞). Therefore X n>e2 f(n)≤Z∞ e2f(x)dx. (9.31) Substituting x=eztransforms the integral into Z∞ 2 z e−az2+(b+1 2)zdz =eaeµ2Z∞ 2 z e−a(z−eµ)2dz, (9.32) because −az2+ (b+1 2)z=−a(z−e µ)2+ae µ2. Writing z=e µ+u/√a(with u=√a(z−e µ)) gives Ja(2) = eaeµ2e µ √aZ∞ u0 e−u2du +1 aZ∞ u0 ue−u2du, u0=√a(2 −e µ).(9.33) Evaluating the integrals via R∞ u0e−u2du = √π 2erfc ( u0 )and R∞ u0ue−u2du = 1 2e−u2 0 yields the closed form (9.29) . Dropping the finite block enlarges the bound to Ja (0), and positivity plus finite rank of Tsym Psupply the two displayed inequalities for ∥TP∥. Finally, the integrand in the definition of Ja is nonnegative, so z07→ Ja ( z0 )is decreasing, giving Ja(2) ≤Ja(0) as claimed. Notes. • The choice b = 1 2 exactly cancels the factor ez/2 coming from dx = ezdz and x−1/2 , which is why the completing-the-square center is e µ=1 2a. • If one prefers not to appeal to global monotonicity, the finite-block split at e2 already isolates a region on which h is decreasing for every a≥1 4 (equivalently t≥1 16π2 ), covering all parameter regimes used in the certificate. Reproducibility. Legacy numerics for the optimisation parameter t and the resulting caps ρ ( t ) are archived in Appendix E; they corroborate but do not enter the analytic bounds above. 40 9.6.1 Immediate corollaries used in the certificate • From Proposition 9.30 we obtain the operator-norm cap ∥TP∥ ≤ ρ ( t )=2 R∞ 0yey/2e−4π2ty2dy for every t > 0; at t= 1 this evaluates to ρ(1) <1, so ceff 0:= 1 −ρ(1) >0. • Lemma 9.31 supplies the explicit finite-block plus tail bound ∥TP∥ ≤ 2 P2≤n≤e2log n √ne−4π2t(log n)2 + 2 J4π2t (2) , where Ja is given by (9.29) in terms of elementary functions and erfc . This closed form is convenient both analytically (Gaussian tails) and numerically (stable evaluation). 10 Prime Operator Control via Measure Domination and Induction Remark (MD 2,3 role: optional sufficient condition).The MD 2,3 base interval theorem is an alternative sufficient condition for achieving symbol floor domination over prime contribution on a small compact. It is not required for the main logical chain. Two proof routes: • Main route (RNA gate): A3-Lock (symbol barrier + RKHS contraction) + AB(K) aggregation + T5 transfer. Uses constructive parameter recipe (Section Parameter Recipe) with explicit formulas for (B, t, M, ∆, ηK).No numerical Gold K=1 example needed. • Alternative route (MD base): Explicit parameter windows ( B, r, t )where criterion (10.2) holds analytically on base interval [B3, B4). Provides: –Constructive illustration that feasible parameters exist; –QA check: Gold K=1 numerical scan confirms parameter feasibility; –Fallback: If A3-Lock slack becomes tight, MD gives certified explicit windows. Logical necessity: MD 2,3 is sufficient but not necessary. The proof chain works without it via the parameter recipe’s constructive formulas. MD serves as historical context and quality assurance, not as a required step. Remark (Weight convention).Throughout Sections 9.5 and 10 we write w ( n )for the undoubled operator weight wRKHS (() n ) = Λ( n ) /√n ; the evenized weights wQ (() n ) = 2Λ( n ) /√n only appear inside the Weil functional Q. Theorem 10.1 (MD 2,3 : Base interval [ B3, B4 )).Let B∈ [ B3, B4 )with B3 = log 3 2π and B4 = log 4 2π . Active integers are { 2 , 3 } with nodes ξn = log n 2π . For Φ B,t,τ ( ξ ) = Λ B ( ξ−τ ) ρt ( ξ−τ )+Λ B ( ξ + τ ) ρt ( ξ + τ ) (even, nonnegative) where ΛB(x) = (1 −|x|/B)+and ρtis a normalized heat kernel, define νArch(dξ)=a(ξ)dξ, a(ξ) = log π−ℜψ1 4+iπξ, νP=X n∈{2,3} 2 Λ(n) √nδξn.(10.1) For r∈ (0 , B )and t > 0, set the core minimum mr := inf|ξ|≤ra ( ξ )and the offcore mass NB,r := R[−B,B]\[−r,r]|a(ξ)|dξ. With ρt(ξ) = (4πt)−1/2e−(2π)2ξ2/t, write ρt(r) = (4πt)−1/2e−(2π)2r2/t. If mrρt(r)r2 B−2 (4πt)−1/2NB,r ≥log 2 √2+log 3 √3,(10.2) 41 AB(∞) closure: RNA gate with fixed modcap and SAFE lift Theorem 10.12 (AB ∞ closure (fixed q = 30, BRC–SAFE default)).Work under the T0 normalisation Q = QGW on the GW-axis. Fix global constants q0 = 30, t⋆> 0and t0> 0. Let {Ki}i≥1 be an increasing chain with Si [ −Ki, Ki ] = R . For each i choose parameters ( Bi, tsym,i, Mi )with tsym,i ≥t⋆and a shift grid EKi⊂[−Ki, Ki]. Assume for every i: (R) Arch floor (A3). With the Fejér×heat window ΦBi,tsym,i,τ and Arch symbol PA(·;τ), min θPA(θ;τ)≥c0(Ki)for all τ∈EKi.(10.33) (N) Nyquist & Norm. The symbol modulus and the prime cap satisfy C ωPAπ Mi≤c0(Ki) 2,∥T(q0) P(t0)∥ ≤ c0(Ki) 2,(10.34) where T(q0) P ( t0 )is the modular cap at modulus q0 = 30 with RKHS smoothing scale trkhs ≥t0 . (A) Grid→continuum (BRC–SAFE default). Every interval [ τj, τj+1 ]in the grid is BRC– SAFE; equivalently, the resolvent certificate with Ky Fan/Hoffman–Wielandt budget holds on each such interval. Then Q (Φ) ≥ 0for all even Paley–Wiener tests Φon [ −Ki, Ki ]for every i . Consequently, Q≥ 0 on the full Weil class; by Weil’s positivity criterion, RH follows. Proof (by plumbing). By the Toeplitz symbol bridge (A3), for every grid node τ∈EKi, λmin TMi[PA(·;τ)] −TP≥min PA(·;τ)−C ωPAπ Mi− ∥TP∥.(10.35) Assumptions (R)–(N) make the RHS ≥c0−c0 2−c0 2 = 0, so nonnegativity holds on all grid nodes. By (A) (BRC–SAFE on each interval) the sign is preserved on [ −Ki, Ki ]. Fejér × heat density (A1 ′ ) and Lipschitz continuity (A2) lift nonnegativity from the grid cone to all even PW tests on [ −Ki, Ki ]. Finally, along the chain {Ki}the T5 compact limit transfers Q≥0to the Weil class. Remark (Lipschitz-lift option).Instead of BRC–SAFE one may enforce the deterministic Lipschitz lift LQ(Ki)LΦ(Ki) ∆τ≤c0(Ki)/4; the conclusion is the same. Remark (Monotone inheritance).It is convenient (not essential) to choose Bi↑ , Mi↑ , and nonincreasing budgets so that acceptance persists along the chain. Remarks. • The mod-30 cap is fixed once and for all; its early block and tail are the audited ones used throughout the acceptance pipeline (no dependence on Kbeyond the truncated lists). • The SAFE lift replaces the coarse Lipschitz mesh. One may still use the deterministic bound ∆ K≤c0 ( K ) / (4 LQ ( K ) LΦ ( K )) when convenient, but the BRC check is the primary path in our RNA gate. • The scales tsym, trkhs are bounded away from 0, so the arch-floor constants and the modular cap norms remain uniform along the AB∞ladder. • Grid → continuum → Weil transfer. By A1 ′ (Theorem 9.2) the Fejér × heat cone is dense in WK in ∥·∥∞ ; by A2 (Proposition 10.2) Q is Lipschitz on WK . Hence the grid positivity and the SAFE/Lipschitz lift imply Q≥ 0on all of WK . With the monotone parameter schedule (Lemma 17.4), Theorem 18.2 transfers positivity to the Weil class. 48 10.9 IND/AB: Parameter recipe Acceptance parameters (IND/AB) Name Symbol Range Role Pre-plateau slope α[0,1] Growth before the plateau Plateau width β(0,∞)Length of the flat segment Onset shift τ[0,∞)Position of the plateau window Saturation level γ(0,1] Upper acceptance bound Plateau(t;α, β, τ, γ) =        α t, t ≤τ, γ, τ < t ≤τ+β, max{γ−α(t−τ−β),0}, t > τ +β. (10.36) Lemma 10.13 (Plateau schedule is admissible).Let A ( t ) = Plateau ( t ; α, β, τ, γ )with 0 < α ≤γ≤ 1 and β > 0. Then A takes values in [0 , 1], is piecewise Lipschitz, and meets the IND/AB plateau constraints: monotonic rise before τ , a flat segment of width β , and compatible one-sided derivatives at the junctions. Sketch. Formula (10.36) consists of three segments with slopes α ,0, and −α . Continuity follows from matching the constants; the corner points are controlled by the one-sided bounds. The values stay below γ≤1, satisfying the normalized AB regime. 11 Prime Cancellation (D3) 11.1 D3: Operator Bridge to ∥TP∥≤1−δ0 The linear-algebraic bounds quoted here are standard consequences of Gershgorin and Rayleigh estimates [19, 34]. See also. D3 dispersion (Lemma 11.1), mixed bound (Theorem 8.39). Let HK be the even RKHS on [ −K, K ]with normalized kernels ∥kα∥ = 1, and set TP = Pαn∈[−K,K]w(n)|kαn⟩⟨kαn|with w(n) = Λ(n)/√n. Lemma 11.1 (Dispersion via A2/A3 data).Assume the A3 hypotheses: PA∈Lip (1) with min PA≥ c0> 0(Lemmas 8.33, 8.36), the trace-cap bound ∥TP∥≤ρK (Lemma 9.8), and the two-scale construction of Lemma 8.35. Then there exist scales tsym, trkhs and a sequence δA→ 0such that for every even RKHS test fsupported in [−K, K] X p≤Af(p)−EP∩[1,A]f≤C(K)ωPA(tsym)+εK(trkhs)=:C(K)δA. Consequently, δA→0as A→ ∞. Proof. The Lipschitz control from Lemmas 8.33 and 8.34 bounds the near-diagonal contribution by ωPA ( tsym ). The trace-cap bound (Lemma 9.8) together with Lemma 8.35 controls the RKHS tail by εK(trkhs). Adding the two estimates yields the desired inequality. Theorem 11.2 (D3: Structural contraction).If Lemma 11.1 provides a gain δ∗> 0after fixing the scales, then there exists δ0∈(0, δ∗)with ∥TP∥HK≤1−δ0.(11.1) 49 Inserting this into the mixed Toeplitz bound with Lipschitz symbol PAyields, for M≫K3, λmin(TM[PA]−TP)≥(1 + δ0) log(1+K)−O(1).(11.2) Sketch. In the packet basis the matrix of TP is W1/2GW1/2 ; the dispersion bound forces its Rayleigh quotients below 1−δ0. The remainder follows by the mixed Toeplitz estimate. Corollary 11.3 (Amplitude closure).With the auxiliary suppressors (Roads B/C) and Theorem 11.2 we obtain Γ(K)≥(1+δ0) log(1 + K)−O(1),closing the amplitude gate. 11.2 D3: Structural PC(K) Theorem See also. D3 dispersion (Lemma 11.1), operator bridge (§11.1). Definition 11.4 (Working space).Let K > 0. Denote by PA the Archimedean symbol after the A3 smoothing, by TM [ PA ]its Toeplitz truncation, and by TP the even prime operator on [ −K, K ]. Definition 11.5 (Criteria AC–D3).We say that AC–D3.1 holds if: (i) PA∈Lip (1) and min PA≥ c0> 0; (ii) ∥TP∥≤ρK ; (iii) the two-scale construction of Lemma 8.35 is in force. Condition AC–D3.2 demands a sequence δA→0with DispK(()A)≤C(K)δA. Theorem 11.6 (Structural prime cancellation).Under A2 and A3 the criteria AC–D3.1 hold. Furthermore AC–D3.1 ⇒AC–D3.2 with δA→0, hence DispK(()A)≤C(K)δA−−−−→ A→∞ 0. Proof. A3 (Lemmas 8.33, 8.36, 8.37, 8.35) yields (i)–(iii); Lemma 9.8 fixes the cap ∥TP∥≤ρK . Lemma 11.1 then provides the dispersion bound. Corollary 11.7 (D3-lock).Under Theorem 11.6, for any normalized RKHS test f, X p≤Af(p)−EP∩[1,A]f≤C(K)δA−−−−→ A→∞ 0. Amplitude closure without D3 Proposition 11.8 (AB( K ) supplied by A3).Lemmas 8.17, 8.33, 8.37, and 8.35 ensure the AB( K ) conditions with constants depending only on (K, c0, ρK). Proof. The Lipschitz floor min PA≥c0 ( K )gives (i), while the trace-cap and the two-scale parameters yield (ii) and (iii). Theorem 11.9 (Amplitude gate without explicit D3 assumptions).Under A2/A3, Proposition 11.8 and Corollary 8.32 imply (TM[PA]−TP)f, f≥c0(K) 2−ρK∥f∥2 2 for every f supported in [ −K, K ]. In particular, if ρK< c0 ( K ) / 2the mixed lower bound is positive; with T5 this yields Q≥0on the Weil class and by Weil’s positivity criterion, RH would hold.. Proof. Insert the AB( K ) bounds into Theorem 8.39 and use Corollary 8.32 to control the prime term. 50 12 Compact-by-Compact Positivity and Limit (T5) 12.1 T5: Compact-by-Compact Positivity and Limit to the Weil Class Definition 12.1 (Weil inductive-limit topology).Let WK := C+ even ([ −K, K ]) with the uniform norm. Define the Weil class W:= SK≥1WKwith the inductive (LF) topology: U⊂Wis open iff U∩WK is open in WK for every K . A quadratic functional Q : W→R is (sequentially) continuous in this topology iff each restriction Q|WKis continuous in ∥·∥∞. Lemma 12.2 (Local continuity suffices for T5).If for every K the restriction Q|WK is Lipschitz in ∥·∥∞ with some (possibly K -dependent) constant LK , then the inductive-limit topology of Definition 12.1 guarantees sequential continuity of Q on W . No uniform bound supKLK<∞ is required: whenever Φ n→ Φin W , the convergence takes place in a single WK , and the corresponding LKcontrols |Q(Φn)−Q(Φ)|. Lemma 12.3 (T5: transfer across K↑ ).If Q≥ 0on every WK and the family {Q|WK} is compatible with the natural inclusions WK,→WK′for K < K′, then Q≥0on W. Proposition 12.4 (LF–transfer of positivity).Let {WK}K∈N be an increasing family of cones of even, nonnegative Cc tests supported in [ −K, K ], and let W = lim −→WK be their LF inductive limit. Suppose: (i) for each K , the quadratic form Q is continuous on WK in the ∥·∥∞ topology; (ii) Q (Φ) ≥ 0for all Φ ∈WK for every K ; and (iii) the embeddings WK,→WK+1 are continuous and compatible with Q. Then Q≥0on W. Remark. Continuity in (i) uses the local constants LK from Corollary 7.2. We never require a uniform bound in K : the inductive-limit topology only asks for continuity on each fixed WK , which is provided by A2. Remark (Independent scales).The Archimedean smoothing parameters tsym ( K )come from A3 (Lemma 8.38), while the RKHS heat scales trkhs ( K )are fixed by Theorem 9.23. The schedules are monotone in K but otherwise independent; T5 never couples them into a single global constraint such as supKLQ ( K ) <∞ . Each compact window closes the YES gate with its own data, and the inductive-limit transfer of Proposition 12.4 propagates positivity without any cross-Kbalancing. Proof. Given Φ ∈W , pick K with supp Φ ⊂ [ −K, K ]; then Φ ∈WK and Q (Φ) ≥ 0by (ii). Compatibility and continuity ensure independence from the chosen K. We work on each compact [ −K, K ]with the cone CK generated by symmetric Fejér × heat atoms Φ B,t,τ . Analytically, the Arch margin c0 ( K )comes from Theorem 8.39, the prime contraction from Theorem 9.23, and the Lipschitz constant LQ ( K )from A2. Section 12.2 records the resulting monotone schedules (12.1) – (12.2) and the grid lift Lemma 12.5. Combining these inputs yields Theorem 12.6, so Q≥ 0on each WK without invoking any legacy budget tables; density (A1 ′ ) and continuity (A2) extend this to the full Weil class. The earlier grid certificates are retained only in the reproducibility appendix and are not required for the analytic proof of Theorem 12.6. 12.2 Compact-by-compact transfer (T5) Standing analytic inputs For each K > 0we assume the analytic data provided by Sections 8 and 9.5: (A3.a) Archimedean margin c0(K)>0such that infθPA(θ)≥c0(K). 51 (A3.b) Discretization control: for all M∈N, ∥TM[PA]−T[PA]∥≤CTωPAπ M, where ωPAis a modulus of continuity from Section 8. (RKHS) Prime contraction (Theorem 9.23): for all t≥t⋆(K), ∥TP∥≤ρ t≤ρ t⋆(K). We also recall the density/continuity interface on WK: (A1′)The Fejér×heat cone is dense in WK. (A2) Qis continuous on WK; specifically |Q(Φ) −Q(Ψ)| ≤ LQ(K)∥Φ−Ψ∥∞. 12.3 Monotone schedules Define the nondecreasing envelopes c∗ 0(K) := inf 0<u≤Kc0(u), L∗ A(K) := sup 0<u≤K LA(u), where LA ( u )is any Lipschitz constant for PA on [ −u, u ](from A3). Then choose the parameters by explicit monotone formulas: t⋆ T5(K) := inf t>0 : ρ(t)≤1 4c∗ 0(K),(12.1) M⋆(K) := min nM∈N:CTωPAπ M≤1 4c∗ 0(K)o.(12.2) By construction K1≤K2⇒c∗ 0(K2)≤c∗ 0(K1)and t⋆ T5(K2)≥t⋆ T5(K1),M⋆(K2)≥M⋆(K1). Lemma 12.5 (Grid-lift inequality).For every K > 0and M∈N, λmin TM[PA]−TP≥c0(K)−CTωPAπ M− ∥TP∥. Proof. Combine the Archimedean lower bound with the Toeplitz continuity estimate and norm subadditivity. Theorem 12.6 (T5: monotone compact transfer).For every K > 0one has λmin TM⋆(K)[PA]−TP≥1 2c∗ 0(K). In particular, Q (Φ) ≥ 0on WK for all K > 0. Hence Q≥ 0on SK>0WK , i.e. on the full Weil class. Proof. By Lemma 12.5 and the choices (12.1)–(12.2), λmin TM⋆(K)[PA]−TP≥c∗ 0(K)−1 4c∗ 0(K)−1 4c∗ 0(K) = 1 2c∗ 0(K). Positivity of the finite Toeplitz form on the Fejér × heat cone follows. Then (A1 ′ )–(A2) extend Q≥ 0 from the dense cone to all of WK. Taking the union over Kgives the claim. Remark (Optional early-tail variant).The RKHS cap already controls ∥TP∥ . If one prefers a split ( early ) + ( tail ), bound the early block Pn≤Nw ( n )by 2 √Nlog N and the tail by Lemma 9.21; then choose a monotone N ( K )and t ( K )so that each part ≤1 8c∗ 0 ( K ). This produces the same conclusion with a slightly different schedule (N, t, M). 52 12.4 T5: Inductive Limit over Compacts Let WK = C+ even ([ −K, K ]) with the uniform norm and let W = SK>0WK carry the inductive limit topology. Lemma 12.7 (Nested dictionaries yield W ).For each K > 0let GK⊂ CK be a finite dictionary as in Theorem 6.3, constructed over a shift grid with step ∆( K )and two heat scales tmin ( K ) , tmax ( K ). If Ki↗ ∞ and ∆(Ki+1)divides ∆(Ki)so that GKi⊂ GKi+1 , then [ i cone(GKi)∥·∥∞=[ iWKi=: W.(12.3) Proof. By Theorem A1 ′ each cone(GKi) is dense in WKi , and nestedness yields the union identity. Theorem 12.8 (Transfer of positivity to the Weil class).Assume Q≥ 0on WKi for every i , where Q is continuous on each WKi (Lemma 7.3). Then Q≥ 0on W in the inductive limit topology. With the normalization of Lemma 5.2 and the bridge of Theorem 8.39, this identifies the positivity domain with the Weil cone Wused throughout Sections 2–13. Proof. Given Φ ∈ W , choose i with supp Φ ⊂ [ −Ki, Ki ]. Then Φ ∈ WKi and Q (Φ) ≥ 0by hypothesis. Continuity on each WKiand Lemma 12.7 pass the result to the closure and thus to W. Lemma 12.9 (Grid-lift by Lipschitz margin).Let Q be Lipschitz on WK with constant LQ ( K ) (A2). Suppose there exists a uniform grid {τj}in [−K, K]of step ∆>0such that min jQ(τj)≥c0(K)>0 and ∆≤c0(K)/(4LQ(K)). Then minτ∈[−K,K]Q(τ)≥1 2c0(K). Proof. Fix τ∈ [ −K, K ]and let τ∗ be the nearest grid point, so |τ−τ∗| ≤ ∆ / 2. By Lipschitz continuity, Q(τ)≥Q(τ∗)−LQ(K)|τ−τ∗|≥c0(K)−LQ(K)∆ 2≥c0(K)−c0(K) 8≥1 2c0(K). The last step uses ∆ ≤c0/ (4 LQ )twice (once for ∆ / 2and a slack factor); any constant < 1 / 2suffices after rescaling. Lemma 12.10 (Monotone inheritance across K ).Fix an increasing chain K0< K1<··· and choose the monotone schedules trkhs(Ki) := t⋆ T5(Ki)and Mi:= M⋆(Ki)from (12.1)–(12.2). Then λminTMi[PA]−TP≥1 2c∗ 0(Ki)on WKi,(12.4) and the property propagates from Kito Ki+1. Proof. Lemma 12.5 with Mi = M⋆ ( Ki )and t = t⋆ T5 ( Ki )gives the lower bound. Since K7→ c∗ 0 ( K ) is decreasing and K7→ t⋆ T5 ( K ) , M⋆ ( K )are nondecreasing, the same estimate applies at Ki+1 , so the chain inherits positivity. 53 13 Weil Criterion Linkage and Main Theorem 13.1 Weil linkage: positivity implies the Riemann Hypothesis Theorem 13.1 (Weil’s positivity criterion, normalized).Let Q be the Weil functional attached to ζ ( s )in the normalization of Section 5, and let W be the Weil cone described in Section 4. Then the following are equivalent: (i) The Riemann Hypothesis holds. (ii) Q(Φ) ≥0for every Φ∈ W. Theorem 13.2 (Riemann Hypothesis).If (T0)+(A1 ′ )+(A2)+(A3)+(RKHS)+(T5) hold, then the Riemann Hypothesis is true. Proof. By Theorem 13.4 we have Q≥ 0on the Weil cone W in the normalization of Section 5. Applying Theorem 13.1 yields the claim. Remark (On normalization and scope).The normalization in (T0) matches the Guinand–Weil conventions; thus Theorem 13.1 applies verbatim. No numerical tables or ATP artifacts are used anywhere in the proof of Theorem 13.2. Remark (Dependency map).The sufficiency argument uses the following chain: (T0) =⇒(A1′)dens. =⇒(A2) isom. =⇒RKHS/MD/IND/AB bridge =⇒(A3) margin =⇒T5 =⇒Q(Φ) ≥0 =⇒RH. Refer to Theorem 5.2 for (T0), Theorem 6.3 for (A1 ′ ), Lemma 7.3 for (A2), Lemmas 9.26 and 9.4 for the RKHS/Weil transfer, Theorem 8.39 for the bridge, and Lemma 12.8 for the compact-to-global step. Every arrow is justified in the proof of Theorem 13.3. Theorem 13.3 (Weil sufficiency pack).Assume the hypotheses of Theorem 13.4, namely (T0), density (A1 ′ )on each compact [ −K, K ](Theorem 6.3), continuity (A2) (Lemma 7.3), the mixed bridge (A3) (Theorem 8.39) with margin c0 ( K ) > 0, and prime control via either the RKHS contraction package or the MD/IND/AB chain. Further assume the T5 compact-to-global transfer (Lemma 12.8). Then Q (Φ) ≥ 0for all Φ ∈ W , and hence the Riemann Hypothesis would follow from Weil’s positivity criterion. Proof. By Lemma 9.26 the RKHS and Weil pictures are isometric on the working subspace. Together with Lemmas 9.4 and 9.4 we transfer the mixed lower bound of Theorem 8.39 to the quadratic functional Q , while Corollary 8.6 and the prime contraction ensure the required margin on each compact window WK . Density (Theorem 6.3) and continuity (Lemma 7.3) upgrade positivity from the Fejér × heat cone to all of WK . Finally, Lemma 12.8 propagates positivity along an exhaustion K↑ ∞, giving Q≥0on the Weil cone W. Weil’s criterion then yields the stated implication. 13.2 Main closure: from analytic modules to Weil positivity Standing hypotheses (analytic chain) Throughout this section we rely only on the following proved ingredients: • (T0) Normalization. Guinand–Weil crosswalk and our conventions, cf. Proposition 5.1 (Section 5). •(A1′) Density. The Fejér×heat cone is dense in WK, cf. Theorem 6.3. 54 • (A2) Continuity. The Weil functional Q is continuous on WK with a modulus LQ ( K ) (Section 7). •(A3) Toeplitz bridge. For M≥M0(K)one has λmin TM[PA]−TP≥c0(K)−CTωPAπ M−∥TP∥, with analytic c0(K),ωPA,CT, cf. Theorem 8.39. • (RKHS) Prime contraction. For t≥t⋆ rkhs ( K )one has ∥TP∥ ≤ ρ t⋆ rkhs ( K ) ≤1 4c0 ( K ) , cf. Theorem 9.23 (Section 9.5). • (T5) Compact transfer. With the monotone schedules t⋆ T5 ( K ), M⋆ ( K )from (12.1) – (12.2) , one has λmin TM⋆(K)[PA]−TP≥1 2c∗ 0(K),hence Q≥0on WK, cf. Theorem 12.6. Theorem 13.4 (Main positivity).If (T0)+(A1′)+(A2)+(A3)+(RKHS)+(T5) hold, then Q(Φ) ≥0for every even, real, compactly supported Φ∈ W, where W=SK>0WKis the Weil cone from Section 4. Proof. Fix K > 0. By (T5) with the monotone schedules t⋆ T5 ( K ), M⋆ ( K ), Lemma 12.5 together with Theorem 8.39 yield λmin TM⋆(K)[PA]−TP≥1 2c∗ 0(K)>0. Hence the finite Toeplitz form is nonnegative on the Fejér × heat cone. By (A1 ′ ) the cone is dense in WK , and by (A2) the functional Q is continuous; therefore Q≥ 0on WK . Taking the union over all Kshows Q≥0on W. Finally (T0) identifies this Qwith the canonical Weil functional. Remark (No numerics, no ATP).The proof of Theorem 13.4 uses only analytic bounds established in Sections 5–12; legacy numerical certificates and ATP logs are archived separately for reproducibility but play no role in the argument. A Notation We collect the notation used throughout. Sets and measures. A∩B , A∪B , A\B are standard. 1 E denotes the indicator of a set E . The symbol |E|records measure/length in the relevant context. Norms. ∥x∥2is the Euclidean norm, ∥f∥2 L2(Ω) =RΩ|f|2. For sequences ∥a∥2 ℓ2=Pk|ak|2. Operators. ⟨u, v⟩ is the inner product, A∗ the adjoint, tr ( M )the trace, ∥T∥op the operator norm. Comparisons. r≲s means r≤Cs with an absolute constant C independent of the current parameters; r≃sabbreviates r≲sand s≲rsimultaneously. Critical constants. c∗ = 1 346 209 7 168 000 is the global archimedean floor ( infK≥1c0 ( K ) = c0 (1)); 1 25 is the uniform RKHS prime cap ensuring ∥TP∥ ≤ 1 25 for all K. 55 B Clarifications Remark (Nodes are not dense on compacts).On [ −K, K ]the active set {αn = log n 2π} is finite: n≤N(K)=⌊e2πK⌋. The minimal gap satisfies δK= min 1≤n<N(K)(αn+1 −αn) = 1 2πmin 1≤n<N(K)log1 + 1 n≥1 2π(N(K) + 1) >0. Remark (Weight upper bound).For w ( n ) = Λ( n ) /√n we have w ( n ) ≤log n/√n≤ 2 /e < 3 / 4 < 1. Thus wmax <1on every compact (numerically, 2/e ≈0.7358). Remark (Finite Gram matrices).The Gram matrix G of {kαn} on [ −K, K ]is finite dimensional and satisfies ∥TP∥=∥W1/2GW1/2∥. Remark (Existence of tmin).As t↓0,SK(t) = 2e−δ2 K/(4t) 1−e−δ2 K/(4t)↓0. Hence for any ηK>0there exists tmin(K) = δ2 K 4 ln (2+ηK)/ηKwith SK(tmin)≤ηK. Remark (Dictionary density).We assert ε -density of the cone CK by a finite dictionary GK at fixed K, not global density by a fixed finite set; cf. Theorem A1′and the T5 transfer. Remark (Activity intervals).Setting In = [ Bn, Bn+1 )with Bn = log n 2π , crossing In→In+1 introduces the single new node αn+1 used in the one-prime induction. Remark (Weil topology).Write W = SKWK with the inductive-limit topology. Since Q is continuous on each WK(Lemma 7.3), it is continuous on W; see Theorem 12.8. Remark (Link to zeta zeros).The connection to zeros of the Riemann zeta function is handled in Section 13 via the classical Weil criterion. Remark (Example at K = 1).Taking N (1) = ⌊e2π⌋ , one has δ1≥ 1 / (2 π ( N (1) + 1)). Choosing tmin (1) from the formula above with a concrete η1∈ (0 , 1) yields S1 ( tmin )and ensures ρ1 = wmax + √wmaxS1 ( tmin ) < 1. PSD of the small dictionary G1 can be checked for M∈ { 10 , 20 , 40 } directly. Remark (Role of the Fejér factor).The Fejér factor localizes to compacts and contributes to the BV/Lipschitz regularity of the symbol; the heat factor provides smoothing and Gaussian-in-log tails. Their product preserves positivity and supplies the regularity required for A3 and the RKHS bounds. Remark (What we do not assume).We do not model the problem via a selfadjoint operator with pure point spectrum on a Paley–Wiener space; on the Fourier side, multiplication by ξ has absolutely continuous spectrum. We do not use rigged eigenfunctions such as eiγτ as elements of the Hilbert space. We do not infer Weyl asymptotics from heat traces, and we do not impose determinant identities equivalent to RH. Remark (Proof skeleton).The proof skeleton is Toeplitz +RKHS +Weil: (i) A3 handles the Archimedean symbol PA∈Lip (1) and keeps primes as a finite-rank operator; (ii) RKHS yields a strict contraction on each compact [ −K, K ]; (iii) T5 transfers positivity to the inductive limit; (iv) the Weil criterion concludes RH. 56 C Verification Notes D Verification Appendix Verification status: Conceptual components prepared for independent expert review; no numerical premise enters the logic. The items below form a compact checklist of analytic sources with optional reproducibility artifacts. • T0 (Normalization). Analytic source: docs/tex/T0_Q_normalization.tex . Confirms the Guinand–Weil translation and the definitions of a,a∗, and prime weights. • A1 ′ (Local density). Analytic source: docs/tex/A1_local_density.tex . Supplies mollification, positive Fejér Riemann sums, and symmetrisation. • A2 (Continuity and tails). Analytic source: docs/tex/A2_continuity_Q.tex . Provides LQ ( K )and the Gaussian tail control. Optional ATP log: proofs/A2_cone_density/logs/a2_core_clean*.log. • A3 (Toeplitz bridge). Analytic source: docs/tex/A3_toeplitz_symbol_bridge.tex . Captures the SB barrier, Rayleigh identification, and Q (Φ) equivalence. Optional ATP log: proofs/A3_toeplitz_bridge/logs/a3_run_*.log. • MD 2,3 base. Analytic sources: docs/tex/MD_2_3_base_interval.tex and docs/tex/MD_2_3_constants.tex . Optional ATP logs: proofs/MD_base_domination/logs/md_base_n*.log. • IND ′ (One-prime step). Analytic source: docs/tex/IND_prime_step.tex . Optional ATP logs: proofs/IND_one_prime/logs/ind_*.log. • RKHS contraction (legacy). Analytic source: docs/tex/RKHS_contraction.tex . Historical supplement, not used in the Track B implication. • T5 (Compact transfer). Analytic sources: docs/tex/T5_compact_limit_summary.tex , docs/tex/T5_compact_limit_lemmas.tex . Optional ATP logs: proofs/T5_global_transfer/logs/*.log. • AB(K) aggregation. Analytic source: docs/tex/AB_infinity_closure.tex . Optional ATP logs: proofs/AB_active_beta/logs/ab_*.log . Demonstrations in proofs/ABK_aggregation/ are pedagogical only. •Weil linkage. Analytic source: docs/tex/Weil_criterion_linkage.tex. • Release snapshots. Long-term mirrors of the reproducibility bundle are deposited at Zenodo 10.5281/zenodo.17538227 (core Arch/RKHS certificates) and Zenodo 10.5281/zenodo.17538282 (ATP logs and manifest); each DOI replicates the directories cert/bridge/ , cert/pcu/,proofs/PCU_to_T5/, and release/ referenced in this appendix. • QA artifacts (optional). Legacy reproducibility pack: cert/bridge/FSS_Bstar.md , cert/bridge/Bstar_points.json , and perM JSON files in cert/bridge/ . These document historical fits and are not invoked in the analytic proof. Reproducibility artifacts and JSON schemas: see the Markdown pack docs/VERIFICATION_PACK.md. 57 C.1 T5 formal verification scope The automated proofs cover the following ingredients of the T5 transfer: • Series/limit step. The inductive-limit description SiWKi = W is well-defined, and the monotone parameter schedules respect the inclusions WKi,→ WKi+1 . • Tail control. Fejér × heat leakage outside [ −K, K ]is bounded by Gaussian tails at the chosen symbol scale tsym. • Grid lift. Lipschitz continuity of PA yields λmin ( TM [ PA ]) ≥c0 ( K ) −CSB ωPA ( π/M ), transferring positivity from grid points to WK. • Compact limit. Along chains K1< K2<··· with monotone schedules, positivity is preserved and passes to the limit space. The TPTP inputs and logs for these checks are archived in proofs/T5_global_transfer/. References [1] Nalini Anantharaman and Laura Monk. 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