RESULTS: theory lane, hunt `oob_envelope`

Branch teal-sea/oob-cert-theory. Author: theory worker (Claude Code, Opus). Every statement below carries its grade on the ladder of AGENTS.md. A proof here is an ordinary derivation; none is kernel-checked. The hunt's independent referee lane (a different model family) reviewed the proofs as of theory commit b447f0d: hunts/oob_envelope/referee/REVIEW.md at commit b65ef69 on branch teal-sea/oob-cert-referee. Its verdict per result is in the status table below. That review does not move any claim up the ladder, and outside verification is pending. Numbers from float64 scripts are measured.

Summary (five lines)

1. Task 1: terms at frequencies ≥ 2L (boundary included) leave Weil's window form unchanged for every complex f ∈ L²[−L, L], so Zhu's reduction runs with any bound S ≥ sup(P_L − H) in place of A_L, but no H brings positivity below an H-free floor T_res(L) ≥ 2π e^{S*_L} (§1). [ordinary derivation; referee PASS, REVIEW §2–§3 and repair readback]
2. Task 2: for every L the best out-of-band constant is exactly λ_max of the windowed comb operator, reached by explicit trigonometric H at rate N^{−2} (§2). [ordinary derivation; referee PASS after the b447f0d repair, REVIEW §4.1; float checks at L = 0.6, 0.8 measured, not reproduced]
3. Task 3: that constant is e^L(1 + o(1)), against 2e^L per prime and Zhu's 4e^L, so the threshold stays doubly exponential, asymptotically the fourth root of Zhu's (§3). [ordinary derivation; referee inspected it, no gap, prime number theorem input stated, REVIEW §4.4; table measured]
4. Task 4: the device is not new (Burnol 2000 adds a support-edge cosine; Liu's Theorem B is its operator form), and the window record in half-width of supp f is (log 2)/2 refereed, 17/16 (Liu) and 0.8 (Zhu) unrefereed (§4). [literature reading, scope stated; not independently cleared by the referee]
5. Task 5: Liu's obstruction concerns a unit-window localisation and does not touch the out-of-band reduction, whose reduced form is β* I plus a compact operator (§5). [ordinary derivation from the stated theorem; the Liu-source translation was not independently cleared by the referee]

Referee status (REVIEW.md at b65ef69)

Corrections after the independent referee (REVIEW §3–§4, 2026-09-27) are marked Correction in §1.3, §1.5, §1.7, §2.3 and §2.4; none changes a theorem's conclusion.

Expanded summary

1. Task 1, lemma and modified reduction. For every complex f ∈ L² with supp f ⊆ [-L, L] and every H = μ̂ with μ a finite measure carried by |λ| ≥ 2L (boundary included), ∫|F|²H = 0; Zhu's Theorem 1.1 holds with A_L replaced by any bound S ≥ sup_{t≥T#}(P_L − H), and four computational constants of his §4–§5 change (§1.1–1.5). The binding height need not be the envelope threshold 2π e^S (≈ 30 at L = 0.8): every out-of-band reduced form is dominated by an H-free archimedean-capped form, so no H can bring positivity below a floor T_res(L) ≥ 2π e^{S*_L} (§1.7). Ordinary derivation; referee PASS (REVIEW §2–§3; the §1.7 repairs in its repair readback). The numerics lane's sampled N = 200 blocks put a candidate operating point near 65 (measured; no full-form positivity proved).
2. Task 2, duality. For every L > 0, inf_H sup_t(P_L − H) = λ_max(P), the top of the windowed comb operator: weak duality for every finite-measure H, strong duality by an explicit trigonometric H_N within (π²/(4(N+1)²)) Σ m² log p/p^{m/2} of the floor (a completion argument on the order k ↦ k·log p); continuous parts of μ do not lower the constant even on tails t ≥ T (§2.1–2.6). Ordinary derivation; referee PASS after the b447f0d repair (REVIEW §4.1). Measured (float64): λ_max(P) = 0.9009276 at L = 0.6 and 1.2191380 at L = 0.8 (finite 4- and 12-vertex components), and the construction reaches 0.90719 and 1.23295 at N = 20 against per-prime 1.12441 and 1.52205 (§2.4–2.5).
3. Task 3, asymptotics. S*_L = e^L (1 + O(e^{−c√L})) from the prime number theorem with the classical error term (constant not effective), against S_sep ~ 2e^L and Zhu's A_L ~ 4e^L. So every constant-envelope out-of-band reduction still needs T# > 2π exp((1 + o(1)) e^L): doubly exponential, asymptotically the fourth root of Zhu's threshold (§3.1–3.4), plus an explicit lower bound ℓ(L) valid for every L. Ordinary derivation; the referee inspected it and found no gap (REVIEW §4.4); the table of §3.5 is measured (float64, one route; S*_L/e^L = 0.811, 1.079, 1.082 at L = 1.19, 2, 4).
4. Task 4, prior art and record. The device is not new: Burnol 2000 (C. R. Acad. Sci. Paris 331, 423–428, arXiv:math/0101068, Théorème 3.7) adds a cosine at the support-edge frequency to Weil's symbol, and Liu's Theorem B realises the λ_max(P)-level constant as an operator split. No prior instance found of Theorems 1′, 2, 3 or §1.7 (search scope in §4.2). Record, half-width of supp f: refereed (log 2)/2 (Yoshida 1992); unrefereed computer-assisted 0.8 (Zhu) and 1, 17/16 (Liu) (§4.1). Literature reading, scope stated; not a proof; not independently cleared by the referee (REVIEW §4.4).
5. Task 5, Liu's obstruction. It does not touch the out-of-band route: it concerns a unit-window localisation that loses the prime-power-8 interaction, whereas Theorem 1′ never localises and its reduced form is β* I plus a compact operator, so Liu's modulated two-bump tests give R_H → β* > 0 (§5). Ordinary derivation from Liu's stated theorem; the referee did not independently clear the Liu-source translation (REVIEW §4.4).

0. Conventions

Zhu = arXiv:2608.24827v2 (X. Zhu, 2026). Section, equation and lemma numbers below are those of v2's LaTeX source (sections: 1 intro, 2 the Weil form, 3 envelope lemmas, 4 proof of Theorem 1.1, 5 the computation at support 1.6, 6 parity, 7 the support-2.38 computation, 14 barrier, 15 failed routes).

• F(z) = f̂(z) = ∫ f(u) e^{izu} du.
• Window class W_L = { f ∈ L²(ℝ; ℂ) : f = 0 a.e. outside [-L, L] }.
• f̃(x) = conj f(-x), g = f ⋆ f̃, so g(x) = ∫ f(y) conj f(y − x) dy and ĝ(t) = |F(t)|² for real t.
• Zhu's form, for real even f ∈ W_L (his eq. (2), (3)): Q(f) = 2F(i/2)² + (1/2π) ∫_ℝ |F(t)|² Ψ_L(t) dt, Ψ_L(t) = a(t) − P_L(t), a(t) = Re ψ(1/4 + it/2) − log π, P_L(t) = Σ_{log n < 2L} (2Λ(n)/√n) cos(t log n), A_L = P_L(0). For general complex f the pole term is ĝ(i/2) + ĝ(−i/2) = 2 Re[F(i/2) conj F(−i/2)] = 2|∫f cosh(u/2)du|² − 2|∫f sinh(u/2)du|² (derived and checked in §1.3); the multiplier term is the same integral.
• a is even, continuous, bounded below (a(0) = ψ(1/4) − log π ≈ −5.37) and a(t) = log(|t|/2π) + O(1/|t|). P_L is bounded. So Ψ_L is bounded below and the multiplier integral is well defined in (−∞, +∞] for every f ∈ W_L; it is finite when ∫|F|² log(2 + |t|) dt < ∞. Nothing below needs more than this.

1. Task 1: the out-of-band lemma and the modified reduction

1.1 Statement

Lemma 1 (out-of-band orthogonality). Let L > 0, f ∈ W_L, and let μ be a finite complex Borel measure on ℝ with |μ|((−2L, 2L)) = 0. Put H(t) = ∫ e^{iλt} dμ(λ). Then |F|² H ∈ L¹(ℝ) and

∫_ℝ |F(t)|² H(t) dt = 0.

In particular this holds for H(t) = Σ_w b_w cos(λ_w t) with Σ_w |b_w| < ∞ and every |λ_w| ≥ 2L (take μ = Σ_w (b_w/2)(δ_{λ_w} + δ_{−λ_w})), including frequencies exactly equal to ±2L.

No reality, parity or smoothness of f is assumed. H may be complex; for Q to stay real one takes μ real and symmetric, so that H is a real cosine transform.

Corollary 1 (invariance of Q). For every f ∈ W_L and every real admissible H,

Q(f) = [pole term] + (1/2π) ∫_ℝ |F(t)|² (Ψ_L(t) + H(t)) dt,

as an identity in (−∞, +∞]. The pole term is not touched.

1.2 Proof of Lemma 1

(i) g is continuous, bounded, and vanishes on |x| ≥ 2L. Write g(x) = ⟨f, τ_x f⟩ with (τ_x f)(y) = f(y − x). Translation is continuous from ℝ into L², so g is continuous, and |g| ≤ ‖f‖₂². The integrand f(y) conj f(y − x) vanishes unless y ∈ [−L, L] ∩ [x − L, x + L]. For x > 2L that set is empty. For x = 2L it is the single point {L}. A point has Lebesgue measure zero and f(y) dy is absolutely continuous, so g(2L) = 0. The case x ≤ −2L is symmetric.

(ii) Inversion holds pointwise. f ∈ L² with compact support is in L¹, so g ∈ L¹ ∩ C(ℝ), and Fubini gives ĝ(t) = F(t) conj F(t) = |F(t)|² ≥ 0. By Plancherel ∫ ĝ = 2π‖f‖₂² < ∞, so ĝ ∈ L¹. For g ∈ L¹ ∩ C with ĝ ∈ L¹, Fourier inversion holds at every point: g(x) = (1/2π) ∫ ĝ(t) e^{−itx} dt for all x ∈ ℝ.

(iii) Fubini against μ. ∫∫ |ĝ(t)| d|μ|(λ) dt = ‖ĝ‖₁ · |μ|(ℝ) < ∞, so

∫ ĝ(t) H(t) dt = ∫ ( ∫ ĝ(t) e^{iλt} dt ) dμ(λ) = 2π ∫ g(−λ) dμ(λ).

μ is carried by {|λ| ≥ 2L} and g(−λ) = 0 there by (i), boundary included. Hence the integral is 0, and |F|²H ∈ L¹ because |H| ≤ |μ|(ℝ). ∎

Proof of Corollary 1. Ψ_L is bounded below and H is bounded, so ∫|F|²(Ψ_L + H) is defined in (−∞, +∞], and it equals ∫|F|²Ψ_L + ∫|F|²H (a bounded-below integral plus an absolutely convergent one). The second term is 0 by Lemma 1. ∎

1.3 Remarks that the referee brief asks about

• Boundary frequency |λ| = 2L. Handled in step (i): the overlap of [−L, L] and [L, 3L] is {L}, a null set. This is the only place the proof uses that f is a function rather than a measure. It is sharp: for f = δ_{−L} + δ_L (not in L²) one gets |F(t)|² = 2 + 2cos(2Lt), whose mean against cos(2Lt) is 1, not 0. So the lemma needs "no atoms at ±L", which every f ∈ L² satisfies. Equivalently: g is continuous and supported in [−2L, 2L], hence g(±2L) = 0.
• Complex and odd f. No symmetry is used. For real H the multiplier term stays real for complex f (|F|² is real).
• The pole term. Lemma 1 is a statement about the multiplier integral only. Adding H to the symbol is licensed because the pole term is a separate finite-rank term evaluated at ±i/2; H is never evaluated there and never enters it.
• The pole term for complex f, checked. g is continuous with compact support, so ĝ is entire, and for every z ∈ ℂ ĝ(z) = F(z) · (f̃)^(z) with (f̃)^(z) = ∫ conj f(−x) e^{izx} dx = conj( ∫ f(u) e^{i z̄ u} du ) = conj F(z̄). Hence ĝ(i/2) = F(i/2) conj F(−i/2) and ĝ(−i/2) = conj ĝ(i/2), and the pole term is the real number 2 Re[F(i/2) conj F(−i/2)]. With c = ∫ f(u) cosh(u/2) du and s = ∫ f(u) sinh(u/2) du one has F(i/2) = c − s, F(−i/2) = c + s, and (c − s)(conj c + conj s) = |c|² − |s|² + 2i Im(c conj s), so

pole term = 2|c|² − 2|s|²,

a Hermitian form of rank at most 2 and signature (1, 1), with c seeing only the even part of f and s only the odd part. It reduces to 2F(i/2)² for real even f (Zhu (2)), to −2(∫ f sinh(u/2))² for real odd f (Zhu Lemma 6.1), and it splits as pole(Re f) + pole(Im f) because cosh and sinh are real, which is the pole half of Zhu's Q(f) = Q(Re f) + Q(Im f). Two formulas that are wrong for complex f and easy to type by accident: 2F(i/2)² (complex-valued) and 2|F(i/2)|² (loses the negative odd-sector sign). Both terms ĝ(±i/2) enter with a plus sign in Zhu's normalisation, as his (2) shows for real even f. A measured check of the identity against a direct time-domain computation of ĝ(±i/2) from g = f ⋆ f̃ is in theory/pole_check.py.

• Polarised form. For f_1, f_2 ∈ W_L, ∫ F_1 conj F_2 · H = 0 (same proof with g = f_1 ⋆ f̃_2). So in any basis {T_n} of W_L, every full-line matrix entry ∫_ℝ H T̂_n conj T̂_m dt vanishes. This is a free numerical control for K3.
• Odd parts of H never help. For u = e + o (even plus odd), sup u ≥ sup e, because max(u(t), u(−t)) ≥ e(t). So one may take H even without loss.
• S_L(H) ≥ 0. P_L − H has Bohr mean 0: every frequency of P_L is log n > 0, and the mean of μ̂ is μ({0}) = 0. A bounded continuous function with mean 0 has sup ≥ 0. So S_L(H) := sup_t (P_L − H)(t) ≥ 0 for every admissible H.
• Almost periodic H. If H is Bohr almost periodic (for instance the absolutely summable cosine series of the brief), P_L − H is almost periodic, and then sup_{t ≥ T}(P_L − H) = sup_ℝ (P_L − H) for every T: near-maxima recur at arbitrarily large t. So for this class, restricting the envelope to t ≥ T# gains nothing. (A non-almost-periodic H, i.e. a measure μ with a continuous part, is outside this remark; see §1.6.) Correction (scope). Lemma 1 is proved for transforms of finite measures, which include the absolutely summable cosine series of the brief. A general Bohr almost periodic H with spectrum in |λ| ≥ 2L would need an approximation argument that is not given here; no construction in this hunt needs it.

1.4 The modified one-stroke reduction

Theorem 1′ (Zhu's Theorem 1.1 with an out-of-band correction). Let L > 0 and let H be admissible, real and even. Let S be any number with

S ≥ sup_{t ≥ T#} (P_L(t) − H(t)),

and fix T# ≥ 15/4 with

β* := log(T#/2π) − 1/T# − S > 0.

(If S ≥ sup_ℝ(P_L − H), then S ≥ 0 by §1.3, so β* > 0 already forces T# > 2π > 15/4.) Then for every real even f ∈ W_L,

Q(f) ≥ R_H(f) := 2F(i/2)² + (1/π) ∫_0^{T#} (Ψ_L(t) + H(t) − β*) |F(t)|² dt + β*‖f‖₂².

The Legendre-basis statement of Theorem 1.1 holds verbatim with C_nm = (1/π) ∫_0^{T#} (Ψ_L + H − β*) T̂_n T̂_m dt and with the constant changes listed in §1.5. The odd-sector version (Zhu §6) holds with pole term −2(∫ f sinh(x/2) dx)².

Proof. By Corollary 1, Q(f) = 2F(i/2)² + (1/2π)∫_ℝ |F|²(Ψ_L + H). For |t| ≥ T#, Zhu's Lemma 3.1 (valid for |t| ≥ 15/4) and the choice of S give

Ψ_L(t) + H(t) = a(t) − (P_L − H)(t) ≥ log(|t|/2π) − 1/|t| − S ≥ β*,

the last step because t ↦ log(t/2π) − 1/t increases. For real even f, |F|² and Ψ_L + H are even, and ∫_0^∞ |F|² = π‖f‖², so

(1/2π) ∫_{|t|≥T#} |F|²(Ψ_L + H) ≥ (β*/π) ∫_{T#}^∞ |F|² = β*(‖f‖² − (1/π)∫_0^{T#}|F|²),

which rearranges to Q(f) ≥ R_H(f). If Q(f) = +∞ there is nothing to prove, and R_H(f) is always finite (a bounded symbol on [0, T#]). ∎

For the per-prime construction of probes/separable.py the bound S is supplied prime by prime: P_L(t) = Σ_p φ_p(t log p) exactly, with φ_p(θ) = Σ_{k log p < 2L} (2 log p / p^{k/2}) cos kθ, and a pointwise identity Φ_p = M_p − φ_p + h_p ≥ 0 in θ, with h_p supported on m_p < |k| ≤ D, gives (P_L − H)(t) ≤ Σ_p M_p for all t. The frequencies of h_p(t log p) are k log p ≥ (m_p + 1) log p ≥ 2L by the definition of m_p, so H = Σ_p h_p(t log p) is admissible (a finite cosine sum).

1.5 Line-by-line check against Zhu §3–§7

"Unchanged" means the argument and its constants carry over word for word. "Changed" means the logic holds but a constant or an assembly step must be redone. Nothing below breaks the inequality Q ≥ R_H.

(b) The quadrature bound, in numbers. Lemma 5.1 continues the integrand into the strip |Im t| ≤ 0.4. There |cos(λt)| ≤ cosh(0.4 λ), so the bound |Ψ_L − β*| ≤ 20 becomes |Ψ_L + H − β*| ≤ 20 + Σ_w |b_w| cosh(0.4 λ_w). With Fejér degree D = 64 as in probes/separable.py, at L = 0.8 the largest frequency is 64 log 3 ≈ 70.3 (prime 3; prime 2 reaches 64 log 2 ≈ 44.4), and cosh(0.4 · 70.3) = cosh(28.1) ≈ 8 × 10^11. The ellipse constant M in the Gauss error bound therefore grows by up to about eleven orders of magnitude, weighted by the size of the top coefficients. This is a budget break, not a logic break: the numerics lane must either shrink the strip to δ ≈ c/λ_max with correspondingly narrower panels, or split C = C^Ψ + C^H and integrate C^H with a rule matched to its bandwidth, or keep D small. H also oscillates at frequency up to λ_max, which raises the node count per panel.

Correction (scope of (b)). Lemma 1, Theorem 1′ and the real-axis tail and coupling bounds need only a finite measure μ. The ellipse method of Lemma 5.1 needs more: H must extend analytically to the strip used and be bounded there. Finite cosine sums, which is every construction in this hunt including the numerics lane's sine-kernel witnesses, are entire. A general μ qualifies if ∫ e^{δ|λ|} d|μ|(λ) < ∞, and then |H(t)| ≤ ∫ cosh(δλ) d|μ|(λ) on |Im t| ≤ δ. Finite total variation alone gives neither (referee, REVIEW §4.2).

(d) An incidental constant in Zhu §4 (not caused by H). As printed, the row bound |C_nm| ≤ max|Ψ_L − β*| · 2√(Lν_n) (T# L)^n/(2n+1)!! · 2√(Lν_m) has no factor from the length of [0, T#]. Bounding (1/π)∫_0^{T#} by (T#/π) · sup and |j_m| ≤ 1 gives the same expression times T#/π (about 64 at T# = 200). Against Zhu's margins of 10^{-100} this is immaterial; it is recorded so the referee does not have to find it.

(e) T# ≥ 15/4. Automatic whenever S bounds the global supremum, because then S ≥ 0 (§1.3). If one uses a bound valid only on t ≥ T#, the hypothesis must be checked; for almost periodic H the two suprema coincide.

1.6 The load-bearing caveat for the numerics lane

The inequality Q ≥ R_H is valid for every admissible H. Whether the reduced form is useful is a different question, and here the out-of-band route moves into a regime Zhu never ran.

• Zhu's run at L = 0.8 used T# = 200, about six times the window's resolution height T* = 2π e^{2L} ≈ 31.1.
• With the seed constants of MISSION.md, 2π e^S is 29.6 (separable, S = 1.549) and 21.3 (comb floor, S = 1.219), both below T*; with β* = 0.5 the split point is T# ≈ 49 or ≈ 35, just above T*.
• Zhu §15, failed route (2): replacing Ψ_L beyond a height the window resolves by a constant minorant made the form indefinite (measured λ_min ≈ −2.7). R_H is exactly such a replacement above T#, by the constant β*. It remains a valid lower bound, but λ_min(R_H) is not bounded below by anything in this lane, and may be negative or far below Zhu's λ_min(R) ≈ 0.54 λ* at L = 0.8. The loss Q(f) − R_H(f) = (1/π)∫_{T#}^∞ (Ψ_L + H − β*)|F|² is paid on the minimiser, whose mass reaches up to about T*.
• A better objective. For fixed T# and S, R_H is affine in H (and in (H, S) jointly, since β* is affine in S), and the envelope constraint sup(P_L − H) ≤ S is convex. So "maximise λ_min of the leading N × N block of R_H over H, subject to the envelope" is a concave maximisation over a convex set: after truncating H to finitely many frequencies it is an SDP (an LMI for λ_min, a sum-of-squares constraint on the torus for the envelope). Minimising S alone controls the matrix size, not the loss. This is a proposal for the numerics lane, not a measured result.

1.7 Envelope threshold versus positivity operating point

Two different heights govern the reduction.

• Envelope threshold T_env(H): the T# with β* = 0, i.e. log(T#/2π) − 1/T# = S, about 2π e^S.
• Operating point T_op(H) := inf{ T# : R_{H,T#} ⪰ 0 }: where the reduced form is nonnegative. A positivity proof with a positive constant needs more, coercivity R_{H,T#} ≥ c‖·‖² with c > 0, which by Proposition 1.2 forces T# > T_env(H).

What the numerics lane has sampled (relayed by the supervisor on 2026-09-27, not read here). The evidence is a finite N = 200 Legendre block of R_H, assembled with an Arb quadrature rule whose quadrature error, tail and coupling are not yet bounded, so everything in this paragraph concerns that finite block and is measured. Per-prime H at L = 0.8: T_env ≈ 30; LDL inertia of the block shows negative eigenvalues at the sampled split points up to T# = 60 and none at the sampled T# = 65; at T# = 200 the block's leading eigenvalue is about 1.42 × 10^{−17}, below the K1 ceiling 2.27 × 10^{−17}. So the finite block has a candidate operating point between the sampled values 60 and 65, about 2 T* (T* = 2π e^{1.6} ≈ 31.1), while T_env ≈ T*. No positivity of the full reduced form at T# = 65 is proved, and a negative eigenvalue of the block is a negative Rayleigh quotient of the full form only once its entries are enclosed.

Proposition 1.2 (the operating point is a threshold). Fix H with envelope bound S, and let T# ≥ 15/4 be arbitrary, so that β* = log(T#/2π) − 1/T# − S may have either sign.

1. Q ≥ R_{H,T#}: the proof of Theorem 1′ uses only Ψ_L + H ≥ β* on |t| ≥ T#, which holds for every such T#.
2. T# ↦ R_{H,T#} is pointwise nondecreasing (the argument of the table entry for Zhu §5.5(b), which needs the same inequality on [T#, T#′]).
3. R_{H,T#} − β* I is compact on W_L (a pole form of rank at most 2 plus an integral operator with a continuous kernel band-limited to [−T#, T#]), so σ_ess(R_{H,T#}) = {β*}.

Consequently the split points where R_{H,T#} ⪰ 0 form a (possibly empty) interval unbounded above, starting at T_op(H) ≥ T_env(H); nothing proved here says it is nonempty for a given H, so T_op(H) = +∞ is allowed. Wherever β* > 0, the negative spectrum consists of finitely many eigenvalues, which is what an inertia count sees. The endpoint β* = 0 (T# = T_env): the reduced form is then a compact form, the infimum of its Rayleigh quotients is at most 0, and it yields no positive constant. Coercivity needs β* > 0, i.e. T# > T_env(H), and no nonpositive eigenvalue. Ordinary derivation; the §1.7 repairs pass the referee's ordinary analytic review (REVIEW, repair readback).

Proposition 1.3 (every out-of-band reduction is dominated by two H-free forms). Fix T# ≥ 15/4, put κ = log(T#/2π) − 1/T#, and let H be admissible with envelope bound S and β* = κ − S (either sign). For every f ∈ W_L,

R_{H,T#}(f) ≤ R′{T#}(f) ≤ B{T#}(f) ≤ Q(f),

R′{T#}(f) := pole(f) + (1/2π) ∫{|t|<T#} (a(t) − κ) |F|² dt + κ‖f‖² − ⟨Pf, f⟩, B_T(f) := pole(f) + (1/2π) ∫_ℝ (Ψ_L(t) − log(|t|/T)_+) |F|² dt.

Correction (definitions). Both are finite for every f ∈ W_L: the first has a bounded retained symbol, and the symbol of the second is bounded because a(t) − log(|t|/T)_+ → log(T/2π) at infinity. When Q(f) < ∞ they equal Q(f) − (1/2π)∫_{|t|≥T#}(a − κ)|F|² and Q(f) − (1/2π)∫_{|t|≥T} log(|t|/T)|F|². The first version of this section defined them that way on all of W_L, which is ∞ − ∞ when ∫|F|² log(2 + |t|) = ∞ (referee, REVIEW §4.4).

R′ is the operator split: the comb is kept exactly, as the time-domain operator P, and only the archimedean term is cut at T#. B_T is Weil's form with the archimedean growth removed above T, a capping of the kind in Zhu §15, failed route (2).

Proof. The first two differences below are integrals of bounded functions against |F|², and the third is an extended nonnegative quantity, so no infinite quantities are subtracted (wording corrected after the referee's repair readback). With R_H in its full-line form pole + (1/2π)∫_{|t|<T#}(Ψ_L + H − β*)|F|² + β*‖f‖², and ⟨Pf, f⟩ = (1/2π)∫ P_L|F|² = (1/2π)∫ (P_L − H)|F|² by (2.1) and Lemma 1,

R′ − R_H = (1/2π) ∫_{|t|≥T#} (S − (P_L − H)) |F|² ≥ 0, B_T − R′T = (1/2π) ∫{|t|≥T} (a(t) − κ − log(|t|/T)) |F|² ≥ 0, Q − B_T = (1/2π) ∫_{|t|≥T} log(|t|/T) |F|² ∈ [0, +∞].

The first uses P_L − H ≤ S on |t| ≥ T#; the second uses Zhu's Lemma 3.1 and 1/T ≥ 1/|t|, which give a(t) − κ ≥ log(|t|/T) for |t| ≥ T ≥ 15/4. ∎

Proposition 1.4 (an H-independent floor). R′_T and B_T are nondecreasing in T, so with T′_op := inf{T : R′_T ⪰ 0} and T_res := inf{T : B_T ⪰ 0},

T_op(H) ≥ T′_op ≥ T_res ≥ 2π e^{λ_max(P)} = 2π e^{S*_L} for every admissible H.

Proof of the last inequality. The symbol a(t) − log(|t|/T)_+ − log(T/2π) is bounded, continuous and tends to 0 at ±∞; a Fourier multiplier with such a symbol, compressed to W_L, is compact (a norm limit of compressions with compactly supported symbols, which are Hilbert–Schmidt). So B_T = log(T/2π) I − P + (compact), and by Weyl's theorem inf σ_ess(B_T) = log(T/2π) − sup σ_ess(P). And sup σ_ess(P) = λ_max(P): if ⟨Pφ, φ⟩ > λ_max − ε with ‖φ‖ = 1, choose τ_k → ∞ by Dirichlet's simultaneous approximation with τ_k log p ∈ 2πℤ + o(1) for the finitely many primes p < e^{2L}; then φ_k = e^{iτ_k x} φ is weakly null and ⟨Pφ_k, φ_k⟩ → ⟨Pφ, φ⟩. So B_T ⪰ 0 forces log(T/2π) ≥ λ_max(P). ∎ (Ordinary derivation. On the even sector it needs the correction below, which the referee's repair readback accepts.)

Correction (the even sector). Theorem 1′ and the numerics work on real even f, while Propositions 1.2 to 1.4 were stated on complex W_L. All of them hold on the even sector too. Every form here splits as f ↦ form(Re f) + form(Im f) (even symbols, the pole splitting of §1.3, and P real and symmetric), so nonnegativity on real even and on complex even functions agree, and the chain T_op ≥ T′_op ≥ T_res holds sector by sector with the same proofs. The last step needs sup σ_ess(P|_even) = λ_max(P). Proof: with φ and τ_k as above, ψ_k(x) = e^{iτ_k x} φ(x) + e^{−iτ_k x} φ(−x) is even. P commutes with x ↦ −x, so the two diagonal terms of ⟨Pψ_k, ψ_k⟩ are equal and tend to ⟨Pφ, φ⟩. The cross terms are integrals ∫ e^{2iτ_k x} g(x) dx of L¹ functions and tend to 0 (Riemann–Lebesgue), as does the cross term of ‖ψ_k‖², so ‖ψ_k‖² → 2‖φ‖²; and ψ_k ⇀ 0. (Ordinary derivation; this closes the gap noted in PROGRESS.md, and the referee's repair readback accepts it.)

A proved bracket at L = 0.8, conditional on Zhu's computation. Zhu's H = 0 runs at T# = 150 cover both sectors: the even one in his §5.5 (a) (β* = 0.2241, λ_min ≥ 1.2 × 10^{−18}, with the tail bound at T# = 150 quoted in his §5.3) and the odd one in his §6.2 (Q ≥ 8.2 × 10^{−15}‖f‖² on real odd f from the odd reduced matrix at T# = 150). So R_{0,150} ⪰ 0 on both sectors, hence on W_L; on the even sector alone §5.5 (a) suffices. (Correction: the first version cited only §5.5 (a) for "the whole window".) Proposition 1.3 with H = 0 gives R_{0,150} ≤ B_{150}, so B_{150} ⪰ 0 and T_res(0.8) ≤ 150. With Proposition 1.4 and the Rayleigh value λ_max(P) ≥ 1.2186 (Galerkin, float), T_res(0.8) ∈ [≈ 21.3, 150]. Grades: the inclusion is an ordinary derivation; its upper end rests on Zhu's computer-assisted result (unrefereed), its lower end on a float Rayleigh value (measured).

A tighter bracket on the even sector. The referee has since reproduced the numerics lane's T# = 100 result independently (REVIEW §5: R_H ≥ 1.1579 × 10^{−17} on the full even sector, hardened, two CC/Arb resolutions). With Proposition 1.3 this gives B_{100} ⪰ 0 there, so on the even sector T_res(0.8) ∈ [≈ 21.3, 100]. Grades: the upper end is an ordinary derivation from a hardened premise, the lower end rests on the same float Rayleigh value (measured). The heuristic 2T* ≈ 62 below and the numerics lane's sampled candidate near 65 both lie inside.

This recovers the barrier of Corollary 3 by a second route, and shows the floor applies to all H at once: the out-of-band freedom lowers T_env, but the operating point can never go below T_res(L), which depends on Q and the archimedean growth only.

Why T_res should track T* (heuristic, not proved). Under RH, Q(f) = Σ_γ 2|F(γ)|², and a Riemann-sum reading gives the zeros above T weight about (1/π) log(t/2π) dt; B_T then looks like the same sum with the zero density above T frozen at ν(T) = log(T/2π)/2π. A function of exponential type L can hide between zeros only where the density is below its Nyquist density L/π, that is below T* = 2π e^{2L}. So B_T should admit "fake zeros" (Zhu's phrase for his route (2)) unless T exceeds T* with some margin. The finite-block candidate near 2 T* at L = 0.8 is consistent with this reading. It is not a bound on T_res: that would need the full reduced form shown nonnegative at some T#, after which Proposition 1.4 gives T_res ≤ T#.

Consequences.

• Cost estimates must use N ≈ eL · max(T_env e^{β*}, T_op)/2, not eL · T_env/2. At L = 0.8 the sampled finite blocks suggest that the operating point, not the envelope, binds (candidate T# near 65, where N ≈ 71). Whatever T_res turns out to be, no H can go below it. Correction: the first version added that a better joint H "helps only where the envelope is the binding height", which is too strong. Above T_res a better H can still lower T_op(H) toward T′_op when the gap R′ − R_H is what keeps R_H indefinite, and it raises λ_min at a fixed T#: at L = 0.8, T# = 200 the numerics lane has λ_min ∈ [1.02, 1.028] × 10^{−17} with H = 0 (enclosure-carrying by that lane's account) against 1.42 × 10^{−17} with per-prime H (measured, N = 200).
• At L = 1.19, T* ≈ 68. If the ratio T_op/T* ≈ 2 carried over (not measured), the per-prime envelope (2π e^{S_sep + 0.5} ≈ 450) would bind, and the joint optimum (≈ 150) would sit near the resolution floor.
• R′ is never worse than any R_H and needs no H, so none of the quadrature cost of §1.5 (b). Its drawback is structural: P is a sum of shifts and does not localise in Legendre order (Zhu §15, failed route (3), O(1) coupling across any order cut), so passing from a finite block to the whole space needs a Schur complement with κI − P inverted on the complement. Liu's Theorem B does exactly that with κ replaced by 7/2 (§4.2). The out-of-band R_H keeps Zhu's frequency split, with super-exponentially small coupling, at the price of the gap R′ − R_H = (1/2π)∫_{|t|≥T#} (S − (P_L − H)) |F|² ≥ 0.
• At fixed T# the out-of-band gain can be spent on β* instead of on matrix size (§3.6). At T# = 200 the numerics lane's leading eigenvalue with per-prime H is 1.42 × 10^{−17}, against a true floor of about 1.66 × 10^{−17} (Zhu, measured); Zhu's H = 0 run at the same T# passed a Cholesky shift at 9 × 10^{−18}, which bounds its λ_min only from below. Whether H raised λ_min at fixed T# needs the H = 0 value from the same pipeline; the heuristic of §3.6 predicts it did.
• Request for the numerics lane: λ_min(B_T) (or λ_min(R′_T)) on a large Legendre basis at L = 0.8 for T ∈ [30, 70]. The crossing is a finite-block estimate of T_res, the floor for every H; if it sits near the per-prime candidate, the per-prime H is already near the floor for positivity at L = 0.8. Both would remain candidates until the blocks' entries, tails and couplings are enclosed.

2. Task 2: the optimal constant (weak and strong duality)

2.1 The two problems

Write w_n = Λ(n)/√n and let n range over prime powers with log n < 2L. The windowed comb operator on L²[−L, L] is

(Pφ)(x) = Σ_n w_n [φ(x − log n) + φ(x + log n)], x ∈ [−L, L],

with φ = 0 outside [−L, L]. It is bounded, self-adjoint and positivity preserving, with ‖P‖ ≤ A_L. Put λ_max(P) = sup spec(P). For φ ∈ W_L with transform Φ, (1/2π)∫|Φ|² e^{iat} dt = g_φ(−a) (step (ii) of §1.2), which gives the identity

⟨Pφ, φ⟩ = (1/2π) ∫_ℝ |Φ(t)|² P_L(t) dt. (2.1)

The out-of-band problem is S*_L := inf_H S_L(H), the infimum over real even admissible H of S_L(H) = sup_t (P_L − H)(t). Three classes of H give the same infimum (by 2.2 and 2.3 below): finite real even trigonometric polynomials with frequencies |λ| ≥ 2L, absolutely summable cosine series, and transforms of finite real symmetric measures carried by |λ| ≥ 2L.

2.2 Weak duality

Proposition 2.1. For every admissible H, S_L(H) ≥ λ_max(P).

Proof. By (2.1) and Lemma 1, for φ ∈ W_L, ⟨Pφ, φ⟩ = (1/2π)∫|Φ|²(P_L − H) ≤ S_L(H) · (1/2π)∫|Φ|² = S_L(H)‖φ‖². ∎

So S*_L ≥ λ_max(P), and the seed column S_opt of MISSION.md (a piecewise-constant Galerkin value, which approximates λ_max(P) from below) is a floor for every out-of-band construction.

2.3 Strong duality

Let p_1 < … < p_r be the primes below e^{2L}, ℓ = (log p_1, …, log p_r) and s(k) = k · ℓ for k ∈ ℤ^r. Unique factorisation makes the log p_i linearly independent over ℚ, so s is injective. Every comb frequency is log p_i^m = s(m e_i), and K := { k : |s(k)| < 2L } (the slab) contains all of them.

Theorem 2 (strong duality). For every L > 0,

S*_L = inf_H sup_t (P_L − H)(t) = λ_max(P),

the infimum taken over real even trigonometric polynomials whose frequencies lie in { s(k) : k ∈ ℤ^r, |s(k)| ≥ 2L }. Quantitatively: for every t ≥ λ_max(P) and every integer N ≥ 1 there is such an H_N, with frequencies s(k) for k ∈ [−2N, 2N]^r, satisfying

S_L(H_N) ≤ t + Σ_{p^m < e^{2L}} (2 log p / p^{m/2}) (1 − ρ_N(m)) ≤ t + (π² / (4(N+1)²)) Σ_{p^m < e^{2L}} m² log p / p^{m/2},

with ρ_N(m) the explicit lag-m autocorrelation of Step 3. When 2N + 1 ≥ m for every exponent m in the comb, the middle term is also at most Σ (2 log p/p^{m/2})(1 − cos(πm/(2N+2))). Correction: the first version printed that cosine form for every N ≥ 1. It fails once some exponent exceeds 2N + 1: at N = 1, m = 8 the weights live on {−1, 0, 1}, so ρ_1(8) = 0 against cos 2π = 1 (such exponents occur once L > 4 log 2). The quadratic bound holds for every N ≥ 1 by the referee's argument, now in Step 3 (REVIEW §4.1). The identity of the infima and the N^{−2} rate are unaffected.

The infimum is in general not attained (for a single prime it needs a Fejér limit, D → ∞ in probes/separable.py).

Proof. Weak duality is Proposition 2.1. For the other direction fix t ≥ λ_max(P) and define ψ : K → ℝ by ψ(0) = t, ψ(±m e_i) = −w_{p_i^m}, and ψ(k) = 0 for every other k ∈ K.

Step 1 (the window gives positivity on strips). Let F ⊂ ℤ^r be finite with diam s(F) < 2L. Then the matrix [ψ(j′ − j)]_{j, j′ ∈ F} is positive semidefinite. Indeed, choose x_0 with x_0 + s(F) ⊂ (−L, L) and for ε > 0 put φ_ε = Σ_{j∈F} a_j ε^{−1/2} 1_{[x_0 + s(j), x_0 + s(j) + ε)}. The points x_0 + s(j) are distinct (injectivity), and a shifted bump overlaps another only if s(j′) − s(j) = ±log n up to ε. For ε below the least nonzero value of |s(j′) − s(j) ∓ log n| over the finitely many triples, only exact coincidences remain, and s(j′ − j) = s(m e_i) forces j′ − j = m e_i. Hence 0 ≤ ⟨(t − P)φ_ε, φ_ε⟩ = Σ_{j,j′} a_j conj a_{j′} ψ(j′ − j).

Step 2 (completion along the order). Let F ⊂ ℤ^r be finite, listed as j_1, …, j_M in increasing order of s. Consider the partial symmetric matrix with entries ψ(j_a − j_b) specified exactly when |s(j_a) − s(j_b)| < 2L. It has a positive semidefinite completion M̃. Proof by induction on M: suppose the first k points are completed to M_k ⪰ 0. The earlier points specified against j_{k+1} form a contiguous block B = {j_a : s(j_{k+1}) − s(j_a) < 2L}, and B ∪ {j_{k+1}} has s-diameter < 2L, so by Step 1 the fully specified matrix [[M_BB, v], [v*, t]] is positive semidefinite; in particular v ∈ range(M_BB) and t − v* M_BB⁺ v ≥ 0. Fill the unspecified entries against the remaining earlier points U by u = M_UB M_BB⁺ v. The new column is w = M_k E_B M_BB⁺ v ∈ range(M_k) (with E_B the coordinate inclusion of B), and w* M_k⁺ w = v* M_BB⁺ v ≤ t, so the extended matrix is positive semidefinite. (This is the standard completion argument for chordal patterns: R. Grone, C. R. Johnson, E. M. Sá and H. Wolkowicz, Positive definite completions of partial Hermitian matrices, Linear Algebra Appl. 58 (1984) 109–124; bibliographic data and statement checked against a secondary summary, paper not re-read. The pattern here is a unit interval graph on the real line through s, hence chordal, and the proof above is self-contained.)

Step 3 (averaging over a box). Take F = F_N = {−N, …, N}^r with weights α_j = Π_i cos(π j_i/(2N+2)) > 0, Z = Σ α_j², and set

c(k) = (1/Z) Σ_{j − j′ = k} α_j α_{j′} M̃_{j j′}.

For finitely supported a, Σ_{k,k′} a_k conj a_{k′} c(k − k′) equals (1/Z) times the Frobenius product of M̃ with the positive semidefinite matrix [α_j β(j − j′) α_{j′}], β = a ⋆ ã; so c is positive definite on ℤ^r, supported in [−2N, 2N]^r. For k ∈ K every pair with j − j′ = k was specified with value ψ(k), so c(k) = ψ(k) ρ(k) with ρ(k) = Σ_{j−j′=k} α_j α_{j′} / Z. For k = m e_i this is the one-dimensional ratio ρ_N(m) = [(n−1−m) cos(πm/n) + sin(π(m+1)/n)/sin(π/n)]/n, n = 2N + 2, valid for 0 ≤ m ≤ 2N + 1; for m ≥ 2N + 1 there are no pairs at that lag and ρ_N(m) = 0. For m ≤ 2N + 1, sin((m+1)x)/sin x = Σ_{q=0}^m cos((m − 2q)x) ≥ (m+1) cos(mx) because |m − 2q| x ≤ mx < π, so ρ_N(m) ≥ cos(πm/n). For every m ≥ 0 (the referee's argument, REVIEW §4.1): let v be the one-dimensional weight vector extended by zero and S the unit shift on ℓ²(ℤ); then 2(1 − ρ_N(m))‖v‖² = ‖v − S^m v‖² ≤ m²‖v − Sv‖² = 2m²(1 − ρ_N(1))‖v‖², and ρ_N(1) = cos(π/n), so 1 − ρ_N(m) ≤ m²(1 − cos(π/n)) ≤ π²m²/(8(N+1)²).

Step 4 (the nonnegative polynomial). μ(θ) := Σ_k c(k) e^{ik·θ} is a real even trigonometric polynomial on T^r, and μ(θ) = (1/Z) Σ_{j,j′} (α_j e^{ij·θ}) M̃_{jj′} (α_{j′} e^{−ij′·θ}) ≥ 0 because M̃ ⪰ 0. Split μ = μ_K + H̃_N, where μ_K collects the coefficients with k ∈ K and H̃_N the rest. By Step 3,

μ_K(θ) = t − Σ_{p^m} 2 w_{p^m} ρ_N(m) cos(m θ_i) (i the index of p),

so, with Φ(θ) = Σ_{p^m} 2 w_{p^m} cos(m θ_i) (the comb on the torus, P_L(t) = Φ(tℓ)), the inequality μ ≥ 0 reads

Φ(θ) − H̃N(θ) ≤ t + Σ{p^m} 2 w_{p^m} (1 − ρ_N(m)).

Step 5 (back to the line). Put H_N(t) = H̃_N(tℓ), a real even trigonometric polynomial whose frequencies are s(k) with k ∉ K, i.e. |s(k)| ≥ 2L: admissible. Evaluating Step 4 at θ = tℓ gives the bound on sup_t (P_L − H_N), and 1 − ρ_N(m) ≤ π²m²/(8(N+1)²) for every m (Step 3). Letting N → ∞ and t ↓ λ_max(P) gives S*_L ≤ λ_max(P). ∎

Remarks.

• Kronecker's theorem is not needed for Theorem 2: Step 5 only uses that t ↦ tℓ maps into the torus. (It does show the torus problem and the line problem have the same value, which is how the probes are organised.)
• For one prime (r = 1) the theorem is the Carathéodory–Toeplitz statement of MISSION.md: λ_max(P) is then the top eigenvalue of the (m+1) × (m+1) Toeplitz matrix with first row (0, t_1, …, t_m), because the window graph is a union of arithmetic progressions of step log p with at most m + 1 terms. S_sep = Σ_p λ_max(T_p) is Theorem 2 applied one prime at a time, and weak duality gives S_sep ≥ λ_max(P).
• Why a Perron–Frobenius argument alone does not close the gap: the dual objects are vectors on ℤ^r whose autocorrelation vanishes off the slab, and taking absolute values destroys that cancellation. What closes it is that s totally orders ℤ^r and the slab is an interval for that order, so the specification pattern is an interval graph (Step 2). This is a Krein-type extension property for the ordered group (ℤ^r, s): a function positive definite on every strip of width 2L extends to a positive definite function on ℤ^r. In two or more dimensions with box-shaped patterns the analogous extension is known to fail (W. Rudin, The extension problem for positive-definite functions, Illinois J. Math. 7 (1963) 532–539; bibliographic data checked, content as recalled, not re-read); the irrational order makes the problem one-dimensional.
• The construction converts any upper bound t on λ_max(P) into an explicit H. Where λ_max(P) is not known in closed form, a Collatz–Wielandt bound t = sup_x (Ph)(x)/h(x) for a positive trial h (valid because P preserves positivity) supplies it.

2.4 Exact values while the window graph has finite components

Join x, y ∈ [−L, L] when y − x = ±log n for a comb term n. P is the weighted adjacency operator of this graph against Lebesgue measure.

Proposition 2.2. If every component has at most K vertices for some K, then up to null sets [−L, L] splits into finitely many families of translated base intervals, one family per combinatorial type τ with k_τ vertices and weighted adjacency matrix A_τ, and P ≅ ⊕_τ A_τ ⊗ I_{L²(B_τ)}. Hence spec P = ∪_τ spec A_τ and λ_max(P) = max_τ λ_max(A_τ), attained with infinite multiplicity. (The type changes only where some x + s(k) crosses ±L, finitely many breakpoints, so each base set is a finite union of intervals.) Ordinary derivation; the referee accepts the uniform-size hypothesis and notes that the census below is sampled (REVIEW §1, §4.4).

Measured census (float64 breadth-first search, points merged at 1e-9; 4000 random base points plus a grid of 160 001; one route):

At L = 0.6 the top type is the path x, x − log 2, x − log 2 + log 3, x − 2 log 2 + log 3 with weights (a, b, a), a = w_2, b = w_3, and λ_max² = [(2a² + b²) + √((2a² + b²)² − 4a⁴)]/2, λ_max = 0.9009276 (closed form; the float value agrees with the census). At L = 0.8 the top type has 12 vertices and 13 edges (two triangles from log 2 + log 2 = log 4) and occupies 71.9% of the window; a 13-vertex "type" seen once in a random sample was two float copies of one point 3e−17 apart. Correction: the census is sampled, so it does not prove that no other component type occurs (referee, REVIEW §4.4); a proof would enumerate the breakpoints of Proposition 2.2 exactly. Given the types, the two values are eigenvalue problems of size 4 and 12, which ball arithmetic can enclose; that is for the numerics lane.

The regime ends soon after 5 enters the comb at L = log 5 / 2 ≈ 0.8047. Measured (200 base points, cap 2000 vertices): largest finite component 24 vertices at L = 0.81, 92 at L = 0.83; at L = 0.85 89.5% of samples exceed the cap, at L = 0.88 all do. Without the prime 5 the components stay finite to at least L = 0.88. So at L = 1.0 and L = 1.19 the window operator has (numerically) infinite quasi-periodic components, Proposition 2.2 does not apply, and λ_max(P) is available only as a bracket: Galerkin or Rayleigh quotients from below, Collatz–Wielandt from above.

2.5 Measured check of the construction

theory/duality_check.py runs Steps 2 to 5 literally (float64, one route, t = λ_max(P) + 10^{−9}, the torus sampled on a 256 × 256 grid). Output (hunts/oob_envelope/theory/duality_check.py, about 5 s):

The in-slab coefficients of μ other than those of t − Φ vanish to machine precision, and the completed matrices have least eigenvalue about 1e−10 (the margin 10^{−9} in t). The joint out-of-band constant beats the per-prime one by 0.22 at L = 0.6 and 0.29 at L = 0.8, and the gap to the floor closes like N^{−2}, as the proof bound says.

The price is visible in the last two columns: the H that approaches the floor has large coefficients and high frequencies, which is exactly what inflates the quadrature constant of §1.5 (b). The matrix size gained by a smaller S has to be weighed against that, and against §1.6.

2.6 Finite-measure H outside the almost periodic class

The supervisor asked whether transforms of finite measures with a continuous part change the duality. Answer: not for the constant envelope that Theorem 1′ uses, even restricted to a tail t ≥ T; they could matter only for a pointwise envelope, and there they pay exponentially in total variation.

Proposition 2.3 (continuous parts do not lower the constant). Let μ be a finite real symmetric Borel measure carried by {|λ| ≥ 2L}, H = μ̂, and T ∈ ℝ. Then

sup_{t ≥ T} (P_L − H)(t) ≥ λ_max(P).

Proof. Split μ = μ_d + μ_c into its atomic and continuous parts. μ_d is again finite, real, symmetric and carried by {|λ| ≥ 2L}, so u := P_L − μ̂_d is Bohr almost periodic and, by Proposition 2.1, sup_ℝ u ≥ λ_max(P). Fix ε > 0. u is uniformly continuous and its ε-almost periods are relatively dense, so E = {t : u(t) > sup u − ε} contains an interval of fixed length 2δ > 0 inside every interval of some fixed length ℓ_ε; hence |E ∩ [T, T + X]| ≥ c X for large X, with c > 0. By Wiener's theorem, (1/2X) ∫_{−X}^{X} |μ̂_c|² → Σ_x |μ_c({x})|² = 0, so |{t ∈ [T, T + X] : |μ̂_c(t)| ≥ ε}| ≤ ε^{−2} ∫_T^{T+X} |μ̂_c|² = o(X). For large X the two sets meet, and at a common point (P_L − H)(t) = u(t) − μ̂_c(t) > sup u − 2ε. ∎

So over every class between trigonometric polynomials and finite measures the constant-envelope optimum is the same number, λ_max(P): Proposition 2.1 bounds all of them below, Theorem 2 attains it with trigonometric polynomials, and Proposition 2.3 closes the tail loophole.

Where a non-almost-periodic H could still act. The proof of Theorem 1′ uses only the pointwise condition Ψ_L(t) + H(t) ≥ β* for |t| ≥ T#, which is weaker than a constant bound because a(t) increases. Zhu §7 already exploits this for H = 0 (interval evaluation of Ψ_L on a finite range lowers T# at L = 1.19 from about 1.2 × 10^4 to about 8.5 × 10^3). The same applies with any H, and a localised out-of-band wave packet (a measure with a smooth density on |λ| ≥ 2L, modulated to sit at a chosen height) can in principle lift Ψ_L + H where an almost periodic envelope dips. The cost is exponential in the length lifted:

Proposition 2.4 (cost of a localised lift). Let μ be a finite complex measure carried by {|λ| ≥ a}, a > 0, H = μ̂, and suppose Re H ≥ h_0 > 0 on an interval I of length ℓ ≥ 2e/a. Then

‖μ‖ ≥ h_0 · exp(⌊aℓ/(2e)⌋), i.e. ‖μ‖ ≥ h_0 · exp(⌊Lℓ/e⌋) for a = 2L.

Proof. Put q = ⌊aℓ/(2e)⌋ ≥ 1 and let k be the q-fold convolution of the uniform probability density on an interval of length ℓ/q, translated into I. Then k ≥ 0, ∫ k = 1, supp k ⊆ I, and |k̂(λ)| = |sin(λℓ/2q)/(λℓ/2q)|^q ≤ (2q/(aℓ))^q ≤ e^{−q} for |λ| ≥ a. By Fubini, h_0 ≤ Re ∫ H k = Re ∫ k̂ dμ ≤ ‖μ‖ e^{−q}. ∎

At L = 0.8 a lift over a stretch of length 10 costs a factor at least e^2 ≈ 7.4 in total variation, over length 100 at least e^{29}. Since ‖H‖_∞ ≤ ‖μ‖ enters the retained matrix on [0, T#] and the quadrature bound of §1.5 (b), long lifts are not a practical route. Short lifts at the isolated near-alignment spikes of an almost periodic envelope are not excluded by this bound. The best pointwise threshold over finite-measure H is unresolved here; it is a different optimisation problem from the one Theorem 2 solves, and no duality statement is claimed for it.

2.7 Cross-check against the numerics lane

The numerics lane reports enclosure-carrying per-prime constants at Fejér degree D = 128 (sine-kernel decomposition plus an adaptive Arb scan in θ): S(0.8) = 1.52268, S(1.0) = 2.97835, S(1.19) = 3.76891 (numerics/PROGRESS.md on teal-sea/oob-cert, relayed by the supervisor; not read here). The per-prime infimum over all degrees is Σ_p λ_max(T_p) (Theorem 2 with r = 1, one prime at a time), which float64 evaluation gives as 1.522051, 2.977298, 3.767084 (measured, one route). Every reported value lies above its limit, by 6 × 10^{−4}, 1.1 × 10^{−3} and 1.8 × 10^{−3}, as it must: a per-prime enclosure below these numbers would contradict weak duality and would signal a bug.

3. Task 3: asymptotics of S*_L

3.1 Statement

By Theorem 2, S*_L = λ_max(P). Let κ_L be the unique root > 1/2 of κ tanh(κL) = 1/2 and put λ_mod(L) = 1/(κ_L² − 1/4). Expanding κ_L = 1/2 + δ gives δ e^{2κ_L L} = 1 + δ, so δ = e^{−L}(1 + O(L e^{−L})) and

λ_mod(L) = e^L + 2L − 2 + O(L² e^{−L}).

Theorem 3. As L → ∞,

S*_L = λ_max(P) = λ_mod(L) · (1 + O(e^{−c√L})) = e^L (1 + O(e^{−c√L}))

for an absolute constant c > 0. The only arithmetic input is the prime number theorem with the de la Vallée Poussin error term, ψ(u) = u + O(u e^{−c₀ √(log u)}); with only ψ(u) ~ u the conclusion is S*_L ~ e^L. The constant is not made effective here.

Companion facts (same input): A_L = (4 + o(1)) e^L (Zhu Theorem 1.4) and S_sep = Σ_p λ_max(T_p) = (2 + o(1)) e^L (§3.4). So the three envelope constants run 4 : 2 : 1 in units of e^L.

Corollary 3 (the barrier with out-of-band freedom). Every application of Theorem 1′ with a constant envelope, for any finite-measure H, needs T# > 2π e^{S*_L} = 2π exp((1 + o(1)) e^L). The threshold stays doubly exponential in L. Asymptotically it is the fourth root of Zhu's T_1 = 2π exp((4 + o(1)) e^L) and the square root of the per-prime threshold. Proof: Proposition 2.1 (all finite-measure H), Proposition 2.3 (tails), Theorem 3.

3.2 The model operator

Let W(y) = Σ_{log n ≤ y} Λ(n)/√n (a right-continuous step function, W = 0 on [0, log 2)). Partial summation with ψ(u) = u + E(u) gives

W(y) = 2e^{y/2} − 1 + R(y), R(y) = E(e^y) e^{−y/2} + ½ ∫_1^{e^y} E(u) u^{−3/2} du,

so with the model W_0(y) = 2(e^{y/2} − 1), whose density is e^{y/2}, the difference D = W − W_0 = 1 + R satisfies D(0) = 0, |D(y)| ≤ C_0 e^{y/2} for all y ≥ 0 (Chebyshev), and η(Y) := sup_{y ≥ Y} |D(y)| e^{−y/2} = O(e^{−c₁√Y}) (de la Vallée Poussin).

The model operator (Kh)(x) = ∫_{−L}^{L} e^{|x − x′|/2} h(x′) dx′ replaces the comb by its density. Since (d²/dx² − 1/4) e^{|u|/2} = δ(u), Kh solves (Kh)″ − (Kh)/4 = h on (−L, L) with (Kh)′(L) = (Kh)(L)/2 and (Kh)′(−L) = −(Kh)(−L)/2. For h_L(x) = cosh(κ_L x), the function λ_mod h_L satisfies the same equation and boundary conditions (that is the defining equation of κ_L), and the difference of two solutions is αe^{x/2} + βe^{−x/2}, which the boundary conditions force to 0. So K h_L = λ_mod(L) h_L exactly.

3.3 Proof of Theorem 3

For x ∈ [−L, L] write (Ph)(x) = ∫_{(0, x+L]} h(x − y) dW(y) + ∫_{(0, L−x]} h(x + y) dW(y) and the same with W_0 for Kh. With h = h_L and κ = κ_L ∈ [1/2, 1] (true for L ≥ 0.55), integration by parts against D (D(0) = 0) gives

(Ph)(x) − λ_mod h(x) = E_1(x) + E_2(x), E_1(x) = h(−L) D(x + L) + ∫_0^{x+L} D(y) h′(x − y) dy,

and E_2 symmetric. Bound each piece relative to h(x) ≥ e^{κ|x|}/2, using |D(y)| ≤ η(Y) e^{y/2} for y ≥ Y and ≤ C_0 e^{y/2} below:

• boundary term: ≤ 2η(Y) e^{(κ + 1/2)L} when x + L ≥ Y, and ≤ 2C_0 e^{(κ + 1/2)Y} otherwise (then |x| ≥ L − Y);
• the part y > x of the integral: ≤ 2η(Y) e^{(κ+1/2)L}/(κ + 1/2) plus 2C_0 e^{(κ+1/2)Y}/(κ + 1/2) from y < Y;
• the part y ≤ x: ≤ 2κ C_0 L.

With e^{(κ_L + 1/2)L} = e^L e^{δL} = e^L (1 + o(1)) and Y = L/2, all of this is O(η(L/2) e^L + e^{L/2 + o(L)} + L) = e^L · O(e^{−c√L}). Hence |(Ph)(x)/h(x) − λ_mod(L)| ≤ λ_mod(L) · O(e^{−c√L}) uniformly on [−L, L]. Then:

• upper bound (Collatz–Wielandt, valid because P preserves positivity): λ_max(P) ≤ sup_x (Ph)(x)/h(x);
• lower bound (Rayleigh): λ_max(P) ≥ ⟨Ph, h⟩/‖h‖² ≥ inf_x (Ph)(x)/h(x).

Both are λ_mod(L)(1 + O(e^{−c√L})). ∎ (Ordinary derivation; the referee inspected it and found no gap, subject to the stated prime number theorem input, REVIEW §4.4.)

3.4 Explicit bounds for every L, and the per-prime constant

• Lower bound, fully explicit. The Rayleigh quotient of h(x) = cosh(x/2) is an exact finite sum:

S*L ≥ ℓ(L) := 2 Σ{log n < 2L} (Λ(n)/√n) [sinh(L − ½log n) + (L − ½log n) cosh(½log n)] / (L + sinh L),

valid for every L > 0 (Rayleigh principle plus Theorem 2), with ℓ(L) ~ e^L. By Corollary 3, 2π e^{ℓ(L)} is a lower bound for the split point of every constant-envelope out-of-band reduction.

• Upper bounds. S*_L ≤ S_sep(L) for every L (weak duality one prime at a time), and S*_L ≤ sup_x (Ph)(x)/h(x) for any positive h; with h = cosh(κ_L x) this is a finite expression whose supremum over x can be enclosed, and it is e^L(1 + o(1)).
• S_sep ~ 2e^L. Interlacing with the leading 2 × 2 block gives λ_max(T_p) ≥ log p/√p, so S_sep ≥ Σ_{p < e^{2L}} log p/√p ~ 2e^L. For p > e^L only k = 1 occurs and λ_max(T_p) = log p/√p exactly; for p ≤ e^L the row-sum bound λ_max(T_p) ≤ 2 log p/(√p − 1) sums to O(e^{L/2}). Hence S_sep = (2 + o(1)) e^L: the per-prime correction halves every large prime's cosine and no more, while the joint optimum halves the total again.

3.5 Measured table

theory/asymptotics_check.py (float64, one route, about 10 s). All columns divided by e^L; gal is the Galerkin value of λ_max(P) from below (M = 800 cells), cw the Collatz–Wielandt ratio on a 20 001-point grid (an estimate of an upper bound, not an enclosure).

The ratio gal/e^L rises to about 1.12 and then falls toward 1, as λ_mod/e^L = 1 + (2L − 2)e^{−L} + … predicts; the explicit bound ℓ(L) is within 1% of the Galerkin value from L = 2 on. At L = 1.19 the implied thresholds are 2π e^{A_L} ≈ 7.4 × 10^3, 2π e^{S_sep} ≈ 272 and 2π e^{S*} ≈ 90, against the resolution height T* = 2π e^{2L} ≈ 68.

3.6 What this decides

• The doubly exponential barrier keeps its shape (Corollary 3), as the mission expected. The out-of-band freedom buys exactly a factor 4 in the exponent's constant, asymptotically, and no constant envelope of any finite-measure H can buy more.
• The factor 4 → 1 splits as 4 → 2 (per prime, each large prime's cosine halved) and 2 → 1 (joint: the optimal window function is ≈ cosh(x/2), concentrated at both ends of the window, and correlates all primes at once).
• Practical note for L = 1.19. With β* = 0.5 the per-prime split point is T# ≈ 2π e^{S_sep + 0.5} ≈ 450, about 6.6 T*, the same ratio as Zhu's successful run at L = 0.8 (200/31). So the §1.6 risk is smaller at L = 1.19 with the per-prime envelope than at L = 0.8 with the joint one. At L = 0.8 the gain can instead be spent on β* at fixed T#: at T# = 200, β* = log(200/2π) − 1/200 − S is 0.513 with A_L, 1.933 with S_sep and 2.236 with S*, which shrinks the discarded tail. That is a suggestion, not a measured result.

4. Task 4: prior art and the current record

4.1 The record, in one normalization

Normalization: additive variable u = log x, and L is the half-width of supp f. A multiplicative interval [λ^{−1}, λ] (measure dx/x) becomes L = log λ; the autocorrelation is supported in [−2L, 2L], and Zhu's "support 1.6" means 2L = 1.6. Liu uses the same form as Zhu (his eq. (1), normalised against Connes–Consani 2021, Appendix B, (149)–(154)).

Current record. Refereed: L = (log 2)/2 (Yoshida 1992). Unrefereed and computer-assisted: L = 17/16 (Liu), then L = 0.8 (Zhu). A valid positivity bound at L = 1.19 would exceed every window claim found, refereed or not. Liu's subtitle is quoted as the brief gives it; the full title contains the reserved word and is not reproduced.

Two normalization traps met on the way (both from a search summary, both wrong on reading the sources): "Bombieri proves L ≤ log 2" (his hypothesis |I| < log 2 is on the total length of supp F, so L < (log 2)/2, and positivity needs |I| small) and "Burnol proves L ≤ √2" (√2 is his multiplicative endpoint c, so L = log c ≤ (log 2)/2, and he proves existence of some c > 1 only).

4.2 Has the out-of-band freedom been used before?

Yes, in its simplest form: Burnol 2000. In the English proof of Théorème 3.7 (arXiv:math/0101068, p. 2), with ε = 2 log c the edge of the autocorrelation support, Burnol writes the symbol as α(τ) = 8√2 cos(log(2)τ)/(1 + 4τ²) + h_+(τ), notes that A_ε cos(ετ) + α(τ) ≥ 0 for all τ once ε is small and A_ε suitable, and uses ∫ cos(ετ)|ĝ(s)|² dτ/2π = Re(e^{ε/2} k(e^ε)) = 0 for g supported in [1/c, c]. That is Lemma 1 for a single cosine at the boundary frequency 2L, used to make the archimedean-plus-pole symbol pointwise nonnegative (no primes are present at that support). The same proof also rewrites the pole term as a multiplier by an identity valid on the support [1/2, 2]. So the device is not novel.

Also related, but not the same use.

• Liu 2026, Theorem B keeps the comb as the bounded operator M = (7/2)I − Σ_n c_n J_{log n} = (7/2)I − P and proves mI ⪯ M with m = 9617/10400, that is λ_max(P) ≤ 2.5753 at L = 17/16 (our Galerkin value there is 2.1665, measured, consistent). Only the archimedean term is cut, at Ω = 256 > 2π e^{7/2} ≈ 208. This is the operator split R′ of §1.7 with κ replaced by 7/2, and it realises the λ_max(P)-level constant without any out-of-band H. Compare his Theorem A at L = 1, which uses Zhu's pointwise envelope with the full comb mass and needs Ω_0 = 2560. Liu does not phrase this as a symbol modification or compare it with pointwise envelopes; Theorem 2 and Proposition 1.3 say how the two formulations relate.
• Connes and van Suijlekom, arXiv:2511.23257, define their quadratic forms by a distribution on the interval itself (the time-domain form of the same fact) and use the Carathéodory–Fejér structure of positive Toeplitz matrices for zero localisation, not for envelope constants. Connes–Consani–Moscovici, arXiv:2511.22755, use an extension of Carathéodory–Fejér for self-adjointness (abstract).
• Suzuki, arXiv:2606.09096v3: screw-function framework; its Theorem 1.4 is positivity for sufficiently small a; a text search found no out-of-band modification.
• Beurling–Selberg and Fourier optimisation on the zero side (Carneiro, Chandee and Milinovich, Math. Ann. 356 (2013) 939–968, arXiv:1309.1526; Carneiro, Milinovich and Soundararajan, Comment. Math. Helv. 94 (2019) 533–568, arXiv:1708.04122): the same Paley–Wiener duality used the other way round, choosing band-limited test functions so that the prime side is a finite sum. Not a modification of a Weil symbol outside the window's band. (Abstracts only.)
• Zhu's barrier (Lemma 3.2, Theorem 1.4, Remark 1.6) is stated for pointwise envelopes of P_L itself and does not consider out-of-band corrections.

Original versus novel. Original to this lab: the systematic use of out-of-band corrections against the prime comb inside Zhu's reduction (Theorem 1′), strong duality with the windowed comb operator (Theorem 2), the asymptotic constant e^L (Theorem 3), and the domination chain and floor of §1.7. Novelty was searched as follows, and within that scope no prior instance of those four was found: one Perplexity query; three web searches; full texts of Bombieri 2000, Burnol 2000 (math/0101068), Connes– Consani (2006.13771), Connes–Consani–Moscovici (2511.22755), Connes–van Suijlekom (2511.23257), Suzuki (2606.09096v3), Liu (GitHub manuscript), Zhu (2608.24827v2 source); a keyword scan of Burnol 1998 (math/9809119); abstracts only for Desogus and the Carneiro papers; Yoshida 1992 not read. The basic device (a boundary-frequency cosine) is Burnol's.

5. Task 5: does Liu's obstruction touch the out-of-band route?

What Liu proves (§5, "Obstruction Theorem"). Fix a real χ ∈ D_1 (smooth, supported in (−1, 1), ‖χ‖ = 1) and localise f ∈ D_L as f_y(u) = χ(u) f(u + y). Put L_χ(f) = ∫ Q(f_y) dy, E_χ = L_χ − Q (the localisation error), M_χ(f) = ∫ R_1(f_y) dy with R_1 the unit-window comparison of his Theorem A (tail above Ω_0 = 2560 dropped to β_0 = 1/16), and D_χ = M_χ − E_χ, so that Q = D_χ + S_χ with S_χ ≥ 0. If L > (log 8)/2, there is a fixed two-bump g (bumps at ±(log 8)/2) such that f_k = e^{iT_k u} g, T_k = 2πk/log 8, gives D_χ(f_k)/‖f_k‖² → 1/16 − log 2/(2√2) < −1/8; no fixed bounded compact self-adjoint K_c makes D_χ + K_c ≥ 0. The mechanism: every localised piece lives on a window of length 2 < log 8, so none sees the prime-power-8 interaction between the two bumps, and E_χ carries the whole term c_8 (1 − ρ_χ(log 8)) H_f(log 8) with ρ_χ(log 8) = 0; modulation aligns the phase and kills everything compact. Liu's own scope: "This obstruction concerns the specified lower comparison. It does not give a negative Weil test ... It is also not an obstruction to every possible noncompact tail supplier or every positivity method."

Answer: no. Three separate reasons.

1. No localisation. Theorem 1′ works on the whole window. The envelope S ≥ sup(P_L − H) contains every prime power with log n < 2L, including 8 once L > (log 8)/2. The quantity that fails in Liu's theorem, the localisation error E_χ, does not occur.
2. The opposite high-frequency behaviour. R_H − β* I is compact (Proposition 1.2), so along every weakly null normalised sequence, Liu's f_k included, R_H(f_k)/‖f_k‖² → β* > 0. H is part of a bounded symbol, not a compact correction of a failing comparison; above T# it is absorbed into the pointwise bound Ψ_L + H ≥ β*.
3. Where the prime power 8 does enter our route: through the window comb operator, as the requirement T# > 2π e^{S} with S ≥ λ_max(P) (Corollary 3, Proposition 1.4). It raises a threshold; it does not create a negative limit.

Scope: this settles whether Liu's stated obstruction applies to the out-of-band reduction. It does not exclude other failure modes; the constraint met so far is the sampled candidate operating point of §1.7. Liu's own Theorem B works at L = 17/16 > (log 8)/2 with a non-localised decomposition, consistent with this answer. Ordinary derivation from the paper's stated theorem; the referee did not independently clear the Liu-source translation (REVIEW §4.4).