results register.

RH holds if and only if |u_n| ≤ u₀ for every n, where u_n = Σ θ_ρ T_n(1 − 2θ_ρ) and θ_ρ = −27/(4q(ρ)²); equivalently (u_n) is bounded, or limsup |u_n|^{1/n} ≤ 1.

RH ⇔ zeros of K in (−∞, −27/4] ⇔ Hankel positivity ⇔ Hausdorff/beta ⇔ compact Jacobi ⇔ completely monotone boundary ⇔ |u_n| ≤ u₀ ⇔ bounded ⇔ generating function holomorphic on the disk ⇔ Carathéodory ⇔ Schur.

RH ⇔ the single unshifted Hankel tower [m_{i+j}] built from the cubic descent moments is positive semidefinite for every size; the cubic geometry supplies the support restriction.

Unconditionally u₀ = (4/3)(ℓ(2) + ℓ′(2)) + (1/24)ℓ′(½) = 0.15496752570039314..., with ℓ = ξ′/ξ; every u_n is a rational linear form in finitely many derivatives of ξ′/ξ at 2 and ½.

Digits certified by ball arithmetic.

Within the finite local reflection-invariant polynomial class generated by h₀...h_n, the γ⁻² escape exponent of a high off-line zero is optimal (Bernstein–Walsh); e-folding scale n ≍ γ²/|η|.

Under RH, limsup u_n = u₀ exactly (Dirichlet simultaneous approximation), with near-extremal indices of positive lower density; no bound |u_n| ≤ (1 − δ)u₀ can hold.

The trace measure is supported in |φ| ≤ φ₁ = 0.36601674861975873..., sharpening the Hausdorff normalization 27/4 to B(γ₁)² = 203.805426808998669...

An off-line zero escapes at rate g_ρ = 2|Re arcsinh(√(27/4)/q(ρ))| = √27|η|/γ²(1 − (η² + 21/8)/γ² + ...).

F(X) = XK′/K = (√X/4)log(√X/2π) + O(log X) and log K(X) = πN(√X) + O(log² X), unconditionally.

Open Problem 11.1: the completed cubic positivity / completed trace bound left open by the descent papers.

The proven frontier is a hard constant δ₀ ≈ 1.87×10⁻³; the heuristic prediction is four orders of magnitude better. Close the gap.

Near s = 2 the two orders of operation split; the regularized seed is g₀ = (2μ/3)(E_{2,2}(z²) − 1) with E_{2,2}(z²) = sinh z/z, and successive diagonals g_j = μ^{−j}p_j(D)g₀ with p_j(n) = (−1)^j(2n+1)_j L_j(n).

In χ = (s − 2)/(s + 1), the trace has a Schur factorization with multiplier φ = ψ², φ(0) = 16/27, and the uniform bound ∫e^{−3x}|Π_n|² = O(1); the remaining obstruction is localized to interior poles of ξ′/ξ in the critical disk.

|φ| = 1 on the arc {1 + 2cos θ ≥ 0}, so lim‖Gφ^m‖² > 0: inner/outer factorization cannot upgrade the uniform bound to decay.

With v_ρ = −(ρ − ½)⁻², RH ⇔ [τ_{i+j+1}] ⪰ 0 for all sizes, with no simplicity hypothesis; Pick-kernel realization.

Disc(F_Z)·E_M^{4M−1} = D_M^{4M+1}; the finite statements are checked in Lean 4 / Mathlib with no sorry, and the certification boundary is stated explicitly.

A vantage point at horizontal offset b from the critical line carries exactly 2π·min(b, δ) of visibility along Re s = ½ + δ; to first order the visibility rate is δ times the vantage's angular speed.

Fixed vantages are parabolically blind (rate of order δ/γ²); for any m vantage points some height below T is seen at rate at most πmδ/T.

For γ ≥ 12, |θ(ρ)| ≤ 27/(4γ²) and g(ρ) = 3√3·δ/γ²·(1 + O(γ⁻²)): √3 times its own vantage s = 2 and 3√3 times Li's criterion (√3 per Taylor coefficient).

With RH verified to 3·10¹², |u_n| ≤ u₀ + 2.1·10⁻¹¹ for every n ≤ 10²⁴, the trace's Hardy-space function has no poles in |χ| < 1 − 1.7·10⁻²⁵, and the tail bound holds for every n.

Σ u_n zⁿ = (27/8)[L(0) + ((1 + z)/(1 − z))·L(27z/(1 − z)²)] with L = K′/K: a log-derivative read along a Koebe curve. RH ⇔ it is holomorphic on the disk ⇔ radius exactly 1.

Σ θ^{k+1} T_h(y) = ((−¼Δ²)^k u)_h; on generating functions −¼Δ² is multiplication by θ(X(z)) = −(1 − z)²/(4z). Under RH each zero's wave carries its own Laplacian eigenvalue sin²(Θ/2).

Under RH, lim (1/N)Σ u_n u_{n+h} = ½Σ m_γ² θ² cos(hΘ); RH and simplicity ⇔ mean u_n² = (u₀ − u₁)/4 = 0.000821184005...; if RH fails the mean square is unbounded.

Weights θ = −tan²(k·arctan(c/(s − ½))); k = 3, c = √3/2 is the cubic trace. Even k: invariant under γ ↦ c²/γ, angle collisions exactly when γγ′ = c²; large c gives prime formulas (Weil explicit formula instances).

u₀ ... u_{100000} computed from values of ξ with proved radius ≤ 6.6·10⁻¹⁹ (Krawczyk roots, ball arithmetic, rigorous DFT, unconditional aliasing bound); under RH the spectrum locates the first 71 zero ordinates.

Reproduced byte for byte from a clean copy; a second run on a 2¹⁸ grid gives intersecting enclosures.

Under RH, simple zeros and four-term nonresonance, mean u_n⁴ = 3A(0)² − (3/8)Σθ⁴; data over 10⁴ ≤ n < 10⁵ give 1.5512·10⁻⁶ against 1.5451·10⁻⁶.

A full descent exists at every odd order m (only ξ(1 − s) = ξ(s) is arithmetic); escape g^{(m)} = (ma|η|/γ²)sech(ma/2γ)(1 + O(γ⁻¹)), optimal at λ = ma/2 ≈ 1.19967864γ with value 1.32548684|η|/γ: the finite-local exponent is γ⁻¹, not γ⁻².

For gcd(m, m′) = 1 the two spurious families meet only on the critical line, so the pair of criteria is unconditionally RH-equivalent with no auxiliary zero-free region.

Σ_{γ>0} tanh²(λ/γ) = (16λ/π³) Σ_{j≥0} (2j+1)⁻³[ℓ(p_j) + (λ/(π(2j+1)))ℓ′(p_j)], p_j = ½ + 2λ/(π(2j+1)); λ → 0 recovers Σγ⁻² = ½ℓ′(½).

Two scales share a spurious component iff λ/λ′ ∈ ℚ, so RH ⇔ the criterion at two scales of irrational ratio, unconditionally.

By a dimension count, no finite set of scales and depths certifies an RH-true configuration: the tower is a pure falsification instrument, and any proof must be an identity, not an inequality with slack.

The harmonic lattice accumulates at s = ½, which is χ = −1 where the Schur multiplier vanishes; conjecture: the rotation tower discretizes the Hardy-space contraction.

λ∂θ/∂λ = V(θ) = 2√θ(1 − θ)artanh√θ, one autonomous holomorphic field whose conserved quantity is the zero itself; RH ⇔ every nontrivial zero lies on the heteroclinic orbit (0, 1).

H(λ) = Σ_ρ θ_λ(ρ) has double poles exactly on π(ℤ + ½)·{ρ − ½}; H̃(w) = M(w)Z(w) with Z the secondary zeta function and M(w) built from ζ(w + 2).

Find a positivity, Schwarzian or transfer-operator property of V equivalent to RH; and decide whether the multiplicative dilation divisor interacts with the Euler product.

Under RH, (u_n) has a limiting distribution (and RH ⇔ it exists). If the trace angles are linearly independent, it is the law of Σ θ_γ cos(2πU_γ), with characteristic function ∏ J₀(θ_γ t), bounded support [−u₀, u₀], and every cumulant local: κ_{2r} is a constant times Σθ^{2r}.

Against 10⁶ certified terms: κ₂, κ₄, κ₆, κ₈ match the local predictions to 0.01%, 0.2%, 0.3%, 0.4%; skewness −0.0004; tail frequencies match the model down to 8·10⁻⁵.

Excess kurtosis of the trace = −(3/2)Σθ⁴/(Σθ²)² = −0.708682 from local data alone; measured −0.70727 over 10⁶ certified terms.

−log P(u_n ≥ u₀ − ε) ~ A²·log²(1/ε)/ε with A = 0.575191; the refined form sup_s[A√s(log s + b) − εs], b = −3.152912, matches the exact model to 0.01%.

Conditional on RH and trace linear independence; de Bruijn's Tauberian theorem plus Riemann–von Mangoldt. Needs a full writeup.

u₀ − max_{n≤N} u_n ~ A²(log log N)²/log N: how fast the trace's records creep toward its threshold.

Under RH the third moment equals (3/4)Σ over Θ_c = Θ_a + Θ_b of the weight products; the trace's skewness vanishes iff no trace angle is the sum of two others.

Measured skewness −0.004; closest near-relation among the first 3000 angles misses by 7.9·10⁻¹⁰.

If the zero ordinates are algebraically independent over ℚ, then 1 and the trace angles Θ_γ/2π are linearly independent over ℚ; the same holds for every algebraic angle map (all finite tower levels, the dihedral family).

Proof by a derivation argument; written out, not yet formalized.

With multiplicities, each even cumulant exceeds its local value by a multiple of Σ(m^{2r} − m)θ^{2r} ≥ 0, with equality iff all zeros are simple: an infinite family of simplicity tests extending the mean-square test.

exp(Σ u_n zⁿ/n) = ∏_γ (1 − 2cos Θ_γ·z + z²)^{−θ_γ/2}; RH ⇔ every Satake-type parameter e^{±iΘ} is unimodular. At z = 1: Σ u_n/n = −½Σθ log(4θ) = 0.36746.

The first zero of the trace's empirical characteristic function sits at j₀,₁/θ(γ₁): from the histogram of u_n alone, with the order of n thrown away, γ₁ ≈ 14.1386 from 10⁶ terms (true 14.134725).

u₀ ... u_{1000000} computed from values of ξ on a 2²² grid, every term with proved radius ≤ 6.6·10⁻¹⁹; the enclosures intersect the independent 2¹⁹-grid run at all 100,001 shared terms. 1229 seconds on two cores.

New record u_{844054} = 0.10039 = 0.648·u₀ (previous record 0.567·u₀). The first five zero angles line up almost exactly (cos nΘ_j = 0.999, 0.997, 1.000, 0.996, 0.976), and the LI model puts this at roughly a one-in-several-hundred event for 10⁶ terms.

Either luck or Diophantine structure among the first trace angles; worth an LLL look.

An unconditional quartic spectral certificate, with a conditional 13/18 theorem aimed at the simple-zero proportion record.

Third moments of prime gaps through an exponential-kernel projection.

g(q) = min |Re L(1, χ) − ½| decays like q^{−α}, α ≈ 1.87, with Tracy–Widom GUE edge statistics.

The automorphic product F(s) = ζ(s)ζ(w(s))ζ(w²(s)) satisfies a floor bound (Conjecture 10.7), claimed equivalent to RH.

A Pascal-type array with left border 1 and right border (−1)ⁿBₙ, a unique spectral zero at ω = π, four laws, and a prime constant invariant at stabilization depth 5.

For cross-intersecting Sperner families, the binary equilibrium gap satisfies μ(U) ≥ L^{−C}.

Minimal binding sets are minimal covers of the counterworld set by exclusion sets; conjecture on discovering them.

P = NP ⇔ every efficiently recognizable, polynomially bounded relation has a polynomial-time selector.

Exposure cost Σ_R = b_R + log T_R = log(T_R/p_R) of a reduced description; conjecture Σ_R ≥ c·q(L).