BOUNDS: P1 to P4, the a priori quadrature and tail bounds of the rebuild

Written 2026-10-26 for PREREG s1 (with amendment 1). Code: bounds.py; tests: test_bounds.py. Grade of everything in this file: derived (ordinary arguments), unreviewed, resting on the two cited inequalities of s0. Where a test checks a bound against data, the check is measured: it can show a bound is violated, never that it holds. Nothing here is a claim about RH.

0. Objects, conventions, cited inequalities

• Modes (PREREG s1): φ̃_n = Σ_r c_r P_{k_r} on [0, 1] (even k_r ≤ 648 at 280 modes), ∫_0^1 φ̃_n² = 1, read as exact decimals from kernel/'s prolate_vectors(40, 0, nvec). ξ_n is φ̃_n on [−1, 1], so ‖ξ_n‖_2 = √2 and ‖ξ_n‖_1 ≤ 2. For w ≥ 1, ζ_n(w) = norm_n η_n(w), η_n(w) = ∫_{−1}^{1} ξ_n(t) e^{2πiwt} dt = Σ_r c_r 2(−1)^{k_r/2} j_{k_r}(2πw), with norm_n = (1 − λ_n²)^{−1/2}. b_n(v) = Σ_{k≥0} ζ_n(2^k v) for v ≥ 1 (α = 1).
• Hats: ŵ(s) = ∫_1^∞ w(v) v^{−1/2−is} dv = ∫_0^∞ k(u) e^{−isu} du with k(u) = w(e^u) e^{u/2}.
• (Landau) |J_ν(x)| ≤ c x^{−1/3} for ν ≥ 0, x > 0, c = 0.7857468704… (L. J. Landau, J. London Math. Soc. (2) 61 (2000) 197–215); used with c ≤ 0.7858. Hence |j_k(x)| ≤ JB x^{−5/6}, JB = √(π/2)·0.7858 ≤ 0.98487, and |ζ_n(v)| ≤ C_n v^{−5/6}, C_n = 2·JB·norm_n·Σ_r|c_r|·(2π)^{−5/6}; |b_n(v)| ≤ C_n v^{−5/6}/(1 − 2^{−5/6}).
• (Kapteyn) |J_ν(νt)| ≤ t^ν e^{ν√(1−t²)}/(1 + √(1−t²))^ν for 0 < t ≤ 1 (DLMF 10.14.5). Used only to bound j_k(x) for k + ½ ≥ x beyond the order where the ball recurrence stops (bessel_table).
• (Gauss) For f analytic in the Bernstein ellipse E_ρ of [−1, 1] with |f| ≤ M there, the n-point Gauss-Legendre error is at most (64/15) M ρ^{−2n}/(ρ² − 1) (Trefethen, SIAM Rev. 50 (2008) 67–87, Thm 4.5). On a panel [a, b] the error scales by (b − a)/2 and E_ρ maps to the ellipse with semi-axes (b − a)(ρ ± 1/ρ)/4. Nodes and weights are arb's legendre_p_root balls, so the rule is the exact Gauss rule. Corrected 2026-10-27 (s5): that is the rate of the (n + 1)-point rule. The n-point bound is ρ² larger, (64/15) M ρ^{2−2n}/(ρ² − 1).
• test_bounds.py spot-checks Landau and Kapteyn against arb's J_ν on grids (measured).

1. P1: the Gauss panels in w (E3)

The w-integral of ζ̂_n(s) on [1, 2^K] is split into two_adic/'s panels (ta_mellin.w_nodes): width h ≤ 2π/(2π + S/left), never straddling a power of 2. Endpoints are exact floats, each panel ending where the next begins (s1.1).

Prop P1. For |s| ≤ S and every panel [a, b] with ρ chosen so that x_min = (a + b)/2 − (b − a)(ρ + 1/ρ)/4 > 0, with β = (b − a)(ρ − 1/ρ)/4, the integrand g(w) = η_n(w) w^{−1/2−is} satisfies, on the image ellipse,

|g| ≤ 2 e^{2πβ + Sβ/x_min} x_min^{−1/2},

so the panel error is at most (b − a)/2 · (64/15) · that · ρ^{−24}/(ρ² − 1) for 12 nodes, and the hat error is norm_n times the sum over panels (e3_hat_bound). Corrected 2026-10-27 (s5): ρ^{−24} is the 13-node rate; for 12 nodes it is ρ^{−22}, and production now runs 13 nodes.

Proof. |e^{2πizt}| = e^{−2π Im(z) t} ≤ e^{2π|Im z|} for |t| ≤ 1, so |η_n(z)| ≤ e^{2πβ}‖ξ_n‖_1 ≤ 2e^{2πβ}. |z^{−1/2−is}| = |z|^{−1/2}e^{s arg z}, with |z| ≥ x_min and |arg z| ≤ arctan(β/x_min) ≤ β/x_min. Then (Gauss). ∎

Since h(2π + S/a) ≤ 2π, the exponent is at most (π/2)(ρ − 1/ρ)(a/x_min), so the panel error falls like e^{(π/2)ρ}ρ^{−24}: ρ = 15 where the ellipse fits (1.8·mid/h ≥ 15), smaller near w = 1. Value: at the toy grid (K = 6, S = 16) and at the production grid (K = 14, S' = 10^4) the totals are pinned in test_bounds.py.

For b̂. b̂ = Σ_{k<K} r^k T(2^k) + …, with T(2^k) = Σ_{j≥k} lev_j + tail and |r| = 2^{−1/2}, so the level errors enter b̂ with weight at most Σ_k 2^{−k/2} = 1/(1 − 2^{−1/2}).

1.1 A defect this caught

The first G_z run (s4) used panel midpoints and widths computed in float64 (mid = left + h/2). Adjacent panels then overlap or leave gaps of one ulp, and the balls cannot see it: G_z of four prolates came out as 0.9999999999999998 ± 6.5e−17 on the diagonal and 2e−16 ± 5e−17 off it, a ball excluding the exact value 1 (the modes are orthonormal on [0, 1] to 1e−62). With exact shared endpoints, every entry contains δ_ij (test_bounds.py::test_gram_z_is_the_identity). The same rule now holds for the w-panels of the hats.

2. P2: the tail beyond W (E4)

For W ≥ 2^K, T_n(W; s) = ∫_W^∞ ζ_n(w) w^{−1/2−is} dw is computed from the M-term integration-by-parts form of η_n and, per term, the J-term asymptotic series of ∫_W^∞ e^{iaw} w^β dw.

Prop P2. (i) η_n(w) = Σ_{m<M} (−1)^m φ̃_n^{(m)}(1)[e^{iω}/(iω)^{m+1}

• e^{−iω}/(−iω)^{m+1}] + R_M(w), ω = 2πw, with

|R_M(w)| ≤ ‖φ̃n^{(M)}‖{L1[−1,1]}/ω^M ≤ 2Σ_r|c_r|P_{k_r}^{(M)}(1)/ω^M. (ii) For real a ≠ 0 and Re β < −1, ∫W^∞ e^{iaw}w^β dw = −e^{iaW}Σ{j<J}(−1)^j (β)_j W^{β−j}/(ia)^{j+1} + ρ_J, (β)_j the falling factorial, |ρ_J| ≤ |(β)_J| a^{−J} W^{Re β−J+1}/(J − Re β − 1). Hence |T_n − expansion| ≤ norm_n[‖φ̃^{(M)}‖_1(2π)^{−M}W^{1/2−M}/(M − ½)

• Σ_{m<M} 2|φ̃^{(m)}(1)|(2π)^{−(m+1)}|ρ_J(β_m)|], β_m = −(m + 1) − ½ − is

(e4_tail_bound).

Proof. (i) M integrations by parts on [−1, 1]; ξ even gives φ̃^{(m)}(−1) = (−1)^m φ̃^{(m)}(1); |P_k^{(m)}| ≤ P_k^{(m)}(1) on [−1, 1] (the derivatives are positive multiples of Gegenbauer polynomials, maximal at 1). Integrate |R_M| w^{−1/2} from W. (ii) J integrations by parts; the remainder is (−1)^J(β)_J(ia)^{−J}∫_W^∞ e^{iaw}w^{β−J}dw. ∎

The k-series of b̂ (k ≥ K): terms K ≤ k < k_b by the same expansion at W = 2^k, the rest by Landau: |T_n(2^k; s)| ≤ ∫_{2^k}^∞ C_n w^{−4/3} = 3C_n 2^{−k/3}, so Σ_{k≥k_b} 2^{−k/2}|T| ≤ 3C_n 2^{−5k_b/6}/(1 − 2^{−5/6}) (b_kseries_tail).

An observation about the stored route (not a defect claim). two_adic/'s hats_modes sums the k ≥ Kmax terms with only 5 derivatives and 8 series terms (_tail_modes(..., mmax=4, terms=8)). By (i), the remainder of that truncation at W = 2^14 for the highest modes is not small a priori. The value is pinned in test_bounds.py. Its effect on the stored rows is not measured here.

3. P3: the s-rule (amendment 1)

3.1 Why the Gauss s-panels did not close

A Gauss bound needs |ρ(z)| on an ellipse in the strip |Im z| < ½, i.e. the hats continued off the line: |ŵ_i(s + iτ)| ≤ ∫|w_i| v^{−1/2+τ}. With the pointwise bounds available (Landau, or one integration by parts), this is of order 10^3 per hat at 280 modes, so |ρ| ≲ ‖G^{−1}‖ Σ_i|ŵ_i|² ~ 10^10 per panel against ρ_B^{−16} ~ 10^{−6} at panel width ½. It does not close.

3.2 The trapezoid rule and its aliasing (Prop P3a)

With s_k = kΔ, k ∈ ℤ: if F(s) = (1/2π) φ̂_1(s) conj(φ̂_2(s)), with φ̂(s) = ∫ φ(u)e^{−isu}du and φ_1, φ_2 ∈ L¹ ∩ L², then (Poisson summation)

Δ Σ_k F(kΔ) − ∫ F ds = Σ_{μ≠0} ∫ φ_1(y + 2πμ/Δ) conj(φ_2(y)) dy.

The window entry (m, n) of M_Q is Σ_ij G^{−1}_ij times such an F with φ_1 = k_i ∗ U_n and φ_2 = k_j ∗ U_m: ŵ_i V̂_n is the transform of the convolution, and V̂ is the transform of U on [−L/2, L/2]. From Landau, |k_i(u)| ≤ C_i e^{−u/3} on u ≥ 0, so |φ(u)| ≤ L^{1/2}C_i e^{−(u−L/2)/3} on u ≥ −L/2 (zero below), and for U > 0

∫ |φ_1(y + U)||φ_2(y)| dy ≤ 1.5 L C_i C_j e^{2L/3} e^{−U/3}.

With q = e^{−2π/(3Δ)}: |alias(m, n)| ≤ Σ_ij|G^{−1}_ij|C_iC_j · 1.5 L e^{2L/3} · 2q/(1 − q), and the spectral norm is at most (2N + 1) times that, for each family (p3_alias_entry). The same argument without U gives the aliasing of the Gram sums: (Δ/2π)Σ_k ŵ_i conj(ŵ_j)(kΔ) = G_ij + alias_ij, |alias_ij| ≤ 1.5 C_iC_j · 2q/(1 − q).

Sharpness check. For k(u) = e^{−u/3}: (Δ/2π)Σ_k|k̂(kΔ)|² − ‖k‖² = (3/2)(coth(π/(3Δ)) − 1) = 3q/(1 − q), exactly the bound (test_bounds.py::test_poisson_bound_is_sharp_on_the_model).

At Δ = 1/16, q = e^{−33.5} = 2.8e−15. With Σ_n C_n² = 5.2e4 at 280 modes (RUNS.md), the ζ family's aliasing is of order 1e−6 per entry. The b family's G_b^{−1} enters only once G_b exists (P4).

3.3 The truncation |s| > S' (Prop P3b)

Since ρ_Q ≥ 0 pointwise and |g_v(s)|² ≤ κ(S') for |s| > S' (bound_quad/ Prop 2), the omitted nodes contribute at most κ(S')·(Δ/2π)Σ_{all k}ρ(kΔ) = κ(S')(nvec + Σ_ij G^{−1}_ij alias_ij) in each family, and so at most that to ΔT (a difference of two such non-negative forms). P3 total = (2N + 1)(alias_z + alias_b) + κ(S')(nvec + max alias_ρ) (p3_total).

4. P4: the Gram matrices in v-space

4.1 The quadrature region

∫_a^b ζ_i(u)ζ_j(2^m u) du by 12-node Gauss panels of width at most min(1/(1 + 2^m), 1.8a/15), with exact shared endpoints. On the panel ellipse, |ζ_i(z)ζ_j(2^m z)| ≤ 4 norm_i norm_j e^{2πβ(1+2^m)} (as in P1), so the (Gauss) bound applies (quad_pair). Values ζ_n at the nodes come from bessel_table. It runs the upward recurrence in balls at raised precision (nodes and 2πw recomputed at that precision), stops at the first order whose ball radius passes 2^{−(prec+8)}, and from there uses [0 ± Kapteyn]. The extra precision doubles until the Kapteyn bound at the hand-over order is below 2^{−(prec+8)}.

4.2 G_z: closed (Prop P4z)

G_z = ∫_1^{2^K} ζ_iζ_j + ∫_{2^K}^∞ ζ_iζ_j. The first part is s4.1 with m = 0. For the second, multiply the two M-term expansions of P2 (i). The products are e^{i(σ+τ)ω}ω^{−(p+q+2)} terms: σ + τ = 0 integrates exactly to W^{−(p+q+1)}/(p + q + 1), and σ + τ = ±2 by P2 (ii) with its remainder. The expansion remainders contribute at most ∫(C_i u^{−5/6} r_j + r_i C_j u^{−5/6}

• r_i r_j), r(u) = norm‖φ̃^{(M)}‖_1(2πu)^{−M} (gram_z_tail). M is the

smallest with norm‖φ̃^{(M)}‖_1(2πW)^{−M} < 1e−30 for every mode (choose_M). Check: on four prolates at K = 10, every entry contains δ_ij, with radius 6.2e−13 at (0, 0) and below 1e−15 elsewhere. That is an independent check of the tables, the panels, the expansion and its bounds, since the identity is the prolates' double orthogonality, used nowhere in the computation.

4.3 G_b: the decomposition, and the region that is built

With l ≥ k and u = 2^k v, ∫_1^∞ ζ_i(2^k v)ζ_j(2^l v) dv = 2^{−k}∫_{2^k}^∞ ζ_i(u)ζ_j(2^{l−k}u) du. Write P_ij(m, t) = ∫_{2^t}^{2^{t+1}} ζ_i(u)ζ_j(2^m u) du. Summing over k ≤ t gives the weight w_t = Σ_{k≤t}2^{−k} = 2 − 2^{−t}, so

G_b = Σ_{t≥0} w_t [P(0, t) + Σ_{m≥1}(P(m, t) + P(m, t)ᵀ)].

The (m, t) plane splits into three regions:

• A (t + m + 1 ≤ K): both factors are below the asymptotic threshold; quadrature, s4.1. Built (gram_b_region_a). At most 12·2^K nodes per piece, about K²/2 pieces.
• B (t < K ≤ t + m + 1): the second factor is in its expansion range, the first is not. The piece is ∫ ζ_i(u) e^{±2πi2^m u}(2^m u)^{−q−1} du at frequency 2π2^m. For m ≤ 3 it is quadrature as in A (at most 9·12·2^K nodes). For m ≥ 4, R integrations by parts with endpoint derivatives ζ_i^{(a)} (a linear map of the Bessel table) and remainder ∫|f^{(R)}|/Ω^R, the ratio per step being about 2^{−m}. Not built.
• C (t ≥ K): both factors are expansions. That is s4.2 with σ2^0 + τ2^m frequencies; only m = 0 has non-oscillating terms. Not built for m ≥ 1.
• The m-tail: by Landau, |P(m, t)| ≤ 1.5 C_iC_j 2^{−5m/6}(2^{−2t/3} − 2^{−2(t+1)/3}), so Σ_{m≥m₀} contributes at most 10.9 C_iC_j 2^{−5m₀/6}. That is about 5.7e5·2^{−5m₀/6} in norm at 280 modes, which forces m₀ ≈ 50, so regions B and C must be built for m up to about 50.

4.4 Status

P4 is not closed: G_z is, G_b's region A is, and regions B and C are derived in outline (s4.3) and not built. bounds.gram_b raises NotImplementedError, and a run with the exact provider stops before any block, which is PREREG row D0 (coordinator, 2026-10-26: 'go' waits for P4). The toy provider toy-region-a (G_b from region A only) exists for the pipeline tests and is never read as a bound: rebuild.assemble marks such a run D0.

4.5 G_b, regions B and C (2026-10-26, third unit)

bounds.ibp_pieces, bounds.gram_b; tests test_p4.py. Grade: derived (ordinary argument), unreviewed; each region checked against direct quadrature at toy size (measured), each with a planted fault that the check catches.

The fast factor. For m ≥ 1 and u in a piece with 2^m u ≥ 2^K, write ζ_j(2^m u) by its M-term expansion (P2 (i)): ζ_j(λu) = Σ_{p<M} D_jp Σ_{σ=±1} (σi)^{−(p+1)}(2πλu)^{−(p+1)} e^{σi2πλu} + R_j, D_jp = norm_j (−1)^p φ̃_j^{(p)}(1), |R_j| ≤ norm_j‖φ̃_j^{(M)}‖_1 (2πλu)^{−M}. This holds in regions B (t < K ≤ t + m) and C (t ≥ K, m ≥ 1) alike.

Prop P4b (one term). For f(u) = ζ_i(u)u^{−(p+1)}, Ω = 2πλ, λ = 2^m ≥ 2 and an exact interval [a, b] ⊂ [1, ∞):

∫a^b f e^{iσΩu} du = Σ{r<R} (−1)^r [f^{(r)}(u) e^{iσΩu}/(iσΩ)^{r+1}]_a^b + ρ_R, |ρ_R| ≤ (b − a) · 2 norm_i Σ_{k≤R} C(R, k)(2π)^k (p + 1)↑_{R−k} a^{−(p+1+R−k)} / Ω^R.

Proof. R integrations by parts. Leibniz with |ζ_i^{(k)}(u)| = norm_i|∫ξ_i(y)(2πiy)^k e^{2πiuy}dy| ≤ 2 norm_i (2π)^k and |d^b u^{−(p+1)}| = (p + 1)↑_b u^{−(p+1+b)}. ∎ The ratio of successive terms is about (2π + (p + R)/a)/Ω ≈ 1/λ: the slowest is m = 1 (ratio ½).

Endpoint derivatives. f^{(r)} at the dyadic endpoints follows by Leibniz from ζ_i^{(a)}(u) = Σ_r CM[r, i] (d/du)^a j_{k_r}(2πu). The Bessel derivatives come from j_k' = (k j_{k−1} − (k + 1) j_{k+1})/(2k + 1) applied a times to a ball table j_0 .. j_{kmax+R} (zeta_derivs, checked by a Taylor expansion and by the bound |ζ^{(a)}| ≤ 2 norm (2π)^a).

Assembly. P_ij(m, t) = Σ_{p,σ} D_jp (σi)^{−(p+1)}(2πλ)^{−(p+1)} J^σ_i(p)

• the expansion remainder, which is at most

C_i norm_j‖φ̃_j^{(M)}‖_1(2πλ)^{−M} ∫_a^b u^{−5/6−M}du (Landau for ζ_i). In matrix form this is one product per piece, the endpoint sums being shared by every piece that meets the endpoint.

Region C, m = 0. Σ_{t≥K} w_t P(0, t) = 2 Z(2^K) − Σ_{t≥K} 2^{−t}(Z(2^t) − Z(2^{t+1})), Z(W) = ∫_W^∞ ζ_iζ_j (the G_z tail of s4.2, a ball matrix).

The remainders beyond m_max and t_max (Landau, s0): Σ_{m>m_max} ≤ 6 C_iC_j 2^{−5(m_max+1)/6}/(1 − 2^{−5/6}); Σ_{t>t_max, m≥1} ≤ 6 C_iC_j W^{−2/3}/(1 − 2^{−5/6}), W = 2^{t_max+1}; the m = 0 correction beyond t_max ≤ 2^{−(t_max+1)}·1.5 C_iC_j W^{−2/3}. gram_b picks m_max and t_max from these for a target radius and adds them to every entry.

Checks (toy, 2 modes, K = 6; test_p4.py):

• Region B: pieces (m, t) = (3, 4), (2, 5) and (5, 1) overlap direct quadrature. Planted: R = 2 with the remainder suppressed does not.
• Region C, m ≥ 1: (1, 6), (3, 6) and (2, 7) overlap. Planted: a 2-term expansion with its remainder suppressed does not.
• Region C, m = 0: Z(64) − Z(128) overlaps quadrature on [64, 128]. Planted: Z(64) alone does not.
• The whole G_b at target 1e−8: symmetric, positive definite in balls, m_max = 53, t_max = 66, radius below 1e−6.
• Two routes (slow test): G_b in v-space against the hats' Plancherel sum (Δ/2π)Σ|b̂|² on |s| ≤ 200 (w-quadrature, P1, P2). The gap is a tail, non-negative up to aliasing, a few percent, and the off-diagonal gap obeys Cauchy-Schwarz.

4.6 Status against PREREG row D0

P4 is closed as a derivation with an implementation: G_z (s4.2) and G_b (s4.3 region A, s4.5 regions B and C, with all truncations bounded). With P1 to P3 (s1 to s3), every prerequisite of PREREG s1 now has a derivation, code and a test. PREREG row D0's precondition is lifted. A run with the exact provider no longer stops at setup. Whether any rebuilt row closes (its ε_q finite) is decided by the run, not by this file. All of it is ordinary argument, unreviewed, on the cited inequalities of s0. The production cost of the Gram at 280 modes is an estimate, not a measurement (RUNS.md s9).

5. Correction, 2026-10-27: the Gauss bound was the (n + 1)-point bound

An adversarial review of PR #362 found that s0's (Gauss) inequality, used for n-point rules in P1 (s1) and in P4's quadrature (s4.1), is the bound of the (n + 1)-point rule. This section verifies that independently and records the fix. s0 to s4.6 above stay as written, apart from two dated pointers. The text in question, as written in s0:

(Gauss) For f analytic in the Bernstein ellipse E_ρ of [−1, 1] with |f| ≤ M there, the n-point Gauss-Legendre error is at most (64/15) M ρ^{−2n}/(ρ² − 1) (Trefethen, SIAM Rev. 50 (2008) 67–87, Thm 4.5).

and in s1: "so the panel error is at most (b − a)/2 · (64/15) · that · ρ^{−24}/(ρ² − 1) for 12 nodes". bounds.gauss_panel_error coded exactly that, and e3_hat_bound and _quad_pair_core called it with n = 12.

The bound, derived here. Let f = Σ a_k T_k on [−1, 1], analytic in the open ellipse E_ρ with |f| ≤ M there, so |a_k| ≤ 2Mρ^{−k}. The n-point rule G_n is exact to degree 2n − 1 and symmetric, so odd T_k contribute nothing to I or G_n. For even k ≥ 2n, |I(T_k) − G_n(T_k)| ≤ 2/(k² − 1) + Σ_j w_j = 2 + 2/(k² − 1), the weights being positive with sum 2. Hence

|I − G_n| ≤ Σ_{k even ≥ 2n} 2Mρ^{−k}(2 + 2/(k² − 1)) ≤ 4M · 4n²/(4n² − 1) · ρ^{2−2n}/(ρ² − 1) ≤ (64/15) M ρ^{2−2n}/(ρ² − 1) (n ≥ 2).

The old formula is this bound at n + 1 nodes. It agrees with the cited theorem if that theorem's rule has n + 1 points, as an index counted from 0 would give. The source was not re-read here (its PDF did not extract on the box), so the code now rests on the derivation above, not on the citation. The derivation is an ordinary argument and is unreviewed, like the rest of this file.

The counterexample (measured, 256-bit balls; test_bounds.py). T_24 is entire and is the first even Chebyshev polynomial the 12-point rule misses; its maximum on E_ρ is (ρ^24 + ρ^{−24})/2. Its 12-point error is 1.54. The old formula gives 0.267 at ρ = 3 and 0.0095 at ρ = 15; the corrected one gives 2.40 and 2.14. A non-polynomial integrand also breaks it: exp(3x) at 12 points, ρ = 15, errs by 9.1e−20 against an old bound of 7.4e−21 (corrected: 1.7e−18). Over 90 cases (T_2n, exp(3x), exp(10x) and cos(20x), n = 2, 4, 12, 13, 16, ρ from 1.2 to 40) the old formula fails in 25 and the corrected one in none (test_gauss_bound_holds_on_entire_functions, test_gauss_old_formula_fails_on_entire_functions). A check of this kind can refute a bound, never prove it.

What it changed. Every Gauss term the code computed was too small by ρ² of its panel, up to 225 at ρ = 15. At the production grid (K = 14, S' = 2,700), measured with the panel loops of e3_hat_bound and _quad_pair_core:

So the radii of the committed gram.json (run 40342895119) are not bounds. At (0, 0), where the quadrature term dominates, they are 9.97e−12 (G_z) and 5.97e−11 (G_b), against about 2.2e−9 and 1.3e−8 under the corrected bound at 12 nodes (the review's figures). Every ball built from them, and so ε_q of run 40371181784, is not established as a bound (RESULTS.md, correction of 2026-10-27). s1's remark that the panel error "falls like e^{(π/2)ρ}ρ^{−24}" should read ρ^{−22} at 12 nodes. The E3 claim "below 1e−15 per hat" (RESULTS.md, second unit) holds at 13 nodes, not at 12.

The fix. gauss_panel_error returns (h/2)(64/15) M ρ^{2−2n}/(ρ² − 1) and refuses n < 2. Production now uses 13 nodes per panel, in the hat table (request key per_panel) and in the Gram (gram_per_panel). Panels, ρ and M do not depend on the node count, so the corrected 13-node bound equals the old 12-node formula term by term (test_corrected_bound_at_n_plus_one_is_the_old_formula_at_n). The rerun's radii are therefore the sizes run 40371181784 carried, now as bounds, at 13/12 of the hat products, the phases and the region-A quadrature. Requests without the keys mean 12 (what the earlier runs used).

Observed with it. Under the corrected bound at 12 nodes, G_z of four prolates at K = 10 has radius 1.4e−10 at (0, 0), and the one-ulp float endpoint fault of s1.1 hides inside it. At 13 nodes the radius is 6.2e−13 again and the planted fault is caught (test_gram_z_is_the_identity, test_planted_float_endpoints_exclude_the_identity).

What it did not change. P2 (the integration-by-parts and series remainders), P3 (the trapezoid in s and its Poisson aliasing), P4's regions B and C (integration by parts) and C with m = 0 (the G_z tail), and the Landau tails: no Gauss rule enters them.

Also added with the rerun (PREREG items the first run did not meet; RESULTS.md lists them as deviations): ball_pd decides positive definiteness of each Gram by a ball LDLᵀ (every pivot > 0) and the rebuild stops otherwise (in the merge job and in every process's setup); ρ is still formed with the ball inverse, not by triangular solves. And every replay block keeps its own window sum in the checkpoint (PREREG s6).