jensen_wedge: the two open inputs H1 and H3 (2026-10-19)

Nothing here bears on RH. This file closes, by derivation, the two inputs that RESULTS.md section 12.5 left open in the conditional six-coefficient wedge of hunt #135. Both derivations are self-reviewed only, not externally reviewed, and are not theorems; this file does not call them theorems. No novelty search was run. Every numerical statement is measured (probe_h1_h3.py, results_h1_h3.json) and pinned in test_h1_h3.py, except the rational constants of section 4.3, which are exact.

Measured beside the derivations: the explicit contraction inequality of section 4.3 holds on the measured inputs at every measured n from 1e7 to 1e20 and fails at n <= 1e6; the measured Pade configuration lies inside the radius it predicts at every n where it holds; and the first-order corrections the derivation predicts (sections 2 and 4.4) are approached by the data to n = 1e20 (route B) and n = 1e50 (Holland's main term).

1. Source and hypotheses

Holland, arXiv:2608.08682v1, PDF SHA-256 3fc31ba84fb113bc0b3109fb0e569bd1d7183018485aca1eb1dab565c839b49d (the file phase 1 read), re-downloaded on 2026-10-19 and extracted with pdftotext -layout into a scratch directory outside the tree. Equation numbers below are his.

Notation as in RESULTS.md sections 1 and 11: h(z) = log M_z, b = n + 1/2, N = 2z - 2, L_N from his (16), L_x := L_(2x-2) written L when n is clear, Q_N = (1 + L_N) N - (3/4) L_N^2, and G_0 his (17). His (18) reads h(z) = G_0(N) + E_l(z) + c_sad + R(z) with E_l(z) = log(32 z (2z - 1)) - (2z + 2) log 2.

The derivations use exactly these inputs from the paper:

• (P1) Holland's Proposition 4.1 in the case r = 0 of (19): there are 0 < theta < pi/2, x_0, 0 < eps_0 < 1/2 and C_R such that on S = {|z| >= x_0, |arg z| <= theta} the function h is holomorphic (real on the positive axis), (18) holds, and |R(z)| <= C_R |z|^(-1+eps_0). Holland obtains it from the complex form of GORTTW (3.2). Neither paper makes C_R, theta, x_0 explicit.
• (P2) On {N = 2z - 2 : z in S} the branch L_N of (16) continued from the real axis ("the branch of the holomorphic continuation used in [3, Section 3]") is holomorphic with |L_N| >= 4 and |L_N/N| <= 1/8. Holland uses this without separate statement.
• (P3), route 1 of H3 only: Holland's (25), |h^(5)(x + w)| <= C/(x^4 log x) for real x >= x_* and |w| <= eta x.

Nothing else from the paper is used. In particular Holland's Lemma 6.1 is not used: H1 follows its method (a contraction in scaled variables), not its result, which is about the D-fixed system.

Two facts used throughout. Disk geometry: a disk of radius rho x about a real x lies in {|arg z| <= arcsin rho, |z| >= (1 - rho) x}; with eta := min(sin theta, 1/4)/2 and x >= 2 x_0 + 2, every disk |zeta - (x + w)| <= eta x with |w| <= eta x lies in S. Cauchy's inequality: if F is holomorphic on |zeta - z| <= rho with |F| <= M there, then |F^(m)(z)| <= m! M rho^(-m).

2. Derivatives of h of every order

Lemma S. Assume (P1), (P2). For every k >= 2 there are C_k, C'_k, x_k such that for real x >= x_k:

(a) |h^(k)(x + w)| <= C_k/(x^(k-1) L_x) whenever |w| <= eta x;

(b) h^(k)(x) = (-1)^k 2 (k-2)!/(x^(k-1) L_x) (1 + e_k(x)) with |e_k(x)| <= C'_k/L_x;

(c) e_k(x) = (H_(k-2) - 1)/L_x + O(1/L_x^2), H_m the m-th harmonic number (H_0 = 0).

For k = 2, 3, 4, (b) is Holland's (21) to (23), and (a) at k = 4, 5 is his (24), (25). New here: the asymptotic (b) at k = 5 (his (25) is an upper bound only), every order k >= 6, and the first-order constants (c). H1 needs (b) at k = 2..5; H3 is (a) at k = 6. Status: derivation, self-reviewed. The Cauchy factors are explicit; C_k, C'_k, x_k are not, because they inherit C_R, theta, x_0 from (P1).

Proof. Differentiate (18) k times: h^(k)(z) = 2^k G_0^(k)(N) + E_l^(k)(z) + R^(k)(z).

The remainder. On the disk of radius eta x about x + w, |zeta| >= (1 - 2 eta) x, so |R^(k)(x + w)| <= k! (eta x)^(-k) C_R ((1 - 2 eta) x)^(-1+eps_0). Relative to the claimed main term this is O(L x^(-2+eps_0)).

The elementary terms. E_l^(k)(z) = (-1)^(k-1) (k-1)! (z^(-k) + (z - 1/2)^(-k)), relative size O(L/x).

The main term, in the variable N. From (16), L' = L/Q (Holland's own identity). Substituting it into the derivative of (17) and using Q = N + L N - (3/4) L^2 collapses the result to

    G_0'(N) = log L + rho_G(N),        rho_G = (1 + L)/Q - Q'/(2Q),

(pinned numerically at 24 points, defect below 1e-60). So G_0^(k) = (1/Q)^(k-2) + rho_G^(k-1) for k >= 2. Write Q = N L (1 + u), u = 1/L - 3L/(4N); (P2) gives |u| <= 1/4 + 3/32. Then Q' = 1 + L + (1 - 3L/(2N))/(1 + u) and |rho_G| <= 2/|N|, so Cauchy gives rho_G^(k-1) = O(N^(-k)), relative size O(L/N). Next 1/Q = phi (1 + v) with phi = 1/(N L) and v = (1 + u)^(-1) - 1 = O(1/|L|).

• (1/L)' = -1/(L Q) = -(1 + v)/(N L^2); along a disk of radius rho |N|, |L_zeta - L_N| <= 2 rho/(1 - rho), so |L| changes by a factor within [5/6, 7/6] for rho <= 1/4. Cauchy on (1/L)' gives (1/L)^(i) = O(N^(-i) L^(-2)) for i >= 1.
• Leibniz: phi^(m) = sum_i C(m,i) (N^(-1))^(m-i) (1/L)^(i) = (-1)^m m!/(N^(m+1) L) (1 + O(1/L)).
• |phi v| = O(1/(N L^2)) on disks, so Cauchy gives (phi v)^(m) = O(m!/(N^(m+1) L^2)).

Hence G_0^(k)(N) = (-1)^k (k-2)!/(N^(k-1) L_N) (1 + O(1/L_N)), uniformly on the disks, which is Holland's (26) to (29) sharpened at k = 5 and continued to every k.

Back to z. 2^k (-1)^k (k-2)!/((2z - 2)^(k-1) L_(2z-2)) = (-1)^k 2 (k-2)!/((z - 1)^(k-1) L_z), and (z - 1)^(-(k-1)) = z^(-(k-1)) (1 + O(k/|z|)). With |L_(x+w)| >= (5/6) L_x on the disk this gives (a) and (b).

First order (c). The same induction with the main terms kept: (1/L)^(i) = (-1)^i (i-1)!/(N^i L^2) (1 + O(1/L)), so phi^(m) = (-1)^m m!/(N^(m+1) L) (1 + H_m/L + O(1/L^2)); and v = -1/L + O(1/L^2 + L/N), so (phi v)^(m) = -(-1)^m m!/(N^(m+1) L^2) (1 + O(1/L)). Adding, (1/Q)^(m) = (-1)^m m!/(N^(m+1) L) (1 + (H_m - 1)/L + O(1/L^2)), and m = k - 2. QED.

Measured (results_h1_h3.json, G0 and h_circle):

• Holland's main term alone, by Cauchy's formula on 64 points: the ratio of G_0^(k)(N) to (-1)^k (k-2)!/(N^(k-1) L_N) is 1.127 (k = 6, N = 1e3), 1.025 (1e20) and 1.0097 (1e50), and on the complex rays arg N = 0.25 and 0.5 its modulus agrees with the real value, over k = 2..6, to relative 1.4e-3 at |N| = 1e3, 3.2e-6 at 1e20 and 1.9e-7 at 1e50. (ratio - 1) L_N at N = 1e50, k = 2..6: -0.991, -0.018, 0.477, 0.810, 1.062, against (c)'s -1, 0, 0.5, 0.833, 1.083; the gap closes like 1/L at every k.
• The full h on a complex circle: h computed from Holland's (15) by quadrature at 64 points of |z - n| = n/8 (n = 400 and 3200; trapezoid coarse/fine below 5e-41) has Taylor coefficients at n that agree with route B's real-axis derivatives for k = 2..7 to relative 4e-20 to 1e-17 (n = 400) and 5e-26 to 3e-23 (n = 3200). On |z - n| = n/16, max |h^(6)| is 1.52 (n = 400) and 1.54 (n = 3200) times 48/(n^5 L_n); and Holland's remainder R varies on the circle by at most 0.028/n and 0.038/n. This measures (P1) at two n; it does not establish it.

3. H3: the sixth derivative

H3. There are eta_6 > 0, C_6, x_6 with |h^(6)(x + w)| <= C_6/(x^5 log x) for real x >= x_6 and complex |w| <= eta_6 x.

1/(x^5 L_x) and 1/(x^5 log x) are interchangeable up to a factor 2 for large x, as Holland treats them.

3.1 Route 1: one Cauchy estimate on Holland's (25)

Let |w| <= eta x/2. The disk of radius eta x/2 about x + w lies in {x + w' : |w'| <= eta x}, where (P3) bounds |h^(5)| by C/(x^4 log x). Cauchy with m = 1: |h^(6)(x + w)| <= (2 C/eta)/(x^5 log x). So H3 holds with eta_6 = eta/2 and C_6 = 2C/eta. Status: derivation, self-reviewed, from (P3) alone. It is one line, and it is the step RESULTS.md 12.4 needed.

3.2 Route 2: Holland's own route, (19) at r = 6 and (29) carried once more

(19) at r = 6. For z in S' = {|arg z| <= theta/2, |z| >= 2 x_0} put delta = min(sin(theta/2), 1/2). The disk |zeta - z| <= delta |z| lies in S and |zeta| >= (1 - delta)|z|, so Cauchy gives

    |R^(6)(z)| <= 720 delta^(-6) (1 - delta)^(-1+eps_0) C_R |z|^(-7+eps_0),

which is (19) at r = 6 with its constant explicit in C_R. Nothing restricts r: Holland needed r <= 5 only.

(29) carried once more. Lemma S at k = 6 gives G_0^(6)(N) = 24/(N^5 L_N) (1 + O(1/L_N)) on the disks, the analogue (29') of his (29) (where he stated only G_0^(5) = O(1/(N^4 L_N))). With h^(6) = 64 G_0^(6)(2z - 2) + E_l^(6) + R^(6) this is H3 with C_6 from Lemma S(a), and in addition the real-axis asymptotic h^(6)(x) = 48/(x^5 L_x)(1 + (H_4 - 1)/L_x + O(1/L_x^2)). Status: derivation, self-reviewed, from (P1) and (P2).

3.3 Carried through to the sixth-order residual (RESULTS.md 12.4)

Let (A, B, C, D) be H1's solution, eps = tau_0 (section 4) and E^(6)(z) = h^(6)(n + z) - psi^(5)(b + z) + psi^(5)(B + z) + psi^(5)(D + z) - psi^(5)(A + z) - psi^(5)(C + z). For complex |z| <= eta_6 n:

1. For Re y > 0, |psi^(5)(y) - 24/y^5| <= 120/(Re y)^6. From the integral representation, the difference is int_0^inf t^4 e^(-yt) (t/(1 - e^(-t)) - 1) dt, and 0 <= t/(1 - e^(-t)) - 1 <= t.
2. 24/(p + z)^5 = f_z(n/p) with f_z(w) = 24 w^5/(n + z w)^5 and f_z(0) = 0, so the five leading terms sum to -int_G f_z'(t) dt over the gap set G of section 4.1.
3. |f_z'(t)| = 120 n t^4/|n + z t|^6 <= 120/((1 - eta_6)^6 n^5) for 0 <= t <= 1.
4. H1 gives |G| <= eps (1 + 13r/4) + 1/(2n + 1) (g0 + gx + g1 on the box of section 4.3) and every parameter at least n, so Re(p + z) >= (1 - eta_6) n.

Hence

    sup_{|z| <= eta_6 n} |E^(6)(z)| <= C_6/(n^5 log n) + 120 (eps (1 + 13r/4) + 1/(2n+1))/((1 - eta_6)^6 n^5)
                                       + 600/((1 - eta_6)^6 n^6),

and with eps <= 2/L_n (measured at every n: tau_0 L/2 is 0.87 to 0.977 from 1e4 to 1e20, and 1 - 1/L + O(1/L^2) by Lemma S(c)) the second term is at most 240 (1 + 13r/4)(1 - eta_6)^(-6)/(n^5 L_n) plus an O(1/n^6) part. This is 12.4's bound sup |E^(6)| <= C/(n^5 log n) with every constant explicit except C_6. Status: derivation, self-reviewed.

Measured: item 1 at nine complex and real points (largest ratio to the bound 0.817, at y = 1); item 2 at n = 3200 with the route A Pade parameters at z = 0 (difference 24.8/n^6 against the bound 599/n^6); and tau_4 = n^5 Delta^6 h(n)/24 against 2/L: (tau_4 L/2 - 1) L is 0.86 at n = 1e6 and 1.027 at 1e20, approaching H_4 - 1 = 1.083.

4. H1: the matching system at every large n

4.1 The system, exactly, in moment form

Holland's (52) with D free and k = 0..3, in his logarithmic form (58), with his (31) at every k (log q_k = l(b + k) - delta_k, l(y) = log(1 + 1/y), delta_k = Delta^2 h(n + k)):

    l(B+k) + l(D+k) - l(A+k) - l(C+k) - l(b+k) + delta_k = 0,     k = 0, 1, 2, 3.

Apply (-1)^j n^(j+1)/j! Delta^j at k = 0 (an invertible triangular recombination). Since l(p + k) = Delta log(p + k), with

    lambda_j(p) := (-1)^j n^(j+1)/j! Delta^(j+1) log(p),     tau_j := (-1)^j n^(j+1)/j! Delta^(j+2) h(n),

the system is lambda_j(B) + lambda_j(D) - lambda_j(A) - lambda_j(C) - lambda_j(b) + tau_j = 0, j = 0..3. The model side has a closed form: (d/dp) Delta^(j+1) log p = Delta^(j+1)(1/p) = (-1)^(j+1) (j+1)!/(p (p+1) ... (p+j+1)), so with u_j(w) := lambda_j(n/w)

    u_j'(w) = (j+1) w^j prod_{i=1}^{j+1} (1 + i w/n)^(-1),     u_j(0) = 0.        (4.1)

With w_P := n/P and the signed gap set G = [0, w_A] u [w_B, w_C] u [w_D, w_b], the system is exactly

    tau_j = int_G u_j'(t) dt,     j = 0, 1, 2, 3.                                (4.2)

This is the exact form of RESULTS.md 11.1's moment picture: the targets are moments of the three gaps, with the weight prod (1 + i t/n)^(-1) as the only correction. The targets tau_j are the only place where xi enters.

Scaled variables. Put eps := tau_0 (by Lemma S(b), eps = (2/L)(1 + O(1/L)); this is Holland's (32)) and X = (g0, gx, g1, x) by

    w_A = eps g0,   w_B = x - eps gx/2,   w_C = x + eps gx/2,   w_D = 1 - eps g1,

a bijection onto parameters with 0 < w_A < w_B < w_C < w_D < 1. Write Phi_n(X) for the vector (tau_j - int_G u_j')/eps; the system is Phi_n(X) = 0.

Measured: at the phase 1 solution, Phi_n evaluated through lambda_j vanishes to 2e-60 or better at all 21 route A points (n = 50..3200) and below 2e-68 at all 9 route B points (n = 1e4..1e20), so this reformulation and phase 1's solver describe the same system.

4.2 The decomposition

With u_j'(t) = (j+1) t^j (1 + rhohat_j(t)), rhohat_j = prod (1 + i t/n)^(-1) - 1, and -(j+1)(j+2) t/(2n) <= rhohat_j(t) <= 0:

    Phi_n(X) = Phi_inf(X) + t - eps S(X; eps) + beta - rho(X),
    Phi_inf,j(X) = 1 - [j = 0] g0 - (j+1)(gx x^j + g1),
    t_j  = tau_j/eps - 1                                 (t_0 = 0 by the choice of eps),
    beta_j = (1 - w_b^(j+1))/eps                     in [0, (j+1)/((2n+1) eps)],
    rho_j(X) = ((j+1)/eps) int_G t^j rhohat_j(t) dt,

and eps S_j = P_j(X; eps) - P_j(X; 0) with P_j = [(eps g0)^(j+1) + (x + eps gx/2)^(j+1) - (x - eps gx/2)^(j+1) + 1 - (1 - eps g1)^(j+1)]/eps:

    S_0 = 0,   S_1 = g0^2 - g1^2,   S_2 = -3 g1^2 + eps (g0^3 + gx^3/4 + g1^3),
    S_3 = -6 g1^2 + eps (4 g1^3 + x gx^3) + eps^2 (g0^4 - g1^4).

Phi_inf(X) = 0 says the three-node rule with nodes 0, x, 1 and weights g0, gx, g1 is exact on (j+1) t^j, j = 0..3: its only solution is Simpson's rule X* = (1/6, 2/3, 1/6, 1/2), the Lobatto limit of RESULTS.md 11.3.

The 1/n terms, explicitly. On the box below every point of G lies in [0, 1] and |G| <= eps (g0 + gx + g1) + 1/(2n+1), so |rho_j| <= (j+1)^2 (j+2)/(2n) (g0 + gx + g1 + 1/((2n+1) eps)). Differentiating at the moving endpoints: |d rho_j/d g0| <= (j+1)^2 (j+2) (eps g0)^(j+1)/(2n), |d rho_j/d gx|, |d rho_j/d g1| <= (j+1)^2 (j+2)/(2n) and |d rho_j/d x| <= (j+1)^2 (j+2)/(n eps).

4.3 The contraction, with every constant exact

    J = D Phi_inf(X*) = [[-1, -1, -1, 0], [0, -1, -2, -4/3], [0, -3/4, -3, -2], [0, -1/2, -4, -2]],
    det J = -1,
    J^-1 = [[-1, 5/2, -8/3, 1], [0, -2, 4/3, 0], [0, -1/2, 4/3, -1], [0, 3/2, -3, 3/2]].

J is -diag(1, 2, 3, 4) times phase 1's moment matrix, whose determinant -1/24 is pinned in test_jensen_wedge_phase1.py (-1 = 4! (-1/24)). The norm is the weighted max norm ||y||_w = max_i |y_i|/w_i with w = (3/4, 1, 3/2, 3/4), and the box is B = {||X - X*||_w <= r}, r = 2/25. In the deviations d = X - X*, the matrix I - J^-1 D Phi_inf(X) has two nonzero columns (gx and x):

    row g0: [ 4 dx^3 - 2 dx^2,   12 dgx dx^2 - 4 dgx dx + 8 dx^2 - (8/3) dx ]
    row gx: [ 4 dx^2,             8 dgx dx + (16/3) dx ]
    row g1: [ -4 dx^3 - 2 dx^2,  -12 dgx dx^2 - 4 dgx dx - 8 dx^2 - (8/3) dx ]
    row x:  [ 6 dx^3 - (3/2) dx,  18 dgx dx^2 - (3/2) dgx + 12 dx^2 ]

Bounding each polynomial by the sum of its monomials' absolute values on the box gives the constants (exact rationals, probe_h1_h3.py):

Lemma C (explicit contraction). Let n >= 1, let tau_0 > 0, tau_1, tau_2, tau_3 be given, eps = tau_0 < 2/3, and put

q   := kappa + sigma(eps) + nu_w rho'(n, eps),
d_0 := ||J^-1 (t - eps S(X*; eps))||_w + nu_w max_j (beta_j + rho_j),

with rho' the largest w-weighted row sum of the derivative bounds of section 4.2 and beta_j, rho_j their bounds there. If q < 1 and d_0 <= (1 - q) r, then (4.2) has exactly one solution with X in B, it has A > B > 0 and C > D > 0, and ||X_n - X*||_w <= d_0/(1 - q).

Proof. T(X) = X - J^-1 Phi_n(X) has DT = (I - J^-1 D Phi_inf) + J^-1 D(eps S) + J^-1 D rho, so ||DT||_w <= q on the convex box, and T(X*) - X* = -J^-1 (t - eps S(X*) + beta - rho(X*)) has w-norm at most d_0. For X in B, ||T(X) - X*||_w <= q r + d_0 <= r: T maps B into itself and contracts, so it has one fixed point, which is a zero of Phi_n, and every zero in B is a fixed point. The radius bound is ||X_n - X*|| <= q ||X_n - X*|| + d_0. Admissibility gives the signs. QED. Status: derivation, self-reviewed; every constant exact.

A uniform sufficient form, for reading n_0: with eps <= 1/16 and n eps >= 1e5 one has q <= 0.4574, (1 - q) r >= 0.0434, the eps part of d_0 at most eps/9 <= 0.0070 and the 1/n parts at most 0.0005, so the lemma applies whenever ||J^-1 t||_w <= 7/200.

Remark (a measured symmetry, derived). With constant density, RESULTS.md 11.3 measured a1 + b2 = b1 + a2 = 1 without deriving it. The constant-density moment conditions are invariant under t -> 1 - t, which maps a configuration with gaps at 0, inside and at 1 to another of the same shape, (g0, gx, g1, x) -> (g1, gx, g0, 1 - x). For small eps the solution lies within O(eps) of X*, so it and its reflection both lie in B, and uniqueness in B (Lemma C) forces them to coincide. The x-component of J^-1 (eps S)(X*; eps) above is exactly 0 for the same reason. For every eps it follows from uniqueness of the [2/2] Pade approximant, a classical fact not re-proved here.

4.4 The targets, and the first order

Lemma T. Assume Lemma S(b) for k = 2..5. Then tau_j = (2/L_n) int_{[0,1]^(j+2)} (1 + sigma/n)^(-(j+1)) (L_n/L_(n+sigma)) (1 + e_(j+2)(n + sigma)) ds with sigma = s_1 + ... + s_(j+2), so tau_j L_n/2 - 1 = O(1/L_n), the explicit part of the error being at most (j+1)(j+2)/n + (j+2)/((n-1) L_n) beside C'_(j+2)/L_n, and t_j = O(1/L_n). With Lemma S(c): tau_j L_n/2 = 1 + (H_j - 1)/L_n + O(1/L_n^2) and t_j = H_j/L_n + O(1/L_n^2).

Proof. Delta^(j+2) h(n) = int h^(j+2)(n + sigma) ds; insert Lemma S(b); L_x is increasing with dL_x/dx <= 1/(x - 1). QED. Status: derivation, self-reviewed.

The first-order configuration. Running Lemma C with a box shrinking like d_0 makes q = O(1/L), so X_n = X* - J^-1 Phi_n(X*) + O(1/L^2). With Lemma T, J^-1 (0, 1, 3/2, 11/6) = (1/3, 0, -1/3, -1/4) and the eps part above,

    (X_n - X*) L  ->  (-2/9, -2/9, 4/9, 1/4).

Substituting, with eps L/2 = 1 - 1/L + O(1/L^2):

    A/(nL) = 3 + 7/L + O(1/L^2),        (D - n) L/n = 1/3 + 2/(3L) + O(1/L^2),
    (B - 2n) L/n -> 5/3,   (C - 2n) L/n -> -11/3,   (B - C) L/n = 16/3 - 112/(9L) + O(1/L^2).

This answers the open item of RESULTS.md 11.3: B and C separately sit at 2n(1 + O(1/L)) with the first-order coefficients 5/3 and -11/3, which the constant-density model could not see because the interior node moves by 1/(4L). Status: derivation, self-reviewed (first order); measured agreement below.

4.5 H1

H1. Assume (P1) and (P2). There is n_0 such that for every integer n >= n_0 Holland's four-parameter system with D free (his (52) at k = 0..3) has a solution (A, B, C, D) with A > B > 0, C > D > 0, unique in the box B of the scaled variables, with ||X_n - X*||_w <= d_0/(1 - q) = O(1/L_n). In the original parameters: A = 3nL(1 + O(1/L)), B = 2n(1 + O(1/L)), C = 2n(1 + O(1/L)), D = n(1 + 1/(3L) + O(1/L^2)), B - C = (16n/(3L))(1 + O(1/L)), and the gap total w_A + w_C - w_B + 1 - w_D = (2/L)(1 + O(1/L)): within relative O(1/L) of the leading-order [2/2] Pade configuration.

Proof. By Lemma T, eps -> 0 and t -> 0 like 1/L, and n eps -> infinity; then sigma(eps) -> 0, rho' -> 0 and d_0 -> 0 while kappa = 0.290 < 1, so Lemma C applies for all large n. Its radius bound gives the rest by substitution. QED. Status: derivation, self-reviewed.

What is explicit and what is not. Explicit and exact: the system (4.2), X*, J, J^-1, the weights, r, kappa, nu_w, sigma(eps), every 1/n bound, the admissibility range eps < 2/3, and the uniform condition ||J^-1 t||_w <= 9/250 (with eps <= 1/16, n eps >= 1e5). Not explicit: n_0. The condition is on the data t(n), and the rate at which t(n) -> 0 comes from Lemma S, whose constants inherit C_R, theta, x_0 of (P1), which neither Holland nor GORTTW makes explicit. The missing step for an explicit n_0 is an explicit form of (P1), that is an explicit error term in GORTTW's complex saddle estimate (3.2); that is phase 2's object (MISSION Q2), not H1's. As with Holland's section 10, the finite range n < n_0 is absorbed into the eventual constant of the wedge, so H1 as the chain uses it needs no explicit n_0.

4.6 Measured: the inequality on the measured targets

Route A for n <= 3200 (exact table, validated against zeta/li.py and agreeing with route B on every tau_j to the latter's own M5/M6 spread, relative 1e-18 to 1e-23 at n = 400..3200), route B beyond (spread between five and six Laplace corrections below 1e-24 relative). The inputs are measured, not enclosed, so a "holds" here is a measurement of the inequality at that n, not a proof at that n.

The full table, all 21 route A points and 9 route B points, is in results_h1_h3.json. At every n where the inequality holds, the measured configuration lies inside the radius the lemma predicts, which is the check that could have failed. At n <= 3200 the inequality fails everywhere, by a wide margin (eps = 0.29 at n = 3200 and 0.53 at n = 50, q from 2.1 to 43); existence there rests on phase 1's direct solve (measured), not on this lemma.

The first order, measured (route B; RESULTS.md 11.3's table extends the same columns):

Each column moves monotonically toward its first-order value over all nine route B points 1e4..1e20, with a gap that shrinks like 1/L (like 1/L^2 for A and D, whose 1/L term is included).

5. What this changes in the chain of RESULTS.md 12.5

The independent review (REVIEW_PHASE1.md, merged while this file was written) found no step of sections 11 and 12 broken and listed three open inputs: H1, H3 and R1, the leading constant of h^(5) that H1's limit needs. All three are closed here, R1 as Lemma S(b) at k = 5. The review's other items, as they bear on this file:

• R2 (12.3 needs Holland's (77) for (C, D) and D > 64d, unstated): H1 gives D = n(1 + O(1/L)) and C - D = n(1 + O(1/L)), so D >= 32d, C >= D + d and D > 64d hold for d = o(n) and n large. They still have to be written into 12.3's statement before its constants are made explicit; this file does not edit 12.3.
• R3, R4 (A is of order nL, and A >= 8B holds only from some n): H1 gives A/B = (3/2) L (1 + O(1/L)), so A >= 8B for all large n; at leading order from L = 16/3, and in fact earlier, since the 7/L correction to A/(nL) is positive (the review measured the first n with A/B >= 8 at 346, where L = 4.0).
• The review's warning that H1's limit is singular unless the interior gap is parametrised by its centre and width is met: section 4.1 uses (gx, x), and det J = -1. The review's own coordinates give determinant 1; the sign is orientation.
• The review measured rho_k = h^(k)/(leading term) from differences of log M_x at n = 50 to 1e6; this file's tau_j L/2 are the same quantity in difference form, and the two agree (at n = 1e6: rho_5 = 1.056 there, tau_3 L/2 = 1.0557 here, against Lemma S's first order 1 + 0.833/L = 1.076).

Then:

• H1, H3 and R1 are no longer open inputs: each is a self-reviewed derivation from (P1) and (P2), and H3 also from Holland's (25) alone.
• The conditional wedge n^4 log^2(n+2) >= K d^6 log^12(d+2) therefore rests on: (P1), (P2) and Holland's eq. (79) and Lemma 7.1, which Holland's own Theorem 1.1 also rests on; Martinez-Finkelshtein, Morales and Perales, Propositions 2.7 and 2.17, which it also uses; and the self-reviewed steps of RESULTS.md 12 (Proposition 2.2_6, Lemmas A to C) and of this file. A composite claim takes the grade of its weakest step: it is a derivation, reviewed internally once (sections 11 and 12, by REVIEW_PHASE1.md) and self-reviewed (this file, not yet reviewed), not a theorem, pending external review. This file does not promote it.
• The derivations read xi only through the targets tau_j, that is through (P1). The Davenport-Heilbronn gate (RESULTS.md section 15) found the construction succeeding for DH as well; nothing in H1 or H3 distinguishes the two, which is consistent with it.
• The constants of the wedge stay non-explicit for the same reason n_0 does: (P1). Lemma C and section 3.3 are explicit; the saddle constants are phase 2's job.
• It does not touch RH: the wedge leaves the small-n, large-d corridor exactly as Holland's does, and n = 0 with all d is RH itself (MISSION.md).

6. Reproduce

.venv/bin/python hunts/jensen_wedge/probe_h1_h3.py --time-unit   # about 7 s
.venv/bin/python hunts/jensen_wedge/probe_h1_h3.py               # about 30 s, writes results_h1_h3.json
.venv/bin/python -m pytest -q -n0 hunts/jensen_wedge/test_h1_h3.py