jensen_wedge: independent review of H1_H3.md (H1, H3, R1)

Nothing here bears on RH. This file reviews H1_H3.md and RESULTS.md section 16 of hunt #135 as merged on main at 81ce5921 (PR #311). The reviewer is a separate session that did not write them, started from a blank context, read Holland's PDF itself, and reproduced every numerical claim it checks with its own code (review_h1h3/review_h1h3.py), which imports nothing from probe.py, probe_h1_h3.py or review/. The brief was to find where the derivations break, not to confirm them.

Verdict, piece by piece

Overall: no piece is broken. No counterexample was found to Lemma S, H3, R1, Lemma C, Lemma T or H1; no exact constant is wrong; every measured number in H1_H3.md that this review recomputed agrees with it to the digits printed. Four nits (N1 to N4) change no verdict.

On the question the brief asked: does the conditional six-coefficient wedge now rest only on Holland's own inputs? Yes, with two qualifications. Its external inputs are Holland's Proposition 4.1 (which he derives from the complex form of GORTTW (3.2)), his (79) and Lemma 7.1, and MFMP Propositions 2.7 and 2.17, all of which Holland's own Theorem 1.1 also uses. The first qualification is (P2): it is not Holland's statement, but it follows from his (16) alone by the argument in section 4 below, so it is a missing paragraph, not a missing input. The second is REVIEW_PHASE1.md R2: the two hypotheses of RESULTS.md 12.3 that H1 makes automatic are still not written into 12.3's statement. Every step the chain uses has now had one internal review (sections 11 and 12 by REVIEW_PHASE1.md, H1_H3.md here). The grade of the composite is that of a derivation reviewed internally once, not a theorem, pending external review. This review ran no novelty search. The hunt's own search (NOVELTY.md, RESULTS.md section 17, merged after the commit reviewed here) found earlier, unreviewed deposits of the wedge and of the Pade point; that bears on the delta against Holland, not on whether these derivations hold.

1. What was read, and how the numbers were reproduced

• Holland, arXiv:2608.08682v1, downloaded as PDF on 2026-10-19 (SHA-256 3fc31ba84fb113bc0b3109fb0e569bd1d7183018485aca1eb1dab565c839b49d, the file phase 1 and the phase 1 review recorded), extracted with pdftotext -layout into a scratch directory outside the tree and read as data: section 3 (eq. (10)), section 4 (eqs. (14) to (33), Proposition 4.1, Lemmas 4.2 and 4.3), section 6 (eqs. (49) to (66), Lemma 6.1).
• Four routes this hunt had not used.
• G_0 symbolically. The derivatives of Holland's main term are exact rational functions of (N, L), produced by sympy from D = d/dN + (L/Q) d/dL. The hunt measured them by Cauchy's formula on 64 points.
• The targets from Titchmarsh's kernel. h(z) = log M_z with M_z = int_0^oo Phi(u) u^(2z) du, Phi(u) = sum_k (2 pi^2 k^4 e^(9u) - 3 pi k^2 e^(5u)) exp(-pi k^2 e^(4u)), by tanh-sinh quadrature in the saddle variable, all shifts n..n+6 from one set of integrand values. The u-scale is half Holland's, which adds a linear function of z to h and changes no Delta^m, m >= 2. Against zeta.li's Cauchy route (no Mellin representation) at n = 20..26: Delta^2..Delta^6 log gamma agree to relative 9e-62 or better. A planted 1e-30 error in one log moment moves Delta^5 h(1e7) by 10 percent; two quadratures at different precision and window agree to 1e-20 (test).
• Holland's (15) on a complex circle, F(s) integrated along the real axis with the large phase s log y_c taken out explicitly and the residual phase checked to stay below 1, so no branch jump can pass unseen.
• The matching system solved twice: in the gap-moment form (4.2) through exact finite differences of log, and in Holland's quotient form (52) at k = 0..3 in the unknowns log A..log D. The two share no formula.
• Precision. Working precision 50 + 7 log10 n digits (190 at n = 1e20), chosen so Delta^6 h keeps at least 30 digits after the cancellation in the differences.

Reproduce:

.venv/bin/python hunts/jensen_wedge/review_h1h3/review_h1h3.py             # about 80 s, writes results_review_h1h3.json
.venv/bin/python -m pytest -q -n0 hunts/jensen_wedge/review_h1h3          # 39 tests, about 20 s

2. Lemma S: holds

Checked line by line. The remainder bound (Cauchy on the disk of radius eta x, with |zeta| >= (1 - 2 eta) x), the elementary terms E_l^(k), the identity L' = L/Q (from (16): dx/dL = Q/L), the collapse to G_0' = log L + rho_G, the factorisation Q = N L (1 + u) with |u| <= 11/32 under (P2), the bound |rho_G| <= 2/|N| (it holds only after the cancellation rho_G = [(1 + L) - (1 - 3L/(2N))/(1 + u)]/(2Q), which gives about 1.3/|N|; the term-by-term triangle inequality gives 3.2/|N|, so the write-up should show the combined form), the Leibniz step phi^(m) = (-1)^m m!/(N^(m+1) L)(1 + H_m/L + ...) (the i >= 1 terms sum to m! H_m), (phi v)^(m), and the passage back to z.

Strengthened, not just confirmed. Symbolically, with a = 1/L and b = L/N,

    f_k(a, b) := (-1)^k N^(k-1) L G_0^(k)(N)/(k-2)! = P_k(a, b) / (c_k (4 + 4a - 3b)^(2k)),

with P_k a polynomial and c_k a rational constant, for k = 2..8. Since 4 + 4a - 3b = 4(1 + u), (P2) gives |4 + 4a - 3b| >= 21/8, and bounding P_k by its monomials on |a| <= 1/4, |b| <= 1/8 gives, pointwise at every N where (P2) holds, |G_0^(k)(N)| <= C_k (k-2)!/(|N|^(k-1) |L_N|) with C_2..C_6 = 13.9, 65.9, 294.8, 1287, 5548 (exact rationals in the JSON; crude, since |f_k| near the real axis is about 1.1). The main-term part of (a) therefore needs no disk at all, and its constant is explicit; only R's Cauchy estimate needs disks, and its constant inherits C_R.

(c), exactly. f_k(a, 0) = 1 + c_1(k) a + c_2(k) a^2 + ... with

For h rather than G_0 the difference is O(L/x) (elementary terms), O(1/x) (the factor (x/(x-1))^(k-1)) and O(L x^(-2+eps_0)) (R), all beyond every power of 1/L, so (c) holds for h with the same constants. The b-dependence is likewise beyond every power of 1/L on the real axis, since L/N ~ e^(-L). A planted wrong saddle identity, L' = L/(Q + N), moves the k = 3 constant from 0 to -1 (test).

Measured, reproduced. The ratios of G_0^(k) to the leading term at N = 1e3, 1e20, 1e50 and (ratio - 1) L at 1e50 (-0.9909, -0.0179, 0.4773, 0.8105, 1.0618) agree with section 2's quoted values to every printed digit; the gap to H_(k-2) - 1 is the second-order term (k = 6: 13/12 - (7/3)/109.29 = 1.0620). On the rays arg N = 0.25, 0.5 the modulus agrees with the real value to 1.37e-3 (1e3), 3.3e-6 (1e20) and 1.9e-7 (1e50), as quoted. On the measured targets, (tau_j L/2 - 1 - c_1/L - c_2/L^2) L^3 stays between -6.1 and 4.8 over all 13 values of n and j = 0..4, and at n = 1e20 it reads -0.977, 4.79, 2.57, 0.21, -1.78 against the third-order coefficients -1, 5, 5/2, 0, -49/24: Lemma S's expansion explains the targets through third order, well beyond what H1_H3.md claims.

N1 (radius, harmless). The disk argument says |L| changes by a factor in [5/6, 7/6] along N-disks of radius rho |N| "for rho <= 1/4". Such a disk about N = 2(x + w) - 2 stays inside {2z - 2 : z in S} only for rho up to about eta, which is min(sin theta, 1/4)/2 and can be far below 1/4 when theta is small. Taking rho = eta changes constants only, and with the pointwise bound above the main term does not need the disks.

3. H3 and R1: hold

Route 1. For |w| <= eta x/2 the disk of radius eta x/2 about x + w lies in {x + w' : |w'| <= eta x}, where Holland's (25) applies; Cauchy with m = 1 gives |h^(6)| <= (2C/eta)/(x^5 log x). Correct, and it uses (25) exactly as Holland states it (absolute eta, C, for x large).

Route 2. With delta = min(sin(theta/2), 1/2) the disk |zeta - z| <= delta |z| about z in the half-angle sector stays in S (angle at most theta/2 + theta/2, modulus at least x_0), and the displayed bound 720 delta^(-6)(1 - delta)^(-1+eps_0) C_R |z|^(-7+eps_0) follows. Nothing in Holland's proof of (19) restricts r; he stated r <= 5 because that is all he used.

Section 3.3. Item 1: psi^(5)(y) - 24/y^5 = int_0^oo t^4 e^(-yt)(t/(1 - e^(-t)) - 1) dt and 0 <= t/(1 - e^(-t)) - 1 <= t (equivalent to (1 + t) e^(-t) <= 1); the nine quoted ratios (0.81734306 at y = 1 down to 0.00050523513 at 10 + 30i) reproduce to 8 digits. Item 2: f_z(n/p) = 24/(p + z)^5, f_z(0) = 0, and the five terms are -int_G f_z' over the signed gap set; at n = 3200 the exact sum and the gap integral are -9.2806146 and -9.2728605 (times n^-5), difference 24.813/n^6, all as quoted. The quoted bound 599.44/n^6 is 600 (n/b)^6, every parameter replaced by the smallest, b; with each parameter's own value it is 201.6/n^6. Item 3: f_z'(t) = 120 n t^4/(n + z t)^6. Item 4: |G| = eps(g0 + gx + g1) - 1/(2n + 1) on the box, so the stated eps(1 + 13r/4) + 1/(2n + 1) is a valid, slightly loose bound. (tau_4 L/2 - 1) L is 0.861 at n = 1e6 and 1.027 at 1e20, as quoted.

N2 (asymptotic step, harmless). The explicit second term of the final bound uses eps <= 2/L_n. That holds for n beyond a non-explicit n_1 (the first-order term of Lemma S(c) at k = 2 is -1/L, negative) and is measured at all 13 values of n here (eps L/2 from 0.663 to 0.977), but it is not a derived inequality at any stated n. The bound needs only eps = O(1/L), which Lemma T gives, so nothing breaks.

R1. Lemma S(b) at k = 5 is h^(5)(x) = -12/(x^4 L_x)(1 + 5/(6 L_x) + O(1/L_x^2)). This is the input REVIEW_PHASE1.md R1 asked for, and its first-order constant is now exact.

4. (P1), (P2), (P3) and Lemma 6.1 against the paper

(P1): stated by Holland. Proposition 4.1 gives a sector S_theta on which M_z is nonzero, the branch h = log M_z real on the positive axis, the decomposition (18), and (19): for every fixed 0 < eps_0 < 1/2 and 0 <= r <= 5, R^(r) = O(|z|^(-r-1+eps_0)) uniformly on a slightly smaller closed sector, with constants depending on eps_0, r, theta. (P1) is the case r = 0 on that smaller sector. Holland's proof is a short derivation from the complex form of GORTTW (3.2), which neither the hunt nor this review checked; his own Theorem 1.1 rests on the same input.

Measured support for (P1), all on this review's routes:

• On the real axis, Holland's main term G_0(2z - 2) + log(32 z (2z - 1)) reproduces Delta^m h(n), m = 2..6, to relative 1.8e-4 (n = 50), 1.7e-7 (3200), 7.8e-14 (1e7) and 7.5e-39 (1e20). That is the size L/n^2 expected if R^(m) = O(n^(-m-1)).
• On the circle |z - n| = n/8, from Holland's (15): R + c_sad at the centre is 1.669524288 (n = 400) and 1.669034388 (3200); its largest deviation on the circle is 0.0283/n and 0.0384/n; sup |h^(6)| on |z - n| = n/16 is 1.52200 and 1.54447 times 48/(n^5 L_n) (the same with half the points); the Taylor ratios at the centre for k = 2..7 match. Every one of these equals the hunt's value to the digits it prints. A planted 1e-3 z term in R would vary by about 0.1 n on this circle and is caught (test).
• Holland's (15) against the Titchmarsh-kernel moment: Delta^2..Delta^6 h from the circle's Taylor coefficients agree with the real-axis quadrature to relative 7.5e-33 or better at n = 400 and 3.5e-28 or better at 3200. So (15) is right up to the linear normalisation that Holland's own text discusses.

(P2): not stated by Holland (N4). Off the real axis Holland says only that L_x means the holomorphic continuation of the saddle that GORTTW use in their section 3. The bounds |L_N| >= 4 and |L_N/N| <= 1/8 appear nowhere in the paper. H1_H3.md section 1 is candid about this ("uses this without separate statement"), but it lists (P2) among the inputs "from the paper", and RESULTS.md section 16's closing list of what the wedge rests on leaves it out. It is in fact a consequence of (16) alone:

(P2) from (16) (reviewer's derivation, self-reviewed). Let |arg N| <= pi/2 and l := log |N| >= 13. On D = {L : |L - l| <= l/2} put T(L) = log(N/L - 3/4) - log pi (principal log). For L in D: |L| lies in [l/2, 3l/2], |arg L| <= pi/6, and e := |3L/(4N)| <= (9/8) l e^(-l), so arg(N/L - 3/4) lies within pi/2 + pi/6 + asin e < pi of 0 and T is holomorphic. Then |Re T - l| <= log(3 pi l/2) + log(1/(1 - e)) and |Im T| <= pi/2 + pi/6 + asin e, whose sum is at most l/2 once l >= 13 (slack 0.29 at l = 13; negative, -0.13, at l = 12). Also |T'(L)| = 1/|L (1 - 3L/(4N))| <= (2/l)/(1 - e) <= 0.154. So T is a contraction of D into itself, with a unique fixed point, which solves N = L(pi e^L + 3/4). The iterates of log N (which lies in D) are holomorphic in N and converge uniformly, so the fixed point L_N is holomorphic; it is real for real N, hence equal to the positive root of (16) there, hence GORTTW's continued branch by uniqueness of continuation. Finally |L_N| >= l/2 >= 6.5 and |L_N/N| <= (3l/2) e^(-l) < 1/8.

So (P2) holds on Holland's sector once x_0 >= 2.3e5 (derived); numerically, along every ray of |arg z| <= 1 from |z| = 346 to 1e30, the continued branch has |L_N| >= 4.0010 and |L_N/N| <= 0.0059, while |L_N| >= 4 already fails at |z| = 50 (smallest 2.50). Since (P1)'s x_0 is not explicit anyway, enlarging it costs nothing. Recommendation: write this paragraph into H1_H3.md section 1 and drop (P2) from the list of inputs.

(P3): stated by Holland. His (25): absolute eta, C > 0 with |h^(5)(x + w)| <= C/(x^4 log x) whenever |w| <= eta x, x large.

Lemma 6.1: not used, confirmed. It solves the D-fixed three-equation system (52) in the scaled variables (a, b, c) on a sup-norm cube of radius 3/20, with its own constants (||J^-1||_inf = 66/25, contraction factor below 2/5). H1 is a different, square four-equation system with its own matrix, norm and box; it borrows the method only. The equations H1 does use: (31) at every k follows from (14) (gamma = sqrt(pi) M_z/(4^z Gamma(z + 1/2)) by duplication, so log q_k = l(b + k) - Delta^2 h(n + k)), and (58) is (52) in logs, f_k(U) = -l(U + k). (21) to (23) are Lemma S(b) at k = 2..4, and (24), (25), (29) are consistent with Lemma S(a); R1 sharpens (29) at k = 5 from an O-bound to an asymptotic.

5. H1: holds

4.1, the gap-moment form. Rederived: log qq_k(U, V) = l(V + k) - l(U + k), so (52) at k = 0..3 is the displayed system; applying (-1)^j n^(j+1)/j! Delta^j at k = 0 is triangular and invertible; l(p + k) = Delta log(p + k); Delta^(j+1)(1/p) = (-1)^(j+1)(j+1)!/(p (p+1) ... (p+j+1)); and with p = n/w, u_j'(w) = (j + 1) w^j prod_(i=1)^(j+1) (1 + i w/n)^(-1), u_j(0) = 0. The system is exactly tau_j = int_G u_j'. At all 13 values of n from 50 to 1e20 the root of (4.2) and the root of (52) solved independently agree to relative 1.1e-46 (n = 50, at 61 digits), 2.0e-65 (n = 3200) and 8.9e-115 (n = 1e20), and the root satisfies 0 < w_A < w_B < w_C < w_D < 1.

4.2, the decomposition. S_j derived from the gap endpoints by sympy equals the display for j = 0..3, and Phi_inf = 1 - P_j(X; 0), so the decomposition is exact. The bounds -(j+1)(j+2)t/(2n) <= rhohat_j <= 0 (from prod (1 + a_i)^(-1) >= 1 - sum a_i), on beta_j, on rho_j and on its four partial derivatives each rederive. The x-derivative bound (j+1)^2 (j+2)/(n eps) is crude (it ignores that the two endpoint values nearly cancel), which only costs room.

4.3, the constants. Simpson's rule is the only zero of Phi_inf (sympy solve). J, det J = -1 and J^-1 are as displayed, and J = -diag(1,2,3,4) M with M the matrix pinned at determinant -1/24 in test_jensen_wedge_phase1.py. The matrix I - J^-1 D Phi_inf(X* + d) equals the display entry by entry. Its monomial row bounds are 3472/15625, 177/625, 1736/15625 and 4533/15625 (row x), so kappa = 4533/15625; the largest value on a 21 x 21 grid is 177/625 (row gx at a corner), as quoted. nu_w row sums are 86/9, 10/3, 17/9, 8: nu_w = 86/9. The four row bounds of sigma(eps) are polynomials with nonnegative coefficients and the x-row dominates the others coefficientwise, so sigma(eps) is that one polynomial, 33053/93750 eps^3 + 778139/281250 eps^2 + 12/5 eps; at 1/32, 1/16, 1/8 and 1/6 it equals the hunt's rationals exactly. J^-1 (eps S)(X*; eps) = eps (1 - eps)(1/18, -1/9, 1/18, 0) exactly. Admissibility: on the box, w_A < w_B binds at eps = 11/15 and w_C < w_D at eps = 2/3; an exact corner check passes at 2/3 - 1e-9 and fails at 2/3.

Lemma C, the proof. DT = (I - J^-1 D Phi_inf) + J^-1 D(eps S) + J^-1 D rho has the right signs (D Phi_n = D Phi_inf - D(eps S) - D rho; t and beta do not depend on X); the operator norm of the weighted max norm is the weighted row sum; the mean-value inequality on the convex box, the self-map step q r + d_0 <= r and Banach's theorem are standard. Uniqueness is in B only, as stated.

Lemma C, attacked. The derivative bound is the step most likely to be quietly wrong, so it was tested against the exact Phi_n on measured data. At n = 1e7 (q = 0.688) the largest ||DT||_w over 120 random points of B is 0.550, and at n = 1e20 (q = 0.408) it is 0.357; the largest ||T(X) - X*||_w over the same points is 0.0405 and 0.0195, inside r = 0.08; Newton from 24 random starts in B reaches one root every time (spread 9e-76 and 1e-107 in the weighted norm). With a planted wrong J^-1 (one entry changed from -2 to -3) the sampled ||DT||_w exceeds q, so the probe can fail (test).

N3 (threshold inconsistency, harmless). Section 4.3 says the lemma applies at eps <= 1/16, n eps >= 1e5 whenever ||J^-1 t||_w <= 7/200; section 4.5 says 9/250. With the bounds 4.3 displays (eps part at most eps/9, the rho tail summed over the box) the exact threshold is 0.0359766, so 7/200 holds and 9/250 does not, by 2.3e-5. With the sharper eps (1 - eps)/9 and the tail at X* it is 0.0364727 and both hold. One of the two numbers should change; nothing else depends on it.

4.4, Lemma T and the first order. Lemma T is the integral form of the finite difference with Lemma S(b) inserted; dL_x/dx = 1/((x - 1)(1 + u)) <= 1/(x - 1) since u > 0 for large x. Symbolically: J^-1 (0, 1, 3/2, 11/6) = (1/3, 0, -1/3, -1/4), (X_n - X*) L -> (-2/9, -2/9, 4/9, 1/4), and with eps L/2 = 1 - 1/L the expansions A/(nL) = 3 + 7/L, (D - n) L/n = 1/3 + 2/(3L), (B - 2n) L/n -> 5/3, (C - 2n) L/n -> -11/3, (B - C) L/n = 16/3 - 112/(9L) all come out as displayed. Using X_n - X* = -J^-1 Phi_n(X*) + O(1/L^2) with a shrinking box is legitimate: the entries of I - J^-1 D Phi_inf are linear in the deviation, so q = O(1/L) there.

4.5, H1 itself. Lemma T gives eps, t = O(1/L) and n eps -> oo, so sigma(eps), rho' and d_0 tend to 0 while kappa < 1; Lemma C applies for all large n, and the parameter statements follow from its radius bound. Correct as stated, with n_0 not explicit for the reason given (it inherits C_R, theta, x_0). Absorbing n < n_0 into K is legitimate: for K large the wedge condition admits no d >= 6 with n < n_0.

4.6, the measured table, reproduced on an independent route.

(At n = 50, eps = 0.5296. All 13 rows, n = 50, 200, 800, 3200, 1e4, 1e5, 1e6, 1e7, 1e8, 1e10, 1e12, 1e16, 1e20, are in results_review_h1h3.json.) Against the hunt's ten rows: tau_j and X_n agree to the 10 digits stored here (relative 3e-10; the test pins tau_j at 1e7 to 1e-13), d_0 and the parameter columns to every digit, and the holds/fails pattern is identical (fails at every n <= 1e6, holds from 1e7 to 1e20), as is ||X_n - X*||_w inside the radius at all six n where it holds. q differs by at most 1.2e-6: the hunt evaluates sigma at eps (1 + 1e-6) as a deliberate rational upper bound. With the rho tail at X* instead of over the box the pattern is unchanged. At n = 50, 200, 800 the parameters reproduce RESULTS.md 11.3's table (for example A/(nL) = 7.3916, 5.5461, 4.7632). The first-order columns of section 4.6's second table reproduce to the printed digits and move monotonically toward (-2/9, -2/9, 4/9, 1/4).

6. Nits, collected (none changes a verdict)

• N1 (Lemma S): the N-disk radius must be at most about eta, not 1/4; or use the pointwise main-term bound of section 2 here. Also show rho_G in its combined form, since the term-by-term bound is 3.2/|N|, not 2/|N|.
• N2 (section 3.3): eps <= 2/L_n is asymptotic (beyond a non-explicit n_1), not a stated inequality; eps = O(1/L) suffices.
• N3 (sections 4.3 and 4.5): 7/200 against 9/250; the displayed bounds support 7/200.
• N4 (sections 1 and 5 of H1_H3.md, RESULTS.md 16): (P2) is not Holland's; it follows from (16) by the fixed-point paragraph of section 4 here.

7. What this review did not do

• It did not check the complex form of GORTTW (3.2), from which Holland derives Proposition 4.1, nor Holland's (79), Lemma 7.1 or MFMP Propositions 2.7 and 2.17.
• It did not re-review RESULTS.md sections 11 and 12 (REVIEW_PHASE1.md did) or write R2's hypotheses into 12.3.
• Its measurements are finite: 13 values of n, two circles. Lemma C "closing" at a measured n is a measurement at that n, since the inputs are not enclosed in balls.
• No novelty search here (see NOVELTY.md for the hunt's). The (P2) paragraph above is the reviewer's own derivation and has had no second reader.
• Compute: the script ran locally in about 80 s and the tests in about 20 s; nothing heavy. MISSION.md asks for a runmanifest in RUNS.md for every run, but this review's brief forbade editing existing files, so the run is recorded here instead and RUNS.md was not touched.
• It edited no existing file. Files: this one, review_h1h3/review_h1h3.py, review_h1h3/test_review_h1h3.py, review_h1h3/results_review_h1h3.json.