jensen_wedge: independent review of the phase 1 derivations

Nothing here bears on RH. This file reviews RESULTS.md sections 11 and 12 of hunt #135 as merged on main at d0ce271b (PR #308). 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/wedge_review.py), which does not import probe.py or probe_phase1.py. The brief was to find where the derivations break, not to confirm them.

Verdict, step by step

Overall: no step is broken. No counterexample was found to any lemma, no constant is wrong, and every reproduced number agrees with phase 1. The conditional wedge n^4 log^2(n+2) >= K d^6 log^12(d+2) follows from sections 11 and 12 given H1, H3 and the extra input R1, and the exponent comparison (2/3 against Holland's 3/5) is stated correctly. It remains a conditional derivation, not a theorem: H1 is unproved, H3 and R1 are routine but unwritten, and this is one internal review, not an external one. The weakest link is H1, so the composite claim takes H1's status.

What was read, and how the numbers were reproduced

• Holland, arXiv:2608.08682v1, downloaded as PDF on 2026-10-19 (SHA-256 3fc31ba84fb113bc0b3109fb0e569bd1d7183018485aca1eb1dab565c839b49d, identical to the file phase 1 recorded), extracted with pdftotext -layout into a scratch directory outside the tree and read as data: Proposition 2.2 and its proof, section 4 (eqs. (14) to (29)), Lemmas 4.3 and 5.1 to 6.2, section 7 (eqs. (74) to (80)), Lemmas 8.1 and 9.1, sections 10 and 12.
• Martinez-Finkelshtein, Morales, Perales, arXiv:2309.10970v3 (SHA-256 23f228e0682430c1ae6285be4abed4d730e9a08accad6c3f540fbdf4eb5bb4a3), Propositions 2.7, 2.11 and 2.17, read the same way.
• The target sequence, by a route the hunt did not use. gamma(n) is const * n! M_n/(2n)! (Holland eq. (10)) with M_x the Mellin moment of Riemann's kernel phi(u) = sum_k (2 pi^2 k^4 e^(9u/2) - 3 pi k^2 e^(5u/2)) exp(-pi k^2 e^(2u)), integrated by 48 Gauss-Legendre panels of 40 nodes across +-50 saddle widths. Phase 1's route A integrates u^(2k) e^(u/4) omega(e^u) by a trapezoid rule and route B is a saddle expansion; this route shares no integrand, rule or node with either. Against zeta.li.xi_taylor_coefficients (dps 50) the ratios gamma(n+1)/gamma(n), n = 20..31, agree to 1.5e-51, and a planted 1e-30 error in one log gamma is seen (test). The integrand at the panel ends is below 4e-376 of its peak at every n used.
• Solvers. Newton in the logs of (A, B, C, D) with an analytic Jacobian, started from the leading-order [2/2] Pade configuration, recovers a planted two-factor model to 1e-40 and a planted sixth-order defect of either sign (test).

Reproduce:

.venv/bin/python -m pytest -q -n0 hunts/jensen_wedge/review     # 30 tests, about 15 s
.venv/bin/python hunts/jensen_wedge/review/wedge_review.py       # about 100 s, rewrites review/results_review.json

Compute: everything above ran locally in under two minutes per script (one log gamma window costs about 0.6 s); nothing heavy was run, so no RUNS.md manifest is owed by this review, and that file was not touched.

11.1 The leading-order reduction: holds

Checked. (i) The derivative formula: from Holland (14), log gamma(n+z) = h(n+z) - log Gamma(b+z) + linear, so E^(k) is as written, and it agrees with Holland's (46) and (83) at k = 4, 5. (ii) The bracket: with psi^(k-1)(p) = (-1)^k (k-2)!/p^(k-1) (1 + O(k/p)) and h^(k) as stated, the bracket is beta_(k-1) + O(1/L^2) + O(1/n). (iii) E(m) = E^(m)(xi): when E vanishes at 0..m-1, E(m) is the m-th forward difference, which equals E^(m)(xi) by the mean-value form of divided differences. (iv) The moment and Pade rewritings, term by term, including (1 - z)^(eps - 1) = int dlambda(x)/(1 - xz) with lambda = Beta(1 - eps, eps) (moments (1 - eps)_k/k!). (v) The sign-change count. (vi) Holland's constants: one interval gives D_n ~ 2/(n^2 L); two give alpha = beta = eps/2, A_J ~ nL, B_J ~ n(1 + 1/L), E_J(4) ~ -2/(n^3 L), all as in his (30), (41), (42), (48). The 11.2 first-order family was rederived in exact fractions at omega = 3/2, 2, 5, and at omega = 2 gives (6/5, 1/3, 1/24) (test).

R1, the one input Holland does not supply. 11.1 takes h^(k)(x) = (-1)^k (k-2)! (2/L)/x^(k-1) (1 + O(1/L)) for k = 5 and 6 from "(28) differentiated beyond". Holland states the constant only through k = 4; for k = 5 his (25) and (29) are O-bounds. An asymptotic relation on the real axis cannot be differentiated in general. Here it can, because h is analytic on a sector and the relative O(1/L) error of h'' holds on that sector (G_0 is explicit, R obeys (19)), so Cauchy's inequality carries it to every fixed k. That argument is routine and unwritten. Measured from k-th differences of log M_x on this route (Delta^k h(n) is h^(k)(n + k/2) up to a relative O(k/x^2)), the ratio rho_k of h^(k) to the stated leading term:

(rho_k - 1) L stays between -0.92 and 0.86 at every entry: consistent with a 1/L approach for every k, including the two Holland does not state. Measured, not proved. This matters for 11.2's and 11.4's leading constants and for H1 (see 12.5); the chain's stability step needs only O-bounds.

11.3 The [2/2] Pade point inside Holland's family: holds

Checked. The [2/2] approximant was computed from its Hankel system at eps = 0.9, 0.5, 0.342, 0.1, 0.01, 1e-4, 1e-8. At each: 0 < a1 < b1 < a2 < b2 < 1, beta_1..beta_4 vanish to 1e-45, beta_5 < 0, and beta_5/eps -> -1/24 (Simpson's error on t^4); the gaps tend to eps/6, 2eps/3, eps/6 (Lobatto). Interlacing is classical for Stieltjes functions: [2/2] f = f(0) + z [1/2] g with g the Stieltjes transform of x dlambda(x), so the poles are its two Gauss nodes in (0, 1), and the zeros interlace them (one in each gap between poles, one below the first).

On the finite-n system, every reproduced number agrees (independent route, residuals below 1e-59):

D/B_J at n = 100 is 0.819287720988 here, against phase 0's bisected crossing 0.81929 and section 10's 0.81928772: the identification of phase 0's crossing with the Pade point reproduces. Holland's theta = 1/2 family gives n^4 L E_F(5) = -0.6978 (n = 50) and -4.207 (n = 3200), as section 4 reports. Sign scan: E_S(6) < 0, every parameter positive and 0 < w_A < w_B < w_C < w_D < 1 at all 127 values n = 50, 75, .., 3200 and at n = 1e4, 1e5, 1e6, 1e7, where n^5 L E_S(6) = -1.235, -1.406, -1.517, -1.594, still approaching -2 (review/results_review.json). This is a second route agreeing with phase 1's routes A and B at those points; whether that earns them the hardened grade is the owner's call, not this review's.

R6 (an upgrade, not a defect). 11.3 calls a1 + b2 = b1 + a2 = 1 "measured and not derived". It is derived: the matching conditions are int t^j (mu - tau) dt = 0, j = 0..3, with tau = (1 - eps) 1_[0,1]. The reflection t -> 1 - t fixes tau and maps polynomials of degree 3 to polynomials of degree 3, so the reflected configuration (zeros 1 - b2, 1 - b1; poles 1 - a2, 1 - a1) solves the same problem; the [2/2] approximant of a Stieltjes function is unique, so the two coincide, and the ordering forces a1 = 1 - b2, a2 = 1 - b1. It holds for every eps in the leading-order model (measured: defect below 1e-40 at all seven eps; pinned by a test).

What 11.3 does and does not need. The identification is "D-fixed family at theta_star equals the four-parameter solution", which is a tautology once both are solved; the substance is that the solution reached from the Pade guess is the one Holland's family reaches from theta = 1/2, and that reproduces. "Inside Holland's family" means the parametric form F_j only: the point is far outside his eq. (57) window and outside Lemma 9.1, as 11.3 itself says.

11.5 Near-identity factors: obstruction holds; mesoscopic part reproduces

The obstruction. The first-order equations are rederived exactly as written. With m_k = b - 1/(k+1): m_1 >= 0 needs b >= 1/2 and m_2^2 <= m_1 m_3 is (b - 1/2)(b - 1/4) - (b - 1/3)^2 = (1/6 - b)/12 >= 0, so b <= 1/6; contradiction. A non-negative least-squares check over 400 nodes omega in (0, 10] and b in [-2, 2] finds no solution (residual above 1e-3), and dropping the sign constraint recovers the hole b = 1/6, omega = 1/2, g = -2/3 exactly (tests).

R7, the compactness step, tightened. "A sequence of exact solutions with bounded scaled parameters converges to a first-order solution": the limit has g_j >= 0, not g_j > 0, which still contradicts (all g_j = 0 gives b = 1/2 = 1/3). If the number of auxiliary factors may grow with n, "bounded scaled parameters" must mean bounded total mass sum g_j, and the limit is a positive measure on a compact set of omega; the same two inequalities apply. With both stated the obstruction is sound for its class. The class excludes omega_j -> 0 or infinity and deviations larger than order 1/L, as the text says.

Mesoscopic configurations (leading-order model). An independent continuous optimisation (SLSQP, 400 starts, minimising the larger auxiliary relative gap X subject to the four moment equations) finds X = 0.1717, 0.1809, 0.1437, 0.1073, 0.0749, 0.0531 at eps = 0.342, 0.1, 0.03, 0.01, 0.003, 0.001, each just below the grid upper bounds in 11.5 (0.173, 0.182, 0.145, 0.109, 0.077, 0.055), and the same eps^0.30 fall. Two descriptive additions: at every optimum the two auxiliary factors coincide (one auxiliary Jacobi factor convolved with itself), and the main factor's right end sits at w = 0.42 (eps = 0.342), 0.79 (eps = 0.01) and 0.90 (eps = 0.001), so the main factor is far from Holland's (w = 1 - eps/2). These configurations are not perturbations of Holland's construction in any factor. Existence at finite n is still unexamined, as 11.5 says.

12.1 Proposition 2.2 for m points: holds

Read against the PDF: Holland's proof uses the five vanishing conditions only to start the sum (8) at k = 5. (9) needs only c holomorphic on Omega_r and k <= d, and the bound sum_{k=m}^d 2^(-k) < 2^(1-m) gives eps < 2^(m-1); the sign alternation at 0, the d - 1 critical points and infinity is unchanged. Planted check: p with ten random positive roots, c(z) = 1 + kappa z(z-1)...(z-5) with kappa set so that sup over Omega_r of |c - 1| = 31 (outside Holland's eps < 16, inside 2.2_6's eps < 32), r the exact need of (6): |P/p - 1| at every critical point is below 31/32 and the exact Sturm count (zeta.li.is_hyperbolic, method sturm) gives 10 real roots (test). The verdict "trivial" is right; "same radius" means Proposition 2.2's radius is inherited from (6), which is all 12.1 claims.

12.2 Lemma A: holds

Both claims were checked line by line. |e_k(x)| <= e_k(|x|) <= (sum |x_j|)^k/k! is Maclaurin's inequality for non-negative numbers. In the second claim, the other zeros on each side contribute at most H_d/g, p'(y) = 0 gives 1/delta_1 <= 1/delta_2 + H_d/g because the left-hand terms are positive, and the total is (4 + 3H_d)/g. On ten adversarial sets in the exploratory run (five are pinned in tests) (30 equally spaced roots, a geometric progression, a planted collision at distance 1e-6, two far clusters, Chebyshev nodes, five random sets) the need max_k |T_k|^(1/k), Lemma A's y sum 1/|y - y_j| and the explicit y(4 + 3H_d)/g are ordered at every critical point with positive slack (smallest absolute slack: the two-cluster set, 0.77). T_k is checked against direct differentiation, with a planted wrong point that the check catches (tests).

12.3 Lemmas B and C and their combination

Lemma B: holds. The operator J, the Liouville form, and the bounds on [B/2, 3B/2] rederive: 1 - y/A >= 13/16 from A >= 8B; |P| <= B/2 + 3(d - 1)/16 <= 0.501B from B >= 256d; Q <= 32d/(13B) + (0.501 + 1.5)B/(2 (13B/32)^2) = (2.4615d + 6.062)/B, so 2.47 and 6.07 are correct roundings up. The localisation uses Holland's (79), whose derivation does not need his (57) (checked: it needs only U >= V + d, V >= 32d). Exact rational tests at the edge of the hypotheses, A = 8B and B = 256d, d = 2 to 16: all roots real, inside [B/2, 3B/2] and inside B +- 8 sqrt(Bd), and the smallest gap is 1.98 (d = 2) down to 1.55 (d = 16) times the bound.

R4 (a pin, not a defect): A >= 8B is load-bearing. With A = B + d and B = 32d (Holland's (77) holds, A >= 8B does not), the smallest gap is 0.159, 0.074 and 0.036 times the Sturm bound at d = 4, 8, 16. For the Pade model A/B tends to 1.5 L, so the hypothesis holds only from some n_0: measured A/B = 5.88 at n = 50, 7.36 at 200, 8.003 at 346 (first n with A/B >= 8 in a bisection between 200 and 800), 10.79 at 3200, 21.7 at 1e7.

Lemma C: holds as cited. MFMP Proposition 2.17 (v3, attributed to Katkova, Shapiro and Vishnyakova) reads: for p, q in P(R>0), lmesh(p box q) >= lmesh(p). Holland's convolution is the reciprocal form, and lmesh is invariant under reciprocation. Random check: rational positive-root pairs of degree 2 to 8 in Holland's convention (about 200 in the exploratory run, up to 60 in the test); lmesh(p box q)/max(lmesh p, lmesh q) >= 1 in every case, minimum 1.001.

The combination: holds, with two hypotheses it does not state (R2). The step "zeros of the normalised second factor in [1 - eta, 1 + eta], eta = 8 sqrt(d/D) by (79)" needs Holland's (77) for (U, V) = (C, D), that is D >= 32d and C >= D + d, and the gap bound (1 - eta) g_1/3 is useful only for eta < 1, that is D > 64d. Neither is written in 12.3. Both follow once H1 gives D ~ n and C - D ~ n with d = o(n), so the chain is not affected; they should be written down before the constants are made explicit. The rest of the combination (zeros of p_F in [(B/2)(1 - eta), (3B/2)(1 + eta)] by Lemma 7.1, smallest gap >= (B/2)(1 - eta)(lmesh(p_1) - 1) >= (1 - eta) g_1/3, and the displayed r) rederives exactly.

R5, how loose the explicit constants are, at the route's own Pade parameters, n = 3200 (B = 7540.4):

(radii in units of sqrt(Bd)). The first four columns reproduce 12.6 (its "explicit" uses the largest zero rather than the largest critical point, hence 3.055 and 4.337 there). The displayed r of 12.3 is 72 to 182 times what (6) needs. Lemma A itself costs a factor 1.9 to 2.2 over the need; using the minimum gap costs 2 more; the rest, a factor of about 20, is the gap lower bound (the actual gap is 10 to 19 times (1 - eta) g_1/3: the factor 3 of the lmesh transfer, the Sturm bound's 1.6 to 1.7, and the crude eta = 8 sqrt(d/D), against the second factor's actual spread of 0.034 to 0.086) and the bound on y_max. The exponent does not care; phase 2's effective constant will: a radius 182 times too large is a factor 182^6 in sup |E|, so about 182^12 in K, if left as is. The multiplier sizes of 12.6 also reproduce: max_{j<=d} |c_j - 1| = 5.67e-19, 5.21e-16, 7.53e-14 (Pade, d = 6, 12, 24) and 4.11e-14, 5.40e-12, 2.87e-10 (Holland).

12.4 The sixth-order residual: holds, conditional on H1 and H3

Checked. psi^(5)(p + z) = 24/(p + z)^5 + O(p^(-6)) = f_z(n/p) + O(p^(-6)) for |z| <= eta n; f_z(0) = 0 turns the five psi terms into int f_z'(t)(mu - 1_[0, n/b]) dt; f_z'(w) = 120 w^4 n/(n + zw)^6, so |f_z'| <= 1920/(n^5 (1 - 2 eta)^6) on 0 <= t <= 2. With disjoint intervals inside [0, 1] (true at every n checked, see the sign scan) the L^1 distance is the total gap, which measures 1.366/L at n = 50 rising to 1.858/L at n = 1e7, toward 2/L. Hermite-Genocchi needs E holomorphic on the convex Omega containing 0..5, which holds once d + 2r = o(n). The step needs H1 (parameters within O(1/L) of the Pade configuration, for the L^1 bound) and H3 (h^(6) on the complex neighbourhood), as stated. The bound is the same mechanism as Holland's Lemma 8.1, where the pairs (B, b) and (C, D) are close; here the close pairs are (D, b) and (B, C), and the L^1 form makes that pairing unnecessary.

12.5 The chain: holds as a conditional derivation; open inputs incomplete

The chain was walked end to end: Lemma A to C give (6) with r = O(sqrt(Bd) log d); 12.4 gives sup |E| <= C (d + 2r)^6/(6! n^5 log n) = C d^3 log^6 d/(n^2 log n); at most 1/2 under n^4 log^2(n+2) >= K d^6 log^12(d+2); then |c - 1| <= 1 < 32 and Proposition 2.2_6 applies to the Pade model (simple positive zeros of p_F by MFMP 2.7(iii) and Lemma C, p_F(0) = 1, c(d) > 0). Small n is absorbed into K as in Holland's section 10. The delta against Holland's Theorem 1.1 (exponent 3/5 to 2/3, at the cost of log^12 d) and the m = 5 cross-check (his wedge with a log^10 d loss) are both right.

R1 again, as an omission from the list of open inputs. H1's proof by Holland's Lemma 6.1 contraction scales the four matching equations by nL, n^2 L, n^3 L, n^4 L. The fourth component's target is a fifth difference of log gamma, so its limit requires the leading constant of h^(5), -12/(x^4 L)(1 + O(1/L)), uniformly. Holland supplies that constant only through h^(4) (his Lemma 5.4 is exactly the k = 4 case); for h^(5) he has O-bounds. This input is neither H1 as described nor H3 (which is the sixth derivative bound). It is routine (R1's Cauchy argument) and measured above (rho_5 = 1.069 at n = 3200, 1.056 at 1e6), but it is a third unwritten input and belongs in the list.

R3, a wording error. "Every parameter condition of 12.3 holds (all parameters are of order n and d = o(n))": A is of order n log n, and A >= 8B holds because A/B ~ 1.5 L grows, only from n of a few hundred (R4). The conclusion stands for n >= n_0; the parenthesis is wrong for A.

H1's limiting system. Nonsingular in two coordinate systems: phase 1's quadrature coordinates (determinant -1/24, its test) and this review's (a1 = eps x1, b2 = 1 - eps x4, interior gap eps g centred at nu; determinant 1 at x1 = x4 = 1/6, g = 2/3, nu = 1/2, test). Note that at first order the equations depend on b1 and a2 only through nu and g, so the contraction must use (nu, g), not (b1, a2) scaled separately, or its limit is singular. Not a defect; a constraint on how H1 is written.

Overall verdict on the conditional wedge

The conditional wedge stands. Every step listed holds; none is broken; no counterexample was found to Lemmas A, B or C, Proposition 2.2_6, the reduction or the obstruction. Its status after this review: a conditional derivation with three open inputs, not two,

1. H1 (matching at every large n within O(1/L) of the Pade configuration): unproved; measured here on a second route at 127 integers n in 50..3200 and at 1e4..1e7;
2. H3 (Proposition 4.1 at r = 6): routine, unwritten;
3. R1 (leading constant of h^(5), needed by H1): routine, unwritten, measured;

plus two hypotheses of 12.3 to write down (R2) and one wording fix (R3). One internal review, no external one; no novelty search was run by phase 1 or here. Nothing in it bears on RH.

What this review did not do

• It did not attempt H1, H3 or R1.
• It did not reproduce route B's values from 1e8 to 1e20 (this route stops at 1e7), section 10's theta map, or the d = 48, 96 rows of 12.6.
• It did not review 11.2, 11.4, 12.6 and 12.7 as steps, though their numbers reproduced wherever this code touched them.
• It edited no existing file. Files: this one, review/wedge_review.py, review/test_wedge_review.py, review/results_review.json.