jensen_wedge: independent review of phase 2 (the explicit constants)

Nothing here bears on RH. This file reviews EFFECTIVE.md and RESULTS.md section 18 of hunt #135 as merged on main at 70d72a8e (PR #314). The reviewer is a separate session that did not write them, read the three primary sources itself, and reproduced the four constants with its own code (review_p2/reproduce_p2.py), written from EFFECTIVE.md's formulas and from this review's own derivations of the steps that file states only in words. That code does not import effective.py, probe.py, review/ or review_h1h3/. The brief was to find where phase 2 breaks, not to confirm it.

Verdict

No step is broken. All four constants and the c = 1 remark hold, at the grade "derivation, self-reviewed by phase 2 and now reviewed once by an independent session". That is one internal review, not an external one: none of the four is a theorem, and this file does not promote any of them. One finding is about the checking, not the mathematics: the planted-weakening check covers the claimed constants, but three of four formula faults planted here pass phase 2's entire test suite (item P). The steps it misses were re-derived by hand here and are correct, so the constants stand; the suite's coverage claim should be read as narrower than section 18 of RESULTS.md states it.

Every number in the middle column is an upper end (lower end for n_min ratios) of an Arb ball (python-flint, 256 bits, zeta.rigor.BACKEND checked). The (5a) requirement agrees with phase 2's recomputation to 0.02 percent and (5s) to 2.5 percent (a finer cover here); the two six-coefficient requirements come out a factor 1.95 (6a) and 2.4 (6s) lower, because this review encloses the sixth-order bracket more tightly (R13).

Item by item

What was read

• Holland, arXiv:2608.08682v1, PDF downloaded 2026-10-19, SHA-256 3fc31ba84fb113bc0b3109fb0e569bd1d7183018485aca1eb1dab565c839b49d (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, Proposition 4.1 and its proof, Lemma 4.2, Lemma 6.1 and (52), section 7 ((73) to (80)), Lemmas 8.1 and 9.1 with (82) to (93), section 10.
• GORTTW, arXiv:1910.01227v3, SHA-256 f203487acc45a58808462e897240ce4929c998d9f0230d9a4bedb6d3841c3b2c: Theorem 1.1 (n >= c e^(d/2), c not computed), Theorem 1.2, Corollary 1.3 (d <= 9.36e20, every n >= 0, via Platt's RH verification to 3.06e10), section 3 ((3.1), (3.2), the regularised R*_M, Lemma 3.1).
• Martinez-Finkelshtein, Morales, Perales, arXiv:2309.10970v3, SHA-256 23f228e0682430c1ae6285be4abed4d730e9a08accad6c3f540fbdf4eb5bb4a3: Propositions 2.7 and 2.17.

R1. Lemma V

Rederived. F(v) = v + g(v), g = log v + log pi + log(1 - (a/pi) e^-v), and on H_V = {Re v >= V, |Im v| <= 1}, |g'| <= 1/V + delta_a/(1 - delta_a) = q. The map v -> v_1 + log(s/s_1) - (g(v) - g(v_1)) sends the disk of radius rho' = |log(s/s_1)|/(1 - q) about v_1(s_1) into itself because |log(s/s_1)| + q rho' = rho'. The disk stays in H_V: for s_1 = |s| >= 1.5 x_0, v_1(s_1) - rho' >= V + 1 - b > V, and rho' <= b <= 1 (b is 0.256 to 0.283 at the four thresholds). Uniqueness: |Im F| <= 1 + atan(1/V) + asin(delta_a) < pi on H_V, so two solutions have equal F, and |F(v) - F(v') - (v - v')| <= q |v - v'| on the convex H_V forces v = v'. F' = 1 + g' != 0 gives holomorphy. Nothing missing.

R2. Lemma L: the contour, and the majorants

Geometry. The integrand v^s e^(av - pi e^v) alpha(v) is holomorphic on {Re v > 0, |Im v| < pi/2} (the theta series in alpha converges there because Re e^v > 0), and |Im v_0| <= b < 0.29 < pi/2, so [0, inf) deforms to [0, w_1] followed by v_0 + t, t >= -R/2, with doubly exponential decay at the far end. On the line, phi(v_0 + t) - phi(v_0) = -s g(t/v_0) - P k(t) exactly. The argument bound is sharper than stated: g(t/v_0) = t^2 int_0^1 (1-u)/(v_0 + u t)^2 du, so every integrand has argument -2 arg(v_0 + u t), and Re(v_0 + u t) >= R/2 gives |arg g| <= 2 atan(2b/R) with all contributions on one side. With (i), Re(s g) >= 0 on both tails. Correct.

Majorants. Rederived with my own intermediate bounds (Re Psi >= p cos b instead of phase 2's p cos b - a/R; |Psi| <= p(1 + (1 + a/p)/R)), and C_rho reproduces:

The leading part is (8/3) e^(1/2)/(sqrt(2 pi) cos^2 b) = 1.874 at b = 0.256: the 16/6 is int |t|^3 e^(-c t^2) dt = 1/c^2 at c = Re Psi/4. The other four pieces times sqrt(p) are below 10^-3700 at the smallest threshold (log10 of each: -4438, -5281, -3778, -2.1e5). Each factor of the rho_3 majorant decreases in R on [R_min, inf) (sqrt(R^2 + b^2)/(R - T)^3, 1 + a/p, 1/R, and sqrt(p) a_1, which falls like e^(-R/2)), so evaluating at R_min is valid.

Measured, independently. Quadrature of I(s) on a contour tilted by slope 0.1 (not the lemma's), at x = 789 702, 1e6, 1e8 and five points of each disk boundary: the worst |varrho| is 1.29e-5 at x_0 = 789 702 (z = 0.75 x_0), against lambda = 6.98e-3, slack 540; |varrho| sqrt(p) is 3e-3 to 4e-3, so the true remainder is O(1/p) and the lemma's O(1/sqrt(p)) is the price of bounding the cubic Taylor term without its odd cancellation.

R3. Lemma R

• Uniformity on |z - x| <= x/4. For such z, |arg z| <= asin(1/4) = theta and |2z| >= 1.5 x >= 1.5 x_0 = 2 x_0 (1 - rho), which is exactly Lemma V's range and Lemma L's |arg s| <= theta. Lemma L's majorant depends on s only through R = Re v_0 >= R_min, so one lambda serves the whole union of disks. The union is the sector |arg z| <= theta truncated by the first disk's arc (each disk is tangent to the two rays), which is convex. Correct.
• The branch. G(2z) + log(1 + varrho(2z)) is holomorphic on the union (v_0 holomorphic, Re Psi > 0, |varrho| < 1), real on the axis, and differs from any logarithm of M_z/(2 pi^2 4^(-2z-1)) by a constant multiple of 2 pi i, which the real axis fixes at 0. So it is the branch Holland names. Correct.
• The rate. |R| <= -log(1 - C_rho/sqrt(pi e^R)) and pi e^R ~ |2z|/log|2z|, so |z|^(-1/2) (log |z|)^(1/2): "-1/2 + eps" is right.
• "Only |R| < 1 enters." True in substance, loosely worded. What enters is an explicit finite bound lambda on the disk: Cauchy on radius n/8 turns it into |R^(k)| <= k! lambda (8/n)^k, which is O(n^-k) against main terms of order n^-(k-1)/log n, so the rate in |z| never matters. Lambda itself does enter the numbers, through the remainder part of every H_k enclosure: k(k-1) 8^k lambda v_1/(2n) is 0.031 for k = 5 at the (5s) threshold (3 percent of Theta_5) and below 1e-40 at the assisted thresholds. Neither changes a verdict.

R4 and R5. Lemma D, and the targets

Lemma D(i) was rederived from scratch in sympy (derive_F): F_k(0,0) = 1, first-order constants -1, 0, 1/2, 5/6, 13/12, 77/60 for k = 2..7, F_2(e, 0) = 1/(1+e). Lemma D(ii): for |z - n| <= n/8, the disk of radius n/8 about z lies in |zeta - n| <= n/4, where Lemma R holds with x = n >= x_0. The targets: Delta^m h(n) is the B-spline average of h^(m)(n + sigma), so T_j lies in the hull of H_(j+2) over [0, (j+2)/n]. This review encloses T_j - T_0 as one function of e (mean-value form of F_(j+2) - F_2), which keeps the t_j radii near 5e-5 on a 1e-4 subinterval; without it the Krawczyk boxes on the top (5a) subinterval are about 14 times wider. That is a property of enclosures, not of the lemma.

R6 and R11. The matching systems

Both systems were checked against Holland's own equations on real data, which phase 2's measurements (Newton on the same reduced system) could not do: at n = 1000 and 20 000, with h from quadrature at 80 digits, the solution of the reduced system Phi_j = 0 satisfies Holland's (52), q_k(A,B) q_k(C,D) = q_k (D = n/2, k = 0..2 for the five model; D free, k = 0..3 for the six model), to 1e-76, and a direct solve of (52) agrees with it to 1e-62. The parametric Krawczyk test (my implementation: float Newton at the parameter midpoints, inverse at the midpoint, strict inclusion K(B) inside B, parameter balls for eps, t_j, w_b, 1/n) passes on every subinterval of all four covers; where a 5e-4 subinterval fails near e_1 = 0 in (6s), its halves pass. The positivity conditions need eps > 0, which the ball touching e_1 = 0 cannot show; they hold because e_1 > 0 at every finite n, and C - D >= d is checked as T_0 g_c/w_C >= (d/n) v_1(2n), which uses Lemma W(ii)'s second clause. effective.py does the same (its kappa).

R7. Lemma G and C_loc

Rederived: (78) as an exact identity (sympy), the off-diagonal inequalities (k + beta)(k + H) <= (2k+H)^2 - 1 and (k + H)(H + 1) <= (2k+H)^2 - 1 for k >= 1, both diagonal cases (U >= 2V: a_k - V in [0, (d-1)(2U+V)/H]; U < 2V: the second form), and Gershgorin on the symmetric tridiagonal matrix, whose eigenvalues match the roots of 2F1(-d, U; V; y/U) to 2e-37 at (5405, 400, 10) and (900, 400, 12). The assembly through Lemma 7.1 (MSS Theorem 1.13, an input) gives |y - B| <= max(2 beta_1 (1 - eta_-) + B eta_-, (abar_1 + 2 beta_1)(1 + eta_+) + B eta_+). The reason C_loc tends to 2 rather than to Holland's 2(1 + sqrt 2): the second factor is nearly degenerate, C - D ~ n/(24 L), so its spread is sqrt(d (C - D)), not sqrt(d D). Max over the (5a) cover: 2.09318 (claimed 2.094). One remark: written naively, (1/w_A)/(1/w_A - delta - iota) counts the box of w_A twice; this review's first draft did that with wider boxes and got C_loc = 2.127 and a + b + theta = 1.010 on the top (5a) subinterval. Both codes now write the ratio as 1/(1 - (delta + iota) w_A).

R8. Lemma 9.1-E and K_6 = 2.53

Every identity checked exactly in rational arithmetic at d = 3, 5, 6: Holland's factorisation (86), the expansion of V_m with beta_m, gamma_m, and (90). The backward substitution from U_(d+1) = 0 gives |U_m| <= eps_p c_V M r^(m+2)/(D(1 - r/D)) by induction (the fixed point of Y = (eps_p c_V M + Y r)/D). The bounds behind a, b, theta and c_V were rederived term by term (|m + (d-1-2m) y/A| <= d(1 + y/A), |beta_m| <= A(d+1) + 5d^2 + 4d + 1, |gamma_m| <= (d^2/4)(d + A)). The closing argument needs only a + b + theta < 1: if M > 1 its maximum sits at some k >= 2 and M <= (a + b + theta) M. Holland's 1/4 each is a special case. Max over the (5a) cover: 0.99693 at K_6 = 2.53; at 2.52 the top subinterval gives 1.0015. (5s): 0.99709 at 2.82.

R9. Lemma 8.1-E and Theta_5 <= 1.034

Signs rederived from (82), (83): psi^(4)(P + z) = -(6/n^4) omega(w_P) + r_4, so n^4 E^(5)/(6 eps) = -H_5/T_0 + g_0 w_A^3 (1 + w_A y)^-4 + g_b <omega'>[w_B, w_b] - g_c <omega'>[w_C, 2] + rem, leading value -1 + 8/3 - 8/3 = -1. The remainder bound |psi^(4)(y) + 6/y^4| <= 24/(Re y)^5 holds for Re y >= 5 (Euler-Maclaurin gives 12/(Re y)^5 + 60/(Re y)^6); every P + z here has real part above 3e5. Hermite-Genocchi: the convex hull of {0, ..., 4, z} lies in the convex Omega_r, and |z - j| <= d + 2r there, so C_E = T_0 Theta_5 (...)^5/10 as stated. Max over (5a): 1.03384 (claimed 1.034); (5s): 1.5520 (claimed 1.571).

R10. Assembly and Lemma W

sup |c_F - 1| < 16 from |E| < log 17 and |e^E - 1| <= e^|E| - 1; Proposition 2.2's proof needs exactly eps sum_(k>=5) 2^-k < 1. Lemma W reproved: (i) v_1' = v_1/(n(1 + v_1) + a v_1^2/2) and 1 + v_1 <= log(n + 2) - 1.03 for n >= 1000 (the last step needs n >= 618); (ii) the logarithmic derivatives -(2/5)(1 - 1/log(n+2))/n and v_1'/v_1 <= 1/(5.88 n); (iii) the 0.715/(n log(d_max + 2)) term. K_req is formed per subinterval and then maximised, not assembled from separate maxima. Correct.

R12. Lemmas A to C with R2's hypotheses

Lemma A (p^(k)/p = k! e_k(1/(y - y_j)), Maclaurin, and the critical-point balance 1/delta_1 <= (2 + H_d)/g) and Lemma B (d/R <= 32d/(13B) = 2.4615 d/B, (P R' - P' R)/(2 R^2) <= 2.001 B/(2 (13B/32)^2) = 6.062/B) were rederived; so was the gap (1 - eta) g_1/3 from Lemma C and the zero window [(B/2)(1 - eta), (3B/2)(1 + eta)]. MFMP Proposition 2.17 reads exactly lmesh(p ⊠ q) >= lmesh(p) for p, q in P(R>0), and Holland's convolution is the reciprocal form of theirs, under which lmesh is invariant. H_d <= log(d + 2) + 0.6607 holds for every d >= 1 (from H_d < log d + gamma + 1/(2d)), so c_H = 3 + 5.9821/log(d_min + 2). All five conditions (A >= 8B, B >= 256d, C >= D + d, D >= 32d, D > 64d) hold on every subinterval of both six-coefficient covers.

R13. Theta_6, and a factor 1.95 in K'

The leading value 1 - 25/24 = -1/24 (Simpson's error on 5t^4) and the signs were rederived. Phase 2 encloses <omega_5'> over [w_B, w_C] and [w_D, w_b] by the hull of omega_5' on the interval, which is valid but wide: at the top of the (6a) cover the two hulls add about 0.02 to the modulus, half of the 1/24 being bounded. The exact divided difference, (u_2^5 - u_1^5)/(w_2 - w_1) = sum_i u_2^i u_1^(4-i)/((1 + w_1 y)(1 + w_2 y)) with u = w/(1 + w y), gives Theta_6 <= 0.0477 (6a) and 0.403 (6s), against 0.0661 and 0.538, and K'_req = 3.70e9 for (6a). So K' = 7.21e9 holds with a factor 1.95 to spare, and the same chain supports about 3.7e9. This is an input for phase 3's doors, not a defect.

R14. The small-n hand-off to GORTTW Corollary 1.3

The question was whether the arithmetic covers every (d, n) below the chain's range. It does. Take (d, n) in the (5a) wedge. If d <= 9.36e20, Corollary 1.3 applies for every n >= 0. If d > 9.36e20, then n^3 log^2(n+2) >= 17 900 d^5 > 17 900 (9.36e20)^5, so n > n_min = 1.2534e35 >= 12.647 x 9.91e33, inside the chain. Equivalently, at the chain's start the wedge allows only d <= 2.016e20. For (6a): n_min = 1.0044e38 >= 177.77 x 5.65e35, and at the start d <= 3.35e19. Both sides of each comparison increase in n and in d, so the single comparison is enough. The standalone variants need no hand-off: at n_0 = 789 702 the (5s) wedge allows d <= 0.9996 < 1, and at n_0 = 14 237 025 the (6s) wedge at d = 1 needs n^4 log^2(n+2)/K' >= log^12 3 = 3.0913 but has 3.0877, so both wedges are empty below their n_0. Two remarks: Corollary 1.3 gives real zeros, and the statements (5a), (6a) correctly claim only hyperbolicity there; and the six-coefficient chain handles d >= 6, so (6s) at d <= 5 rests on the model matching every coefficient (P_F = p_F, simple positive zeros by Lemma C), which effective.py's docstring says and EFFECTIVE.md does not (D2).

R15. The c = 1 remark

For d <= 9.36e20, Corollary 1.3 covers every n. For d > 9.36e20 and n >= e^(d/2): n^3 log^2(n+2) > e^(3d/2) (d/2)^2, and e^(3d/2) >= 71 600 d^3 (the log of the ratio is 1.40e21 at d = 9.36e20 and increases), so the pair is in the (5a) wedge. Hence J^{d,n} is hyperbolic for every d >= 1 and n >= e^(d/2): c = 1 is admissible in GORTTW Theorem 1.1 (v3), with (5a)'s grade. Correct; the remark needs nothing beyond (5a).

P. The planted-weakening check, and what it does not see

The kill condition as worded, "a planted weakening of an explicit constant does not turn the effectivity check red", is not met: phase 2's 44 constant plants all turn it red (its own test, green here), and this review's own plants (K at 0.999 of K_req for (5a) and (6a), K_6 = 2.52, a standalone K below the absorption, a chain start where R_min < 10) turn this reproduction red too.

What the check cannot see, by construction, is a wrong formula: it compares claims with a recomputation from the same formulas, and a fault that makes a bound optimistic lowers the recomputation, which a claimed upper bound then still covers. Phase 2 says this and leans on its measurements. Their reach was tested here by planting four formula faults into a copy of effective.py in a temporary directory and running phase 2's own test_effective.py there (review_p2/plant_formula_faults.py; nothing in the tree is edited):

So the steps whose formulas no phase 2 test pins are: Lemma L below a factor of about 8 (the quadrature slack is 785, and the one test that reacts at 10x does so through a slack > 100 threshold); the Cauchy constant of Lemma D(ii) entirely (no measurement sees a remainder of order lambda); Lemmas A to C up to the measured looseness of 36 to 44; and, by extension, Lemma W, c_H and the psi remainders, which are checked only on samples or not at all. This review rederived every one of those steps by hand (R2, R3, R4, R10, R12) and found them correct, so no constant changes. The finding is about what section 8 of EFFECTIVE.md and section 18 of RESULTS.md may claim for the checks: a green suite attests to the constants against their formulas, and to a subset of the formulas against measurement; it does not attest to the derivation.

D. Documentation gaps (none changes a number)

• D1. EFFECTIVE.md Lemma W(ii) defines the six-coefficient d_max with log^12 3; effective.py uses log^12(9.36e20 + 3) for (6a), which is also valid because that chain only needs d > 9.36e20. Question: should Lemma W(ii) state both?
• D2. EFFECTIVE.md section 6 does not say how (6s) handles d <= 5; the code's docstring does (the model matches every coefficient). Question: add the sentence?
• D3. RESULTS.md section 18 says formula-level faults "are caught by independent measurements"; EFFECTIVE.md section 8 adds "by as much as their slack", but its table lists no step as uncovered. Item P names the uncovered steps. Question: add them to that table, and soften the sentence in section 18?

Reproduce

.venv/bin/python hunts/jensen_wedge/review_p2/reproduce_p2.py               # about 2 min, writes results_review_p2.json
.venv/bin/python -m pytest -q -n0 hunts/jensen_wedge/review_p2/test_review_p2.py   # 20 tests, about 2 min
.venv/bin/python hunts/jensen_wedge/review_p2/plant_formula_faults.py       # about 5 min, writes plants_review_p2.json