""" tail.py -- the real support's margin at every P >= P* = 399 2^25 10^20 = 1.3388e30, by one closed form in P. QUESTION. box.py reads the margin at d = P + 1 = 0 mod 4 on boxes of P from 399 to P*, where its range stops. Is there a closed form R(P) >= P R_maj(P) at every P >= P*, with budget(P) - F(c(P), P) - X - R(P)/P > 0 at every such P? Notation as box.py's head (D1..D10), majorant.py (W1, W2), uniform.py (X1, X2). THE ANSWER. At every P >= P*, P R_maj(P) <= C_K + L/P^2 + 8e-40 T(P) + W(P) + P past_T, C_K = 7809.1, L = 1.30e8 (C6), each piece over P at most its value at P* (THE MARGIN), so the margin is least at P*, 0.2497859, and positive at every P >= P*. Conditions: W1 and W2 holding at P = P0 (majorant.py's hand derivations), the pieces K1..K7 by hand as written below, and X <= 2.06e-4 past 1e6 by hand (variation.py). C_K and L are sums in mpmath at 30 digits, not interval arithmetic. THE SCALE. With F <= 1e-5 (K1), X <= 2.06e-4 and budget > 1/4, the margin is positive while R_maj(P) < 0.2497, i.e. P R_maj < 3.3e29 at P*. A piece of P R_maj growing no faster than polylog(P) needs only a crude constant: C_K <= 1e20 costs the margin 7.5e-11. Only pieces linear in P (or faster) are owed sharply, and one is: P past_T. Every other P-dependence is a floor made small by K(P), or the tau range [1/c(P), T(P)] widening at both ends, paid by integrals that converge over (0, inf). THE PIECES (P-dependence named and signed) (K1) c(P) := ceil(16 ln(1600 P)) + 1 (1229 at P*). By hand: g >= (c/16)(1 - eps), g' >= (c/(4P))(1 - eps), eps < c/P, and 1/(1 - e^-x) <= 1 + 1/x, so F <= 4 e^-g (1 + 1/g') <= (1/400)(1/P + 4/c)(1 + delta) <= 8.2e-6 at P*, falling in P (F_box reads 7.62e-6 at P*, 5.79e-6 at 1e40, 2.35e-6 at 1e100). DERIVED. (K2) T(P) = 1 + ln(P/399)/(2 pi^2): past_T(P, T(P)) -> L_T = 8 399^2 e^(-4 pi^2)/(4 pi^2) = 2.30896e-13 (the k = 1 root: e^(-4 pi^2 T) = e^(-4 pi^2)(399/P)^2 against 1/(1 - e^lr) ~ P^3/(4 pi^2)). Read: uniform.past_T/L_T = 1.368 at 399, 1.0142 at 1e4, 1 + 2.2e-8 at 1e10, 1 +- 1.3e-13 at 1.3388e26, 1.3388e30, 1e40. In R_maj it is past_T itself: 2.31e-13, P-free. By hand: d = P + 1, phi_k = 2 pi k/d, x_k = ln Q_k = ln(1 + min(8 sin^2(phi_k/2), 2)), beta = T P^3/(2d), v_k = x_k/(2d); the Q-bound summed over m >= T P^3 is (8/P) sum_(k=1..d/2) e^(-beta x_k)/(1 - e^(-v_k)) (uniform.past_T reads it, larger past k = 2000 by lq_beyond). k = 1, z = pi/d: sin z >= z(1 - z^2/6) and ln(1 + y) >= y - y^2/2 give x_1 >= (8 pi^2/d^2)(1 - eta), eta = 13 pi^2/(3 d^2), so beta x_1 >= 4 pi^2 T (P/d)^3 (1 - eta) >= 4 pi^2 T(1 - 3/d - eta), and e^(-4 pi^2 T) = e^(-4 pi^2)(399/P)^2; v_1 in [(4 pi^2/d^3)(1 - eta), 4 pi^2/d^3] and 1 - e^-v >= v(1 - v/2). So the k = 1 term is at most L_T (d/P)^3 e^(4 pi^2 T(3/d + eta))/((1 - eta)(1 - 2 pi^2/d^3)). k >= 2, phi_k <= pi/3: sin t >= 3t/pi on [0, pi/6] and ln(1 + y) >= y ln3/2 on [0, 2] give x_k >= (9 ln3/pi^2) phi_k^2 >= 4 pi^2 k^2/d^2, so beta x_k >= a k^2, a = 2 pi^2 T (P/d)^3, and 1/(1 - e^-v) <= 1 + 1/v <= 1 + d^3/(2 pi^2 k^2); with k^2 >= 4 + 4(k - 2) they add at most (8/P) e^(-4a)(1 + d^3/(8 pi^2))/(1 - e^(-4a)), over L_T eps_2 = ((d/P)^3/2 + 4 pi^2/P^3) e^(-4 pi^2 T(1 - 6/d))/(1 - e^(-4a)). phi_k > pi/3: x_k >= ln 3, at most d/2 terms, eps_3 = (d/2)(1 + 2d/ln3)(4 pi^2/P^3) e^(4 pi^2 T - T P^3 ln3/(2d)). uniform.past_T's own terms past k = 2000 (R_maj's): x read as lb = min(1.196 phi_2001^2, 0.846), at most d/2 of them, eps_4 = (d/2)(1 + 2d/lb)(4 pi^2/P^3) e^(4 pi^2 T - beta lb), beta lb >= 4.2e7 T past P = 1e4 (0.423 T P^3/d while 0.846 binds, 9.45e7 T (P/d)^3 past d = 1.5e4). So uniform.past_T(P, T(P)) <= L_T(1 + eps_T(P)), 1 + eps_T = (k = 1 factor) + eps_2 + eps_3 + eps_4, each falling in P past 399 (T/d, T eta and e^(-4 pi^2 T) fall, beta lb rises on both branches of its min): eps_T(P*) about 3.7e-28. In R_maj: past_T <= 2.30897e-13 (1 + eps_T(P*)) at every P >= P*. (K3) The floors, every P-dependent term of B: the walk P^3 e^-g/(1 - e^(-P g')), the N sum P(...)e^(-K^2/(4 tau)), phi's modes past K P sum 4 pi k e^(-4 pi^2 k^2 tau), the root tail 4P^2(sum_(k>K) e^(-8.72 tau k^2) + (P/4 + 1)e^(-0.546 tau P^2)). Images below 1/4 with K_img(P) = ceil(sqrt(4 ln P + 100)) + 2, each floor there <= 1e-40/P; modes from 1/64 (K6 reads images to 1/16, modes from 1/64), each floor <= 1e-40/P below 1/4 and, above it, falling as K6 (iv)'s lambda K_lo >= 12.5 e^(-4 pi^2 tau) does (lambda K_lo at T(P*) is 8.0e-72 against 1e-40/P = 7.5e-71, below it past tau 4.15, so a flat floor would not hold o = 0 alone): each floor <= 1e-40 e^(-4 pi^2 tau)/P^2. K_mode(P) = ceil(sqrt(7.34(3 ln P + 95))) (by hand, worst at tau = 1/64), 48 at P*. Counted at EVERY class, the floors add at most 8e-40 T(P) to P R_maj. Bad needs P|K_x| <= b + fl <= 2 max(b, fl), and P|K_x| <= 2 fl holds at o = 0 alone (K6 (iv)), where P|K_x| <= 2b holds too, so [Bad](b + fl) <= b[P|K_x| <= 2b] + fl. By hand, tau < 1/4, K = K_img(P), J = KP, n0 = 2 tau P^2 < P^2/2: g = J^2/(2(n0 + BJ)) >= K^2/(1 + 2BK/P) >= K^2 - 2.01 K^3/P, and K^2 >= 4 ln P + 100, so P^3 e^-g <= e^(-100 + 2.01 K^3/P)/P; g' >= J/(2(n0 + BJ)), so P g' >= K/(4 tau + 2BK/P) >= 0.99 K: the walk <= 3.8e-44/P. The N sum, P(4 pi tau)^(-1/2) sum_(N >= K)(N + 1) tau^-1 e^(-N^2/(4 tau)), rises in tau while K^2/(4 tau) > 3/2: at tau = 1/4 it is at most 2.26 P (K + 1) e^(-K^2)(1 + e^(-2K)) <= 2.3 (K + 1) e^-100/P^3. tau >= 1/64, K = K_mode(P), 8.72 K^2/64 >= 1.00007(3 ln P + 95): the root tail's first term 4P^2 e^(-8.72 tau (K + 1)^2)/(1 - e^(-17.4 tau(K + 1))) falls in tau, at 1/64 at most 4.01 e^-95/P = 2.2e-41/P; over e^(-4 pi^2 tau)/P^2 it is at most 4.1 P^4 e^(-(8.72(K + 1)^2 - 4 pi^2)tau), largest at tau = 1/4, there below e^(-44 ln P - 1400); its second, 4P^2(P/4 + 1) e^(-0.546 tau P^2), and phi's modes past K, at most P 4 pi (K + 1) e^(-4 pi^2 tau (K + 1)^2)(1.01) with 4 pi^2 (K + 1)^2/64 >= 13.5 ln P + 430, lie far below. So four floors, fl <= 4e-40/P (tau < 1/4), 4e-40 e^(-4 pi^2 tau)/P^2 (tau >= 1/4), and over the T P^3 values of m at weight P/P^3 each, 2 fl adds at most 8e-40 T(P) to P R_maj. (K4) The images' arm, P-free: b_img(x, tau) = sum_(|k| <= K_img) G(x + k, tau) Abar(|x + k|/(2 tau), tau) (K6 sums every k, above it), A read at P0 = P (W1 holds at P = P0), Abar = 1.75(1.5 a + a^3 tau) + 1e-20. By hand: core images have a <= (K + 1)c/2, a/P <= 1.1e-26 at P*, so dbar, ybar <= 1.01 a/P; |l| <= 1.5a + a^3 tau; dropping Hbar's subtracted terms, Hbar <= 1.25 a^2 (1 + e), and |Lbar| <= (7/6) a^4 tau (1 + e) + (1 + e)/(6 tau), e <= 1e-25; and 1.25 a^2 <= (a/P)(1.5a) P, (7/6) a^4 tau = (7/6)(a/P) a^3 tau P, so U, |V| <= (1.5a + a^3 tau)(1 + 2/P + 1.17 a/P) + 1.01/(6 tau P). U <= (K + 1)^3 c^2/4 <= P (4.6e9 at P*), so f(U) <= (e - 1)U, and A <= max(f(U), |V|) <= 1.7183(1 + 1e-25)(1.5a + a^3 tau) + 1.75 c(P)/(6P): below Abar, c/(6P) 1.5e-28 at P*. Each P-term (a/P, c/P, (K + 1)^3 c^2/P) falls in P. Core: K_img + 1 <= P/(2c(P)). DERIVED. (K5) The modes' arm, P-free: b_mode = sum_(k >= 1) env_k Abar_k(tau) bounds sum_(k <= K_mode(P)) at every P >= P*. W2 read at P0 = P (it holds at P = P0), theta = 2 pi k/P <= 1e-6 for every k <= K_mode(P) at every P >= P* (W2 asks theta0 <= 0.35, and variation.py's Q6 the disc's root). By hand at theta <= 1e-6: Ghat(r) <= 1.0834 r^4 (W2's exact |g_4| = 13/12, Cauchy past it, 30.5 r^5); sigma = 1.01 is a fixed point's radius (left side <= 1 + 1.0001e-6); mubar = 1.01 theta^2/P; rhobar <= 2.03 theta^3/P + 1.0835 theta^4, zeta0 = rhobar tau P^2 <= 1e-6; E_v <= theta^3/6 + 1.02 theta^2/P; |v|, q <= (1 + 2e-6)theta; |E_D'| <= 1.50001 theta^2 + 2.02 theta/P; |e| <= 2.1667 theta^2 + 3.05 theta/P. So E_c = P|e| is O(k^2)/P, P(e^zeta0 - 1) O(k^4 tau)/P, and A_k <= Abar_k := |beta1 - beta2| + C_k(tau)/P*, C_k = sqrt(beta1 beta2^3/3) + 2.167(2 pi k)^2 + 3.06(2 pi k) + (2.04(2 pi k)^3 + 1.084(2 pi k)^4) tau, each 1/P term falling in P; at beta1 = beta2 (tau = 3/(8 pi^2 k^2)) C_k is the whole of it, about 201.03 k^2 + 38.5 k against W2's own 201 k^2 + 37.7 k. DERIVED. (K6) The window, one inequality on the whole period, no zones. K(x, tau) = theta_3(pi x, q), q = e^(-4 pi^2 tau), and Jacobi's triple product (known) gives -K_x = 4 pi K sin(2 pi x) sum_(m >= 1) q^(2m-1)/|1 + q^(2m-1) e^(2 pi i x)|^2, each denominator at most (1 + q)^2, so at every x |K_x(x, tau)| >= lambda(tau) |sin 2 pi x| K(x, tau), lambda = 4 pi q/((1 + q)^3 (1 - q)) (about 1/(8 pi tau) at 0). A class x = x0 + u, u = o/P, |u| <= 1/4, has |sin 2 pi x| >= 4|u|, so P|K_x| >= 4|o| lambda K(x). With K3's split, Bad needs P|K_x| <= 2b(x), so |o| <= N := r/(2 lambda), r = sup_x b/K, P-free. (i) Images, tau <= 1/16, b = sum_k G0(x + k) w(a_k), G0 the heat Gaussian, a_k = |x + k|/(2 tau), w(a) = a Abar(a, tau), K4's Abar needing only: w rising, w(ca) <= c^4 w(a) at c >= 1. With y the nearest image (|y| <= 1/2), K >= G0(y); image y + j, j >= 1 (j <= -1 alike), z = y + j in [j - 1/2, j + 1/2], weighs w(z/(2 tau)) e^(-j(2z - j)/(4 tau)) <= (2z)^4 w(1/(4 tau)) e^(-j(2z - j)/(4 tau)), falling in z past 8 tau/j <= 1/2: so r <= rho w(1/(4 tau)), rho = 1 + 2 sum_(j>=1) (2j - 1)^4 e^(-j(j - 1)/(4 tau)) (3 + 2e-12 at 1/64). The truth at x = 1/2 is about w(1/(4 tau)): r is 3 high, 4 lambda about pi/(8 tau) low against K_xx(1/2)/K(1/2) (480 at 1/c(P*)), so N is about 0.26/tau^2 against the pair's 8 tau Abar: crude, and C_K affords it. (ii) Modes, tau >= 1/64, b_mode x-free (K5): N = b_mode/(2 lambda K_lo), K(x) >= K(1/2) >= K_lo = max(2 G0(1/2), 1 - 2q). (iii) Bad classes lie in |u| <= u_w := (N + 1/2)/P_eff, P_eff = max(P*, P_min(tau)), P_min(tau) = e^(1/(16 tau) - 1/8)/1600 the least P whose [1/c(P), T(P)] can hold tau (c <= 16 ln(1600 P) + 2); u_w/tau <= 4e-22 at every tau (3.538e-22 at most, at 1/1229.95, where P_min = P*). There, with z over |x0 + Z| less 0, two of each, bloc_x0 := sup b <= (1 + 1e-20) (4 pi tau)^(-1/2) ([x0 = 0] w(u_w/(2 tau)) + 2 w(1/(4 tau)) sum_z (2z)^4 e^(-z^2/(4 tau))), at x0 = 1/2 the pair's b(1/2) = 2 G(1/2) Abar(1/(4 tau)) itself. (iv) The o = 0 clause (K3): |o| >= 1/2 gives P|K_x| >= 2 lambda K_lo, so o = 0 alone has P|K_x| <= 2 fl once lambda K_lo > fl: 4e-7/P against 4e-40/P below 1/4 (e^(-1/(16 tau)) >= e^(-1/8)/(1600 P) at tau >= 1/c(P); lambda falls in tau, d ln/dq = (1 - 2q + 3q^2)/(q(1 - q^2)) > 0, 6.4e-4 at 1/4), 12.5 e^(-4 pi^2 tau) against 4e-40 e^(-4 pi^2 tau)/P^2 above. So no mv, m_far, L1 or far-zone cut is owed. Counted with the wrong-signed class (its 2 UB; its 2P|K_x| by K7's W(P)), at most 2N + 2 classes a side: S_img = 2(2N + 2)(bloc_0 + bloc_1/2), S_mode = 4(2N + 2) b_mode, the smaller on [1/64, 1/16]. Monotone in P: only u_w moves, falling; N, lambda, K_lo are P-free. (K7) The q-lattice: P R_maj = sum_q P^-2 (the class sum at tau_q, P^3 units), tau_q of step P^-2 in [1/c(P), T(P)), and S read at P_eff(tau) bounds the class sum at every P whose range holds tau (only u_w moves, falling in P). Cells [t1, t2) gapless on [tau0, tau1] = [1/2000, 40], cut at 1/64 and 1/16, each holding at most (t2 - t1)P^2 + 1 points: sum <= (t2 - t1 + P^-2) S_sup, S_sup each monotone factor at its worse end: w(1/(4 tau)), (4 pi tau)^(-1/2) and q^(k^2) falling, rho, 1/lambda, e^(-z^2/(4 tau)) and C_k rising, w(a, tau) rising in both, a_w = (N + 1/2)/(2 tau P_eff) at (N_sup, t1, P_eff(t2)), |beta1 - beta2| convex (larger end), G0(1/2) unimodal (max at 1/8: lesser end), 1 - 2q rising; sums cut at z, j <= 20 and k <= 60 (each remainder under e^-1665 its first term; S carries them at most squared, so C_K and L rise by under a factor 1 + e^-1600). Tails, by hand: tau <= tau0, w(1/(4 tau)) <= 0.0072 tau^-3, 1/lambda <= 25.65 tau (q >= 0.98, 1 - q <= 4 pi^2 tau), rho <= 3.0001, so N <= 0.278 tau^-2, 2N + 2 <= 0.557 tau^-2; bloc_1/2 <= 0.0040627 tau^-3.5 e^(-1/(16 tau)), bloc_0 <= 1e-18 tau^-3.5 e^(-1/(16 tau)) (u_w <= 505 tau^-2 e^(-1/(16 tau)) by P_min, w(a) <= 1.0001e-20 a there): S <= f0 = 0.0046 tau^-5.5 e^(-1/(16 tau)), rising below 1/88, int_0^tau0 f0 = 0.0046 16^4.5 Gamma(4.5, 125). tau >= tau1: Abar_1 <= 40 tau^1.5 (Abar_1 below it from 38.4), the modes past 1 under q^3, lambda >= 4 pi q(1 - 3q), K_lo >= 1 - 2q, so S <= 16 pi q (Abar_1^2 + 2 Abar_1)(1 + e^-1000), 16 pi 1600 = 80424.8, the 2 Abar_1 inside the slack 75.2 tau^3 from tau 14.2: S <= f_inf = 80500 tau^3 e^(-4 pi^2 tau), falling. Each tail's lattice sum is its integral plus P^-2 f at its end. The wrong-signed class (odd P): 2P|K_x(1/2 - 1/(2P))| <= sup_x |K_xx| <= F(tau) = sum_k 8 pi^2 k^2 q^(k^2), falling, so its sum is at most int F from 1/c - P^-2 = 2 sum_k q^(k^2) there <= W(P) = 1/(2 sqrt(pi (1/c(P) - P^-2))), 9.9 at P*: polylog, in place of the limit's int Gamma <= 1. So P R_maj(P) <= C_K + P^-2 L + 8e-40 T(P) + W(P) + P past_T, C_K = sum_cells (t2 - t1) S_sup + both tails' integrals, L = sum_cells S_sup + f0(tau0) + f_inf(tau1), P-free. The limit's first orders put int S near 18 (uniform.py C_R); S itself peaks near 1/88 (f0's shape), so C_K sits decades above. THE MARGIN. Each piece of the closed form over P is at most its value at P* (T grows like ln P, W like sqrt(ln P); c(P) steps by 1, first at 1.0029 P*, raising W by 1.0004, its steps e^(1/16) apart in P after; at each step F_K1's 4/(c(1 - c/P)) falls by about c/(c + 1), outweighing its e^(c^2/(16P))'s rise, under 1e-27), so the margin is least at P*: budget - 8.2e-6 - 2.06e-4 - C_K/P* - 2.31e-13 > 0.2497 for any C_K < 1e20. THE CONTROLS, each frozen before its code. The closed form must dominate the TRUE class sum, not box.py's U/Pa, which carries its box's ratio (Pb past_T/Pa = 1e4 L_T on the last box). (C1) S(tau) against the true class sum at P = P* (uniform.py's summand rewritten per (x0, o): u = o/P carried apart from x0 and K_x(x0 + u) evaluated from u (the Fourier form's sin(2 pi k(1/2 + u)) = (-1)^k sin(2 pi k u); below tau = 0.05 the cancelling image pair combined analytically, or extended precision), and the sign s and img_B's x and j formed from (x0, o) too: in doubles an absolute residue near 6.7e29 has spacing 1.4e14, and even 0.5 + 400/P* reads 0.5, so K_x would read 0 and every class Bad; the classes within 400 of each sign change, majorants at P0 = P*) at the centres of every third of 300 geometric tau cells on [1/c(P*), T(P*)]: largest ratio truth/S at most 1, and no Bad class at 300 <= |o| <= 400 (the scan wide enough). K6 has no far zone, so C1 sees it whole. FROZEN BEFORE CODE: the truth is the closed form's own B, images |k| <= K_img(P*) = 22 at W1's A (P0 = P*, the k = 0 image's xi from u, the rest from x0: B moves by under 1e-24 relative), modes k <= K_mode(P*) = 48 at W2's A_k (P0 = P*), the smaller where both run; Bad iff s phi <= B + fl (K3's fl), each Bad class 2(b + fl); the wrong-signed class's 2P|K_x| (odd P alone) printed apart over Gamma (K7's W(P), not S). P = P* (even, every class right- signed or s phi = 0) and P* + 1 (odd: x0 = 1/2 on Z + 1/2, o = -1/2 wrong-signed), S at P_eff = P*. By hand: at large tau Bad runs to |o| about A_1/(2 pi) (165 at T(P*)) against N about A_1/2, the window's 4|u| against sin's 2 pi |u|; so the largest ratio in [0.15, 0.6], past tau = 1; at 1/1229 below 0.01. K_x in mpmath at 60 digits (u near 3e-28). (C2) PLANT: N := 0 (the window collapsed to o = 0 and the wrong-signed class) puts the ratio past 1 at some cell. READ (section_C1, copied from its print): largest truth/S 0.325828 at tau 0.13963 (P*), 0.317892 at 3.5999 (P* + 1), a plateau near 0.31 from tau 0.14 to 3.6 at both, not a peak past 1 (the place predicted, missed); at the first centre, 0.00082535, 1.095e-4; farthest Bad |o| 153 at both x0 (152.5 at 1/2, P* + 1), none at 300..400; wrong- signed 2P|K_x|/Gamma 1 at P* + 1 (within 1e-12, a bar set after this read; the mean value's first order, which no 60-digit compare splits, so the bound is K7's F by hand), 4.9e-32 at P* (o = 0's rounding, s phi 0 there); the plant largest 153.5, past 1 at 85 and 83 of 100 cells. Every bar and the plant hold: S dominates the truth at P*. The rehearsal caught the sign's residue mod P dropped at x0 = 0, o < 0, and Gamma's Fourier series truncated below 0.05 (images now). (C4) K6's inequalities, mpmath at 40 digits (frozen before code): (a) |K_x|/(lambda |sin 2 pi x| K) on 40 geometric tau in [1/1229, 20] by 200 x in (0, 1/2) geometric toward both ends: least at least 1, within 1e-6 of it (x near 0, large tau); at tau = 1/1229, x = 1/2 - 1e-6 in [290, 300] (by hand (1/(16 tau^2) - 1/(2 tau))/(2 pi lambda) = 295.6). PLANT: (1 + q)^3 read as 1 + q puts the least below 1 at every tau <= 1. AFTER THE FIRST RUN: the least read 1 - 2.3e-41 at tau 9.19: near x = 0 the ratio is 1 + 2q(1 - cos 2 pi x) + ..., about 1 + 4e-11 q at x = 1e-6, under 40 digits once q < 2.5e-30 (tau > 1.74), where the least is rounding and its place moves between runs; so (a) reads at least 1 - 1e-35. Where q shows, least - 1 is 0.94 at 1/1229 falling to 2.1e-26 at 0.89, each such row's least at x = 1e-6. (b) (b/K)/(rho w(1/(4 tau))), images |k| <= 6, K4's Abar as written, on 30 geometric tau in [1/1229, 1/16] by 400 x in [0, 1/2]: largest at most 1, in [0.30, 0.40]. PLANT: w at 1/(8 tau) (a = |xi|/(4 tau)) puts it past 1 at every tau. (c) b(x0 + u) at u = +-u_w against bloc_x0, same tau: at most 1, in [0.97, 1] at x0 = 1/2, [0.5, 1] at 0. PLANT: the pair's 2 dropped puts x0 = 1/2 past 1. (d) Printed: N(1/1229) (by hand 0.26 1229^2, in [3.5e5, 4.5e5]), and the largest u_w/tau over the tau read, at most 4e-22. (C5) K4's and K5's majorants against W1's and W2's own at P0 = P* (frozen before code): (a) majorant.image_A (uniform arm, both signs) over Abar on 60 geometric a in [1e-3, (K_img + 1)c/2] by 60 geometric tau in [1/1229, 1/4]: largest at most 1, in [0.55, 0.572] (|l|/(1.75(1.5a + a^3 tau)) near 1/1.75 where w -> 0 or w >> 1). PLANT: Abar's a^3 tau dropped puts it past 1. (b) W2's A_k (majorant.mode_const's constants, mode_A's formula in mpmath at 60 digits) over Abar_k at k = 1..50 (to K_mode(P*) = 48; first run at 1..40, before K3's modes moved to 1/64) on 60 geometric tau in [1/64, T(P*)] and at each tau*_k = 3/(8 pi^2 k^2): largest at most 1, at tau*_k in [0.95, 1]. PLANT: C_k's 2.167(2 pi k)^2 dropped puts tau*_k's ratio past 1 at every k. READ (section_C5, copied from its print): (a) a to 14133.5, largest A/Abar 0.5714285708086326 (doubles read W1's leading order; its 1e-26 terms sit inside 1.75 against 1.7183), the plant 1.902e7; (b) largest A_k/Abar_k 1.0 (below it in 60 digits), at tau*_k 0.996632 (k = 1) to 0.999664 (k = 50; 0.999646 at k = 40 in the first run), the plant's least 1.551. Every bar and plant holds: K4 and K5 stand, K5 within 0.4% of W2 at its corner. READ (section_C4, copied from its second print): (a) least - 1 -3.4438e-41 at tau 15.433 (the wall), 295.728 at 1/1229 by 1/2 - 1e-6, the plant below 1 at every tau <= 1; (b) largest 0.333333 (b/K is w(1/(4 tau)) at 1/2, rho 3), the plant's least 1.894; (c) x0 = 0 in [0.55, 1.0], x0 = 1/2 in [0.985891, 1.0], the plant past 1 at every tau; (d) N(1/1229) 384475, u_w/tau 3.529e-22 (the grid's; the sup is 3.538e-22, K6 (iii)). Every bar and plant holds: K6 stands. (C3) The past_T bound (K2), in mpmath at 120 digits, at P = 1e10, P*, 1e40 (frozen before code): (a) against the k = 1..3 terms summed exactly in mpmath (a lower bound on the truth): bound/sum - 1 in [0, 1e-9]; (b) against uniform.past_T in doubles: in [1 - 1e-12, 1 + 1e-9], the float wall the read's own rounding, 1.3e-13 past 1e26; (c) eps_T(P*) in [3.5e-28, 4e-28]. PLANT: 3/d dropped from the k = 1 exponent puts (a) below 0 at all three P. READ (section_C3, copied from its print): eps_T 2.23657e-8 at 1e10, 3.74746e-28 at P*, 6.38122e-38 at 1e40; bound/sum - 1 2.20657e-18, 2.78233e-58, 6.35122e-78 (the bound second-order tight); bound/read 0.9999999999999845, 1.0000000000000067, 1.0000000000000124 (the float wall); the plant's bound/sum - 1 -2.20657e-8, -3.72505e-28, -6.35122e-38. (a), (b), (c) and the plant hold: K2 stands, and P past_T <= 2.30897e-13 (1 + 3.75e-28) P at every P >= P*. (C6) K7's C_K (frozen before code; section_CK, mpmath at 30 digits, 2000 geometric cells a band-free span, the cuts added): (a) per cell, S_sup at least S (S_closed) at t1, the centre and t2 (every cell); PLANTS, one per arm: e^(-z^2/(4 tau)) read at t1 fails (a) on some image cell; q^(k^2) read at t2 fails it on some mode cell. (b) Known answer: the same cells on f0's shape tau^-5.5 e^(-1/(16 tau)) (tau^-5.5 at t1, the exponential at t2) against 16^4.5 (Gamma(4.5, 1/(16 tau1)) - Gamma(4.5, 1/(16 tau0))), the exact int: ratio in [1, 1.1]. (c) The tails' majorants at their ends: S(tau0)/f0(tau0) and S(tau1)/f_inf(tau1) at most 1; PLANT: f0 with 0.557 for 0.557 tau^-2 puts the first past 1. (d) Printed: C_K (predicted in [1e3, 1e5], the cells near 1/88 most of it), L, both tails, and the cells' sum against the midpoint sum of S at the centres (at most 1.2 times it). MARGIN: m(P) = 1/4 - F_K1(P) - 2.06e-4 - (C_K + L/P^2 + 8e-40 T(P) + W(P))/P - L_T (1 + eps_T(P)), F_K1 = (1/400)(1/P + 4/(c(1 - c/P))) e^(c^2/(16 P)) (K1's with its eps kept), on P = P* 10^j to 1e300: predicted least at P*, in [0.2497, 0.2498]. READ (section_CK, copied from its print): 2002 cells, none with S_sup below S; the plants fail 535 (e at t1) and 1254 (q at t2) cells; known answer 1.0285808; S/f0 0.867988 at tau0, S/f_inf 0.95886 at tau1, the plant's S/f0 3.47e6; C_K 7809.09775 (tails 1.4e-44, 2.0e-678), L 1.30067e8, cells/midpoint 1.03648, the largest cell 68.58 at tau 0.014457, 0.9762 of the cells' sum in [1/176, 1/44], S's largest at 0.011666; the margin 0.249785863303 at P*, least there, rising to 0.249793105305 at 1.3e300 (F_K1 8.1367e-6, C_K/P* 5.8e-27, W/P* 7.4e-30). Every bar and plant holds and the KILL misses: the margin is positive at every P >= P*. (C7) The residues (frozen before code). The bound at d = P + 1 needs floor(H) <= C_d = floor(P/4), and H <= W + E + the larger excess, so the margin at d is at least b(P) - F - X - R_maj, b = C_d + 1 - W, W = P/4 at even P and (P^2 - 1)/(4P) + 1/P^2 at odd (the sawtooth's, classes.py). By hand, at P = 0, 1, 2, 3 mod 4 (d = 1, 2, 3, 0 mod 4): b = 1, 3/4 + 1/(4P) - 1/P^2, 1/2, 1/4 + 1/(4P) - 1/P^2 = budget(P), the least of the four at every P >= 2. So a margin read at budget(P), here and in box.py, holds at every d, not only d = 0 mod 4. Read in exact rationals at P = 399..20000 and P* + j, j = 0..3: b equal to its residue's form, at least budget(P), and equal to it at P = 3 mod 4 alone. PLANT: C_d read as floor((P - 1)/4) puts b below budget(P) at every P = 0 mod 4. READ (section_C7, copied from its print): least b - budget by P = 0, 1, 2, 3 mod 4 0.749381, 0.5, 0.249384, 0; the forms and the bound hold, the plant fires at all 4902 P = 0 mod 4 read. KILL: the closed form's margin at or below 0 at some P >= P* (read on a grid of P to 1e300, with each piece's monotonicity in P proved by hand), or C1's ratio above 1. FINDINGS. The closed form dominates the true class sum at P* and P* + 1 (largest truth/S 0.326, a plateau near 0.31 from tau 0.14 to 3.6), every piece K1..K7 holds against its control and its plant fires, and the margin is positive at every P >= P*, least at P*, 0.2497859. With box.py below P*, the real support's margin is positive at every d = P + 1 = 0 mod 4 from 400 on, under both scripts' conditions, and so (C7) at every d from 400 on, the other residues' budgets being larger. A bare run is the verdict: C3, C4, C5, C1 (with C2), C6, the margin and C7 in turn, each bar and plant a check; it exits 0 iff every check passes. RUN RECORD: 36 of 36 checks, every READ above reprinted to its digits; 121.2 s, peak 45.9 MB. """ import math import os import sys from fractions import Fraction as Fr os.environ.setdefault("OPENBLAS_NUM_THREADS", "1") import mpmath as mp # noqa: E402 import numpy as np # noqa: E402 import majorant as MJ # noqa: E402 import uniform as UN # noqa: E402 from every import T_of # noqa: E402 CHECKS = [] def check(name, ok, detail=""): CHECKS.append(bool(ok)) print(f" [{'PASS' if ok else 'FAIL'}] {name}" + (f" -- {detail}" if detail else ""), flush=True) P_STAR = 399 * 2 ** 25 * 10 ** 20 mp.mp.dps = 120 def L_T(): return 8 * mp.mpf(399) ** 2 * mp.exp(-4 * mp.pi ** 2) / (4 * mp.pi ** 2) def T_mp(P): return 1 + mp.log(mp.mpf(P) / 399) / (2 * mp.pi ** 2) def eps_T(P, plant=False): """(K2) past_T(P, T(P)) <= L_T (1 + eps_T(P)); the plant drops 3/d from the k = 1 exponent.""" P = mp.mpf(P) d, T, pi2 = P + 1, T_mp(P), mp.pi ** 2 eta = 13 * pi2 / (3 * d * d) lin = (0 if plant else 3 / d) + eta k1 = (d / P) ** 3 * mp.exp(4 * pi2 * T * lin) / ((1 - eta) * (1 - 2 * pi2 / d ** 3)) a = 2 * pi2 * T * (P / d) ** 3 e2 = (((d / P) ** 3 / 2 + 4 * pi2 / P ** 3) * mp.exp(-4 * pi2 * T * (1 - 6 / d)) / (1 - mp.exp(-4 * a))) e3 = ((d / 2) * (1 + 2 * d / mp.log(3)) * (4 * pi2 / P ** 3) * mp.exp(4 * pi2 * T - T * P ** 3 * mp.log(3) / (2 * d))) lb = min(1.196 * (2 * mp.pi * 2001 / d) ** 2, mp.mpf(0.846)) e4 = ((d / 2) * (1 + 2 * d / lb) * (4 * pi2 / P ** 3) * mp.exp(4 * pi2 * T - T * P ** 3 * lb / (2 * d))) return k1 - 1 + e2 + e3 + e4 def past_T_mp(P, kmax=3): """The Q-bound's terms k = 1..kmax, exactly, at T = T(P).""" P = mp.mpf(P) d, T = P + 1, T_mp(P) out = mp.mpf(0) for k in range(1, kmax + 1): x = mp.log1p(min(8 * mp.sin(mp.pi * k / d) ** 2, 2)) v = x / (2 * d) out += 8 / P * mp.exp(-T * P ** 3 * x / (2 * d)) / -mp.expm1(-v) return out def section_C3(): print("(C3) the past_T bound (K2) against the truth's first terms and " "uniform.past_T") rows = [] for P in (10 ** 10, P_STAR, 10 ** 40): e = eps_T(P) bound = L_T() * (1 + e) tr = past_T_mp(P) rd = UN.past_T(P, T_of(P)) pl = L_T() * (1 + eps_T(P, plant=True)) ra, rb, rp = bound / tr - 1, float(bound) / rd, pl / tr - 1 rows.append((P, e, ra, rb, rp)) print(f" P {float(P):.4e}: eps_T {mp.nstr(e, 6)}, bound/sum - 1 " f"{mp.nstr(ra, 6)}, bound/read {rb!r}, plant bound/sum - 1 " f"{mp.nstr(rp, 6)}") check("C3 (a) bound/sum - 1 in [0, 1e-9] at all three", all(0 <= r[2] <= 1e-9 for r in rows)) check("C3 (b) bound/read in [1 - 1e-12, 1 + 1e-9] at all three", all(1 - 1e-12 <= r[3] <= 1 + 1e-9 for r in rows)) check("C3 (c) eps_T(P*) in [3.5e-28, 4e-28]", 3.5e-28 <= rows[1][1] <= 4e-28) check("C3 PLANT: (a) below 0 at all three", all(r[4] < 0 for r in rows)) def lam(t, plant=False): """(K6) lambda(tau); the plant reads (1 + q)^3 as 1 + q.""" q = mp.exp(-4 * mp.pi ** 2 * t) return 4 * mp.pi * q / ((1 + q) ** (1 if plant else 3) * (1 - q)) def G0(y, t): return mp.exp(-y * y / (4 * t)) / mp.sqrt(4 * mp.pi * t) def K_Kx(x, t): """K and K_x: Gaussian images |k| <= 6 below tau = 0.05 (the pair at 1/2 cancels in K_x with no loss), the Fourier series above.""" if t < 0.05: ys = [x + k for k in range(-6, 7)] return (sum(G0(y, t) for y in ys), sum(-y / (2 * t) * G0(y, t) for y in ys)) q, K, Kx = mp.exp(-4 * mp.pi ** 2 * t), mp.mpf(1), mp.mpf(0) for k in range(1, 12): K += 2 * mp.cos(2 * mp.pi * k * x) * q ** (k * k) Kx -= 4 * mp.pi * k * mp.sin(2 * mp.pi * k * x) * q ** (k * k) return K, Kx def w_img(a, t): """(K6 (i)) w(a) = a Abar(a, tau), K4's Abar as written.""" return a * (mp.mpf(1.75) * (1.5 * a + a ** 3 * t) + mp.mpf(10) ** -20) def rho(t): return 1 + 2 * sum((2 * j - 1) ** 4 * mp.exp(-j * (j - 1) / (4 * t)) for j in range(1, 21)) def b_img(x, t): return sum(G0(x + k, t) * w_img(abs(x + k) / (2 * t), t) for k in range(-6, 7)) def N_img(t): return rho(t) * w_img(1 / (4 * t), t) / (2 * lam(t)) def u_w(t): P_eff = max(mp.mpf(P_STAR), mp.exp(1 / (16 * t) - mp.mpf(1) / 8) / 1600) return (N_img(t) + mp.mpf(1) / 2) / P_eff def bloc(x0, t, uw, plant=False): """(K6 (iii)) sup of b_img within u_w of x0; the plant drops the pair's 2 (one image of it).""" zs = ([j + mp.mpf(1) / 2 for j in range(20)] if x0 else [mp.mpf(j) for j in range(1, 21)]) terms = [(2 * z) ** 4 * mp.exp(-z * z / (4 * t)) for z in zs] if plant and x0: terms[0] /= 2 s = 2 * w_img(1 / (4 * t), t) * sum(terms) if not x0: s += w_img(uw / (2 * t), t) return (1 + mp.mpf(10) ** -20) * s / mp.sqrt(4 * mp.pi * t) def section_C4(): print("(C4) K6's inequalities, mpmath at 40 digits") mp.mp.dps = 40 lo = mp.mpf(1) / 1229 ts = [lo * (20 / lo) ** (mp.mpf(i) / 39) for i in range(40)] g = [mp.mpf(10) ** (-6 + (mp.log10(0.25) + 6) * i / 99) for i in range(100)] xs = g + [mp.mpf(1) / 2 - v for v in g] least, least_pl, at = mp.inf, [], None for t in ts: row, row_pl = mp.inf, mp.inf for x in xs: K, Kx = K_Kx(x, t) base = abs(Kx) / (abs(mp.sin(2 * mp.pi * x)) * K) r, rp = base / lam(t), base / lam(t, plant=True) if r < least: least, at = r, (t, x) row, row_pl = min(row, r), min(row_pl, rp) least_pl.append((t, row_pl)) K, Kx = K_Kx(mp.mpf(1) / 2 - mp.mpf(10) ** -6, lo) top = abs(Kx) / (abs(mp.sin(2 * mp.pi * (mp.mpf(1) / 2 - mp.mpf(10) ** -6))) * K * lam(lo)) print(f" (a) least - 1 {mp.nstr(least - 1, 5)} at tau {mp.nstr(at[0], 5)}" f", x {mp.nstr(at[1], 5)}; at 1/1229, x = 1/2 - 1e-6: " f"{mp.nstr(top, 6)}") check("C4 (a) least in [1 - 1e-35, 1 + 1e-6] (the 40-digit wall)", 1 - mp.mpf(10) ** -35 <= least <= 1 + 1e-6) check("C4 (a) at 1/1229, x = 1/2 - 1e-6, in [290, 300]", 290 <= top <= 300) check("C4 (a) PLANT: least below 1 at every tau <= 1", all(r < 1 for t, r in least_pl if t <= 1)) tb = [lo * ((mp.mpf(1) / 16) / lo) ** (mp.mpf(i) / 29) for i in range(30)] xb = [mp.mpf(i) / 798 for i in range(400)] big, pl_rows, c_rows = 0, [], [] for t in tb: bound = rho(t) * w_img(1 / (4 * t), t) bound_pl = rho(t) * w_img(1 / (8 * t), t) rows = [b_img(x, t) / K_Kx(x, t)[0] for x in xb] big = max(big, max(rows) / bound) pl_rows.append(max(rows) / bound_pl) uw = u_w(t) c_rows.append((t, uw / t, *(max(b_img(x0 + s * uw, t) for s in (1, -1)) / bloc(x0, t, uw) for x0 in (mp.mpf(0), mp.mpf(1) / 2)), b_img(mp.mpf(1) / 2 + uw, t) / bloc(mp.mpf(1) / 2, t, uw, plant=True))) check("C4 (b) largest in [0.30, 0.40]", 0.30 <= big <= 0.40, mp.nstr(big, 6)) check("C4 (b) PLANT: past 1 at every tau", all(r > 1 for r in pl_rows), f"least {mp.nstr(min(pl_rows), 4)}") c0 = [r[2] for r in c_rows] c12 = [r[3] for r in c_rows] print(f" (c) x0 = 0 in [{mp.nstr(min(c0), 4)}, {mp.nstr(max(c0), 4)}], " f"x0 = 1/2 in [{mp.nstr(min(c12), 6)}, {mp.nstr(max(c12), 6)}]") check("C4 (c) x0 = 1/2 in [0.97, 1]", 0.97 <= min(c12) and max(c12) <= 1) check("C4 (c) x0 = 0 in [0.5, 1]", 0.5 <= min(c0) and max(c0) <= 1) check("C4 (c) PLANT: x0 = 1/2 past 1 at every tau", all(r[4] > 1 for r in c_rows)) n0 = N_img(lo) uwt = max(r[1] for r in c_rows) check("C4 (d) N(1/1229) in [3.5e5, 4.5e5]", 3.5e5 <= n0 <= 4.5e5, mp.nstr(n0, 6)) check("C4 (d) largest u_w/tau at most 4e-22", uwt <= 4e-22, mp.nstr(uwt, 4)) mp.mp.dps = 120 def K_img(P): return math.ceil(math.sqrt(4 * math.log(P) + 100)) + 2 def c_of(P): return math.ceil(16 * math.log(1600 * P)) + 1 def Abar(a, t, plant=False): """(K4) the images' majorant at P0 = P; the plant drops a^3 tau.""" return 1.75 * (1.5 * a + (0 if plant else a ** 3 * t)) + 1e-20 def Abar_k(k, t, plant=False): """(K5) the modes' majorant at P0 = P, P-free; the plant drops C_k's 2.167 (2 pi k)^2.""" w = 2 * mp.pi * k b1, b2 = 3 * mp.pi * k, w ** 3 * t C = (mp.sqrt(b1 * b2 ** 3 / 3) + (0 if plant else 2.167 * w ** 2) + 3.06 * w + (2.04 * w ** 3 + 1.084 * w ** 4) * t) return abs(b1 - b2) + C / P_STAR def section_C5(): print("(C5) K4's and K5's majorants against W1's and W2's own at P0 = P*") P0 = float(P_STAR) amax = (K_img(P0) + 1) * c_of(P0) / 2 a = np.geomspace(1e-3, amax, 60) big = big_pl = 0.0 for t in np.geomspace(1 / 1229, 0.25, 60): for s in (1.0, -1.0): A = MJ.image_A(a, t, s, P0) big = max(big, float(np.max(A / Abar(a, t)))) big_pl = max(big_pl, float(np.max(A / Abar(a, t, plant=True)))) print(f" (a) a in [1e-3, {amax}]: largest A/Abar {big!r}") check("C5 (a) largest A/Abar in [0.55, 0.572]", 0.55 <= big <= 0.572) check("C5 (a) PLANT: past 1", big_pl > 1, f"{big_pl:.4g}") mp.mp.dps = 60 T0 = mp.mpf(T_of(P0)) grid = [mp.mpf(1) / 64 * (T0 * 64) ** (mp.mpf(i) / 59) for i in range(60)] Pm = mp.mpf(P_STAR) worst, star, star_pl = 0, [], [] for k in range(1, 51): th = 2 * math.pi * k / P0 sg, mb, ec, rho_ = MJ.mode_const(th, P0) Ec, zc = mp.mpf(P0 * ec), mp.mpf(rho_ * P0 * P0) b1 = 3 * mp.pi * k def A_W2(t): b2 = (2 * mp.pi * k) ** 3 * t return (mp.sqrt((b1 - b2) ** 2 + b1 * b2 ** 3 / (3 * Pm ** 2)) + Ec * mp.exp(zc * t) + mp.sqrt(1 + 2.25 * mp.mpf(th) ** 2) * Pm * mp.expm1(zc * t)) for t in grid: worst = max(worst, A_W2(t) / Abar_k(k, t)) ts = 3 / (8 * mp.pi ** 2 * k * k) star.append(A_W2(ts) / Abar_k(k, ts)) star_pl.append(A_W2(ts) / Abar_k(k, ts, plant=True)) worst = max(worst, star[-1]) print(f" (b) k = 1..50: largest A_k/Abar_k {mp.nstr(worst, 8)}; at tau*_k " f"in [{mp.nstr(min(star), 6)}, {mp.nstr(max(star), 6)}] (k = 1: " f"{mp.nstr(star[0], 6)}, k = 50: {mp.nstr(star[-1], 6)})") check("C5 (b) largest A_k/Abar_k at most 1", worst <= 1) check("C5 (b) at tau*_k in [0.95, 1]", 0.95 <= min(star) and max(star) <= 1) check("C5 (b) PLANT: past 1 at every k", all(r > 1 for r in star_pl), f"least {mp.nstr(min(star_pl), 4)}") mp.mp.dps = 120 def Kxx_half(t): """Gamma = |K_xx(1/2, tau)|: images |k| <= 6 below tau = 0.05, the Fourier series above (as K_Kx).""" if t < 0.05: return abs(sum(G0(y, t) * (y * y / (4 * t * t) - 1 / (2 * t)) for y in (mp.mpf(1) / 2 + k for k in range(-6, 7)))) q = mp.exp(-4 * mp.pi ** 2 * t) return abs(sum(-8 * mp.pi ** 2 * k * k * (-1) ** k * q ** (k * k) for k in range(1, 12))) def b_true_img(x0, u, t, P0): """(C1) P^3 times the images' arm at x0 + u: W1's A at P0, |k| <= K_img(P0); the k = 0 image's xi from u, the rest from x0.""" K = K_img(P0) ks = np.array([k for k in range(-K, K + 1) if x0 + k != 0], float) xi = np.concatenate([x0 + ks, [u] if x0 == 0 else []]) xi = xi[xi != 0] a = np.abs(xi) / (2 * t) G = (4 * np.pi * t) ** -0.5 * np.exp(-xi * xi / (4 * t)) * a return float((G * np.minimum(MJ.image_A(a, t, np.sign(xi), P0), 1e300) ).sum()) def b_true_mode(t, P0): """(C1) P^3 times the modes' arm: W2's A_k at P0, k <= K_mode(P0).""" if P0 not in MODE_CST: Km = math.ceil(math.sqrt(7.34 * (3 * math.log(P0) + 95))) MODE_CST[P0] = [] for k in range(1, Km + 1): th = 2 * math.pi * k / P0 sg, mb, ec, rho_ = MJ.mode_const(th, P0) MODE_CST[P0].append((k, th, P0 * ec, rho_ * P0 * P0)) return sum(4 * math.pi * k * math.exp(-4 * math.pi ** 2 * k * k * t) * float(MJ.mode_A(k, th, Ec, zc, t, P0)) for k, th, Ec, zc in MODE_CST[P0]) MODE_CST = {} def S_closed(t, plant=False): """(K6) S(tau) at P_eff: S_img to 1/16, S_mode from 1/64, the smaller between; the plant (C2) puts N := 0.""" out = [] if t <= mp.mpf(1) / 16: N = 0 if plant else N_img(t) uw = u_w(t) out.append(2 * (2 * N + 2) * (bloc(0, t, uw) + bloc(1, t, uw))) if t >= mp.mpf(1) / 64: q = mp.exp(-4 * mp.pi ** 2 * t) bm = sum(4 * mp.pi * k * q ** (k * k) * Abar_k(k, t) for k in range(1, 61)) K_lo = max(2 * G0(mp.mpf(1) / 2, t), 1 - 2 * q) N = 0 if plant else bm / (2 * lam(t) * K_lo) out.append(4 * (2 * N + 2) * bm) return min(out) def truth(t, P, scan=400): """(C1) the true class sum at P: per (x0, o), u = o/P apart from x0, Bad iff s phi <= B + fl, each Bad class 2(B + fl); with the wrong- signed classes' 2P|K_x| apart, and the farthest Bad |o| per x0.""" tf = float(t) fl = 4e-40 / P if tf < 0.25 else 4e-40 * math.exp(-4 * math.pi ** 2 * tf) / P ** 2 bm = b_true_mode(tf, float(P)) if tf >= 1 / 64 else math.inf tot, wrong, far = 0.0, 0.0, {} for x0 in (0, 1): # x0 = 0, 1/2 half = x0 and P % 2 os_ = [mp.mpf(j) + (mp.mpf(1) / 2 if half else 0) for j in range(-scan - (1 if half else 0), scan + 1)] bi = (b_true_img(0.5, 0.0, tf, float(P)) if x0 and tf < 0.25 else math.inf) far[x0] = -1.0 for o in os_: u = o / P x = mp.mpf(x0) / 2 + u if x0 == 0 and tf < 0.25: bi = b_true_img(0.0, float(u), tf, float(P)) B = min(bi, bm) r = mp.mpf(x0) * P / 2 + o + (P if x0 == 0 and o < 0 else 0) s = 1 if 2 * r < P - 1 else -1 sp = float(s * (-P * K_Kx(x, t)[1])) if sp <= B + fl: tot += 2 * (B + fl) if sp < 0: wrong += -2 * sp far[x0] = max(far[x0], float(abs(o))) return tot, wrong, far def section_C1(n_cells=300): print("(C1) the true class sum at P*, P* + 1 against K6's S; (C2) N := 0") mp.mp.dps = 60 lo = mp.mpf(1) / c_of(P_STAR) hi = mp.mpf(T_of(P_STAR)) e = [lo * (hi / lo) ** (mp.mpf(i) / n_cells) for i in range(n_cells + 1)] ts = [mp.sqrt(e[i] * e[i + 1]) for i in range(0, n_cells, 3)] rows = [] for P in (P_STAR, P_STAR + 1): for t in ts: tr, wr, far = truth(t, P) S, Sp = S_closed(t), S_closed(t, plant=True) g = Kxx_half(t) rows.append((P, float(t), tr / float(S), tr / float(Sp), float(wr / g) if wr else 0.0, far[0], far[1])) for P in (P_STAR, P_STAR + 1): rp = [r for r in rows if r[0] == P] big = max(rp, key=lambda r: r[2]) print(f" P {'P*' if P == P_STAR else 'P* + 1'}: largest truth/S " f"{big[2]:.6g} at tau {big[1]:.5g}; at tau {rp[0][1]:.5g} " f"{rp[0][2]:.4g}; farthest Bad |o| at 0: " f"{max(r[5] for r in rp):g}, at 1/2: {max(r[6] for r in rp):g}" f"; wrong-signed 2P|K_x|/Gamma largest " f"{max(r[4] for r in rp):.6g}") print(f" plant (N := 0): largest {max(r[3] for r in rp):.4g}, " f"cells past 1: {sum(r[3] > 1 for r in rp)} of {len(rp)}") for r in rows[::10]: print(" ", " ".join(f"{v:.4g}" for v in r)) big = max(r[2] for r in rows) check("C1 largest truth/S at most 1", big <= 1, f"{big:.6g}") check("C1 largest truth/S in [0.15, 0.6]", 0.15 <= big <= 0.6) check("C1 no Bad class at 300 <= |o| <= 400", all(max(r[5], r[6]) < 300 for r in rows)) check("C1 wrong-signed 2P|K_x|/Gamma within 1e-12 of 1 at P* + 1 (first" " order; K7 bounds it by F, not Gamma)", all(abs(r[4] - 1) <= 1e-12 for r in rows if r[0] == P_STAR + 1)) check("C2 PLANT: past 1 at some cell", any(r[3] > 1 for r in rows)) mp.mp.dps = 120 TAU0, TAU1 = mp.mpf(1) / 2000, mp.mpf(40) def P_eff(t): return max(mp.mpf(P_STAR), mp.exp(1 / (16 * t) - mp.mpf(1) / 8) / 1600) def S_sup(t1, t2, plant=None): """(K7) sup of S on [t1, t2], each monotone factor at its worse end; the plants read e^(-z^2/(4 tau)) at t1 ("e") or q^(k^2) at t2 ("q").""" out = [] if t2 <= mp.mpf(1) / 16: w_hi = w_img(1 / (4 * t1), t1) N = rho(t2) * w_hi / (2 * lam(t2)) te = t1 if plant == "e" else t2 a_w = (N + mp.mpf(1) / 2) / (2 * t1 * P_eff(t2)) pref = (1 + mp.mpf(10) ** -20) / mp.sqrt(4 * mp.pi * t1) eh = sum((2 * j + 1) ** 4 * mp.exp(-(j + mp.mpf(1) / 2) ** 2 / (4 * te)) for j in range(20)) e0 = sum((2 * z) ** 4 * mp.exp(-mp.mpf(z) ** 2 / (4 * te)) for z in range(1, 21)) bh = pref * 2 * w_hi * eh b0 = pref * (w_img(a_w, t2) + 2 * w_hi * e0) out.append(2 * (2 * N + 2) * (b0 + bh)) if t1 >= mp.mpf(1) / 64: tq = t2 if plant == "q" else t1 q1 = mp.exp(-4 * mp.pi ** 2 * tq) bm = mp.mpf(0) for k in range(1, 61): b1 = 3 * mp.pi * k far = max(abs(b1 - (2 * mp.pi * k) ** 3 * t1), abs(b1 - (2 * mp.pi * k) ** 3 * t2)) Ck = Abar_k(k, t2) - abs(b1 - (2 * mp.pi * k) ** 3 * t2) bm += 4 * mp.pi * k * q1 ** (k * k) * (far + Ck) h = mp.mpf(1) / 2 K_lo = max(min(2 * G0(h, t1), 2 * G0(h, t2)), 1 - 2 * mp.exp(-4 * mp.pi ** 2 * t1)) N = bm / (2 * lam(t2) * K_lo) out.append(4 * (2 * N + 2) * bm) return min(out) def f0(t, plant=False): """(K7) the tail's majorant below tau0; the plant drops tau^-2 from 2N + 2's bound.""" return (mp.mpf("0.0046") * (t ** 2 if plant else 1) * t ** -5.5 * mp.exp(-1 / (16 * t))) def f_inf(t): return 80500 * t ** 3 * mp.exp(-4 * mp.pi ** 2 * t) def cells_K(n=2000): e = [TAU0 * (TAU1 / TAU0) ** (mp.mpf(i) / n) for i in range(n + 1)] e += [mp.mpf(1) / 64, mp.mpf(1) / 16] return sorted(set(e)) def c_mp(P): return mp.ceil(16 * mp.log(1600 * mp.mpf(P))) + 1 def F_K1(P): P = mp.mpf(P) c = c_mp(P) return (1 / P + 4 / (c * (1 - c / P))) * mp.exp(c * c / (16 * P)) / 400 def W_of(P): P = mp.mpf(P) return 1 / (2 * mp.sqrt(mp.pi * (1 / c_mp(P) - P ** -2))) def section_CK(): print("(C6) K7's C_K by cells, the tails by hand; the margin") mp.mp.dps = 30 e = cells_K() Sv = {t: S_closed(t) for t in e} CK = L = mid = known = near = mp.mpf(0) bad, bad_e, bad_q, top = 0, 0, 0, (0, None) for t1, t2 in zip(e[:-1], e[1:]): tc = mp.sqrt(t1 * t2) Sc = S_closed(tc) s = S_sup(t1, t2) w = t2 - t1 CK += w * s L += s mid += w * Sc if w * s > top[0]: top = (w * s, t1) bad += s < max(Sv[t1], Sc, Sv[t2]) if t2 <= mp.mpf(1) / 16: bad_e += S_sup(t1, t2, "e") < max(Sv[t1], Sc, Sv[t2]) if t1 >= mp.mpf(1) / 64: bad_q += S_sup(t1, t2, "q") < max(Sv[t1], Sc, Sv[t2]) known += w * t1 ** -5.5 * mp.exp(-1 / (16 * t2)) if mp.mpf(1) / 176 <= t1 and t2 <= mp.mpf(1) / 44: near += w * s exact = 16 ** mp.mpf(4.5) * (mp.gammainc(4.5, 1 / (16 * TAU1)) - mp.gammainc(4.5, 1 / (16 * TAU0))) tail0 = mp.mpf("0.0046") * 16 ** mp.mpf(4.5) * mp.gammainc( 4.5, 1 / (16 * TAU0)) a = 4 * mp.pi ** 2 tail1 = 80500 * mp.gammainc(4, a * TAU1) / a ** 4 cells = CK CK += tail0 + tail1 L += f0(TAU0) + f_inf(TAU1) r0, r1 = Sv[e[0]] / f0(e[0]), Sv[e[-1]] / f_inf(e[-1]) r0p = Sv[e[0]] / f0(e[0], plant=True) print(f" {len(e) - 1} cells on [{mp.nstr(TAU0, 4)}, {mp.nstr(TAU1, 4)}]" f"; (a) cells with S_sup below S: {bad}; PLANTS: e at t1 {bad_e}" f", q at t2 {bad_q}") print(f" (b) known answer: cells/exact {mp.nstr(known / exact, 8)}") print(f" (c) S/f0 at tau0 {mp.nstr(r0, 6)}, S/f_inf at tau1 " f"{mp.nstr(r1, 6)}; PLANT S/f0 {mp.nstr(r0p, 6)}") print(f" (d) C_K {mp.nstr(CK, 10)} (cells {mp.nstr(cells, 10)}, tails " f"{mp.nstr(tail0, 4)}, {mp.nstr(tail1, 4)}), L {mp.nstr(L, 6)}, " f"cells/midpoint {mp.nstr(cells / mid, 6)}, largest cell " f"{mp.nstr(top[0], 4)} at tau {mp.nstr(top[1], 5)}; share in " f"[1/176, 1/44] {mp.nstr(near / cells, 4)}, S's largest at tau " f"{mp.nstr(max(e, key=lambda t: Sv[t]), 5)}") check("C6 (a) every cell's S_sup at least S", bad == 0) check("C6 (a) PLANTS fail some cell, each arm", bad_e > 0 and bad_q > 0) check("C6 (b) known answer in [1, 1.1]", 1 <= known / exact <= 1.1) check("C6 (c) both tails' ratios at most 1", r0 <= 1 and r1 <= 1) check("C6 (c) PLANT: past 1", r0p > 1) check("C6 (d) C_K in [1e3, 1e5]", 1e3 <= CK <= 1e5) check("C6 (d) cells/midpoint at most 1.2", cells / mid <= 1.2) mp.mp.dps = 120 eT = eps_T(P_STAR) # eps_T falls in P (K2) rows = [] for j in range(0, 271): P = mp.mpf(P_STAR) * mp.mpf(10) ** j m = (mp.mpf(1) / 4 - F_K1(P) - mp.mpf("2.06e-4") - (CK + L / P ** 2 + mp.mpf("8e-40") * T_mp(P) + W_of(P)) / P - mp.mpf("2.30897e-13") * (1 + eT)) rows.append((j, m)) least = min(rows, key=lambda r: r[1]) mono = all(b[1] >= a[1] for a, b in zip(rows, rows[1:])) print(f" MARGIN: at P* {mp.nstr(rows[0][1], 12)}, least " f"{mp.nstr(least[1], 12)} at P* 10^{least[0]}, at 1.3e300 " f"{mp.nstr(rows[-1][1], 12)}; rising along the grid: {mono}") print(f" pieces at P*: F_K1 {mp.nstr(F_K1(P_STAR), 6)}, C_K/P* " f"{mp.nstr(CK / P_STAR, 4)}, W/P* {mp.nstr(W_of(P_STAR) / P_STAR, 4)}" f", eps_T {mp.nstr(eT, 4)}") check("MARGIN least at P*, in [0.2497, 0.2498]", least[0] == 0 and mp.mpf("0.2497") <= least[1] <= mp.mpf("0.2498")) check("KILL not fired: the margin above 0 at every P read", least[1] > 0) def section_C7(): print("(C7) the residues: b = C_d + 1 - W against budget(P)") def W(P): return Fr(P, 4) if P % 2 == 0 else Fr(P * P - 1, 4 * P) + Fr(1, P * P) def bud(P): return Fr(1, 4) + Fr(1, 4 * P) - Fr(1, P * P) form = {0: lambda P: Fr(1), 2: lambda P: Fr(1, 2), 3: bud, 1: lambda P: Fr(3, 4) + Fr(1, 4 * P) - Fr(1, P * P)} Ps = list(range(399, 20001)) + [P_STAR + j for j in range(4)] ok_form = ok_ge = ok_eq = True least = {r: None for r in range(4)} plant = [P for P in Ps if P % 4 == 0 and (P - 1) // 4 + 1 - W(P) < bud(P)] for P in Ps: b = P // 4 + 1 - W(P) ok_form &= b == form[P % 4](P) ok_ge &= b >= bud(P) ok_eq &= (b == bud(P)) == (P % 4 == 3) r = P % 4 if least[r] is None or b - bud(P) < least[r]: least[r] = b - bud(P) print(" least b - budget by P mod 4 = 0, 1, 2, 3: " + ", ".join( f"{float(least[r]):.6g}" for r in range(4))) check("C7 b equal to its residue's form", ok_form) check("C7 b at least budget(P), equal at P = 3 mod 4 alone", ok_ge and ok_eq) check("C7 PLANT: b below budget at every P = 0 mod 4", len(plant) == sum(P % 4 == 0 for P in Ps), f"{len(plant)} P") if __name__ == "__main__": section_C3() section_C4() section_C5() section_C1() section_CK() section_C7() print(f"{sum(CHECKS)}/{len(CHECKS)} checks passed") sys.exit(0 if all(CHECKS) else 1)