{ "row": 35, "line": { "name": "L2", "x": "17/2", "z": "7/4", "mt2": "1", "variable": "y", "connection": "row35_L2/connection_L2.json (A_entries)" }, "what": "the canonical-basis census of the 35 non-elliptic diagonal blocks of the line connection: each block run through the eps-factoriser by itself; 29 admit a rational rotation T(eps, y) to an eps-factorised dlog form eps * Atilde(y), 6 stop (the engine's stop string verbatim, its class by name); sector 481 (rows 54-59, n = 6): the maximal-cut curve is elliptic (j non-constant in z5); the eps^0 block is reducible to first order -- six exact hyperexponential factors, see the sector_481 entry", "form": { "T_Atilde": "the engine's string matrices in (eps, x), x == y on the line, nested lists COLUMN-major (json[j][i] = M[i][j]); rational literals as p//q; verbatim from the engine's out file", "identity": "T A T^-1 + (dT/dy) T^-1 == eps * Atilde, A the block of the line connection at d = 4 - 2 eps", "residues": "row-major nested lists (json[i][j] = R[i][j]) of exact rationals p/q: the residue matrix of Atilde at each rational pole (letters.poles, the same order), Rinf the residue at infinity, quad_kernels[q] the (U y + V)/q(y) term of the quadratic letter q (U, V row-major)", "dlog_form": "Atilde(y) == sum_p residues[p]/(y - p) + sum_q (U_q y + V_q)/q(y) exactly; Rinf == -(sum_p residues[p] + sum_q U_q / lead(q))", "alphabet_letters": "each rational pole named by the alphabet letter whose restriction to this line vanishes there (quadratic_alphabet_letter: the letter whose restriction is the quadratic factor)", "sector_487_entry": "the sector-487 entry carries, beside the OK-block keys, basis_change (the cut-IBP dictionary C over the served four, the transcribed rotation, S = T_transcribed . C written as this block's T, det C, the receipts by sha256, the engine report) and served_basis_stop (the stop entry of the served basis, kept beside as the record); letters.quadratic_alphabet_letters states the two-variable form of each quadratic factor as a restriction check to this line only", "sector_481": "the sector-481 entry beside blocks[] (the sector is not in blocks[]: not run through the eps-factoriser; census.elliptic_not_run names it): the label, the maximal-cut curve statement in the served text's words, eps0_block_chain = the first-order factorisation receipt of the served block on this line at d = 4 in y (verdict, size, remainder_rank, the six chain factors verbatim -- exponents per singular factor as [factor, exponent] pairs (search labels; the canonical exponents are the residues of rate_u_prime_over_u, rate_partial_fractions), degree_bound, P (the polynomial vector), n_solutions_at_this_exponent, rate_u_prime_over_u, rate_partial_fractions -- the search_log, the bounds EB / NB / MAXCOMBO, the versions, the wall; the input by basename + sha256), rays (the two mass-scaling rays P and E: the ray block receipts and their chain receipts by sha256, verdicts), controls (the S-vs-N block control, the thimble receipts with their digit figures, the built-in C1-C4 of the served script), provenance_scripts (the block extractor and the ray config maker, as text; not runnable from this bundle); every receipt named by its vendored sha256 and its original sha256" }, "check_points": [ [ "1/223", "29/211" ], [ "5/6", "17/139" ] ], "eps_free_shift": "1/97", "census": { "n_blocks": 35, "n_ok": 29, "n_stopped": 6, "ok_sectors": [ 262, 294, 98, 321, 322, 326, 327, 353, 354, 355, 358, 359, 161, 167, 384, 388, 416, 420, 422, 423, 225, 227, 231, 455, 480, 484, 485, 486, 487 ], "stopped_by_class": { "Moser-irreducible": [ 417, 449 ], "eps^0 obstruction": [ 421, 453, 482, 483 ] }, "elliptic_not_run": { "sector": 481, "rows": [ 54, 59 ], "n": 6 }, "line_text": "29 OK / 6 stopped (2 Moser-irreducible + 4 eps^0) / 1 elliptic" }, "sources": { "engine": { "Normalize.jl": "4c5772a76f92d0d291a55c5427d55cc6e55798a1133d0d59a9fd4074bef54ea0", "Fuchsia.jl": "8a8c7369f13e5dcb7e27ecad8cd1bc2d2e07cde3a6d3bef5f8a1841347ef76f2" }, "runner_sha256": "6e497c73cda9a1da9c7d1ab3c1a11f2b0e18558c9be02ebed9ceacbc61b859d5", "verifier_sha256": "857543b36073fe1047e5e62ac16b78f0263dfb753f3d24d94876cf2e510f2a84", "out_files": "the engine's out file per block (blocks[*].source: file name + sha256)", "receipt_sha256_16": "a675806bd6068dec", "alphabet_census_sha256_16": "cd1b7bf207d0d5e0", "p1c_receipt_sha256_16": "875b7217badfe6e2", "live_receipt_sha256_16": "e9ba6a61677fdec5" }, "blocks": [ { "sector": 262, "n": 1, "rows": [ 0 ], "masters": [ "wpairT4[0,1,1,0,0,0,0,0,1]" ], "ok": true, "stop_class": "OK", "stop": "", "T": [ [ "1" ] ], "Atilde": [ [ "0" ] ], "letters": { "engine_string": "(String[], Dict{String, Any}(\"inf\" => [0]))", "poles": [], "alphabet_letters": {}, "algebraic_factors": [], "quadratic_alphabet_letter": null, "residues": {}, "Rinf": [ [ "0" ] ], "quad_kernels": {} }, "source": { "file": "out_b262.json", "sha256": "ccb64f8cd61655a4d9b74376f94e551e55c9ec6897c09266b29cdf2c7ee56fcb" } }, { "sector": 294, "n": 1, "rows": [ 1 ], "masters": [ "wpairT4[0,1,1,0,0,1,0,0,1]" ], "ok": true, "stop_class": "OK", "stop": "", "T": [ [ "1" ] ], "Atilde": [ [ "0" ] ], "letters": { "engine_string": "(String[], Dict{String, Any}(\"inf\" => [0]))", "poles": [], "alphabet_letters": {}, "algebraic_factors": [], "quadratic_alphabet_letter": null, "residues": {}, "Rinf": [ [ "0" ] ], "quad_kernels": {} }, "source": { "file": "out_b294.json", "sha256": "a76bbc5979b1b7c0f80a908438c27d450552d5f7d32c02700b52c9369b5f6cf3" } }, { "sector": 98, "n": 1, "rows": [ 2 ], "masters": [ "wpairT4[0,1,0,0,0,1,1,0,0]" ], "ok": true, "stop_class": "OK", "stop": "", "T": [ [ "1" ] ], "Atilde": [ [ "0" ] ], "letters": { "engine_string": "(String[], Dict{String, Any}(\"inf\" => [0]))", "poles": [], "alphabet_letters": {}, "algebraic_factors": [], "quadratic_alphabet_letter": null, "residues": {}, "Rinf": [ [ "0" ] ], "quad_kernels": {} }, "source": { "file": "out_b98.json", "sha256": "9a05dde16dbee3a658f620758dc37cd67a29688596164a3b1b1b31b6e979e29e" } }, { "sector": 321, "n": 2, "rows": [ 3, 4 ], "masters": [ "wpairT4[1,0,0,0,0,0,1,0,1]", "wpairT4[1,-1,0,0,0,0,1,0,1]" ], "ok": true, "stop_class": "OK", "stop": "", "T": [ [ "(-35//27//(eps^2 - eps + 2//9)*x^3 + (385//36*eps^2 - 385//36*eps + 35//18)//(eps^4 - 2*eps^3 + 11//9*eps^2 - 2//9*eps)*x^2 + 35//18//(eps^2 - eps + 2//9)*x + 35//108//(eps^2 - eps + 2//9))//(x^3 + 3*x^2 + 3*x + 1)", "(26//27//(eps^2 - eps + 2//9)*x^3 + (-364//27*eps^2 + 130//9*eps - 26//9)//(eps^4 - 2*eps^3 + 11//9*eps^2 - 2//9*eps)*x^2 + (26//27*eps)//(eps^3 - 2*eps^2 + 11//9*eps - 2//9)*x)//(x^3 + 3*x^2 + 3*x + 1)" ], [ "(35//9//(eps^2 - eps + 2//9)*x^3 + (-245//18*eps + 35//9)//(eps^3 - eps^2 + 2//9*eps)*x^2 - 35//18//(eps^2 - eps + 2//9)*x)//(x^4 + 5//4*x^3 - 9//4*x^2 - 17//4*x - 7//4)", "(-26//9//(eps^2 - eps + 2//9)*x^3 + (182//9*eps - 52//9)//(eps^3 - eps^2 + 2//9*eps)*x^2)//(x^4 + 5//4*x^3 - 9//4*x^2 - 17//4*x - 7//4)" ] ], "Atilde": [ [ "2//(x + 1)", "-104//35//(x + 1)" ], [ "105//26//(x + 1)", "(-5*x + 1)//(x^2 + x)" ] ], "letters": { "engine_string": "([\"x\", \"x - (-1)\"], Dict{String, Any}(\"0\" => [0 0; 0 1], \"inf\" => [-2 -105//26; 104//35 5], \"-1\" => [2 105//26; -104//35 -6]))", "poles": [ "0", "-1" ], "alphabet_letters": { "0": "y", "-1": "y + 1" }, "algebraic_factors": [], "quadratic_alphabet_letter": null, "residues": { "0": [ [ "0", "0" ], [ "0", "1" ] ], "-1": [ [ "2", "105/26" ], [ "-104/35", "-6" ] ] }, "Rinf": [ [ "-2", "-105/26" ], [ "104/35", "5" ] ], "quad_kernels": {} }, "source": { "file": "out_b321.json", "sha256": "f0b223d27b79ae5ead53d5d00cff1ce4bd739d847f630dc351a3f4369335329f" } }, { "sector": 322, "n": 2, "rows": [ 5, 6 ], "masters": [ "wpairT4[0,1,0,0,0,0,1,0,1]", "wpairT4[-1,1,0,0,0,0,1,0,1]" ], "ok": true, "stop_class": "OK", "stop": "", "T": [ [ "1", "0" ], [ "0", "1//(x - 7//4)" ] ], "Atilde": [ [ "0", "0" ], [ "0", "0" ] ], "letters": { "engine_string": "(String[], Dict{String, Any}(\"inf\" => [0 0; 0 0]))", "poles": [], "alphabet_letters": {}, "algebraic_factors": [], "quadratic_alphabet_letter": null, "residues": {}, "Rinf": [ [ "0", "0" ], [ "0", "0" ] ], "quad_kernels": {} }, "source": { "file": "out_b322.json", "sha256": "82e632dca4faebb8922a171a331fcb8296929ec6cad345e3d701fbbb609662bf" } }, { "sector": 326, "n": 3, "rows": [ 7, 8, 9 ], "masters": [ "wpairT4[0,1,1,0,0,0,1,0,1]", "wpairT4[-1,1,1,0,0,0,1,0,1]", "wpairT4[0,1,1,0,0,-1,1,0,1]" ], "ok": true, "stop_class": "OK", "stop": "", "T": [ [ "1", "(17//4*x + 51//8)//(x + 5//2)", "0" ], [ "0", "1//(x + 5//2)", "0" ], [ "0", "0", "1" ] ], "Atilde": [ [ "0", "0", "0" ], [ "0", "0", "0" ], [ "0", "0", "0" ] ], "letters": { "engine_string": "(String[], Dict{String, Any}(\"inf\" => [0 0 0; 0 0 0; 0 0 0]))", "poles": [], "alphabet_letters": {}, "algebraic_factors": [], "quadratic_alphabet_letter": null, "residues": {}, "Rinf": [ [ "0", "0", "0" ], [ "0", "0", "0" ], [ "0", "0", "0" ] ], "quad_kernels": {} }, "source": { "file": "out_b326.json", "sha256": "911c1fb11aa4c41875de6f92112fbe4c724d956f3a1ec8613a9c4b3f218ced5d" } }, { "sector": 327, "n": 2, "rows": [ 10, 11 ], "masters": [ "wpairT4[1,1,1,0,0,0,1,0,1]", "wpairT4[1,1,1,0,0,-1,1,0,1]" ], "ok": true, "stop_class": "OK", "stop": "", "T": [ [ "((1//3*eps - 1//3)//(eps - 1//3)*x^3 + 1//4*x^2 + (19//2*eps - 5//2)//(eps - 1)*x + (-7//4*eps - 7//4)//(eps - 1))//(x^2 + (25//4*eps - 11//4)//(eps - 1)*x + (21//4*eps - 7//4)//(eps - 1))", "((-5//12*eps + 5//12)//(eps - 1//3)*x^3 + (-7//48*eps + 5//48)//(eps - 1//3)*x^2 + (-39//4*eps + 15//8)//(eps - 1)*x + (21//16*eps + 21//16)//(eps - 1))//(x^2 + (25//4*eps - 11//4)//(eps - 1)*x + (21//4*eps - 7//4)//(eps - 1))" ], [ "(-2*x^2 + 2*x)//(x^2 + (25//4*eps - 11//4)//(eps - 1)*x + (21//4*eps - 7//4)//(eps - 1))", "(5//2*x^2 - 3//2*x)//(x^2 + (25//4*eps - 11//4)//(eps - 1)*x + (21//4*eps - 7//4)//(eps - 1))" ] ], "Atilde": [ [ "(-10*x^2 - 53*x - 127//4)//(x^3 + 6*x^2 + 129//16*x + 49//16)", "(45//4*x - 3//4)//(x^2 + x)" ], [ "-8//(x + 1)", "(9*x^3 + 41*x^2 + 343//16*x - 49//16)//(x^4 + 6*x^3 + 129//16*x^2 + 49//16*x)" ] ], "letters": { "engine_string": "([\"x\", \"x - (-1)\"], Dict{String, Any}(\"_algebraic_factors\" => [\"16*x^2 + 80*x + 49\", \"16*x^2 + 80*x + 49\"], \"0\" => [0 0; 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0 0]))", "poles": [], "alphabet_letters": {}, "algebraic_factors": [], "quadratic_alphabet_letter": null, "residues": {}, "Rinf": [ [ "0", "0" ], [ "0", "0" ] ], "quad_kernels": {} }, "source": { "file": "out_b354.json", "sha256": "2dbb950e708803135500e7f86b309e2fdb3d4ea7778b51b8ba29ca7c29e95005" } }, { "sector": 355, "n": 2, "rows": [ 15, 16 ], "masters": [ "wpairT4[1,1,0,0,0,1,1,0,1]", "wpairT4[1,1,-1,0,0,1,1,0,1]" ], "ok": true, "stop_class": "OK", "stop": "", "T": [ [ "(3145//132//(eps^2 - 1//3*eps)*x^3 + (248455//2904*eps - 292485//3872)//(eps^3 - 5//6*eps^2 + 1//6*eps)*x^2 + (50725705//23232*eps^2 - 65689615//46464*eps + 13639865//46464)//(eps^4 - 11//6*eps^3 + eps^2 - 1//6*eps)*x + (131558495//46464*eps^2 - 104838575//61952*eps + 53254285//185856)//(eps^4 - 11//6*eps^3 + eps^2 - 1//6*eps))//(x^2 + (-611//44*eps + 135//44)//(eps - 1)*x + (-655//44*eps + 179//44)//(eps - 1))", "(-2771//88//(eps^2 - 1//2*eps)*x^3 + (-2771//5808*eps^2 + 2097647//11616*eps - 257703//3872)//(eps^4 - 11//6*eps^3 + eps^2 - 1//6*eps)*x^2 + (-36413711//46464*eps^2 + 22896773//23232*eps - 12017827//46464)//(eps^4 - 11//6*eps^3 + eps^2 - 1//6*eps)*x + (-33454283//23232*eps^2 + 74514961//61952*eps - 46921343//185856)//(eps^4 - 11//6*eps^3 + eps^2 - 1//6*eps))//(x^2 + (-611//44*eps + 135//44)//(eps - 1)*x + (-655//44*eps + 179//44)//(eps - 1))" ], [ "(185//22//(eps^2 - eps)*x^3 + 120805//968//(eps^2 - eps)*x^2 + 2072925//3872//(eps^2 - eps)*x + 7957035//15488//(eps^2 - eps))//(x^2 + (-611//44*eps + 135//44)//(eps - 1)*x + (-655//44*eps + 179//44)//(eps - 1))", "(-163//22//(eps^2 - eps)*x^3 - 106439//968//(eps^2 - eps)*x^2 - 1826415//3872//(eps^2 - eps)*x - 7010793//15488//(eps^2 - eps))//(x^2 + (-611//44*eps + 135//44)//(eps - 1)*x + (-655//44*eps + 179//44)//(eps - 1))" ] ], "Atilde": [ [ "(x^4 - 531//44*x^3 - 15135//176*x^2 - 102511//704*x - 107517//1408)//(x^5 + 155//11*x^4 + 2889//44*x^3 + 10789//88*x^2 + 275273//2816*x + 78057//2816)", "(1793//370*x^2 + 9617//740*x + 104483//5920)//(x^4 + 51//4*x^3 + 777//16*x^2 + 3679//64*x + 1323//64)" ], [ "(-6105//326*x^2 - 32745//652*x - 355755//5216)//(x^4 + 51//4*x^3 + 777//16*x^2 + 3679//64*x + 1323//64)", "(-1//2*x^2 - 28*x + 155//32)//(x^4 + 51//4*x^3 + 777//16*x^2 + 3679//64*x + 1323//64)" ] ], "letters": { "engine_string": "([\"x - (-59//44)\", \"x - (-1)\", \"x - (-27//4)\"], Dict{String, Any}(\"-59//44\" => [-1 0; 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0 0 0 0; 0 0 0 0; -2//3 0 0 -2//3]))", "poles": [], "alphabet_letters": {}, "algebraic_factors": [ "16*x^2 + 80*x + 49", "16*x^2 + 80*x + 185" ], "quadratic_alphabet_letter": "x*y + y**2 - 2*y*z + z**2", "residues": {}, "Rinf": [ [ "8/3", "0", "0", "8/3" ], [ "0", "0", "0", "0" ], [ "0", "0", "0", "0" ], [ "-2/3", "0", "0", "-2/3" ] ], "quad_kernels": { "16*x^2 + 80*x + 49": { "U": [ [ "160/3", "0", "0", "-128/3" ], [ "0", "32", "0", "0" ], [ "0", "0", "0", "0" ], [ "32/3", "0", "0", "32/3" ] ], "V": [ [ "400/3", "0", "-204", "-320/3" ], [ "0", "80", "0", "0" ], [ "32/3", "0", "0", "-64/3" ], [ "80/3", "0", "-102", "80/3" ] ] }, "16*x^2 + 80*x + 185": { "U": [ [ "-96", "0", "0", "0" ], [ "0", "-32", "0", "0" ], [ "0", "0", "0", "0" ], [ "0", "0", "0", "0" ] ], "V": [ [ "-240", "-1020", "0", "0" ], [ "16", "-80", "0", "0" ], [ "0", "0", "0", "0" ], [ "0", "0", "0", "0" ] ] } }, "quadratic_alphabet_letters": { "16*x^2 + 80*x + 49": { "form": "x*y + y**2 - 2*y*z + z**2", "restriction_to_the_line_equals_q_up_to_the_integer_scale": true, "scale": "16", "note": "the served letter of the first quadratic (the alphabet letter whose restriction to this line is this factor)" }, "16*x^2 + 80*x + 185": { "form": "x*y + y**2 - 2*y*z + z**2 + x", "restriction_to_the_line_equals_q_up_to_the_integer_scale": true, "scale": "16", "note": "a restriction check of the two-variable form to this line only" } } }, "source": { "file": "out_b487_g.json", "sha256": "494baa8865deb75e511eb4a3a2a7891d52e65aa6e4b10e4c54e807018b7de6e7" }, "basis_change": { "their_masters": [ 1, 1, 1, -1, -1, 1, 1, 1, 1 ], "in_place_of": [ 1, 1, 1, -2, 0, 1, 1, 1, 1 ], "served_four": [ "wpairT4[1,1,1,0,0,1,1,1,1]", "wpairT4[1,1,1,-1,0,1,1,1,1]", "wpairT4[1,1,1,-2,0,1,1,1,1]", "wpairT4[1,1,1,0,-1,1,1,1,1]" ], "their_basis_order": [ "1,1,1,0,0,1,1,1,1", "1,1,1,-1,0,1,1,1,1", "1,1,1,0,-1,1,1,1,1", "1,1,1,-1,-1,1,1,1,1" ], "C_rows_in_their_basis_order_over_the_served_four": { "1,1,1,0,0,1,1,1,1": [ "1", "0", "0", "0" ], "1,1,1,-1,0,1,1,1,1": [ "0", "1", "0", "0" ], "1,1,1,0,-1,1,1,1,1": [ "0", "0", "0", "1" ], "1,1,1,-1,-1,1,1,1,1": [ "(-240*d*y**2 + 160*d*y + 2325*d + 960*y**2 - 640*y - 9300)/(592*d - 2080)", "(12*d*y - 599*d - 72*y + 2030)/(148*d - 520)", "(18*d - 54)/(37*d - 130)", "(20*d*y + 50*d - 80*y - 200)/(37*d - 130)" ] }, "C_variables": "(d, y): the cut-IBP dictionary rows as the P1 receipt writes them; y == x on the line, d = 4 - 2 eps in T", "det_C": "-18*(d - 3)/(37*d - 130)", "T_transcribed": [ [ "289*x/8 + 4335/32", "0", "17*x/2 + 85/4", "0" ], [ "119/16 - 17*x/4", "0", "17/4", "0" ], [ "0", "17/2", "0", "0" ], [ "0", "289/8", "0", "17/2" ] ], "T_transcribed_source": "the sympy matrix literal of p1c_make_blocks.py with r1 = r2 = 1 and the line's (X, Z) = (17/2, 7/4)", "S": "T = S = T_transcribed . C, the transformation FROM THE SERVED BASIS (written above as this block's T, column-major); det S = 751689*(2*eps - 1)*(16*x^2 + 80*x + 185)/(512*(37*eps - 9))", "identity_asserted_symbolically": "S A S^-1 + (dS/dy) S^-1 == eps * Atilde over Q(eps)(y), A the served block of rows 75-78 at d = 4 - 2 eps", "p1_receipt_sha256": "8aa0eccef9ecac1e0acb050f562606064e0accfe36130758f06d67d0d3a6fba0", "p1c_receipt_sha256": "875b7217badfe6e2540af6689817f094a05833abcbcb28e5ece5e9e6dfe500a8", "live_receipt_sha256": "e9ba6a61677fdec5f35bbab3439a58bb9e700e493b53fbdaa5206f7a5dce19f7", "engine_report": { "already_epsform": "true", "fuchsian_input": "true" }, "engine_t_factor_s": 7.5283238887786865, "engine_t_parse_s": 2.248685121536255, "engine_T_relative_to_the_g_block": "identity", "verifier_verdict_at_the_two_census_points": "PASS", "verifier_points": [ { "eps": "1/223", "d": "890/223", "y": "29/211", "identity_TAT_plus_dTT_eq_eps_Atilde": true, "Atilde_eps_free": true }, { "eps": "5/6", "d": "7/3", "y": "17/139", "identity_TAT_plus_dTT_eq_eps_Atilde": true, "Atilde_eps_free": true } ] }, "served_basis_stop": { "stop_class": "Moser-irreducible", "stop": "input not Fuchsian and Moser reduction failed: all higher-order poles are stuck (no kernel/cokernel shear reduces rank) \u2014 Moser-irreducible or needs full Barkatou pencil", "source": { "file": "out_b487.json", "sha256": "23f44cbab8419e5a2c5a34693d7a41b599d69274224ab545f44cdf3466954f7f" }, "note": "in the basis carrying [1,1,1,-2,0,1,1,1,1]" } } ], "sector_481": { "sector": 481, "n": 6, "rows": [ 54, 55, 56, 57, 58, 59 ], "masters": [ "wpairT4[1,0,0,0,0,1,1,1,1]", "wpairT4[1,-1,0,0,0,1,1,1,1]", "wpairT4[1,0,-1,0,0,1,1,1,1]", "wpairT4[1,0,0,-1,0,1,1,1,1]", "wpairT4[1,0,0,0,-1,1,1,1,1]", "wpairT4[1,-2,0,0,0,1,1,1,1]" ], "label": "sector 481 (rows 54-59, n = 6): the maximal-cut curve is elliptic (j non-constant in z5); the eps^0 block is reducible to first order -- six exact hyperexponential factors on this line, on the other served line and along the two mass-scaling rays; not run through the eps-factoriser (census.elliptic_not_run unchanged)", "maximal_cut_curve": { "statement": "(sector $481$) has an elliptic maximal-cut curve: its fibrewise curve has non-constant $j$ (an order-two Picard--Fuchs operator in the fibre variable $z_5$; $j$ non-constant along the mass-scaling ray as well), while its $\\eps^0$ differential-equation block is reducible to first order.", "source": "the served section text (the statement object STATEMENT_SEC481_20260910T021848Z.json, pair c_L213_is_elliptic, TO)" }, "eps0_block_chain": { "schema": "P2_REDUCIBILITY_v1", "recipe": "report_C3.md L135-142 (sys_factor): hyperexponential vector solutions prod f_i^{e_i} P, quotient by a rational gauge, iterate; L508-511 (blocktri)", "mode": "--served", "line": "L2", "stamp_utc": "2026-09-10T01:59:15Z", "stamp_end_utc": "2026-09-10T02:01:55Z", "script_of_record": { "file": "p2_reducibility.py", "sha256": "2eb65015d73bce5992bb8590e28724f9692caad36d54353d0d76106f5b053ef4", "vendored_as": "vendor_row35_sec481/scripts/p2_reducibility.py (as text; re-cut)", "vendored_sha256": "96e5690908a6c8856f14fa111732a15f5fe48c304bfd8e9946a9708fc4584ea4" }, "served_script": { "file": "qqww-t4-sec481-chain.py", "note": "the succession of the script of record served beside this file (pinned by the bundle MANIFEST.sha256, not here): --line L2 re-runs this chain on the served connection and compares it factor by factor with --reference" }, "bounds": { "EB": 3, "NB": 12, "MAXCOMBO": 20000 }, "sympy": "1.14.0", "python_flint": "0.8.0", "input": { "file": "row35_L2/connection_L2.json", "sha256": "fd83c63b685adb69364b936f26672543581ad0aa45d2f6f8d85d31458596d9bb", "rows": [ 54, 55, 56, 57, 58, 59 ], "masters": [ "wpairT4[1,0,0,0,0,1,1,1,1]", "wpairT4[1,-1,0,0,0,1,1,1,1]", "wpairT4[1,0,-1,0,0,1,1,1,1]", "wpairT4[1,0,0,-1,0,1,1,1,1]", "wpairT4[1,0,0,0,-1,1,1,1,1]", "wpairT4[1,-2,0,0,0,1,1,1,1]" ], "d": 4, "variable": "y", "note": "the SERVED L2 line block in y at d = 4 (L1: (x,z) = (17/2, 5/3); L2: (17/2, 7/4)); NOT the ray object" }, "verdict": "REDUCIBLE_TO_FIRST_ORDER", "size": 6, "remainder_rank": 0, "bounded_search_used": true, "n_factors": 6, "chain": [ { "exponents": [ [ "y + 1", "0" ], [ "y + 4", "0" ], [ "y - 7/4", "-1" ], [ "y + 27/4", "-1" ], [ "y**2 + 5*y + 49/16", "0" ], [ "y**2 + 5*y + 185/16", "0" ], [ "y**2 - 185*y/14 - 7663/784", "0" ], [ "y**2 + 325*y/14 + 63737/784", "0" ], [ "y**8 + 20*y**7 - 1475083*y**6/12544 - 44078245*y**5/12544 - 180993880329*y**4/9834496 - 113314866645*y**3/4917248 + 1070149070517*y**2/22478848 + 16990177326095*y/157351936 + 121236210296673/2517630976", "0" ] ], "degree_bound": 12, "P": [ "1517481/1493161", "145530*y**2/1493161 + 727650*y/1493161 + 6382033/5972644", "145530*y**2/1493161 + 727650*y/1493161 - 19589003/5972644", "-72765*y**2/1493161 - 3005695*y/5972644 + 2959257/2986322", "120491*y/175666 - 37793/87833", "y**2 + 5*y + 11509913/1405328" ], "n_solutions_at_this_exponent": 2, "rate_u_prime_over_u": "0", "rate_partial_fractions": "0", "exponents_note": "search labels; the canonical exponents are the residues of rate_u_prime_over_u (rate_partial_fractions)" }, { "exponents": [ [ "y + 1", "0" ], [ "y + 4", "0" ], [ "y - 7/4", "-1" ], [ "y + 27/4", "-1" ], [ "y**2 + 5*y + 49/16", "0" ], [ "y**2 + 5*y + 185/16", "0" ], [ "y**2 - 185*y/14 - 7663/784", "0" ], [ "y**2 + 325*y/14 + 63737/784", "0" ], [ "y**8 + 20*y**7 - 1475083*y**6/12544 - 44078245*y**5/12544 - 180993880329*y**4/9834496 - 113314866645*y**3/4917248 + 1070149070517*y**2/22478848 + 16990177326095*y/157351936 + 121236210296673/2517630976", "0" ] ], "degree_bound": 12, "P": [ "51*y**2/1147 + 255*y/1147 + 41701/192696", "51*y**2/1147 + 255*y/1147 + 45169/192696", "-51*y**2/2294 - 493*y/5208 - 697/9176", "289*y/13764 + 8381/192696", "y**4 + 10*y**3 + 46386301*y**2/1348872 + 63322505*y/1348872 + 903223289/43163904" ], "n_solutions_at_this_exponent": 1, "rate_u_prime_over_u": "0", "rate_partial_fractions": "0", "exponents_note": "search labels; the canonical exponents are the residues of rate_u_prime_over_u (rate_partial_fractions)" }, { "exponents": [ [ "y + 1", "0" ], [ "y + 4", "0" ], [ "y - 7/4", "-1" ], [ "y + 27/4", "-1" ], [ "y + 29/14", "-1" ], [ "y**2 + 5*y + 49/16", "0" ], [ "y**2 + 5*y + 185/16", "0" ], [ "y**2 - 185*y/14 - 7663/784", "-3/2" ], [ "y**2 + 325*y/14 + 63737/784", "0" ], [ "y**8 + 20*y**7 - 1475083*y**6/12544 - 44078245*y**5/12544 - 180993880329*y**4/9834496 - 113314866645*y**3/4917248 + 1070149070517*y**2/22478848 + 16990177326095*y/157351936 + 121236210296673/2517630976", "0" ] ], "degree_bound": 12, "P": [ "51*y**6/1147 - 47345*y**5/48174 + 7382981*y**4/4046616 + 622188355*y**3/14163156 + 123686569163*y**2/6345093888 - 3692899767785*y/88831314432 - 307242199435/12690187776", "51*y**6/1147 - 731*y**5/777 + 2432173*y**4/2023308 + 4840219991*y**3/113305248 + 268443179777*y**2/6345093888 - 1967166605849*y/88831314432 - 245190735089/11103914304", "-51*y**6/2294 + 93823*y**5/192696 - 1487891*y**4/2023308 - 5148722867*y**3/226610496 - 255053479739*y**2/12690187776 + 190700199583*y/50760751104 + 502128136519/101521502208", "y**8 - 1242718*y**7/72261 - 101566450*y**6/1517481 + 250931218987*y**5/226610496 + 112595410190551*y**4/19035281664 + 265149308101291*y**3/38070563328 - 806003171568101*y**2/240704206848 - 70812734223467683*y/8527806185472 - 51360637182617099/17055612370944" ], "n_solutions_at_this_exponent": 1, "rate_u_prime_over_u": "(-43904*y**2 + 294392*y + 557942)/(10976*y**3 - 122304*y**2 - 407722*y - 222227)", "rate_partial_fractions": "-84*(28*y - 185)/(784*y**2 - 10360*y - 7663) - 14/(14*y + 29)", "exponents_note": "search labels; the canonical exponents are the residues of rate_u_prime_over_u (rate_partial_fractions)" }, { "exponents": [ [ "y + 1", "0" ], [ "y + 4", "0" ], [ "y - 7/4", "-1" ], [ "y + 27/4", "-1" ], [ "y**2 + 5*y + 49/16", "0" ], [ "y**2 + 5*y + 185/16", "0" ], [ "y**2 - 185*y/14 - 7663/784", "-1" ], [ "y**2 + 325*y/14 + 63737/784", "0" ], [ "y**4 - 80*y**3/9 - 201559*y**2/3024 + 6027935*y/296352 + 2358485/42336", "-1" ], [ "y**8 + 20*y**7 - 1475083*y**6/12544 - 44078245*y**5/12544 - 180993880329*y**4/9834496 - 113314866645*y**3/4917248 + 1070149070517*y**2/22478848 + 16990177326095*y/157351936 + 121236210296673/2517630976", "0" ] ], "degree_bound": 12, "P": [ "-289*y**5/495 + 17629*y**4/2772 + 4501753*y**3/129360 - 409306943*y**2/3622080 - 59920308323*y/152127360 - 32021002613/152127360", "289*y**5/4620 - 15028*y**4/4851 + 15999329*y**3/1086624 + 5270859163*y**2/25354560 + 307754212541*y/1217018880 + 14069398271/173859840", "y**8 - 11209*y**7/945 - 16324685*y**6/232848 + 1417130005*y**5/3259872 + 260094201749*y**4/130394880 + 914936171*y**3/3697540 - 242142136156673*y**2/102229585920 + 11452100333741*y/4172636160 + 81191681123369/29208453120" ], "n_solutions_at_this_exponent": 1, "rate_u_prime_over_u": "(-1161699840*y**6 + 19147763712*y**5 - 2923545408*y**4 - 497236207104*y**3 - 790512238586*y**2 - 203719055092*y + 90717904220)/(232339968*y**7 - 4903110912*y**6 + 4398149056*y**5 + 239084503616*y**4 + 331410431226*y**3 - 115369870559*y**2 - 343740891990*y - 126511493885)", "rate_partial_fractions": "-56*(28*y - 185)/(784*y**2 - 10360*y - 7663) - (1185408*y**3 - 7902720*y**2 - 39505564*y + 6027935)/(296352*y**4 - 2634240*y**3 - 19752782*y**2 + 6027935*y + 16509395) + 1/(y + 1)", "exponents_note": "search labels; the canonical exponents are the residues of rate_u_prime_over_u (rate_partial_fractions)" }, { "exponents": [ [ "y + 4", "0" ], [ "y - 7/4", "-1" ], [ "y + 27/4", "-1" ], [ "y**2 + 5*y + 49/16", "0" ], [ "y**2 + 5*y + 185/16", "0" ], [ "y**2 - 185*y/14 - 7663/784", "-1" ], [ "y**2 + 325*y/14 + 63737/784", "-3/2" ], [ "y**3 + 65*y**2/28 - 7509*y/392 - 14459/392", "-1" ], [ "y**8 + 20*y**7 - 1475083*y**6/12544 - 44078245*y**5/12544 - 180993880329*y**4/9834496 - 113314866645*y**3/4917248 + 1070149070517*y**2/22478848 + 16990177326095*y/157351936 + 121236210296673/2517630976", "0" ] ], "degree_bound": 12, "P": [ "357*y**6/6044 + 5355*y**5/6044 - 7284075*y**4/676928 - 48915375*y**3/338464 - 251436785907*y**2/530711552 - 249462179535*y/530711552 - 24909248181/173293568", "y**7 + 35*y**6/2 - 11033699*y**5/74039 - 875605575*y**4/296156 - 14012478814541*y**3/928745216 - 62734077258115*y**2/1857490432 - 249949437770581*y/7429961728 - 170655267982405/14859923456" ], "n_solutions_at_this_exponent": 1, "rate_u_prime_over_u": "(-1686616064*y**7 - 22726290944*y**6 + 195691718656*y**5 + 2913792661248*y**4 + 7623343656824*y**3 - 13371894605874*y**2 - 70116635631420*y - 57897348125007)/(240945152*y**8 + 3932569088*y**7 - 43826202112*y**6 - 722938652352*y**5 - 1922940700632*y**4 + 7962098730782*y**3 + 45838486790363*y**2 + 67988285121465*y + 28248064270516)", "rate_partial_fractions": "-56*(28*y - 185)/(784*y**2 - 10360*y - 7663) - 84*(28*y + 325)/(784*y**2 + 18200*y + 63737) - (1176*y**2 + 1820*y - 7509)/(392*y**3 + 910*y**2 - 7509*y - 14459) + 1/(y + 4)", "exponents_note": "search labels; the canonical exponents are the residues of rate_u_prime_over_u (rate_partial_fractions)" }, { "exponents": [ [ "y - 7/4", "-1" ], [ "y + 27/4", "-1" ], [ "y**2 + 5*y + 49/16", "-1" ], [ "y**2 + 5*y + 185/16", "2" ], [ "y**2 - 185*y/14 - 7663/784", "-1" ], [ "y**2 + 325*y/14 + 63737/784", "-1" ], [ "y**8 + 20*y**7 - 1475083*y**6/12544 - 44078245*y**5/12544 - 180993880329*y**4/9834496 - 113314866645*y**3/4917248 + 1070149070517*y**2/22478848 + 16990177326095*y/157351936 + 121236210296673/2517630976", "1" ] ], "degree_bound": 0, "P": [ "1" ], "n_solutions_at_this_exponent": 1, "rate_u_prime_over_u": "(25353862925258850304*y**17 + 1077539174323501137920*y**16 + 6661037245714044813312*y**15 - 288980690447473888460800*y**14 - 4892503759209832648802304*y**13 - 8577833353219626898554880*y**12 + 411963079312697814499196928*y**11 + 4451268581022461773500907520*y**10 + 20348831220706251775942852608*y**9 + 38194546625569378928746823680*y**8 - 31745724791346849768680849408*y**7 - 272105284727651042629000560640*y**6 - 375047607388115369199224905728*y**5 + 172282253967932544394274918400*y**4 + 909858231099282325595232982528*y**3 + 833977840852355342786619208960*y**2 + 284949350926015295640018511392*y + 26402451210811134668740374480)/(6338465731314712576*y**18 + 285230957909162065920*y**17 + 2376813988847995060224*y**16 - 66558316594605368279040*y**15 - 1397635784957919104073728*y**14 - 5655287300391367585300480*y**13 + 91352490209279786979688448*y**12 + 1372540082039730902338109440*y**11 + 8482105474431016049603248128*y**10 + 27488180405740363319292723200*y**9 + 34197411924612868253857349632*y**8 - 73538654433909900122899087360*y**7 - 355001746046170831706697949184*y**6 - 468958294639805778109160693760*y**5 + 124450809680441322989174964736*y**4 + 1033228405935453231627260687360*y**3 + 1165298573884540944668473977248*y**2 + 560162948077336088629745694240*y + 101450083445868192103012787955)", "rate_partial_fractions": "16*(2*y + 5)*(629407744*y**6 + 9441116160*y**5 - 79113113856*y**4 - 1184510978560*y**3 - 2830526724128*y**2 + 1637203211360*y + 3398035465219)/(2517630976*y**8 + 50352619520*y**7 - 296055058432*y**6 - 8846680084480*y**5 - 46334433364224*y**4 - 58017211722240*y**3 + 119856695897904*y**2 + 271842837217520*y + 121236210296673) + 32*(2*y + 5)/(16*y**2 + 80*y + 185) - 16*(2*y + 5)/(16*y**2 + 80*y + 49) - 56*(28*y - 185)/(784*y**2 - 10360*y - 7663) - 56*(28*y + 325)/(784*y**2 + 18200*y + 63737) - 4/(4*y + 27) - 4/(4*y - 7)", "exponents_note": "search labels; the canonical exponents are the residues of rate_u_prime_over_u (rate_partial_fractions)" } ], "search_log": [ { "size": 6, "singular_factors": [ [ "y + 1", 1 ], [ "y + 4", 1 ], [ "y - 7/4", 1 ], [ "y + 27/4", 1 ], [ "y**2 + 5*y + 49/16", 1 ], [ "y**2 + 5*y + 185/16", 1 ], [ "y**2 - 185*y/14 - 7663/784", 1 ], [ "y**2 + 325*y/14 + 63737/784", 1 ], [ "y**8 + 20*y**7 - 1475083*y**6/12544 - 44078245*y**5/12544 - 180993880329*y**4/9834496 - 113314866645*y**3/4917248 + 1070149070517*y**2/22478848 + 16990177326095*y/157351936 + 121236210296673/2517630976", 1 ] ], "pole_order_at_infinity": 3, "fuchsian": false, "candidate_exponents": [ [ "0", "1" ], [ "0", "1" ], [ "-1", "0" ], [ "-1", "0" ], [ "0" ], [ "0", "2" ], [ "-3/2", "0" ], [ "-3/2", "0" ], [ "0", "1" ] ], "infinity_rational_exponents": "irregular at infinity: degree bound NB", "n_combinations": 256, "combinations_tried": 7 }, { "size": 5, "singular_factors": [ [ "y + 1", 1 ], [ "y + 4", 1 ], [ "y - 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