{ "row": 35, "lines": { "L1": { "x": "17/2", "z": "5/3" }, "L2": { "x": "17/2", "z": "7/4" } }, "what": "the constant gauge relating the two lines' eps-factorised block forms letter by letter: for each of the 29 blocks factorised on both lines, every rational pole of the L1 form is matched to its alphabet letter and to the L2 pole of the same letter (restriction_map: L1 pole -> [letter, L2 pole, present]); G is a constant invertible matrix with G R1 G^-1 == R2 on every matched letter and at infinity (residue_pairs; G = identity where the residues are identical); the six stopped blocks stop with the same string on both lines", "test": "for each block OK on both lines: the engine's dlog residue matrices R_p at the rational poles p of Atilde and at infinity (parsed exactly from the dlog_alphabet string); every L1 pole matched to its alphabet letter through vendored/probe3var/ALPHABET_LINE_CENSUS.json (L1_banked_x17o2_z5o3.per_letter), the L2 pole of the SAME letter predicted from the L2 restriction (L2_shiftz_x17o2_z7o4.per_letter) and required present, no unmatched L2 pole; then (i) the characteristic polynomials of R_p equal letterwise, (ii) an exact search for a constant invertible G with G R1_p = R2_p G for all matched p and infinity: the nullspace of the linear system in the n^2 entries of G by exact rational row reduction, an invertible element sought on an integer grid (a finite search: exists=False would not prove absence; residues identical -> G = identity); the irreducible quadratic letter is matched by its polynomial (18y^2+93y+50 <-> 16y^2+80y+49), not by residue; the rational-pole partial-fraction identity Atilde == sum_p R_p/(y-p) checked exactly at two rational y on the blocks without an algebraic factor; stopped blocks: stop strings compared", "form": { "G": "row-major exact rationals; det_G and nullspace_dim (the dimension of the space of constant G solving G R1 = R2 G) in transformation for the five nontrivial blocks", "residue_pairs": "label ' -> [letter]' (the poles as the engine writes them, p//q); L1_pole / L2_pole the same poles as p/q ('inf' at infinity), R1 the L1 residue, R2 the L2 residue (row-major exact rationals), equal (R1 == R2), G_R1_Ginv_equals_R2 (re-verified exactly at build)" }, "counts": { "n_ok_both": 29, "n_constant_gauge_exists": 29, "n_residues_identical": 24, "n_charpolys_equal": 29, "nontrivial_G_blocks": [ 321, 327, 355, 227, 485 ], "identity_G_blocks": 24, "nullspace_dim_histogram": { "1": 19, "2": 1, "4": 5, "9": 3, "10": 1 }, "Rinf_conjugated_not_equal_blocks": [ 321, 327, 485 ], "n_stopped_both": 6, "n_stop_string_equal": 6, "residue_pairs": 51, "residue_pairs_equal": 38, "residue_pairs_conjugated_verified": 51 }, "sources": { "receipt_sha256_16": "a675806bd6068dec", "compare_receipt_sha256_16": "ef446b1058e1ab79", "alphabet_census_sha256_16": "cd1b7bf207d0d5e0", "blocks_epsform": { "row35_L1/blocks_epsform.json": "edbc1f375f48699bf32231cbd0111febe0f5737a8df212e238b1a32222a596e3", "row35_L2/blocks_epsform.json": "90bc8a28c9fa2cd85b1c38f75d5c5f7253fa58bba69b809c21685f2e42d025b8" } }, "blocks": { "262": { "n": 1, "L1_poles": [], "L2_poles": [], "L1_quadratic": [], "L2_quadratic": [], "restriction_map": {}, "L2_poles_without_partner": [], "residues_identical_letterwise": true, "charpolys_equal_letterwise": true, "constant_gauge_exists": true, "nullspace_dim": 1, "G": [ [ "1" ] ], "pf_certificate_L1_L2": [ true, true ], "residue_pairs": { "inf -> inf [infinity]": { "L1_pole": "inf", "L2_pole": "inf", "letter": "infinity", "R1": [ [ "0" ] ], "R2": [ [ "0" ] ], "equal": true, "G_R1_Ginv_equals_R2": true } } }, "294": { "n": 1, "L1_poles": [], "L2_poles": [], "L1_quadratic": [], "L2_quadratic": [], "restriction_map": {}, "L2_poles_without_partner": [], "residues_identical_letterwise": true, "charpolys_equal_letterwise": true, "constant_gauge_exists": true, "nullspace_dim": 1, "G": [ [ "1" ] ], "pf_certificate_L1_L2": [ true, true ], "residue_pairs": { "inf -> inf [infinity]": { "L1_pole": "inf", "L2_pole": "inf", "letter": "infinity", "R1": [ [ "0" ] ], "R2": [ [ "0" ] ], "equal": true, "G_R1_Ginv_equals_R2": true } } }, "98": { "n": 1, "L1_poles": [], "L2_poles": [], "L1_quadratic": [], "L2_quadratic": [], "restriction_map": {}, "L2_poles_without_partner": [], "residues_identical_letterwise": true, "charpolys_equal_letterwise": true, "constant_gauge_exists": true, "nullspace_dim": 1, "G": [ [ "1" ] ], "pf_certificate_L1_L2": [ true, true ], "residue_pairs": { "inf -> inf [infinity]": { "L1_pole": "inf", "L2_pole": "inf", "letter": "infinity", "R1": [ [ "0" ] ], "R2": [ [ "0" ] ], "equal": true, "G_R1_Ginv_equals_R2": true } } }, "321": { "n": 2, "L1_poles": [ "-1", "0" ], "L2_poles": [ "-1", "0" ], "L1_quadratic": [], "L2_quadratic": [], "restriction_map": { "-1": [ "y + 1", "-1", true ], "0": [ "y", "0", true ] }, "L2_poles_without_partner": [], "residues_identical_letterwise": false, "charpolys_equal_letterwise": true, 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"all_conjugations_verified": true, "conjugated_not_equal": [ "-1 -> -1 [y + 1]", "0 -> 0 [y]", "inf -> inf [infinity]" ], "equal_letters": [] }, "residue_pairs": { "-1 -> -1 [y + 1]": { "L1_pole": "-1", "L2_pole": "-1", "letter": "y + 1", "R1": [ [ "-6", "-50/23" ], [ "138/25", "2" ] ], "R2": [ [ "2", "105/26" ], [ "-104/35", "-6" ] ], "equal": false, "G_R1_Ginv_equals_R2": true }, "0 -> 0 [y]": { "L1_pole": "0", "L2_pole": "0", "letter": "y", "R1": [ [ "4", "50/23" ], [ "-138/25", "-3" ] ], "R2": [ [ "0", "0" ], [ "0", "1" ] ], "equal": false, "G_R1_Ginv_equals_R2": true }, "inf -> inf [infinity]": { "L1_pole": "inf", "L2_pole": "inf", "letter": "infinity", "R1": [ [ "2", "0" ], [ "0", "1" ] ], "R2": [ [ "-2", "-105/26" ], [ "104/35", "5" ] ], "equal": false, "G_R1_Ginv_equals_R2": true } } }, "322": { "n": 2, "L1_poles": [], "L2_poles": [], "L1_quadratic": [], "L2_quadratic": [], "restriction_map": {}, "L2_poles_without_partner": [], "residues_identical_letterwise": true, 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