Henry Cohn made an interesting parallel with equiangular line arrangements in complex space: "These two problems are finely balanced between order and disorder. Any Hadamard matrix or equiangular line configuration must have considerable structure, but in practice they frequently seem to have just enough structure to be tantalizing, without enough to guarantee a clear construction. This contrasts with many of the most symmetrical mathematical objects, which are characterized by their symmetry groups: once you know the full group and the stabilizer of a point, it is often not hard to deduce the structure of the complete object. That seems not to be possible in either of these two problems, and it stands as a challenge to find techniques that can circumvent this difficulty." from "Order and disorder in energy minimization" On Oct 7, 2012, at 3:23 PM, Warren Smith <warren.wds@gmail.com> wrote:
I don't know if I'd agree with NJA Sloane the evidence for it is "overwhelming" but it is pretty good.
I find http://oeis.org/A007299 unconvincing since it has so few entries. And I have strong doubts this sequence is monotonic.
You know Laplace said the probability the sun rises tomorrow based on fact it was true N previous days, is (N+1)/(N+2).
The probability the Hadamard conjecture is true is in my view about 99%.
It previously was conjectured that Williamson-form Hadamard matrices always existed. True for n=1,2,3,...,34. But false for n=35 (no 140x140 Hadamard of Williamson form exists, 140=4*35). So this should serve as a cautionary tale.
I suggest you solve my recently posted question about highly cancelling polynomials. If you do I will be able with probability 99% to make very strong progress on the Hadamard conjecture.
(In fact in unpublished work I already did make strong progress on the Hadamard conjecture...)
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