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3h

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Definition of short proof
In fact, if "there are no short proofs of tautology" then coNP != NP. $\;$ 
6h

awarded  Good Question 
21h

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Random Diophantine polynomials: Percent solvable?
Presumably $\: \log C/C \:$ should be replaced with $\: (\log C)/C \;$. $\;\;\;\;$ 
Apr 19 
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is there a global obstruction for a diffeomorphism to be an isometry?
$x\mapsto x^3 \;$ is not a diffeomorphism from $\mathbf{R}$ to itself, since the inverse of that map is not differentiable at zero. $\;\;\;\;$ 
Apr 16 
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Rate of convergence in the Law of Large Numbers
You'll also need $\: \mathbb{E}X_1\hspace{0.02 in} < \infty \:$ in order to reach the conclusions you claim from the laws of large numbers. $\;\;\;$ 
Apr 4 
revised 
Why is differential Galois theory not widely used?
fixed title's grammar 
Apr 1 
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How large do algebraic representations need to be for packing circles in squares?
I mean "at most $\:r1\:$ from the origin" instead of $\:r+1\;$. $\;\;\;$ I believe the rest of my two previous comments are correct.) $\;\;\;\;\;\;\;$ 
Apr 1 
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How large do algebraic representations need to be for packing circles in squares?
If there is a bound on the bitlengths of the coefficients that is better than $n^{O(n)}$, then that would yield a better bound for my problem (in particular, enough for the bounty), since it would provide an explicit constant for the main factor of the runtime. $\:$ (i.e., $n^{\hspace{.02 in}O(n)}$ becomes $\: n^{\hspace{.02 in}c\cdot n} \cdot \text{something_asymptotically_smaller_than_that} \;$.) $\;\;\;\;\;\;\;$ 
Apr 1 
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How large do algebraic representations need to be for packing circles in squares?
Essentially, "there are $\:2\cdot n\:$ reals such that the $n$ points described by those reals are each at most $\:r+1\:$ from the origin and at least $2$ from each other". $\;\;\;$ Thus, $s$ would be quadratic in $n$. $\:$ Do you know where they address the sizes of the coefficients produced by their global optimization algorithm? $\;\;\;\;\;\;\;$ 
Apr 1 
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How large do algebraic representations need to be for packing circles in squares?
I'm looking at Exercise 11.9 from the "For downloading the .pdf file of the book" link on the page you linked to, and as far as I can see, that will only improve on the result I cited when $\ell$ is $(\Omega(1))^k$ (using the notation of Algorithm 11.16). $\;$ 
Mar 31 
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Proof that no differentiable spacefilling curve exists
@PabloShmerkin : $\:$ How can one show that? $\;\;\;\;$ 
Mar 30 
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Proof that no differentiable spacefilling curve exists
However, a differentiable map (even on a compact interval) is not necessarily Lipschitz. $\hspace{1.27 in}$ 
Mar 27 
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Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?
@npbool : $\;\;\; \sin\hspace{0.04 in}\Big(\hspace{0.03 in}\frac{\pi}2\hspace{0.03 in}\Big) = 1 \not< 1 \;\;\;\;\;\;\;\;\;$ 
Mar 23 
asked  How large do algebraic representations need to be for packing circles in squares? 
Mar 13 
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Does Langton's ant cover every n by 6 gridded torus?
I'm not sure how much this would help, but one could try splitting the grid into congruent pieces and computing a lookup table with [after_colors,exit_edge] for each possible [entry_edge,before_colors]. $\:$ (Such a table could presumably be compressed, and might reveal that some aprioripossible color configurations of the pieces are not actually possible.) $\;\;\;\;$ 
Mar 13 
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Does Langton's ant cover every n by 6 gridded torus?
That would give $\: 24\cdot n_{\hspace{.02 in}0} \cdot 2^{6\cdot n_{\hspace{.02 in}0}} \:$, $\:$ not $\: 4.6n_0.2^{6n_0} \;$. $\;\;\;\;$ 
Mar 13 
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Does Langton's ant cover every n by 6 gridded torus?
Where does 4.6 come from (in your upper remark)? $\;$ 
Mar 9 
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Does $X_n \xrightarrow{d} N(0,1)$ and $X_n/Y_n \xrightarrow{d} N(0,1)$ imply that $Y_n \xrightarrow{d} 1$?
Do you assume that $Y_n$ is independent of $X_n$? $\;$ 
Mar 4 
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Balancing real numbers in one dimension
How can one see that for $\: m = 1 \:$ and $\: d_1 = 3 \;$? $\;\;\;\;$ 
Mar 2 
awarded  Popular Question 