What is the probability that this constant corresponds to real life?

Question

I was surprised last week to find this value:
$m_{p_{0000}}=0.16726218229590580987863882056891582636342622102204{}^{1}$
$m_{p_{1959}}=0.167339(31){}^{2}$
$m_{p_{2012}}=0.1672621777(74){}^{3}$
for a formula I had designed for a completely unrelated purpose.

$$\sum _{m=1}^{\infty } \frac{1}{\left(m^2+1\right){}_{2 m}}$$ where $\left(m^2+1\right){}_{2 m}$ is the Pochhammer symbol.

I posted over onPhysics.SE where they beat me up over the definition of mass. I'm not a physicist so it was a bit confusing. But, I don't want to discuss proton mass in this post. I want to discuss the odds of this happening at all!

We have $0.167$ as $3$ digits of accuracy for $1959$ and $0.1672621$ as $7$ digits of accuracy for $2012$. At $10$ numbers per each of the $4$ digits difference, we beat $9999$ to $1$ odds against that happening.

Is there a point after a number of successful steps that we would have some kind of confidence factor that would indicate that the number corresponds to the proton's mass or not?

Footnotes
1: Mathematica code:
NSum[10/Pochhammer[m^2 + 1, 2 m], {m, 1, [Infinity]}, WorkingPrecision -> 50]
2: CRC Handbook for Chemistry and Physics, 12th edition, (1959), Page 16.
3: http://pdg.lbl.gov/2012/reviews/rpp2012-rev-phys-constants.pdf

Clarification

The number $m_{p_{0000}}$ is a percentage. When we multiply the length of the sides of the gram by 10 and then cube it to get kg, we don't cube the number. We change its exponent by 3. Therefore, the number is dimensionless.

I've added a bounty for the best answer that is back on topic.

Answer 1

The threshold in Physics is colloquially known as "five sigma." See, for example, this article, which might resolve some of your underlying confusion. Note that it is important to go into an experiment with a well-supported hypothesis; you should not (in general) first get the result and then go back and see if it corresponds to something known.

For example, your first number begins: $0.1672621822\ldots$

Next, let us consider $15357 \pi/288442 = 0.1672621822\ldots$

Neither mathematical nor physical reasoning allows us to conclude whether or not they are equal.

In any event, to answer your question directly: there is no chance the constant you have found corresponds to the physics term of interest, and with quantities that can be measured very well, you would need both (1) digits matching as far as they can be computed, and (2) a convincing reason [where the meaning of convincing depends on your discipline] for why they should be equal.