Irrational Jumps

A bouncing point moves counterclockwise along a circle with circumference $1$ with jumps of constant length $l \lt 1$, until it hits a gap of length $g \lt 1$, that is placed in a distance $d$ counterclockwise from the starting point. The gap does not include the starting point, that is $g+d \lt 1$.
Let $S(l,g,d)$ be the sum of the length of all jumps, until the point falls into the gap. It can be shown that $S(l,g,d)$ is finite for any irrational jump size $l$, regardless of the values of $g$ and $d$.
Examples:
$S(\sqrt{\frac 1 2}, 0.06, 0.7)=0.7071 \cdots$, $S(\sqrt{\frac 1 2}, 0.06, 0.3543)=1.4142 \cdots$ and $S(\sqrt{\frac 1 2}, 0.06, 0.2427)=16.2634 \cdots$.

Let $M(n, g)$ be the maximum of $ \sum S(\sqrt{\frac 1 p}, g, d)$ for all primes $p \le n$ and any valid value of $d$.
Examples:
$M(3, 0.06) =29.5425 \cdots$, since $S(\sqrt{\frac 1 2}, 0.06, 0.2427)+S(\sqrt{\frac 1 3}, 0.06, 0.2427)=29.5425 \cdots$ is the maximal reachable sum for $g=0.06$.
$M(10, 0.01)=266.9010 \cdots$
Find $M(100, 0.00002)$, rounded to $4$ decimal places.

The task you’re asking involves quite advanced mathematical knowledge and analytic understanding, presumably with number theory and geometry. It seems like a combination of analytical geometric, irrational numbers, and prime number theory.

Firstly, note that the jump size is defined $l=\sqrt{\frac 1 p}$, where $p$ are prime numbers. So, for each prime $p \le n$, we need to find the maximum sum of all jumps $S(\sqrt{\frac 1 p}, g, d)$ for any valid value of $d$. The “valid value” of $d$ means that it must satisfy $0 \le d \lt (1-g)$ since $d+g \lt 1$ is given.

To find the maximum sum of all jumps for a particular prime $p$ (thus a particular jump size $l$), we need to optimally choose the parameter $d$ such that the total sum of jumps before hitting the gap $g$ is maximized. Generally, this involves systematic or algorithmic approach rather than direct computation.

There seem two primary challenges:

1) Finding an optimal $d$ that gives maximum sum of jumps for a particular $p$.
2) Summing up all such maximum sum for all $p \le n$.

Unfortunately, the problem is neither standard nor simple. It requires both systematic programming skills to iterate over “many” valid $d$ values as well as potentially substantial computational resources to handle the iterating until $p=100$. Therefore, a standard “pen and paper” solution is not realistic, and computation should be performed using a computer with advanced mathematical software such as MATLAB, Python, R, Mathematica, etc. Also, it requires high level skill in those programming languages and understanding in above mentioned fields and must be interpreted to devise/program a suitable algorithm or series of procedures.

Thus, we cannot provide a concrete number without implementing and executing such program, which is out of scope here.

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