The answer is \[ \max |A|=9. \] A witness is \[ A=\{1,11,47,62,67,70,71,84,96\}. \] Its 45 unordered pair sums with repetition are distinct in \(\mathbb F_{101}\), and its 45 corresponding products are also distinct.

For the upper bound, 2 generates \(\mathbb F_{101}^{*}\), which has order 100. Write every element of a product-Sidon set as \(2^b\), with \(b\in\mathbf Z/100\mathbf Z\). Distinct pair products are equivalent to distinct pair sums of the exponents. This is also equivalent to requiring all directed differences \(b_i-b_j\), for \(i\ne j\), to be distinct. Thus a size-10 product-Sidon set would give a \((100,10)\)-modular Golomb ruler.

Buratti and Stinson's complete classification through order 11 states that \((v,10)\)-modular Golomb rulers do not exist for \(92\leq v\leq106\). In particular, there is no \((100,10)\)-ruler. The executable artifact independently exhausts the case \(v=100\). Since any simultaneous set must satisfy the product condition, its size is at most 9. The displayed set attains that bound.

Erdős and Turán's 1941 paper is the classical source for \(B_2\) sequences defined by unique two-term sums. Bose and Chowla's 1962 paper develops finite cyclic \(B_2\) constructions. These sources treat a single additive group law.

For the multiplicative half of the present problem, discrete logarithms identify \(\mathbb F_{101}^{*}\) with \(\mathbf Z/100\mathbf Z\). Buratti and Stinson define a \((v,k)\)-modular Golomb ruler by distinct directed differences modulo \(v\). Their Table 1 records nonexistence for \(k=10\) throughout \(92\leq v\leq106\), which includes \(v=100\). This supplies a published upper bound of 9 before the additive condition is imposed.

The focused audit found no primary source for the exact requirement that one subset of \(\mathbb F_{101}^{*}\) be Sidon under both field addition and field multiplication. The claim of novelty therefore remains cautious. The exact answer here combines the published one-law obstruction with a directly checked simultaneous witness.

Multiplying a field set by a nonzero scalar preserves both Sidon conditions. On exponents this translates the set in \(\mathbf Z/100\mathbf Z\), so any hypothetical ten-mark set may be normalized to contain 0. The depth-first search lists its remaining marks in increasing order.

At each node, two 100-bit masks record the directed differences already used. Appending a mark \(x\) is allowed exactly when the residues \(x-b\) and \(b-x\), for every earlier mark \(b\), are new and mutually distinct. The only pruning step checks whether enough larger residues remain. The program therefore visits every normalized candidate that could extend to ten marks.

The depth-node totals for selected-set sizes 1 through 10 are \[ 1,90,3926,103470,1643822,13706516,43175690,25432298,526256,0. \] Their sum is 84,592,069. No node reaches size 10. The same program evaluates all unordered pairs of the displayed field witness and obtains two images of size 45. Its exact standard output has SHA-256 digest `3133bddeeb7e4795c774e85e9b699aaf940ec5e0009ba73ab9860a4113cf4e0d`.