A table from even numbers under the map x ↦ → 2x − 3. Part III. The coordinate grid of odd numbers

We continue the study of the iteration table T_{n,k} = f^k(2n), f(x) = 2x - 3, introduced in Part I and extended in Part II. In Parts I and II the table was built left to right: starting from an even seed 2n, the iterates of f generate the row, with closed form T_{n,k} = 3 + 2^k(2n - 3). In the present part we look at the same table in the reverse direction. The reverse process is defined by the inverse map g(x) = (x + 3)/2, which is the inverse of f on odd integers. Given an odd m ≠ 3, we apply g repeatedly until the result becomes even. The number of steps k gives the column index, and the final even number 2n gives the row index n. The original m is then recovered by the closed form T_{n,k} = 3 + 2^k(2n - 3) = m. We call this the descent algorithm. The table naturally splits into two parts: the even column k = 0, serving as seeds of rows, and the odd columns k ≥ 1, filling those rows. Thus the table realises a coordinate grid for odd numbers: every odd number m ≠ 3 occurs exactly once in some column k ≥ 1 and row n, and the descent algorithm computes the coordinates (n, k) constructively in exactly v_2(m - 3) steps. The map is proved to be a bijection between odd integers (excluding 3) and pairs (n, k) with k ≥ 1. The exceptional point 3 is the only odd integer without coordinates: it is the fixed point of f and the axis of the mirror symmetry of Part II. The key property is that the location of an odd number in the table depends only on the 2-adic structure of the difference m - 3 and not on the size of m: a small odd number may sit in a distant column, and an arbitrarily large one may sit in the first column. We give several worked examples (including 67, 137, the Fermat number 5, the Mersenne number 7, and the large number 2^{100} + 3 with coordinates (n, k) = (2, 100)) and provide complete Python code implementing the descent algorithm. The main results of this part are: a descent theorem establishing correctness and finiteness of the algorithm; a coordinate bijection between odd integers (excluding 3) and pairs (n, k) with k ≥ 1; and an independence-of-size principle showing that neither the row nor the column is bounded by the magnitude of the input. The material continues the sequence of Parts I and II and prepares the ground for further study of the arithmetic of the table.