No linear-overhead compilation of the pointer machine into the tape machine

The real pointer machine of Definitions.Def_SmaleNinth_RealRAM cannot be simulated by the fixed-address tape machine of Definitions.Def_SmaleNinth_BSSMachine with only a constant-factor loss of time.

Precisely, there is no assignment $R \mapsto (P_R, K_R)$ of a tape program and a constant to every pointer-machine program such that, whenever $R$ decides an input $x$ (vanishing on the negative cells, so that the tape machine has scratch space) with answer $b$ within $T$ steps, $P_R$ decides the same input with the same answer within $K_R (T+1)$ steps.

The counterexample is one-variable linear programming feasibility. On the pointer machine it is decided in $O(m)$ steps, because the coefficient $a_i$ and the right-hand side $b_i$ of a constraint are reached through two independently incremented pointers (SmaleNinth.real_ram_decides_one_variable_lp_linear). On the tape machine no program decides it in $O(m)$ steps: in the encoding encodeLP the two halves of a constraint sit $m$ cells apart, and a crossing-sequence argument over $\mathbb{R}$ forbids the linear budget (SmaleNinth.bss_one_variable_lp_no_linear_program). A linear-overhead compiler applied to the first program would produce a tape program of the forbidden kind.

The statement is the exact sharpness counterpart of SmaleNinth.ram_to_bss_quadratic_nonneg, which achieves overhead $K(T+1)^2$: the quadratic loss there is a genuine feature of the two models, not a defect of that construction. It also justifies stating Smale's ninth problem in the random-access form SmaleNinth.smale_ninth_real_ram, since the two models are provably not interchangeable at the level of constant-factor time.