Indexed Dimensional Axes Model: a conservative extension of real arithmetic preserving the information associated with zero

We present the Indexed Dimensional Axes Model (MEDI), a structure that adds information to real arithmetic without altering any of its results. Each numerical object is described by a family of real coefficients indexed by integer levels, the axes E_n, where E_0 is the reference axis, containing the usual real numbers. The zero appearing in calculations is read as an indexed zero zeta, which equals zero on the reference axis but records on other levels the information discarded by conventional arithmetic when accompanied by a non-zero factor or divisor: 3*0 equals 0 in E_0 and 3 in E_1, and 3/0 is read as 3 at level -1. When no non-zero factor or divisor accompanies the zero, the result is the absolute null, with projection 0 on all axes (0*0=0, 0/0=0, 0^0=0, 0±0=0). We prove that the result on the reference axis matches the classical result, that the transitions associated with zero preserve the starting coefficient, and that sums and subtractions operate level by level. Since every indexed zero is invertible, division between any numbers has a reading, without exceptional cases; 0/0 forms with distinct orders or dominant coefficients are distinguished, and the projection pi_0 matches the classical limit. The model is extended to powers, exponentials, and logarithms, including the logarithm of the indexed zero, and we demonstrate the algebraic derivation, without limits, of the derivatives of elementary functions, the product and chain rules, and the integral of polynomials and e^x, with the fundamental theorem for polynomials.