Equate the Sum to the Annihilated Combined Space

 

Step-by-Step Derivation of the Equation

The equation in question is:

$$\sum_{i=1}^{Z_x} RM_i = 0 \cdot RM\left( \sum_{i=1}^{Z_x} RM_i \right)$$

This appears to be a custom or theoretical expression describing the summation of space-time delays and generations (denoted as $RM_i$), leading to total annihilation or cancellation. Based on the provided context—where the left side represents the total generated spaces and the right side expresses annihilation via an absorbing zero applied to the combined space—here is a structured, step-by-step derivation. I'll assume a mathematical framework where $RM$ is an operator or function that combines or processes space-time contributions, and the equation formalizes a principle of complete neutralization.

Step 1: Define the Individual Components

• Let $RM_i$ represent the $i$-th contribution to space-time, where each $RM_i$ could be a delay (retardement) or generation term.
• The index $i$ ranges from 1 to $Z_x$, where $Z_x$ is the total number of such contributions (e.g., iterations, events, or discrete space-time units in the model).
• These terms might include positive (generations) and negative (delays) values that interact or cancel in the system.

Step 2: Compute the Total Sum

• Calculate the aggregate effect as the sum of all individual contributions: $$S = \sum_{i=1}^{Z_x} RM_i$$
• This $S$ represents the net space-time generated or delayed across all components. In physical or theoretical terms, this could model cumulative effects in a dynamic system, such as balanced expansions and contractions in space-time.

Step 3: Introduce the Combined Space Operator

• Define $RM(\cdot)$ as an operator that computes the "combined" or "effective" space-time from the aggregate. For simplicity, assume $RM$ is a function that processes the sum (e.g., it could be identity if additive, or some transformation in a more complex model).
• The combined space is thus $RM(S) = RM\left( \sum_{i=1}^{Z_x} RM_i \right)$.
• In the context, this represents the holistic or merged space-time structure resulting from all contributions, potentially accounting for interactions not captured in the raw sum.

Step 4: Apply the Principle of Total Annihilation

• According to the described theory, the net effect undergoes complete cancellation via an "absorbing zero." This zero acts as a multiplier that nullifies the combined space.
• Mathematically, this annihilation is expressed as multiplying the combined space by 0: $$0 \cdot RM(S)$$
• This term equals 0 in standard arithmetic (since $0 \times$ anything = 0), symbolizing total neutralization of the generated spaces.

Step 5: Equate the Sum to the Annihilated Combined Space

• Set the total sum equal to the annihilated combined space to formalize the cancellation: $$S = 0 \cdot RM(S)$$
• Substituting back the definition of $S$: $$\sum_{i=1}^{Z_x} RM_i = 0 \cdot RM\left( \sum_{i=1}^{Z_x} RM_i \right)$$
• This is the final equation, expressing that the aggregate contributions balance out to zero through the absorbing mechanism.

Step 6: Analyze the Implication

• Since the right side simplifies to 0 (assuming finite and defined $RM(S)$), the equation implies $S = 0$.
• To arrive at this algebraically:
  • Start with $S = 0 \cdot RM(S)$.
  • The right side is 0, so $S = 0$.
  • This confirms complete cancellation: the delays and generations must pair or balance such that their net sum is zero (e.g., equal positive and negative terms).
• If $RM$ is the identity function (i.e., $RM(S) = S$), the equation becomes $S = 0 \cdot S = 0$, which explicitly requires $S = 0$.

This derivation assumes the equation arises from a balance principle in a custom space-time model. If $RM$ has a specific definition (e.g., non-linear or from a particular theory), additional assumptions might be needed.