Abstract War

Abstract Mathematical Construct
Let 𝒰 be an abstract process space, where elements undergo transformations based on evolving conditions.

1. Core Elements & Definitions
We define a mapping over 𝒰, denoted as:

𝒳:𝒰×𝑅×𝑁→𝒰

where each instance 𝒰ₙ consists of a triple:

𝒰𝑛=(𝒫𝑛,𝒮𝑛,𝒴𝑛)

where:

𝒫𝑛 ∈ ℝ represents a continuous scalar adjustment.
𝒮𝑛 ∈ ℕ represents a discrete state magnitude.
𝒴𝑛 ∈ 𝒰 represents an evolving reference structure.

2. Transformation Rule
A process 𝒜 applies adjustments to 𝒰, evolving it under a conditionally propagated mapping:

𝒳(𝒰𝑛,𝒫𝑛,𝒮𝑛)={ ∅, 𝒮𝑛 ≤0 otherwise (𝒫𝑛,𝒮𝑛,𝒴𝑛) }
This transformation continues under the presence of a binary condition.

3. Conditional Evolution
A transition function 𝒯 is introduced, acting within a probabilistic structure:

𝒰𝑛+1={ 𝒳(𝒰𝑛,𝒫𝑛−𝛿,⌊𝒮𝑛/2⌋) -> 𝑋𝑛=1 otherwise 𝒰𝑛 -> 𝑋𝑛=0 }
​
where:

𝒫𝑛 undergoes a gradual decrement by δ.
𝒮𝑛 undergoes quantized contraction.
𝑋𝑛 ∈ {0,1} is determined by an independent stochastic event.

4. Underlying Structure
The transformation 𝒯 ensures a structured evolution, yet never explicitly defines iteration or recursion.

∃ 𝑛 0∈𝑁, ∀ 𝑛>𝑛0, 𝑃(𝒰𝑛=∅)=1

This ensures that, over an extended progression, the transformation reaches a terminal state, albeit through non-deterministic yet structured steps.

Fundamental Definitions
Let 𝒵 be a class of structures evolving over successive transformative interactions, denoted as:

𝒵𝑛=(𝒫𝑛,𝒬𝑛,𝒵𝑛−1)

where:

𝒫𝑛 ∈ ℝ represents a principal scalar undergoing progressive adjustments.
𝒬𝑛 ∈ ℝ represents an external perturbation affecting state transitions.
𝒵𝑛{n-1} ∈ 𝒵 provides an implicit reference to prior evolutionary states.
A transformation 𝒮 governs the system, dynamically modifying 𝒫𝑛 under structured dependencies.

2. Evolutionary Process: Perturbation-Driven Adaptation
We define an adjustment operator 𝒯 acting over 𝒵, modifying the system based on a decaying propagative rule:

𝒯(𝒵𝑛,𝒫𝑛,𝒬𝑛)={𝒫𝑛 -> 𝑛=0 otherwise 𝒯(𝒵𝑛−1,𝒫𝑛−1,𝒬𝑛−1)+(Δ−𝒵𝑛𝜀)−𝒬𝑛 -> 𝑛>0 }

where:

𝒫𝑛 recursively inherits the prior state 𝒵𝑛{n-1}.
𝒬𝑛 is an external stochastic perturbation, influencing transitions.
Δ represents a structured bias introduced in every step.
𝜀 scales the internal transformation based on prior conditions.
This formulation inherently adapts based on preceding influences while adjusting dynamically due to probabilistic perturbations.

3. Probabilistic Interference Mechanism
A perturbation generator 𝒫𝒳 : ℝ → {0,1} defines interference based on an uncertain external process, akin to selective disruption mechanisms:

𝒬𝑛={ 𝜆,𝑃(𝑋𝑛=1)=𝑝0, otherwise 𝑃(𝑋𝑛=0)=1−𝑝 }

where:

𝒫𝒳 enforces an external intervention with probability p.
The scalar λ dictates the intensity of modification when intervention occurs.
The process introduces non-deterministic fluctuations influencing the evolution.

4. Emergent Behavior & Structured Adaptation
By applying repeated transformations, the structure of 𝒵 evolves in a way that balances prior adjustments while reacting to perturbative influences. The final form expresses a regulated adaptive process, where the outcome reflects both historical dependencies and external interactions.

For sufficiently large n, the process asymptotically stabilizes under:

𝑛
lim⁡ 𝒫𝑛= ∑ (Δ−𝜀𝒵𝑘−𝒬𝑘)
𝑛→∞ 𝑘=1

where the cumulative perturbations regulate the ultimate adjustment.

Who among the minds that wander sees war not as blood and steel,
But as a silent drift of shifting states, where choice and chance congeal?
Who discerns in walls that crumble the weight of forms unseen,
Where every strike, a measured shift, shapes fate's unwritten scheme?

Who shall trace, in veiled equations, the battlefield's silent code,
Where power bends, where fate unfolds, yet none escape the road?

don't try this
because at run time they are completely different.