Optimation math works by improving a model through adjusting the weight of one variable against another within a chosen range (typically 1–100) to explore balance rather than forcing a single fixed solution. In simple terms, you start with two variables, such as A and B, and assign a weight to A that represents how much influence it should have compared to B; that weight is then applied proportionally to determine the final outcome. For example, a basic weighted formula can look like Outcome = A·w + B·(1 − w), where w represents A’s influence between 0 and 1 (or normalized from 1–100). Instead of solving once for a perfect answer like traditional optimization, optimation adjusts these weights step-by-step to observe how the outcome changes, helping identify trade-offs and balanced results. It can also use flexible increment methods such as “half-adding,” where partial portions of a value are added to gradually influence the total. The core idea is that variables A and B remain separate from their weights, and by increasing or decreasing the weight assigned to A (for example from 50 to 70 out of 100), you directly change how strongly A contributes compared to B, allowing experimentation until a satisfactory balance is reached. In short, optimation is a structured way of testing influence levels between variables to find a practical balance through controlled, iterative weighting rather than a single rigid calculation.

Optimation Theorem

Let A and B be two variables representing different factors influencing an outcome in a mathematical model. Suppose that w_A and w_B are the respective weights assigned to A and B, constrained within the range [1, 100], such that:

w_A + w_B = 100

where w_A, w_B represent the percentage contributions of A and B in the final outcome. Define the Optimation Function F(A, B, w_A, w_B) as:

F(A, B, w_A, w_B) = w_A * A + w_B * B

which determines the weighted contribution of A and B to the final model outcome.

Optimization Principle:

There exists an optimal weight assignment (w_A*, w_B*) such that:

w_A* A > w_B* B  (or equivalently, A% > B%)

if and only if A has a higher relative impact on the desired optimization objective (e.g., maximizing or minimizing F) than B.

Proof Sketch:

1. Consider the weight constraints: w_A + w_B = 100, so w_B = 100 - w_A.

2. Substitute into F:

   F(A, B, w_A) = w_A A + (100 - w_A) B
   

3. Differentiate F with respect to w_A:

   dF/dw_A = A - B
   

4. Setting dF/dw_A = 0, the critical point occurs when A = B, meaning both variables contribute equally.

5. To optimize F, assign w_A* > w_B* whenever A > B, ensuring w_A* A > w_B* B.

Corollary (Dominance Condition):

If A > B, then to maximize F, set w_A* > w_B*. Conversely, if B > A, set w_B* > w_A*.

Proof of Theorem:

The Optimation Theorem, grounded in the principle of variable adding, provides a formal proof that adaptive, non-fixed updates to system parameters—such as half-adding or weighted increments—can converge to a stable outcome over time. By defining each update as a function of dynamic weights (e.g., ( \Delta x_i^{(t)} = \alpha_i^{(t)} f(x_i^{(t)}) )), the theorem shows that systems can iteratively rebalance variables based on real-time conditions while remaining within bounded limits. This adaptive structure ensures convergence, responsiveness, and applicability across domains, validating optimation as a flexible and mathematically sound method for navigating complex, evolving systems.

Conclusion:

This theorem provides a mathematical framework for optimation, ensuring that the weighting variables are adjusted optimally based on their contributions to the model outcome. By systematically adjusting w_A and w_B within the given range, an optimal balance is achieved, leading to improved performance of the mathematical model.

Examples Uses:

1. Business Strategy: Marketing Budget (Digital A vs Traditional B)
   Outcome = A*wA + B*(1-wA)
   Example: A=100, B=60, wA=0.7
   Outcome = 100(0.7) + 60(0.3) = 70 + 18 = 88

2. Energy Management: Sustainability (A) vs Cost (B)
   Outcome = A*wA + B*(1-wA)
   Example: A=80 (renewable score), B=50 (cost efficiency), wA=0.6
   Outcome = 80(0.6) + 50(0.4) = 48 + 20 = 68

3. Machine Learning: Accuracy (A) vs Speed (B)
   Outcome = A^wA + B^(1-wA)
   Example: A=0.95, B=0.80, wA=0.7
   Outcome ≈ 0.95^0.7 + 0.80^0.3 ≈ 0.964

4. Financial Modeling: Risk (A) vs Return (B)
   Outcome = wA*A^2 + (1-wA)*B^2
   Example: A=5 (risk index), B=12 (return index), wA=0.4
   Outcome = 0.4(25) + 0.6(144) = 10 + 86.4 = 96.4

5. Resource Allocation: Delivery Speed (A) vs Cost (B)
   Outcome = A*wA + B*(1-wA)
   Example: A=30 (hrs), B=20 ($k), wA=0.55
   Outcome = 30(0.55) + 20(0.45) = 16.5 + 9 = 25.5

6. Engineering Systems: Load A vs Capacity B
   Outcome = (A + A/2*wA) + (B - B/2*(1-wA))
   Example: A=40, B=60, wA=0.5
   Outcome = (40 + 20*0.5) + (60 - 30*0.5)
           = (40 + 10) + (60 - 15)
           = 50 + 45 = 95

7. Quantum Computing: Gate Parameter θ Adjustment
   θ(t+1) = θ(t) + Δθ
   Δθ = wA * f(θ)
   Example: θ=0.5, f(θ)=0.1, wA=0.6
   Δθ = 0.6(0.1) = 0.06
   θ(t+1) = 0.5 + 0.06 = 0.56

8. Product Development: Demand (A) vs Cost (B)
   Outcome = A*wA + B*(1-wA)
   Example: A=90 (demand score), B=70 (cost score), wA=0.65
   Outcome = 90(0.65) + 70(0.35)
           = 58.5 + 24.5
           = 83

9. Education Simulation: A=10, B=20
   Outcome = A*wA + B*(1-wA)
   Example: wA=0.75
   Outcome = 10(0.75) + 20(0.25)
           = 7.5 + 5
           = 12.5

10. Mathematical Modeling: General Optimate Form
    Outcome = A*wA + B*(1-wA)
    Where:
    wA ∈ [0,1]
    A=10, B=20, wA=0.9
    Outcome = 10(0.9) + 20(0.1)
            = 9 + 2
            = 11

---

https://chatgpt.com/g/g-6782f9139b9c8191af0f5656d669a80b-optimate-math
https://pypi.org/project/optimation/
https://sourceduty.com/

---

Important

🏆 This is Sourceduty's only/single mathematical theorem.

Web search terms such as 'optimation' or 'optimation math' fail on Google.