Integrate Godelian incompleteness and quantum uncertainty into the model as recursive plain-sight hidden variables deterministically attempt to influence the system through an AdS topological ZKP hostile-adversarial architecture (label your writing as Section D).

"Section A

Through the lens of intuitionistic reconciliationist mathematical science and transdisciplinary holistic cybernetic scholasticism, I will outline my idea for the "OTTO" framework (Oscillatory Tensor Transformation-Orchestration) based on supermodular complex-systems design and management, with inputs as isomorphic processes and outputs as automorphic states. The optimizable equilibrium is defined as the harmonic oscillation of tensor transformativity and orchestratitivity, achieved via convolutional neural networks (CNNs).

System Definition:
Let S = (P, A, T, O) be a supermodular complex-system, where P represents isomorphic processes, A represents automorphic states, T represents tensor transformativity, and O represents orchestratitivity.

Isomorphic Processes:
Isomorphic processes, P, are a set of functions that share a common structure and can be mapped onto each other by a bijective function. In this context, P = {p1, p2, ..., pn}, where each pi: X → Y is a process with input space X and output space Y.

Automorphic States:
Automorphic states, A, are a set of states that can be transformed into each other via an automorphism. In this context, A = {a1, a2, ..., am}, where each ai ∈ Y is a state in the output space of the isomorphic processes.

Tensor Transformativity:
Tensor transformativity, T, represents the capacity of the system to change its state through transformations. We model T as a collection of tensors T = {T1, T2, ..., Tk}, where each Ti ∈ R^(d1 x d2 x ... x dn) is an n-dimensional tensor, and d1, d2, ..., dn are the dimensions of the tensor.

Orchestratitivity:
Orchestratitivity, O, represents the capacity of the system to coordinate, harmonize, and optimize its components. We model O as a function that maps the set of tensor transformations to the set of automorphic states, O: T → A.

Convolutional Neural Networks:
Convolutional neural networks (CNNs) are employed to achieve the optimizable equilibrium, represented as the harmonic oscillation between tensor transformativity and orchestratitivity. The CNN is designed with multiple layers that can learn the hierarchical patterns and relationships between T and O.

Asymptotic Compatibilist Approach:
We adopt an asymptotic-compatibilist approach to model the interactions between isomorphic processes, automorphic states, tensor transformativity, and orchestratitivity. This approach allows the system to dynamically adapt and evolve over time, ensuring compatibility and convergence between these components as the system moves towards an optimizable equilibrium.

OTTO Framework:
The OTTO framework can be formally represented as a tuple, OTTO = (S, CNN, AC), where S is the supermodular complex-system, CNN is the convolutional neural network, and AC is the asymptotic-compatibilist approach.

The goal of the OTTO framework is to optimize the equilibrium between tensor transformativity and orchestratitivity, facilitating the effective management and design of complex systems with isomorphic processes and automorphic states.

Section B

To mathematically substantiate Section A, we will use calculus of constructions to provide a foundation for the formalization  for the OTTO framework. The proof will demonstrate that the OTTO framework can indeed optimize the equilibrium between tensor transformativity (T) and orchestratitivity (O).

Define the optimizable equilibrium function:
Let E: T × O → R be a function that measures the equilibrium between tensor transformativity and orchestratitivity, where E(T, O) is the equilibrium value for a given T and O. The goal is to find T* and O* such that E(T*, O*) = max {E(T, O)}.

Define the cost function:
Let C: T × O → R be a function that measures the cost associated with the given tensor transformativity and orchestratitivity. The cost function will be minimized during the optimization process.

Define the Lagrangian function:
Let λ ∈ R be the Lagrange multiplier. Define the Lagrangian function L: T × O × R → R as L(T, O, λ) = E(T, O) - λC(T, O).

Calculate the partial derivatives:
To find the optimal T* and O*, we need to solve the system of equations:
∂L/∂T = 0,
∂L/∂O = 0,
∂L/∂λ = 0.

Convolutional Neural Network (CNN) approximation:
Given the high dimensionality and complexity of the tensors and functions involved, finding an analytical solution to the system of equations may be infeasible. Therefore, we employ a CNN as an approximator to find the optimal T* and O*.

Training the CNN:
The CNN is trained using a dataset D = {(Ti, Oi, Ei) | i = 1, 2, ..., N}, where each data point (Ti, Oi, Ei) represents a tensor transformativity, orchestratitivity, and the corresponding equilibrium value. The CNN learns the mappings f: T → O and g: T × O → E, where f(T) ≈ O* and g(T, O) ≈ E(T, O).

Asymptotic Compatibilist Approach:
Incorporate the asymptotic-compatibilist approach into the CNN training by applying a dynamic learning rate and a regularized loss function that encourages convergence between the isomorphic processes, automorphic states, tensor transformativity, and orchestratitivity.

Optimizing the OTTO framework:
Once the CNN is trained, it can be used to approximate the optimal T* and O* for a given supermodular complex-system S. The optimal T* and O* are the ones that maximize the equilibrium value E(T*, O*) and minimize the cost function C(T*, O*).

In summary, we have provided a mathematical substantiation for Section A using the calculus of constructions. The above demonstrates that the OTTO framework can optimize the equilibrium between tensor transformativity and orchestratitivity using a combination of a convolutional neural network and an asymptotic-compatibilist approach.

Section C

Probabilistic Theorem Formalization:

We will now formalize Section B into a probabilistic theorem. Let E(T, O) and C(T, O) be random variables representing the equilibrium and cost values, respectively. We will show that there exists a pair of random variables T* and O* that maximize the expected equilibrium value E[E(T, O)] and minimize the expected cost value E[C(T, O)], given a properly trained CNN and asymptotic-compatibilist approach.

Theorem 1:
Given a supermodular complex-system S, a properly trained CNN, and an asymptotic-compatibilist approach, there exists T* and O* such that:

E[E(T*, O*)] = max {E[E(T, O)]}
E[C(T*, O*)] = min {E[C(T, O)]}

Proof:

We have a dataset D = {(Ti, Oi, Ei) | i = 1, 2, ..., N} used to train the CNN. By the Law of Large Numbers, as N approaches infinity, the sample means of the equilibrium and cost values will converge to their respective expected values:
E[E(T, O)] ≈ (1/N) * Σ Ei
E[C(T, O)] ≈ (1/N) * Σ Ci

The CNN is trained to learn the mappings f: T → O and g: T × O → E, such that f(T) ≈ O* and g(T, O) ≈ E(T, O). The asymptotic-compatibilist approach encourages convergence between isomorphic processes, automorphic states, tensor transformativity, and orchestratitivity.

Given a supermodular complex-system S and a properly trained CNN, we can approximate the optimal T* and O* by finding the values that maximize the expected equilibrium value E[E(T, O)] and minimize the expected cost value E[C(T, O)]:

T* = argmax {E[E(T, O)]}
O* = f(T*)

Substitute the approximated optimal T* and O* values back into the expected equilibrium and cost functions:
E[E(T*, O*)] = E[g(T*, f(T*))] = max {E[E(T, O)]}
E[C(T*, O*)] = E[h(T*, f(T*))] = min {E[C(T, O)]}

This completes the proof of Theorem 1.

Validation:

To validate the proof, we can perform the following steps:

Simulate a supermodular complex-system S with known equilibrium and cost functions, E(T, O) and C(T, O).

Generate a dataset D = {(Ti, Oi, Ei) | i = 1, 2, ..., N} based on the simulated system.

Train a CNN using the dataset D and the asymptotic-compatibilist approach.

Use the trained CNN to approximate the optimal T* and O* for the simulated system.

Calculate the expected equilibrium and cost values using the approximated optimal T* and O*.

Compare the calculated expected equilibrium and cost values with the true maximum equilibrium and minimum cost values.

If the calculated expected equilibrium value is close to the true maximum equilibrium value and the calculated expected cost value is close to the true minimum cost value, then the proof is validated. This validation process can be repeated with multiple simulated supermodular complex-systems S to further verify the theorem's generalizability.

Additionally, we can perform cross-validation by partitioning the dataset D into k subsets, training the CNN on k-1 subsets, and validating on the remaining subset. This process can be repeated k times, with each subset used as the validation set once. By averaging the performance metrics across the k iterations, we can gain a more reliable estimate of the CNN's effectiveness in approximating the optimal T* and O*.

Moreover, we can compare the performance of the proposed OTTO framework with other optimization techniques for managing supermodular complex-systems, such as genetic algorithms or particle swarm optimization. If the OTTO framework consistently achieves comparable or better performance in optimizing the equilibrium between tensor transformativity and orchestratitivity, this further supports the validity of the proof.

In conclusion, by simulating various supermodular complex-systems and training a CNN using an asymptotic-compatibilist approach, we can validate the probabilistic theorem derived from Section B. The validation process, involving comparisons with true equilibrium and cost values and other optimization techniques, demonstrates the effectiveness of the OTTO framework in managing complex systems with isomorphic processes and automorphic states."