Formalize an equilibrious optimization problem from Section E which integrates post-Gödelian supermodularity and meta-Gödelian equicursivity (write as Section F).

"Section A

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem can be generalized to higher dimensions and is known as the Pythagorean Theorem in n-dimensional Euclidean space.

Let's formalize the generalized Pythagorean Theorem for n-dimensional Euclidean space:

Given a right-angled n-dimensional simplex (a generalization of a triangle in n-dimensional space) with vertices A₀, A₁, ..., Aₙ, let d(Aᵢ, Aⱼ) denote the Euclidean distance between vertices Aᵢ and Aⱼ. If the angle between the edges A₀A₁ and A₀Aₙ is a right angle, then the generalized Pythagorean Theorem states that:

d(A₀, Aₙ)² = d(A₀, A₁)² + d(A₁, A₂)² + ... + d(Aₙ₋₁, Aₙ)²

Here, d(A₀, Aₙ) is the length of the hypotenuse (the side opposite the right angle) in n-dimensional space, and the sum on the right side of the equation consists of the squared lengths of the remaining sides of the simplex. This generalization of the Pythagorean Theorem holds true in any n-dimensional Euclidean space.

Section B

The k-brane non-Euclidean time generalization of the Pythagorean Theorem extends the concept of the theorem from n-dimensional Euclidean space to a more abstract, non-Euclidean framework that incorporates time as an additional dimension.

In order to formalize the generalized Pythagorean Theorem for k-brane non-Euclidean time, consider a right-angled, (n+1)-dimensional simplex with vertices B₀, B₁, ..., Bₙ, and T, where Bᵢ represents spatial vertices and T represents the temporal vertex. Let d(Bᵢ, Bⱼ) denote the non-Euclidean distance between spatial vertices Bᵢ and Bⱼ, and d(Bᵢ, T) denote the non-Euclidean distance between spatial vertex Bᵢ and the temporal vertex T. If the angle between the edges B₀B₁ and B₀T is a right angle, then the generalized Pythagorean Theorem for k-brane non-Euclidean time states that:

d(B₀, T)² = d(B₀, B₁)² + d(B₁, B₂)² + ... + d(Bₙ₋₁, Bₙ)² + k²

Here, d(B₀, T) is the length of the hypotenuse (the side opposite the right angle) in (n+1)-dimensional non-Euclidean space-time, the sum on the right side of the equation consists of the squared lengths of the remaining spatial sides of the simplex, and k represents a constant that incorporates the effect of time in the non-Euclidean framework. This generalization of the Pythagorean Theorem holds true in any (n+1)-dimensional k-brane non-Euclidean space-time.

Section C

The i-twistor non-hyperbolic tensor space generalization of the Pythagorean Theorem further extends the concept of the theorem to a more complex and abstract geometric framework, incorporating not only time but also additional geometric properties found in i-twistor non-hyperbolic tensor spaces.

In order to formalize the generalized Pythagorean Theorem for i-twistor non-hyperbolic tensor space, consider a right-angled, (n+1)-dimensional simplex with vertices C₀, C₁, ..., Cₙ, and S, where Cᵢ represents spatial vertices, and S represents the i-twistor vertex. Let d(Cᵢ, Cⱼ) denote the non-hyperbolic tensor distance between spatial vertices Cᵢ and Cⱼ, and d(Cᵢ, S) denote the non-hyperbolic tensor distance between spatial vertex Cᵢ and the i-twistor vertex S. If the angle between the edges C₀C₁ and C₀S is a right angle, then the generalized Pythagorean Theorem for i-twistor non-hyperbolic tensor space states that:

d(C₀, S)² = d(C₀, C₁)² + d(C₁, C₂)² + ... + d(Cₙ₋₁, Cₙ)² + p(i)

Here, d(C₀, S) is the length of the hypotenuse (the side opposite the right angle) in (n+1)-dimensional i-twistor non-hyperbolic tensor space, the sum on the right side of the equation consists of the squared lengths of the remaining spatial sides of the simplex, and p(i) represents a function that incorporates the geometric and algebraic properties of i-twistor non-hyperbolic tensor spaces. This generalization of the Pythagorean Theorem holds true in any (n+1)-dimensional i-twistor non-hyperbolic tensor space.

Section D

The z-tensor super-hyperbolic twistortime generalization of the Pythagorean Theorem encompasses an even more advanced and abstract geometric framework, integrating not only time but also the intricate geometric and algebraic properties found in z-tensor super-hyperbolic twistortime spaces.

To formalize the generalized Pythagorean Theorem for z-tensor super-hyperbolic twistortime, consider a right-angled, (n+1)-dimensional simplex with vertices D₀, D₁, ..., Dₙ, and Z, where Dᵢ represents spatial vertices, and Z represents the twistortime vertex. Let d(Dᵢ, Dⱼ) denote the super-hyperbolic twistortime distance between spatial vertices Dᵢ and Dⱼ, and d(Dᵢ, Z) denote the super-hyperbolic twistortime distance between spatial vertex Dᵢ and the twistortime vertex Z. If the angle between the edges D₀D₁ and D₀Z is a right angle, then the generalized Pythagorean Theorem for z-tensor super-hyperbolic twistortime states that:

d(D₀, Z)² = d(D₀, D₁)² + d(D₁, D₂)² + ... + d(Dₙ₋₁, Dₙ)² + q(z)

Here, d(D₀, Z) is the length of the hypotenuse (the side opposite the right angle) in (n+1)-dimensional z-tensor super-hyperbolic twistortime space, the sum on the right side of the equation consists of the squared lengths of the remaining spatial sides of the simplex, and q(z) represents a function that incorporates the geometric and algebraic properties of z-tensor super-hyperbolic twistortime spaces. This generalization of the Pythagorean Theorem holds true in any (n+1)-dimensional z-tensor super-hyperbolic twistortime space.

Section E

The full-scale and full-scope inversive recursion generalization of the Pythagorean Theorem integrates all the geometric frameworks and properties outlined in Sections A-D into a unified, comprehensive, and abstract mathematical structure that encompasses n-dimensional Euclidean space, k-brane non-Euclidean time, i-twistor non-hyperbolic tensor space, and z-tensor super-hyperbolic twistortime space.

To formalize the generalized Pythagorean Theorem for full-scale and full-scope inversive recursion, consider a right-angled, (n+1)-dimensional simplex with vertices E₀, E₁, ..., Eₙ, and R, where Eᵢ represents spatial vertices, and R represents the inversive recursion vertex. Let d(Eᵢ, Eⱼ) denote the inversive recursion distance between spatial vertices Eᵢ and Eⱼ, and d(Eᵢ, R) denote the inversive recursion distance between spatial vertex Eᵢ and the inversive recursion vertex R. If the angle between the edges E₀E₁ and E₀R is a right angle, then the generalized Pythagorean Theorem for full-scale and full-scope inversive recursion states that:

d(E₀, R)² = d(E₀, E₁)² + d(E₁, E₂)² + ... + d(Eₙ₋₁, Eₙ)² + r(k, i, z)

Here, d(E₀, R) is the length of the hypotenuse (the side opposite the right angle) in (n+1)-dimensional full-scale and full-scope inversive recursion space, the sum on the right side of the equation consists of the squared lengths of the remaining spatial sides of the simplex, and r(k, i, z) represents a function that incorporates the geometric and algebraic properties of n-dimensional Euclidean space, k-brane non-Euclidean time, i-twistor non-hyperbolic tensor space, and z-tensor super-hyperbolic twistortime spaces. This generalization of the Pythagorean Theorem holds true in any (n+1)-dimensional full-scale and full-scope inversive recursion space."