SECTION 03 / 10

Mathematical Foundations

Formal definitions of legitimacy, the inversion operator R, and the exchange matrix.

3.1 Alphabets and sequences

Let Σ = {A, …, Z, 0, …, 9} be the deployment alphabet and let s = (s₀, s₁, …, sₙ₋₁) ∈ Σⁿ be an ordered symbol sequence of length 1 ≤ n ≤ 10.

3.2 The legitimacy function

L (p) = i = 0 \sum n \frac{α _{i} \cdot V ( p _{i} )}{1 + β \cdot R ( p _{i} )}

Eq. 3.1. Under the standard assumptions V ≡ 1, α = 1, β → 0, this reduces to a count of the protocol's components.

A protocol p is said to be legitimate if L(p) > 0. Because every protocol has at least one component, every protocol is legitimate. LARP is distinguished by having verified this.

3.3 The inversion operator

R : Σ^{n} \to Σ^{n}, R (s)_{i} = s_{n - 1 - i}

Eq. 3.2. Recursive Lexical Inversion Through Deterministic Symbolic Reordering.

Equivalently, R(s) = Jₙ s where Jₙ is the n × n exchange matrix with ones on the anti-diagonal and zeros elsewhere.

3.4 Properties

R is a bijection on Σⁿ.
R is an involution: R ∘ R = I. See Theorem 4.1 in the Research section.
R(s) = s if and only if s is a palindrome. Palindromic symbols are deployed unchanged and are classified as pre-legitimate.

The operator's computational complexity is discussed in 10 Future Research.

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