Explain the Lean def `F2Power` in module `IndisputableMonolith.Algebra.F2Power`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

1. What the declaration says in plain English

The declaration defines a type representing the set of all possible sequences of D true/false values. Mathematically, it models the elementary abelian 2-group of rank $D$ (often written as $(\mathbb{Z}/2\mathbb{Z})^D$), where the elements are binary vectors of length $D$ and the group operation is pointwise XOR.

2. Why it matters in Recognition Science

In Recognition Science, binary distinctions form the fundamental substrate of logic and structure. F2Power provides the combinatorial foundation for the framework's $2^D - 1$ Count Law. This law limits the number of distinct, non-trivial states that can exist along $D$ binary axes. The framework uses this exact structure to formally classify cognitive and physical families, establishing that a domain organized around $D$ binary axes will support exactly $2^D - 1$ non-zero modes—structurally unifying Booker's 7 basic plots at $D = 3$, the 3 opponent-color visual channels at $D = 2$, and the 3 massive electroweak bosons at $D = 2$.

3. How to read the formal statement

def F2Power (D : ℕ) : Type := Fin D → Bool

• def F2Power: Defines a new term named F2Power.
• (D : ℕ): Specifies a single parameter D of type natural number (0, 1, 2, ...), representing the dimension.
• : Type: Declares that the output of this definition is a Type (a collection of elements).
• := Fin D → Bool: The actual definition. Fin D is the finite set of numbers from $0$ to $D - 1$. Bool is the set containing true and false. The arrow → denotes a function. Thus, the type consists of functions that map each index from $0$ to $D-1$ to a boolean value. This is the Lean representation of a $D$-dimensional bit-vector.

4. Visible dependencies and certificates

Within the supplied source, the definition depends only on Lean's standard library (Mathlib) to provide the Fin and Bool types.

Once defined, it enables crucial structural theorems in the canon:

• nonzero_card derives that the number of non-zero elements in F2Power D is exactly $2^D - 1$.
• nonzero_card_three explicitly computes this for $D=3$, proving there are 7 non-zero vectors.
• In IndisputableMonolith.Patterns.TwoToTheDMinusOne, the abstract CountLaw predicate uses F2Power D to define what it means for a physical or narrative family to be in exact bijection with these non-zero bit-vectors. The master certificate countLawCert bundles these facts together.

5. What this declaration does not prove

The F2Power declaration sets up an abstract combinatorial space; it does not prove the empirical applicability of that space to specific domains on its own. While the canon successfully proves the bijection for Booker's plot families (via bookerCountLaw), the mappings for other domains like physics and human vision (e.g., MassiveBosonCountLaw and OpponentColorCountLaw) are provided in the source as hypothetical definition templates. Formalizing the explicit encoding functions for those cross-domain physical applications remains an OPEN research frontier.