1. What the declaration says in plain English
The declaration defines a constant named Dspatial to be exactly the natural number 3.
2. Why it matters in Recognition Science
In Recognition Science, the number 3 represents the spatial dimension of the universe. In this specific cross-domain module, Dspatial acts as a primitive generator. The framework shows that various canonical domain sizes across the theory (the "RS Cardinality Spectrum") are not arbitrary, but decompose cleanly into small primitive constants like Dspatial (3) and Dconfig (5).
3. How to read the formal statement
def Dspatial : ℕ := 3
def: Instructs Lean to create a new definition.Dspatial: The name of the constant.: ℕ: Declares the type as a natural number (a non-negative integer).:= 3: Assigns the exact value 3.
4. Visible dependencies and certificates As a bare definition, it has no upstream dependencies. Downstream, it is visibly used to construct other spectrum generators and their verification proofs:
- eightTick_eq: The 8-tick processing cadence is $2^{D_{spatial}}$.
- gap45_eq: The gap dimension is $D_{spatial}^2 \times D_{config}$.
- cubeFaces_eq: The faces of a cube are $2 \times D_{spatial}$.
Its status as exactly 3 is also formalized in the module's overarching certificate structure, CardinalitySpectrumCert, via the field Dspatial_is_3.
5. What this declaration does not prove
This declaration is merely a MODEL: a definitional choice used to build the cardinality spectrum list. It does not prove that physical space must have 3 dimensions. The actual THEOREM forcing $D=3$ occurs in a different part of the framework. Specifically, the canon proves that non-trivial circle linking in a $D$-sphere exists if and only if $D=3$ via alexander_duality_circle_linking. The Dspatial definition simply takes that established result and uses the number 3 as a structural baseline for algebraic decomposition.