D
D is defined as the integer 3, the spatial dimension selected by the linking argument in Foundation.DimensionForcing. Researchers studying the integration gap D²(D+2) and its coprimality with recognition periods cite this constant when restricting dimensions to odd values consistent with orderly cycles. The declaration is a direct constant assignment with no computation or lemmas.
claimLet $D$ denote the spatial dimension forced by linking. Then $D = 3$.
background
The IntegrationGap module examines the integer $D^2(D+2)$, called the integration gap. At $D=3$ this equals 45, with $D^2=9$ as the parity count and $D+2=5$ as the configuration dimension. The module proves that gcd$(2^D, D^2(D+2))=1$ if and only if $D$ is odd, which combines with the linking argument to restrict $D$ to 3 among odd candidates. Upstream structures include SpectralEmergence, which derives exactly three particle generations and 24 chiral fermions from $D=3$, and PhiForcingDerived, which supplies the J-cost minimization underlying the dimension selection.
proof idea
The declaration is a direct constant definition assigning the natural number 3. No lemmas or tactics are invoked; it functions as the base value for sibling definitions such as configDim_at_D3 and integrationGap_at_D3.
why it matters in Recognition Science
This supplies the concrete value required by the linking argument in Foundation.DimensionForcing and by the coprimality result that selects D=3 among odd dimensions. It aligns with framework landmarks including the eight-tick octave (period $2^3$), the emergence of SU(3)×SU(2)×U(1) gauge content, and the restriction to three spatial dimensions. The definition closes the combinatorial foundation for the integration gap calculations in the same module.
scope and limits
- Does not prove that D must equal 3; that step occurs in DimensionForcing.
- Does not evaluate the integration gap or coprimality; those reside in sibling definitions.
- Does not address physical interpretations of the gap beyond the combinatorial setting.
formal statement (Lean)
40def D : ℕ := 3
proof body
Definition body.
41
42/-- The configuration dimension of a recognition event:
43 `D` spatial degrees of freedom + 1 temporal + 1 ledger balance. -/