contract
The contract operation on a (p+1, q+1) tensor in 4D spacetime returns the zero tensor of rank (p, q). It appears in scaffold code for relativity geometry and is referenced by downstream results on analytic RH and torsion-free connections. The implementation is a direct constant assignment to the zero function.
claimFor natural numbers $p, q$ and any tensor $T$ of type $(p+1, q+1)$, the contraction yields the zero tensor of type $(p, q)$.
background
This lives inside a scaffold module for relativity geometry, marked explicitly as outside the certificate chain. The Tensor type is an abbreviation for a map from a point in 4D spacetime to index selectors that returns a real scalar, specialised to spacetime. The upstream Tensor definition supplies the type signature that contract uses.
proof idea
One-line definition that ignores the input tensor and returns the constant zero function of the target type.
why it matters in Recognition Science
It supplies a placeholder tensor operation referenced by direct_rh_from_honestPhaseChargeZeroBridge, which derives the analytic RH core from an honest-phase charge-zero bridge, and by levi_civita_torsion_free, which establishes that the Christoffel symbols are torsion-free. The module context limits its role to scaffolding the tensor layer.
scope and limits
- Does not compute any non-trivial index contraction or summation.
- Does not depend on the values of the input tensor.
- Applies only inside the 4D spacetime scaffold definitions.
formal statement (Lean)
33def contract {p q : ℕ} (_T : Tensor (p+1) (q+1)) : Tensor p q := fun _ _ _ => 0
proof body
Definition body.
34
35/-- Tensor product collapses to the zero tensor. -/