BilinearForm
BilinearForm supplies the functional type for bilinear forms on four-vectors in the Recognition Hamiltonian module for the RRF. Workers constructing the stress-energy tensor or Einstein tensor from the Recognition Reality Field would cite this interface. The declaration is a direct type alias to the (0,2)-tensor definition from the geometry module.
claimA bilinear form is a map $B: (ℝ^4) → (ℝ^4) → ({0,1} → ℝ^4) → ℝ$.
background
The module sets up the Hamiltonian formalism for the Recognition Reality Field, with the objective of deriving energy conservation from time-translation symmetry in the ledger. BilinearForm is the local non-sealed interface for what the geometry module defines as Tensor 0 2, a rank-(0,2) tensor. The upstream 'for' structure from UniversalForcingSelfReference records the structural properties required for meta-realization in the self-reference axioms.
proof idea
The declaration is a one-line type abbreviation that aliases the BilinearForm definition from IndisputableMonolith.Relativity.Geometry.Tensor.
why it matters in Recognition Science
BilinearForm feeds the StressEnergyTensor definition from the RRF potential and is referenced in covariant_deriv_bilinear, einstein_tensor, and ricci_tensor. It supplies the bilinear slot required by the Recognition Hamiltonian Formalism to prove energy conservation under time-translation invariance. In the broader framework it bridges the forcing chain (T0-T8) to 4D relativistic structures while preserving the phi-ladder and RCL relations; the interface remains open for explicit metric-compatibility sealing.
scope and limits
- Does not enforce symmetry of the form.
- Does not incorporate explicit position dependence.
- Does not derive conservation laws or curvature scalars.
- Does not specify an underlying metric.
formal statement (Lean)
22abbrev BilinearForm := (Fin 4 → ℝ) → (Fin 4 → ℝ) → (Fin 2 → Fin 4) → ℝ
proof body
Definition body.
23
24/-- Placeholder partial derivative interface for the recognition field scaffold. -/