TotalHamiltonian
The TotalHamiltonian definition computes global recognition energy by summing the local density over a discrete 3D integer lattice of voxel centers, inserting time t into each coordinate and weighting by the metric volume element. Researchers deriving Noether theorems for the Recognition Reality Field cite this when establishing time-independent energy. The construction uses a Finset sum with a placeholder universal domain and multiplies each term by the square root of the absolute metric determinant.
claimLet $ψ$ be a recognition reality field and $g$ a metric tensor. The total Hamiltonian at time $t$ equals $∑_{ijk ∈ ℤ³} ℋ(ψ, g, x) √|det g(x)|$, where the four-vector $x$ sets its time component to $t$ and its spatial components to the integer triple $ijk$.
background
The module develops the Hamiltonian formalism for the Recognition Reality Field with the objective of proving energy conservation from time-translation symmetry in the ledger. The recognition reality field is an abbrev for maps from four-dimensional coordinates to the reals. The metric tensor is a structure supplying a bilinear form on those coordinates for volume computations. Upstream, the voxel is defined as the fundamental length quantum in RS-native units where the speed of light equals one.
proof idea
The definition declares spatial centers as a Finset of Fin 3 → ℤ triples. For each triple it builds the four-vector x by placing t in the time slot and the triple entries in the spatial slots. It then evaluates the Hamiltonian density at x, multiplies by the square root of the absolute metric determinant at x, and sums over the Finset.
why it matters in Recognition Science
This definition supplies the total energy functional that enters the energy conservation theorem, which asserts constancy under time-translation invariance of the field and metric. It implements the spatial integral of the density required for the Noether argument in the Recognition Science ledger. The construction supports the hypothesis that total recognition energy remains constant for stationary sections of the potential.
scope and limits
- Does not specify the explicit functional form of the local Hamiltonian density.
- Does not restrict the spatial domain beyond the placeholder Finset sum.
- Does not address convergence of the infinite lattice sum.
- Does not incorporate Recognition Science constants such as the phi-ladder or native units for energy.
Lean usage
theorem conservation (h : H_EnergyConservation psi g) (psi : RRF) (g : MetricTensor) : IsTimeTranslationInvariant psi g → (∀ t, TotalHamiltonian psi g t = TotalHamiltonian psi g 0) := hformal statement (Lean)
49noncomputable def TotalHamiltonian (psi : RRF) (g : MetricTensor) (t : ℝ) : ℝ :=
proof body
Definition body.
50 -- The cubic voxel centers are at integer coordinates (i,j,k).
51 -- In 3D, the ledger state consists of a set of active boundaries.
52 let spatial_centers : Finset (Fin 3 → ℤ) :=
53 -- Trivial domain: all integers in 3D.
54 Finset.univ -- placeholder for finite bounding set
55 Finset.sum spatial_centers (fun ijk =>
56 let x : Fin 4 → ℝ := fun i =>
57 if i.val = 0 then t
58 else (ijk (match i.val with | 1 => 0 | 2 => 1 | _ => 2) : ℝ)
59 HamiltonianDensity psi g x * Real.sqrt (abs (metric_det g x))
60 )
61
62/-- **DEFINITION: Recognition Energy-Momentum Tensor**
63 The stress-energy tensor derived from the RRF potential Ψ. -/