IsTimeTranslationInvariant
The predicate requires that a recognition field and its metric take identical values at four-vectors differing only in the time coordinate. Researchers deriving Noether-type conservation laws from the Recognition Hamiltonian cite this predicate when establishing time-independent energy. It is introduced directly by universal quantification over all real times and fixed integer spatial indices.
claimA recognition field $ψ : (ℝ^4 → ℝ)$ and metric tensor $g$ are time-translation invariant if $ψ(x_1) = ψ(x_2)$ and $g(x_1) = g(x_2)$ whenever the four-vectors $x_1, x_2$ differ only in their time component, for all $t_1, t_2 ∈ ℝ$ and all spatial indices $ijk ∈ ℤ^3$.
background
The Recognition Hamiltonian Formalism module derives the Hamiltonian for the Recognition Reality Field with the explicit objective of proving energy conservation as a consequence of time-translation symmetry in the ledger. The RRF is the local interface given by functions from four-dimensional coordinates to the reals. The MetricTensor is the structure supplying the metric components on those coordinates. TotalHamiltonian is defined as the spatial integral of the Hamiltonian density over the cubic voxel lattice at fixed time.
proof idea
As a definition the predicate is introduced directly by the displayed universal quantification over times and spatial indices together with the pointwise equality conditions on the field and metric.
why it matters in Recognition Science
This definition supplies the antecedent hypothesis for the energy conservation theorem, which concludes that the total recognition energy is independent of time whenever the invariance holds. It thereby implements the Noether correspondence between time-translation symmetry and energy conservation inside the Recognition framework, linking the Hamiltonian construction to the forcing chain steps that fix D = 3 and the eight-tick octave.
scope and limits
- Does not supply an explicit formula for the Hamiltonian density.
- Does not address spatial translation invariance.
- Does not incorporate Berry creation thresholds or phi-ladder mass assignments.
- Does not calibrate constants such as alpha or G.
Lean usage
theorem energy_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)
80def IsTimeTranslationInvariant (psi : RRF) (g : MetricTensor) : Prop :=
proof body
Definition body.
81 ∀ t₁ t₂ : ℝ, ∀ ijk : Fin 3 → ℤ,
82 let x₁ : Fin 4 → ℝ := fun i => if i.val = 0 then t₁ else (ijk (match i.val with | 1 => 0 | 2 => 1 | _ => 2) : ℝ)
83 let x₂ : Fin 4 → ℝ := fun i => if i.val = 0 then t₂ else (ijk (match i.val with | 1 => 0 | 2 => 1 | _ => 2) : ℝ)
84 psi x₁ = psi x₂ ∧ g.g x₁ = g.g x₂
85
86/-- **HYPOTHESIS**: Total recognition energy is conserved under time-translation
87 invariant dynamics.
88 STATUS: EMPIRICAL_HYPO
89 TEST_PROTOCOL: Verify that dH/dt = 0 for stationary sections of the RRF potential.
90 FALSIFIER: Discovery of a time-translation invariant system where TotalHamiltonian
91 is not conserved. -/