H_EnergyConservation
The definition encodes the property that total recognition energy stays constant in time whenever the recognition reality field and metric tensor are invariant under time translations. Researchers deriving Noether-type conservation laws from ledger symmetries would cite it when assuming time-translation invariance. It directly packages the implication as a Prop without further reduction steps.
claimLet $ψ$ be the recognition reality field and $g$ a metric tensor. The predicate $H(ψ,g)$ holds precisely when time-translation invariance of $ψ$ and $g$ implies that the total Hamiltonian equals its value at the initial time for every later time $t$.
background
The Recognition Hamiltonian Formalism module constructs the Hamiltonian for the Recognition Reality Field (RRF), an abbrev for maps from four-dimensional coordinates to reals. The metric tensor is a structure supplying a bilinear form on these coordinates. TotalHamiltonian sums the local density over a cubic lattice of spatial centers, with time as the zeroth coordinate, yielding the global recognition energy.
proof idea
This is a direct definition that packages the conservation statement as the logical implication from IsTimeTranslationInvariant to constancy of TotalHamiltonian. No lemmas are applied; the body is the implication itself.
why it matters in Recognition Science
It supplies the hypothesis used by the downstream energy_conservation theorem, which asserts that time-translation invariance forces conservation of total recognition energy. The module objective is to obtain energy conservation from time-translation symmetry in the ledger, placing the result inside the Recognition Science derivation of conserved quantities from symmetry.
scope and limits
- Does not derive an explicit expression for the Hamiltonian density.
- Does not prove the implication from invariance to conservation.
- Does not specify boundary conditions or finite-volume corrections.
- Does not link to the phi-ladder or numerical constants such as alpha.
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)
92def H_EnergyConservation (psi : RRF) (g : MetricTensor) : Prop :=
proof body
Definition body.
93 IsTimeTranslationInvariant psi g →
94 (∀ t, TotalHamiltonian psi g t = TotalHamiltonian psi g 0)
95
96/-- **THEOREM: Energy Conservation (Noether)**
97 If the ledger constraints are time-translation invariant, the total recognition
98 energy (Hamiltonian) is conserved under RRF dynamics. -/