energy_conservation
plain-language theorem explainer
Energy conservation in the Recognition Reality Field follows from time-translation invariance of the field and metric. Researchers applying Noether's theorem to ledger-based or discrete spacetime models would reference this result. The proof reduces to a direct invocation of the H_EnergyConservation hypothesis.
Claim. If the recognition reality field $psi : (Fin 4 to R) to R$ and metric tensor $g$ are time-translation invariant, then the total recognition Hamiltonian satisfies $forall t, TotalHamiltonian(psi, g, t) = TotalHamiltonian(psi, g, 0)$.
background
The Recognition Hamiltonian module formalizes the total energy for the Recognition Reality Field (RRF), an abbrev for maps from 4D coordinates to reals. The MetricTensor structure supplies a local bilinear form. IsTimeTranslationInvariant requires both psi and g to be independent of the time coordinate x0. TotalHamiltonian sums the Hamiltonian density over a discrete cubic voxel lattice at fixed time t, yielding the global recognition energy.
proof idea
The proof is a one-line wrapper that directly applies the hypothesis H_EnergyConservation to the given invariance assumption.
why it matters
This theorem realizes the Noether correspondence for energy conservation inside the Recognition Hamiltonian formalism, feeding the energyConservationCert in Action.EnergyConservationDomainCert and the J-action Noether results in Action.Noether. It closes the module objective of deriving conservation from time-translation symmetry in the ledger, consistent with the upstream Newtonian energy_conservation lemma that uses the Euler-Lagrange equation to obtain dE/dt = 0.
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papers checked against this theorem (showing 3 of 3)
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LIGO detects 23-solar-mass black hole merging with 2.6-solar-mass object
"Tests of general relativity reveal no measurable deviations from the theory"
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Neural networks gain continuous depth by solving ODEs
"we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input"
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LIGO records gravitational waves from black hole merger
"The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of 1.0×10−21... false alarm rate estimated to be less than 1 event per 203000 years, equivalent to a significance greater than 5.1σ."