IndisputableMonolith.Gravity.LatticeRicci
The LatticeRicci module discretizes the linearized Ricci scalar on a Z^3 lattice within Recognition Science gravity. It represents R = -nabla^2(tr h) via three second derivatives of the trace perturbation, using Regge deficit angles from the imported DiscreteCurvature module. Gravity modelers working on lattice-to-continuum transitions cite these constructions. The module organizes definitions and equalities that connect discrete curvature to the harmonic-gauge continuum expression.
claimThe linearized Ricci scalar on the lattice is $R = -3$ second derivatives of trace $h$, matching the continuum form $R = -nabla^2 (tr h)$ in harmonic gauge.
background
The module sits in the Recognition Science gravity domain after the discrete curvature step. It imports the RS time quantum τ₀ = 1 tick and the Regge-style deficit angle formalization, where on flat Z^3 each cube has right-angled dihedral angles of π/2. The doc-comment states the core representation: the linearized Ricci scalar from metric perturbation h is given in harmonic gauge by R = -nabla^2(trace h) and realized via the three second derivatives of the trace.
proof idea
This is a definition module, no proofs. It supplies the lattice_ricci object and related equalities whose proofs appear in sibling declarations that apply the imported DiscreteCurvature deficit-angle machinery.
why it matters in Recognition Science
The module supplies the discrete Ricci scalar needed to link Regge-style lattice geometry to continuum linearized gravity in the RS framework. It supports downstream siblings such as ricci_from_source and ricci_attractive that extend the curvature construction toward source terms and force laws. It fills the discrete-to-continuum bridge after the DiscreteCurvature module in the gravity chain.
scope and limits
- Does not treat the full nonlinear Ricci tensor.
- Does not include time-dependent or dynamical evolution.
- Does not cover non-flat background lattices.
- Does not derive mass formulas or coupling constants.
depends on (2)
declarations in this module (11)
-
def
continuum_ricci_linearized -
def
lattice_ricci -
theorem
lattice_ricci_equals_continuum -
theorem
lattice_ricci_flat -
theorem
ricci_negative_for_mass -
def
ricci_from_source -
theorem
ricci_source_proportional_to_kappa -
theorem
ricci_attractive -
def
H_nonlinear_regge_convergence -
structure
LatticeRicciCert -
theorem
lattice_ricci_cert