IndisputableMonolith.Gravity.CubicReggeConvergence
Defines the RS J-cost action on the cubic lattice Z³ and packages weak-field O(a²) convergence of that action to the continuum Dirichlet energy. Gravity and continuum-limit workers cite it when specializing Regge convergence to the regular cubic case. The module supplies lattice/continuum actions, error estimates, UV cutoffs, and cubic shape-quality hypotheses consumed by the cubic-lattice limit theorem.
claimOn the cubic lattice $\mathbb{Z}^3$ with spacing $a$, the RS lattice action of a scalar field $\varepsilon$ is $S_{\mathrm{RS}}(\varepsilon,a)=a^3\sum_x\sum_{\mu=1}^3 J\bigl(e^{\varepsilon(x+a\hat{e}_\mu)-\varepsilon(x)}\bigr)$. It expands to the discrete Dirichlet energy $\frac{a^3}{2}\sum_x|\nabla_a\varepsilon|^2$ and, under weak-field cubic conditions, converges to $\frac12\int|\nabla\varepsilon|^2\,d^3x$ at order $O(a^2)$.
background
Recognition Science gravity is built from the J-cost $J(x)=(x+x^{-1})/2-1$ on discrete edges. ContinuumLimit shows that discrete J-cost dynamics on $\mathbb{Z}^3$ produce, in the long-wavelength limit, second-order continuum equations matching Klein-Gordon structure. ReggeCalculus formalizes the full nonlinear Regge framework on the RS lattice; ReggeConvergence and NonlinearConvergence record the lattice-to-Einstein-Hilbert programme and the CMS-style convergence inputs (with the documented correction that CMS gives weak curvature-measure convergence, not a plain action estimate).
This module specializes to the cubic mesh and the weak-field scalar regime. It introduces the RS lattice action $S_{\mathrm{RS}}(\varepsilon,a)$ and its continuum counterpart, UV cutoffs, $\varphi$-exponential growth controls from GrowthBounds, and shape-quality hypotheses that keep the cubic lattice regular enough for an $O(a^2)$ comparison.
proof idea
Not a single theorem: a definition-and-estimate package. It defines the cubic RS lattice action and continuum action, then records quartic remainder control and weak-field error estimates under named cubic conditions. A weak-field convergence structure bundles those hypotheses with shape quality and an explicit convergence bound. UV-cutoff positivity and $\varphi$-exponential growth supply the analytic side conditions inherited from GrowthBounds and ContinuumLimit. Downstream ReggeCubicLatticeLimit consumes the package to isolate the plain $O(a^2)$ cubic case from the weaker general CMS statement.
why it matters in Recognition Science
General Regge convergence yields a weak curvature-measure limit, not a plain $O(a^2)$ action estimate. Downstream ReggeCubicLatticeLimit states that the $O(a^2)$ claim belongs to the regular weak-field cubic-lattice case and imports this module to isolate that special case for the canonical second-order Regge action. Within the RS gravity chain this pins continuum recovery of the J-cost action on $\mathbb{Z}^3$ (the spatial lattice forced by T8, $D=3$) to the Dirichlet energy that seeds Einstein-Hilbert recovery in the weak field. It sits between ContinuumLimit / ReggeCalculus and the cubic-lattice limit theorem.
scope and limits
- Does not prove full nonlinear Regge-to-Einstein-Hilbert convergence off the cubic lattice.
- Does not replace the general CMS weak curvature-measure theorem.
- Does not treat strong-field regimes or highly irregular triangulations.
- Does not derive the Einstein equations from the lattice action alone.
- Does not claim continuum convergence without the packaged weak-field and shape-quality hypotheses.
used by (1)
depends on (6)
declarations in this module (15)
-
def
rs_lattice_action -
def
continuum_action -
theorem
quartic_error_controlled -
theorem
weak_field_error_estimate -
structure
WeakFieldConvergence -
def
weak_field_convergence -
structure
RSCubicConvergenceConditions -
theorem
rs_cubic_shape_quality -
def
rs_convergence_bound -
def
uv_cutoff -
theorem
uv_cutoff_pos -
theorem
phi_exponential_growth -
theorem
exponential_defeats_cubic -
structure
CubicConvergenceCert -
theorem
cubic_convergence_cert