Pith. sign in
module module moderate

IndisputableMonolith.Relativity.Geometry.DiscreteBridge

show as:
view Lean formalization →

Module formalizing the discrete-to-continuum bridge in RS relativity: lattice spacing on a finite box, flat and weak-field chains, and a Regge-style convergence hypothesis linking the phi-ladder lattice to continuum metric geometry. Relativists and RS auditors cite it when connecting discrete recognition ticks to Levi-Civita curvature. Structure is definitions plus certificates and an end-to-end chain, not a single theorem.

claimOn a box of side $L$ with $N$ sites, lattice spacing is $a = L/N$ (positive, $a \to 0$ as $N \to \infty$). The module packages a flat-chain identity, a weak-field bridge with coupling from $\phi$, invertibility of the metric matrix, a Regge convergence hypothesis, and a discrete-continuum bridge certificate assembling an end-to-end chain from the lattice to continuum geometry.

background

Recognition Science forces continuum spacetime from a discrete recognition lattice (eight-tick octave, $D=3$ from the T0–T8 chain). Continuum geometry in this stack already has metrics, Christoffel symbols, the unique torsion-free metric-compatible connection (Levi-Civita), and Riemann symmetries. Metric unification identifies the RS-derived Minkowski $\eta$ with the stack's minkowski_tensor.

This module sits at the interface: it defines lattice spacing for $N$ sites in a box of side $L$, records positivity and the continuum limit $a \to 0$, and states the structural bridges (flat chain, weak-field bridge, coupling extracted from $\phi$) needed to pass from discrete curvature (Regge-like) to the continuum curvature already formalized upstream. Constants supply the RS time quantum $\tau_0$.

proof idea

Definition-and-certificate module rather than a single deep proof. Lattice spacing is introduced as $L/N$ with elementary positivity and limit lemmas. Flat and weak-field bridges are packaged as named structures or props; coupling is read off from $\phi$. Metric-matrix invertibility is a local nondegeneracy fact. Regge convergence is stated as a hypothesis interface. The discrete-continuum bridge and bridge certificate assemble these pieces; EndToEndChain wires the path from lattice data through the weak-field/flat layers into the continuum geometry stack.

why it matters in Recognition Science

Feeds the Geometry aggregator, which re-exports all geometry components for the relativity stack. Without a discrete-continuum bridge, RS remains stuck at the lattice while curvature, Levi-Civita uniqueness, and metric unification live in the continuum. The module is the natural home for closing Regge-style convergence and for citing the weak-field and flat chains when matching continuum GR limits to the phi-ladder and eight-tick discrete skeleton. Landmarks in play: T7 (eight-tick), T8 ($D=3$), and the RS-derived $\eta$ already unified upstream.

scope and limits

used by (1)

From the project-wide theorem graph. These declarations reference this one in their body.

depends on (8)

Lean names referenced from this declaration's body.

declarations in this module (13)