IndisputableMonolith.Gravity.MetricFromDefect
Defines how a spatial defect field on the ledger lattice induces a weak-field metric perturbation: a symmetric 2-tensor proportional to the RS coupling, reducing to flat space when the defect vanishes. Gravity workers cite it when linking G-001 defect curvature to continuum geometry. The module is mostly definitions plus elementary algebraic identities and a certificate wrapper.
claimIn $D$ dimensions a symmetric $2$-tensor is stored as a symmetric matrix. A defect field $\delta$ induces a metric perturbation $h_{\mu\nu}(\delta)$ that is symmetric, vanishes when $\delta=0$ (flat spatial metric), and scales linearly with the RS coupling $\kappa$. Under a weak-field bound on $\delta$, $\|h\|\ll 1$.
background
Recognition Science treats gravity as emergent ledger curvature, not a fundamental force. Upstream ZeroParameterGravity (G-001) states that large-scale geometry is induced by defect distributions on the recognition lattice; the continuum limit should recover a Lorentzian metric whose weak-field piece tracks those defects.
This module supplies the linear map from defect data to that metric piece. A SymmetricTensor is a symmetric bilinear form on $\mathbb{R}^D$ (indices ${0,1,2,3}$ in $3+1$, or ${1,2,3}$ spatially), with the usual trace. The flat spatial metric is the Euclidean background. A defect field is a scalar (or density) assignment on the lattice whose continuum avatar sources curvature.
Constants enter only through the RS-native coupling that sets the overall scale of the perturbation; the time quantum $\tau_0$ is ambient context from the Constants import, not a free parameter here.
proof idea
Definition-heavy module. Symmetric tensors, flat spatial metric, trace, and the defect field are introduced as data. The core map sends a defect field to a metric perturbation by a $\kappa$-proportional formula; symmetry of the output is immediate from the construction. Vanishing defect yields the flat background by direct substitution. Weak-field hypotheses bound the defect so the perturbation norm is small. A certificate bundles these facts for downstream import; no deep analytic estimates live here.
why it matters in Recognition Science
Closes the local dictionary between ledger defects and continuum $h_{\mu\nu}$ needed by the gravity stack. Downstream UnifiedLatticeManifoldCorrespondence packages the global deformed-cubic-lattice $\leftrightarrow$ curved-manifold statement: sequences of lattices with prescribed edge lengths and dihedral angles whose Regge action converges to $S_{\mathrm{EH}}[g]$ and whose equations converge to the EFE. Without a concrete defect-to-metric map, that correspondence has no weak-field anchor.
In the RS forcing picture this sits under G-001 (gravity as defect-induced curvature) and feeds any later identification of Einstein dynamics with ledger balance. It does not itself derive the field equations or fix $G$; it only builds the geometric intermediary.
scope and limits
- Does not derive the Einstein field equations or the Einstein-Hilbert action.
- Does not treat strong-field or fully nonlinear metric reconstruction.
- Does not prove continuum limits of Regge calculus or lattice convergence.
- Does not fix Newton's constant or post-Newtonian parameters beyond $\kappa$-scaling.
- Does not address matter coupling or stress-energy matching.
used by (1)
depends on (2)
declarations in this module (12)
-
structure
SymmetricTensor -
def
flat_metric_spatial -
def
trace -
structure
DefectField -
def
metric_perturbation_from_defect -
theorem
metric_perturbation_symmetric -
theorem
zero_defect_flat_space -
theorem
perturbation_proportional_to_kappa -
def
weak_field_condition -
theorem
weak_field_small_perturbation -
structure
MetricFromDefectCert -
theorem
metric_from_defect_cert