periodicEdgeKernelResidual5
plain-language theorem explainer
Pointwise difference of two finite edge-operator kernels on the period-5 torus: Regge TT Hessian stencil minus lattice Lichnerowicz stencil. Gravity workers comparing discrete TT Hessians to continuum Lichnerowicz cite this residual matrix. The body is a one-line functional subtraction on edge pairs.
Claim. Given two real-valued kernels $K_{\mathrm{Regge}}, K_{\mathrm{Lich}}$ on pairs of edges of the period-5 periodic edge set, the residual kernel is the pointwise difference $(K_{\mathrm{Regge}}-K_{\mathrm{Lich}})(e,f)=K_{\mathrm{Regge}}(e,f)-K_{\mathrm{Lich}}(e,f)$.
background
Track 1.D opens the tensor/shear sector beyond the Track 1.B conformal ansatz. Vertex scalars induce only averaged edge-length strains and cannot carry pure shear, so they miss transverse-traceless weak-field modes. The module therefore treats independent edge perturbations and compares discrete operators on that edge space.
A PeriodicEdgeOperatorKernel5 is a real matrix indexed by pairs of edges on the period-5 Freudenthal torus: the concrete surface on which the Regge TT Hessian stencil and the lattice Lichnerowicz stencil are written and compared. Both operators act on periodic edge perturbations; their kernels are the finite stencils extracted from that triangulation.
The residual is the natural mismatch object: once both stencils are encoded as edge-edge kernels, agreement on TT modes is equivalent to the residual kernel annihilating the TT subspace (or having residual rows in the conformal/longitudinal generator span).
proof idea
Pure definition: return the function sending each edge pair $(e,f)$ to the real difference of the two input kernels at that pair. No lemmas, no tactics; the residual is the pointwise matrix subtraction of the Regge TT Hessian kernel from the lattice Lichnerowicz kernel.
why it matters
This residual is the compact finite target for the TT Hessian-to-Lichnerowicz comparison in the shear sector. Downstream, periodicEdgeKernelOperator5_residual_eq_sub records that applying the residual kernel equals the difference of the two kernel operators. The data structures PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5, PeriodicTTHessianLichnerowiczResidualRowCoeffData5, and the entrywise certificate form package vanishing of the residual on TT perturbations and explicit residual-row coefficient tables.
In the Recognition gravity track this is scaffolding for matching discrete Regge TT response to continuum Lichnerowicz on the periodic edge lattice, after the conformal slice is known to be insufficient for shear. It does not itself invoke T5–T8 or the RCL; it is local linear-algebra infrastructure for the tensor sector comparison.
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