PeriodicTTHessianLichnerowiczKernelEntryData5
plain-language theorem explainer
Bundles two finite edge-operator kernels on the 5×5×5 periodic Freudenthal torus—the Regge TT Hessian stencil and the lattice Lichnerowicz stencil—together with the demand that every matrix entry agrees. Gravity Track 1.D cites it as the strongest stencil-level certificate that the discrete TT Hessian matches the spin-2 Lichnerowicz operator. As a structure it is pure data packaging: no proof obligations beyond the equality field.
Claim. A data package consisting of two real-valued kernels $K_{\mathrm{Regge}}, K_{\mathrm{Lich}} : E_5 \times E_5 \to \mathbb{R}$ on the edge set $E_5$ of the encoded $5\times 5\times 5$ periodic Freudenthal torus, together with the entrywise identity $K_{\mathrm{Regge}}(e,f) = K_{\mathrm{Lich}}(e,f)$ for all edges $e,f \in E_5$.
background
Track 1.D isolates pure shear (transverse-traceless) edge perturbations from the vertex-conformal scalar slice of Track 1.B. The conformal ansatz averages endpoint potentials into edge-length changes and therefore cannot represent pure shear or gravitational-wave modes; this module separates independent edge perturbations and records the elementary rectangle obstruction.
On the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus, operators on edge perturbations are represented as finite kernels $E_5\to E_5\to\mathbb{R}$. That matrix surface is where the Regge second-variation TT Hessian stencil is compared to the discrete spin-2 Lichnerowicz stencil.
Entrywise kernel equality is a stricter certificate than rowwise agreement on the TT subspace or abstract operator matching on TT perturbations: if every matrix entry coincides, both weaker match statements follow immediately.
proof idea
No proof body: the declaration is a structure (definition). It packages two fields of type PeriodicEdgeOperatorKernel5 (functions $E_5\times E_5\to\mathbb{R}$) and a Prop field asserting pointwise equality of those kernels on every pair of periodic edges. Instantiation later supplies concrete stencils and discharges the equality.
why it matters
In the Recognition gravity stack, matching the Regge TT Hessian to the lattice Lichnerowicz operator is the discrete stand-in for the continuum fact that the second variation of the Einstein–Hilbert action on transverse-traceless modes is the Lichnerowicz Laplacian. This structure is the strongest finite-calculation target on that path: residual-kernel vanishing by entrywise identity.
Downstream, Track1DTTHessianLichnerowiczKernelEntryReductionEndpoint treats inhabitation of this structure as a sufficient condition that yields nonempty rowwise kernel data, nonempty operator-match data, and the bilinear reduction endpoint. Sibling structures PeriodicTTHessianLichnerowiczKernelRowData5 and PeriodicTTHessianLichnerowiczMatchData5 record the weaker rowwise-on-TT and operator-on-TT match goals that entrywise equality implies.
It does not itself close the continuum limit or force $D=3$; it only organizes the stencil comparison on the fixed $5^3$ torus used for Track 1.D.
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