periodicLongitudinalGaugeMap5
plain-language theorem explainer
Defines the finite longitudinal gauge map on the N=5 periodic Freudenthal torus: real coefficients on vertex-vector basis indices are sent to edge-length perturbations. Anyone working the Track 1.D TT Gram split cites it as the concrete generator of pure longitudinal gauge modes. It is a one-line specialization of the generic gauge-generator map to the longitudinal generator family.
Claim. The longitudinal gauge map on the $N=5$ periodic complex is the linear map that takes a real coefficient assignment on pairs $(\text{vertex},\, i)$ with $i\in\{0,1,2\}$ and returns an edge perturbation on the typed periodic edges, by evaluating the fixed longitudinal gauge generator family against those coefficients.
background
Track 1.D isolates the tensor/shear sector of weak-field Regge gravity. Track 1.B only supplies a conformal (vertex-scalar) ansatz: one potential per vertex, edge strains by endpoint averaging. That slice cannot carry pure shear, so it cannot represent transverse-traceless gravitational-wave modes. This module therefore treats independent edge perturbations as the ambient space and carves out conformal and longitudinal gauge subspaces inside it.
The domain index type is one spatial vector component at one periodic vertex (vertex $\times$ $\mathrm{Fin},3$). The codomain is real-valued edge perturbations on the typed periodic Freudenthal edges. The map is the concrete finite realization of longitudinal gauge: vertex-vector delta generators induce edge strains supported only at the two endpoints of each edge.
The ambient geometry is the periodic Freudenthal torus at $N=5$, imported from the periodic-torus and first-variation Regge infrastructure. Longitudinal modes sit beside the conformal family; together they form the gauge part of the eventual TT projector split.
proof idea
One-line definitional wrapper. It applies the generic periodic gauge-generator map to the fixed longitudinal gauge generator family (vertex-vector delta basis). No extra algebra or tactics: the body is pure specialization of that generator map.
why it matters
This map is the concrete longitudinal half of the finite gauge family used throughout Track 1.D handoff endpoints. Downstream MasterTheoremHandoffIntegration reductions (Gram kernel criterion, kernel-generator-map-zero, load image, load solver, range closed, range criterion, longitudinal coefficient projector) all thread through the combined conformal-plus-longitudinal generator data built on this map.
In the Recognition gravity program it supplies the missing pure-gauge directions that the conformal ansatz alone cannot span, so the residual after gauge projection can be identified with the TT/shear sector. It does not itself close the Gram range or kernel criteria; those are separate finite-linear-algebra endpoints that consume this generator. Framework-wise it is scaffolding for the weak-field $D=3$ tensor sector on the eight-tick-compatible discrete geometry, not a forcing-chain (T0–T8) step.
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