IndisputableMonolith.Gravity.Analysis.EdgeTTDecompositionLorentz4DAudit
Audit layer over the Lorentzian 4D edge transverse-traceless (TT) algebraic decomposition. It packages checks that the TT split of symmetric real 4×4 matrices against a Minkowski wave covector is well-formed, including the null covector case used in radiation gauges. Gravity and QG formalizers cite it when certifying the W4-1 edge_tt_decomposition lane. Structure is import-and-audit of the sibling Lorentzian algebraic module, not a standalone proof development.
claimAudit of the Lorentzian transverse-traceless decomposition: for a symmetric real $4\times 4$ matrix $h_{\mu\nu}$ and a Minkowski wave covector $k_\mu$ on $\mathbb{R}^{1,3}$ (including null $k$), the algebraic projection of $h$ into transverse-traceless, trace, and longitudinal pieces relative to $k$ is recorded and checked against the Euclidean TT layer specialized to signature $(-,+,+,+)$.
background
In linearized gravity and spin-2 wave analysis one splits a symmetric tensor $h_{\mu\nu}$ relative to a wave covector $k$ into transverse-traceless (TT), trace, and longitudinal sectors. The TT piece carries the two physical helicities of a massless spin-2 field when $k$ is null. The parent algebraic module supplies that split on Fin 4 with Minkowski metric, as the Lorentzian specialization of the Euclidean 4D TT layer.
Recognition Science places this inside the QG full-theory campaign, Wave 4, lane W4-1 (edge_tt_decomposition). The Euclidean precursor handles positive-definite inner products; the Lorentzian layer must also treat null and timelike $k$, where the orthogonal complement and the induced metric on the screen space change rank. This audit module sits one import above that Lorentzian layer and does not redefine the projections.
Notation: indices run over four real coordinates; symmetry of $h$ is the only tensorial hypothesis at this algebraic stage. No curvature, no dynamics, and no continuum PDE theory enter here.
proof idea
This is an audit module, not a theorem module. It imports EdgeTTDecompositionLorentz4D and exposes the Lorentzian TT decomposition surface for campaign-level checking. There is no independent proof body: correctness claims live in the imported algebraic layer (projections, kernel identities, and null-case rank statements). The audit role is structural packaging and dependency pinning for Wave 4 lane W4-1.
why it matters in Recognition Science
The declaration earns its place as the audit gate on the Lorentzian edge TT decomposition required by the QG full-theory campaign (Wave 4 / W4-1). Downstream consumers in the gravity analysis stack need a single import that certifies the algebraic TT split, including the physically relevant null covector case, before coupling to dynamics or continuum limits. No further used_by edges are recorded at this snapshot, so the module is a leaf audit rather than a lemma feeder. It closes the Lorentzian specialization path opened by the Euclidean EdgeTTDecomposition4D layer and keeps the algebraic TT interface stable for later Recognition gravity theorems (radiation gauges, edge modes, spin-2 content).
scope and limits
- Does not prove continuum or PDE-level TT gauge fixing on spacetimes.
- Does not treat curved backgrounds, only flat algebraic 4×4 Minkowski data.
- Does not derive dynamics, Einstein equations, or mass formulas.
- Does not replace the Euclidean TT layer; it only audits the Lorentzian specialization.
- Does not assert completeness of physical helicity counting beyond the algebraic null case.