Mat4
plain-language theorem explainer
Type alias for real 4×4 matrices on Fin 4 indices. Used throughout the exact midpoint Bloch m² TT identity development in 4D Regge analysis whenever a matrix of edge or mode coefficients is needed. Pure abbreviation; no proof content.
Claim. Write $\mathrm{Mat}_4$ for the space of $4\times 4$ real matrices, i.e. maps $(\mathrm{Fin}\,4)\times(\mathrm{Fin}\,4)\to\mathbb{R}$.
background
The module closes the exact midpoint Bloch $m^2$ transverse-traceless identity in four-dimensional Regge calculus. Companion scripts certify the kernel and assemble the numerical $m^2$ coefficient.
In that setting one repeatedly manipulates real matrices indexed by four discrete labels (edges, modes, or Bloch components). The abbreviation packages Mathlib's Matrix (Fin 4) (Fin 4) ℝ under a short local name so later statements about Frobenius norms, wave vectors, and coupling coefficients stay readable.
No deeper Recognition-Science structure is encoded here; the alias is purely notational scaffolding for the 4D Hessian and TT-decomposition lemmas imported by the module.
proof idea
Definitional abbreviation only. Expands by unfolding to Mathlib's Matrix type with index type Fin 4 and value type ℝ. No tactics, no lemmas.
why it matters
Local notational convenience inside the Gravity.Analysis stack that targets the exact midpoint $m^2$ TT identity. Downstream siblings (Wave4, frobeniusNormSq, couplingS, m2Coeff, etc.) are stated in terms of this matrix type, keeping the certified kernel and numerical assembly scripts aligned with the Lean development. It does not itself advance a forcing-chain step or a physical constant; it simply standardizes the 4D matrix language used to finish that identity.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.