canonicalFiniteH
plain-language theorem explainer
Defines the reduced A2 canonical finite second-variation value of the Regge TT action on a periodic Freudenthal torus of side N, for polarization E and integer mode m. Continuum-limit and Bloch-assembly theorems cite it as the finite symbol whose normalized limit is the isotropic Einstein-Hilbert coefficient -1/4. The body is an explicit scaled double sum of slot sqrt and angle derivatives over periodic tetrahedra.
Claim. For lattice side $N\ge 1$, TT polarization matrix $E\in\mathbb{R}^{3\times 3}$, and integer wavevector $m\in\mathbb{Z}^3$, the reduced A2 canonical finite value is $$H_N(E,m)=\frac{2}{N^3}\Biggl(-\sum_{\tau}\sum_{f=0}^{5}\partial_{\mathrm{sqrt}}\,\partial_{\mathrm{angle}}\Biggr),$$ summed over periodic Freudenthal tetrahedra $\tau$ and six face slots $f$, with derivatives evaluated at the commensurate plane-wave momentum of $m$.
background
This module is the C11 Regge TT Bloch interface audit (attempt 2). It keeps the A2 gate non-tautological: the reduced finite second variation is named separately from the literal triple stencil it equals, so the equality is a proved distribution identity rather than a definitional alias.
PeriodicTet N N N indexes the six Freudenthal tetrahedra in each cell of the $N\times N\times N$ periodic cubic torus. The factors flatSlotSqrtDeriv and flatSlotAngleDeriv are the slot-wise pieces of the Schläfli-reduced second variation of the Regge action on a flat background, evaluated on the commensurate momentum of the integer mode $m$. The overall prefactor $2/N^3$ normalizes the torus sum to a density.
The sign convention matches the live A2 theorem: the reduced second variation is the negative of the Schläfli-reduced contraction. Continuum and assembly modules treat this quantity as the finite TT Bloch symbol before taking $N\to\infty$.
proof idea
Definitional, not a proof. The right-hand side multiplies the global density factor $2/N^3$ by the negated double sum, over all periodic tetrahedra and six slots, of the product of the flat-slot square-root derivative and the flat-slot angle derivative at commensurate momentum. No lemmas are applied; downstream theorems unfold this name and rewrite the sum.
why it matters
This is the named finite object that the continuum and Bloch pipelines close on. Downstream, a2_reduced_eq_rawCellStencil identifies it with the literal triple raw stencil; canonicalFiniteH_eq_rawCosineBlochFold and its eventual form assemble it into the raw cosine Bloch fold under non-aliasing; canonicalFiniteH_TTBlochSymbolIs packages it as a fixed-$N$ TT Bloch symbol.
The algebraic and continuum closers then prove that $H_N(E,m)/|k_m|^2\to -1/4$ for every nonzero integer mode and every TT polarization (canonicalFiniteH_div_momentumNormSq_tendsto_isotropy, reggeTTContinuumIsotropyTarget_closed). That is the 3D isotropic Einstein-Hilbert TT coefficient, superseding C10/C8 numerics by kernel proof.
Within RS gravity analysis this is the finite hinge of the Regge TT continuum isotropy target. Gates still open in this file (full rational bucket aggregation, hinge-aware zero-mode, spike convention bridge) sit above this definition, not inside it.
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