sym2exC
plain-language theorem explainer
Pair-exchange average of a six-index rational coefficient table: half the sum of C at (a,b,c,d,i,j) and at the swapped pair indices (c,d,a,b,i,j). Gravity analysts cite it when building the order-8 bi-quadratic symmetrization used in the exact midpoint Bloch m² TT identity in 4D Regge calculus. The body is a one-line arithmetic mean; no proof content.
Claim. Given a rational six-index array $C$ on $\mathrm{Fin}\,4$, the pair-exchange average at indices $(a,b,c,d,i,j)$ is $\frac{1}{2}\bigl(C_{abcdij}+C_{cdabij}\bigr)$. This is the average of $C$ under the exchange of the two matrix-index pairs $(a,b)\leftrightarrow(c,d)$, leaving the wave indices $(i,j)$ fixed.
background
The module closes the exact midpoint Bloch $m^2$ TT identity in 4D Regge calculus (script-backed kernel certificates). The ambient objects are six-index rational coefficient tables $C(a,b,c,d,i,j)$ that enter a bi-quadratic form pairing a $4\times 4$ matrix mode $H$ with a 4-component wave $k$.
Pair exchange $(a,b)\leftrightarrow(c,d)$ is one generator of the order-8 symmetry group of that bi-quadratic form (together with flips on each pair). Averaging over the exchange produces a coefficient table that is symmetric under that swap while leaving the contracted form unchanged when the contraction is fully summed.
Sibling infrastructure in the same file includes the full symmetrizer (order-8 average), Frobenius and wave norms, and the explicit $m^2$ coefficient tables that the identity equates after symmetrization.
proof idea
Pure definition: return the rational mean of the two pair-exchanged evaluations of $C$. No lemmas, no tactics. Downstream theorems then show that contracting the averaged table against $H\otimes H\otimes k\otimes k$ recovers the same bi-quadratic value as the unsymmetrized table.
why it matters
This is the pair-exchange half of the full bi-quadratic symmetrization used to close exact_midpoint_m2_tt_identity. Downstream, symFull is defined as this average applied after the four-index flip symmetrizer, and biquad_sym2ex proves that the bi-quadratic form is invariant under the average: contracting the symmetrized table equals contracting the original. A companion identity relates the rational full symmetrizer to an integer table divided by 32.
In the broader gravity analysis stack this sits under the 4D Regge TT / midpoint Hessian work that certifies the Bloch $m^2$ coefficient against the closed-form kernel. It is bookkeeping symmetry, not a dynamical claim, but without it the pointwise 4096-entry certificate comparing explicit and closed-form $m^2$ tables would not be well-posed on a unique representative.
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