e_130302
plain-language theorem explainer
For the six-index slot (1,3,0,3,0,2) on Fin 4, the folded numerator coupling equals eight times the tabulated explicit kernel integer. Gravity analysts cite it as one concrete cell in the 256-case kernel that certifies the Regge midpoint M2 TT identity in 4D. The proof is a single kernel decide on closed integer arithmetic.
Claim. With indices in $\{0,1,2,3\}$, the numerator coupling $m_2^{\mathrm{num}}(1,3,0,3,0,2)$ equals $8$ times the explicit kernel value $Z(1,3,0,3,0,2)$.
background
This module is chunk 7 of a 256-cell kernel certification that the folded numerator coupling equals eight times an explicit integer table, written $m_2^{\mathrm{num}}=8\cdot Z$, in the 4D Regge exact-midpoint M2 TT identity analysis.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the fold of a fixed coupling list: each term contributes an integer depending on the six Fin-4 indices, and the accumulator starts at 0. The companion $Z(a,b,c,d,i,j)$ is a sparse pattern-matched table of small integers (entries such as $4$, $-2$, and so on) that is meant to be the closed form of that fold after a universal factor of 8.
The ambient setting is discrete gravity: midpoint Regge calculus in four dimensions, where TT-sector mass-matrix numerators must match an explicit algebraic kernel before continuum or continuum-limit claims are made.
proof idea
One-line kernel proof: decide evaluates both sides as concrete Int expressions for the fixed indices $(1,3,0,3,0,2)$ and checks equality. No lemmas are invoked; the fold defining the numerator and the pattern match defining $Z$ reduce by computation.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the full universal statement $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhausting Fin-4 cases. This declaration is one named cell in that exhaustion (chunk 7 of the 256 kernel decides).
In the Recognition gravity stack the identity is infrastructure, not a forcing-chain landmark: it locks the discrete TT numerator against the explicit kernel so later continuum or phenomenological gravity claims rest on a fully decided algebraic match rather than an unchecked fold. It does not itself touch T0–T8, RCL, or the $\phi$-ladder; it is a pure 4D Regge bookkeeping certificate.
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