e_130200
plain-language theorem explainer
Pointwise identity: the Regge midpoint m2 numerator at multi-index (1,3,0,2,0,0) equals eight times the explicit Z-table entry. Gravity analysts cite it when assembling the global 4D kernel certificate m2Num = 8·explicitZ. The proof is a single kernel decide on concrete Fin-4 integers.
Claim. For indices $(a,b,c,d,i,j)=(1,3,0,2,0,0)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module is chunk 7 of a 256-way split of the 4D Regge midpoint identity $m_2^{\mathrm{num}}=8\cdot Z$. Each chunk discharges a block of concrete multi-indices in $(\mathbb{F}_4)^6$ by kernel decision.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution term for each coupling triple at the six indices. The table $Z$ is an explicit case-split function $\mathbb{F}_4^6\to\mathbb{Z}$ listing the closed-form integer values expected after that fold (typical entries $\pm 2,\pm 4$, and zeros elsewhere).
The ambient goal is an exact algebraic identity in the discrete Regge calculus kernel used for the midpoint $M_2$ / TT sector in four dimensions, not a continuum GR statement.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers once the six Fin 4 arguments are fixed at $(1,3,0,2,0,0)$, so the kernel equality checker closes the goal with no lemmas and no case split inside this declaration.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states the identity for every $(a,b,c,d,i,j):\mathbb{F}_4$ by exhausting indices. That global certificate is the analytic backbone of the exact midpoint $M_2$ TT identity in the 4D Regge kernel.
In the Recognition gravity stack this is bookkeeping infrastructure: it locks the discrete curvature/coupling numerator to a sparse explicit integer table so later mass and force derivations can quote a proved algebraic identity rather than a numerical fit. It does not itself invoke the forcing chain (T5–T8), $\phi$-ladder masses, or the RCL; it only certifies one kernel cell used by those layers.
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