e_012011
plain-language theorem explainer
Pointwise identity: the Regge midpoint mass-squared numerator at index tuple (0,1,2,0,1,1) equals eight times the explicit Z-table entry there. Gravity analysts cite it when assembling the full 4D kernel identity m2Num = 8·explicitZ. The proof is a single kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,1,2,0,1,1)$ in $(\mathrm{Fin}\,4)^6$, the integer numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit coupling table value $Z(a,b,c,d,i,j)$.
background
This module is chunk 1 of a 256-case kernel certification that the 4D Regge-exact midpoint mass-squared numerator equals eight times a closed-form integer table. The ambient setting is discrete gravity analysis: couplings on a 4-index simplex lattice with values in $\mathrm{Fin},4$.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed contribution list couplingZList over the six indices, summing each term's integer contribution. The companion table explicitZ is a pattern-matched function $\mathrm{Fin},4^6\to\mathbb{Z}$ that records the expected closed form (e.g. $4$, $-2$, and other small integers on the nonzero support).
The local claim is one concrete instance of $m_2^{\mathrm{num}}=8\cdot Z$ at the multi-index $(0,1,2,0,1,1)$. Sibling theorems cover the other index tuples in the same chunk.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are fixed, so the kernel checks integer equality with no further lemmas. No algebraic rewriting or induction is involved; the work is pure evaluation of m2Num (the fold) against explicitZ (the table match).
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$ and discharges the universal claim by exhaustive fin_cases on all six indices. Each pointwise decide such as this one closes one residual goal in that case split.
In the Recognition gravity stack this identity is bookkeeping infrastructure for the Regge-exact midpoint $M_2$ TT sector in 4D: it certifies that the summed coupling numerator matches the explicit integer kernel used downstream. It does not itself invoke the forcing chain (T0–T8), RCL, or $\varphi$-ladder mass formula; those enter at higher layers that consume the certified kernel.
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