e_311211
plain-language theorem explainer
One of 256 concrete kernel cases: the midpoint Regge mass-squared numerator at multi-index (3,1,1,2,1,1) equals eight times the explicit integer kernel value. Gravity analysts assembling the 4D TT midpoint identity cite these case facts. The proof is a single kernel decide on fixed Fin-4 indices.
Claim. For indices $(a,b,c,d,i,j)=(3,1,1,2,1,1)$ with each coordinate in $\{0,1,2,3\}$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 13 of a 256-case kernel certification that the midpoint Regge mass-squared numerator matches eight times a closed-form integer table in four dimensions.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: it sums a local contribution at each coupling triple over the six Fin-4 indices. The explicit kernel $Z$ is a pattern-matched integer table on those same indices (sample entries include $4$, $-2$, and so on for distinguished index patterns).
The ambient goal is an exact algebraic identity for the 4D transverse-traceless midpoint sector in the Regge analysis stack, not a continuum limit statement.
proof idea
Pure computational discharge. After the six indices are specialized to the concrete values $3,1,1,2,1,1$, both sides reduce to closed integers (the fold for the numerator versus the pattern match for $Z$), and decide checks equality in Int. No lemmas beyond the two defining defs are required.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in $(\mathrm{Fin},4)^6$ by exhausting cases. That global equality is the certified numerator form used downstream in the Regge exact midpoint M2 TT identity stack.
Within Recognition gravity analysis, these kernel chunks turn a symbolic six-index identity into a finite, machine-checked table comparison. They do not themselves invoke the forcing chain (T5–T8) or the Recognition Composition Law; they sit in the discrete geometric bookkeeping layer that supports later continuum or phenomenological claims.
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