e_311112
plain-language theorem explainer
Pointwise identity: the midpoint Regge mass-squared numerator at multi-index (3,1,1,1,1,2) equals eight times the explicit Z-table entry there. Gravity analysts building the 4D midpoint M2 TT identity cite these 256 kernel chunks. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,1,1,1,2)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8\,Z(a,b,c,d,i,j)$, where $Z$ is the explicit integer lookup table on six $\mathbb{F}_4$ indices.
background
This module is chunk 13 of a 256-way split of the kernel identity $m_2^{\mathrm{num}}=8\cdot Z$ on $(\mathbb{F}_4)^6$. The setting is 4D Regge calculus at the exact midpoint: discrete curvature couplings are tabulated, then folded into a mass-squared numerator.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list, accumulating each term's contribution at those six indices. The companion $Z$ is an explicit case-table $\mathbb{F}_4^6\to\mathbb{Z}$ (sample entries include $4$ on diagonal-type pairs and $-2$ on crossed pairs).
The full statement is the universal quantification over all six indices; each chunk discharges one concrete tuple so the assembler can finish by exhaustive fin_cases.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers once the six Fin 4 arguments are fixed at $(3,1,1,1,1,2)$, so the kernel decision procedure closes the equality with no lemmas beyond the definitions of the folded numerator and the explicit $Z$ table.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall(a,b,c,d,i,j),, m_2^{\mathrm{num}}=8Z$ by six nested fin_cases over the 256 kernel points. That universal identity is the algebraic backbone of the exact midpoint M2 TT relation in the 4D Regge analysis stack.
Within Recognition Science gravity work, these kernel equalities certify that the discrete mass-squared numerator matches the closed-form $Z$ table used downstream in curvature and continuum-limit arguments. Chunk 13 is pure scaffolding closure: one more decide in the 256-pack, not a new physical hypothesis.
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