e_223322
plain-language theorem explainer
At multi-index (2,2,3,3,2,2) the Regge midpoint mass-squared numerator equals eight times the explicit integer Z-kernel. Gravity analysts cite it as one of the 256 concrete kernel checks that assemble into the global m2Num = 8·explicitZ identity. The proof is a single decide on fixed Fin-4 integers.
Claim. For indices $(a,b,c,d,i,j)=(2,2,3,3,2,2)$ 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 kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 10 of a 256-way case split proving $m_2^{\mathrm{num}}=8\cdot Z$ on the 4D Regge midpoint kernel. Indices run over $\mathbb{F}_4$ (four discrete directions).
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: sum the local contributions of each coupling term at the six indices. The comparison target $Z$ is an explicit integer-valued kernel on the same six indices, tabulated by pattern match (e.g. $(0,0,1,1,2,2)\mapsto 4$, mixed pairs $\mapsto -2$).
The local setting is pure integer arithmetic on a finite domain: once the six coordinates are fixed, both sides reduce to concrete Int values.
proof idea
One-line computational proof: decide. With all six arguments literal Fin 4 values, both m2Num (a fold of integer contributions) and explicitZ (a closed pattern match) evaluate to closed integers, and the equality $n=8\cdot z$ is discharged by the kernel decision procedure. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states the identity for every $(a,b,c,d,i,j):(\mathbb{F}_4)^6$ by exhaustive fin_cases. Each chunk theorem such as this one closes one concrete cell of that 4^6 grid.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge mass-squared kernel in 4D: it certifies that the folded coupling numerator is uniformly eight times the explicit Z table, so later continuum or continuum-limit arguments can quote a single closed form rather than a fold. It is infrastructure inside the Gravity.Analysis layer, not a forcing-chain (T0–T8) step.
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