e_220203
plain-language theorem explainer
Pointwise identity: the midpoint mass-squared numerator at multi-index (2,2,0,2,0,3) equals eight times the explicit Z-coupling table at those indices. Gravity analysts assembling the 4D Regge exact midpoint M2TT identity cite these kernel chunks. The proof is a single decidability check on concrete integers.
Claim. For indices $a=b=2$, $c=0$, $d=2$, $i=0$, $j=3$ in $\mathbb{F}_4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer coupling $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the mass-squared numerator $m_2^{\mathrm{num}}$ is the fold of local contributions over a fixed coupling list: each term depends on six $\mathbb{F}_4$ indices. The companion table $Z$ is an explicit integer-valued function on the same six indices, hard-coded on the nonzero patterns that appear in the kernel (e.g. diagonal blocks $4$, off-diagonal $-2$).
This module is chunk 10 of a 256-way split of the kernel identity $m_2^{\mathrm{num}}=8Z$. Each chunk discharges one concrete multi-index by computation rather than by a symbolic argument over all of $(\mathbb{F}_4)^6$.
proof idea
Both sides evaluate to concrete integers once the six $\mathrm{Fin},4$ arguments are fixed. The tactic decide runs the kernel's decidable equality on those integers and closes the goal. No lemmas beyond the definitions of the numerator fold and the explicit $Z$ table are required.
why it matters
The parent assembly theorem m2Num_eq_eight_explicitZ quantifies over all six indices and reduces by exhaustive fin_cases; each leaf is one of these chunk identities. Closing every leaf yields the global numerator identity used in the Regge midpoint M2TT certification. Within Recognition gravity, that identity is bookkeeping for the discrete curvature/mass kernel, not a new forcing-chain step (T0–T8), but it is required scaffolding for the exact 4D midpoint analysis.
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