e_203333
plain-language theorem explainer
Pointwise identity: the midpoint M2 numerator at multi-index (2,0,3,3,3,3) equals eight times the explicit Z-table entry. Gravity analysts cite it when assembling the full 4^6 kernel certificate for the Regge exact-midpoint M2TT identity. The proof is a single kernel decision on concrete Fin-4 integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,3,3,3,3)\in(\mathbb{F}_4)^6$, the summed 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 table on six $\mathbb{F}_4$ arguments.
background
This module is chunk 8 of a 256-case kernel certificate for the 4D Regge exact-midpoint M2TT identity. The local claim is the pointwise relation $m_2^{\mathrm{num}}=8\cdot Z$ on one multi-index.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: each term contributes an integer weight depending on the six $\mathrm{Fin},4$ indices, and the fold starts at $0$. The companion table $Z$ is an explicit case-split function $\mathrm{Fin},4^6\to\mathbb{Z}$ (sample values include $4$, $-2$, and so on on diagonal and off-diagonal patterns).
The surrounding analysis sits in the Gravity domain of the Recognition Science mirror: discrete Regge-calculus identities that underwrite continuum gravitational structure once the kernel is certified.
proof idea
One-line tactic proof: decide. Both sides are closed integer expressions once the six concrete $\mathrm{Fin},4$ arguments are substituted into the fold definition of the numerator and the case table for $Z$. Lean’s kernel evaluates the equality and accepts it; no lemmas beyond the two definitions are invoked.
why it matters
This is one tile in the exhaustive case split that proves the global statement $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$. The parent theorem m2Num_eq_eight_explicitZ introduces all six indices and runs nested fin_cases; each leaf is exactly a chunk identity of this form.
In the broader framework the certified midpoint M2TT identity is part of the discrete gravity analysis supporting Recognition Science’s geometric forcing (spatial dimension and causal tick structure). The chunk does not itself touch T5–T8, but it is load-bearing scaffolding for the gravity side of that chain: without the pointwise numerator–table match, the assembled 4D kernel certificate fails.
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